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abstract: |
In [@Kim] the author generalized the Conway algebra and constructed the invariant valued in the generalized Conway algebra defined by applying two skein relations to crossings, which is called a generalized Conway type invariant. The generalized Conway type invariant is a generalization of Homflypt polynomial.
In this paper we show that an example of links, which have the same value of Homflypt polynomial, but have different values of the generalized Conway type invariant. We study a properties of Conway type invariant related to Vassiliev invariant. In section 3 we discuss about further researches.
address: |
Department of Fundamental Sciences, Bauman Moscow State Technical University, Moscow, Russia\
[email protected]
author:
- Seongjeong Kim
title: Remarks on the invariants valued in the generalization of Conway algebra
---
Introduction
============
In [@PrzytyskiTraczyk] J.H.Przytycki and P.Traczyk introduced an algebraic structure, called [*a Conway algebra,*]{} and constructed an invariant of oriented links, which is a generalization of the Homflypt polynomial invariant. In 2017 L. H. Kauffman and S. Lambropoulou [@KauffmanLambropoulou] constructed new 4-variable polynomial invariants, which are generalized from the Homflypt polynomial, the Dubrovnik polynomial and the Kauffman polynomial. In [@Kim] the author generalized the Conway algebra and constructed the invariant valued in the generalized Conway algebra, which is constructed by applying two skein relations to crossings.
In this paper we show that an example of links, which have the same value of Homflypt polynomial, but have different values of the generalized Conway type invariant. We study a properties of Conway type invariant related to Vassiliev invariant. In section 3 we discuss about further researches.
Some remarks on the Conway type invariant
=========================================
We introduce a generalization of the Conway algebra and invariant valued in the generalized Conway algebra.
[@Kim]\[def\_genConaltype1\] Let $\widehat{\mathcal{A}}$ be a set with four binary operations $\circ,*,/$ and $//$ on $\widehat{\mathcal{A}}$. Let $\{a_{n}\}_{n=1}^{\infty} \subset \widehat{\mathcal{A}}$. The hexuple $( \widehat{\mathcal{A}}, \circ,/,*, //,\{a_{n}\}_{n=1}^{\infty})$ is called [*a generalized Conway algebra*]{} if it satisfies the following conditions:
(A)
: $(a \circ b) / b = (a / b) \circ b = a = (a* b) // b = (a // b) * b$ for $a,b \in \widehat{\mathcal{A}}$,
(B)
: $ a_{n} = a_{n} \circ a_{n+1}$ for $n = 1,2,\cdots,$
(C)
: $(a \circ b) \circ (c \circ d) = (a \circ c) \circ (b \circ d)$ for $a,b,c,d \in \widehat{\mathcal{A}}$,
(D)
: $(a * b) * (c * d) = (a * c) * (b * d)$ for $a,b,c,d \in \widehat{\mathcal{A}}$,
(E)
: $(a \circ b) \circ (c * d) = (a \circ c) \circ (b * d)$ for $a,b,c,d \in \widehat{\mathcal{A}}$,
(F)
: $(a * b) * (c \circ d) = (a * c) * (b \circ d)$ for $a,b,c,d \in \widehat{\mathcal{A}}$,
(G)
: $(a \circ b) * (c \circ d) = (a * c) \circ (b * d)$ for $a,b,c,d \in \widehat{\mathcal{A}}$.
Let $( \widehat{\mathcal{A}}, \circ,/,*,//,\{a_{n}\}_{n=1}^{\infty})$ be a generalized Conway algebra. If two operations $\circ$ and $*$ are same, then the generalized Conway algebra is a Conway algebra, and hence the Conway type invariant can be defined on $(\widehat{\mathcal{A}}, \circ, /,\{a_{n}\}_{n=1}^{\infty})$.
[@Kim]\[Main\_thm\] Let $\mathcal{L}$ be the set of equivalence classes of oriented link diagrams modulo Reidemeister moves. Let $(\widehat{\mathcal{A}}, \circ,/,*, //,\{a_{n}\}_{n=1}^{\infty})$ be a generalized Conway algebra. Then there uniquely exists the invariant of classical oriented links $\widehat{W} : \mathcal{L} \rightarrow \widehat{\mathcal{A}}$ satisfying the following rules:
1. For self crossings $c$ the following relation holds: $$\label{selfConwayrel}
\widehat{W}(L_{+}^{c}) = \widehat{W}(L_{-}^{c}) \circ \widehat{W}(L_{0}^{c}).$$
2. For mixed crossings $c$ the following relation holds: $$\label{mixedConwayrel}
\widehat{W}(L_{+}^{c}) = \widehat{W}(L_{-}^{c}) * \widehat{W}(L_{0}^{c}).$$
3. Let $T_{n}$ be a trivial link of $n$ components. Then $$\widehat{W}(T_{n}) = a_{n}.$$
We call $\widehat{W}$ [*a generalized Conway type invariant valued in $(\widehat{\mathcal{A}}, \circ,/,*,//,\{a_{n}\}_{n=1}^{\infty})$.*]{}
**Construction of $\widehat{W}$.** First we will define $\widehat{W}$ for ordered oriented link diagrams. Let $L = L_{1} \cup \cdots \cup L_{r}$ be an ordered oriented link diagram of $r$ components. Fix a base point $b_{i}$ on each component $L_{i}$. Suppose that we walk along the diagram $L_{1}$ according to the orientation from the base point $b_{1}$ to itself, then we walk along the diagram $L_{2}$ from the base point $b_{2}$ to itself and so on. If we pass a crossing $c$ first along the undercrossing(or overcrossing), we call $c$ [*a bad crossing*]{}(or [*a good crossing*]{}) with respect to the base points $b = \{b_{1}, \cdots, b_{r}\}$. Now we perform the crossing change or splicing for all bad crossings. Denote the value of $\widehat{W}$ for $L$ corresponding to the base points $b$ by $\widehat{W}_{b}(L)$. Suppose that we meet the first bad crossing $c$. If it is a self crossing, we apply the skein relation on $c$ with the following property: $$\widehat{W}_{b}(L_{+}^{c}) = \widehat{W}_{b}(L_{-}^{c}) \circ \widehat{W}_{\tilde{b}}(L_{0}^{c}),$$ where $\tilde{b} = \{b_{1}, \cdots, b_{j-1},b_{j},b'_{j},b_{j+1}, \cdots, b_{n},b_{n+1}\}$ and $b'_{j}$ is a chosen base point near the place of the crossing $c$ on the component, which is appeared by splicing the self crossing $c$ of $L_{j}$, see Fig. \[1-2splicing\].
If it is a mixed crossing between two components $L_{i}$ and $L_{j}$, we apply the skein relation on $c$ with the following property: $$\widehat{W}_{b}(L_{+}^{c}) = \widehat{W}_{b}(L_{-}^{c}) *\widehat{W}_{\tilde{b}}(L_{0}^{c}),$$ where $\tilde{b} = \{b_{1}, \cdots, b_{j-1},b_{j+1}, \cdots, b_{n}\}$. As an abuse of notation we write $\tilde{b} = b$, if it does not cause confusion. Notice that if $c$ is a positive crossing, then the number of bad crossings of $L_{-}^{c}$ is less than the number of bad crossings of $L_{+}^{c}$ and the number of crossings of $L_{0}^{c}$ is less than the number of crossings of $L_{+}^{c}$. We apply those relations to the first bad crossings of $L_{-}^{c}$ and $L_{0}^{c}$ inductively until we switch all bad crossings. If $c$ is a negative crossing, then we apply those relations to the first bad crossings of $L_{+}^{c}$ and $L_{0}^{c}$ inductively until we switch all bad crossings. If $L = L_{1} \cup \cdots \cup L_{r}$ has no bad crossings, then we define $\widehat{W}_{b}(L) = a_{n}$.
\[exa\_gen\_Conway\] Let $\widehat{\mathcal{A}} = \mathbb{Z} [p^{\pm 1 }, q^{\pm 1}, r]$. Define the binary operations $\circ,*,/$ and $//$ by $$\begin{aligned}
a \circ b = pa + qb,& a / b = p^{-1} a - p^{-1}qb,\\
a * b = pa + rb,& a // b = p^{-1} a - p^{-1}rb.\end{aligned}$$ Denote $a_{n} = (\frac{1-p}{q})^{n-1}$ for each $n$. Then $(\widehat{\mathcal{A}}, \circ,/,*, //,\{a_{n}\}_{n=1}^{\infty})$ is a generalized Conway algebra.
\[exa\_gen\_Homflypt\] Let $\widehat{\mathcal{A}} = Z[v^{\pm 1}, z, w^{\pm 1}]$ be an algebra. Define binary operations $\circ,/,*$ and $//$ by $$\begin{aligned}
a\circ b = v^{2}a+vwb,& a/b = v^{-2}a-v^{-1}wb,\\
a* b = v^{2}a+vzb,& a//b = v^{-2}a-v^{-1}zb.\end{aligned}$$ Put $a_{n} = (\frac{v^{-1}-v}{w})^{n-1}.$ Then $(\widehat{\mathcal{A}},\circ,/,*,//,\{a_{n}\}_{n=1}^{\infty})$ is a generalized Conway algebra. In fact, this is obtained from the generalized Conway algebra in example \[exa\_gen\_Conway\] by substituting $p=v^{2}$, $q = vw$ and $r = vz$. Moreover if $w=z$, then the Conway type invariant valued in the generalized Conway algebra $(\widehat{\mathcal{A}},\circ,/,*,//,\{a_{n}\}_{n=1}^{\infty})$ is the Homflypt polynomial.
Let $(\widehat{\mathcal{A}},\circ,/,*,//,\{a_{n}\}_{n=1}^{\infty})$ be the generalized Conway algebra in the previous example. It is well known that two links $L_{1} = L11n418\{0,0\}$ and $L_{2} = L11n358\{0,1\}$ have the same value of Homflypt polynomial. The values of Conway type invariant valued in $(\widehat{\mathcal{A}},\circ,/,*,//,\{a_{n}\}_{n=1}^{\infty})$ from $L_{1}$ and $L_{2}$ are calculated as follow. $$\begin{aligned}
\widehat{W}(L_{1}) &=& \frac{1}{p^3 q^2}-\frac{2}{p^2 q^2}+\frac{1}{p q^2}+\frac{r}{q}+\frac{r}{p^4 q}
-\frac{6 r}{p^3 q}+\frac{7 r}{p^2 q}-\frac{3 r}{p q}-\frac{4r^2}{p^4}+\frac{9 r^2}{p^3}\\
&&-\frac{4 r^2}{p^2}+\frac{r^2}{p}+\frac{q r^2}{p^3}-\frac{q r^3}{p^5}+\frac{5 q r^3}{p^4}-\frac{2 q r^3}{p^3}+\frac{q^2
r^4}{p^5}\end{aligned}$$ $$\begin{aligned}
\widehat{W}(L_{2}) &=&\frac{1}{p^3 q^2}-\frac{2}{p^2 q^2}+\frac{1}{p q^2}-\frac{r}{p^5 q}+\frac{4 r}{p^4 q}-\frac{8 r}{p^3 q}+\frac{5 r}{p^2 q}\\
&&+\frac{3 r^2}{p^5}-\frac{8r^2}{p^4}+\frac{8 r^2}{p^3}-\frac{r^2}{p^2}-\frac{3 q r^3}{p^5}+\frac{4 q r^3}{p^4}-\frac{q r^3}{p^3}+\frac{q^2 r^4}{p^5}\end{aligned}$$
That is, $\widehat{W}(L_{1}) \neq \widehat{W}(L_{2})$ and the generalized Conway type invariant is stronger than Homflypt polynomial.
\[exa\_non\_linear\_gen\_Conway\] Let us fix a natural number $k$. Let $\widehat{\mathcal{A}}$ be the smallest commutative ring with identity containing $\mathbb{Z}[p^{\pm1},q^{\pm1},r^{\pm1}]$ such that $\sqrt[k]{f} \in \widehat{\mathcal{A}}$ for each $f \in \widehat{\mathcal{A}}$, where $\sqrt[k]{f}$ is the formal $k$-th root, that is, $(\sqrt[k]{f})^{k} = \sqrt[k]{f^{k}} =f$. Define binary operations $\circ,/,*$ and $//$ by $$\begin{aligned}
a\circ b = \sqrt[k]{pa^{k}+qb^{k}}, & a/b = \sqrt[k]{p^{-1}a^{k}-p^{-1}qb^{k}} \\
a * b = \sqrt[k]{pa^{k}+rb^{k}}, & a//b = \sqrt[k]{p^{-1}a^{k}-p^{-1}rb^{k}},\end{aligned}$$ for $a,b \in \widehat{\mathcal{A}}$. Let $\{a_{n}\}$ be the sequence defined by the following recurrence relation $$a_{1} = 1, a_{n+1}^{k} = \frac{(1-p)}{q}a_{n}^{k}.$$ We can show that $(\widehat{\mathcal{A}}, \circ,/,*, //,\{a_{n}\}_{n=1}^{\infty})$ is a generalized Conway algebra.
The generalized Conway type invariant valued in the generalized Conway algebra in example \[exa\_gen\_Homflypt\] weaker than two variable Vassiliev invariant.
Let $\widehat{\mathcal{A}} = Z[v^{\pm 1}, z, w^{\pm 1}]$ be an algebra with binary operations $\circ,/,*$ and $//$ defined by $$\begin{aligned}
a\circ b = v^{2}a+vwb,& a/b = v^{-2}a-v^{-1}wb,\\
a* b = v^{2}a+vzb,& a//b = v^{-2}a-v^{-1}zb.\end{aligned}$$ and $a_{n} = (\frac{v^{-1}-v}{w})^{n-1}.$ By substituting $v = e^{x}e^{y}e^{z}$, $w=e^{-x}-e^{x}$ and $z=e^{-y}-e^{y}$ we obtain that $$\widehat{W}(L^{C}_{+}) - \widehat{W}(L^{C}_{-}) = xf(x,y,z),$$ for a self crossing $C$ and $$\widehat{W}(L^{C}_{+}) - \widehat{W}(L^{C}_{-}) = yg(x,y,z)$$ for a mixed crossing $C$, therefore the generalized Conway type invariant is weaker than two variable Vassiliev invariant.
Further research
================
When we define the Conway type invariant, we applied two different skein relations to self crossings and mixed crossings. It is well known that the division of crossings to [*self/mixed crossings*]{} is a parity, which is introduced by V.O.Manturov in [@Ma]. We would like to generalized Conway type invariant by applying two different skein relations to even crossings and odd crossings. On the above purpose we define [**the link parity**]{} as follow.
A link parity $p$ on diagrams of a link $\mathcal{L}$ with coefficients in $\mathbb{Z}_{2}$ is a family of maps $p_{L}: \mathcal{V}(L) \rightarrow A,$ $L \in ob(\mathcal{L})$ is an object of the category, such that for any elementary morphism $f : L \rightarrow L'$ the following hols:
1. $p_{L'}(f_{*}(v)) =p_{L}(v)$ provided that $v \in \mathcal{V}(L) $ and there exists $f_{*}(v) \in \mathcal{V}(L')$, if $f$ is a Reidemeister moves;
2. $p_{L}(v_{1}) + p_{L}(v_{2}) = 0$ if f is a decreasing second Reidemeister move and $v_{1}, v_{2}$ are the disappearing crossings;
3. $p_{L}(v_{1}) + p_{L}(v_{2}) + p_{L}(v_{3}) = 0$ if $f$ is a third Reidemeister move and $v_{1}, v_{2}, v_{3}$ are the crossing participating in this move;
4. $p_{L}(f_{*}(v)) = p_{L}(v)$ if $f$ is a splicing of $v'$ and $p_{L}(v) + p_{L}(v') = 1$;
5. If $f$ is a splicing of $v'$, $p_{L}(v) + p_{L}(v') = 0$ and $p_{L}(f_{*}(v)) + p_{L}(v) =1$, then $p_{L}(f'_{*}(v')) + p_{L}(v') =1$, where $f'$ is a splicing of $v$.
It is easy to show that the division of crossings to self/mixed crossings satisfies the above conditions. But the (link) parity, which gives non-trivial parity for knots, is not known. Now we define the descending diagram for virtual link diagrams.
Let $L = L_{1} \cup \cdots \cup L_{n}$ be an ordered oriented virtual link diagram of $n$ components. Fix a point $b_{i}$ on each component $L_{i}$, but it is not a crossings. Now we walk along the component $L_{1}$ from the fixed point $b_{1}$ in the given direction of $L_{1}$, and then we walk along the component $L_{2}$ from the fixed point $b_{2}$ in the given direction of $L_{2}$ and so on. If while walking along the diagram each classical crossing is first passed over and then under, then we call the diagram $L$ [*descending diagram with respect to $\{b_{i}\}_{i=1}^{n}$ and the order of components*]{}. For an oriented virtual link diagram $L$, if we can give an order of components and fix a point on each component of $L$ for $L$ to be a descending diagram, then we call $L$ is a descending diagram. It is well known that if $L$ is a classical descending link diagram, then it is equivalent to the trivial link diagram.
We would like to define [*the virtual Conway algebra $(\widetilde{\mathcal{A}}, \circ,/,*, //,\{a_{T}\}_{T \in I} \subset \widetilde{\mathcal{A}})$,*]{} which is a generalization of the generalized Conway algebra $(\widehat{\mathcal{A}}, \circ,/,*, //,\{a_{n}\}_{n \in \mathbb{N}})$ to define invariant for oriented virtual links as follow:\
Let $\widetilde{\mathcal{L}}$ be the set of equivalence classes of oriented virtual link diagrams modulo generalized Reidemeister moves such that every diagram is equipped by link parity. Let $(\widetilde{\mathcal{A}}, \circ,/,*, //,\{a_{T}\}_{T \in I} \subset \widetilde{\mathcal{A}})$ be a virtual Conway algebra, where $I$ is the set of equivalence classes of all descending virtual link diagrams. We would like to define the invariant of oriented virtual links $\widetilde{W} : \widetilde{\mathcal{L}} \rightarrow \widetilde{\mathcal{A}}$ satisfying the following rules:
1. For even crossings $c$ the following relation holds: $$\label{selfConwayrel}
\widetilde{W}(L_{+}^{c}) = \widetilde{W}(L_{-}^{c}) \circ \widetilde{W}(L_{0}^{c}).$$
2. For odd crossings $c$ the following relation holds: $$\label{mixedConwayrel}
\widetilde{W}(L_{+}^{c}) = \widetilde{W}(L_{-}^{c}) * \widetilde{W}(L_{0}^{c}).$$
3. Let $L_{T}$ be a virtual link, which has a descending diagram $T$. Then $$\widetilde{W}(L_{T}) = a_{T}.$$
But when we define the virtual Conway algebra, we meet the following difficulty: it is well-known that there are infinitely many nontrivial free links. It follows that there are infinitely many nontrivial descending virtual link diagrams, because by forgetting under/over information and the underlying surface of virtual link diagram, we obtain free link diagram. But we don’t know how to list all such virtual link diagrams. Moreover, to define the virtual Conway algebra $(\widetilde{\mathcal{A}}, \circ,/,*, //,\{a_{T}\}_{T \in I} \subset \widetilde{\mathcal{A}})$ we need to find relations $$\label{needed_rels}
a_{T_{+}} = a_{T_{-}} \circ a_{T_{0}}, a_{T_{+}} = a_{T_{-}} * a_{T_{0}}$$
for $T_{*} \in I$, where $T_{+},T_{-}$ and $T_{0}$ are the Conway triple.\
**Questions**\
1. Is there a (link) parity, which gives non-trivial parity for classical links?
2. How to find the set $\{a_{T}\}_{T \in I}$ and the relations \[needed\_rels\] for the invariant $\widetilde{W}$ is well-defined.
[0]{}
L.H.Kauffman,S.Lambropoulou, [*New invariants of links and their state sum models*]{}, arXiv:1703.03655v2 \[math.GT\] 15 Mar 2017.
S.Kim, [*On the generalization of Conway algebra*]{}, arXiv:1707.02416v3 \[math.GT\] J.H.Przytyski, P.Traczyk, [*Invariants of links of Conway type,*]{} Kobe Journal of Mathematics, 4 (1989) 115-139.
V.O.Manturov, [*Free knots and parity,*]{} Sbornik : Mathematics, 2010, vol. 201, no. 5, p. 65-110.
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abstract: 'The theoretical solid-state physicist Walter Kohn was awarded one-half of the 1998 Nobel Prize in Chemistry for his mid-1960’s creation of an approach to the many-particle problem in quantum mechanics called density functional theory (DFT). In its exact form, DFT establishes that the total charge density of any system of electrons and nuclei provides all the information needed for a complete description of that system. This was a breakthrough for the study of atoms, molecules, gases, liquids, and solids. Before DFT, it was thought that only the vastly more complicated many-electron wave function was needed for a complete description of such systems. Today, fifty years after its introduction, DFT (in one of its approximate forms) is the method of choice used by most scientists to calculate the physical properties of materials of all kinds. In this paper, I present a biographical essay of Kohn’s educational experiences and professional career up to and including the creation of DFT. My account begins with Kohn’s student years in Austria, England, and Canada during World War II and continues with his graduate and post-graduate training at Harvard University and Niels Bohr’s Institute for Theoretical Physics in Copenhagen. I then study the research choices he made during the first ten years of his career (when he was a faculty member at the Carnegie Institute of Technology and a frequent visitor to the Bell Telephone Laboratories) in the context of the theoretical solid-state physics agenda of the late 1950’s and early 1960’s. Subsequent sections discuss his move to the University of California, San Diego, identify the research issue which led directly to DFT, and analyze the two foundational papers of the theory. The paper concludes with an explanation of how the chemists came to award “their” Nobel Prize to the physicist Kohn and a discussion of why he was unusually well-suited to create the theory in the first place.'
author:
- Andrew Zangwill
title: |
The education of Walter Kohn\
and the creation of density functional theory
---
Introduction
============
The 1998 Nobel Prize in Chemistry recognized the field of quantum chemistry, a theoretical enterprise where quantum mechanics is used to study molecules and address chemical problems (Barden and Schaefer 2000, Gavroglu and Simões 2012). The prize was shared equally by “Walter Kohn for his development of density functional theory and John Pople for his development of computational methods in quantum chemistry” (Nobel 2013). The award to Pople surprised no one.[^1] Ten years earlier, an international conference, “Forty Years of Quantum Chemistry”, had honored Pople’s career-long commitment to developing semi-empirical and first-principles methods to predict the structure and properties of molecular systems (Handy and Schaefer 1990). A notable early contribution was the 1970 free release of his group’s GAUSSIAN computer program which solved the Schrödinger equation for molecules in the Hartree-Fock approximation. Today, after decades of improvements in accuracy and functionality, it is estimated that 90% of all quantum chemistry calculations are performed using the (now commercial) GAUSSIAN suite of programs (Crawford [*et al.*]{} 2001).
By contrast, the fact that Walter Kohn earned a share of the Nobel Prize in [*chemistry*]{} surprised many because his international reputation was gained as a theoretical solid-state physicist. Indeed, his body of work had previously earned him the Oliver E. Buckley Prize (1961) and the Davisson-Germer Prize (1977) of the American Physical Society for his contributions to, respectively, “the foundations of the electron theory of solids” and “understanding the inhomogeneous electron gas and its application to electronic phenomena at surfaces” (APS 2013a). A card-carrying chemist could reasonably ask: what do these things have to do with quantum chemistry? The answer lies in the research Kohn conducted in the period 1963-1965 which created the density functional theory (DFT) cited by the Nobel committee. This theory focuses on the density of electrons in an atom, molecule, or solid rather than on the many-electron wave function that is the focus of traditional quantum chemistry.
The connection between the work of the two 1998 Chemistry Nobel laureates can be understood from the ‘hyperbola of quantum chemistry’’ which Pople published in the proceedings of a 1965 symposium on atomic and molecular quantum theory (Pople 1965). The horizontal axis of this graph (see Fig. 1) indicates the number of electrons $N$ in the system of interest. The vertical axis lists a sequence of quantum mechanical methods (in order of increasing accuracy) used to determine the system’s ‘electronic structure’, [*i.e.*]{}, its many-electron wave function, electronic charge density, electron energy levels, and other properties calculable from these. Pople suggested that the activities of most quantum chemists tended to cluster around the extremities of the hyperbolic solid line. Those interested in the highest accuracy were forced by computational constraints to focus on small molecules (small $N$) while those interested in large molecules (large $N$) were forced by computational constraints to use methods that were capable of only low accuracy. He noted that progress would occur by moving off the hyperbola either horizontally from left to right or vertically from bottom to top.
Fig. 1. The applicability range of Kohn’s density\
functional theory placed on Pople’s hyperbola of\
quantum chemistry. Adapted from Pople (1965).
The density functional theory created by Walter Kohn represented a dramatic move away from Pople’s hyperbola, particulary for systems with a large number of electrons (see the arrow in Fig. 1). Many chemists were wary (or dismissive) of this theory at first because it made no use of the $N$-electron wave function, a quantity thought to be indispensable for a proper description of any atom or molecule. Eventually, improvements to Kohn’s theory made by chemists themselves led to quantitative successes that could not be denied. Today, it is the method of choice used by most scientists who wish to calculate the properties of real materials.
Unlike many other scientific achievements, the technical question which led Kohn to create DFT in the mid-1960’s was not “in the air” among physicists, chemists, or anyone else. It is entirely possible that the theory would be unknown today if Kohn’s background, technical skills, and scientific experiences differed very much from what they were. Accordingly, this article (i) recounts the unusual personal and intellectual journey which led to Walter Kohn’s success as a theoretical physicist; (ii) identifies the scientific issue which motivated the creation of DFT in the context of theoretical solid state physics research in the 1960’s; (iii) summarizes the key elements of the founding papers of DFT; (iv) argues that Kohn’s scientific background made him particularly well-suited to create density functional theory; and (v) explains in brief how Kohn came to win a share of the Nobel Prize in Chemistry for the creation of DFT.
An Unsentimental Education
==========================
In 1933, ten-year-old Walther Kohn began the eight-year course of study at the Akademische Gymnasium, the oldest and one of the best secondary schools in his home city of Vienna.[^2] In doing so, he was not unlike the children of many middle-class Jewish parents who were actively engaged in the intellectual and artistic life of their city.[^3] His father Salomon owned an art publishing company that specialized in the manufacture and distribution of high-quality art postcards. Despite a worldwide clientele, it was a struggle to keep the business going in the face of a global economic depression which hit Austria particulary hard. Nevertheless, there was a tacit understanding that Walther would eventually run the family business. Kohn’s mother Gittel was a highly educated woman who spoke four languages and it was she who chose the humanistically oriented Akademische Gymnasium to educate her son. Walther excelled at Latin and ancient Roman history but showed no apparent aptitude for mathematics. The only grade of C he ever received was in that subject (Kohn 1998).
Besides marking the beginning of Kohn’s secondary school education, 1933 was also the year that Adolf Hitler and his Nazi party took power in neighboring Germany. The Nazi party was illegal in Austria, but its many sympathizers worked tirelessly to undermine the existing democratically elected government. Finally, in May 1934, the country succumbed to a form of authoritarian rule known to historians as Austrofascism (Berkley 1988). Four years later, cheering crowds welcomed Hitler when the German army crossed the border and marched into Vienna. The [*Anschluss*]{} (political union) of Germany and Austria was a [*fait accompli*]{}. These events were vivid in Kohn’s memory many years later (Kohn 1998, Kohn 2013a):
> The \[March 1938\] Anschluss changed everything. The family business was confiscated but my father was required to continue its management without any compensation. . . . He wrote to a London art distributor and business client named Charles Hauff (whom he had never met) and asked if he and his wife would temporarily accept me and my older sister Minna into their home. Hauff replied affirmatively and Minna emigrated to England very soon thereafter. For reasons of their own, the Nazis made it much more difficult for young Jewish boys to leave. I remained in Vienna, but was expelled from my school.
Many expelled students never went to school again. Kohn was lucky because he was permitted to finish the school year at a segregated high school for Jews. Then, in August 1938, he was one of a few hundred high-achieving Jewish students from the various Viennese secondary schools who were given the opportunity to continue their education at the Zwi Perez Chajes Gymnasium, a private all-Jewish high school.[^4] This experience was transformative for Kohn because his interest in science was ignited by the physics teacher Emil Nohel and the mathematics teacher Victor Sabbath.[^5] He later recalled that (Hollander 2000, Kohn 2004)
> Nohel was a tall, quiet, noble man who devoted himself to his students. It was really a combination of my admiration for this man as a person and his deep knowledge of physics that started me off . . . Though I was only fifteen going on sixteen years old, I already understood—due to Nohel’s role model and by comparing myself to others—what it meant to really comprehend something in physics. This is one of the most important insights for a future theoretical physicist.
>
> Sabbath was also a fantastic guy. The thing I remember about him is that while he was teaching us he told us about a new book he was reading by the great French physicist \[Louis\] de Broglie called [*Matter and Light*]{}. . . Sabbath was a teacher with great enthusiasm and it was very exciting what he told us.
Kohn’s inspirational teachers Emil Nohel and Victor Sabbath did not survive the Holocaust.
November 10 1938 was particulary memorable at the Chajes school because it was the day following the night of the infamous, state-sanctioned orgy of destructive violence against Jewish homes, businesses and synagogues known as [*Kristallnacht*]{} (the night of broken glass). The principal dismissed the students early to avoid attracting attention, but Kohn and a classmate were arrested on their way home. They were released after several terrifying hours in the police station, but Kohn returned home to find “our apartment absolutely vandalized by a group of hooligans, including the person who had taken over my father’s business” (Hanta 1999). The classes at Chajes got smaller and smaller in the months after Kristallnacht. Emigration was on everyone’s mind, but it was difficult and expensive to make it happen (Ehrlich 2003). Kohn was again lucky. He escaped from Austria to England just three weeks before Germany invaded Poland and World War II began. His parents were unable to leave. Both were eventually deported, first to the Terezin concentration camp in Czechoslovakia and then to Auschwitz, where they were murdered in 1944.
Emil Eliezer Nohel (ca. 1938) was a high school physics\
teacher who inspired Kohn to become a scientist.\
Courtesy of the Yad Vashem Photo Archive.
Sixteen-year old Walther was saved from probable extermination by the Nazis by the [*Kindertransport*]{}, an organized effort to evacuate Jewish children under the age of 18 from Germany, Austria, and Czechoslovakia. Between December 1 1938 and September 1 1939, this collaboration between British Quakers, World Jewish Relief, the Refugee Children’s Movement (a British rescue organization), and the Jewish agencies in the affected countries made it possible for almost 10,000 unaccompanied children to board special trains bound for port cities in the Netherlands. Boats carried them from there to safety across the English Channel (Fast 2011). By agreement with the British government, the children were to be placed with sponsor families or volunteer foster parents for a period of up to two years, whereafter repatriation to their home countries was anticipated. The Jewish community of Britain bore full financial responsibility for the transport. The host families were expected to arrange educational opportunities for their guest children.
One quarter of the Kindertransport children came from Austria and the Jewish Community of Vienna \[Israelitsche Kultusgemeinde Wien (IKG)\] was responsible for deciding which children would receive exit visas from amongst the great many applications they received (Curio 2004). Initially, the IKG focused on urgent cases, including stateless children under threat of expulsion and children in orphanages. Priority was given to children whose parents had been arrested or deported. Later, preference was given to children with guaranteed sponsors in Britain, but this alone did not ensure selection. The children were interviewed and the additional criteria of good health, a pleasant personality, likely success in school, and the ability to “fit in” were used to make the final choices. Kohn apparently met all these criteria and thereby earned a seat on one of the last Kindertransport trains. He arrived in England safely with the pre-arranged plan to live with Charles and Eva Hauff, the same couple who had previously welcomed his sister Minna. In fact, Minna met Walther at London’s Victoria station and they traveled together by train to West Sussex where the Hauff’s lived (Kohn 2013a).
Before leaving Vienna, Kohn and his parents had agreed that he should learn to be a farmer in England (Kohn 1998). They had seen too many unemployed intellectuals in pre-war Vienna and farming seemed like an occupation that would make him less subject to economic dislocations.[^6] Moreover, Salomon Kohn was 66 in 1939 and it was understood that Walther would soon be responsible for his parents’ financial security. All of this was communicated to the Hauffs ahead of time and they were able to arrange for Kohn’s formal English education to begin at a training farm in Sittingbourne, Kent, about one hour away from the Hauff’s home by car. There, Kohn recalls (Hollander 2000),
> I pulled carrots and looked after piglets. . . Unfortunately, on the farm, I contracted what turned out to be meningitis and so was very ill. Sulfa drugs had just been invented, so I pulled through, but it was touch and go. After that, I was very weak and going back to the farm was out of the question.
The Hauffs dealt swiftly with this setback and, in January 1940, Walther entered the nearby East Grinstead County Grammar School. His limited English skills led the headmaster, Thomas W. Scott, to estimate that 2-3 years could be required for this new student to earn the school certificate needed to enter college. Scott’s method to redress this situation was to enroll Kohn in the lower $6^{\rm th}$ form (to avoid extra English requirements) and instruct his teachers to overlook their new student’s deficiencies in English. He then created a daily German class where Walther was the only student. The instructor taught English to Kohn for half the class and Kohn taught advanced German to the instructor for the other half (Kohn 2000, Ford 2013).
Kohn’s preparation in math and science was equal to or exceeded that of his fellow 16-year-old English classmates (Kohn 2013a). The level of physics he was exposed to at East Grinstead can be judged from two of his textbooks, both intended for students preparing for University Scholarship Examinations. [*Heat*]{} (1939) by R.G. Mitton is a thorough introduction to the thermal properties of matter, the kinetic theory of gases, heat engines, entropy, and the laws of thermodynamics. The prose is brisk, yet clear, and the author assumes familiarity with the laws of algebra and the geometrical meaning of the derivative. The latter occurs in a section devoted to the Clausius-Clapeyron equation, an advanced topic not often found in present-day American high school textbooks. The oddly-named [*Properties of Matter*]{} (1937) by D.N. Shorthose is a textbook of particle and continuum mechanics which includes chapters devoted to ballistic motion, circular motion, simple harmonic motion, rigid body motion, hydrostatics, friction, elasticity, and viscosity. The exposition moves freely between geometrical and algebraic reasoning and the reader is expected to understand first derivatives, second derivatives, and the law of integration which connects them. This is material found in present-day American textbooks intended for first-year college students.
Kohn’s life at East Grinstead was happy and peaceful for five months. Then, on May 10 1940, Germany invaded Holland, Belgium, and Luxembourg. Winston Churchill replaced Neville Chamberlain as Prime Minister of Great Britain and the British newspapers became filled with war hysteria and reports of ‘fifth columnists’.[^7] The British War Office feared that an invasion was imminent and recommended to the Home Office that the government “intern all enemy aliens in areas where German parachute troops are likely to land" (Gilman & Gilman 1987). Arrests begin immediately and, when Italy entered the war on June 10, Churchill demanded that police officials ‘collar the lot’. This terse order expanded internment to all parts of the country and to all enemy aliens, [*i.e.*]{}, to all holders of passports from Italy or Nazi-occupied countries age 17 or older. It took Scotland Yard less than six weeks to intern 24,000 men and 4000 women (Cesarini and Kushner 1993). Those arrested included not just foreign nationals of quite short residency in England like university students but also highly trained scientists, engineers, doctors, artists, and musicians who had lived in England for many years.[^8]
Kohn had turned 17 on March 9 and thus was subject to the order of internment. He was arrested and shipped by train to a large camp that had been hastily constructed in the town of Huyton on the outskirts of Liverpool. It was at Huyton that Walther met and began a lifelong friendship with Josef Eisinger, a boy who had been one year behind him at the Akademische Gymnasium in Vienna.[^9] A week or so later, both boys were transferred to an internment camp on the Isle of Man. The Hauffs arranged for Walther’s East Grinstead teachers to send him his physics textbooks because they were told he would return home soon (Hanta 1999). Instead, he spent a month on the Isle of Man where there was no work and little food. He lost 30 pounds.
The burden of warehousing this great mass of civilian internees and the belief that many German prisoners of war would arrive soon led Churchill’s government to seek help from the British Commonwealth nations. Accordingly, on July 4 1940, Kohn and his friend Eisinger found themselves bound for Canada aboard the [*Sobieski*]{}, a Polish cruise ship that had been captured by the British and converted to a troop transport. The ship arrived in Quebec City two weeks later as part of a convoy that zigzagged across the Atlantic to avoid German submarines (Koch 1980, Auger 2005).[^10] By the time the British government discontinued its policy of internments and deportations in early August 1940, nearly 4400 civilian internees and 1950 prisoners of war had been relocated to Canada. What the Canadians did not know—because the British did not tell them—was that the majority of the civilian internees were not Nazi sympathizers but Jewish refugees from Nazi barbarism.
Kohn spent the next eighteen months in four different Canadian internment camps, a burden lessened only slightly by the Red Cross, which made it possible for him to exchange letters with his parents every two weeks or so. He was also able to engage in learning much of the time, albeit not exactly of the sort he had experienced before. For example, on the evening of his arrival in Quebec City, he and 710 others internees were moved eighty miles down the St. Lawrence river to the town of Trois Rivières, where barbed wire had been strung around the perimeter of a local agricultural exhibition ground to create Camp T. Barracks were built in an arena designed to house livestock (Jones 1988). However, a baseball field adjacent to the arena was the home of the Trois Rivières Renards of the Quebec Provincial Baseball League. By standing on several tables stacked on top of one another to reach the windows, Walther and others could watch the games. They pieced together the rules of baseball during the month they were interned there (Koch 1980, Kohn 2013a).
Walther Kohn in Canada at age 18 (1941).\
Courtesy of Josef Eisinger.
Camp T was closed on August 12, 1940 and its population was transferred to Camp B, a 15 acre compound built in the heavily wooded Acadia forest twenty miles east of Fredericton, the capital of the province of New Brunswick.[^11] Kohn spent nearly a full year at Camp B and a vivid portrait of the camp written by Ted Jones (1988) makes it possible to reconstruct life there in some detail. For our purposes, a salient fact is that the internees organized themselves into a highly-structured mini-society composed of sixteen distinct groups based on age, education, geography, past friendships, religious practices, and political views. Kohn belonged to a group of academically-minded boys between 16 and 20. A different group was populated largely by academics from Cambridge University (graduate students, recent PhD’s, post-doctoral fellows, and junior faculty), another group self-identified as communists, and another group was composed entirely of ultra-orthodox Jews. Even before the construction of the camp was complete, each group elected a leader and a deputy leader, as well as chairmen for the various departments of the internee camp Cabinet.
The camp’s Canadian officials had a clear idea of how the able-bodied internees would spend most of their time: lumberjacking in the forest outside the camp for 20 cents a day. Kohn was not unhappy to do this because the sheer physical labor of chopping trees warded off the Canadian winter cold. However, much more important to him were the many hours he spent at the ‘camp school’ where internees with special technical or academic knowledge offered classes for the benefit of anyone who wanted to learn. Each class met at a designated table in the camp’s Recreation Hut and internees sought and received time off from their work details to attend. Appeals to humanitarian organizations and letters written by the instructors to acquaintances around the world produced donations of textbooks, writing materials, exercise books, blackboards, painting supplies, musical instruments, etc. Two months after it opened in November 1940, the camp library boasted a collection of nearly 1000 volumes.
The Camp B school was organized by the chair of the Education Department, Alfons Rosenberg, a 39-year-old former gymnasium teacher from Berlin who had taught briefly at the Cranbrook School in Kent, England before he was ‘collared’ and deported to Canada. His motivation to create the camp school was simple: 60% of the internees were under 20 years old and in desperate need of organized instruction. The many Cambridge men in the camp offered their services and, over time, courses were given in accounting, acting, anthropology, architecture, art history, astronomy, biology, composition, chemistry, economics, engineering, English, French, geography, German, history, Latin, law, literature, mathematics, music theory, philosophy, physics, physiology, political theory, psychology, sex education, Spanish, and typing.
In a brilliant stroke, Rosenberg arranged with the camp commandant for official examination booklets to be brought into the camp so internees could sit for the nation-wide McGill University Junior Matriculation Exams—a necessary step to enter a Canadian college. Thomas Cassirer, a future professor of French at the University of Massachusetts, Amherst characterized Rosenberg as (Jones, 1988)
> a schoolmaster in the good sense: he could listen to others, he had a wide outlook, and he gave suitable advice. . . . He was always at everybody’s disposal. At discussions, and we had many with him, he would often solve everything with just a short sentence.
Walther Kohn attended a daily physics class offered by Kurt Guggenheimer, a physical chemist who anticipated the shell model of the nucleus by pointing out similarities in the systematics of the binding energies of molecules and nuclei.[^12] Kohn and future McGill University mathematics professor Joachim Lambek were the only students in a set theory course taught by Fritz Rothberger.[^13] According to Lambek (1980),
> Rothberger was an outstanding teacher with an inimitable style of lecturing. He gave unsparingly of his time and managed to bring the most abstract concepts down to earth. He instilled a love of mathematics in numerous young people.
A similar impression was left on Kohn (Kohn 1998, Kohn 2013a):
> Rothberger normally taught us out-of-doors where he wore shorts and boots and nothing else. He used a stick and a sandy area as a blackboard to teach us about the different types of infinities . . . . He was a most kind and unassuming man whose love for the intrinsic depth and beauty of mathematics was gradually absorbed by his students.
Walther “took the classes very seriously because I felt a huge responsibility to support my parents after the war” (Kohn 2013a). The level of his commitment is not difficult to demonstrate. First, he earned all passing marks when Rosenberg and his camp staff administered the McGill Junior Matriculation Exam in June of 1941 (Giannakis 2013).[^14] Second, Kohn took the 20 cents a day he earned lumberjacking and used the “princely sum, carefully saved” to order two books which were sent to him at the camp: [*A Course in Pure Mathematics*]{} (1938) by G.H. Hardy and [*Introduction to Chemical Physics*]{} (1939) by J.C. Slater (Kohn 1998). The Cambridge-educated cohort at Camp B would have been very familiar with the Hardy book because its author was a Cambridge mathematician and the book was in its seventh edition at the time. By contrast, a brand new book by the American physicist Slater would have been completely unknown to his fellow internees. Kohn purchased it entirely on the basis of a catalogue description (Kohn 2013b). In an uncanny way, the contents of both books foreshadow the mathematical rigor, taste for foundational issues, and deep interest in the properties of matter that characterize much of this subsequent work.
Hardy[^15] and Slater[^16] were both first-rate researchers with a strong interest in the pedagogy of their fields. Hardy wrote the first edition of his book in 1908 for the express purpose of making the teaching of mathematics more rigorous at British universities.[^17] It was aimed specifically at first-year University students of ‘scholarship standard’, [*i.e.*]{}, the top 10-20% in ability. The preface to the first edition is explicit: this is “a book for mathematicians: I have nowhere made any attempt to meet the needs of students of engineering or indeed any class of students whose interest are not primarily mathematical." Thirty years later, in the preface to Kohn’s edition, Hardy remarks that “the general plan of the book is unchanged" but “I have inserted a large number of new examples from the papers of the Mathematical Tripos". Kohn would have learned from his camp-mates that these ‘new examples’ were drawn from the demanding examinations used by Cambridge University to evaluate its students who hoped to earn a BA degree in Mathematics.
The Slater book that Walther bought and read (doubtless cover to cover) was very unusual for its time. The preface states that the author worked hard “to make it intelligible to a reader with a knowledge of calculus and differential equations, but unfamiliar with the more difficult branches of mathematical physics". For that reason “the quantum theory used is of a very elementary sort . . . and it has seemed desirable to omit wave mechanics.” On the other hand, the content is far from elementary. Slater notes that “it is customary to write books either on thermodynamics or on statistical mechanics; this one combines both." Moreover, “atomic and molecular structure are introduced, together with a discussion of different types of substances, explaining their interatomic forces from quantum theory and their thermal and elastic behavior from our thermodynamic and statistical methods.” The preface does not warn the reader about the author’s frequent use of kinetic theory, which is a non-trivial subject of its own. All told, the first 100 pages cover “Thermodynamics, Statistical Mechanics, and Kinetic Theory”, the second 200 pages discuss “Gases, Liquids, and Solids”, and the final 200 pages concern themselves with “Atoms, Molecules and the Structure of Matter.” This material would be challenging for a good American college student. It must have been even more so for a 17-18 year old student with a twice-interrupted high school career who was still learning English.[^18]
The Canadian military dissolved Camp B in the spring and summer of 1941. Some Jewish internees were returned to England and released when the British government finally acknowledged that they posed no threat to the war effort.[^19] Kohn was not so lucky. At the end of July, he and Josef Eisinger were transferred to Camp A in the southern Quebec town of Farnham, 60 km southeast of Montreal (Eisinger 2013). Kohn was assigned to the knitting shop where he spent hundreds of hours making woolen socks and camouflage nets. On the other hand, Farnham boasted a camp school every bit as good as the one at Fredericton and the many hours of quiet repetitive work allowed him to think deeply about his schoolwork (Kohn 2013a, Auger 2005).
The Camp A school was organized by William Heckscher, a non-Jewish native of Hamburg who was working as an art historian in England when he was interned (Sears 1990). Kohn singles out Heckscher for special praise in his autobiography (Kohn 1998) and in the first history of the Canadian internment camps, Eric Koch (1980) reports that
> William Heckscher was a remarkable figure. He was the ideal headmaster. The adjective with which several ‘old boys’ of the Farnham camp school described him was ‘elegant’. He had grace, style, and patience.
Heckscher told Koch that the Farnham camp commandant, Major Eric D.B. Kippen, once said to him, “You know, Heckscher, I wish I could send my two sons to your school." At the end of September 1941, Heckscher escorted Kohn and a group of other internees when they traveled to Camp S on St. Helen’s Island in Montreal to sit for McGill University’s Senior Matriculation Examination. The records show that Kohn did well in all the subjects tested: algebra, geometry, trigonometry, physics, chemistry, and coordinate geometry (Giannakis 2013).
Salvation for Kohn and Eisinger came in October 1941 when they received a letter from the wife of a faculty member at the University of Toronto. She had heard about them from a former internee and offered to sponsor them to come live with her family after their release (Eisinger 2011).[^20] Letters were exchanged with Scotland Yard and, by the end of January 1942, the boys found themselves sharing a comfortable attic space in the home of Hertha and Bruno Mendel.[^21] The Mendels were refugees from the Nazis themselves who had earlier helped young Jewish couples escape Germany and start new lives in Canada (Feldberg 1960).[^22] Their new mission was to arrange educational opportunities for recently released Jewish internees. Within days, Eisinger was enrolled in a Toronto high school. Kohn, who was one year older, needed help from two complete strangers to get started at the University of Toronto.[^23] He first received some valuable advice (Kohn 2013a),
> Dr. Mendel worked at the university and his good friend, Leopold Infeld, came to their home very soon after I arrived. Infeld questioned me about my plans and I told him I wanted to be an engineer (another practical profession). He asked “Is that your main interest?” and I said no, it was mathematics and physics. He told me that engineering at the university was good but that the Math-Physics program was superb and that I should pursue a degree there. With the training I received, I could always do engineering.
At that time, the University of Toronto had a Mathematics department, a Physics department, and a small (five-person) department of Applied Mathematics (Robinson 1979). Leopold Infeld was a member of the Applied Mathematics faculty, having come to Toronto in 1938 after working with Albert Einstein for two years at the Institute for Advanced Study in Princeton.[^24] The Math-Physics program recommended to Kohn by Infeld was an honors curriculum where all physics, mathematics, and applied mathematics students took the same courses for the first two years and then specialized in their final two years (Allin 1981).
Walther attempted to enroll in Toronto’s Math-Physics program, but was rebuffed by the University registrar because he lacked some of the mandatory prerequisites (Kohn 2013a). Mendel and Infeld arranged a meeting for Kohn (and five other camp boys with a similar problem) with the Dean of the College of Arts & Science, Samuel Beatty, who was also the chair of the Mathematics department.[^25] Beatty was sympathetic, but he was unable to move the inflexible registrar. Therefore, he proposed to admit Walther and the others as ‘special students’ who did not need the prerequisites. This artful maneuver required only the assent of the Department chairs whose departments were involved in the Math-Physics curriculum. This time, it was the chair of the Chemistry department, Frank Kenrick, who threw up a roadblock by refusing to allow a foreign national from any Triple Axis country to enter his chemistry building where war research was being conducted.[^26] Beatty arranged for Kohn to plead his case in person, first with Kenrick (who refused to acknowledge that Kohn was a refugee) and then with the University Chancellor, Rev. Henry John Cody (who was unwilling to overrule one of his department chairs).
Dean Samuel Beatty (ca. 1953) bent the admission rules\
so Kohn could enter the University of Toronto.\
Courtesy of the University of Toronto Archives.
In the end, the creative Beatty simple redefined the meaning of ‘special student’ to constitute the normal Math-Physics curriculum minus the usual chemistry requirement. In this way, Kohn and his compatriots entered the Math-Physics program several weeks after the beginning of the spring 1942 term. Relatively little time was lost because Beatty had permitted the group to audit his gateway mathematics course the entire time the admission negotiations were going on. Even after the formalities were completed, Beatty tutored the group privately for a month to bring them up to speed with the rest of the class.
Kohn’s undergraduate transcript shows that he completed the two-year Math-Physics “common core” in three semesters.[^27] Besides algebra, analytic geometry, differential and integral calculus, differential equations, physics laboratory, mechanics, properties of matter, dynamics, electricity & magnetism, and light & acoustics, one finds two required courses in actuarial science, two required courses in French and German replaced by English courses, and two elected courses in ‘oriental literature’ where texts in ancient Egyptian, Arabic, Hebrew, Persian, and Turkish were read in translation. Kohn took twelve more advanced courses during the 1943-1944 academic year, including algebraic geometry, differential geometry, partial differential equations, the theory of functions, group theory, thermodynamics, classical dynamics, quantum mechanics, variational principles in physics, English literature, and modern ethics. These two semesters turned out to be his last as an undergraduate because he was inducted into the Canadian army in September 1944.[^28] He served until the war ended in August 1945 and was awarded his BA in Applied Mathematics at the summer 1945 convocation ceremony while still on active duty.[^29]
The bare list of courses Walter took during his five undergraduate semesters does not communicate the elite quality of the instructors who taught and mentored him.[^30] Leopold Infeld, who lectured to upperclassmen only, had been invited to join the Applied Mathematics Department by its chair, the eminent Irish mathematician and theoretical physicist, J.L. Synge.[^31] Synge and his Applied Mathematics colleague Bernard Griffith were the authors of [*Principles of Mechanics*]{}, the introductory but quite sophisticated textbook used by all the students in the Math-Physics program. Synge was always eager to add talent to his faculty and in 1941 he succeeded to recruit the Russian mathematical physicist Alexander Weinstein, a mature scientist with a strong reputation for his work on free boundary problems and variational principles.[^32]
It is significant to our story that variational methods were something of a Toronto speciality at the time. Besides Weinstein, Synge, and Griffith, one should include Gilbert Robinson (Mathematics) and Arthur Stevenson (Applied Mathematics) because Cornelius Lanczos thanks them in the preface to his now-classic 1949 text [*The Variational Principles of Mechanics*]{} because they together “revised the entire manuscript”. In later years, Kohn singled out the algebraist Richard Brauer and the non-Euclidean geometer H.S.M. (Donald) Coxeter as “luminous faculty members whom I recall with special vividness” (Kohn 1998). He also recalled the first-year electricity and magnetism lectures given by Lachlan Gilchrist, a 1913 PhD student of Robert Millikan, because Gilchrist told his Toronto students that it was he who had purchased the oil used by Millikan in his famous oil-drop experiment (Kohn 2003).
Weinstein’s influence on the undergraduate Kohn is apparent from Kohn’s first two published scientific papers. The first, submitted in July 1944, is a two-page report on an exact solution for the oscillations of a spherical gyroscope which generalizes a method presented in the Synge and Griffith book but thanks Weinstein for “his advice and criticism" (Kohn 1945). The second paper was completed and submitted in November 1944 at a time when Kohn was engaged in advanced basic training at Camp Borden, Ontario. This substantial piece of work (Kohn 1946) establishes bounds on the motion of a heavy spherical top using a contour integration method used by Weinstein (1942) to study a spherical pendulum. The text makes clear that Kohn had at least some familiarity with [*Über die Theorie des Kreisels*]{} (1898), the great treatise on tops by Felix Klein and Arnold Sommerfeld.
In the 1945-1946 academic year (Kohn 1998),
> after my discharge from the army, I took an excellent crash masters program, including some senior courses I had missed, graduate courses, and a master’s thesis consisting of my paper on tops and a paper on scaling atomic wave functions.
The atomic wave functions paper, “Two Applications of the Variational Method to Quantum Mechanics” (Kohn 1947) was the first of many papers to come (including the density functional papers) where Kohn exploits a variational principle. He first learned about such principles from an advanced undergraduate course where Weinstein discussed the Lagrangian and Hamiltonian formulations of classical mechanics. Weinstein regarded Kohn as a potential PhD student and thus shared with him his recent work on variational methods to study the vibrations of clamped plates and membranes (Aronszajn & Weinstein 1941). A review paper by Weinstein (1941) summarized the original contributions to this subject by Lord Rayleigh and Walter Ritz.
Notwithstanding the foregoing, it was Applied Mathematics Professor Arthur Stevenson who broadened Kohn’s perspective to include quantum problems and the use of variational methods to study them. He is thanked in Kohn (1947) “for his kind advice and interest".[^33] Stevenson’s early research concerned quantum mechanical methods to calculate the energy levels of electrons in atoms and he had performed variational calculations for the helium atom in collaboration with a colleague in the Toronto physics department (Stevenson and Crawford 1938). Kohn surely read this paper because the helium atom figured into his work also.
Kohn assumed that the readers of his paper were familiar with the variational method to find approximate solutions to the Schrödinger equation. Indeed, most textbooks of quantum mechanics written between 1930 and 1945 devoted more than passing attention to this topic because Egil Hylleraas (1929) had used it with spectacular success to calculate the ionization energy of the helium atom. This provided the first convincing evidence that quantum mechanics could achieve quantitative success for a system of more than one electron.
For future reference, I sketch here a simple form of the Rayleigh-Ritz variational method appropriate to an $N$-electron system with ground state energy $E_0$ and Hamiltonian operator $H$. If ${\bf r}=(x,y,z)$, the starting point is a trial wave function, $\psi({\bf r}_1, {\bf r}_2,\dots, {\bf r}_N)$, which depends on the Cartesian coordinates of all the electrons. One then computes a $3N$ dimensional integral with the configuration space volume element $d\tau = d{\bf r}_1 \cdots d{\bf r}_N$ and exploits the inequality,[^34] $$\label{one}
0\le E[\psi]=\int d\tau \psi^* (H-E_0) \psi = \int d\tau \psi^* H \psi - E_0.$$ The exact ground state wave function $\psi_0$ satisfies the Schrödinger equation, $H\psi_0 = E_0 \psi_0$. Using the latter in Eq. (\[one\]) shows that $E[\psi]=0$ when $\psi=\psi_0$ and suggests a strategy to find an upper bound to $E_0$: write the trial function $\psi$ as a linear combination of a set of basis functions and minimize the integral on the right side of Eq. (\[one\]) with respect to the expansion coefficients. Increasing the number of basis functions generally lowers the bound obtained for $E_0$. An important feature of this procedure emerges if we consider a trial function of the form $\psi = \psi_0 + \delta \psi$ where $\delta \psi$ is “small” by some measure. Inserting this trial function into Eq. (\[one\]) gives the variation $\delta E=E[\psi_0+\delta \psi]$ as
$$\label{two}
\delta E = \int d \tau (\psi_0 + \delta \psi)^* (H-E_0) (\psi_0 + \delta \psi) = \int d \tau \delta \psi^* (H-E_0) \delta \psi = O(\delta \psi^2).$$
One says that the energy functional $E[\psi]$ is “stationary” in the sense that a trial function that differs from $\psi_0$ by a small amount (first order in $\delta \psi$) produces an energy which differs from $E_0$ by an amount that is [*very*]{} small (second order in $\delta \psi$). For that reason, minimizing $E[\psi]$ with respect to a set of variational parameters produces a much better estimate for the ground-state energy than one might have supposed. The elegance and generality of this technique must have appealed powerfully to the young Kohn, because he “read many of the old papers on the subject and variational methods became the first tool in my theoretical physics toolbox” (Kohn 2013b).
In the spring of 1946, Kohn completed his MS studies, taught calculus and analytic geometry as an Instructor for the Mathematics Department, and applied to a dozen or so PhD programs with the clear idea to study theoretical physics. Offers of admission with financial support came from Rudolf Peierls at Birmingham and Eugene Wigner at Princeton, among others. Kohn accepted the Birmingham offer on the advice of Infeld, who knew Peierls personally (Kohn 2013b). One day later, an offer arrived from Harvard which included a prestigious Arthur Lehman Fellowship. Kohn again consulted Infeld, who without hesitation told him to communicate his regrets to Peierls, accept the offer from Harvard, and try to work for the young physics superstar Julian Schwinger.[^35]
Accordingly, a somewhat insecure twenty-three year old Walter Kohn arrived on the Harvard campus in the fall of 1946 as one of a group of about thirty first-year graduate students. The twenty-eight year old Schwinger had joined the Harvard faculty the previous spring and immediately began teaching a three-semester sequence of courses on special topics in theoretical physics (Schweber 1994). Kohn and his cohort stepped into the middle of this sequence as a supplement to their required courses in classical mechanics, electrodynamics, quantum mechanics, and statistical mechanics. In principle, Schwinger’s course was devoted to nuclear physics. In practice, he devoted part of the time to a highly personal exposition of quantum mechanics in the style of Dirac (1935) and the rest of the time was given over to (Anderson 1999)
> essentially everything Schwinger knew about. All about Green functions, all about nuclear physics and so on. All the numerical tricks he had devised to solve quantum mechanical problems and nuclear physics problems. . . there was a lot of physics and there were a lot of variational techniques, for example to solve the deuteron. . . . He was also starting to build the machinery that was going to solve the problems of quantum electrodynamics. We were treated to a lot of that machinery.
Kohn himself has given one of the best descriptions of Schwinger’s teaching style (Kohn 1996):
> Attending one of his formal lectures was comparable to hearing a new major concert by a very great composer, flawlessly performed by the composer himself. . . Old and new material were treated from fresh points of view and organized in magnificent overall structures. The delivery was magisterial, even, carefully worded, irresistible like a mighty river. He commanded the attention of his audience entirely from the content and form of his material, and by his personal mastery of it, without a touch of dramatization.
Quite early on, the chairman of the Harvard Physics department, John Van Vleck, approached Kohn and asked him whether he would like to work with him on a solid state physics problem.[^36] Kohn was not interested in solid state physics and instead presented himself to Schwinger as Infeld had urged him to do.[^37] He described to Schwinger his experiences in Toronto and (Kohn 1998, 2001b)
> Luckily for me, we shared a common interest in the variational methods of theoretical physics. . . . He accepted me within minutes as one of his 10 PhD students. He suggested I should try to develop a Green function variational principle for three-body scattering problems, like low-energy neutron-deuteron scattering, while warning me ominously that he himself had tried and failed.
Schwinger was an acknowledged expert in the use of both variational principles and Green functions to solve a wide variety of problems. In Eq. (\[one\]), an energy functional $E[\psi]$ with the stationary property $\delta E=0$ made it possible to estimate the ground state energy of a bound-electron system like helium. For a scattering problem, the energy is known and one is led to seek stationary variational functionals for other quantities. An example is the Green function, an energy-dependent operator defined in terms of the system Hamiltonian by $$\label{three}
G(E)={1\over E-H}.$$ For two-body scattering, one writes $H=H_f + V({\bf r})$ where $H_f \varphi = E_f \varphi$ is the Schrödinger equation for a free particle and $V({\bf r})$ is the potential responsible for the scattering. Using a coordinate-space representation of the corresponding free-particle Green function $G_f$, the scattered wave function $\psi$ satisfies the integral equation,[^38] $$\label{four}
\psi({\bf r})= \varphi({\bf r}) + \int d^{\,3}r' G_f({\bf r},{\bf r'}|E_f) V({\bf r'}) \psi({\bf r'}).$$
Kohn worked on the three-body Green function problem for half a year before abandoning it.[^39] Instead, he generalized Eq. (\[one\]) and developed a variational principle for the two-body scattering phase shift, a quantity which characterizes the final state when two particles interact via a short-range potential.[^40] He also derived a variational principle for the scattering amplitude for two-particle scattering with an arbitrary interaction potential. For both cases, Kohn borrowed from the Rayleigh-Ritz method and expanded the trial scattering wave function in a set of basis functions with the correct long-distance behavior. Finally, he derived a variational principle for the elements of the scattering matrix for the special case of nuclear collisions where multiple disintegrations are energetically forbidden. It is interesting that Kohn made no use of Schwinger’s “beloved Green functions” (Kohn 1998) in his thesis, “Collisions of light nuclei”, or in the published version of his thesis (Kohn 1948). He did, however, use his scattering amplitude variational principle to [*rederive*]{} an alternative variational principle for the phase shift that Schwinger had derived in his spring 1947 theoretical physics class using Eq. (\[three\]) and reported at a meeting of the American Physical Society (Schwinger 1947). The latter is commonly called “Schwinger’s variational principle for scattering" (Adhikari 1998, Nesbet 2003).
Julian Schwinger (ca. late 1940’s) was Kohn’s PhD\
supervisor at Harvard University.\
Source unknown.
Kohn did not have a close personal relationship with his advisor.[^41] None of Schwinger’s students did, in part because it was notoriously difficult to schedule a personal meeting with him. Kohn met with him only “a few times a year” (Kohn 1996) and according to John David Jackson, an MIT graduate student who listened to Schwinger’s lectures at Harvard, there was one occasion where “Kohn was miffed by Julian’s unavailability. He completed his thesis, wrote up a paper, and submitted it to [*Physical Review*]{} without ever consulting him” (Mehra and Milton 2000). Such feelings must have passed quickly because, in a moving tribute at a memorial symposium after Schwinger’s death, Kohn makes it clear that (Kohn 1996)
> It was during these meetings, sometimes more than 2 hours long, that I learned the most from him. . . . to dig for the essential; to pay attention to the experimental facts; to try to say something precise and operationally meaningful, even if one cannot calculate everything [*a priori*]{}; not to be satisfied until one has embedded his ideas in a coherent, logical, and aesthetically satisfying structure. . . . I cannot even imagine my subsequent scientific life without Julian’s example and teaching.
Besides Schwinger, Kohn benefitted from members of his graduate student cohort who either contributed materially to his education at Harvard or who played an important role subsequently. One group consisted of fellow Schwinger students: Kenneth Case, Frederic de Hoffmann, Roy Glauber, Julian Eisenstein, Ben Mottelson, and Fritz Rohrlich. Another group did their PhD work in other areas of physics: the theorists Thomas Kuhn, Rolf Landauer, and Philip Anderson, and the experimentalists Nicolaas Bloembergen, George Pake, and Charles Slichter. Joaquin Luttinger, another MIT graduate student who made the short trip to Harvard to audit Schwinger’s classes, later became a close friend and a scientific collaborator. All these members of Kohn’s student network went on to have successful scientific careers.[^42] Glauber, Mottelson, Anderson, and Bloembergen won Nobel Prizes themselves.
Walter’s life changed profoundly in two important ways when he accepted an offer by Schwinger to remain at Harvard as a post-doctoral fellow. First, the income from this job permitted him to bring to Boston and marry Lois Mary Adams, a former nursing student he had met at the University of Toronto who had been working in New York City while he finished his PhD (SDUT 2010). A baby daughter soon arrived and family responsibilities were added to the research and teaching responsibilities that came with his position as Schwinger’s assistant. The research project he undertook was an investigation of the electromagnetic properties of mesons done in collaboration with fellow Schwinger post-doc Sidney Borowitz. His teaching consisted of an introductory physics course in the summers of 1949-1950 and a junior/senior level classical mechanics course in the summer of 1950.[^43]
The second profound change in Kohn’s life occurred through the good offices of John Van Vleck, the solid state theorist he had rebuffed as a thesis supervisor. Van Vleck re-enters the story because Kohn supplemented his summer 1949 income by working for the Polaroid Corporation at their Cambridge, Massachusetts research laboratory. His job was to discover the mechanism whereby high-energy charged particles produce an image when they impinge on photographic plates.[^44] This task required a knowledge of solid state physics, which he acquired by reading Frederick Seitz’ [*The Modern Theory of Solids*]{} (1940) and consulting with Van Vleck when necessary.[^45]
Harvard’s John Van Vleck facilitated Kohn’s\
transition from nuclear physics to solid state physics.\
Courtesy of the UW-Madison Archives.
By this time, Kohn’s first paper with Borowitz had appeared (Borowitz and Kohn 1949) and he had applied to the National Research Council for a Fellowship to spend the 1950-1951 academic year with Wolfgang Pauli at the Eidgenössische Technische Hochschule in Zürich. Nevertheless, he was painfully aware that he had so far made only a “very minor contribution” to field-theory research. This fact, and the stunning quality of the latest achievements by Schwinger and Richard Feynman, made it easy for him when (Kohn 1998)
> Van Vleck explained to me that he was about to take a leave of absence and ‘since you are familiar with solid state physics’, he asked me if I could teach a graduate course on this subject he had planned to offer \[for the fall 1949 semester\]. This time, frustrated with my work on quantum field theory, I agreed.
Kohn not only taught the solid-state physics course, he collaborated with Harvard graduate student Richard Allan Silverman to find approximate numerical solutions of the Schrödinger equation for the purpose of calculating the cohesive energy of metallic lithium (Silverman and Kohn 1950).[^46] Using the same numerical data, he and Bloembergen estimated the Knight shift for lithium, the latter being a measure of the electron wave function amplitude at the atomic nucleus accessible to experimenters using magnetic resonance techniques (Kohn and Bloembergen 1950). It is notable that Silverman and Kohn conclude with the remark, “One of us (W. Kohn) is investigating the cohesive energy by means of a variation iteration procedure based on the integral equation \[Eq. (\[four\]) of this paper\] and a Green function appropriate to a periodic lattice."
Kohn, a naturalized Canadian citizen since 1943, looked everywhere in Canada and the United States for an entry-level academic position. Nothing turned up, but an early 1950 interview trip to the Westinghouse Research Laboratory in Pittsburgh, Pennsylvania bore fruit even though his foreign citizenship precluded a job offer from Westinghouse. In Pittsburgh, Kohn stayed at the home of Alfred Schild, a friend from Toronto who had found a job teaching mathematics at the Carnegie Institute of Technology. Schild told him that the chairman of the physics department, Frederick Seitz, had just resigned and was moving his solid state group to the University of Illinois (Seitz 1994). Perhaps there was an opportunity at Carnegie Tech itself. The new chair, nuclear physicist Edward Creutz, interviewed Kohn and offered him a job as an Assistant Professor 48 hours later.[^47] It turned out that Creutz needed someone who could teach solid state physics and mentor a few graduate students who had lost their advisors when the Seitz group left (Kohn 1998). Kohn was delighted to accept.
Portrait of the Physicist as a Young Man
========================================
Walter Kohn was thinking about the start of his academic career at Carnegie Tech when, in March 1950, the National Research Council approved his application for a fellowship to spend a year in Europe. Ed Creutz agreed to a one-year leave of absence, but only if Walter agreed to teach solid state physics for the fall 1950 semester. At the same time, Kohn was having second thoughts about his original plan to work with Wolfgang Pauli. This led him to seek and secure the approvals needed to switch the venue for his fellowship year from Zürich to Niels Bohr’s Institute for Theoretical Physics in Copenhagen. Accordingly, Kohn moved to Pittsburgh, taught his course, and left for Copenhagen at the end of the term.[^48] When he arrived at the Institute in January 1951, Walter Kohn was an expert in scattering theory who had begun to think of himself as a solid state physicist. Unfortunately, “no one in Copenhagen, including Niels Bohr, had even heard the expression “solid state physics’" (Kohn 1998).
Kohn managed to publish two papers that year, but more important to his future was the fact that the Institute attracted a steady stream of short-term and long-term visitors from around the world from whom he could learn new physics.[^49] It was good luck for Kohn that post-war freedom of movement motivated Bohr to organize a meeting for all foreign physicists who had ever worked at the Institute (Rozental 1967). The resulting Conference on Problems in Quantum Physics (July 6-10 1951) was attended by an outstanding collection of theoretical physicists, many of whom Walter was able to incorporate into his expanding professional network.[^50] A few weeks later, Kohn was tapped to lecture on solid state physics for two weeks at the first Summer School of Theoretical Physics organized by Cécile DeWitt at Les Houches, near Chamonix in the French Alps.[^51]
At the end of 1951, Bohr wrote a formal evaluation which concluded that (Bohr 1951)
> Dr. Kohn has proved himself a highly qualified theoretical physicist with great knowledge of a wide field of problems. His ability to stimulate others in their work and his willingness to assist them with his knowledge has been of great value to the many members of our group.
This good opinion led Bohr to arrange a Rask [Ø]{}rsted Foundation fellowship for Kohn so he could remain in Copenhagen through the summer of 1952. This was welcome news because Walter and his family enjoyed living in Denmark. Moreover, he had just begun a scientific collaboration with Res Jost, a Swiss mathematical physicist five years his senior who had lectured on quantum field theory at the Les Houches summer school.[^52] Jost was interested in scattering theory and his “predilection for mathematical rigor” (Enz 2002) struck a responsive chord in Kohn. Together, the two theorists completed three papers (including an ‘inverse scattering problem’ where one deduces characteristics of the scattering potential from phase shift information) before Kohn returned to Carnegie Tech to begin the 1952-1953 academic year (Jost and Kohn, 1952a, 1952b, 1953). On his own, Kohn studied the validity of the Born expansion for scattering (Kohn 1952a) and a non-Green function variational principle for electron waves in a periodic potential (Kohn 1952b).
Back in Pittsburgh, the Physics Department had changed somewhat during Walter’s absence. The senior experimentalist Immanuel Estermann had left to head the physics section of the Office of Naval Research (ONR) and his last PhD student, Simeon Friedberg, had taken over his low-temperature physics laboratory. A senior theorist, Gian-Carlo Wick, had joined the faculty from Berkeley and Roman Smoluchowski, an expert in the theory of defects in solids, had transferred to the Physics Department from the Metallurgy Department. A young experimenter, Jacob Goldman, and a young theorist, Paul Marcus, had joined the solid state physics group to complement the senior experimentalist Emerson Pugh. One familiar face was Norman Rostoker, a Toronto native who had graduated from his hometown university as a physics major one year behind Kohn and then received his PhD at Carnegie Tech under Pugh’s supervision (Rostoker 2013). Kohn and Rostoker had become friends during the fall 1950 semester and Norman was still working in the Physics Department as a post-graduate research scientist when Kohn returned from Europe.
The fall 1952 semester found Kohn teaching thermodynamics to undergraduates and nuclear physics to graduate students. He was also named as co-principal investigator with Jack Goldman on an ONR contract to conduct solid state research.[^53] His main research project was to develop a Green function method to calculate the energy band structure for crystalline solids. In other words, he wanted to use Eq. (\[three\]) to solve the Schrödinger equation to find the energy eigenfunctions and eigenvalues for electrons moving in a periodic potential. A distraction arose in the spring 1953 semester when Carnegie Tech learned that Walter had received job offers from the Department of Mathematics at McGill University in Montreal and the Physical Research Department at Bell Telephone Laboratories in Murray Hill, New Jersey. Evidently, others besides Niels Bohr had formed a very positive impression of this new Assistant Professor. In the end, Carnegie Tech retained his services by promoting him to Associate Professor (WKP 1953a) after only three semesters of academic service.
Kohn had brought to Carnegie Tech the germ of his Green function method to solve the electron band-structure problem. He recruited Norman Rostoker to help with the numerical calculations and that activity continued (part-time) while Walter was in Copenhagen.[^54] The work accelerated when Kohn returned to Pittsburgh and he reported their still-unpublished results at two invited talks, one at the June 1953 Summer Meeting of the American Physical Society and one at a July 1953 Gordon Research Conference devoted to the Physics and Chemistry of Metals. The latter was a particularly prestigious venue and it is notable that of the seven theorists invited to speak, the three youngest (by far) were Walter Kohn, Jacques Friedel, and David Pines (WKP 1953b). Friedel, an expert on the theory of metals and alloys, and Pines, an expert on electron-electron interactions in solids, had both published half a dozen papers in their fields by the time of the Gordon Conference. It is an indication of Kohn’s rising reputation that he had published only one full-length paper in solid state physics by this time. Kohn’s Green function paper (Kohn and Rostoker 1954) finally appeared in the June 1 1954 issue of the journal [*Physical Review*]{}. Therein, he and Rostoker (Kohn 1998)
> developed a theory for the energy band structure of electrons in solids harking back to my earlier experience with scattering, Green functions, and variational methods. We showed how to determine the band structure from a knowledge of purely geometric structure constants and a small number ( $\sim 3$) of scattering phase shifts of the potential in a single sphericalized cell.
It happens that the same basic idea had been published several years earlier by the Dutch physicist Jan Korringa. However, Korringa (1947) included no numerical applications and his paper went largely unnoticed.[^55] Kohn and Rostoker illustrated their method by calculating the energy as a function of wave number for the 2s conduction band of lithium metal and comparing their results with previous calculations in the literature. It is entirely characteristic of Kohn that he did [*not*]{} take his band structure formalism and begin applying it to one material after another.[^56] Instead, he made one use of his lithium results (Kohn 1954) and then proceeded to expand his personal research activities into other areas of solid state physics. To understand the choices he made, we interrupt our narrative briefly to survey the research agenda of solid state physics in the mid-1950’s.
Wartime developments in computers, instrumentation, and materials processing had a profound effect on the issues addressed by solid state physicists at the mid-point of the twentieth century.[^57] In June 1954, the National Science Foundation and the American Society for Engineering Education sponsored a meeting at Carnegie Tech attended by representatives from forty-five colleges and universities and several industrial and government laboratories. According to the conference co-chair, Professor Jack Goldman,[^58] the purpose of the meeting was to “make more definitive the state of knowledge of solid state physics and the levels at which various parts of it may be expected to be integrated into engineering education" (Goldman 1957). To this end, the conferees identified six broad areas of active solid state physics research: the structure of crystalline matter, metals and alloys, surfaces, magnetism, semiconductors and dielectrics, and non-crystalline materials.
One needed to attend conferences reserved for specialists to learn the cutting-edge issues in each area. Happily, the same purpose was soon served by the articles published in [*Solid State Physics*]{}, a series of volumes initiated by Frederick Seitz and David Turnbull in 1954 to provide “broad surveys of fields of advanced research that serve to inform and stimulate the more experienced investigator” (Seitz and Turnbull 1955). The inaugural volume contained articles devoted to five issues: the band structure problem, the properties of valence semiconductors, electron-electron interactions, cohesion in solids, and the theory of order-disorder phase transitions. Kohn had already made a significant contribution to band structure theory and he now added to his repertoire research projects devoted to semiconductors and to the effects of the electron-electron interaction (soon relabeled many-body physics). The total energy (cohesion) problem became a central concern when he developed density functional theory a decade later.[^59]
Kohn became interested in semiconductors because his flirtation with permanent employment at Bell Telephone Laboratories led to a summer consulting arrangement that lasted from 1953 to 1966. His first summer project, a theoretical study of the damage done to germanium crystals after bombardment by energetic electrons, was motivated by experimental results obtained at Bell Labs by Walter Brown and Robert Fletcher (Brown [*et al.*]{} 1953). These experiments, in turn, were part of an enormous in-house effort to investigate the properties of the elemental semiconductors germanium and silicon after the 1947 invention of the transistor at Bell Labs by John Bardeen, Walter Brattain, and William Shockley (Millman 1983). Kohn was happy to return to Bell Labs summer after summer, both to gain access to exciting experimental results and for the opportunity to interact with senior theorists on the Bell Labs staff like Conyers Herring and Gregory Wannier and junior theorists closer to him in age like Peter Wolff and his Harvard classmate Philip Anderson. He would later remark that “I owe this institution my growing up from amateur to professional" (Kohn 1998).
Walter’s 1954 ‘summer vacation’ at Bell Labs was particulary important because he began a long-lasting scientific collaboration with Joaquin Luttinger, another consultant to the semiconductor group.[^60] They worked together to create a rigorous “effective mass theory” for the electronic energy levels produced when impurity atoms are purposely substituted for germanium or silicon atoms in pure crystals of the latter. The crucial importance of these impurities and their quantum mechanical states to the extraordinary electrical properties of semiconductors had been explained qualitatively by William Shockley in his seminal treatise, [*Electrons and Holes in Semiconductors*]{} (1950). However, careful electron spin resonance and cyclotron resonance experiments at Bell Labs and elsewhere demanded a quantitative theory. Not for the first time and not for the last time, Kohn combined the creation of a novel and sophisticated theory with variational calculations designed to produce numbers for comparison with measurements for specific material systems. That fall, Kohn and Luttinger completed three substantial papers in semiconductor physics (Luttinger and Kohn 1955, Kohn and Luttinger 1955a,b) and thereby finished in a virtual dead heat with Berkeley solid state theorist, Charles Kittel, who published similar work independently (Kittel and Mitchell 1954, Dresselhaus [*et al.*]{} 1955).[^61]
Joaquin Luttinger was Kohn’s principal\
scientific collaborator in the 1950’s.\
Courtesy of Walter Kohn.
In the spring of 1955, Walter worked hard to convince the British theoretical solid state physicist Harry Jones to accept an offer of a chaired position at Carnegie Tech. Jones had co-authored (with Nevill Mott) the influential book [*The Theory of the Properties of Metals and Alloys*]{} (1934) and he had spent the spring 1954 semester as a visiting professor in Kohn’s department. Kohn wrote to Jones and pointed out that “all of us in solid state physics, as well as all the people in metallurgy, would be delighted to see you come here. With your field of interest, I honestly think that probably no other school in this country could offer you better opportunities for creative work along research and teaching lines" (WKP 1955). Jones ultimately declined for personal reasons. The undeterred Kohn pursued the physics of semiconductors and metals simultaneously and submitted two manuscripts to the [*Physical Review*]{}. The first paper, co-authored with his first Carnegie Tech Ph.D. student, Daniel Schechter, reported calculations for the wave functions and energy levels associated with shallow (weakly bound) impurity states in germanium.[^62] The second paper reported a Knight shift calculation for metallic sodium with Terje Kjeldaas, a full-time employee of the Westinghouse Research Laboratories in East Pittsburgh (Kohn and Schechter 1955, Kjeldaas and Kohn 1956).[^63]
For later reference, it is important to note that Kohn acted as an informal consultant to the transition-metal magnetism groups of his faculty colleagues Emerson Pugh and Jack Goldman. Goldman’s PhD student Anthony Arrott recalls Kohn’s surprise when Arrott successfully used a simple energy band model to analyze his magnetic data for concentrated Cu-Ni alloys. At Arrott’s oral thesis defense, Kohn asked him a question that foreshadowed his motivation to invent density functional theory ten years later: “how can you use a band model when the potential felt by the electrons is not periodic” (Arrott 2013)?
As 1955 turned into 1956, Walter found himself thinking more and more about the effective mass equation he had derived with Luttinger for the energy levels of impurity states in silicon. Their “one-particle” method treated the impurity atom as unaware of its silicon host except for whatever influence could be captured by two numbers: an effective mass $m^\ast$ which parameterized the arrangement of atoms in the silicon crystal and an effective dielectric constant $\kappa^\ast$ which parameterized the ability of the silicon conduction electrons to “screen” or “shield” the Coulomb potential produced by a positively charge impurity embedded in the semiconductor.[^64] Why then did the energy levels calculated using the effective mass equation agree so very well with the energy levels measured in the laboratory? Surely, he reasoned, it must be that “this equation can be derived from some very general properties of the entire many-electron wave function without any recourse to the one-particle picture" (Kohn 1957). For the first time, Kohn attacked the quantum-mechanical “many-body problem” in solid state physics where the repulsive Coulomb interaction between all pairs of electrons is taken seriously. Working alone, he managed to demonstrate his assertion for a hypothetical situation where the charge on the impurity nucleus exceeds that of the other nuclei by an infinitesimal amount. He announced this result in a comprehensive review of the Kohn-Luttinger theory written for Seitz and Turnbull’s [*Solid State Physics*]{} series (Kohn 1957a). A full account appeared later (Kohn 1957b).
Meanwhile, back at Bell Labs, the resident theoretical physicists had successfully convinced the vice-president for research, William Oliver Baker, to create (Anderson 2011)
> a separate ‘super-department’ for theorists . . . with post-doctoral fellows, a rotating boss on whose identity we were consulted, sabbaticals, a travel budget under our control, and a spectacular summer visitor program. . . . One of the reasons for our success with management was the fact that for several years we had had Walter Kohn and Quin Luttinger as regular summer visitors and they had become so useful that our bosses desperately wanted to attract them permanently.
The advent of a ‘spectacular summer visitor program’ meant that an unusually large number of theoretical physicists passed through and interacted with Kohn and Luttinger during their 1956 summer stay at the Labs.[^65] A hot topic was the effect on the properties of semiconductors when one systematically increased the number of impurities present. When the impurity concentration is low and the temperature is low, it was well known that electron scattering from impurities is the main source of a solid’s electrical resistance. What happens when the concentration of impurities is high? Luttinger had been thinking about the general subject of electric current flow in solids already in connection with his studies of the Hall effect in ferromagnets and it was not difficult to convince Kohn to work with him to produce as rigorous a theory of electrical conductivity as they could.[^66] After all, Luttinger’s understanding of the experimental facts for the Hall effect came directly from a review paper written by Kohn’s Carnegie Tech colleagues Emerson Pugh and Norman Rostoker (Pugh and Rostoker 1951). The fruits of that summer’s labors were two long papers on the quantum theory of electrical transport in solids[^67] (Kohn and Luttinger 1957, Luttinger and Kohn 1958). In contrast to their effective mass theory work, which more reflected Kohn’s style to address important physical questions with intuition, a good idea, and mathematical elegance, the transport theory papers more reflected Luttinger’s preference for general formalism and mathematical rigor. In this way, the two young theorists enlarged each others’ perspectives of their craft.
For Walter Kohn, professor of physics, the calendar change from 1956 to 1957 meant little more than a change in his teaching assignment from statistical mechanics for physics majors to classical physics for engineers. However, for Walter Kohn, solid-state physicist, the new year saw changes in his field that had a profound effect on his future research efforts. In the words of Canadian physicist Allan Griffin, 1957 was a “magic year” when “the way all theoretical physicists thought about interacting many-body systems underwent a revolution” (Griffin 2007). The key event was the realization that the methods of quantum field theory could be applied with equal success to study the many-electron problem in solid-state physics.[^68] In particular, diagrammatic methods like those invented by Richard Feynman (1949) to study quantum electrodynamics made it possible to define a perturbation theory that remained consistent as the number of particles in a system increased. Feynman diagrams posed no problem for a Harvard PhD like Kohn who was both well-trained in quantum mechanics and familiar with quantum field theory from Julian Schwinger’s lectures. He also had a ready-made problem: his own desire to understand the success of the Kohn-Luttinger effective mass equation from a many-body point of view. A breakthrough paper by Jeffrey Goldstone (1957) provided all the technical details he needed.
Walter became a naturalized citizen of the United States in 1957 and he had arranged a sabbatical leave from Carnegie Tech for the 1957-1958 academic year. He spent the fall of 1957 at the Physics Department of the University of Pennsylvania and it was there that he wrote up Kohn (1958), his first contribution to the many-body revolution–a diagrammatic analysis of the static dielectric constant of an insulator. At the end of this paper, Kohn acknowledges “stimulating conversations” with Keith Brueckner, a senior member of Penn’s Physics Department whose own thinking about the quantum many-body problem had stimulated Goldstone’s work.[^69] Brueckner, in turn, led an effort by his Penn colleagues to hire Kohn away from Carnegie Tech. A similar effort was mounted by the Physics Department of the University of Chicago (WKP 1957). Kohn took these overtures seriously and Carnegie Tech responded by awarding him tenure and promoting him to the rank of Professor with a substantial increase in salary (WKP 1958). Walter made his decision to return to Pittsburgh while completing his sabbatical and spending the spring 1958 semester with Harry Jones and his group at the Department of Mathematics of the Imperial College of Science and Technology in London.
The many-body revolution introduced new ideas into solid state physics like the quasi-particles of Lev Landau (1956) and new objects for study like the one-particle and two-particle Green functions exploited by Victor Galitskii and Arkday Migdal (1958).[^70] The physical and mathematical elegance of this subject caused some solid state theorists to focus their attention exclusively on problems where many-body effects dominate the physics.[^71] Kohn did not follow this particular path because not every problem that interested him demanded a many-body analysis. For example, a theorem due to Felix Bloch (1928) demonstrated that the eigenfunctions and energy eigenvalues of the Schrödinger band for a spatially periodic system like a crystal have the form $$\label{five}
\psi_{\bf k}({\bf r}) = \exp(i{\bf k}\cdot {\bf r})u_{\bf k}({\bf r})~~~~~~~~~~{\rm and}~~~~~~~~~~E({\bf k}),$$ where the three quantum numbers collected in the vector ${\bf k}=(k_x,k_y,k_z)$ are real numbers confined to a finite volume of the three-dimensional ${\bf k}$-space called the Brillouin zone, and the function $u_{\bf k}({\bf r})$ has the spatial periodicity of the crystal. While at Imperial College, Kohn performed an extensive study of the properties of the Bloch solutions when ${\bf k}$ is a complex-valued vector. This allowed him to make precise statements about the exponential decay of a class of spatially localized wave functions first introduced by Wannier (1937). In a separate project, Kohn analyzed the motion of Bloch electrons in a magnetic field with more rigor than had been done previously. He succeeded to show that an approximation for this problem first made by Peierls (1933) had a much broader range of validity than previously thought.
Kohn’s Imperial College projects in mathematical physics did not disengage him from the more practical aspects of solid state physics.[^72] For example, he was invited to the 1958 International Conference on Semiconductors in Rochester, New York, to report his many-body analysis of the effective mass equation for shallow impurity states. While there, he attended a session devoted to the calculation of energy bands and listened to a talk by fellow-theorist James Phillips. Kohn asked Phillips whether he was doing “physics or magic” because Phillips’ ‘pseudopotential method’ reproduced the results of much more elaborate band structure calculations for silicon and germanium using only three parameters for each (Bassani and Tosi 1988). Similarly, Kohn paid careful attention when the [*phonon spectrum*]{} of a crystal was measured for the first time (Brockhouse and Stewart 1958) and also when the [*Fermi surface*]{} of a metal was measured for the first time (Pippard 1957, Gold, 1958).[^73]
These experimental breakthroughs stimulated Walter’s scientific imagination and he soon completed a simple and elegant analysis which predicted that the phonon spectrum of a metal possesses observable “anomalies” which depend only on the existence and shape of the Fermi surface. More precisely, the Fermi surface locates a singularity in a linear response function which describes the ability of the conduction electrons to screen the ions which move during a lattice vibration. The short communication which described what came to be known as “Kohn anomalies” was one of four manuscripts he submitted for publication during the 1958-1959 academic year at Carnegie Tech (Kohn 1959a,b,c, Ambegaokar and Kohn 1959). Two longer papers described the results of the projects begun at Imperial College and the paper co-authored by his PhD student Vinay Ambegaokar reported a new sum rule for insulators. The Kohn anomaly and Ambegaokar papers appeared in the same issue of [*Physical Review Letters*]{}, a journal spun-off from the [*Physical Review*]{} to provide “speedy publication” of “new discoveries of major importance and for significant contributions to highly active and rapidly changing lines of research in basic physics" (Goudsmit and Trigg 1964).
Ambegaokar (2004, 2013) recalls that
> Before going on leave \[to Pennsylvania\], Walter advised me to take a second course in quantum mechanics taught by Gian-Carlo Wick even though I had not finished a first course. . . . Upon his return \[from England\], he suggested a research project that was very much to my taste. We met for at least an hour a week and his supervision was both precise and constructive. He thought hard during our meetings to keep the project moving along. . . . \[Walter\] could be formal as a person, but he opened up considerably with people he respected. He got me a summer job at Bell Labs and we played tennis together there frequently.
Another student, James Langer, was an undergraduate physics major at Carnegie Tech from 1951-1955. Langer never took a formal course from Kohn, but in his senior year, “Walter somehow became my private instructor for a year-long supervised reading course. We went through the first edition of Leonard Schiff’s classic text [*Quantum Mechanics*]{} essentially cover to cover” (Langer 2003). Langer won a Marshall Scholarship to attend graduate school in Great Britain and Kohn directed him to Rudolf Peierls at the University of Birmingham.[^74] Langer earned his PhD for a problem in nuclear physics and then returned to Carnegie Tech as an Instructor in the fall of 1958. For the next year, he and Seymour Vosko (a recent PhD student of Gian-Carlo Wick) functioned as post-doctoral fellows in Kohn’s theoretical solid-state physics group.
The problem Kohn set for Langer and Vosko was the shielding of a positively charged impurity embedded in a metal host. This is the analog of the problem Walter had studied previously for the case of a non-metal host. Kohn’s many-body perturbation theory calculation for the non-metal case confirmed the classical result that the Coulomb potential $q/r$ at distance $r$ from a point charge $q$ in vacuum changes to $q/\kappa r$ when the point charge is embedded in an insulator with dielectric constant $\kappa$. Nevill Mott (1936) studied the screening of a point charge in a metal in connection with a calculation of the electrical resistivity of dilute metal alloys. He used semi-classical Thomas-Fermi theory (March 1957) and showed that the potential $q/r$ in vacuum changes to $(q/r)\exp(-r/\ell)$ in a metal. The screening length $\ell$ depends on the density of conduction electrons and takes the value $1$-$2~ {\rm \AA}$ in a typical metal. Jacques Friedel (1958) revisited this problem using a scattering theory method and found that the disturbance of the electronic charge density at a distance $r$ from the impurity charge, $\delta n(r)$, varied as $$\label{fivehalf}
\delta n(r) \propto {\cos(2k_F r + \Delta) \over r^3},$$ where $k_F$ and $\Delta$ are two constants. The $1/r^3$ decay of this function falls off much more slowly with distance than the exponential variation predicted by Mott’s theory and thus implies that the effect of isolated impurities might not be completely screened at the position of nearby atoms. The oscillatory behavior in Eq. (\[fivehalf\]) is an intrinsically quantum effect. Kohn did not fully believe Friedel’s result and therefore asked his post-docs to attack the problem themselves (Kohn 2012b). Much to his surprise, Langer and Vosko (1959) fully confirmed Friedel’s formula using diagrammatic many-body perturbation theory.[^75] Kohn promptly applied their results to a quantitative calculation of the magnitude of the nuclear magnetic resonance signal in copper metal when small amounts of impurity atoms are introduced (Kohn and Vosko 1960). Confirmation of the Kohn-Vosko theory came from comparison with experiments performed by Theodore Rowland at the Union Carbide Metals Company (Rowland 1960). Kohn knew Rowland from his Harvard days when Rowland was a PhD student of Nicolaas Bloembergen.
The summer of 1959 reunited Walter and Quin Luttinger at Bell Laboratories. Once again, the pair produced an interesting paper (Kohn and Luttinger 1960) and once again, Kohn grappled with an offer from Keith Brueckner to leave Carnegie Tech. This time, however, Brueckner was not soliciting on behalf of the University of Pennsylvania. He had resigned from Penn a few months previously and his mission now was to convince Walter to help him create the Physics Department at the soon-to-open University of California at La Jolla (later San Diego). Earlier in the year, Brueckner had flown Kohn to the beautiful site of the proposed campus to meet and hear the vision of its principal advocate, Roger Revelle, the Director of the Scripps Institute for Oceanography. The salary was attractive and La Jolla seemed like an ideal place to relocate his wife and two elementary school-aged daughters.[^76] Moreover, Ed Creutz, the man who had hired Kohn at Carnegie Tech and who was now Vice-President of Research at the General Atomics division of the General Dynamics Corporation in San Diego, had recently concluded a consulting contract with him. This time, the allure was too great and Walter agreed to sign on. His only condition was that Keith Brueckner must serve as chair of the new department (Brueckner 2013).
Walter’s research group had grown to include three PhD students and four post-doctoral fellows by the fall of 1959 when he submitted his resignation to the President of Carnegie Tech. His senior student, Vinay Ambegaokar, was one semester from graduation. His junior students, Larry Glasser and Edwin Woll, Jr. were not too far from the beginning of their research so Walter invited them both to join him in La Jolla. Woll chose to accompany his advisor; Glasser remained in Pittsburgh and finished his degree with Assistant Professor J. Michael Radcliffe (Glasser 2013). Post-docs Hiroshi Hasegawa and Robert Howard had arrived the previous fall from Tokyo and Oxford, respectively, and worked together on a problem motivated by Kohn’s experience with shallow donor states in semiconductors (Hasegawa 2004). Hasegawa accompanied Kohn to San Diego while Howard moved on to a permanent position at the National Bureau of Standards in Washington, D.C. Post-docs Emile Daniel and Anthony Houghton were former PhD students of Jacques Friedel in Paris and Geoffrey Chester (in the group of Rudolf Peierls) in Birmingham, respectively. Both began research projects in metal alloy physics before moving to the west coast. Kohn himself taught a graduate course in advanced solid state physics and gave a talk on “The Electron Theory of Solids” at a one-day “Solid State Symposium” in New York City sponsored by the American Institute of Physics for the benefit of science writers from national magazines and newspapers.[^77]
Even before leaving Pittsburgh, Walter worked hard to recruit faculty members to his new Physics Department in San Diego. He had immediate luck with three Bell Laboratories scientists, the statistical mechanician Harry Suhl, the nuclear magnetic resonance experimenter George Feher, and the superconductivity experimenter Bernd Matthias.[^78] In an October 26 1959 letter to Keith Brueckner, Kohn laments Quin Luttinger’s decision to choose Columbia University over UCSD and suggests several solid state and/or low-temperature physicists whom Brueckner should approach (WKP 1959b). On the theoretical side, he proposed J. Robert Schrieffer, the junior author of a breakthrough paper on the theory of superconductivity published in 1957, Volker Heine, a specialist in the electronic structure of metals from the University of Cambridge, and Philippe Nozières, an expert in many-body theory trained at Princeton by David Pines. On the experimental side, he suggested his old Harvard friend and magnetic resonance practitioner, Charles Slichter, and the liquid helium experts William Vinen and Russell Donnelly. As it turned out, none of these people joined the UCSD faculty.[^79]
Kohn’s arrival in San Diego in January 1960 coincided with his election as a Fellow of the American Physical Society (APS), the professional organization of American physicists.[^80] Resettlement and administrative issues dominated his time at first, so Kohn used his students, post-docs, and short-term visitors to pursue a research agenda now focused primarily on the physics of metals and alloys.[^81] A new post-doctoral fellow, Stephen Nettel, studied the spatial arrangement of electron spins in the ground state of a [*homogeneous and non-interacting electron gas*]{}, a much-studied hypothetical system composed of a collection of mobile electrons (with their mutual Coulomb interaction turned off) distributed throughout a uniform and static distribution of electrically neutralizing positive charge (Kohn and Nettel 1960). Emile Daniel and former post-doc Seymour Vosko used many-body perturbation theory to study the sharpness of the Fermi surface for a fully interacting electron gas at zero temperature and Tony Houghton studied the specific heat and spin susceptibility of a dilute alloy (Daniel and Vosko 1960, Houghton 1961).[^82] The numerical calculations of Daniel and Vosko confirmed Quin Luttinger’s analytic demonstration that electron-electron interactions do not destroy the sharpness of the Fermi surface of an electron gas. This result was reported by Luttinger at a (retrospectively famous) meeting attended by Kohn and Vosko on “The Fermi Surface” held at Cooperstown, New York on August 22-24, 1960. (Luttinger 1960, Harrison & Webb 1960).
The La Jolla campus of the University of California opened for business in the fall of 1960. There were no undergraduates (until 1964), but sixteen physics graduate students arrived and began taking classes. Walter Kohn taught a course from 4:30 PM - 6:00 PM on Thursdays and from 9:30 AM - 11:00 AM on Saturdays using the textbooks [*Thermodynamics*]{} by Herbert Callen (1960) and [*Elements of Statistical Mechanics*]{} by Dirk ter Haar (1954).[^83] The smallness of the new department and the setting just steps from the Pacific Ocean fostered considerable informality in the early days. Arnold Sherwood, a member of the first cohort of graduate students, recalled that “Kohn had a charming European formality even when he was trying to be informal” (Sherwood 2013). According to faculty member George Feher (2002):
> Occasionally a student would come to class in a bathing suit or scantily dressed. We didn’t mind it, except perhaps Walter Kohn, who was a bit more formal than the rest of us. But he couldn’t tell the student off lest he be called a stuffy professor. He finally solved this problem by telling the students that he didn’t mind their behavior but did not want them to acquire bad habits because some stuffy professor might take offense.
Kohn flew to New York City on February 1 1961 to attend the annual joint meeting of the American Physical Society and the American Association of Physics Teachers. He was there to accept the 1961 Oliver Buckley Solid State Physics Prize “for having extended and elucidated the foundations of the electron theory of solids" (PT 1961). This prize, endowed in 1952 by Bell Telephone Laboratories and named in honor its former president and board chairman, is awarded each year by the APS to “recognize and encourage outstanding theoretical or experimental contributions to solid state physics”. Some measure of the esteem carried by this honor may be judged from the fact that four of the eight persons who won the Buckley Prize before Kohn later won a Nobel Prize (William Shockley, John Bardeen, Clifford Shull, and Nicolaas Bloembergen).
A few months later, Kohn submitted for publication his forty-eighth scientific paper in sixteen years. The first paragraph of this paper gives a good indication of his style, perspective, and level of engagement at this point in this career (Kohn 1961):
> There has been considerable interest in recent months in the effects of the electron-electron interaction on the cyclotron resonance frequency and de Haas-van Alphen oscillations of a gas of electrons. As some of the theoretical treatments of these problems use very sophisticated methods, and others are based on incorrect qualitative reasoning, we wish here to present some simple considerations which we think shed some light on what has been a rather confusing situation.
The paper Walter cited as bringing “very sophisticated methods” to the issue at hand was written by his good friend and collaborator, Quin Luttinger (1961). A more precise statement would be that Luttinger’s work employed very sophisticated [*mathematical*]{} methods. Kohn’s paper exploited much simpler mathematics but rather sophisticated [*physical*]{} reasoning to reach the same conclusions.
At UCSD, Kohn initiated a weekly Theoretical Solid State Lunch where faculty members, post-docs, and long-term visitors made presentations to anyone who brought a bag lunch and cared to listen. If a short-term visitor addressed the group, Kohn would take the visitor out to lunch and invite his personal research group to come along. After lunch, everyone was invited to take a walk around the campus. Twenty-five years later, this behavior was parodied in a skit performed at a celebration to commemorate the anniversary of the founding of the UCSD Physics Department. The relevant dialog involves a visitor to the campus and a professor in the department (LJPS 1985):
> Visitor: And what are those figures I see in the misty distance? It looks like a giant duck followed in a straight line by giant ducklings.
>
> Professor: Nothing too strange. That’s Walter Kohn taking his students for a walk after lunch.
Sometime in the spring of 1961, Keith Brueckner announced that he was stepping down as the chair of the Physics Department after one year of service. He had accepted the position of vice-president and technical director of the Institute for Defense Analyses in Washington, D.C. (Brueckner 2013) Therefore, against his expressed desire and much to his chagrin, Walter Kohn found himself in the position of chair for the 1961 fall semester. As a department head at a start-up university, Kohn enjoyed opportunities and faced challenges that do not ordinarily arise for administrators at established universities. He was permitted to hire a dozen new faculty members (which doubled the size of his department), but the offices and laboratories he could offer new recruits occupied temporary space that would soon revert to the Scripps Institution of Oceanography. One interesting hire was his friend Norman Rostoker, who had turned himself into a plasma physicist at General Atomics since he and Walter had collaborated at Carnegie Tech. The senior faculty members recruited by Brueckner and Kohn had excellent research records, but many came from industrial or government laboratories with no teaching experience. It was Walter’s responsibility to ensure that competent instructors staffed the courses offered to the first few classes of graduate students. At the same time, he maintained a research group of never less than five persons (graduate students, post-doctoral fellows, and visitors), served as an Associate Editor of the Journal of Mathematical Physics, team-taught a course on “Advanced Solid State Physics”, presented a university-wide lecture on “New Viewpoints in the Theory of Matter”, and submitted four research papers.[^84]
Within his group, Edwin Woll, Jr., Kohn’s PhD student from Carnegie Tech was making good progress with semi-quantitative calculations of Kohn’s phonon anomaly in the metals aluminum, sodium, and lead. Walter also began working with Michael Greene and Max Luming, two PhD candidates from the 1960 crop of UCSD graduate students, and Chanchal Majumdar, a student from the 1961 class. Greene was tasked to use scattering theory to compute the resistivity of liquid alkali metals. Max Luming involved himself in calculations of the orbital susceptibility of dilute metal alloys but switched to theoretical particle physics after the publication of Kohn and Luming (1963).[^85] Majumdar began a project on the theory of positron annihilation in metals. Overall, Kohn acquired a reputation among potential theory students as a supervisor with very high standards who could assign a thesis problem that might take a very long time to complete (Feibelman 2012). This did not deter Michael Greene (PhD 1965), who joined Kohn’s group (despite an initial lack of interest in solid state physics) because he was impressed by the thoughtful questions Walter asked at seminars. Greene wanted to learn to [*think*]{} like Kohn (Greene 2013).
Walter Kohn at age 39 (1962).\
Courtesy of Walter Kohn and\
the John Simon Guggenheim Foundation.
At the beginning of the fall 1962 semester, Walter learned that his colleague Norman Kroll would take over as Physics chair beginning in the fall 1963 semester. Right away, he made an application to the John Simon Guggenheim Memorial Foundation for funds to support a recuperative leave for the fall 1963 semester (Hohenberg, Kohn, and Sham 1990). His plan was to spend that time at the physics department of the École Normale Supérieure in Paris (Kohn, 1962). This was an ideal place to get back to full-time research. It was also an ideal place to renew his personal and scientific ties with three specialists in his own field of theoretical solid state physics: Jacques Friedel, Pierre-Gilles de Gennes, and Philippe Nozières. It was a bonus that the physics department was just a short walk from the Luxembourg Gardens, his favorite place in the whole world (Kohn 2001a).
Kohn’s fellowship application proposed that he would study the interaction of electrons and phonons in metals (Kohn, 1962). This was a hot topic. The collisions between electrons and the particle-like phonons had long been recognized as important for a proper description of the electrical conductivity of metals (Ziman, 1960). However, only five years earlier, John Bardeen, Leon Cooper, and J. Robert Schrieffer had proposed a many-electron wave function for a [*superconductor*]{} based on a model where the electron-phonon interaction mediates an effective attractive interaction between pairs of electrons with opposite spin (Bardeen [*et al.*]{} 1957). Moreover, barely a year earlier, neutron scattering experiments had confirmed Kohn’s own prediction of anomalies in the phonon spectra of metals (Kohn, 1959c). Accordingly, Kohn proposed to spend the fall of 1963 generalizing the theory of Kohn anomalies. For good measure, he also proposed to develop a theory of the effect of electron-phonon interactions on the optical properties of metals.
The Guggenheim Foundation responded positively to Walter’s application in the spring of 1963. This news must have been a great stimulant because he quickly completed a calculation which achieved “a new and more comprehensive characterization of the insulating state of matter” (Kohn 1964). The fundamental difference between the conducting state and the insulating state had been an issue for Kohn since his Bell Labs-inspired work with Luttinger on electrical transport (Kohn and Luttinger 1957, Luttinger and Kohn 1958). Subsequent papers on the behavior of a point charge in a dielectric, the nature of Wannier’s spatially localized states in solids, and the electromagnetic properties of insulators continued this theme (Kohn 1958, Kohn 1959a, Ambegaokar and Kohn 1960). His new work took seriously a suggestion by Nevill Mott (1949) that the many-body wave function in an insulator should be fundamentally different from the many-body wave function in a conductor. Kohn exploited a ground-breaking paper that focused attention on the gauge principle for the electromagnetic vector potential in quantum mechanics (Aharonov and Bohm 1959) and used a characteristically elegant method to calculate the electrical conductivity of a ring threaded by a line of magnetic flux. The result was a proof that the spatial organization of the electrons in an insulator corresponds to a many-body wave function that breaks up into disconnected regions that do not overlap with one another. The published paper, Kohn (1964), has been called a “a mine of ideas and methods” by no less an expert than Walter’s old Harvard classmate and Bell Laboratories colleague Philip Anderson (Anderson 2012).[^86]
Now 40 years old, Walter Kohn was a mature solid state physicist whose scientific talent and taste in problems had produced results that were highly valued by his peers. Two of those peers, David Pines and Charles Kittel, highlighted Kohn’s work four and eight times, respectively in their (now classic) 1963 graduate level textbooks [*Elementary Excitations in Solids*]{} and [*Quantum Theory of Solids*]{}. By the end of that summer, Walter’s manuscript on the “Theory of the Insulating State” was ready for submission and he had only to review some professional correspondence before he could depart for Paris. In retrospect, the most important letter on his desk came from Lu-Jeu Sham, a graduating PhD student from John Ziman’s group at the University of Cambridge whom Kohn had earlier recruited to become a post-doctoral fellow. Kohn had written to Sham to inform him about his Paris sabbatical and to urge him to come to San Diego as originally planned. Kohn proposed that Sham work on liquid metals with graduate student Mike Greene until he (Kohn) returned to campus. The return letter from Sham agreed to this plan (WKP 1963a).
Alloys in Paris
===============
Walter Kohn’s base of operations in Paris was the Ecole Normale Supérieure, one of the elite [*grandes écoles*]{} of the higher education system in France. His host was the 31-year old Philippe Nozières, an expert in many-body theory who had just collaborated with Joaquin Luttinger to derive Landau’s theory of the Fermi liquid using diagrammatic perturbation theory (Nozières and Luttinger, 1962, Luttinger and Nozières 1962).[^87] The setting inside the Physics Department at 24 rue Lhomond in the Latin Quarter was not typical. According to Nozières (Cheetham 1992, Nozières 2012b),
> I had inherited a very magnificent office with an old desk made from an oak tree. It was gigantic, eight times the size of a normal desk, and I put Walter on one side and me on the other side, facing each other. The office had a very high ceiling, maybe 15 feet high, and an upper level balcony had been installed for a secretary, which I did not have. Instead, Pierre Hohenberg settled there, watching Walter and I from above.
Pierre Hohenberg was a newly-arrived post-doc in Nozière’s group. He places himself on the main floor of “Philippe’s own very large office . . . and I remember it to have been a general meeting place and thoroughfare, a little like trying to think deep thoughts in the middle of Times Square” (Hohenberg 2003, 2012).
Kohn began his research activities, but he did not work on the electron-phonon interaction as he had proposed to the Guggenheim Foundation. Some months earlier, he had changed his mind and decided to think more deeply about the electronic structure of disordered metal alloys. More precisely, he asked himself how one might best describe the behavior of the electrons in a bulk metal composed of different types of atoms where there is at least some randomness in the identities of the atoms that occupy the sites of the underlying periodic lattice.[^88] Unlike most theoretical solid state physicists in the United States, Kohn had followed developments in alloy physics for more than a decade because of the intense experimental interest in this subject by his faculty colleagues in the Physics and Metallurgy departments at Carnegie Tech. On the other hand, his personal contribution to the field consisted of only two published papers and both concerned [*dilute*]{} alloys like ${\rm A}_x{\rm B}_{1-x}$ where the fraction $x$ of A-type atoms dissolved in a host metal made of B-type atoms was very small (Kohn and Vosko 1960, Kohn and Luming 1963). He now turned his attention to [*concentrated*]{} alloys where the populations of A-type atoms and B-type atoms could be comparable. Philippe Nozières was not particulary interested in alloys (he was working on liquid helium at the time), but a short 30 km train ride took Kohn to the suburban campus of the University of Paris in Orsay where his old friend Jacques Friedel maintained his research group. Friedel was an acknowledged expert in the theory of metals and alloys.[^89] Also present in Orsay at the time were André Guinier, an experimentalist renowned for his x-ray diffraction studies of alloys, and Pierre-Gilles de Gennes, a theorist working on a set of problems he would soon collect and discuss in his book, [*Superconductivity in Metals and Alloys*]{} (1966).[^90]
Kohn immersed himself in the literature of metals and alloys and soon discovered that two seemingly contradictory points of view dominated discussions of their electronic structure. I pause here to sketch the field as he found it, because his desire to reconcile these points of view was the immediate trigger for the creation of density functional theory. The fundamental problem was to calculate the eigenfunctions and energy eigenvalues for a binary alloy where A-type atoms replace a fraction of the atoms in a perfect B-type crystal. If the replaced B-type atoms are chosen randomly, the resulting structure is no longer periodic and the Bloch theorem which underlies conventional band structure theory is no longer valid.[^91] By the end of the 1950’s, approximate ways to analyze this situation had been proposed by Lothar Nordheim (1931), Harry Jones (1934), and Jacques Friedel (1954). All of them acknowledge a debt to the eminent English physical metallurgist William Hume-Rothery and his 1931 book, [*The Metallic State*]{}.
The first half of Hume-Rothery (1931) reviews years of experimental effort to systematize the electric, thermo-electric, and thermionic properties of metals and alloys. The second half reviews the classical and quantum mechanical theories that had been devised to explain some of these properties. A typical result reported in [*The Metallic State*]{} was the observation that many disordered substitutional alloys ${\rm A}_x{\rm B}_{1-x}$ exhibit an electrical resistivity that varies with the A-type atom concentration as $x(1-x)$. Nordheim (1931) explained this by replacing the real alloy, where dissimilar potentials $V_A(r)$ and $V_B(r)$ act on the valence electrons near lattice sites occupied by A-type atoms and B-type atoms, respectively, by a fictitious [*virtual crystal*]{} where the valence electrons near every lattice site feel the same average potential, $\bar{V}(r)=xV_A(r)+(1-x)V_B(r)$. By construction, the potential energy function for the virtual crystal is periodic and any band structure method becomes applicable to find the eigenfunctions and energy eigenvalues (Muto 1938).
Jones (1934) was concerned with some empirical ‘rules’ deduced by Hume-Rothery which related the crystal structure of certain alloys to their ‘electron concentration’, [*i.e.*]{}, the ratio of the total number of valence electrons to the total number of atoms in the entire crystal. Jones focused on the dilute limit and made two independent assumptions. First, he used a [*nearly-free electron*]{} description where the energy spectrum and wave functions of the host B-type metal was presumed to differ only slightly from the energy spectrum and wave functions of a collection of completely free electrons. Second, he made a [*rigid-band*]{} approximation which supposed that the sole effect of the A-type atoms was to contribute their valence electrons to the pre-existing ‘sea’ of valence electrons contributed by the B-type atoms. This implied that the electronic structure of the alloy was identical to the electronic structure of the host metal except that a few energy states were occupied (empty) in the alloy compared to the host if the valence of the A-type atoms was larger (smaller) than the valence of the B-type atoms.[^92] Using quantum mechanical perturbation theory, Jones estimated the change in total electronic energy when the Fermi surface of the rigid-band alloy contacts the Brillouin zone boundary for different crystal structures. In this way, he was able to rationalize the Hume-Rothery’s electron concentration rules in a semi-quantitative way.
The virtual crystal and rigid-band approximations share a ‘delocalized’ view of the electrons in a metal alloy. This means that each electron in the conduction band occupies an eigenstate whose Bloch-like wave function has a non-zero amplitude on every atomic site of the alloy \[see the left side of Eq. (\[five\])\]. This perspective gained popularity among practitioners because its successes were detailed in the first two research monographs devoted exclusively to metal physics: [*The Theory of Metals*]{} (1936) by Alan Wilson and [*The Theory of the Properties of Metals and Alloys*]{} (1936) by Nevill Mott and Harry Jones.
Jacques Friedel pioneered a spatially local des-\
cription of the electronic structure of alloys.\
Courtesy of the AIP Emilio Segrè Visual Archive.
A rather different, ‘localized’ point of view was developed by Jacques Friedel (1954). He considered an alloy where $\Delta Z_V$ is the valence difference between the solute A-type atoms and the solvent B-type atoms and recognized that each A-type atom with its valence electrons removed amounts to a point-like impurity with charge $\Delta Z_V$ with respect to the host crystal. The screening of this impurity \[discussed earlier in connection with the work of Langer and Vosko (1959)\] by the entire sea of conduction electrons implies that electronic charge density accumulates in the immediate vicinity of the A-type atoms. The de-localized electrons of the host scatter from these local charge accumulations and Friedel used perturbation theory to show that the energy shift of each scattering state (with respect to the Fermi energy) was the same as the energy shift predicted by the rigid-band model. He went on to use this ‘local’ perspective to rationalize other experimental trends summarized in Hume-Rothery’s book. An even more localized, covalent-bond approach to metals was advocated by Linus Pauling (1949).
The semi-quantitative nature of all existing theories of alloy electronic structure was criticized by John Slater at an October 1955 conference devoted to the theory of alloy phases.[^93] Speaking to an audience of physicists, physical chemists, metallurgists, and crystallographers, Slater opened the conference by remarking that (Slater 1955)
> The metallurgist expects the physicist to be able to apply wave mechanics to the problem of the cohesive energy of metals. By this one means the energy of the metal as a function of the positions of the nuclei. . . . Unfortunately, the errors in our present calculations of the energies of the isolated atoms and of the atoms combined into a metallic crystal are both considerably larger than the energy difference between the two, which is the cohesive energy which we hope to find. . . . This theory is not yet in a position to make calculations of the accuracy which the metallurgists need and which they have been led to believe that they have been getting. Metallurgists have been understandably anxious to get real guidance from physicists regarding their problems. A few papers written by theoreticians have led them to think that this guidance could be actually given in a quantitatively satisfactory form. For instance, a large literature has grown up as a result of papers \[published\] in the middle 1930’s by Jones . . . and Pauling has discussed metallic cohesion and ferromagnetism using methods that seem simple and quantitative. I wish to state my very firm opinion that these theories, as far as they pretend to be quantitative, are based on approximations which are not really justified. They may have qualitative truth in them, but they do not represent quantitative conclusions firmly based on fundamental theory.
It is perhaps unsurprising that Slater’s research group at the time was exploring the quantitative accuracy of his own ‘augmented plane wave’ method for band structure calculations (Slater 1953).
In Paris in the fall of 1963, Walter Kohn was in an excellent position to learn the latest developments in both the localized and de-localized approaches to alloy theory. The preceding January, Jacques Friedel and André Guinier had completed editing [*Metallic Solid Solutions*]{}, a book which documented the proceedings of an international symposium on the electronic and atomic structure of alloys held in Orsay in July 1962.[^94] In one invited paper, Stanley Raimes of Imperial College noted that Fermi surface measurements for the noble metals invalidated the nearly-free electron assumption used by Jones (1934), but did not quash the rigid-band approximation itself (Raimes 1963). Similarly, an invited paper by Frank Blatt of Michigan State University asserted that a broad range of transport measurements showed that “the rigid-band model is a good approximation to the electronic structure of dilute alloys” (Blatt 1963). On the other hand, Blatt continued, “it is difficult to overstate the importance” of Friedel’s local screening model for the interpretation of not only resistivity and thermoelectric data for alloys, but also for data obtained from measurements of impurity diffusion, positron annihilation, and the Knight shift.” Friedel himself reported an extension of his previous work to the case of transition-metal atom impurities where screening occurs by the occupation of atomic-like orbitals spread out in energy into ‘virtual bound state resonances’ (Friedel 1963).
The most general conclusion to be drawn from the papers collected in [*Metallic Solid Solutions*]{} was that some observations were best understood assuming that the conduction electrons of an alloy are delocalized through the volume of the crystal while other observations were best understood assuming that the most relevant electrons are localized in the immediate vicinity of the solute atoms. At least two prominent metallurgists regarded these points of view as complementary rather than contradictory. In the $4^{\rm th}$ edition of their [*Structure of Metals and Alloys*]{}, William Hume-Rothery and Geoffrey Vincent Raynor write (Hume-Rothery and Raynor 1962)
> The covalency interpretation and the Brillouin zone picture each express a part of the truth. In the case of the diamond structure, for example, the covalency theory . . . gives the more correct picture of the probable cloud density of valence electrons in the crystal. This concept by itself ignores the fact that electrons are free to move in the crystal, and this freedom is emphasized by the zone theories, which in their turn ignore the variation of the electron-cloud density in space. Both concepts are required to express the whole truth.
The complementarity expressed by this paragraph was well-known to many solid-state physicists, and particularly so to Walter Kohn who had published a fundamental paper on the relationship between the delocalized Bloch functions and the localized Wannier functions (Kohn 1959a). Kohn had also kept up with band structure calculations and could well appreciate the point John Slater had made about the absence of quantitative calculations in alloy theory (Callaway and Kohn 1962).
Kohn has written about his 1963 survey of the alloy literature on several occasions. His 1990 account, which is closest in time to the actual events, recalls the “rough and ready” rigid-band model and then recounts a calculation of the total energy of a disordered alloy (Hohenberg, Kohn, and Sham 1990). This is interesting because the total energy of a real metal was not a focus of research at the time. Indeed, the only paper in [*Metallic Solid Solutions*]{} concerned with total energy begins with the statement (Cohen 1963),
> In the early history of the theory of metals, calculation of the cohesive energy was a central concern. Apart from very considerable development of the theory of the electron gas, recent effort, both experimental and theoretical, has been focused primarily on properties of one-electron character. In the present work, we have returned to the study of the cohesive energy of metals.
The remainder of this paper, written by Walter’s friend Morrel Cohen from the University of Chicago, displays an exact formula for the total energy of a uniform (constant density) electron gas due to Nozières and Pines and then generalizes it to the case of an arbitrary “non-uniform system.” Both formulae involve an integral over a parameter $\lambda$ with the quantities in the integrand computed assuming a charge on the electron of $\lambda e$. The limits of the integral extend from $\lambda=0$ (the non-interacting electron gas) to $\lambda=1$ (the fully interacting electron gas).
In his 1990 reconstruction, Kohn wrote a similar integral to compute $\Delta E$, the [*difference*]{} in total energy between a real ${\rm A}_x{\rm B}_{1-x}$ alloy and the same alloy in the virtual crystal approximation. If $Z^{\rm A}$ and $Z^{\rm B}$ are the atomic numbers of the A-type and B-type atoms, every ion in the virtual crystal has nuclear charge $\bar{Z}=xZ^{\rm A}+(1-x)Z^{\rm B}$. Then, replacing $\Delta Z=Z^{\rm A}-Z^{\rm B}$ by $\lambda \Delta Z$, Kohn evaluated the integral of $dE/d\lambda$ from $\lambda=0$ to $\lambda=1$ to second order in $\Delta Z$ (keeping $\bar{Z}$ fixed). The resulting expression for $\Delta E$ depended on two quantities only: the electron density distribution of the virtual crystal, $\bar{n}({\bf r})$, and the electron density distribution of the real alloy, $n({\bf r})$. At this point, “the question occurred to Kohn whether a knowledge of $n({\bf r})$ alone determined–at least in principle–the total energy” (Hohenberg, Kohn, and Sham 1990). In other words, could the many qualitative successes of the Friedel point of view, which put great emphasis on the space-varying electronic charge density inside an alloy, be elevated to show that the exact charge density uniquely predicts the exact total energy?[^95]
It is important to understand the revolutionary nature of this question. Notwithstanding the work of Cohen mentioned above, most solid state theorists in the early 1960’s thought about the total energy of a solid along the lines laid out by Frederick Seitz in his widely admired textbook, [*The Modern Theory of Solids*]{} (1940). One starts with the nuclear charge $eZ_k$ and the fixed position ${\bf R}_k$ of each of the $M$ nuclei in the system. The Coulomb potential energy of interaction between pairs of nuclei is a classical quantity which poses no problem to compute. The remaining energy terms all contribute to the Schrödinger equation for the $N$-electron wave function: the kinetic energy of the electrons, the Coulomb potential energy of interaction between every pair of electrons, and the Coulomb potential energy of interaction between every electron and every nucleus.
It will be convenient to define $v({\bf r})$ as the potential energy of interaction between an electron at the point ${\bf r}$ and all the fixed nuclei: $$\label{six}
v({\bf r})=-e^2\sum\limits_{k=1}^M {Z_k \over |{\bf r}-{\bf R}_k|}.$$ This potential energy appears in the Schrödinger equation, as do the operators for the electron kinetic energy and the electron-electron potential energy, $$\label{seven}
\hat{T}=- {\hbar^2 \over 2m}\sum\limits_{k=1}^N \nabla_k^2~~~~{\rm and}~~~~\hat{U}= {1\over 2}\sum\limits_{k=1}^N \sum\limits_{m=1}^N {e^2 \over |{\bf r}_k -{\bf r}_m|}.$$ The Schrödinger equation determines the $N$-electron wave function, $\psi({\bf r}_1, {\bf r}_2, \dots, {\bf r}_N)$, and then the electron density distribution, $$\label{eight}
n({\bf r})=\int \psi^\ast({\bf r}, {\bf r}_2, \dots, {\bf r}_N)\, \psi({\bf r}, {\bf r}_2, \dots, {\bf r}_N)\, d{\bf r}_2 \cdots d{\bf r}_N.$$ If we omit the classical ion-ion energy, which does not involve the electrons, the total energy is $$\label{nine}
E = \int n({\bf r})\,v({\bf r})\, d{\bf r} + \langle \psi |\hat{T} |\psi \rangle + \langle \psi |\hat{U}|\psi \rangle,$$ where the last two terms are expressed as averages (expectation values) with respect to the ground state wave function.
The conventional perspective outlined in the previous paragraph shows that the external potential, $v({\bf r})$, is the only term in the Schrödinger equation which distinguishes one alloy from another. Therefore, $v({\bf r})$ determines the wave function from the Schrödinger equation, which in turn determines the electron density $n({\bf r})$ from Eq. (\[eight\]) and the total energy $E$ from Eq. (\[nine\]). From this point of view, the energy amounts to a [*functional*]{} of the external potential. Of course, the accuracy of the computed energy depends on the quality of the choice made for the form of the $N$-electron wave function used to solve the Schrödinger equation. Kohn now contemplated a radical inversion of this procedure. Was it possible that the total energy depended only on the electron density $n({\bf r})$? That is, could the total energy be a functional of the density alone? If so, knowledge of $n({\bf r})$ was sufficient to determine (implicitly) the external potential, the $N$-particle wave function, and all ground state properties, including the Green functions of many-body theory! This was a very deep question. Walter realized he wasn’t doing alloy theory anymore (Kohn 2012b).
The work of Hohenberg and Kohn
==============================
Kohn’s proposition that the total energy of an electron system was a functional of the density seemed preposterous on the face of it. How could knowledge of the electron density, which is a function of three Cartesian variables, be sufficient to compute the total energy when the last two terms in Eq. (\[nine\]) depend explicitly on the many-electron wave function, which is a function of $3N$ Cartesian variables? Looking for support, Walter asked himself if he knew [*any*]{} examples where complete knowledge of $n({\bf r})$ implied complete knowledge of $v({\bf r})$ (Hohenberg, Kohn, and Sham 1990). One such example was the elementary Schrödinger equation, $$\label{eleven}
-{\hbar^2 \over 2m}\nabla^2 \psi+ \left [v({\bf r}) -E\right ] \psi({\bf r})=0.$$
One can always choose $\psi({\bf r})$ real for this equation. Therefore, the electron density is $n({\bf r})=\psi^2({\bf r})$ and the inversion we seek is a matter of algebra: $$\label{twelve}
v({\bf r})=E+{\hbar^2 \over 2m}{\nabla^2 \sqrt{n({\bf r})} \over \sqrt{n({\bf r})}}.$$
Unfortunately, Eq. (\[eleven\]) applies only to a one-particle system and thus does not shed any light on the many-body problem. A more relevant example known to Kohn was the Thomas-Fermi method, a semi-classical but self-consistent approximation to the quantum theory of a many-electron system with a non-uniform density $n({\bf r})$ (March 1975, Zangwill 2013). Nevill Mott had used this method to calculate the screening of a point charge by the conduction electrons in a metal. For the problem considered here, Thomas-Fermi theory method shows that $n({\bf r})$ determines $v({\bf r})$ through the relation[^96] $$\label{fifteen}
v({\bf r})=E-{1\over 2m}\left({3h^3\over 8\pi}\right)^{\hspace{-.25em} 2/3}\hspace{-1em} n^{2/3}({\bf r}) - e^2\int d{\bf r'} {n({\bf r'})\over |{\bf r}-{\bf r'}|}. \vspace{0.15in}$$
Encouraged by these examples, Kohn sought to prove that [*the ground state electron density $n({\bf r})$ uniquely determines the external potential $v({\bf r})$.*]{} Such a simple result (if true) could not depend on the details of the many-electron wave function. Therefore, he looked for a very general idea to exploit and found it with the Rayleigh-Ritz variational principle (Hohenberg, Kohn, and Sham 1990). The reader will recall that this was the first tool Walter had placed in his theoretical toolbox while still a student at the University of Toronto. The [*reductio ad absurdum*]{} proof is “disarmingly simple” (Parr and Yang 1989).
Assume that a Hamiltonian (energy) operator $\hat{H}=\hat{v}+\hat{T}+\hat{U}$ produces a ground state energy $E_0$, a ground state wave function $\psi({\bf r}_1, \dots, {\bf r}_N)$ and an electron density $n({\bf r})$. Contrary to what we wish to prove, assume that another Hamiltonian $\hat{H'}=\hat{v}'+\hat{T}+\hat{U}$ produces a ground state energy $E'_0$, a ground state wave function $\psi'({\bf r}_1, \dots, {\bf r}_N)$, but the [*same*]{} electron density $n({\bf r})$. The idea is to use $\psi'$ as a trial function in the variational principle for $\hat{H}$ in Eq. (\[one\]). Ignoring the possibility of degeneracy, this gives the strict inequality[^97]
$$\label{sixteen}
E_0 < \langle \psi' | \hat{H} | \psi' \rangle = \langle \psi' | \hat{H}' |\psi' \rangle + \langle \psi' |\hat{H}-\hat{H}' |\psi' \rangle =E'_0 + \int n({\bf r}) [v({\bf r})-v'({\bf r})] \, d{\bf r}.$$
Similarly, we may use $\psi$ as the trial function in the variational principle for $H'$. The result is the same as Eq. (\[sixteen\]) with the primed and unprimed variables exchanged:
$$\label{seventeen}
E'_0 < \langle \psi | \hat{H}' | \psi \rangle = \langle \psi | \hat{H} |\psi \rangle + \langle \psi |\hat{H}'-\hat{H} |\psi \rangle =E_0 + \int n({\bf r}) [v'({\bf r})-v({\bf r})] \, d{\bf r}.$$
Adding Eq. (\[sixteen\]) to Eq. (\[seventeen\]) yields $E_0 + E'_0 < E'_0 + E_0$, which is impossible. Hence, our assumption that two external potentials correspond to the same electron density is false.
Pierre Hohenberg in 1965, soon after his work\
with Kohn on density functional theory.\
Courtesy of Pierre Hohenberg.
Kohn was exhilarated by his proof, but “it seemed such a remarkable result that I did not trust myself” (Kohn 1998). He looked around for help and found it in the person of Pierre Hohenberg, the recently arrived post-doctoral fellow who was also ensconced in Philippe Nozières’ enormous office. Hohenberg had just completed a post-doctoral year doing many-body theory with Alexei Abrikosov and Lev Gor’kov in Moscow, but was having some trouble getting the attention of Nozières, his new post-doctoral supervisor (Hohenberg 2012). Walter proposed that they work together and Pierre agreed. Like Kohn’s previous collaborators, Res Jost and Quin Luttinger, Hohenberg was a rather formal theoretical physicist. He had been trained at Harvard by Paul Martin and his 1962 PhD thesis, “Excitations in a dilute condensed Bose gas”, used density matrix and Green function methods to provide microscopic justification for phenomenological theories that had been offered by Lev Landau and others to explain the properties of superfluid helium.
Hohenberg’s first task was a literature search to discover if the theorem Kohn had proved was already known. Apparently not. However, it had been known for about a decade that a computation of the total energy of a system of nuclei and electrons did not really require complete knowledge of the entire $N$-particle wave function (Löwdin 1959, McWeeny 1960, Coleman 1963). It was sufficient to specify a quantity derived from the wave function called the [*second-order density matrix*]{}.[^98] In principle, there exists a variational principle for the energy where the second-order density matrix is varied rather than the wave function. Unfortunately, it was not known what properties the density matrix had to possess to ensure that it was derivable from an $N$-particle wave function. This became known as the ‘$N$-representability problem’.
In a similar way, Kohn’s theorem reminded Hohenberg of “formal work on stationary entropy and renormalization which had just been completed by Paul Martin and Cyrano de Dominicis, also working together in Paris” (Hohenberg, Kohn, and Sham 1990). These authors had studied the grand partition function of an arbitrary many-body system and used the mathematical technique of Legendre transformations to effect the desired ‘renormalizations’. The latter eliminated the functional dependence on any one-body potential (like the external potential above) in favor of functional dependence on a one-particle distribution function. It also eliminated the functional dependence on any two-particle potential (like the Coulomb interaction between pairs of electrons) in favor of functional dependence on a two particle distribution function (de Dominicis 1963, de Dominicis and Martin 1964). Fifty years later, Hohenberg and Kohn differ slightly in their recollection of whether Legendre transformations played any role in their work together. Kohn recalls that he and Hohenberg recognized that his theorem could be interpreted as a Legendre transformation only after the fact and notes that the final published paper makes no mention of it (Kohn 2013b). Hohenberg (2012) recalls that
> we were certainly thinking in the language of Legendre transformations, but we did not need that idea in the end. It was characteristic of Walter’s style to introduce \[in print\] only those theoretical ideas needed to solve the problem at hand.
Martin had returned to Harvard, but de Dominicis worked at the Centre d’Etudes Nucléaires in nearby Saclay. According to Hohenberg, “it took a number of intense but informative discussions with de Dominicis and his colleague Roger Balian to convince them that the procedure worked with the density rather than the distribution function (Hohenberg, Kohn, and Sham 1990). Philippe Nozières finally joined the conversation when he learned of the larger theoretical issue that now engaged Hohenberg and Kohn. As he later recalled (Nozières 2012b),
> The three of us discussed it a lot. But I was not fully convinced. In my opinion, putting all the emphasis on the density did not account properly for exchange and correlations. I did not share the enthusiasm of Walter and Pierre and I stayed aside.
Accordingly, Hohenberg and Kohn proceeded on their own to the natural next step: a reformulation of the Rayleigh-Ritz variational principle in terms of the density rather than the many-body wave function.
Kohn’s theorem implies that the many-body wave function is a functional of the ground state electron density $n({\bf r})$. The same is true of the exact kinetic energy $T[n]$ and the exact electron-electron potential energy $U[n]$. Therefore, if $$\label{exactT}
F[n]=T[n] + U[n]$$ is the sum of these two, the total energy in Eq. (\[nine\]) is $$\label{eighteen}
E[n] = \int n({\bf r})\,v({\bf r})\, d{\bf r} + F[n].$$
Now let $n_T({\bf r})\ge 0$ be a [*trial density*]{} which produces the correct total number of electrons, $N=\int n_T({\bf r})\, d{\bf r}$. By Kohn’s theorem, this density determines its own external potential, Hamiltonian, and ground state wave function $\psi_T$. In that case, the usual wave function variational principle expressed by Eq. (\[one\]) tells us that
$$\label{nineteen}
\langle \psi_T |\hat{H} | \psi_T \rangle = \int n_T({\bf r})\,v({\bf r})\, d{\bf r} + F[n_T]=E[n_T] \ge E[n].$$
The variational principle in Eq. (\[nineteen\]) establishes that the energy in Eq. (\[eighteen\]) evaluated with the true ground state density is minimal with respect to all other density functions for the same number of particles.
The proofs given just above appear early in Hohenberg and Kohn (1964), the first foundational paper of density functional theory. In connection with Eq. (\[nineteen\]), Hohenberg and Kohn (HK) use a footnote to warn the reader of a possible ‘v-representability problem’ because “we cannot prove whether an arbitrary positive definite distribution $n({\bf r})$ which satisfies $\int n({\bf r})\, d{\bf r}=N$ can be realized for [*some*]{} external potential.” They express confidence that all “except some pathological distributions” will have this property but the mere fact that they draw attention to this point demonstrates how carefully the authors looked for holes in their arguments. [^99]
HK make a point to remark that the functional $F[n]$ in Eq. (\[exactT\]) is “universal” in the sense that it is valid for any number of particles and any external potential. Even more significant for later work, they define yet another functional $G[n]$ by extracting from $F[n]$ the classical Coulomb energy of a system with charge density $en({\bf r})$. Doing this renders Eq. (\[eighteen\]) in the form
$$\label{twenty}
E[n] = \int n({\bf r})\,v({\bf r})\, d{\bf r} + {e^2\over 2}\int
\int {n({\bf r})\, n({\bf r'}) \over
|{\bf r}-{\bf r'}|}\,d{\bf r}\, d{\bf r'} + G[n].$$
The functional $G[n]$ in Eq. (\[twenty\]) includes the exact kinetic energy of the electron system and all the potential energy associated with the electron-electron interaction that is [*not*]{} already counted by the classical electrostatic energy. Needless to say, HK could not write down $G[n]$. Doing so would imply that they had solved the entire many-electron problem.
The approach Hohenberg and Kohn took to analyze $G[n]$ reflects the way Kohn chose to frame the final published paper. There is no mention of the alloy problem or even of any desire to re-formulate the electronic structure problem for solids. Instead, the title of the HK paper is simply “Inhomogeneous electron gas” and the first line of the abstract announces that “this paper deals with the ground state of an interacting electron gas in an external potential $v({\bf r})$.” The Introduction goes on to note that “during the last decade there has been considerable progress in understanding the properties of a homogeneous, interacting electron gas.”A footnote refers the reader to David Pines’ book [*Elementary Excitation in Solids*]{} (1963) for the details. HK then remind the reader about the Thomas-Fermi method,
> in which the electronic density $n({\bf r})$ plays a central role. . . This approach has been useful, up to now, for simple though crude descriptions of inhomogeneous systems like atoms and impurities in metals. Lately, there have been some important advances along this second line of approach. . . . The present paper represents a contribution in the same area.
HK confirmed that the Thomas-Fermi model of electronic structure follows from Eq. (\[twenty\]) by using an expression for $G[n]$ which accounts approximately for the kinetic energy of the electrons but takes no account of the non-classical electron-electron potential energy. Specifically, the kinetic energy density at a point ${\bf r}$ of the real system is set equal to the kinetic energy density of an infinite gas of [*non-interacting*]{} electrons with a uniform density $n=n({\bf r})$. The latter is computed using elementary statistical mechanics and one finds (March 1975, Zangwill 2013) $$\label{twentyonehalf}
G_{\rm TF}[n]= {3 \over 10m}\left({3h^3\over 8\pi}\right)^{2/3}\int n^{5/3}({\bf r})\, d{\bf r}.$$ After inserting Eq. (\[twentyonehalf\]) into Eq. (\[twenty\]) to produce $E_{\rm TF}[n]$, the variational principle in Eq. (\[nineteen\]) directs us to minimize $E_{\rm TF}[n]$ with respect to density. This is done using Lagrange’s method to ensure that the total particle number, $N=\int n({\bf r})\, d{\bf r}$, remains constant. The final result is the Thomas-Fermi expression in Eq. (\[fifteen\]).
The Thomas-Fermi model never went out of fashion as a quick and easy way to gain qualitative information about atoms, molecules, solids, and plasmas. By 1957, the British solid state physicist Norman March was able to publish a 100-page review article surveying the successes and failures of the model and its generalizations (March 1957). Earlier, I labeled 1957 as the ‘magic year’ when many-body Green functions and diagrammatic perturbation theory transformed the study of many-electron systems. Therefore, it is not surprising that several physicists–including Kohn’s old post-doctoral colleague Sidney Borowitz–applied these methods with the aim to systematically generalize the Thomas-Fermi model to include the effects of electron correlation and spatial inhomogeneity (Baraff and Borowitz 1961, DuBois and Kivelson 1962). These papers are among the “important advances” noted by HK in the passage quoted just above. [^100]
As trained solid-state physicists, HK knew that the entire history of research on the quantum mechanical many-electron problem could be interpreted as attempts to identify and quantify the physical effects described by $G[n]$. For example, many years of approximate quantum mechanical calculations for atoms and molecules had established that the phenomenon of [*exchange*]{}—a consequence of the Pauli exclusion principle—contributes significantly to the potential energy part of $G[n]$. Exchange reduces the Coulomb potential energy of the system by tending to keep electrons with parallel spin spatially separated. The remaining potential energy part of $G[n]$ takes account of short-range [*correlation*]{} effects. Correlation also reduces the Coulomb potential energy by tending to keep [*all*]{} pairs of electrons spatially separated. The effect of correlation is largest for electrons with anti-parallel spins because these pairs are not kept apart at all by the exchange interaction. I note for future reference that the venerable Hartree-Fock approximation takes account of the kinetic energy and the exchange energy exactly but (by definition) takes no account of the correlation energy (Seitz 1940, Löwdin 1959, Slater 1963).
HK devoted the remainder of their time together to studying $G[n]$ for two cases: (i) an interacting electron gas with a nearly constant density; and (ii) an interacting electron gas with a slowly-varying density. For the nearly constant density case, HK wrote $n({\bf r}) = n_0 + \tilde{n}({\bf r})$ with $\tilde{n}({\bf r})/n_0 \ll 1$ and pointed out that $G[n]$ admits a formal expansion in powers of $\tilde{n}({\bf r})$: $$\label{twentytwo}
G[n]= G[n_0] + \int K(|{\bf r}-{\bf r'}|) \tilde{n}({\bf r})\tilde{n}({\bf r'}) \, d{\bf r} \,d{\bf r'} + \cdots$$ $K(|{\bf r}-{\bf r'}|)$ is a linear response function for the uniform and interacting electron gas which had been studied intensely by experts in many-body theory (Pines 1963). In particular, the derivative of its Fourier transform $\tilde{K}(q)$ was known to diverge at a certain value of $q$ (Pines 1963). Walter knew this divergence well: it was responsible for the oscillatory algebraic form of the Friedel density disturbance formula in Eq. (\[fivehalf\]). It was also responsible for the “Kohn anomalies” in the phonon spectrum of metals. HK point this out and remark in passing that “the density oscillations in atoms which correspond to shell structure . . . are of the same general origin.”
The divergence in $\tilde{K}(q)$ disappears if one retains only a finite number of terms in its power series expansion. The corresponding expansion of $K(|{\bf r}-{\bf r'}|)$ reduces Eq. (\[twentytwo\]) to a [*gradient expansion*]{}, $$\label{nodiverge}
G[n]=G[n_0] +a\int \tilde{n}({\bf r})\, d{\bf r} + b\int |\nabla \tilde{n}({\bf r})|^2 \, d{\bf r} + \cdots,$$ where $a$ and $b$ are constants related to $\tilde{K}(q)$. Therefore, as HK pointed out, any ‘quantum oscillations’ produced by the divergence in $\tilde{K}(q)$ cannot be captured by any low-order gradient expansion of $G[n]$ like Eq. (\[nodiverge\]). This explained the failures of the many generalizations of the Thomas-Fermi approximation which involved adding gradient terms.
Walter and Pierre took some time off in the first week of December 1963 to attend an All-Union Conference on Solid State Theory in Moscow. Kohn joined John Bardeen and George Vineyard (Brookhaven National Laboratory) as the only senior Americans in attendance. Hohenberg had left Moscow in July and he relished the opportunity to visit his old friends there. The topics discussed at the meeting were superconductivity, the theory of metals, semiconductors and dielectrics, lasers, the magnetic properties of rare earths, neutron scattering, the Mössbauer effect, phonons, dislocation theory, and the theory of radiation damage in crystals (Agranovich 1964).
Upon their return to Paris, Hohenberg and Kohn focused on $G[n]$ for a system with a slowly-varying density function. This assumption precluded variations with short (spatial) wavelengths, but allowed for the possibility of substantial variations in the overall magnitude of the density. For this case, the appropriate gradient-type expansion is $$\label{twentythree}
G[n] = \int g_0(n({\bf r})) \, d{\bf r} + \int g_2(n({\bf r}))\, |\nabla n({\bf r})|^2\, d{\bf r} + \cdots,$$ where $g_0$ and $g_2$ are functions (not functionals) of $n({\bf r})$. By specializing Eq. (\[twentythree\]) to the previously studied case of a nearly uniform electron gas, HK were able to express these functions in terms of the properties of the uniform and interacting electron gas. For example, $g_0(n)$ is the sum of the kinetic energy density, the exchange energy density, and the correlation energy density for an interacting electron gas with uniform density $n$. With this information, HK performed a partial (infinite) summation of the entire gradient expansion in Eq. (\[twentythree\]). Their final expression had the great virtue of recovering the singularity in $\tilde{K}(q)$ needed to describe quantum oscillations.
Kohn left Paris in January 1964 and visited physicists in London, Cambridge, and Oxford before returning to California. He wrote up a first draft of a manuscript and sent it to Hohenberg for his review. Pierre made several suggestions, all of which were incorporated into the final version (WKP 1964). The paper was sent off to the [*Physical Review*]{} in the second week of June 1964 and published in the November 9 issue. It is notable that the ‘Concluding Remarks’ section does not remind the reader of the two theorem’s proved at the beginning of the paper. Instead the authors remark that they “have developed a theory of the electronic ground state which is exact in two limiting cases.” The importance Hohenberg and Kohn ascribed to the ability of their theory to capture quantum oscillations may be judged from the fact that their paper ends with the statement that ‘the most promising formulation of the theory . . . appears to be that obtained by partial summation of the gradient expansion’’. They say this despite warning the reader in the previous sentence that “actual electronic systems” have neither nearly constant densities nor slowly-varying densities, [*i.e.*]{}, the situations where the partial summation was expected to be valid.
Any reader of Hohenberg and Kohn (1964) cannot help but be struck by its understated and rather formal tone. The Introduction is succinct, the basic theorems are proved quickly, most of the paper is taken up with the gradient expansions, and no applications are discussed or proposed. Earlier, I quoted Kohn to the effect that he understood that the basic theorem which drives the paper was “remarkable”. If so, he and Hohenberg made a conscious decision not to ballyhoo the truly revolutionary idea at its core: that the ground state electron density, in principle, determines all the properties of an electronic system. That being said, we learn from their 1990 reminiscence that the authors spent at least a little time talking about applications (Hohenberg, Kohn, and Sham 1990):
> The question arose as to what the method might be good for, and Kohn suggested that one could try using it to improve current techniques for calculating the band structure of solids. Hohenberg’s immediate reaction was to say, “But band structure calculations are horribly complicated, isn’t that the sort of stuff better left to professionals?” To this, Kohn simply replied, “Young man, I am the Kohn of Kohn and Rostoker (1954) !”
A recommendation from Kohn helped Hohenberg win a job at Bell Telephone laboratories in the fall of 1964. Recently, he recalled that “I gave a talk on my Paris work during my first few months at Bell Labs. Phil Anderson, Bill McMillan, and Phil Platzman were in the audience and there was no enthusiasm. They correctly understood that our results would not help them solve the difficult many-body problems they were struggling with" (Hohenberg 2012). In the event, Hohenberg turned to hydrodynamics and phase transitions as topics for research and worked productively in those areas over a thirty-year career at Bell. He was elected to the US National Academy of Sciences in 1989 and moved to Yale University in 1995 to accept the position of Deputy Provost for Science and Technology. Since 2004, he has served at New York University as a professor of physics and Vice Provost for Research.
The Work of Kohn and Sham
=========================
Kohn returned to La Jolla in early 1964 “to find a group of postdocs and visitors eagerly awaiting a first-hand account of the work” (Hohenberg, Kohn, and Sham 1990). According to one of those post-docs, Vittorio Celli (2013),
> Walter gave a full departmental colloquium rather than just a technical seminar after his return from Paris. Keith Brueckner (who had returned from Washington and was Dean at the time) said that colloquia were usually reserved for “foreign stars” but that today we have “our own star” to give a talk. I remember thinking that the theory with Hohenberg was cute but would not have many consequences. I certainly did not think it compared in significance with the many-body calculations for the electron gas that had been obtained by Brueckner and his collaborators.
Kohn proposed to another post-doc, N. David Mermin, that he exploit his knowledge of statistical mechanics to generalize the Hohenberg-Kohn results to non-zero temperature. Mermin realized almost immediately that “a strange variational principle for the free energy that I had formulated for an utterly unrelated purpose” was perfectly suited for the job. As a result, “it took me less than an hour to check that the HK proof went through” almost without change (Mermin 2004). Kohn was skeptical, but the simplicity of the proof could not be assailed. On the other hand, the significance of Mermin’s work was not very clear and he was disinclined to write it up. It took six months, his imminent departure from La Jolla, and Kohn’s insistence that “it someday might be important” for the manuscript of Mermin (1965) to get written (Mermin 2013).
Kohn’s colloquium featuring the Hohenberg-Kohn theorems generated skepticism from several of the many-body theorists who were in La Jolla to work with Keith Brueckner. Most notably, “Nobuyuki Fukuda, visiting from the University of Tokyo, constructed counter-example after counter-example purporting to demonstrate the non-uniqueness of the potential given a density distribution. The job of resolving these fell to Lu Jeu Sham” (Hohenberg, Kohn, and Sham 1990).
Lu Sham was the graduating PhD student to whom Kohn had written requesting that he begin his post-doctoral fellowship at UCSD while Kohn was still in Paris. Sham, a native of Hong Kong, arrived in San Diego by way of Imperial College and Cambridge University where he had earned, respectively, an undergraduate degree in Mathematics and a graduate degree in physics. His 1963 PhD thesis, “The electron-phonon interaction”, was supervised by John Ziman, a distinguished theorist whose books, [*Electrons and Phonons*]{} (1960) and [*Principles of the Theory of Solids*]{} (1964), helped train a generation of solid-state physicists. In La Jolla, Sham used the months before Kohn returned to write a paper on the phonon spectrum of sodium metal. This quantitative calculation used a pseudo-potential for the sodium ionic potential and a self-consistent modification of the Hartree-Fock method to take approximate account of correlation effects which act to screen (reduce) the exchange interaction (Sham 1965).
Lu-Jeu Sham (circa early 1980’s) was a\
post-doc with Walter Kohn from 1963-1966.\
Courtesy of Lu-Jeu Sham.
Quantum density oscillations were much on Walter’s mind when he related the details of his Paris experience to Sham. He was acutely aware that no oscillations would emerge from Eq. (\[nodiverge\]), the most natural choice for $G[n]$ when the external potential was slowly-varying function of position. Therefore, in the same letter where Walter informed Pierre Hohenberg that the HK manuscript had been submitted for publication, he related that “Lu Sham and I have started looking at situations like a heavy atom where one has a localized and rapidly-varying potential” (WKP 1964). In principle, their goal was to develop a general theory of quantum density oscillations for use in situations where the electron density is strongly non-uniform. In practice, they developed a method to find the leading quantum corrections to the Thomas-Fermi electron density for a collection of non-interacting electrons moving in a one-dimensional potential. The published paper, Kohn and Sham (1965a), reports the results of an elegant Green function calculation which related the electron density (which did exhibit the desired quantum oscillations) to the potential in a way which generalized Eq. (\[fifteen\]). Unfortunately, the extension of their method to three-dimensional periodic potentials presented a daunting numerical challenge which did not lend itself to practical calculations for real solids. Accordingly, Kohn and Sham dropped this line of investigation and moved in a different direction, albeit one still motivated by the basic results obtained by Hohenberg and Kohn.[^101]
By the late fall of 1964, Kohn was thinking about alternative ways to transform the theory he and Hohenberg had developed into a practical scheme for atomic, molecular, and solid state calculations. Happily, he was very well acquainted with an approximate approach to the many-electron problem that was notably superior to the Thomas-Fermi method, at least for the case of atoms. This was a theory proposed by Douglas Hartree in 1923 which exploited the then just-published Schrödinger equation in a heuristic way to calculate the orbital wave functions $\phi_k({\bf r})$, the electron binding energies $\epsilon_k$, and the charge density $n({\bf r})$ of an $N$-electron atom (Park 2009, Zangwill 2013). Hartree’s theory transcended Thomas-Fermi theory primarily by its use of the exact quantum-mechanical expression for the kinetic energy of independent electrons. The [*Hartree equations*]{} which define the theory for an atom with nuclear charge $Z=N$ are $$\label{H1}
-{\hbar^2 \over 2m}\nabla^2 \phi_k+ \left [v_{\rm eff}({\bf r}) -\epsilon_k\right] \phi_k({\bf r})=0,~~~~~k=1,\dots N,$$ where $$\label{H2}
v_{\rm eff}({\bf r})= v({\bf r})+e^2\int d{\bf r'} {n({\bf r'})\over |{\bf r}-{\bf r'}|}$$ and $v({\bf r})=-Ze^2/r$. The electron density $n({\bf r})$ in Eq. (\[H2\]) is calculated assuming that the electrons occupy thet $N$ lowest energy eigenfunctions of the Schrödinger equation in Eq. (\[H1\]). Therefore, if $k=1,2,\dots, N$ labels these $N$ lowest energy orbitals, $$\label{H3}
n({\bf r})=\sum_{k=1}^N |\phi_k({\bf r})|^2.$$ The effective potential energy function in Eq. (\[H2\]) shows that every electron interacts with the charge of the ‘external’ nucleus and with the charge of the entire atomic electron cloud taken as a whole. [^102] Hartree stressed that these equations must be solved [*self-consistently*]{}. That is, an iterative numerical method is required to ensure that the $\phi_k({\bf r})$ generated by Eq. (\[H1\]) are the same as the $\phi_k({\bf r})$ used in Eq. (\[H3\]) to construct the particle density $n({\bf r})$.
Slater (1930) and Fock (1930) had provided a rigorous derivation of the Hartree equations. They used the variational principle in Eq. (\[one\]) and evaluated the total energy using a many-electron wave function of the form $\psi({\bf r}_1, {\bf r}_2, \dots, {\bf r}_N)=\phi_1({\bf r}_1)\phi_2({\bf r}_2)\cdots \phi_N({\bf r}_N)$. Minimizing this energy with respect to different choices for the $\phi_k$ functions generates the Hartree equations. Slater and Fock also evaluated the total energy using a more sophisticated many-body wave function (called a Slater determinant) which combines the same $N$ orbital functions $\phi_k$ in such a way that the Pauli exclusion principle is obeyed automatically. With this choice, minimizing the total energy with respect to choices for the $\phi_k$ functions generates what are called the Hartree-Fock equations. Hartree-Fock theory is superior to Hartree theory because the kinetic energy [*and*]{} the exchange energy are treated exactly. Unfortunately, the Hartree-Fock equations are significantly harder to solve than the Hartree equations.
Kohn suggested to Sham that he try to derive the Hartree equations from the Hohenberg-Kohn formalism. Walter had good reason to believe this could be done (Kohn 2001a, Kohn 2012b). On the one hand, his work with Hohenberg had established the central role of the electron density $n({\bf r})$ for a complete description of any electronic system. On the other hand, the Hartree equations could be read as a self-consistent scheme to deduce an approximate expression for $n({\bf r})$. Therefore, it should be possible to derive the Hartree equations as an [*example*]{} of the HK variational principle. Specifically, the variational minimization of some approximate form of the total energy functional $E[n]$ should lead to Hartree’s equations. Sham set to work with enthusiasm. His state of mind at that particular moment was noted by Philip Taylor, a fellow former-graduate student of John Ziman’s at Cambridge University. Taylor had taken a job at the Case Institute of Technology in Cleveland, Ohio and had recently published a quantitative analysis of Kohn anomalies in the phonon spectrum of metals (Taylor 1963). It was during a visit to La Jolla to consult with Kohn and visit with Sham that Sham remarked to Taylor that he was “thrilled” by the research project Kohn had given to him (Taylor 2013).
Kohn and Sham recognized that the Hartree method regards each electron as moving independently in an effective potential $v_{\rm eff}({\bf r})$ which does not recognize the individual identity of the other electrons. Consistent with this, the kinetic energy implied by Eq. (\[H1\]) is correct only for independent and non-interacting electrons. This was the key to progress because the Hohenberg-Kohn analysis implied that the kinetic energy of a strictly non-interacting system of electrons is also a functional of the density. If we call this functional $T_S[n]$, ordinary quantum mechanics specifies that[^103] $$\label{H4}
T_S[n]=\sum_{k=1}^N \int \phi_k^\ast({\bf r})\,\left[-{\hbar^2 \over 2m}\nabla^2 \right] \phi_k({\bf r}) \, d{\bf r}.$$
The path was now open for Sham to derive the Hartree equations from a density functional point of view (Kohn 2001a). He chose the approximate total energy functional
$$\label{H6}
E_H[n] = T_S[n] + \int v({\bf r}) \, n({\bf r}) \, d{\bf r} + {e^2\over 2}\int
\int {n({\bf r})\, n({\bf r'}) \over
|{\bf r}-{\bf r'}|}\,d{\bf r}\, d{\bf r'},$$
and minimized it with respect to a density assumed to have the form in Eq. (\[H3\]). The latter transforms the density variation of the first term in Eq. (\[H6\]) into variations with respect to the $\phi_k$ functions. The density variation of the remaining terms in Eq. (\[H6\]) is straightforward and the final result is exactly the Hartree equations (\[H1\]) and (\[H2\]).
Kohn and Sham (KS) now knew how to move forward with the general many-electron problem. Motivated by Eq. (\[H6\]), they [*defined*]{} a functional $E_{xc}[n]$ by the partition $G[n]=T_S[n] +E_{xc}[n]$. This puts the exact total energy functional in the form $$\label{E2}
E[n]=E_H[n] + E_{xc}[n]. \vspace{0.4em}$$ The great virtue of Eq. (\[E2\]) is that it has exactly the same structure as Eq. (\[H6\]), even if we revert to Eq. (\[six\]) for $v({\bf r})$. Therefore, the interacting electron density $n({\bf r})$ which minimizes the original total energy in Eq. (\[nine\]) is precisely equal to the non-interacting electron density $n({\bf r})$ which minimizes Eq. (\[E2\]). Carrying out the latter minimization explicitly produces the [*Kohn-Sham equations*]{}, which are identical to the Hartree equations Eqs. (\[H1\]) and (\[H3\]) with Eq. (\[H2\]) replaced by $$\label{KS1}
v_{\rm eff}({\bf r}) = v({\bf r})+e^2\int d{\bf r'} {n({\bf r'})\over |{\bf r}-{\bf r'}|} + v_{xc}({\bf r})$$ where $$\label{KS2}
v_{xc}({\bf r})={\delta E_{xc}[n]\over \delta n({\bf r})}.$$ The exchange-correlation potential energy, $v_{xc}({\bf r})$, obtained in Eq. (\[KS2\]) from the functional derivative of $E_{xc}[n]$, is a function of ${\bf r}$ and not a functional of $n({\bf r})$. Therefore, according to KS, a numerical procedure no more difficult that Hartree’s original method is sufficient to compute the ground state electron density and thus the ground state total energy of an arbitrary many-electron system subject to an external potential. If $E_{xc}[n]$ was known exactly, one could calculate $n({\bf r})$ and $E[n]$ exactly as well.
The “exchange-correlation” energy functional $E_{xc}[n]$ in Eq. (\[E2\]) is similar to $G[n]$ in Eq. (\[twenty\]) in the sense that it accounts for all the energy associated with the Coulomb interaction between electrons [*not*]{} already counted by the classical Coulomb self-energy. However, while $G[n]$ had also to account for the total kinetic energy of the real interacting electron system \[called $T[n]$ in Eq. (\[exactT\])\], $E_{xc}[n]$ has only to account for the [*difference*]{} between the kinetic energy of an interacting electron system and the kinetic energy of a non-interacting electron system with exactly the same density $n({\bf r})$. Of course, the exact and universal functional $E_{xc}[n]$ is no better known that $G[n]$ for the interacting electron problem.
Unlike the Slater-Fock methodology sketched earlier, the foregoing derivation of the Hartree-like Kohn-Sham equations [*does not introduce a many-electron wave function at any stage.*]{} Instead, Kohn and Sham replace the true interacting electron system with a non-interacting electron reference system which has exactly the same ground state electron density. The wave function of the reference system is unambiguously Hartree-like, so it is correct to use the exact kinetic energy for non-interacting electrons in Eq. (\[H4\]) and represent the density function as in Eq. (\[H3\]). On the other hand, the eigenfunctions $\phi_k({\bf r})$ and the eigenvalues $\epsilon_k$ in Eq. (\[H1\]) have [*no*]{} direct physical meaning for the true interacting electron system.
Finally, Kohn and Sham proposed an approximation for $E_{xc}[n]$ which has come to be known as the [*local density approximation*]{} (LDA). Namely, $$\label{KS3}
E_{xc}[n]=\int n({\bf r})\, \epsilon_{xc}(n({\bf r}))\, d{\bf r},$$ where $\epsilon_{xc}(n)$ is the exchange and correlation energy per electron of a fully interacting electron gas with uniform density $n$. Hohenberg and Kohn had introduced a similar approximation for $G[n]$ in their paper and it was reasonable for Kohn and Sham to “regard $\epsilon_{xc}(n)$ as known from theories of the homogeneous electron gas.”[^104] When Eq. (\[KS3\]) is used for $E_{xc}[n]$, the exchange-correlation potential in Eq. (\[KS2\]) becomes $$\label{KS4}
v_{xc}({\bf r}) = {d\over dn}\left[n\epsilon_{xc}(n)\right].$$
The foregoing results were reported in a short manuscript, “Exchange and correlation effects in an inhomogeneous gas” which Kohn and Sham submitted to [*Physical Review Letters*]{} in May of 1965. Samuel Goudsmit, one of the editors of [*Physical Review Letters*]{} at that time, informed Sham by letter that the Kohn-Sham manuscript “deserves publication as an Article in the [*Physical Review*]{}, but it is not of such urgency to warrant speedy publication in [*Physical Review Letters*]{}” (Goudsmit 1965). The authors responded by withdrawing the manuscript and, three weeks later, submitted to the [*Physical Review*]{} a longer and more detailed paper with a new title, “Self-consistent equations including exchange and correlation effects”. The published version, Kohn and Sham (1965b) is the second foundational paper of density functional theory. It is also one of the most highly cited papers in the history of physics.[^105] Interestingly, it was only at the page-proof stage of the longer paper that the authors realized that their Hartree-like equations with Eq. (\[KS2\]) represented a formally exact statement of the complete many-body problem (Sham 2014). For that reason, Eq. (\[KS2\]) appears only in a “Note Added in Proof” while Eq. (\[KS4\]) appears in the main exposition.
Kohn and Sham knew it was straightforward to use Eq. (\[KS4\]) and write down an explicit and analytic formula for $v_{xc}({\bf r})$ and incorporate it seamlessly into existing computer programs to calculate the electronic structure of atoms, molecules, and solids. Briefly, the separation $\epsilon_{xc}(n)=\epsilon_x(n) + \epsilon_c(n)$ known for the interacting and uniform electron gas implies that the exchange-correlation potential Eq. (\[KS4\]) separates similarly into $ v_{xc}({\bf r})=v_x({\bf r}) + v_c({\bf r})$. The exact exchange energy density, $\epsilon_x(n)$, had been calculated years earlier by Dirac (1930) for the purpose of improving the Thomas-Fermi approximation. Using Dirac’s formula, KS reported their result for the LDA exchange potential:[^106] $$\label{x}
v_{x,{\rm LDA}}({\bf r}) = -e^2\left[{3\over \pi} n({\bf r})\right]^{1/3}.$$ This expression was consequential at the time because, when Eq. (\[x\]) replaces $v_{xc}({\bf r})$ in Eq. (\[KS1\]), the Kohn-Sham equations become almost identical to a set of equations John Slater had proposed in 1951 as a local approximation to the non-local Hartree-Fock equations. I say ‘almost’ identical because the [*ad hoc*]{} local exchange potential proposed by Slater was $$\label{Slater}
v_{x,{\rm Slater}}({\bf r})=-{3\over 2}e^2\left[{3\over \pi} n({\bf r})\right]^{1/3}.$$ KS argue for the correctness of their proposed exchange potential and it is notable that the abstract of Kohn and Sham (1965b) devotes a sentence to announcing the factor of $3/2$ difference between Eqs. (\[x\]) and (\[Slater\]). The authors’ motivation to do this was surely their awareness that the so-called ‘Hartree-Fock-Slater’ method was in wide use by physicists performing band structure calculations for real solids (Callaway 1958, Herman 1964).
The Kohn-Sham equations \[with and without the local density approximation for $v_{xc}({\bf r})$\] are the reason for the enduring importance of Kohn and Sham (1965b). The paper itself differs in tone from Hohenberg and Kohn (1964) in the sense that the abstract notion of an “inhomogeneous electron gas” disappears from the title and from most of the text. Instead, there is the practical promise of “self-consistent equations” appropriate to “real systems (atoms, molecules, solids, etc.) \[where\] the electronic density is nonuniform." The Introduction is even more specific and makes the point that
> most theoretical many-body studies have been concerned with elementary excitations and as a result there has been little progress in the theory of cohesive energies, elastic constants, etc. of real metals and alloys. The methods proposed here offer the hope of new progress in the latter area.
That being said, KS did not themselves report any calculations of the cohesive energy or elastic constants (or any other measurable quantity) for any ‘real’ electronic system. Indeed, they did not even bother to write down an explicit form for the correlation part of $v_{xc}({\bf r})$ in the local density approximation. This was a straightforward exercise for anyone familiar with the electron gas literature. As for the LDA itself, KS remark that it should “give a good representation of exchange and correlation effects . . for metals, alloys, and small-gap insulators.” On the other hand, they warn the reader that the LDA should have “no validity \[at\] the ‘surface’ of atoms and the overlap regions of molecules. . . We do not expect an accurate description of chemical binding.”
Walter left for his annual visit to Bell Labs after the June 1965 submission of the longer Kohn-Sham manuscript. He collaborated with Quin Luttinger as usual and their efforts produced a prediction for a new mechanism for superconductivity based on a presumed oscillatory interaction between pairs of electrons (Kohn and Luttinger 1965). Meanwhile, back in La Jolla, Lu Sham began work on two density functional projects. One of these, which became the final paper he and Kohn would publish together, examined the one-body Green function of many-body theory (Sham and Kohn 1966). It was important for them to study this quantity because its properties determine the energy, lifetime, and spatial extent of single-particle-like excitations out of the ground state of a many-body system. At the same time, the Hohenberg-Kohn theory implies that the Green function is as a functional of the ground state electron density.
Sham, who did most of the calculations, demanded that the Green function satisfy the requirements of particle-number conservation and charge neutrality and deduced thereby that an electron at a point ${\bf r}$ in an atom, molecule, or solid responds to the electrostatic potential at that point and to exchange and correlation effects which depend on the electron density distribution in the immediate vicinity of ${\bf r}$ only. For a slowly-varying density, this conclusion justifies a local density approximation for the Green function, which in turn provides an approximation for the energy spectrum and an independent justification for using the LDA with the Kohn-Sham equations to calculate the ground state electron density $n({\bf r})$.
Sham’s second post-Kohn-Sham density functional project was done in collaboration with Bok Yin Tong, a thirty-year old graduate student who had begun to work with Kohn. Their goal was to solve the Kohn-Sham equations numerically for several atoms and ions. Luckily for them, Frank Herman and Sherwood Skillman had just published a computer program which solved the Hartee-Fock-Slater equations for atoms (Herman and Skillman 1963). Tong and Sham needed only to replace Eq. (\[Slater\]) by Eq. (\[x\]) in the program and add some code for the correlation part of $v_{xc}({\bf r})$ in the LDA. For this they used an interpolation formula for the correlation energy derived from the information given in Pines (1963). The published paper, Tong and Sham (1966), focused on total energies, total energy differences, and charge densities. The final results with correlation omitted were gratifying, giving “slightly better results for energies and substantially better results for densities that Slater’s method." The correlation correction worsened the results, “presumably because the electronic density in atoms has too rapid a spatial variation.”
Lu Sham published three articles unrelated to density functional theory before beginning an Assistant Professorship at the University of California at Irvine in the fall of 1966. Two years later, he accepted an offer to return to La Jolla as an Associate Professor. He was promoted to Professor in 1975, served as a Dean from 1985-1989, and is currently Emeritus Professor of Physics. He was elected to the US National Academy of Sciences in 1998. At UCSD, Sham developed a broad research program with a particular expertise in the theory of the electronic and optical properties of semiconductor heterostructures. However, the twenty papers he published on density functional theory over the years show that he never completely abandoned the main subject of his post-doctoral work.
Walter Kohn remained at San Diego until 1979, when he accepted the position as founding Director of the Institute for Theoretical Physics (ITP), a research facility established and supported by the US National Science Foundation at the University of California at Santa Barbara. He served as Director for five years and continued as a Professor of Physics at UCSB until 1991 when he gained Emeritus status. Kohn and his collaborators published over 150 papers between 1965 and 2006. One third of these explore some aspect of density functional theory, particulary its application to solid surfaces. An equal number of papers concern Kohn’s pre-DFT interests including disordered states of matter, superconductivity, Bloch and Wannier functions, scattering theory, and the transition between the conducting and insulating states of matter. Kohn was elected to the National Academy of Sciences (1969) and was a recipient of a National Medal of Science (1998) before winning a share of the 1998 Nobel Prize in Chemistry.
Discussion and Conclusion
=========================
In March 2001, Kohn addressed a symposium on “The History of the Electronic Structure of Atoms, Molecules, and Solids” at a meeting of the American Physical Society. With an audience well-schooled to interpret the lower case Greek letter $\epsilon$ as an infinitesimally small quantity, Walter was understood immediately when he characterized the initial reception of density functional theory as “$+\,\epsilon$ by theoretical physicists and zero by theoretical chemists” (Kohn 2001a). The small reaction by physicists reflected the fact that Kohn’s theory did not directly address the “big” issues that occupied many solid-state physicists at the time: superconductivity, the Kondo effect, superfluid helium, disorder-induced localization, the metal-insulator transition, and quasi-one-dimensional conductors. The positive response was limited mostly to band structure theorists who were well-positioned to carry out the numerical work needed to solve the Kohn-Sham equations for real materials.[^107] The lack of response (or negative response) from chemists came mostly from their near-universal belief that no theory of electronic structure based on the particle density alone could possibly be correct. In a future publication, I will trace the evolving response of both the physics and chemistry communities to DFT. For the present, it suffices to sketch very briefly the path which led from virtually no response by chemists to a Nobel Prize in Chemistry.
An important point mentioned earlier is that a computer program written to solve the local exchange potential equations of the Hartree-Fock-Slater (HFS) method (Slater 1951) was easily adapted to solve the Kohn-Sham equations in the local density approximation. Therefore, after the 1965 publication of the Kohn-Sham paper, systematic calculations for atoms began to reveal that Eq. (\[x\]) was superior to Eq. (\[Slater\]) as a local approximation to the exact, non-local exchange potential (Herman, Van Dyke, and Oretenburger 1969). HFS and LDA calculations for molecules and solids were more difficult to evaluate because computational exigencies encouraged the use of a ‘muffin-tin approximation’’ where the effective potential in Eq. (\[KS1\]) was replaced by its spherical average inside a set of touching spheres centered at the atoms. A constant potential was used outside the spheres.
Beginning in the 1970’s, a small group of scientists committed themselves to carrying out HFS and LDA calculations for small molecules without imposing the muffin-tin (or any other) constraint on the self-consistent potentials (Baerends, Ellis, and Ros 1973, Gunnarsson, Harris, and Jones 1977, Becke 1982). Over time, HFS calculations disappeared in favor of LDA calculations and the basic conclusion was that one obtained reasonable binding energies and predictions for molecular structures that agreed well with experiment (Jones 2012). On the other hand, the LDA was [*not*]{} capable of the kind of ‘chemical accuracy’ ($\pm 2~ {\rm kcal/mole}$) that the best [*ab initio*]{} methods of traditional quantum chemistry could achieve. Then, in the 1980’s, efforts to go beyond the local density approximation led to the proposal and testing of various non-local approximations for the exchange and correlation functional (Langreth and Mehl 1983, Perdew 1986, Becke 1988, Lee, Yang, and Parr 1988). These, so-called “generalized gradient approximations” (GGA) replaced the electron gas exchange-correlation energy density in Eq. (\[KS3\]) with much more complicated functions of both the local density $n({\bf r})$ and the local density gradient $\nabla n({\bf r})$.[^108] Systematic Kohn-Sham calculations for atoms and molecules using GGA, particulary a hybrid approach introduced by Axel Becke (1993), quickly began to approach chemical accuracy.
A turning point occurred in 1991 at the VII$^{\rm th}$ International Congress of Quantum Chemistry in Menton, France (Kohn 2001a). John Pople gave the final talk and summarized the achievements of ‘G2’ theory, his most comprehensive [*ab initio*]{} attempt to improve the Hartree-Fock approximation using perturbation theory (Curtiss, Raghavachari, Trucks, and Pople 1991). On the other hand, earlier in the week, the Congress had given its triennial “outstanding young scientist” award to Axel Becke for his manifestly non-[*ab-initio*]{} work with DFT. Specifically, “for unique advances in numerical methods in density functional theory as applied to molecules, and for important developments in the understanding of the exchange-correlation functional that enters density functional theory” (IAQMS 2014). Pople made a point in his talk to remark that he found Becke’s results “stimulating and intriguing” (Pople 1991).
Barely a year later, Pople’s group published a systematic comparison of the best quantum chemical calculations with DFT calculations performed using a variety of exchange-correlation potentials for 32 molecules. They concluded that the most sophisticated non-local functionals “outperformed correlated [*ab initio*]{} methods, which are computationally more expensive. Good agreement with experiment was obtained with a small basis set” (Johnson, Gill, and Pople 1992). DFT was promptly incorporated into Pople’s widely-used GAUSSIAN computer program and, with this endorsement, the popularity of DFT calculations among chemists began to grow exponentially (see Fig. 2). Accordingly, when the Nobel Chemistry Committee decided it was time to honor quantum chemistry with a Prize, it was not difficult for them to split the award between John Pople and Walter Kohn.
At the beginning of this paper, I suggested that density functional theory might be unknown today if Walter Kohn had not created it in the mid-1960’s. Such a counter-factual claim can never be proved. However, it is interesting to examine the evidence that supports it. We do this with full awareness of a long tradition which examines great scientific discoveries from the [*personalistic*]{} and [*naturalistic*]{} points of view (Boring 1950). The former focuses on the specific attributes of an individual whose “exceptional insight may lead to an original discovery which has not been anticipated by others and which is relatively independent of the times.” The latter posits that “the [*zeitgeist*]{} (scientific climate of the time) determines the great discovery and that he who makes the discovery is great merely because the times employed him." Here, I begin with the zeitgeist of electronic structure theory in Kohn’s lifetime and then turn to his particularism for the case of DFT.
Fig. 2. Numbers of papers that mention\
DFT as found by the Web of Science.\
The modern concept of the electronic structure of atoms, molecules, and solids began when the old quantum theory of Bohr gave way to the new quantum theory of Schrödinger (Jammer 1966). From 1925 to 1960, no fewer than 1000 papers in the scientific literature concerned themselves with the ‘many-electron’ problem.[^109] No fewer than 1000 more papers focused on this topic between 1961 and 1965. The abstracts of these papers reveal five principal approaches to this issue: the Thomas-Fermi method, electron gas models, many-electron wave function methods, density matrix methods, and quantum field theory methods. Of these, only the approximate Thomas-Fermi method singles out the ground state electron density $n({\bf r})$ as the primary quantity for study. Attempts to build “quantum corrections” into the Thomas-Fermi model were current in the early 1960’s, but none of these suggested that the theory could be “exactified” to reveal the density as a truly fundamental quantity. A paper inspired by Friedel’s alloy work which ignores correlation and derives a formal perturbation series for the density in term of the external potential is perhaps closest in this regard (March and Murray 1961).
In the late 1950’s and early 1960’s, the ground state electron density was not very interesting to many theoretical solid state physicists. They focused instead on the excited states of solids and the powerful new methods of quantum field theory which made their study possible (Hoddeson [*et al.*]{} 1992). Those who did appreciate the general importance of the charge density—primarily the practitioners of band structure calculations—saw no reason and had no motivation to elevate it to the lofty status of the many-electron wave function (Herman 1958, Pincherle 1960). The latter attitude was shared by the quantum chemistry community who devoted enormous efforts to solving the Schrödinger equation for molecules with greater and greater accuracy (Barden and Schaefer 2000, Gavroglu and Simões 2012). An interesting exception is the Canadian Richard Bader, perhaps the greatest champion of $n({\bf r})$ among theoretical chemists. Just a year before the publication of Hohenberg and Kohn (1964), he wrote (Bader and Jones 1963):
> The manner in which the electron density is disposed in a molecule has not received the attention its importance would seem to merit. Unlike the energy of a molecular system, which requires a knowledge of the second-order density matrix for its evaluation, many of the observable properties of a molecule are determined in whole or in part by the simple three-dimensional electron density distribution.
Despite his fondness for the density, even Bader could not deny the primacy of the second-order density matrix (Löwdin 1959). This was the rock-solid quantum chemical view that Hohenberg had discovered in Paris when he reviewed the literature of many-electron theory.
To my knowledge, the only work in the pre-1964 electronic structure literature where the electron density plays a fundamental role is a one-page paper by the distinguished theoretical chemist, E. Bright Wilson, Jr. (Wilson 1962). Wilson asks the rhetorical question, “Does there exist some procedure for calculating $n({\bf r})$ \[for an $N$-electron system\] which avoids altogether the use of $3N$ dimensional space?” He then uses a device mentioned earlier (see Section IV) and defines $n({\bf r},\lambda)$ to be the exact ground state electron density of a many-electron system where the charge of every electron is taken to be $\lambda e$ rather than $e$. Using just a few lines of calculation, Wilson shows that the total energy $E$ in Eq. (\[nine\]) can be written as an integral from $\lambda=0$ to $\lambda=1$ of the sum of the classical Coulomb potentials produced by $n({\bf r},\lambda)$ at the positions of all the nuclei.[^110]
From the foregoing, I conclude that no scientists before Walter Kohn in 1963 were even vaguely thinking about using the ground state charge density as a fundamental quantity from which to build an exact theory of a many-electron system. The idea was not “in the air” and all eyes were riveted either on the many-electron wave function, the first- and second-order density matrices, or the Green functions of many-body field theory. The zeitgeist of electronic structure theory was simply not moving in the direction of the electron density function for some particulary well-prepared scientist to exploit and earn the accolades of discovery.
I now turn to Kohn himself. Section IV detailed how Walter chose to focus his fall 1963 sabbatical leave on a problem that was [*not*]{} under active investigation by many of his theoretical colleagues. Namely, how might one calculate the electronic structure of a three-dimensional disordered metal alloy, a system with no underlying spatial periodicity?[^111] His study of the metallurgy and metal physics literature inspired him to ask whether the electron density $n({\bf r})$ was sufficient to completely characterize a many-electron system. It must be admitted that Kohn was a product of his scientific milieu as much as any other electronic structure theorist working at the time. Therefore, the mere fact that this question came to his mind [*and he took it seriously*]{} must be regarded as a legitimate ‘eureka moment’ which few are privileged to experience. That being said, I wish to argue further that his particular history, style of research, and scientific tastes made him unusually well-suited to exploit this insight and create from it the edifice of density functional theory.
Two aspects of Kohn’s pre-college years (surveyed in Section II) bear on the narrow question of his future life as a physicist. First, the cataclysm of the [*Anschluss*]{} put the budding classics scholar into contact with Emil Nohel and Victor Sabbath, two high school teachers whose passion for their subjects converted him to an enthusiastic student of mathematics and physics. Second, the camp schools Kohn attended while interned in Canada exposed him to sophisticated one-on-one instruction from professional scientists. A pedagogical experience of this kind is barely imaginable at a conventional high school, then or now. In a normal setting, there is little chance that the teen-aged Kohn would have encountered (much less devoured) books like Hardy’s [*A Course in Pure Mathematics*]{} and Slater’s [*Introduction to Chemical Physics*]{}.[^112]
Walter’s undergraduate and master’s level classroom experiences at the University of Toronto were probably typical of first-rate academic institutions at the time. What was not typical was the unusually high calibre of the individuals who mentored him and who (because of their own professional interests) repeatedly emphasized variational principles for both general proofs and for detailed numerical calculations. It is true that all well-trained theoretical physicists learn about variational principles and many use them occasionally in their professional work. However, very few physicists who learn about them as undergraduates go on to work with a doctoral supervisor like Julian Schwinger who attacked almost every problem from a variational point of view, and then write a thesis where variational principles are used (again) both to prove a general result and to obtain numerical results for a specific quantum situation. Fundamental and numerical variational calculations appear over and over again in Kohn’s solid state work at Carnegie Tech and Bell Labs through the 1950’s. It is little wonder that he turned quickly to this powerful tool when he sought to prove the first Hohenberg-Kohn theorem, which states that the many-body wave function and everything calculable from it are functionals of the ground state electron density $n({\bf r})$. The second Hohenberg-Kohn theorem, which states that the total energy takes its minimum value when $n({\bf r})$ is the exact ground state density, is explicitly a variational result.
The Hohenberg-Kohn paper is austere, elegant, and deep. Like several other papers in his oeuvre, it demonstrates a characteristic of his work that a Kohn-watcher of fifty years tenure summarized in this way (Langer 2003):
> He always has loved mathematical elegance, but he reserves it for situations where it is truly necessary. His emphasis \[is always\] on the most important physical questions and the ways in which they could be answered with insight and confidence.
This observation is interesting and important, but it does not distinguish Kohn from a number of other theoretical physicists with a taste for proving theorems. However, I believe it is unlikely that any of them would have had either the interest or the inclination to derive the Kohn-Sham equations and suggest the local density approximation for practical calculations. To make this case, I have surveyed the publications of the most active theoretical solid state physicists working between 1950 and 1980. As a matter of personal taste, three broad activities engage them: formal calculations and proofs of theorems, analyses of model Hamiltonians, and numerical calculations for specific materials systems. If an individual was active in more than one of these, it was most often the first and second activities or the second and third activities. Kohn is unusual among his peers simply because he followed up a paper which asks and answers a deep theoretical question with a paper which constructs a practical tool to perform calculations for specific systems.
Kohn’s own early research history demonstrates a willingness to compute actual numbers for direct comparison with experiment. The Kohn-Sham equations are the vehicle for this activity in the context of electronic structure theory. In practice, Kohn turned over much of the explicit numerical work on DFT to his students and post-docs, but there is complete agreement among this cohort that he never regarded this activity as a less important part of his group’s research. This is the reason that a senior quantum chemist could remark that Kohn is “the least arrogant of the deep physicists” in the sense that he does not “give a lower standing to those parts of physics that deal with the complexities of phenomena governed by known laws” (Baerends 2003). The ‘complexities’ mentioned here are discovered only by carrying out numerical computations for specific systems using the ‘known laws’ described by the Kohn-Sham formalism.
In summary, Walter Kohn earned one-half of the 1998 Nobel Prize in Chemistry by asking himself a simple (yet deep) scientific question about the electronic structure of matter. He answered that question in an elegant and thought-provoking manner and then exploited his result to re-formulate the quantum many-electron problem in a manner which made calculations for real systems computationally cheap and surprisingly accurate. These things may have been achieved by someone else if Kohn and his post-doctoral associates had not done so in the years 1963-1965, but that person would probably look very much like Walter Kohn himself.
The physicist Walter Kohn learns a new trade after\
winning one-half the 1998 Nobel Prize in Chemistry.\
Drawing by Peter Meller. Courtesy of Walter Kohn.
Acknowledgments
===============
First and foremost, I thank Walter Kohn, who graciously granted me two interviews, answered my follow-up questions by email, and gave me access to several private documents and photographs. Prof. Kohn’s administrative assistant, Ms. Chris Seaton, deserves special mention for her unfailing help. I am also grateful to Pierre Hohenberg and Lu Jeu Sham, who shared with me both their memories and their personal photographs. Dozens of people communicated with me about their experiences with Kohn and his work and I collectively thank all of them here. Ms. Pia Otte provided excellent translations from German into English and I acknowledge Andre Bernard (John Simon Guggenheim Foundation), Carlo Siochi (University of Toronto), Gregory Giannakis (McGill University), and Felicity Pors (Niels Bohr Archive) for sending me copies of original documents. The staffs of the Department of Special Collections of the Library of the University of California at Santa Barbara and of the Vancouver Holocaust Education Centre provided essential help. Special thanks go to my Georgia Tech colleague, Glenn Smith, who read every word and offered many helpful suggestions.
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[^1]: John Anthony Pople (1925-2004) earned his BA (1946) and PhD (1951) degrees in Mathematics from Cambridge University. His PhD thesis, “Lone pair orbitals” was supervised by John Lennard-Jones, head of the Department of Theoretical Chemistry. A native Englishman, Pople spent more than a decade teaching and conducting research at Cambridge before moving to the United States in 1964 to take a position as Professor of Chemical Physics at the Carnegie Institute of Technology in Pittsburgh, PA. During his thirty-year career at Carnegie Tech, Pople made a gradual transition for the development of semi-empirical methods in molecular orbital theory to the development of computer codes to solve the Schrödinger equation for small molecules at the Hartree-Fock level and beyond. He moved to Northwestern University in 1993 and earned one-half of the 1998 Nobel Prize in Chemistry for his achievements in computational quantum chemistry (Buckingham 2006).
[^2]: Walther Kohn began to use the name Walter Kohn in 1940.
[^3]: Beller (1989) paints a cultural portrait of the Jewish community in early twentieth century Vienna.
[^4]: This would be Kohn’s sixth (and last) year of secondary school in Vienna. Quite unusually for Austria at that time, boys and girls were not separated for instruction in Walther’s small class at the Chajes school (Neuhaus 2003).
[^5]: Emil Eliezer Nohel grew up on a small farm in Bohemia. He studied mathematics at the Karl-Ferdinand (German) University in Prague and served as an assistant to Albert Einstein when Einstein was a professor there from 1911 to 1914. Nohel’s descriptions of the difficult conditions endured by Jews in Bohemia awakened Einstein’s concern for the plight of his co-religionists. For most of his career, Nohel taught mathematics at the Handelsakademie Wien, a business-oriented secondary school in Vienna. The Anschluss precipitated his dismissal and he found work as the physics teacher (and then the principal) of the Zwi Perez Chajes Gymnasium. Nohel was arrested on December 12 1942 and spent two years at the Theresienstadt labor camp before he was transferred to the Auschwitz concentration camp and murdered by the Nazis (Frank, 1947, Pais 1982).
[^6]: The unemployment rate in Austria stood at 20 percent at the time of the Anschluss (Senft 2003).
[^7]: The term ‘fifth column’ refers to the secret supporters of an enemy who live openly within the territory being defended.
[^8]: Vienna native Otto Frisch was working in the physics laboratory of Prof. Mark Oliphant at the University of Birmingham when the roundup of aliens began. He avoided internment only because his employers made the case that he was engaged in important war work (Frisch 1979).
[^9]: Eisinger’s family smuggled him onto a Kindertransport train out of Vienna after failing to find an official sponsor or a foster family for him in England. He fended for himself, working on a farm in Yorkshire and washing dishes in a hotel in Brighton, until the internment roundup of 1940 (Eisinger 2013).
[^10]: The [*Sobieski*]{} was the last of four ships assigned the task to transport internees to Canada. The [*Ettrick*]{} and the [*Duchess of York*]{} crossed the Atlantic safely. The [*Arandora Star*]{} was torpedoed by a German submarine on July 2 and sunk with a loss of more than 800 lives. Public outcry over this incident contributed to the cessation of the internments and the first releases of detainees from the Isle of Man (Gilman & Gilman 1980).
[^11]: Camp B was built on the site of an abandoned, depression-era, unemployment relief facility.
[^12]: Kurt Martin Guggenheimer (1902-1975) studied chemistry at the University of Munich and physics at the University of Berlin. He earned his PhD in 1933 for work on the ultraviolet absorption spectra of zinc, potassium, and cesium under the direction of Fritz Haber. Guggenheimer pursued post-doctoral work in Paris under the direction of Paul Langevin and published his speculations about nuclei while there. He returned to Munich in 1935 but was arrested following Kristallnacht and spent several months at the Dachau detention center. He emigrated to England and was working at King’s College (Cambridge) when he was interned as a enemy alien. After the war, Guggenheimer worked as a Lecturer at the University of Bristol and the University of Glasgow before retiring from academic life in 1967 (Rürup 2008, Fernandez 2013).
[^13]: Fritz Rothberger (1902-2000) was a native of Vienna who graduated from the Akademisches Gymnasium and earned BS and PhD degrees in mathematics from the University of Vienna. He came under the influence of Waclaw Sierpinksi in Warsaw and began his lifelong work in combinatorial set theory. Rothberger emigrated to England just before the start of World War II and he was working as a scholar at Trinity College (Cambridge) when he was interned. After the war, he served as a professor on the mathematics faculties of several Canadian universities: Acadia, Fredericton, Laval, and Windsor (Swaminathan 2000).
[^14]: The subjects tested were English literature, English composition, general history, elementary algebra, elementary geometry, physics, Latin authors, Latin composition, German grammar, German composition, intermediate algebra, intermediate geometry, and trigonometry. By his own account, Kohn performed better in Latin than in German and his worst subject was Canadian history (Kohn 2013a).
[^15]: Godfrey Harold Hardy (1877-1947) was a leading British mathematician of his time. He was a child prodigy who trained at Trinity College (Cambridge) and served as a professor at both Cambridge and Oxford over the course of his career. He authored or co-authored eleven books and over 300 research papers, mostly in the fields of analysis and number theory. He was particularly well-known for his collaborations with Srinivasa Ramanujan and John Edensor Littlewood, and for his disdain for any kind of “applied” work (Titchmarsh 1949).
[^16]: John Clarke Slater (1900-1976) was the chair of the physics department at the Massachusetts Institute of Technology (MIT) from 1930-1951. He wrote an experimental PhD thesis at Harvard, but then traveled to Europe where he made several important theoretical contributions to the early development of quantum mechanics. Dissatisfaction with ‘formal theory’ led him to develop a large research group at MIT devoted to solving the Schrödinger equation numerically to calculate the physical properties of atoms, molecules, and solids. He authored or co-authored thirteen books and over 150 research papers. An important paper he published in 1951 turned out to bear directly on Kohn’s density functional theory (Morse 1982).
[^17]: [*A Course in Pure Mathematics*]{} is an elegant and rigorous introduction to mathematical analysis for serious first-year college students. The subject matter includes the notions of limit and convergence applied to series, sequences, functions, derivatives, and integrals. All the main theorems of the calculus of a real variable are discussed, as is the general theory of logarithmic, exponential and sinusoidal functions.
[^18]: The undergraduate course Slater taught at MIT using [*Introduction to Chemical Physics*]{} proved to be too difficult for most of its intended audience. In later years, a separate “modern physics” course devoted to atoms, molecules, and the structure of matter became a prerequisite for a senior-level “thermal physics” course which retained Slater’s idea to combine thermodynamics and statistical mechanics in a single presentation. A textbook written specifically for the latter by two of Slater’s MIT colleagues expanded his treatment of thermodynamics and contracted his treatment of statistical mechanics (Allis and Herlin 1952).
[^19]: The future Nobel laureate Max Perutz (1914-2002) spent only six months as an internee in Canada. His memoir [*Enemy Alien*]{} recounts how he was arrested by the British just four months after earning his PhD under the direction of Sir Lawrence Bragg. He was deported to the Cove Fields internment camp in Quebec City (Camp L) where he organized a camp school with a faculty that included the physicist Klaus Fuchs and the future astrophysicists Hermann Bondi and Thomas Gold. Fuchs was recruited to the atomic bomb project after his release and gained notoriety in 1950 when it was discovered that he had betrayed the secrets of the Los Alamos laboratory to the Soviet Union (Perutz 1985).
[^20]: The Canadian government announced in May 1941 that any internee under the age of 21 cleared by Scotland Yard would be released and given the opportunity to continue his education in Canada if he could find a sponsor willing to pay a fee of two thousand dollars (Jones 1988).
[^21]: Eisinger was released in early January. Kohn was released a few weeks later after a short stay at Camp N, an internment camp outside the town of Sherbrooke, Quebec, about 130 km east of Montreal (Eisinger 2011).
[^22]: Bruno Mendel (1897-1959) was the son of a research-active medical doctor who trained in Berlin and became a research-active physician himself. His medical practice slowly became less important as he increased the time he spent researching the metabolism of the cancer cell in his small private laboratory. Mendel read the political situation correctly and he took his wife and three children to Holland when Hitler came to power in 1933. In 1937, he emigrated to Canada and became (at first) an unpaid faculty member at the Banting Institute for Cancer Research of the University of Toronto. He returned to Europe in 1950 to accept a chair in Pharmacology at the University of Amsterdam (Feldberg 1960).
[^23]: Eisinger followed Kohn to the University of Toronto the following year (Eisinger 2013).
[^24]: Leopold Infeld (1898-1968) earned the first PhD in theoretical physics awarded by a Polish university from the Jagellonian University in his native city of Krákow. He taught physics at Jewish high schools for nearly a decade before finding a senior assistantship in theoretical physics at Lwow University. Infeld knew the importance of contacts with foreign physicists and successfully gained two-year visiting positions with Max Born in Cambridge and Albert Einstein in Princeton. With the latter, he co-authored [*The Evolution of Physics*]{}, a popular account of the history of ideas in physics. In 1938, Infeld accepted a position at the University of Toronto where he worked on a variety of theoretical problems in general relativity and cosmology. He returned to Poland in 1950 to found an Institute of Theoretical Physics at the University of Warsaw (Infeld 1980).
[^25]: Samuel Beatty (1881-1970) earned the first PhD in mathematics awarded by a Canadian university at the University of Toronto. He remained at Toronto and from 1911-1959 rose from Lecturer in Mathematics to Professor and chair of Mathematics, to Dean of the College of Arts & Science, and finally to Chancellor of the entire University. He published 30 research papers, mostly in the field of algebraic functions, and devoted the bulk of his energy to teaching and to building the Mathematics department (Robinson 1979).
[^26]: Frank B. Kenrick (1874-1951) trained as a physical chemist with Wilhelm Ostwald in Leipzig. He served as chair of the Chemistry Department at the University of Toronto from 1937-1944. Like his mentor and predecessor as department chair, William Lash Miller (1866-1949), Kenrick favored an approach to chemistry that denied the reality of individual atoms and molecules. The war-related work carried out in Kenrick’s department included experiments to develop the new explosive RDX as a replacement for TNT and the development of detectors for poison gas (Brook and McBryde 2007, Avery 1998).
[^27]: Dean Beatty apparently “regularized” Kohn’s ‘special student’ status at some point because his official transcript inaccurately states that he completed an introductory chemistry course and a chemistry laboratory course (Siochi 2013).
[^28]: Kohn and Josef Eisinger had volunteered for (and been rebuffed by) the Canadian Air Force immediately after their release from internment (Eisinger 2013).
[^29]: Kohn worked the summers of 1942-1943 for the Sutton-Horsley Company, a Toronto x-ray equipment manufacturer which began producing signalling lamps and instrument panels for fighter and bomber aircraft after the war began. His specific task was the design and testing of compensation circuits to ensure that cockpit instruments gave accurate readings when operated at unusually high and low temperatures (Kohn 1998, Horsley 2013, Kohn 2013b). Kohn worked the summers of 1945-1946 for the mineral surveying and exploration geophysics company Koulomzine, Geoffroy, Brossard & Company of Val D’Or, Quebec. His job was to conduct magnetic field surveys in suspected gold-bearing regions of northern Ontario. A typical survey consisted of a grid of about 1000 magnetometer measurements with a grid-spacing of 100 meters (Geoffroy 1946, Kohn 1998, Kohn 2013b).
[^30]: See footnote 2.
[^31]: John Lighton Synge (1897-1995) is often regarded as the greatest mathematician of Irish descent since Sir William Rowan Hamilton. Synge studied mathematics at Trinity College Dublin and accepted a position as Assistant Professor at the University of Toronto in 1920. There he began a lifelong interest in Einstein’s theory of relativity and in geometrical methods to analyze dynamical systems. The peripatetic Synge subsequently held positions in Dublin, Toronto (again at the time Kohn was there), Ohio State, Carnegie Tech, and finally the Dublin Institute for Advanced Studies. He published 11 books and over 200 hundred papers (Florides 2008).
[^32]: Alexander Weinstein (1897-1979) was a PhD student of Hermann Weyl, who considered him to be his most talented student. He worked with Tullio Levi-Cività in Rome and Jacques Hadamard in Paris before the German occupation of France in 1940 drove him from Europe permanently. Weinstein was a member of the Applied Mathematics faculty of the University of Toronto from 1941-1946, worked for some time at the US Naval Ordnance Laboratory, and spent 18 productive years at the Institute for Fluid Dynamics and Applied Mathematics at the University of Maryland (Diaz 1978).
[^33]: Arthur Francis Chesterfield Stevenson (1899-1968) accepted a position in the Mathematics department of the University of Toronto immediately after earning his BA from Trinity College, Cambridge in 1922. He returned to Cambridge in 1928 where he worked under the supervision of Ralph Fowler on a problem in theoretical spectroscopy which eventually led to his PhD. He returned to Toronto where he published original research and lectured on atomic physics, quantum mechanics, electromagnetic theory, scattering theory, and the differential equations of mathematical physics. He spent the last dozen years of his academic career on the faculty of Wayne State University in Detroit, Michigan (Duff, 1969).
[^34]: It is necessary here that trial function satisfies $\int d\tau \psi^* \psi =1$ and that the integral $\int d\tau \psi^* H \psi$ converges.
[^35]: Julian Seymour Schwinger (1918-1994) was one of the greatest theoretical physicists of the $20^{\rm th}$ century. By the age of 21, he had earned his PhD under the (nominal) supervision of Isador Rabi and published ten research papers in quantum mechanics and nuclear physics. He spent the war years working on waveguides for radar applications before turning his attention to quantum electrodynamics. This work earned him a one-third share of the 1965 Nobel Prize in physics. He supervised 73 PhD students over a long academic career at Harvard (1945-1972) and UCLA (1972-1994) (Martin & Glashow 1995, Mehra & Milton 2000).
[^36]: John Hasbrouk Van Vleck (1899-1980) was an eminent theoretical physicist who contributed widely to the fields of chemical physics, quantum electronics, solid state physics, and magnetism. He published 169 scientific articles and two books and served on the faculty at Harvard for forty-six years. In 1977, he was awarded a one-third share of the Nobel Prize in physics for his work in magnetism (Anderson 1987).
[^37]: Kohn had learned from his fellow graduate students about Wolfgang Pauli’s famously negative view that solid-state physics was insufficiently fundamental and too approximate to attract the attention of a serious young theoretical physicist (von Meyenn 1989).
[^38]: It is necessary to replace $E_f$ by the complex number $ E_f + i\epsilon$ in Eq. (\[three\]) and let $\epsilon \to 0$ at the end to ensure that $\psi$ behaves like an outgoing spherical wave (Baym 1969).
[^39]: This difficult and subtle problem was solved in the early 1960’s by the Russian mathematical physicist Ludvig Faddeev (Faddeev 1965).
[^40]: At some point before he wrote up his thesis, Kohn learned that the Swedish physicist Lamek Hulthén had independently derived a variational principle for the scattering phase shift very similar to his own. Hulthén (1946) and Kohn (1948) begin with the same variational functional but propose slightly different variational procedures. The Kohn-Hulthén variational principle later found wide application in atomic, molecular, and nuclear scattering problems (Adhikari 1998, Nesbet 2003).
[^41]: This did not prevent Schwinger from later characterizing Kohn as “my most illustrious student” to one of Kohn’s former graduate students (Rudnick 2003).
[^42]: Josef Eisinger remained a close friend. He did his graduate work at MIT, just two miles down the Charles river from Harvard, and earned his PhD in physics in 1951 for an experimental determination of the magnetic moment of ${\rm K}^{40}$. He spent thirty years at Bell Laboratories where he made a transition from solid state physics to biophysics. From 1985 until his retirement in 1998, he taught and conducted research at the Mount Sinai School of Medicine in New York City (Eisinger 2013).
[^43]: An evaluation of Kohn written by Ms. Norine T. Casey provides insight into Kohn’s teaching style and philosophy of physics. Ms. Casey was a 1949 Wellesley graduate pursuing an MA in teaching at Harvard. As part of her curriculum, she attended Kohn’s summer 1950 class (devoted to introductory optics, electricity, magnetism, atomic physics, and nuclear physics) and wrote a four-page evaluation of her experience. Kohn received a copy of her report (Casey 1950), which states that “Dr. Kohn’s lectures were clear and concise. Demonstrations accompanied every lecture and were given with great enthusiasm. . . . It was obvious from the beginning that \[Dr. Kohn’s\] interest was not his own mastery of the mathematics, but in our understanding of the physics. . . . It was not infrequent that he read from source material giving direct quotes \[such\] as Newton’s relating his discovery of the diffraction of light.”
[^44]: This technique had recently been introduced to study cosmic rays using plates produced by another company and Polaroid wanted to enter the business (Powell and Occhialini 1947).
[^45]: Frederick Seitz (1911-2008) was one of the creators and intellectual leaders of the American solid state physics community. He was Eugene Wigner’s first PhD student at Princeton and built influential research groups at four different universities between 1935 and 1965. Later, he served as president of Rockefeller University and president of the National Academy of Sciences. Seitz worked on a wide range of materials problems during World War II and was the author or co-author of more than 100 scientific papers (Slichter 2010).
[^46]: Silverman’s PhD thesis states that “The author wishes to express his indebtedness to Professor Walter Kohn for suggesting the problem and for invaluable guidance during a large portion of the work (Silverman 1951). A generation of English-speaking solid-state physicists have Silverman to thank for his translation into English of [*Methods of Quantum Field Theory in Statistical Physics*]{} by A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski (1963).
[^47]: Edward Chester Creutz (1913-2009) earned his PhD in experimental nuclear physics from the University of Wisconsin in 1938. He moved to Princeton as an Instructor and used their cyclotron for nuclear physics projects until the Manhattan Project redirected his efforts to the synthesis of plutonium and the triggering of the plutonium bomb. After the war, Creutz joined the faculty at Carnegie Tech to direct the construction of a proton synchrocyclotron and to build an experimental nuclear physics group. After nine years (seven as chair), Creutz moved to San Diego, California to help found the General Atomics division of General Dynamics Corporation. He spend 15 years there before concluding his career as Associate Director of the National Science Foundation (Hinman and Rose 2010).
[^48]: One former Carnegie Tech graduate student remembers Kohn’s solid state physics lectures as well-prepared, clearly delivered, and mathematically precise. He was surprised when the final exam avoided mathematical issues and focused entirely on qualitative aspects of the subject (Arrott 2013).
[^49]: Aage Bohr and Christian M[ø]{}ller were already fixtures at the Institute. Kohn’s Harvard classmate Ben Mottelson arrived in the fall of 1950 and never left. A partial list of visitors who overlapped with Kohn for at least some time includes Hendrik Casimir, Freeman Dyson, Ugo Fano, Nicolas Kemmer, Louis Michel, Wladyslaw Swiatecki, Jean Valatin, Nicolaas van Kampen, and Arthur Wightman (NBA 1951a).
[^50]: A partial list of attendees includes Hans Bethe, Homi Bhabha, Léon Brillouin, Richard Dalitz, Paul Dirac, Maria Mayer, Dirk ter Haar, Werner Heisenberg, Walter Heitler, Léon van Hove, Lamek Hulthén, Egil Hylleraas, Hendrik Kramers, Ralph Kronig, Jens Lindhard, Lise Meitner, Wolfgang Pauli, Rudolf Peierls, Léon Rosenfeld, John Slater, Ernst Stueckelberg, Victor Weisskopf, Gregor Wentzel, John Wheeler, and Gian-Carlo Wick (NBA 1951b).
[^51]: Kohn was an emergency replacement for Mario Verde, a nuclear physicist who had fallen ill (DeWitt 1951, 2013).
[^52]: Res Jost (1918-1990) wrote his PhD thesis under the supervision of Gregor Wentzel and spent three years as the principal assistant to Wolfgang Pauli. He was a senior fellow at the Institute for Advanced Study in Princeton for six years (1949-1955) before accepting a professorship at the Eidgenössische Technische Hochschule in Zürich. Jost’s research focused on mathematical physics and quantum field theory, particularly axiomatic versions of the latter. (Kohn [*et al.*]{} 1992, Pais 1996).
[^53]: For fifteen years after World War II ended, most solid state physics research in the United States was funded by the Office of Naval Research. Most nuclear physics research was supported by the United States Atomic Energy Commission (Old 1961, Sapolsky 1990).
[^54]: The actual computing was performed by a ‘computress’ named Alice Watson who operated a Friden Model STW-1 Electro-Mechanical Calculator. Although her equipment changed to an IBM 650 digital computer in 1956, she continued to do computing tasks for Kohn the entire time he worked at Carnegie Tech (Rostoker 2003, Young 2013).
[^55]: Korringa (1994) relates that “computers were rare in the Netherlands in 1946 and a cost estimate \[for a numerical application\] exceeded the annual research budget of our theory group."
[^56]: Kohn left the development and application of the Korringa-Kohn-Rostoker (KKR) method to others. It eventually became a standard method of band structure calculation (Zabloudil [*et al*]{}. 2005).
[^57]: Hoddeson [*et al.*]{} (1992) is a history of solid state physics up to about 1960. The autobiography of Frederick Seitz (1994) provides a broad view from the perspective of a major player in the development of solid state physics as the discipline matured through the 1950’s and 1960’s.
[^58]: Jacob E. Goldman (1921-2011) was born in Brooklyn, New York and studied physics at Yeshiva University and the University of Pennsylvania. His expertise in magnetism led him to the Westinghouse Research Laboratory in 1945 before he joined the faculty of the Carnegie Institute of Technology in 1951. He moved to the nascent Scientific Research Laboratory of the Ford Motor Company in 1955 and eventually became head of all Ford’s corporate research and development. He joined the Xerox corporation in 1969 and one year later founded their Palo Alto Research Center (PARC). The first modern personal computer and the first graphical user interface were invented at PARC a few years later (Markoff 2011).
[^59]: Kohn never worked personally in the field of conventional thermal phase transitions. Nevertheless, he often supported a post-doctoral fellow trained in statistical mechanics to work on this class of problems. See Domb (1996) for a history of this subject.
[^60]: Joaquin Mazdak Luttinger (1923-1997) earned his BS in physics at the Massachusetts Institute of Technology and remained there to complete a PhD thesis (1947) in statistical physics under the supervision of Lazslo Tisza. He worked on quantum electrodynamics as the first American post-doctoral fellow of Wolfgang Pauli but reverted to problems in many-body theory, solid state physics and statistical mechanics for the rest of his career as a professor, primarily at Columbia University (1960-1993) (Anderson [*et al.*]{} (1997).
[^61]: Sixty years later, the Bell Laboratories experimenter Robert Fletcher recalled that “Walter was a very kind and thoughtful person to work with. I never had the impression he looked down on us experimenters as some theorists are inclined to do" (Fletcher 2013). Another Bell Laboratories experimenter who published papers related to the Kohn-Luttinger theory was Walter’s old friend Josef Eisinger (Eisinger and Feher 1958).
[^62]: Another graduate student, James Montague, never quite finished a thesis devoted to deep (strongly bound) impurity levels in semiconductors (Glasser 2013).
[^63]: Kjeldaas pursued his Ph.D. part-time at the University of Pittsburgh. His 1959 thesis thanks Kohn and Westinghouse solid state theorist Theodore Holstein for acting as co-supervisors (Kjeldaas 1959). Westinghouse was a lively place for solid-state physics in the mid-1950’s under the leadership of its Director of Science, Clarence Zener, himself a solid-state theorist. The theorists he recruited to complement Holstein included Edward Neufville Adams, Petros Argyres, William Mullins, and Yako Yafet. The experimenters hired by Zener included Raymond Bowers, Robert Keyes, Colman Goldberg, and John Rayne. Walter Kohn had a consulting arrangement with Westinghouse. Holstein, Adams, and Yafet occasionally taught classes at Carnegie Tech (Ambegaokar 2013, Arrott 2013).
[^64]: By “screening” or “shielding”, we mean that the electrons nearest to the positive charge are attracted to it and thereby partially neutralize the Coulomb force exerted by the positive charge on distant electrons and ions.
[^65]: Visitors to the theory group that summer included Elihu Abrahams, Kerson Huang, David Pines, J. Robert Schrieffer, and Philippe Nozières (Anderson 1978).
[^66]: The Hall effect refers to a voltage that appears across a current-carrying sample when a magnetic field is applied in a direction perpendicular to the direction of current flow (Chien and Westgate 1980).
[^67]: Kohn and Luttinger did not ultimately address the problem of the effect of a large concentration of impurities on the electrical conductivity of a semiconductor. This was done by Phil Anderson (1958). Simultaneous with Kohn and Luttinger’s work on quantum transport, the Japanese physicist Ryogo Kubo proposed a theoretical approach to the same problem which ultimately became standard (Kubo 1957).
[^68]: There is no definitive history of this revolution. Different points of view can be found in Pines (1961), Hoddeson [*et al.*]{} (1992), Gell-Mann (1996), Brueckner (2000), and Kaiser (2005). Percus (1963) is the proceedings of a January 28-29 1957 meeting convened at the Stevens Institute of Technology “for the purpose of bringing together workers in the numerous rapidly moving fields of many-particle physics.” The contributions to this volume (particularly the roundtable discussions) paint a vivid picture of the first months of the revolution. The Bardeen-Cooper-Schrieffer theory of superconductivity appeared later the same year, but Bardeen [*et al.*]{} (1957) makes no explicit use of field theoretic methods.
[^69]: Keith Allen Brueckner (1924 - ) studied mathematics at the University of Minnesota before earning his Ph.D in physics (1950) from the University of California (Berkeley) under the supervision of Robert Serber. As a professor at Indiana University and the University of Pennsylvania in the 1950’s, Brueckner made many significant contributions to the theories of nuclear matter and the electron gas. In 1959, he became the first member of the physics department at the newly created University of California, San Diego. Brueckner divided his professional activities between academia, industry, and the government until his retirement from UCSD in 1991.
[^70]: The Green functions used in many-body theory are the ground state expectation values (averages over the ground state many-body wave function) of various quantum mechanical operators. The Green function for the Schrödinger partial differential equation, Eq. (\[two\]), is related to some of the Green functions used in many-body theory when all the forces between the particles are turned off.
[^71]: Samuel F. Edwards, another Schwinger PhD student who switched from nuclear physics to solid state physics, has remarked that ‘the Green function formalism is very good to write down solutions in abstract exact form, which gives unassailable answers when used in comparatively simple situations’ (Edwards 1998).
[^72]: Kohn became an Associate Editor of the [*Journal of Mathematical Physics*]{} in 1961.
[^73]: A phonon is a quantized lattice vibration in a crystal. The Fermi surface is the constant energy surface in the Bloch ${\bf k}$-space for the most energetic electrons in a metal.
[^74]: In this way, Langer became the Peierls PhD student Kohn would have been if he had not gone to Harvard to work with Julian Schwinger (Kohn 2013).
[^75]: Langer and Vosko used a formulation of many-body perturbation theory due to John Hubbard (1957).
[^76]: The regents of the University of California committed unprecedented financial resources so UCSD could recruit senior scientists like Kohn to its nascent faculty. In its first few years, 50 percent of new faculty hires at UCSD were made at the full professor level or above, as opposed to 15 percent for the University of California system as a whole (Kerr 2001).
[^77]: The Symposium was organized by Conyers Herring from Bell Telephone Laboratories. Besides Kohn, the lecturers were John Fisher from the General Electric Research Laboratory, Jack Goldman from (by then) the Ford Motor Company Research Laboratory, and Frank Herman from RCA Laboratories (WKP 1959a).
[^78]: Suhl and Feher arrived in San Diego in 1960. Matthias waited a year because “during the first year there will be too many administrative chores” (Feher 2002).
[^79]: Offers were also proffered to (and declined by) Kohn’s old Harvard friend Ben Mottelson, a nuclear physicist at Bohr’s Institute for Theoretical Physics, and the French magnetic resonance expert, Anatole Abragam (WKP 1959c, Abragam 1989).
[^80]: An APS Fellow is judged by his peers to have made “exceptional contributions to the physics enterprise". In 1960, the total number of Fellows was 1653 out of a total Society membership of 16,157 (APS 2013b).
[^81]: A visiting French scientist, Jacques des Cloizeaux, arrived in the fall of 1960 and worked on a statistical mechanics problem.
[^82]: A “sharp” Fermi surface has the property that a quantum state labeled by the quantum number wave vector ${\bf k}$ is occupied by an electron if that wave vector lies inside the volume of the ${\bf k}$-space enclosed by the Fermi surface and unoccupied if ${\bf k}$ lies outside that volume.
[^83]: The unusual hours were chosen for “the convenience of students employed in industry” (UCSDA 1960).
[^84]: Kohn taught semiconductor physics and the transport and optical properties of metals for five weeks in the fall of 1962 (Bruch 2013). His April 1962 campus-wide lecture was part of a series delivered by senior UCSD professors from various departments.
[^85]: Luming (later Luming Ren) switched his PhD supervisor from Kohn to Assistant Professor David Wong because he felt that particle physics was “more fundamental” than solid state physics. He came to rue his decision when he was unable to find permanent employment as a particle physicist and made his career as a systems engineer for the Hughes Aircraft Company (Ren 2013).
[^86]: Philip Warren Anderson (1923 - ) is generally regarded as one of the preeminent theoretical physicists of the second half of the twentieth century. He attended Harvard University as an undergraduate, worked at the U.S. Naval Research Laboratory during World War II and returned to Harvard to earn his PhD in 1949 under the supervision of John van Vleck. Anderson spent a long and productive career at Bell Telephone Laboratories before moving to Princeton University in 1984. He shared the 1977 Nobel Prize in Physics with Nevill Mott and John Van Vleck for ‘fundamental theoretical investigations of the electronic structure of magnetic and disordered systems’ (Anderson 1977).
[^87]: Philippe Nozières (1932- ) graduated from the Ecole Normale Supérieure (ENS) in 1955 and earned his PhD two years later from the University of Paris, albeit under the supervision of David Pines at Princeton University. He spent a decade at the ENS before moving to the Institut Laue-Langevin (ILL) in Grenoble. He also lectures at the Collège de France in his capacity (since 1983) as Professor of Statistical Physics (Nozières 2012a).
[^88]: In July, Kohn had written to an editor at Academic Press confirming his interest to contribute to a book about “Impurities in Metals” and indicating that he would be “working in this field” during his stay in Paris (WKP 1963b).
[^89]: Jacques Friedel (1921 - ) is a fourth-generation French scientist who was educated at the École Polytechnique (1944-46) and the École Nationale Supérieure des Mines (1946-48) before earning his PhD in 1952 under the supervision of Nevill Mott at the University of Bristol. Friedel began his academic career at the Sorbonne, but moved in 1959 to the Orsay campus of the University of Paris, now the University of Paris-Sud. Friedel’s life-long interests in metallurgy and the physics of metals resulted in over 200 theoretical publications, most of them characterized by the use of simple models and elementary mathematics (Bréchet 2008). Kohn and Friedel met at a July 1953 Gordon Conference in Laconia, New Hampshire devoted to the Chemistry and Physics of Metals (Kohn 2012a).
[^90]: Pierre-Gilles de Gennes (1932-2007) changed fields after the publication of his superconductivity book and won the 1991 Nobel Prize in Physics for his work on the statistical physics of liquid crystals and polymers.
[^91]: Alloys of this kind are called [*disordered*]{}. The Bloch theorem remains valid for [*ordered alloys*]{} where the A-type atoms form a spatially periodic structure of their own.
[^92]: This follows from the eigenstate occupation rules of quantum mechanics which dictate that states are populated by electrons in order of increasing energy beginning with the lowest.
[^93]: This is the same John Slater whose book [*Introduction to Chemical Physics*]{} was purchased and read by Walter Kohn while he awaited release from internment in wartime Canada. See Section II.
[^94]: The unedited proceedings of this conference were also published in the October 1962 issue of [*Journal de Physique et le Radium*]{}. Reading them may have been the trigger for Kohn to change the subject of his Paris research from the electron-phonon interaction to alloy physics.
[^95]: In later recollections, Kohn emphasizes the concept of charge transfer in alloys. His Nobel Prize autobiography reports that he read some “metallurgical literature in which the concept of an effective charge $e^\ast$ of an atom in an alloy was prominent, which characterized in a rough way the transfer of charge between atomic cells” (Kohn, 1998). His Nobel Prize lecture notes similarly that “there is a transfer of charge between . . . unit cells on account of their chemical differences. The electrostatic interaction energies of these charges is an important part of the total energy. Thus in considering the energetics of this system there was a natural emphasis \[in the literature\] on the electron density distribution $n({\bf r})$" (Kohn 1999). Despite these remarks, I have been unable to find any significant discussion of charge transfer or “effective charge” in the pre-1963 literature of metallurgy or metal physics. Indeed, if they mention this type of charge transfer at all, review articles of the period consistently refer to the same two papers—one by Nevill Mott (1937) which concerns ordered alloys and one by John Henry Oliver Varley (1954) which demonstrates that the electrostatic energy associated with charge transfer is negligible in disordered alloys. On the other hand, less than ten years after the events narrated here, charge transfer became an important variable in two proposed theories of binary alloy formation (Hodges and Stott 1972, Miedema, de Boer, and de Chatel 1973). I have found no literature of the time that emphasizes the spatially-varying electron density distribution $n({\bf r})$ in alloys except in the most qualitative terms (see the passage by Hume-Rothery and Raynor quoted earlier in this section).
[^96]: The coefficient of the kinetic energy term in Eq. (\[fifteen\]) includes the electron mass $m$ and Planck’s constant $h$.
[^97]: Later analysis showed that removing the assumption of no degeneracy does not invalidate the final result (Parr and Yang 1989).
[^98]: Another necessary quantity, the first-order density matrix, is derivable from the second-order density matrix.
[^99]: The term ‘v-representability problem’ was coined in 1975. See Parr and Yang (1989) and Kryachko and Lude$\tilde{\rm n}$a (1990) for extensive discussion of this issue.
[^100]: It is curious that Hohenberg and Kohn do not refer to papers by Hubbard (1958), Bellemans and de Leener (1961), and Jones, March, and Sampanthar (1962), all of whom studied the energy of an electron gas in the presence of a lattice of positive charges using methods superior to the Thomas-Fermi approximation.
[^101]: The Web of Science database (accessed January 2014) lists 230 citations to the quantum density oscillations paper, Kohn and Sham (1965a). A review of these citations shows that the vast majority of the citing papers $(>90\%)$ seem unaware of its actual content. They incorrectly cite it along with Hohenberg and Kohn (1964) and Kohn and Sham (1965b) as one of the foundational papers of density functional theory.
[^102]: In the actual equations written by Hartree (1923), each electron feels the classical electrostatic potential produced by every electron except itself. Therefore, in contrast to Eq. (\[H2\]), each electron in Hartree’s theory feels a slightly different electrostatic potential.
[^103]: The correctness of Eq. (\[H4\]) requires that the functions $\phi_i$ and $\phi_j$ be orthonormal, which means that the integral $\int \phi^*_i({\bf r})\, \phi_j({\bf r}) \, d{\bf r}$ is one when $i=j$ and zero when $i\neq j$.
[^104]: The local density approximation for $G[n]$ retains only the first term on the right hand side of Eq. (\[twentythree\]).
[^105]: The Web of Science database (accessed January 2014) lists 21,372 citations to Kohn and Sham (1965b). Only four physics-related papers have more citations and all four of them are density functional papers which owe their existence to the Kohn-Sham paper.
[^106]: Kohn and Sham were unaware that the Hungarian physicist Rezsö Gáspár had derived Eq. (\[x\]) ten years earlier by similarly computing the variational derivative of Dirac’s exchange energy with a local density approximation (Gáspár 1954).
[^107]: Despite the warning in Sham and Kohn (1966) that the $\epsilon_k$ parameters should not be interpreted as one-electron energies, “the temptation to use \[them\] as band structures in solids proved irresistible” (Hohenberg, Kohn and Sham 1990).
[^108]: The generalized gradient approximations go far beyond the simple gradient expansions analyzed by HK and KS.
[^109]: Data collected from Google Scholar in January 2014. The true number of papers is higher because not all authors used the term ‘many-electron’ in their writing.
[^110]: According to Musher (1966), Wilson’s method goes back to Pauli.
[^111]: Early work on this problem published by Korringa (1958) and Beeby (1964) blossomed into a full-scale theory of the electronic structure of disordered alloys in the late 1960’s and early 1970’s (Ehrenreich and Schwartz 1976).
[^112]: It is impossible to know how the frightful loss of his home and parents to Nazi terror motivated Kohn to succeed in later life. A statistical study of Viennese children who had similar experiences during World War II and then emigrated to America shows that those who entered the sciences achieved success (by conventional measures) more than twice as often as native-born American scientists of the same generation. The study quotes several participants who said “they felt a great responsibility to make the most of their lives because their survival was such a rare and unlikely event.” (Sonnert and Holton 2006)
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abstract: |
We propose a novel approach to multi-fingered grasp planning leveraging learned deep neural network models. We train a voxel-based 3D convolutional neural network to predict grasp success probability as a function of both visual information of an object and grasp configuration. We can then formulate grasp planning as inferring the grasp configuration which maximizes the probability of grasp success. In addition, we learn a prior over grasp configurations as a mixture density network conditioned on our voxel-based object representation.
We show that this object conditional prior improves grasp inference when used with the learned grasp success prediction network when compared to a learned, object-agnostic prior, or an uninformed uniform prior. Our work is the first to directly plan high quality multi-fingered grasps in configuration space using a deep neural network without the need of an external planner. We validate our inference method performing multi-finger grasping on a physical robot. Our experimental results show that our planning method outperforms existing grasp planning methods for neural networks.
author:
- 'Qingkai Lu$^{1}$, Mark Van der Merwe$^{1}$, Balakumar Sundaralingam$^{1}$ and Tucker Hermans$^{1}$ [^1] [^2]'
bibliography:
- 'grasp\_ref.bib'
title: '**Multi-Fingered Grasp Planning via Inference in Deep Neural Networks**'
---
Introduction and Motivation {#sec:intro}
===========================
Grasp Planning as Probabilistic Inference {#sec:grasp_inference}
=========================================
Voxel-Based Deep Networks for Multi-fingered Grasp Learning {#sec:grasp_learning}
===========================================================
Robotic Grasp Inference Experiments {#sec:exp}
===================================
Discussion & Conclusions {#sec:conclusions}
========================
[^1]: Q. Lu and B. Sundaralingam were supported in part by NSF Award 1846341.
[^2]: $^{1}$Qingkai Lu, Mark Van der Merwe, Balakumar Sundaralingam and Tucker Hermans are with the School of Computing and the Robotics Center, University of Utah, Salt Lake City, UT 84112, USA. [[email protected]; [email protected]; [email protected]; [email protected]]{}
|
---
abstract: 'We present a discovery of definitive large-scale structures around RXJ0152.7–1352 at $z=0.83$ based on spectroscopic redshifts. In our previous papers, we reported a photometric identification of the large-scale structures at $z\sim0.8$. A spectroscopic follow-up observation was carried out on 8 selected regions covering the most prominent structures to confirm their association to the main cluster. In six out of the eight fields, a well isolated peak is identified in the distribution of spectroscopic redshifts at or near the cluster redshift. This is strong evidence for the presence of large-scale structures associated to the main cluster at $z=0.83$. It seems that there are two large filaments of galaxies at $z\sim0.837$ and $z\sim0.844$ crossing in this field. We then investigate stellar populations of galaxies in the structures. The composite spectra are constructed from a number of red member galaxies on the colour-magnitude sequence. We consider three representative environments – cluster, group, and field – to investigate the environmental dependence of their star formation histories. We quantify the strengths of the 4000$\rm\AA$ break ($D_{4000}$) and the H$\delta$ absorption features and compare them with model predictions. The “cluster” red galaxies do not show any sign of on-going or recent star formation activities and the passive evolution can naturally link them to the present-day red sequence galaxies in the [*Sloan Digital Sky Survey*]{}. In contrast, the red galaxies in “groups” and in the “field” tend to show signs of remaining and/or recent star formation activities characterized by weak \[OII\] emissions and/or strong H$\delta$ absorptions. Our current data seem to favour a scenario that star formation is truncated in a short time scale ($<$1Gyr). This would imply that galaxy-galaxy interactions are responsible for the truncation of star formation.'
author:
- |
\
$^{1}$Department of Astronomy, School of Science, University of Tokyo, Tokyo 113–0033, Japan\
$^{2}$National Astronomical Observatory of Japan, Mitaka, Tokyo 181–8588, Japan\
$^{3}$European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748, Garching, Germany\
$^{4}$Astronomical Institute, Tohoku University, Aoba-ku, Sendai 980–8578, Japan
title: 'Spectroscopically Confirmed Large-Scale Structures Associated to a $z=0.83$ Cluster'
---
galaxies: evolution — galaxies: clusters: individual RXJ0152.7$-$1357
Introduction {#sec:intro}
============
Distribution of galaxies in the local Universe is highly non-uniform as known from various redshift surveys (e.g. CfA redshift survey; @delapparent86, 2dF; @colless01, and SDSS; @york00). Based on these extensive spectroscopic surveys, a map of the local Universe is drawn with an unprecedented accuracy. Galaxies tend to be clustered by gravity and form large filamentary and clumpy structures with voids in between the filaments. At intersections of filaments, gravitationally bound concentrations of galaxies, which we call clusters of galaxies, are often seen. These filaments and galaxy concentrations are cast all over the Universe forming network structures.
It is of great interest to know how such large scale structures of galaxies have been built up in the history of the Universe. The large-scale structures have been extensively explored even at high redshifts based on various photometric/spectroscopic surveys (e.g. @kodama01 [@kodama05; @shimasaku03; @gal04]). These surveys suggest that filamentary structures of galaxies seem to have appeared since an early epoch of the Universe. However, it is not well known how these structures developed with time and became matured to the present-day structures.
The cosmic large-scale structures inversely mean that galaxies live in various environments – some galaxies live in low density filaments and some live in dense clusters. It is now a widely accepted fact that galaxy properties such as star formation rates depend on environment that surrounds galaxies (see @tanaka05 and references therein). However, the origin of such strong environmental dependence of galaxy properties is not well understood. It must be closely related to the process of structure formation, but physical mechanisms that actually drive the observed environmental dependence remain unidentified. To improve the situation, we are conducting a panoramic imaging and spectroscopic surveys of distant clusters (PISCES; @kodama05) from their cores and to the surrounding filamentary structures.
During the course of this wide area survey, we have identified for the first time the environment where star formation activity drops sharply. That is “groups of galaxies” located in the filaments extending out from the cluster cores. @kodama01 and @tanaka05 have shown that the colour distribution of galaxies changes at relatively low projected density of galaxies that corresponds to groups of galaxies in the filaments. However, the actual physical process behind the truncation of star formation is yet unidentified. In order to further investigate what is happening in groups, spectroscopic information is essential such as \[OII\] emission and Balmer absorption lines. The spectroscopic information resolves the on-going star formation activity and recent star formation history of the galaxies, which cannot be resolved by colours (e.g. @couch87 [@dressler92; @balogh99; @poggianti99]).
In this paper, we focus on one of the clusters from our PISCES sample, RXJ0152.7–1357 at $z=0.83$, for which we already have a complete optical data-set with Suprime-Cam [@miyazaki02]. The cluster RXJ0152.7–1357 (RXJ0153 for short) is one of the most X-ray luminous distant ($z>0.7$) clusters known. It was discovered by various X-ray surveys [@rosati98; @scharf97; @ebeling00; @romer00]. Since these discoveries, the cluster has been investigated in various ways: X-ray observations [@dellaceca00; @maughan03], the Sunyaev-Zeldovich effect [@joy01], near-IR imaging [@ellis04], and the weak lensing analysis [@huo04; @jee05]. @demarco05, @homeier05, and @jorgensen05 presented spectroscopic studies on this cluster. Recently, @girardi05 discussed dynamical properties of this cluster in detail.
We photometrically discovered large-scale structures around RXJ0153 based on the Subaru data as reported in @kodama05. However, the photometric identification of the structures is subject to projection effects. The discovered structures might be chance projections of galaxies along the line of sight. To confirm the structures against the projection effects, a spectroscopic follow-up observation was carried out. In this paper, we first present spectroscopic confirmation of the structures at $z\sim0.8$. We then discuss detailed star formation histories of galaxies from the spectra and constrain physical processes that trigger the truncation of star formation.
The outline of this paper is as follows. In section \[sec:obs\], we briefly review our photometric identification of large-scale structures at $z\sim0.8$ around RXJ0153. Our spectroscopic observations of RXJ0153 and data reduction are described as well. Then we present results of the spectroscopic observation in section \[sec:lss\], and we address star formation histories of galaxies in section \[sec:sfh\]. Implications of our results are discussed in section \[sec:discussion\]. Finally, the paper is summarized in section \[sec:conclusion\].
Throughout this paper, we assume a flat Universe of $\Omega_{\rm M}=0.3,\ \Omega_{\rm \Lambda}=0.7$ and $H_0=70\kms \rm Mpc^{-1}$. Magnitudes are on the AB system.
Observation and Data Reduction {#sec:obs}
==============================
Photometric Identification of Large-Scale Structures at $z\sim0.8$
------------------------------------------------------------------
We briefly review the photometric identification of large-scale structures at $z\sim0.8$. Readers are refereed to @kodama05 for details.
We observed RXJ0153 with Suprime-Cam in $VRi'z'$ under excellent conditions. The seeing was $\sim0''.6$ in all bands. Since the cluster lie at a high redshift and our imaging is deep (we can reach $M^*_V+4$ galaxies at the cluster redshift), galaxies at the cluster redshift are heavily contaminated by fore-/background galaxies. To reduce the contamination, we applied the photometric redshift code of @kodama99 and extracted galaxies around the cluster redshift. The distribution of photo-$z$ selected galaxies revealed a highly disturbed shape of RXJ0153 – the cluster is elongated along the NE–SW direction and it accompanies clumpy subgroups. In particular, the wide field of view (abbreviated as FoV in what follows) of the Suprime-Cam has enabled us to discover a number of groups/clusters surrounding the central cluster.
However, it is unclear if these groups and clusters are physically associated to the central cluster due to the wide redshift slice adopted ($0.78<z_{phot}<0.86$). This wide range was chosen so that we do not miss galaxies at the cluster redshifts due to the error in the photometric redshifts. The discovered systems could lie at different redshifts within the above redshift range. Moreover, the projection effect is a concern. Some of these systems could be chance projections of galaxies along the line of sight. Therefore, more accurate redshift measurements based on spectroscopic observations are essential to identify real physical associations to the main body of the cluster.
Spectroscopic Observation {#sec:spec_obs}
-------------------------
We conducted a spectroscopic follow-up observation during 11–14 October 2004 with FOCAS [@kashikawa02] in MOS mode. We used a 300 lines $\rm mm^{-1}$ grating blazed at 5500 $\rm \AA$ with the order-cut filter SY47. The wavelength coverage was between 4700$\rm\AA$ and 9400 $\rm\AA$ with a pixel resolution of $1.4\rm \AA\ pixel^{-1}$. A slit width was set to $0\farcs 8$, which gave a resolution of $\lambda/\Delta\lambda\sim500$.
We selected 8 FOCAS fields which efficiently cover the large-scale structure traced by photometry as shown in Fig. \[fig:target\_fields\]. Target galaxies were primarily selected on the basis of photometric redshifts. Galaxies at $0.76\leq z_{phot}\leq 0.88$ were given the highest priority in the mask design. This broad and asymmetric redshift range was adopted so that we do not miss the true cluster members whose photometric redshifts are slightly wrong. Roughly 60 percent of the targeted galaxies fall in this photo-$z$ range. Each mask had about 40 slits. Target galaxies are located towards one side of each FoV of FOCAS. This is to optimize the wavelength coverage of a spectrum so that we can observe the most important spectral features of galaxies at the cluster redshift (e.g. \[OII\] and CaII H+K).
Observing conditions were variable : moderate to poor seeing and non-photometric conditions for some nights. Total on-source exposures are listed in Table \[tab:obs\_summary\]. Data reduction is performed in a standard manner using [IRAF]{}. We correct for the A-band telluric extinction by creating an absorption profile from a composite spectra of high-$S/N$ galaxies. We do not correct for the Galactic extinction. All the reduced 1d/2d spectra are visually inspected using purpose written software. Rough redshift estimates and redshift confidence flags ($z_{conf}$) are then assigned to each object. Objects with secure/probable redshift estimates are flagged as $z_{conf}=0$, and objects with likely redshifts are flagged as $z_{conf}=1$. The flag $z_{conf}=1$ is basically assigned to low-$S/N$ galaxies. Important spectroscopic features (e.g. CaII H+K) are sometimes heavily contaminated by the night sky emissions and in this case we assigned $z_{conf}=1$. We present in Fig. \[fig:spec\_example\] examples of our spectra.
Based on the rough redshift estimates, we fit a Gaussian profile to emission/absorption features and redshifts are accurately measured. The Gaussian fitting is repeated 100,000 times by resampling pixels around the features and randomly adding noise, which is estimated directly from each spectrum. A 68 percentile interval of the redshift distribution is quoted as an error. We note that this error should be considered as a rough estimate. Note as well that it does not include an error in the wavelength calibration, which is typically $\sim0.3\rm\AA$. We obtain 161 secure redshifts ($z_{conf}=0$) out of 310 observed galaxies. We present our spectroscopic catalogue in Table \[tab:spec\_data\]. Astrometry is performed using the USNO2-B catalogue [@monet03]. An accuracy of our coordinates is roughly $\sim0''.2$.
We cross-match our spectroscopic objects with those from @demarco05. Six objects are matched. Redshifts of 5 objects out of 6 are matched within errors, and the median of $z_{spec,demarco}-z_{spec,tanaka}$ is $-0.00004$ and the dispersion around it is 0.00068. No systematic difference is seen. However, one object has a deviant redshift: $z_{spec,demarco}=0.8230$, while $z_{spec,tanaka}=0.7612$. This object (Field ID: F1, Slit ID:9) is flagged $z_{conf}=1$ in our catalogue, and hence our measurement may be wrong (photo-$z$ is consistent with the Demarco et al.’s measurement). In what follows, we do not use $z_{conf}=1$ galaxies.
We combine our spectroscopic catalogue with those from @demarco05 and @jorgensen05. This makes a large spectroscopic sample of 347 galaxies around the cluster. Based on this large sample, we present large-scale structures surrounding the cluster in the following section.
1.0
1.0
Field ID Exposures
---------- --------------------------------------------------
F1 1800s $\times$ 4 shots
F2 1800s $\times$ 4 shots $+$ 1345s $\times$ 1 shot
F3 1800s $\times$ 1 shot $+$ 1540s $\times$ 1 shot
F4 1800s $\times$ 4 shots
F5 1800s $\times$ 3 shots
F6 1800s $\times$ 4 shots
F7 1800s $\times$ 3 shots
F8 1800s $\times$ 2 shots
: Log of the spectroscopic observation. []{data-label="tab:obs_summary"}
ID R.A. Dec. $m_{z', tot}$ $V-R$ $R-i'$ $i'-z'$ $z_{phot}$ $z_{spec}$ $z_{spec, min}$ $z_{spec, max}$ $z_{conf}$
----------- ------------ -------------- --------------- ------- -------- --------- ------------ ------------ ----------------- ----------------- ------------ -- --
F1-3 1 52 51.66 -13 57 12.35 24.02 0.30 0.20 0.22 1.52 0.3150 0.3150 0.3151 0
F1-4a 1 52 51.14 -13 57 29.78 22.75 1.22 -2.06 0.22 0.00 0.5165 0.5165 0.5166 0
F1-4b 1 52 50.96 -13 57 30.10 21.78 1.03 1.01 0.72 0.88 0.8455 0.8448 0.8462 0
F1-5 1 52 50.51 -13 57 12.09 21.87 1.15 1.00 0.66 0.85 0.8350 0.8344 0.8354 0
F1-6 1 52 50.06 -13 57 34.89 22.64 1.17 1.13 0.63 0.80 0.8406 0.8404 0.8408 0
F1-7 1 52 49.29 -13 57 27.79 21.47 1.18 1.03 0.84 0.93 0.9588 0.9577 0.9601 0
F1-9 1 52 47.98 -13 55 31.11 22.17 0.73 0.89 0.49 0.83 0.7612 0.7606 0.7616 1
F1-10 1 52 47.45 -13 56 25.81 22.85 0.26 0.47 0.37 1.11 1.0540 1.0540 1.0541 0
F1-11 1 52 46.86 -13 56 42.64 23.49 0.26 0.59 0.39 1.02 1.0552 1.0551 1.0553 0
F1-14 1 52 45.35 -13 57 7.94 23.69 0.86 -0.06 -0.09 0.36 3.9280 3.9279 3.9280 0
F1-$15^*$ 1 52 45.18 -13 57 4.26 23.59 0.96 0.09 0.19 – 3.9276 3.9275 3.9277 0
F1-16 1 52 44.45 -13 56 56.02 23.95 0.90 -0.08 -0.05 0.38 3.9277 3.9277 3.9278 0
F1-21 1 52 41.13 -13 57 43.03 21.31 1.29 1.20 0.79 0.87 0.8323 0.8318 0.8328 0
The format of ID is “Field ID – Slit ID”. Total magnitudes ($m_{z',tot}$) are measured in Kron-type apertures ([MAG\_AUTO]{}), while colours are measured within $2''$ apertures. Objects marked $^*$ are missing photometric redshifts due to strong blending with nearby objects.
Large-Scale Structures {#sec:lss}
======================
We first present distribution of spectroscopically observed galaxies. We then point to the question how accurate our photometric redshifts are. In particular, we assess the colour dependence of the accuracy of photometric redshifts.
Large-Scale Structures at $z\sim0.8$ {#sec:lss_z08}
------------------------------------
Fig. \[fig:close\_up\_views\] shows our results in each field. Note that only our spectroscopic objects are plotted in Fig. \[fig:close\_up\_views\], and the data from @demarco05 and @jorgensen05 are not used. It is impressive that the redshift distribution of spectroscopically observed galaxies show a sharp redshift spike in all the fields. Although galaxies are selected primarily by $z_{phot}$ in the mask design, the photo-$z$ selection range ($0.76\leq z_{phot}\leq0.88$) is much wider than the observed redshift spikes. Therefore, the redshift spikes are not a product of selection effects but are real structures in each field. Furthermore, the peaks are located at or near the cluster redshift of $z=0.837$ [@demarco05], except for F7 and F8 (these turn out to be foreground systems). This is strong evidence for large-scale structures at $z\sim0.83$. Table \[tab:vel\_disp\] presents the redshift centres of the redshift spikes. Fig. \[fig:target\_fields\] shows the distribution of galaxies at $0.82<z_{spec}<0.85$. The spectroscopic samples from @demarco05 and @jorgensen05 are included in this plot. As shown by the triangles, @demarco05 and @jorgensen05 focused only on the central part of the cluster, while we mainly explore the surrounding regions avoiding duplication of the spectroscopic targets. We note that 154 galaxies out of the 347 galaxies with secure redshifts lie at $0.82<z_{spec}<0.85$.
The redshift peaks in F1, F3, and F4, are located very close to the redshift of the main cluster ($z=0.837$), and forming a large filament over 12 Mpc (comoving) in the N–S direction. Interestingly, there seems to be another large scale filament at $z\sim0.844$ in the direction of NE–SSW traced by F5, F2, and F6. This second filament includes a subclump at $z=0.845$ located at $\sim3$ arcmin eastward of the main cluster discovered by the previous studies [@demarco05; @jorgensen05; @girardi05]. It turns out that this second filament also extends to more than 20 Mpc (comoving). Therefore, it seems that there are two large sheets/filaments of galaxies crossing in this field. The relative velocity in the line of sight of these two sheets is 1000 km s$^{-1}$, and it is likely that they are physically connected. It is interesting to recall that the main cluster itself is in the process of mergers [@maughan03; @demarco05; @jee05]. This cluster is therefore currently experiencing the vigorous assembly of the surrounding systems along the filaments and will experience more mergers in the near future. Unfortunately, however, the large errors in the velocity dispersion estimates (i.e. mass estimates) with the current data set do not allow us to perform detailed dynamical analysis.
Based on $z_{phot}$, @tanaka05 identified galaxy groups in F4, F5, F6, and F7, and a galaxy cluster in F8. We examine these fields in detail. We measure velocity dispersions of the redshift spikes using the gapper method [@beers90]. The results are shown in Table \[tab:vel\_disp\] and the bottom-right corner of each panel in Fig. \[fig:close\_up\_views\]. We confirm the biweight estimator gives consistent estimates [@beers90]. Errors are estimated from the jackknife resampling of the spectroscopic members. From these velocity dispersions, we evaluate virial radii [@carlberg97] as plotted in the figure. Here we adopt $r_{200}$ as a virial radius ($r_{200}$ is the radius within which the mean interior density is 200 times the critical density of the Universe). The spatial centre of groups/clusters are determined from the distribution of spectroscopic galaxies weighted by an inverse of redshift difference from the central redshift of the groups/clusters.
Galaxies (including photo-$z$ selected galaxies) in F4, F6, and F7 show clear spatial clustering (see Fig. \[fig:target\_fields\] as well) and most of the galaxies are distributed within the virial radius of each group. This suggests that these are gravitationally bound systems. On the other hand, the spatial clustering of galaxies in F5 is less strong. This may not be a bound system. In F8, galaxies are clearly clustered. However, the estimated virial radius is too small for its apparent extent. This is probably because the spectroscopic sampling of galaxies in this field is too sparse for us to make an accurate measure of their velocity dispersion. In fact, a number of photometric members in this system is the largest among others, excluding the main cluster at $z=0.83$. We regard systems in F4, F6, and F7 as groups and the one in F8 as a cluster. We do not include F5 in groups since it is not clear if this system is gravitationally bound or not. The fact that F5 is dominated by blue galaxies may suggest that this is not a bound system. But, we have confirmed that the inclusion of F5 does not alter our conclusions. We recall that this classification of group and cluster environments is based on 2-dimensional galaxy density on both local and global scales [@tanaka05]. We cannot define groups and clusters based on the spectroscopic data in hand (e.g. velocity dispersions) due to the poor statistics. Star formation histories of galaxies in these regions are examined in section \[sec:sfh\].
We note that we serendipitously discover three Ly$\alpha$ emitters at $z=3.928$ in F1. These galaxies show very similar broadband colours and lie at the same redshift, and it turns out that these are originally a single galaxy strongly lensed by the main cluster at $z=0.83$. Details of the galaxies and the lensing models are reported in @umetsu05.
0.44 0.44\
0.44 0.44\
0.44 0.44\
0.44 0.44\
Field ID $N_{member}$ $z$ $\sigma\rm\ [km\ s^{-1}]$ environment
---------- -------------- ------------------- --------------------------- --------------------
F1 10 $0.8368\pm0.0027$ $756\pm145$ main cluster
F2 8 $0.8416\pm0.0013$ $277\pm30 $ filament
F3 6 $0.8385\pm0.0047$ $490\pm449$ filament
F4 8 $0.8348\pm0.0009$ $413\pm241$ group
F5 14 $0.8439\pm0.0017$ $362\pm159$ group or filament
F6 12 $0.8443\pm0.0046$ $457\pm46 $ group
F7 8 $0.7823\pm0.0012$ $313\pm113$ foreground group
F8 7 $0.7453\pm0.0006$ $210\pm98 $ foreground cluster
Accuracy of Photometric Redshifts {#sec:photoz}
---------------------------------
With the 347 spectroscopic redshifts now available in this field, we assess the accuracy of our photometric redshifts. The errors in photometric redshift ($\Delta z =z_{phot}-z_{spec}$) is plotted against $z_{spec}$ in Fig. \[fig:photoz\_accuracy\]. Although our photometric redshifts are relatively accurate especially around the cluster redshift, some clear deviations can be seen. In particular, some galaxies at $0.7\lesssim z_{spec}\lesssim0.8$ significantly deviate ($\Delta z\sim-0.6$). Also, galaxies at $z\sim0.6$ show a lower redshift tail in their $z_{phot}$ distribution. At $z_{spec}\lesssim0.6$, there is a systematic offset between $z_{spec}$ and $z_{phot}$.
We find that the deviations are related to intrinsic SEDs of galaxies. Fig. \[fig:photoz\_colour\_dep\] shows $\Delta z$ as a function of galaxy colours. The figure demonstrates that $z_{phot}$ is fairly accurate for red galaxies at all redshifts under study, while it is less accurate for blue galaxies. At $0.8<z<1.1$, only the bluest galaxies strongly deviate, but at $0.6<z<0.7$, blue-intermediate colour galaxies also deviate. In Fig. \[fig:photoz\_accuracy\], we find that the galaxies at $z_{spec}\lesssim0.6$ have systematically lower $z_{phot}$. It turns out that most of these galaxies are blue galaxies as shown in Fig. \[fig:photoz\_colour\_dep\]. A few very red galaxies show deviant redshifts (e.g. three red galaxies at $0.6<z_{spec}<0.7$). We find that these galaxies show slightly redder colours than our reddest model colours. These are likely to be heavily dust obscured galaxies, though such population is rare.
The deviation of the bluest galaxies can be explained by the fact that we lack model templates for such very blue galaxies [@kodama99]. The bluest galaxies cannot be fit by any of the templates and thus the photo-$z$ outputs very deviant redshifts. As for the deviation of the galaxies with blue-to-intermediate colours at $z\lesssim0.7$, we speculate that our filter set ($VRi'z'$) is no longer optimal. We do not cover a wide enough wavelength range (particularly in UV) to reliably estimate redshifts of blue galaxies that do not show prominent SED features [@kodama99]. Adding $B$-band will improve the situation.
From the results presented in our previous paper (see Fig. 1 of @tanaka05), we have expected that only the galaxies with intermediate colours would show deviant photometric redshifts at the cluster redshift. We do not, however, observe this trend in our large spectroscopic sample. At the cluster redshift, even the intermediate colour galaxies are given reasonably accurate photometric redshifts, and only the bluest galaxies tend to deviate. It may be the case that the deviation of intermediate colour galaxies could be characteristic of $z\sim0.5$ galaxies (for which we saw the trend in @tanaka05). Given the small number of spectroscopic galaxies at that redshift in our current sample, we cannot pursue this issue further at this point.
The colour dependence tells us that we have to be careful when discussing a fraction of red/blue galaxies (e.g. Butcher-Oemler effect) based on photo-$z$. It is expected to be a general trend that the photometric redshifts are less accurate for blue galaxies than for red galaxies due to the less strong SED features. This colour dependent photo-$z$ accuracy would tend to underestimate a fraction of blue galaxies, and could weaken evolutionary trends. We note, however, that this colour dependence does not change the primary conclusions presented in @tanaka05. They reported the build-up of the colour-magnitude relation from $z=0.83$ to $z=0$ in field, group, and cluster environments. Their results are not strongly affected since they focused on red galaxies whose photometric redshifts are accurate. Further discussions can be found in that paper.
Finally, we revisit the distribution of photo-$z$ selected galaxies with more optimal photo-$z$ and colour cuts. Following the true error distribution shown in Fig. \[fig:photoz\_colour\_dep\], we select galaxies with $z_{cl}-0.03<z_{phot}<z_{cl}+0.05$ ($z_{cl}=0.837$; @demarco05) and $R-z'>1$ and plot their distribution in Fig. \[fig:photoz\_distrib\]. The filamentary structure to the NE direction is more pronounced, and it emerge from the SE of the main cluster and continuously extends towards NE out to 10 Mpc. Galaxy groups at South and SW of the main cluster are also confirmed. They are likely to infall onto the cluster in the near future. We stress that this is one of the most spectacular and reliable large-scale structures of the $z\sim1$ Universe reaching down to very faint magnitude ($M_V^*$+4) traced by spectroscopically calibrated photometric redshifts.
1.0
0.45 0.45
1.0
Star Formation Histories {#sec:sfh}
========================
The concentrations of galaxies, clusters and groups, are spectroscopically identified in section \[sec:lss\_z08\]. In the following, we infer star formation histories of galaxies from their spectroscopic signatures as a function of environment. Here we use our spectra only. @demarco05 and @jorgensen05 focused only on the main cluster and data for other environments (groups in particular) are not available.
Composite Spectra {#sec:composite_spectra}
-----------------
A typical $S/N$ of our spectra is not high enough to quantify the spectroscopic properties of galaxies on an individual basis. We thus composite a number of spectra to make a high $S/N$ spectrum as was done by, for example, @dressler04. Here we restrict ourselves to the red galaxies (those having $|\Delta (R-z')|<0.2$ with respect to the colour-magnitude relation). We do not examine blue galaxies because photometric redshifts are not accurate for blue galaxies, and we cannot avoid a selection bias (e.g. we miss very blue galaxies). Each spectrum is normalized to unity at $\rm4000-4200\AA$ in the restframe, where a spectrum is relatively flat. A composite spectrum is then made by taking 2$\sigma$ clipped mean. We combine the red galaxies in F1 and F8 to make a representative spectrum of “cluster” galaxies. A composite “group” spectrum is made from the red galaxies in F4, F6, and F7. We also make a composite “field” spectrum from the isolated red galaxies outside of the redshift spikes in each field but within $0.74<z_{spec}<0.88$. The cluster/group/field spectra are constructed from 11/16/8 galaxies, respectively.
Before comparing the composite spectra, we discuss the similarity in the photometric properties of the red galaxies among different environments that are used to construct the cluster/group/field composite spectra. After correcting for the small k-correction and passive evolution effects for galaxies at slightly different redshift [@kodama97], we apply the 2-dimensional K-S test [@fasano87] for the distribution of galaxies on the colour-magnitude diagram. It does not reject the hypothesis that the colours and magnitudes of the “cluster” red galaxies and those of the “group” red galaxies are drawn from the same parent population ($80\%$). Therefore, the red galaxies in clusters and those in groups share the common photometric properties and the only obvious difference is their environment. However, the K-S probability is found to be smaller between “cluster” and “field” ($20\%$) and “group” and “field” ($20\%$). Hence, field red galaxies may have slightly different photometric properties.
Fig. \[fig:spec\_comb\] compares the composite spectra. The continua of the cluster and the group spectra are very similar and are typical of red passively evolving galaxies, consistent with the common photometric properties as discussed above. However, the group spectrum shows a small amount of residual star formation as characterized by a weak \[OII\] emission. The strength of the emission is estimated to be EW\[OII$]=4\rm\AA$. Furthermore, a stronger H$\delta$ absorption is seen in the group spectrum than in the cluster spectrum. The field spectrum shows an even stronger \[OII\] emission (EW\[OII$]=13\rm\AA$). Although the continuum of the field spectrum is similar to those of cluster and group galaxies, differences can be seen, e.g. at $\lambda_{rest}\sim3850\rm\AA$, which leads to the difference in the strength of 4000Å break (section \[sec:spec\_diag\]).
Spectral Diagnostics {#sec:spec_diag}
--------------------
### Observational Data
We now quantify the differences in the composite spectra by measuring the strengths of 4000$\rm\AA$ break ($D_{4000}$) and H$\delta$ absorption. The definition of $D_{4000}$ is taken from @bruzual83. H$\delta$ absorption lines echo star formation activities that ended $0.1-1$ Gyr prior to the observed epoch. We adopt the H$\delta_F$ index defined by @worthey97. This index uses a narrower window than that of H$\delta_A$. Since our spectra are not of very high $S/N$, a narrower window that excludes the tails of absorption line features is better suited. We take the median fluxes in the continuum windows to determine a pseudo-continuum level. Appendix \[sec:concerns\] discusses possible sources of uncertainties in our measurements in detail.
We measure $D_{4000}$ and H$\delta_F$ from the composite spectra of the cluster/group/field red galaxies at $z\sim0.8$. In deriving H$\delta_F$, we correct for an H$\delta$ emission from gaseous nebulae. We evaluate a typical amount of the H$\delta$ emission filling from the galaxies at $z=0$ (SDSS) that have identical $D_{4000}$ and EW\[OII\] to those of the $z\sim0.8$ sample. We then correct the H$\delta_F$ index for the emission filling. Note that transforming an \[OII\] flux into an H$\alpha$ flux, then into an H$\delta$ flux gives a consistent amount of the emission filling according to the SDSS data.
A measurement error is estimated as follows. We perform the bootstrap resampling of the input spectra, and a composite spectra is generated in each resampling. Then $D_{4000}$ and H$\delta_F$ are calculated. This procedure is repeated 5,000 times, and a 68 percentile interval of the distribution is quoted as an error. Note that it is difficult to quantify an error in the correction for the H$\delta$ emission filling, and its error is not included in the error quoted below.
We utilize the SDSS data for a local counterpart of our $z\sim0.8$ data. A spectroscopic sample is drawn from the public DR2 [@abazajian04]. For details of the SDSS data, refer to @stoughton02 and @strauss02. For a fair comparison with galaxies $z\sim0.8$, we select the SDSS galaxies with $M_V\lesssim M_V^*+1$ from the @tanaka05 catalogue (Main Galaxies at $0.005<z<0.065$). Note that this magnitude cut corresponds to $m_{z'}\lesssim22$ at $z\sim0.8$ where our spectroscopic galaxies at $z\sim0.8$ fall. Note as well that we include both blue galaxies and red galaxies to show that our models, which we describe later, well reproduce various star formation histories of galaxies at $z=0$. The spectra are smoothed to our instrumental resolution, and then $D_{4000}$ and H$\delta_F$ are measured. We correct for an H$\delta$ emission from a strength of an H$\alpha$ emission assuming the case-B recombination (H$\alpha=11$H$\delta$ ; @osterbrock88).
Measurement errors are estimated from a Monte-Carlo simulation. A noise is randomly added at each wavelength based on the error estimated by the SDSS pipeline, and we re-measure $D_{4000}$ and H$\delta_F$. This procedure is repeated 10,000 times and a 68 percentile interval of the distribution is quoted as an error. The SDSS spectroscopic survey is performed with 3 arcseconds diameter fiberes, which are much smaller than a typical size of galaxies in our $z=0$ sample [@tanaka04]. We argue in Appendix \[sec:aperture\_bias\] that our conclusions are robust to this aperture bias.
A fundamental question here is the evolutional and environmental connection between galaxies at $z\sim0.8$ and those at $z=0$. Do the field red galaxies at $z\sim0.8$ evolve to group/cluster galaxies at $z=0$? Or are they still in the field at $z=0$? Fortunately, we find no convincing evidence that the $D_{4000}$ and H$\delta_F$ indices of the $z=0$ galaxies depends on environment – red field/group/cluster galaxies all have similar $D_{4000}$ and H$\delta_F$ distribution. That is, the red galaxies at $z\sim0.8$ should evolve to, as shown below, $D_{4000}\sim2.3$ and H$\delta_F\sim0$ at $z=0$ regardless of environment. We therefore do not separate the $z=0$ galaxies into various environments.
We present in Fig. \[fig:hd\_d4000\] the distribution of $D_{4000}$ and H$\delta_F$ indices of galaxies at $z\sim0.8$ (large symbols) and $z=0$ (contours). For $z\sim0.8$ galaxies, the $D_{4000}$ indices of cluster and group galaxies agree within the errors. This is consistent with the fact that the colours of group and cluster red galaxies are indistinguishable (see section \[sec:composite\_spectra\]). Interestingly, there is a hint that the H$\delta_F$ absorption is stronger for group red galaxies than for the cluster one. This may suggest that the group galaxies have had recent star formation at $0.1-1$ Gyr prior to the observed epoch. The field red galaxies show a smaller $D_{4000}$, and a stronger H$\delta_F$ absorption, suggesting their younger ages compared to the red galaxies in denser regions.
Galaxies at $z=0$ show a well-defined sequence on the H$\delta_F$-$D_{4000}$ diagram. Blue galaxies (small $D_{4000}$) tend to show a strong H$\delta$ absorption, indicating their active star formation over the past $\sim1$ Gyr. Red galaxies (large $D_{4000}$) are strongly peaked at $D_{4000}\sim2.3$ and H$\delta_F\sim0$. Their weak H$\delta$ absorption suggests that they have not actively formed stars for a long time ($\gtrsim1$ Gyr).
We compare the $D_{4000}$ and H$\delta_F$ indices of galaxies at $z\sim0.8$ with those at $z=0$. Red galaxies at $z\sim0.8$ and those at $z=0$ show quite different distribution in their $D_{4000}$ and H$\delta_F$. Galaxies at $z\sim0.8$ show a smaller $D_{4000}$ and a stronger H$\delta_F$. This is qualitatively consistent with the passive evolution model, which predicts bluer colours (i.e. smaller $D_{4000}$) and a stronger H$\delta$ absorption at higher redshifts. However, galaxies at $z\sim0.8$ tend to show a strong H$\delta_F$ for their $D_{4000}$ compared to local galaxies (group and field galaxies, in particular). This may suggest that a mode of star formation of group and field red galaxies at $z\sim0.8$ is different from that of normal star forming galaxies at $z=0$. We discuss these trends quantitatively in the next subsection.
### Model Predictions
We compare the observed $D_{4000}$ and H$\delta_F$ with those from the population synthesis model by @bruzual03. As a fiducial parameter set, we adopt the following: Padova 1994 evolutionary tracks, Chabrier initial mass function between $0.1-100\rm M_\odot$, solar metallicity, and no dust extinction (see @bruzual03 and references therein). Here we consider three star formation histories to represent the star formation histories of red galaxies. One is an instantaneous starburst model, and another is exponentially decaying model with a decay time scale of $\tau=1$ and 2 Gyr. The other is a constant star formation model with a starburst at $t=4$ Gyr since the onset of star formation, and the star formation is truncated immediately after the burst. We consider the three starburst strengths; stars that are newly born in the burst amount to 0%, 10%, and 100% of the mass of the existing stars. The case for 0% burst means that star formation is truncated without a burst. We refer to the above three star formation histories as “SSP”, “tau”, and “burst” models, respectively. Model spectra are smoothed to our instrumental resolution and spectral features are measured. Note that our aim here is not to explore a comprehensive range of star formation histories with many parameter sets such as various extinctions and metallicities. However, our simple models should represent the typical paths of evolution in $D_{4000}$ and H$\delta_F$.
We now put the arguments in the previous subsection on a more quantitative ground with a help of the model predictions. As shown in Fig. \[fig:hd\_d4000\], the various models reasonably cover the distribution of the observed $z=0$ galaxies. At $z\sim0.8$, the cluster red galaxies are on the passive evolution model within the error, and they would naturally evolve into the red galaxies at $z=0$. However, the group and the field red galaxies cannot be reproduced by the simple evolution models such as “SSP” or “tau” models. Their H$\delta$ absorptions are too strong for their $D_{4000}$. We find that SDSS galaxies with EW\[OII\]$=4\rm\AA$ and EW\[OII\]$=13\rm\AA$ typically have $-0.8<{\rm H}\delta_F<0.8$ and $-0.6<{\rm H}\delta_F<2.6$, respectively (group/field composite spectra show EW\[OII\]$=4\rm\AA/13\AA$, respectively). Therefore, the H$\delta_F$ indices of group and field galaxies are clearly larger compared to normal star forming galaxies. Field red galaxies could be fit by a burst+sharp truncation model. However, they still show a sign of star formation and their star formation is not completely quenched yet. It is interesting to note that 0%, 30%, and 80% of individual cluster, group, and field red galaxies show \[OII\] emissions.
The group red galaxies cannot be reproduced by any of our model. In fact, few local galaxies populate around the group galaxies on the H$\delta_F$–$D_{4000}$ plane. One may suspect that the correction for the emission filling is not reliable. However, the correction applied for the red group galaxies is small ($\Delta$H$\delta_F=0.6$), and whether or not we apply the correction does not significantly alter the result. It could be due to the dust extinction which makes their colours red [@wolf05], although we do not attempt to perform detailed modeling of dust extinction in this paper. As the “tau” model demonstrates, a gradual truncation of star formation does not enhance the H$\delta_F$ absorption and fail to reproduce the group red galaxies. The observed strong H$\delta$ absorption favours a scenario that the star formation is suppressed in a short time scale rather than a slow decline. We further discuss the physical driver of the truncation of star formation in groups in section \[sec:sfr\_truncation\].
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Discussion {#sec:discussion}
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Comparisons with Previous Studies {#sec:comparison}
---------------------------------
@demarco05 reported that the colours of RXJ0153 cluster red galaxies are consistent with the passive evolution model with $z_f\sim1.9$. We find that our result is consistent with theirs (see the “SSP” plot in Fig. \[fig:hd\_d4000\]). They also reported that red \[OII\] emitting galaxies are preferentially located in the outskirts of the cluster. In our analyses, we find that many red \[OII\] emitting galaxies in groups and field, and none of cluster red galaxies show \[OII\] emissions. Therefore, red \[OII\] emitters tend to avoid the dense cluster core in consistent with Demarco et al.’s result.
@jorgensen05 performed a detailed analysis of absorption lines of galaxies in the RXJ0153 cluster. Their results are somewhat different from ours – they suggested that the $D_{4000}$ does not evolve from $z=0.83$ to $z=0$, although they also reported that the evolution of $B$-band luminosity and high order Balmer lines (H$\delta$ and H$\gamma$) are consistent with the passive evolution since $z_f\sim4$. We find that this difference in the evolution of $D_{4000}$ comes from a difference in $z=0$ data. There is no significant difference between our $D_{4000}$ measurements and their $D_{4000}$ measurements for the $z=0.83$ sample, $D_{4000,z=0.83}\sim2.1$. However, for red galaxies at $z=0$, we obtain $D_{4000,z=0}\sim2.3$ from the SDSS data, while they obtained $D_{4000,z=0}\sim2.1-2.2$ from their own Perseus and Abell 194 data. We note that the value $D_{4000,z=0}\sim2.1-2.2$ seems to be inconsistent with model predictions for evolved red galaxies at $z=0$. However, we cannot conclusively figure out the cause of the difference in $D_{4000}$ in the $z=0$ samples at this stage. We note that @jorgensen05 also reported that some scaling relations for metal indices, which we do not explore in this paper, are also inconsistent with the passive evolution.
Epoch of the Truncation of Star Formation Activities {#sec:sfr_truncation}
----------------------------------------------------
@tanaka05 suggested that the truncation of star formation starts from massive galaxies and fainter galaxies stop their star formation later, and galaxies follow ’down-sizing’ evolution [@cowie96; @kodama04]. They showed that, at $z=0.83$, a clear colour-magnitude relation is seen down to $M_V^*+4$ in cluster environments. On the other hand, in group environments, only the bright-end ($M_V<M_V^*+2$) of the relation is seen. Faint red galaxies do not form a tight relation. In field environments, no clear relation is seen even at the bright-end. They therefore suggested that the down-sizing is likely to depend on environments – down-sizing is delayed in low-density environments.
Our results in the previous section support the picture that the truncation of star formation is delayed in lower-density environments. While the colour-magnitude distributions of the cluster and the group red galaxies are identical, the group red galaxies show a weak emission with a strong H$\delta$ absorption, which are not seen in the cluster spectrum. This means that the group red galaxies must have had recent star formation activities, while the cluster red galaxies have not formed stars for a long time. The truncation of field galaxies would occur even later since the field red galaxies are still actively forming stars. It may be reasonable to consider that cluster red galaxies have stopped star formation well in advance, while group red galaxies are just in the process of truncation, and the field red galaxies have not even stopped star formation.
Star formation activities reflect an evolutionary stage of the colour-magnitude build-up. The red cluster and group galaxies show no/little star formation activities, and the bright-end of the colour-magnitude relation in these regions is already built-up. On the other hand, field red galaxies show rather active star formation and they do not form a tight relation. @tanaka05 reported that the bright-end of the field colour-magnitude relation is built-up down to $z\sim0$. We expect that the star formation activities of field red galaxies become significantly weaker by $z=0$.
What physical mechanism truncates star formation activities? A hint for this question lies in groups. Group galaxies form a colour-magnitude relation at the bright-end, but not at the faint-end. This should indicate that the bright-end of the relation is built-up recently; the star formation rates of bright galaxies in groups dropped in a recent past. A close inspection of group galaxies will therefore provide a clue to the physical driver of the truncation.
There are at least two possible physical mechanisms that can truncate star formation activities in groups: galaxy-galaxy interactions (e.g. @mihos96) and strangulation [@larson80; @balogh00]. Galaxy-galaxy interactions trigger starbursts and star formation is truncated after the burst. Strangulation gradually truncates star formation over $\sim1$ Gyr. These processes can be differentiated by the H$\delta$ absorption – starburst+truncation enhances the H$\delta$ absorption after the burst, while strangulation does not trigger the enhancement. This is shown by the model predictions in Fig. \[fig:hd\_d4000\] (strangulation should follow a similar track to the “tau” model). It is interesting to note that a sudden truncation of star formation without burst (0% burst) also enhances the H$\delta$ absorption. Ram-pressure stripping [@gunn72; @abadi99] will truncate star formation in a relatively short time scale, and the model suggests ram-pressure stripping could explain the observed H$\delta$ absorption. However, ram-pressure stripping is not effective in groups, and this is not a viable mechanism. The observed strong H$\delta$ absorption in groups then lends support to the scenario that star formation is quenched in a short time scale, though the group spectra cannot be reproduced by our fiducial burst model. It could be that group galaxies experience dusty starbursts, but the exploration of the dust effects awaits a larger number of higher-$S/N$ spectra. It seems that galaxy-galaxy interactions would be a viable mechanism. Interestingly, we find that a composite spectrum of red galaxies in the filamentary structures (F2 and F3 in Fig \[fig:target\_fields\]) shows a very similar properties to the group composite spectra, in the sense that $D_{4000}$ and H$\delta_F$ agree within the errors. Strangulation cannot be effective in filaments and this would further support galaxy-galaxy interactions.
Further discriminations between the physical drivers can be made with the information of morphology. On one hand, strong interactions destroy a disk structure and transform spiral galaxies into elliptical galaxies. On the other hand, strangulation do not strongly disturb a disk structure, and it would transform spiral galaxies into S0 galaxies. Therefore, the dominant population in groups will give an independent clue to the physical driver of the truncation of star formation. Although the mergers between red galaxies may erase this important dynamical footprint [@faber05; @tran05; @vandokkum05], the morphological information is still essential in order to identify the physical driver more conclusively.
In this paper, we confirm the environmental dependence of star formation of bright red galaxies at $\sim0.8$. However, the down-sizing phenomenon remains unexplored. To spectroscopically confirm the down-sizing, we should go fainter than $M_V^*+2$. To reach such faint magnitudes is difficult even with an 8-m telescope. In this paper, we reach only $M_V^*+1$ and a number of spectra in each environment is still small. For these reasons, we do not attempt to discuss the down-sizing at this bright magnitude range. We expect that faint red galaxies show more active star formation than bright red galaxies in the same environment. That is, an epoch of star formation truncation of faint galaxies comes later than that of bright galaxies. Another forecast is that bright red galaxies in groups show similar spectra to faint red galaxies in clusters due to the environmental dependence of the down-sizing effect. Data from intensive spectroscopic observations will be able to address these points.
Finally, we recall that our results are based only on one cluster field at $z\sim0.8$. A larger sample is essential to step forward. A typical $S/N$ of the individual spectrum presented in this paper is low, and the spectra may be affected by systematic uncertainties. A larger number of higher $S/N$ spectra are clearly needed to confirm our results and to further explore star formation histories of galaxies with various metallicity and dust extinctions as a function of environment and address the down-sizing effect.
Summary and Conclusions {#sec:conclusion}
=======================
We conducted a spectroscopic observation of RXJ0153 at $z=0.83$ with FOCAS on Subaru, and we obtained 161 secure redshifts. By combining with redshifts from @demarco05 and @jorgensen05, we constructed a spectroscopic sample of 347 galaxies. Our primary conclusions can be summarized as follows. Note that conclusions 1 and 3 are drawn from our own sample, and conslusion 2 is from the whole sample.\
1 – We spectroscopically confirmed the large-scale structures at $z\sim0.8$, which were first identified photometrically by @kodama05 based on the panoramic imaging data. Spectroscopic redshifts show a sharp spike at $0.82<z<0.85$, which is much narrower than the redshift width used for our photometric selection of member candidates. This is strong evidence for large-scale structures associated to the central cluster. There seems to be two large sheets of galaxies: one at $z\sim0.837$ hosting the main body of the cluster extending from North to South, and the other at $z=0.844$ extending from NE to SSW. They are likely in the process of interaction.\
2 – The accuracy of our photometric redshifts depends on galaxy SEDs. Photometric redshifts are fairly accurate for red galaxies, while they are less accurate for blue galaxies. We therefore have to be careful when we discuss a fraction of red/blue galaxies (e.g. Butcher-Oemler effect) based on phot-$z$.\
3 – We infer star formation histories of $z\sim0.8$ red galaxies in the field, group, and cluster environments. We quantify the $D_{4000}$ and H$\delta_F$ indices, and compare them with those of SDSS galaxies at $z=0$ and those from model predictions. Red cluster galaxies at $z\sim0.8$ have not formed stars for $>1$ Gyr in the past, and the passive evolution can naturally link them to present-day red galaxies. Red galaxies in groups and in the field at $z\sim0.8$ show evidence for recent activities of star formation as indicated by their relatively strong H$\delta$ absorptions. Our current data seem to favour a scenario that star formation is quenched in a short time scale in groups, and galaxy-galaxy interactions would be a viable mechanism. However, higher $S/N$ data are needed to confirm this.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Kentaro Aoki, Takako Hoshi, Nobunari Kashikawa, and Youichi Oyama for their help during the FOCAS observation and data reduction, and Yoshihiko Yamada for helpful discussions. We thank the anonymous referee for reviewing the paper. M.T. acknowledges support from the Japan Society for Promotion of Science (JSPS) through JSPS research fellowships for Young Scientists. This work was financially supported in part by a Grant-in-Aid for the Scientific Research (No. 15740126, 16540223) by the Japanese Ministry of Education, Culture, Sports and Science. This study is based on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.
Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Korean Scientist Group, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
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Uncertainties in the Measurements of the $D_{4000}$ and H$\delta_F$ Indeces {#sec:concerns}
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In this Appendix, we discuss possible sources of uncertainties in our $D_{4000}$ and H$\delta_F$ measurements. We address effects of the flux calibration, Galactic extinction, and telluric extinction.
Flux calibration:\
No absolute flux calibration is required for the measurements of $D_{4000}$ and H$\delta_F$, and the question here is how accurate the relative flux is over the spectral features. The H$\delta_F$ index adopts a pseudo-continuum around the absorption feature, and thus a fluxing error should not affect H$\delta_F$ in a significant way. As for $D_{4000}$, an accuracy of SDSS flux at the bluest wavelengths is on the order of a few percent [@abazajian04]. A relative flux is more accurate than this, and hence $D_{4000}$ of the SDSS galaxies is expected to be accurate to less than a few per cent. The flux calibration for RXJ0153 should be reliable since the $D_{4000}$ feature lies around the centre of our spectral wavelength coverage. We therefore expect the flux error does not affect our $D_{4000}$ and H$\delta_F$ measurements in a significant way.
Galactic extinction:\
Both our spectra and the SDSS spectra are not corrected for the Galactic extinction. We estimate E($B-V$)=0.015 for RXJ0153 from the dust map of @schlegel98. A typical E($B-V$) of SDSS galaxies is 0.03 to 0.04. Since the $D_{4000}$ index is measured in a relatively small wavelength window, the dust extinction affects our $D_{4000}$ measurements by only $\sim1$%. We therefore expect our results are robust to the Galactic extinction.
Telluric extinction:\
For $z\sim0.8$ galaxies, strong telluric extinction (A-band) lies around the spectral features in interest such as H$\delta$. We have corrected for this extinction (section \[sec:spec\_obs\]), but we speculate that we have over/under corrected by $\sim$10%. The composite spectra are generated from galaxies at slightly different redshifts, and this uncertainty should be reduced to some extent. However, it is difficult to evaluate this uncertainty in our composite spectra. The $D_{4000}$ and H$\delta_F$ indices measured from the SDSS galaxies are free from the strong telluric extinctions.
The remaining concern is the effect of systematic uncertainties, particularly in the $z\sim0.8$ sample. It is a difficult task to evaluate them. However, the fact that our $D_{4000}$ measurements are in good agreement with those by @jorgensen05 suggests that systematic uncertainties do not dominate the overall error budget.
Aperture Bias {#sec:aperture_bias}
=============
The SDSS spectroscopic survey is performed with the fibre-fed spectrographs. Each fibre subtends $3''$ on the sky, which is much smaller than a typical size of galaxies at $z<0.065$ [@tanaka04]. In contrast, a typical extraction aperture adopted in our spectroscopic data reduction for $z\sim0.8$ galaxies is roughly 1/2 to 1/3 of a size of a galaxy. Thus, an extraction aperture adopted in the SDSS is smaller compared to that for $z\sim0.8$ galaxies in a physical size. This may artificially make differences between $z=0$ and $z\sim0.8$ in the spectral features explored in section \[sec:spec\_diag\]. We argue in the following, however, our conclusions are robust to this aperture bias.
In order to quantify the aperture bias, we use spectroscopic data of nearby galaxies from @jansen00. An important feature of the data is that the spectra are obtained for both the nuclear region and the entire region of a galaxy (’nuclear’ and ’integrated’ spectra, respectively). The nuclear spectra typically include 10% of the light encluded in the $B_{26}$ isophote, while the integrated spectra include 80% on average. Although the extraction aperture adopted in @jansen00 is different from that of SDSS, comparisons between the nuclear and integrated spectra will give an estimate of how the aperture bias affects our measurement, at least qualitatively. The spectra are smoothed to our instrumental resolution, and $D_{4000}$ and H$\delta_F$ are measured (again the emission filling is corrected). We do not apply any magnitude cut to the @jansen00 sample to gain the statistics. Our aim here is not to compare the integrated spectra with our spectra of $z\sim0.8$ galaxies, but to investigate the aperture bias.
Comparisons between the nuclear and integrated spectra are shown in Fig. \[fig:aperture\_bias\]. While the overall correlation is encouragingly good, deviations are seen. The integrated spectra show systematically smaller $D_{4000}$ than the nuclear spectra at $1.6\lesssim D_{4000,nuclear}\lesssim2.1$. That is, these galaxies show bluer colours at the outer parts of the galaxies compared to the inner parts. This is expected since active star formation is often observed in a disk rather than in a bulge. Very red/blue galaxies do now show such deviation. As for H$\delta_F$, the correlation between the integrated and nuclear spectra is relatively good, but a tail toward higher H$\delta_{F, integrated}$ is seen. Despite these tails, however, the distribution of galaxies on the H$\delta_F$–$D_{4000}$ plane is basically the same for the nuclear and integrated spectra. Furthermore, for our SDSS sample, spectroscopic fibre typically collects 20% of the total $g$-band light of a galaxy. Thus the aperture bias should be smaller than that we see in Fig. \[fig:aperture\_bias\]. We therefore conclude that the aperture bias has no significant effect on our conclusions. Note that the integrated spectra seem to show a smaller scatter on the $D_{4000}$–H$\delta_F$ plane than the nuclear spectra. Thus, if the indices are measures for the entire part of galaxies, the scatter of SDSS galaxies will be smaller than we observe in Fig. \[fig:hd\_d4000\].
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abstract: 'We address the problem of shaping deformable plastic materials using non-prehensile actions. Shaping plastic objects is challenging, since they are difficult to model and to track visually. We study this problem, by using kinetic sand, a plastic toy material which mimics the physical properties of wet sand. Inspired by a pilot study where humans shape kinetic sand, we define two types of actions: *pushing* the material from the sides and *tapping* from above. The chosen actions are executed with a robotic arm using image-based visual servoing. From the current and desired view of the material, we define states based on visual features such as the outer contour shape and the pixel luminosity values. These are mapped to actions, which are repeated iteratively to reduce the image error until convergence is reached. For pushing, we propose three methods for mapping the visual state to an action. These include heuristic methods and a neural network, trained from human actions. We show that it is possible to obtain simple shapes with the kinetic sand, without explicitly modeling the material. Our approach is limited in the types of shapes it can achieve. A richer set of action types and multi-step reasoning is needed to achieve more sophisticated shapes.'
author:
- Andrea Cherubini Valerio Ortenzi Akansel Cosgun Robert Lee Peter Corke
bibliography:
- 'library.bib'
title: 'Model-free vision-based shaping of deformable plastic materials'
---
Introduction {#sec:introduction}
============
Many tasks such as cooking, folding clothes and gardening require the manipulation of soft objects and deformable materials. The same applies to industrial tasks such as inserting cables, packaging food and excavating soil and to medical ones such as injection, physiotherapy and surgery. Some of these tasks involve the manipulation of granular materials such as dough, sand, soil and salt. The capability of manipulating such materials would enable robots to perform a plethora of new applications that could increasingly help and substantially assist humans with their chores. Despite the potential impact of successful manipulation of deformable materials, decades of robotics research have focused primarily on rigid objects. The reason behind this is the task difficulty. Adding to the inherent complexity of physical interaction with deformable materials are additional challenges, such as modeling the deformations. A direct consequence is that visual tracking of deformable materials is usually very demanding. Visual features of rigid objects can be consistently detected and tracked by exploiting a prior 3D model of the object. In contrast, features on deformable materials change over time and that can mislead both model-based or feature-based visual trackers. Feedback from force and tactile sensors could be beneficial, although this also requires deformation and contact models that map force/tactile signals to the corresponding displacements of the object surface. This mapping is commonly complicated to obtain.
With reference to the taxonomy given in the recent survey by [@BoCoMe:18], our focus is on *solid* or *volumetric* objects, *i.e.*, objects with the three dimensions having comparable length. Sponges, plush toys, and food products fall in this category. In particular, a deformation occurs when an external force applied to an object changes its shape and appearance. Depending on the response of the object once the external force is removed, the deformation can be plastic, elastic, or elasto-plastic. Precisely, a *plastic* deformation entails a permanent deformation, that is, an object maintains the shape caused by the applied force even when that force is removed.
![Illustration of the *shape servoing* problem that we address in this paper. Our goal is to find a sequence of non-prehensile manipulation actions a robot has to perform with a tool (blue in the figure) to mold a plastic material from an initial shape (top left) to a desired shape (top right). The bottom image shows the robot in action while it molds the material.[]{data-label="Fig:introFigure"}](introFigure.pdf){width="\columnwidth"}
Humans often use non-prehensile actions, such as pushing or tapping, to move objects or modify the environment (see the work of [@bullock2011classifying]). Such actions become even more common when the object of interest is plastic. For instance, the manipulation of clay or dough is typically non-prehensile. These observations motivated us to target non-prehensile actions.
In this work, we address the problem of *shape servoing* plastic materials (see the works of [@smolen2009deformation] and of [@Na14]). Our goal is to *find a sequence of non-prehensile actions a robot has to perform using a tool to bring plastic materials from an initial to a desired shape*. This problem is illustrated in Fig. \[Fig:introFigure\].
To study plastic deformation, we use kinetic sand[^1], a toy material which mimics the physical properties of wet sand. It is made out of 98% regular sand and 2% polydimethylsiloxane (a viscoelastic silicone) and it can be molded into any desired shape. Kinetic sand does not stick to any materials other than itself and does not dry out.
The paper is organized as follows. After reviewing the related literature (Sect. \[Sect:relWork\]) and presenting our contribution (Sect. \[Sect:contributions\]), we provide the results of a pilot study performed to understand how humans manipulate plastic materials (Sect. \[sect:userstudy\]). Then, in Sect. \[Sect:problem\] we define the shape servoing problem. Section \[sect:method\] describes our method, which is validated in the experiments of Sections \[sect:setup\] and \[sect:experiments\]. We conclude and propose directions for future work in Sect. \[sect:conclusion\].
Related Work {#Sect:relWork}
============
In this section, we review the literature on the manipulation of deformable objects, which is not as common as that of rigid objects. We also briefly review the literature on two tools needed in our work: visual servoing and action selection.
As observed in the survey on manipulating deformable objects by [@BoCoMe:18], much of the body of work focuses on 1D and 2D objects. Here, we mainly review works on solid 3D objects, as these are the ones addressed in our work. We first review shape estimation and tracking of deformable objects in Sect. \[sec:rel1\], then shape control with and without a deformation model respectively in Sections \[sec:rel2\] and \[sec:rel3\]. Then, since in this paper we draw inspiration from Visual Servoing, we briefly recall the main works in this field (Sect. \[sec:visServo\]). Finally, we review methods on action selection (Sect. \[sec:actionSel\]).
Shape Estimation/Tracking of deformable objects {#sec:rel1}
-----------------------------------------------
The works that estimate and track deformable objects shape can be classified in three categories: those that require a model (generally mechanical), the data-driven ones (model-free) and the ones that combine the two.
Mechanical model-based trackers use a priori knowledge of the object’s physical model. In one of the earliest papers on vision-based deformable shape estimation by [@sarata2004trajectory], stereo vision is used to obtain the 3D volumetric model of a pile for robotic scooping. [@petit2017tracking] use a Finite Element Method (FEM) for mesh-fitting. The goal is to track the shape of a pizza dough in real time. A volumetric FEM model is also used by [@FrScSt:14]. An object is probed and the error between observed and simulated deformation is minimized to estimate its Young’s modulus and Poisson ratio. [@guler2015estimating] estimate the deformations of elastic materials using optical flow and mesh-less shape matching.
Data-driven shape estimation methods rely on data gathered by sensors. [@khalil2010visual] track the surface of an object to be grasped by a robot using tactile and visual data. [@cretu2010deformable] address a similar application, by extracting the foreground before detecting the object contour. [@staffa2015segmentation] train a neural network for visual segmentation of non-rigid deformable objects (a pizza dough in their work).
Some approaches combine mechanical (model-based) and data-driven (model-free) estimation methods. [@caccamo2016active] use tactile and RGB-D data to create a mapping of how the surface is deformed, using Gaussian Processes. [@arriola2017multi] propose a generative model that uses force and vision to predict not only the object deformation, but also the parameters of a spring-mass model that represents it.
Model-based Deformation Control {#sec:rel2}
-------------------------------
Some approaches take advantage of the deformable object’s physical model, whenever it is available. [@howard2000intelligent] train a neural network on a physics-based model to extract the minimum force required for 3D object manipulation. [@gopalakrishnan2005d] focus on 2D objects and use a mesh model with linear elastic polygons. Their approach, called “deform closure” is an adaptation of “form closure”, a well-known method for rigid object grasping. [@DaSa:2011] use a mass-spring-damper model to simulate a planar object so that its shape, described by a curve, can be changed into another desired shape. Their method, however, requires a significant number of actuation points (over 100), and is only validated in simulations. [@higashimori2010active] use a four-element model and present a two-step approach where the elastic parameters are estimated by force sensing, and then the force required to reach the desired shape is calculated based on the plastic response. [@cretu2012soft] monitor shape deformation by tracking lines that form a grid on the object. A feed-forward neural network is used to segment and monitor the deformation. [@arriola2017multi] predict the object behavior by first classifying the material, then using force and computer vision to estimate its plastic and elastic deformations. In [@CoDuFi:18], a method for dexterous in-hand manipulation of 3D soft objects for real-time deformation control is presented, relying on Finite Element Modeling. However, the authors assume the object to be purely elastic.
Model-free Deformation Control {#sec:rel3}
------------------------------
Some approaches explored deformable object manipulation without explicitly modelling the deformations. In their pioneering work, [@WaHiKaKa01] designed a PID controller that can manipulate 2D objects in the absence of a prior model. [@smolen2009deformation] use a mesh-less model of the object where a set of points are controlled on the surface. A Jacobian transform is derived and used to control the robot motion. [@berenson2013manipulation] uses the concept of diminishing rigidity to compute an approximate Jacobian of the deformable object. In his work, human and robot simultaneously manipulate the object (a 2D cloth).
More recent model-free approaches rely on machine learning. [@gemici2014learning] learn haptic properties such as plasticity and tensile strength of food objects. Other researchers propose to predict the next state given the current state and a proposed action. For instance, [@elliott2018robotic] use some defined primitive tool action for rearranging dirt and [@schenck2017learning] present a Convolutional Neural Network for scooping and dumping granular materials. [@li2018learnParticle] propose to learn a particle-based simulator for complex control tasks. This enables the simulator to quickly adapt to new environments or to unknown dynamics within a few observations.
Visual servoing {#sec:visServo}
---------------
Visual servoing is a technique which uses visual features (e.g., points, lines, circles, etc.) extracted from a camera to control the motion of a robot. Visual servoing methods are commonly used in robot manipulation and can be classified as position-based or image-based. In position-based visual servoing, the feature is reprojected in the 3D space and the robot is controlled in Cartesian coordinates as done by [@drummond1999visual]. Instead, the image-based approach regulates an error defined in the image space. This is done via the interaction matrix, which relates the dynamics of the camera (i.e., its velocity) to those of the visual feature to be controlled, as explained by [@ChHu06a]. Image-based visual servoing is robust to calibration errors as shown by [@HuHaCo96]. It can be used with various features, including image moments as done by [@ChTa:05], and mutual information by [@Sh:48], as done by [@DaMa:2009]. The interested reader can refer to the works of [@kragic2002survey] for a detailed survey, and to those of [@ChHu06a; @ChHu07] and [@HuHaCo96] for tutorials on visual servoing.
Visual servoing can also be used for deformation control. For instance, [@Na13; @Na14] actively deform compliant objects using a novel visual servoing scheme that explicitly deals with elastic deformations, by estimating online the interaction matrix relating tool velocities and optical flow. Their controller is model-free, but focuses mainly on shape control, *i.e.*, on manipulating the object to a desired configuration, without dealing with its global deformation over a long time window. The latest approach by [@navarro2018fourier] uses a Fourier series as visual feature for representing the object contour. [@zhu:iros18] use a similar feature for dual arm shaping of flexible cables.
Action selection methods {#sec:actionSel}
------------------------
When manipulation requires various types of actions (as is often the case with deformable objects) an automatic action selection method is needed. Although we did not implement any of these methods (in our work, the operator manually selects the action at each iteration), it is worth reviewing them, as they could be integrated in our framework. Such methods are often based on machine learning or on motion planning.
Learning-based methods are frequently used for choosing within a finite number of actions. An example is deep reinforcement learning (see the works of [@dulac2015deep] and of [@isele2018navigating]), which relies on the Markov Decision Process (MDP) formulation (refer to [@puterman2014markov]). [@laskey2017learning] learn a policy for choosing actions to manipulate cloth using imitation learning.
[@KaMo:06] present a planning method for finding intermediate states and transitioning between these states to shape a flexible wire. Similarly, [@zhu:ral18] plan the sequence of actions for shaping a cable using environmental contacts.
{width="\textwidth"}
Contributions {#Sect:contributions}
=============
Our approach is model-free, because we do not model the deformations of the plastic material, like some of the other work reviewed in Sect. \[Sect:relWork\]. We instead focus on altering the visible characteristics of the material as in [@navarro2018fourier]. Differently from the cited work, we make use of the data collected from humans for both the heuristic and learning-based algorithms explored in this paper. Most works in the field focus on a single deformation action, whereas our algorithmic choices are driven by the sequential nature of the task – typically a sequence of non-prehensile actions are required to achieve a given desired shape. We also have a particular emphasis on the real-world implementation.
In our previous work (see [@ChCoOr:18]) we ran a user study on kinetic sand manipulation, presented the corresponding image dataset and designed an image processing algorithm for extracting features from the dataset. We also proposed an optimization-based algorithm for controlling the robot, along with a neural network architecture for mapping the state of the material to the pushing action to be applied. There were no robot experiments in that paper.
The first three points (user study, dataset and image processing algorithm) constitute the foundations of the present work (see Sections \[sect:userstudy\] and \[sect:gettingTriplets\]). Nevertheless, [@ChCoOr:18] did not address real experiments. As soon as we implemented them, the proposed tools/methods were confronted to their limitations. First, the user dataset was inappropriate for transferring manipulation capabilities from humans to robots, and we had to enrich it with extra data (see Sect. \[Ref:userDis\], point (c)). Second, we had to completely redesign the problem statement and method (see Sect. \[Sect:problem\]). For instance, we had to address the pushing problem locally (see Sect. \[sect.localShaping\]), since the global method failed. Third, we modified the neural network architecture, and, given its limitations, compared it with two other strategies (*maximum* and *average*) which were not present in the previous work. Finally, we realized a series of unprecedented robotic experiments on shape servoing of plastic materials.
In summary, the contributions of this work are:
1. We exploit the human demonstration data to compensate for the lack of a physical material model. The human data is utilized to design a local strategy for robotic kinetic sand manipulation and also to train a machine learning-based action model.
2. We propose two classes of non-prehensile actions to reduce the complexity of the deformable object manipulation problem: *pushing* to modify the outer contours, and *tapping* to adjust the kinetic sand height.
3. We report experiments where a robot manipulator successfully molds the kinetic sand into various desired shapes. In the experiments, we compare various approaches for realizing the pushing and tapping actions.
In a nutshell, while Sections \[sect:userstudy\] and \[sect:gettingTriplets\] had been presented in our previous work (see [@ChCoOr:18]), the rest of the present paper reports unpublished, original research.
User study {#sect:userstudy}
==========
Materials and methods
---------------------
To understand how humans manipulate plastic materials, we ran a pilot user study with 9 volunteers (age range: 20-40; 6 male, 3 female). Each participant was asked to shape kinetic sand in a sandbox, while being recorded with a fixed RGB-D camera (Intel RealSense SR300, resolution $640 \times 480$). We opted for such a low cost and easily available sensor to make the setup reproducible and inexpensive. The RealSense was pointing at the sandbox from above, with its optical axis perpendicular to it, as shown in Fig. \[Fig:setup\] (left).
Each participant was requested to produce a shape of their own choice, repeating the task three times: a) using both hands, b) using only one hand and finally c) using one of the two provided tools (Fig. \[Fig:setup\], left). Four examples of shapes formed by the participants are shown in Fig. \[Fig:setup\] (center). Figure \[Fig:setup\] (right) shows some of the images acquired while a user was shaping the kinetic sand with both hands.
All participants gave their consent to be recorded. We obtained ethics clearance to make the dataset (almost 214000 RGB and depth images) publicly available[^2]. Afterwards, the participants filled out a questionnaire, to help us infer their “sculpting strategy”. None of the participants had previous direct experience with sculpting, although 5 declared to have previously manipulated deformable objects such as dough or clay.
Results {#Sect:results}
-------
Six participants stated that they had performed a clear sequence of different actions while using their hands, while the number raised to eight when using the tool. In particular, the participants clearly identified *pushing* as an action type. In our opinion, this is due to the fact that pushing has a very clear outcome (the kinetic sand moving and consequently changing its outer contours). Two other actions commonly identified were *tapping* and *incising*. The questionnaire included the following four questions:
- “How much did you rely on vision while hand modeling? grade between 1 (very little) and 5 (very much)”
- “How much did you rely on haptic feedback while hand modeling? grade between 1 (very little) and 5 (very much)”
- “How much did you rely on vision while sculpting with the tool? grade between 1 (very little) and 5 (very much)”
- “How much did you rely on haptic feedback while sculpting with the tool? grade between 1 (very little) and 5 (very much)”
All of the participants, whether using their hands or the tool, reported having relied on vision either “very much” or “much”. Haptic feedback was only partially important during the trials (7 participants reported having relied “little” on haptic feedback). This result led us to believe that – although the kinetic sand offers some force resistance while sculpting – haptics is not as valuable a feedback as vision. In our opinion, it is more natural to rely on visual feedback as a measure of how distant the current shape is from the imagined/desired shape. Haptic feedback is valuable *locally* in understanding the force required to overcome the kinetic sand resistance, but offers little *global* information on which action to perform next, to obtain the desired shape. In summary, visual and haptic feedback are complementary. To further investigate the importance of haptics, one could blindfold the participants (so they cannot rely on vision) or have them teleoperate a molding robot without haptic feedback (so they can only rely on vision). These experiments could be the object of future work.
Since we focus on non-prehensile kinetic sand manipulation with a robotic tool, within the dataset we manually labeled only the images where participants used either tool. Within these, we only labeled the RGB images, since the RealSense depth image quality is insufficient for feature extraction (see Fig. \[Fig:setup\], bottom right). Among the 53,100 RGB images of tool shaping, 13.3% were labeled as “pushing”, 9.3% as “tapping” and 8.7% as “incising”. The other images were left unlabeled (68.7%), as they present: retreating actions, transitions between actions, occlusions (*i.e.*, the tool was not visible), simultaneous “push-and-tap” actions (the participant modifies the contour, while also yielding a smoothing effect on the surface) and unclear actions.
Discussion {#Ref:userDis}
----------
The results of the user study motivated us to adopt the following choices for transferring shape servoing capabilities from humans to robots:
a) The importance of visual feedback prompts us to use a camera as the only source of feedback. This choice also enables us to use *Image-Based Visual Servoing* (IBVS, see the works of [@HuHaCo96] and [@ChHu06a; @ChHu07]) to control the robot directly in the image space, even with possibly coarse camera calibration.
b) The results highlight that participants commonly rely on – and can clearly recognize – a *finite set of actions* to model the kinetic sand. This motivates us to implement the two actions that are most heavily identified and chosen in the user study: *pushing* and *tapping*. Pushing shapes the outer contours of the kinetic sand, whereas tapping regularizes the surface by leveling it. Both are tool motions along a vector going towards the kinetic sand from the free space. For pushing, the vector is parallel to the sandbox plane, whereas for tapping it is orthogonal to the sandbox plane and pointing down. We discarded *incising*, since it is the least popular among users. In our opinion, incising can be achieved by tracking a desired trajectory (the incision) with the tool.
c) The user dataset is inappropriate for transferring the mentioned pushing and tapping capabilities from humans to robots. Only 12,000 labeled images (22.6% of those in the tool dataset) are available. This quantity is insufficient for learning such complex actions. Hence, we decided to enrich the dataset with ad-hoc trials where participants performed only one of the two actions at a time. In each trial, the participant performed a sequence of “clean” (without occlusions and with clear movements) pushing or tapping actions. This simplified the image processing steps needed to produce the training dataset. In the following, we denote the complete collection of pushing and tapping images as the *PT-Dataset*. The dataset contains 24,000 images (12,000 from the users and 12,000 from these ad-hoc trials), of which 14,900 labeled as “pushing” and 9,100 as “tapping”.
Problem Statement {#Sect:problem}
=================
In this section, we define the shape servoing task and the work assumptions. The variables defined in this section are shown in Fig. \[Fig:actions\_states\].
The shape servoing task {#sect:shapeservo}
-----------------------
We consider the same setup as in the user study (Sect. \[sect:userstudy\]): a camera points at the object (here, the kinetic sand) from above, with a vertical optical axis. Given a desired image $\mathbf{I}^*$ (e.g., one of those in Fig. \[Fig:setup\], center), the *shape servoing task* consists of shaping the kinetic sand until it appears as in $\mathbf{I}^*$.
We model the robot and object shape as a discrete-time system. The robot tool center point (TCP) can be moved in the workspace to modify the object shape. At each iteration $k \in \left[ 1, K \right] \subset \IntSet$, the robot observes the shape in image $\mathbf{I}_k$ (of width $w$ and height $h$), and then modifies it by moving its TCP to perform some *action* $\mathbf{a}_k$. ** We define $e_k \left(\mathbf{I}_k, \mathbf{I}^*\right) \geq 0$ as a scalar function that measures the *image error* between $\mathbf{I}_k$ and $\mathbf{I}^*$, so that $e_k = 0 \iff \mathbf{I}_k = \mathbf{I}^*$. Then, the task of shaping an object to $\mathbf{I}^*$ with an accuracy $\bar{e} \geq 0$, consists in applying a finite sequence of actions $\mathbf{a}_1, \dots, \mathbf{a}_K$ such that after $K$ iterations: $e_K \leq \bar{e}$. We tolerate such an upper bound on the accuracy, since even a human would be incapable of perfectly reproducing ($e_K = 0$) a given image with the kinetic sand. Nevertheless, the sequence of actions should make $e_K$ decrease. More formally, it must be possible to apply, at each iteration $k$, an action $$\mathbf{a}_k = \mathbf{a}\left( \mathbf{I}_k, \mathbf{I}^* \right)
\label{eq:actionImg}$$ that will reduce $e_k$: $$e_{k+1} < e_k.
\label{eq:errorDecrease}$$
Inspired by the results of the user study, we consider a set of two actions, defined below.
** The robot can execute one of two actions, namely *pushing* or *tapping*: $$\mathbf{a} = \left\{\mathbf{p},\mathbf{t}\right\},
\label{eq:setActions}$$ and each action is parameterized within a parameter set: $\mathbf{p} \in {\cal{A}}_P$, $\mathbf{t} \in {\cal{A}}_T$. We define the *pushing* action ${\mathbf{p}}$ as a translation of the TCP between two points on the sandbox base. Indeed, if the contact between TCP and material is approximated by a point or by a sphere, the effect of pushing will be invariant to the tool orientation. Since the sandbox base is parallel to the image plane, ${\mathbf{p}}$ can be defined via the image coordinates $\left(u , v\right) \in \left[0, w\right] \times \left[0, h\right]$ of the start and end pixels (denoted $S$ and $E$, see top of Fig. \[Fig:actions\_states\]) of the TCP motion: $$\begin{array}{l}
{\mathbf{p}} = \left[ u_S \; v_S \; u_E \; v_E \right]^\top,\\
{\cal{A}}_P\in \left[0, w\right]\times \left[0, h\right]\times\left[0, w\right]\times \left[0, h\right].
\end{array}
\label{eq:toolActionPushing}$$ Similarly, we define the tapping action ${\mathbf{t}}$ as a vertical translation of the TCP, perpendicular to the sandbox base, between 2 points. Since the heights of these points are fixed, ${\mathbf{t}}$ can be parameterized only by the final position of the TCP in the sandbox plane, which corresponds to a pixel $T$ (see bottom of Fig. \[Fig:actions\_states\]) in the image: $$\begin{array}{l}
{\mathbf{t}} = \left[u_T \; v_T \right]^\top,\\
{\cal{A}}_T\in \left[0, w\right] \times \left[0, h\right].
\end{array}
\label{eq:toolActionTapping}$$ Other setups – with the camera optical axis not perpendicular to the sandbox – would require different representations.
![The two non-prehensile actions that we use and their effect on the kinetic sand. Top: the pushing action ${\mathbf{p}} = \left[ u_S \; v_S \; u_E \; v_E \right]^\top$ modifies the outer contours (yellow) from ${\mathbf{x}}^P_k$ (on image $\mathbf{I}_k$) to ${\mathbf{x}}^P_{k+1}$ (on image $\mathbf{I}_{k+1}$). Bottom: the tapping action ${\mathbf{t}} = \left[u_T \; v_T \right]^\top$ adjusts the height and compactness of the kinetic sand; it also affects the resampled images $\tilde{\mathbf{I}}_k$ and $\tilde{\mathbf{I}}_{k+1}$ at the pixel $T$ where the tapping action is applied.[]{data-label="Fig:actions_states"}](actions_states.pdf){width="\columnwidth"}
Assumptions {#sec:assumptions}
-----------
In accordance with the user study (Sect. \[Ref:userDis\]), we make the following hypotheses.
** Each action type $\mathbf{a} = \left\{\mathbf{p},\mathbf{t}\right\}$ regulates only some *features* of image $\mathbf{I}_k$. This hypothesis follows from the user study, where the effects of the pushing and tapping actions were separate and distinct. Users push the object to shape its outer contour. In our setup, this feature is observable, since its perspective projection is also a contour, visible in the image and outlined in yellow in the top of Fig. \[Fig:actions\_states\]. Then, the *push-controlled feature* is the list of image coordinates of the $N$ pixels sampled along the shape’s outer contour: $$\mathbf{x}^P_k = \left[
u_1 \; v_1 \dots u_N \; v_N \right]^\top_k.
\label{eq:pushState}$$ Similarly, when tapping, the users smooth and flatten the kinetic sand surface. This could be interpreted as regulating its height, a feature that is not measurable without accurate 3D sensing. To observe the effect of tapping on a monocular image, we draw inspiration from [@ArRoSi:15], where a robot flattens cloth by closing a feedback loop on the cloth wrinkles. Similarly, we measure the effect of tapping directly on image $\mathbf{I}_k$. We account for the tool resolution (*i.e.*, the size of its contacting surface) by scaling $\mathbf{I}_k$ to a smaller $\tilde{\mathbf{I}}_k$, with pixels having the size of the tool’s image when it contacts the kinetic sand: $w_{\mbox{TCP}} \times h_{\mbox{TCP}}$ pixels (of the original image). The *tap-controlled feature* will correspond to this image $\tilde{\mathbf{I}}_k$ (shown in the bottom of Fig. \[Fig:actions\_states\]) which can be obtained by linearly sampling ${\mathbf{I}}_k$ to a reduced size $w / w_{\mbox{TCP}} \times h / h_{\mbox{TCP}}$.
** Given the initial image $\mathbf{I}_1$, the desired image $\mathbf{I}^*$, and the set of actions defined in (\[eq:setActions\]), there exists at least one sequence of actions that solve the shape servoing problem defined above. Practically, this hypothesis means that the desired shape can be realized using the set of actions defined in (\[eq:setActions\]), e.g., matter cannot be added nor pulled, etc.
Even if a solution exists, finding the best sequence of actions $\mathbf{a}^*_k$ that verifies (\[eq:errorDecrease\]) requires solving – at each iteration $k$ – two subproblems. First, one must find the best action within the set $\left\{\mathbf{p},\mathbf{t}\right\}$ and then, for the chosen action $\mathbf{a}$, one must find the best parameters within the action’s parameter set – either ${\cal{A}}_P$ or ${\cal{A}}_T$.
** In this work, we only address the second subproblem: we assume that at each iteration either a cognition layer or a human operator has selected the best action type (*push* or *tap*), and we focus on finding the best parameters for *that* selected action (*where* to push or tap).
Despite these hypotheses, two non-trivial issues are still to be addressed: the choice of image error measure $e_k$ and the visual control strategy for defining $\mathbf{a}_k = \mathbf{a}\left( \mathbf{I}_k, \mathbf{I}^* \right)$. These choices will be the discussed in the next Section.
Proposed method {#sect:method}
===============
Image error
-----------
Since images $\mathbf{I}_k$ and $\mathbf{I}^*$ are spatially aligned into the same geometric base, we can design image error $e_k$ using a global similarity measure (see [@Mitchell:2010]). These measures include mean square error, mean absolute error, cross-correlation, and mutual information (defined by [@Sh:48]). Among the four, the latter is best in terms of robustness to light variations and occlusions. The mutual information between images $\mathbf{I}_k$ and $\mathbf{I}^*$ is: $${\mbox{MI}}_k = {\mbox{MI}} \left(\mathbf{I}_k, \mathbf{I}^*\right) = \sum_{i_k,i^*} p\left(i_k, i^*\right) \log \left( \frac{p\left(i_k, i^*\right)}{p\left(i_k\right) p\left(i^*\right)} \right),
\label{eq:mutInfo}$$ with:
- $i_k$ and $i^*$ the possible pixel values in images $\mathbf{I}_k$ and $\mathbf{I}^*$ (e.g., in the case of 8-bit grayscale images these are the luminances and $\left(i_k, i^*\right) \in [0, 255] \times [0, 255]$),
- $p\left(i_k\right)$ and $p\left(i^*\right)$ the probabilities of value $i_k$ in $\mathbf{I}_k$ and $i^*$ in $\mathbf{I}^*$ (respectively),
- $p\left(i_k, i^*\right)$ the joint probability of $i_k$ and $i^*$ computed by normalization of the joint histogram of the images.
${\mbox{MI}}_k$ is maximal and unitary when $\mathbf{I}_k \equiv \mathbf{I}^*$: ${\mbox{MI}} \left(\mathbf{I}^*, \mathbf{I}^*\right) = 1$. Then, to obtain $e_k = 0$ for $\mathbf{I}_k = \mathbf{I}^*$ and $e_k$ monotonically increasing with the image dissimilarity, we should set: $$e_k = {\mbox{MI}} \left(\mathbf{I}^*, \mathbf{I}^*\right) - {\mbox{MI}}_k = 1 - {\mbox{MI}}_k.
\label{eq:mutInfoError}$$ In the following, we denote this expression of $e_k$ as the *mutual information error*.
Image-based control {#sec:image_based_control}
-------------------
Having chosen $e_k$, the second issue is the definition of a robot control input $\mathbf{a} \left( \mathbf{I}_k, \mathbf{I}^* \right)$ that ensures (\[eq:errorDecrease\]) when $e_k = {\mbox{MI}}_k$. [@DaMa:2009] address such problem (Mutual Information-based visual servoing) by applying Levenberg-Marquardt optimization to maximize ${\mbox{MI}}$. Yet, they can derive analytic expressions of both the Gradient and Hessian of ${\mbox{MI}}$ with respect to $\mathbf{a}$, since in their work $\mathbf{a}$ is the pose of the camera looking at a photography (the image). Conversely, in our application we do not have an accurate model of the dynamics of the scene, which evolves, is not rigid and cannot be assumed Lambertian, as in the work of [@DaMa:2009]. Hence, we cannot derive an analytic relationship between the applied action $\mathbf{a}$ and the corresponding dynamics of ${\mbox{MI}}$ and therefore find some $\mathbf{a}^*$ that guarantees (\[eq:errorDecrease\]). Thus, while ${\mbox{MI}}$ is an objective metric for assessing the controller convergence, it is not straightforward to use in the design of $\mathbf{a}$.
Instead, based on , we design $\mathbf{a}$ based on the visual feature that it affects, rather than on the whole image $\mathbf{I}$. At each iteration $k$, we will apply $${\mathbf{p}}_k = {\mathbf{p}} \left( {\mathbf{x}}^P_k, {\mathbf{x}}^{P*} \right)
\text{ or }
{\mathbf{t}}_k = {\mathbf{t}} \left( \tilde{\mathbf{I}}_k, \tilde{\mathbf{I}}^* \right),
\label{eq:actionState}$$ depending on the pre-selected () action type (push or tap). In (\[eq:actionState\]), $\mathbf{x}^{P*}$ and $\tilde{\mathbf{I}}^*$ respectively indicate feature $\mathbf{x}^P$ and $\tilde{\mathbf{I}}$, extracted from the desired image $\mathbf{I}^*$. Since the visual features are related to the image $\mathbf{I}$, the dependency of ${\mathbf{a}}$ from $\mathbf{I}$ in (\[eq:actionImg\]) is maintained. Note however that since the relationship between features and image is not bijective (e.g., different images may have the same shape’s outer contour), different images may lead to the same action.
The design of (\[eq:actionState\]) is inspired by classic IBVS, where a feature vector ${\mathbf{x}}_k$ is regulated to ${\mathbf{x}}^{*}$ by applying action $${\mathbf{a}}_k = \lambda {\mathbf{L}}_k^\dagger \left( {\mathbf{x}}^{*} - {\mathbf{x}}_k \right)
\label{eq:ibvsControl}$$ with $\lambda$ a positive scalar gain and ${\mathbf{L}}_k$ the interaction matrix, *i.e.*, the matrix such that: ${\mathbf{x}}_{k+1} - {\mathbf{x}}_k = {\mathbf{L}}_k {\mathbf{a}}_k$.
Even if we apply (\[eq:ibvsControl\]), there is no guarantee that $e_k$ will decrease as indicated in (\[eq:errorDecrease\]). First, the relationship ${\mathbf{L}}$ between dynamics of the feature ${\mathbf{x}}$ and the action ${\mathbf{a}}$ must be known and invertible, as in (\[eq:ibvsControl\]). Second, even if this was the case, (\[eq:ibvsControl\]) would only guarantee convergence of feature ${\mathbf{x}}$ to ${\mathbf{x}}^*$, which by no means implies convergence of $\mathbf{I}$ to $\mathbf{I}^*$ (since the mapping between $\mathbf{x}$ and $\mathbf{I}$ is not bijective).
Therefore, model-based approach (\[eq:ibvsControl\]) seems inappropriate for defining (\[eq:actionState\]) to solve our problem. In the rest of this section, we examine various alternative design choices for defining (\[eq:actionState\]) for both tapping and pushing actions, without knowledge of the interaction matrix ${\mathbf{L}}$. In the experimental section, we will compare these design choices through the evolution of the mutual information when the actions are applied sequentially to realize a desired shape $\mathbf{I}^*$.
Tapping {#subsec:methtap}
-------
The design of the tapping action is simpler than that of the pushing action. The reason is that we have assumed (see Sect. \[sec:assumptions\]) that each tapping action ${\mathbf{t}}$ levels a well-defined portion of the kinetic sand surface, without affecting the other areas. More specifically, since we rescale ${\mathbf{I}}$ to a smaller $\tilde{\mathbf{I}}$, with pixels having the same size $w_{TCP} \times h_{TCP}$ as the tool, each ${\mathbf{t}}$ should change only one pixel of $\tilde{\mathbf{I}}$.
Since by we assume that the shape is feasible and that we can only lower (not raise) the kinetic sand level with the tool, the height of the desired shape is lower or equal than that of the current shape. Then, at each pixel, the difference between the image luminosities should be a monotonically increasing function of the difference between the shape heights.
Therefore, we decide to apply the tapping action on the spot (in resampled image $\tilde{\mathbf{I}}$) where the image difference is the highest. This breaks down to writing (\[eq:actionState\]) as: $$\begin{split}
{\mathbf{t}} \left( \tilde{\mathbf{I}}_k, \tilde{\mathbf{I}}^* \right) &=
\left[
\begin{array}{c}
u_T \\
v_T
\end{array}
\right]\\
& = \left[
\begin{array}{cc}
w_{TCP} & 0\\
0 & h_{TCP}
\end{array}
\right]
\underset{\left(u,v\right) \in \tilde{\mathbf{I}}_k}{\mathrm{argmax}}\left\| \tilde{\mathbf{I}}_k - \tilde{\mathbf{I}}^*\right\|.
\label{eq:toolActionTappingMax}
\end{split}$$ If $\left\| \tilde{\mathbf{I}}_k - \tilde{\mathbf{I}}^*\right\|$ has multiple maxima, we randomly choose one. Eventually – at the following iterations – the other maxima will be selected.
{width="\textwidth"}
Pushing {#subsec:methpush}
-------
In contrast with tapping, in the case of pushing, the affected contours do not always have the same size and the state-action mapping is not as clear as for tapping. This has motivated us to devise a more complex strategy, which was based on the human data recorded in the *PT-Dataset*. In the following, we explain the various steps of this strategy.
### Extracting ground truth state-action triplets from the PT-Dataset {#sect:gettingTriplets}
The first step in designing the pushing actions from (\[eq:actionState\]) is to represent the relationship where a push action ${\mathbf{p}}_{mn}$ causes the kinetic sand contour shape ${\mathbf{x}}^P_m$ to become ${\mathbf{x}}^P_n$. For this, we define *push state-action triplets* of the form: $$\left\{ {\mathbf{p}}_{mn}, {\mathbf{x}}^P_m, {\mathbf{x}}^P_n \right\}.
\label{eq:triplets}$$
We extract these triplets from the *PT-Dataset* images (see Sect. \[Ref:userDis\]). We have two motivations in using these triplets. First, they can help design a local control strategy – inspired by humans – which we will explain in Section \[sect.localShaping\]. Second, we can use them to train a neural network to estimate – at each iteration $m$ – the pushing action ${\mathbf{p}}_{mn}$ required to change the kinetic sand contour from ${\mathbf{x}}^P_m$ to ${\mathbf{x}}^P_n$.
To extract the triplets (\[eq:triplets\]) from the *PT-Dataset*, we look for all the kinetic sand contours that have been obtained after a “sufficiently large” change in both contour and tool positions. Between images $\mathbf{I}_m$ and $\mathbf{I}_n$, we define the change in contour as $\left\| {\mathbf{x}}^P_{n}-{\mathbf{x}}^P_{m}\right\|$ and the change in tool position as $\sqrt{\left(u_n-u_m\right)^2+\left(v_n-v_m\right)^2}$, with $\left( u_{i} , v_{i} \right) \in \left[0, w\right] \times \left[0, h\right]$ the image coordinates of the tool centroid in image $\mathbf{I}_i$, $i = \left\{m,n\right\}$. Therefore, we take all state-action triplets such that: $$\left\{\begin{array}{l}
n > m\\
\sqrt{\left(u_n-u_m\right)^2+\left(v_n-v_m\right)^2} > \tau_u \\
\left\| {\mathbf{x}}^P_{n}-{\mathbf{x}}^P_{m}\right\| > \tau_x.
\end{array}
\right.
\label{eq:desiredContour}$$ where $\tau_u$ and $\tau_x$ are constant, hand-tuned thresholds. Higher values of these thresholds will make criterion (\[eq:desiredContour\]) more selective, requiring bigger changes between the two images.
Let us now detail the image processing pipeline that we use to extract triplets (\[eq:triplets\]) from the *PT-Dataset* images. We process images in the HSV space since it facilitates object segmentation (the tool tip is blue, the users wear black gloves and black long sleeved shirts, the kinetic sand is light brown and the sandbox is black). Among the 14,900 images “pushing” images in the *PT-Dataset*, we choose 3,976 (from four users) where the action and its effect on the kinetic sand are clear. The pipeline, shown in Fig. \[Fig:imgProc\], is as follows (left to right):
1. The 3,976 images are loaded sequentially for each of the users.
2. For each image ${\mathbf{I}}$, we perform a *tool detection*, that yields the tool position as the centroid of a blue blob, segmented in the HSV space. We discard images where the tool is not detected, and output *tool position* $\left( u , v \right)$ for all other images.
3. After processing all images, the tool positions are first smoothened (we apply a weighed centered average filter of size $3$ and weights $\left[1 \quad 2 \quad 1\right]$) and then differentiated to obtain the *tool velocity* on image ${\mathbf{I}}$: $\left( \dot{u} , \dot{v} \right)$.
4. The original image sequence is reduced and broken into smaller sets, each corresponding to a clear tool motion. This is done by detecting images where the tool has either stopped (velocity norm $\sqrt{\dot{u}^2 + \dot{v}^2} < 1$ pixel/image) or changed direction (negative scalar product between consecutive velocities). We use these images as *breakpoints* to split the sequence into smaller sets. Within each of these sets, the tool velocity does not change direction. We also discard images where the tool has stopped. At the end of this step, we obtain 139 sets of images.
5. Each of these 139 sets is processed by a *contour detection* algorithm. First, we subtract final (${\mathbf{I}}_{fin}$) from initial (${\mathbf{I}}_{in}$) image in each set, to generate a difference image. In the luminosity channel of this image, lighter pixels correspond to higher differences. Then, we segment the largest blob of gray pixels in this image, and find its enclosing rectangular bounding box. This defines a ROI (Region of Interest) wherein the kinetic sand configuration has changed the most. Within this ROI, we detect *on each image* ${\mathbf{I}}$ of the set the kinetic sand contour. This is done via the border following algorithm proposed by [@suzuki1985topological], and implemented in the OpenCV function. We then sample the kinetic sand contour with constant $N$ (here, $N = 10$), to obtain $\mathbf{x}^P$ as in (\[eq:pushState\]). We discard images where the contour is not detected.
6. At this stage, we have obtained 2,749 sample images ${\mathbf{I}}$ distributed over 139 sets. The number has diminished from the original 3,976 since we have removed all images where the contour is not detected, or where the tool is not moving. For each ${\mathbf{I}}$, we now have the tool position $\left( u \; v \right)$ (from step 2) and contour $\mathbf{x}^P$ (from step 5). Each set is now explored to find all pairs (2-combinations) of images $\left( {\mathbf{I}}_{m}, {\mathbf{I}}_n \right)$ with sufficient contour and tool change, *i.e*, pairs that comply with (\[eq:desiredContour\]). To this end, we must first match the sample points in ${\mathbf{x}}^P_n$ to those in ${\mathbf{x}}^P_m$; we do this by reordering the points in ${\mathbf{x}}^P_n$, so that the sum of distances between matched pixel pairs (in ${\mathbf{x}}^P_m$ and ${\mathbf{x}}^P_n$) is minimal. After this, we can check if pair $\left( {\mathbf{I}}_{m}, {\mathbf{I}}_n \right)$ complies with (\[eq:desiredContour\]). The output of this last step is the set ${\cal{T}}$ of triplets of the form (\[eq:triplets\]). As mentioned above, each of the 139 sets corresponds to a clear tool motion, roughly 50% of which are pushing actions and the rest ‘retiring’ actions (the user moves the tool away from the kinetic sand between one push and the next). By checking the contour change in (\[eq:desiredContour\]), we discard the images of all these ‘retiring’ actions. By checking the tool position change on all combinations of pairs ${\mathbf{I}}_{m}$, ${\mathbf{I}}_{n}$ ($n>m$) in (\[eq:desiredContour\]) we augment the data: from a single human pushing action between ${\mathbf{I}}_{in}$ and ${\mathbf{I}}_{fin}$ (*i.e.*, one triplet), we can generate many consistent pushes. Ideally (*i.e.*, if the three conditions (\[eq:desiredContour\]) are always met) for a set containing $N_p$ images, we will generate $\frac{N_p!}{2! \left(N_p-2\right)!}$, instead of just 1 triplet. For instance from $N_p=50$ images, we can generate 1,225 triplets. On the other hand, this is an ideal case, since consecutive images generally are too similar to meet (\[eq:desiredContour\]). Using $\tau_u = 5$ and $\tau_x = 3$ pixels in (\[eq:desiredContour\]), from the original 139 sets, we obtain ${\cal{T}}$, with $\dim\left({\cal{T}}\right)=$ 3,539 triplets.
Finally, we derive two important statistical metrics from the set of triplets ${\cal{T}}$. These are useful to characterize the users’ actions, and therefore design the local shaping strategy that will be outlined in Sect. \[sect.localShaping\].
The first metric is the mean of the distance between pairs of pixels matched on all contours. For a pair of contours $\mathbf{x}^P_m$ and $\mathbf{x}^P_n$, the distance is: $$d\left( \mathbf{x}^P_m, \mathbf{x}^P_n \right) = \frac{1}{N} \sum^N_{i=1} \sqrt{\left(u_{i,n}-u_{i,m}\right)^2+\left(v_{i,n}-v_{i,m}\right)^2}.$$ Considering all pairs $\left\{\mathbf{x}^P_m, \mathbf{x}^P_n\right\} \in {\cal{T}}$, its mean is: $$\mu\left(d\right) = \frac{1}{\dim\left({\cal{T}}\right)} \sum_{\mathbf{x}^P_m, \mathbf{x}^P_n \in {\cal{T}}} d\left( \mathbf{x}^P_m, \mathbf{x}^P_n \right).$$ We obtain $\mu\left(d\right) = 42$ pixels, with standard deviation $\sigma\left(d\right) = 10$ pixels.
The second metric is the height (dimension along the image $v$-axis) of the smallest bounding box enclosing both contours $\mathbf{x}^P_m$ and $\mathbf{x}^P_n$, averaged over all contours in ${\cal{T}}$. For a pair of contours $\mathbf{x}^P_m$ and $\mathbf{x}^P_n$, this height is: $$\begin{array}{ccc}
\Delta v \left( \mathbf{x}^P_m, \mathbf{x}^P_n \right) &=& \max \{v_{m,1} \dots, v_{m,N}, v_{n,1}\dots, v_{n,N}\} - \\ && - \min \{v_{m,1} \dots, v_{m,N}, v_{n,1}\dots, v_{n,N}\}.
\end{array}$$ Considering all pairs $\left\{\mathbf{x}^P_m, \mathbf{x}^P_n\right\} \in {\cal{T}}$, its mean is: $$\mu\left(\Delta v\right) = \frac{1}{\dim\left({\cal{T}}\right)} \sum_{\mathbf{x}^P_m, \mathbf{x}^P_n \in {\cal{T}}} \Delta v\left( \mathbf{x}^P_m, \mathbf{x}^P_n \right).$$ We obtain $\mu\left(\Delta v \right) = 100$ pixels, with standard deviation $\sigma\left(\Delta v \right) = 22$ pixels.
### From global to local shaping: {#sect.localShaping}
As mentioned above, we restrict the pushing actions to be along a line segment in the image plane. Although this choice limits the action space, there are still many possible choices as to where on the kinetic sand to start acting. An algorithmic approach is needed to choose the parts of the contour where to push.
Since plastic deformations of the kinetic sand are local, we opt for a local strategy. Instead of defining the push action ${\mathbf{p}}_k$ as a function of the desired contour ${\mathbf{x}}^{P*}$ (*i.e.*, a function of the desired image $\mathbf{I}^*$) as indicated in (\[eq:actionState\]), we focus on local areas of the image, by reducing the workspace along both the $u$ and $v$ axes of the image, to define an alternative desired contour ${\mathbf{x}}^{P\star}$.
![Local strategy for realizing the pushing action, given the current (${\mathbf{I}}_k$) and desired (${\mathbf{I}}^*$) images. a) Indentification of the rectangular ROI (blue) where the robot should act on the kinetic sand. This is obtained from the subtraction of current and desired images. b) Selection of a random ROI of lower height, resized according to the users dataset (green) and detection of sampled contours within this ROI. The contours are detected and sampled on both current (blue) and desired (red) images. c) Interpolation between current (blue) and desired (red) contours according to the users dataset, to obtain the *near* contour ${\mathbf{x}}^{P\star}$ (yellow).[]{data-label="fig:locStrategy"}](locStrategy.pdf){width="\columnwidth"}
This is done through the following steps (see Fig. \[fig:locStrategy\]):
1. *ROI Indentification:* This step consists in identifying a rectangular ROI (blue in Fig. \[fig:locStrategy\]a) where the robot should modify the most the kinetic sand. First, we subtract desired (${\mathbf{I}}^*$) from current (${\mathbf{I}}_k$) images. The following operations (blob segmentation and bounding box derivation) are identical to those presented in step 5 of the pipeline of Sect. \[sect:gettingTriplets\], to identify a ROI on images in the user dataset.
2. *ROI Clipping:* Especially at the beginning of shape servoing, since images $\mathbf{I}$ and $\mathbf{I}^*$ (and therefore features $\mathbf{x}$ and $\mathbf{x}^*$) are most likely very different, the state error can be very large. In other words, it is unlikely that a single action would lead to the desired shape, jeopardizing convergence. Our approach is to operate in a local window by limiting the maximum ROI height (*i.e.*, its dimension along the image $v$-axis). Clipping the ROI limits the state space – the possible combinations of current and desired contours – hence makes it easier to find an action. To determine the size of the clipped ROI, we draw inspiration from the human behavior (the *PT-Dataset*). From the original ROI, we define the set of all smaller rectangular ROIs which have the same width as the original ROI, but height equal to that of the average bounding box in the dataset, $\mu\left(\Delta v \right) = 100$ (see Section \[sect:gettingTriplets\]). If the ROI $v$-axis size is smaller than this threshold, we do not clip the ROI. An example of clipped ROI is shown in green in Fig. \[fig:locStrategy\]b.
3. *ROI random selection:* The natural question that arises from the previous step is which of the clipped ROIs should be selected. Our user study did not indicate any clear preference in humans as to how they choose where to apply the push actions. This means that there seems to be no clear predefined or preferred strategy as to where they start pushing, given multiple options. Moreover, in general it is safe to assume that there is more than one sequence of actions that brings the material to the desired shape. Given these observations, we decided to randomly choose the ROI within the set of clipped ROIs output at the previous step. The randomization strategy also has the benefit of getting the algorithm out of local minima. We noticed that if a fixed heuristic strategy is used here (for example always choosing the topmost among all clipped ROIs), then the same action is likely to be repeated indefinitely, without inducing much change in the kinetic sand shape. ROI randomization also helps the robot exit situations where some actions do not induce progress, e.g., due to servoing or actuation errors.
4. *Contour detection:* The selected clipped ROI is applied to both the current image ${\mathbf{I}}_k$ and the desired image ${\mathbf{I}}^*$, to detect – once again using the OpenCV function – the kinetic sand contour in both images: ${\mathbf{x}}^P_k$ and ${\mathbf{x}}^{P*}$ (with the same number of sample points, $N=10$ as used for processing the dataset). These contours are respectively blue and red in Figure \[fig:locStrategy\].
5. *Contour interpolation:* Although the ROI has been reduced by clipping, the difference in contours can still be large, especially at the start of shape servoing. This makes it hard or impossible to reach the desired kinetic sand shape with a single push. We therefore scale the distances between contour samples by interpolating between the current and desired contours. If the distance between ${\mathbf{x}}_k$ and ${\mathbf{x}}^*$ is higher than the dataset average distance $\mu\left(d\right) = 42$ pixels (see Sect. \[sect:gettingTriplets\]), we scale it to obtain the *near* desired kinetic sand contour: $${\mathbf{x}}^{P\star} = \left\{
\begin{array}{l}
{\mathbf{x}}^{P*} \hspace{2.1cm} \text{if } d\left( \mathbf{x}^P_k, \mathbf{x}^{P*} \right) \leq \mu\left(d\right),\\
{\mathbf{x}}^P_k + \frac{\mu\left(d\right)}{d\left( \mathbf{x}^P_k, \mathbf{x}^{P*} \right)} \left({\mathbf{x}}^{P*} - {\mathbf{x}}^{P}_k \right)\quad \text{otherwise. }
\end{array}
\right.$$ Contour ${\mathbf{x}}^{P\star}$ is shown in yellow in Fig. \[fig:locStrategy\]c. In our work, there is no easy way for the robot to bring the kinetic sand back if the applied push is deeper than intended. A shorter push has the advantage of reducing the uncertainty on the successive kinetic sand shape after the push is applied. This design choice, however, comes at the cost of execution time: the robot will need to apply more pushing actions since the maximum push distance is limited.
![Alternative strategies for defining the push action (red arrow) given the current (blue) and near (yellow) contours. Left to right: maximum, average, and learning-based strategies.[]{data-label="fig:pushingMethods"}](strategies.pdf){width="\columnwidth"}
The interpolated contours ${\mathbf{x}}^{P\star}$ that are extracted from the clipped and randomized ROIs are used as input to the three different pushing strategies explained in the next section.
{width="\textwidth"}
### Strategies {#sect:strategies}
We devise three methods to design the pushing action based on ${\mathbf{x}}^{P}_k$ and ${\mathbf{x}}^{P\star}$: $${\mathbf{p}}_k = {\mathbf{p}} \left( {\mathbf{x}}^P_k, {\mathbf{x}}^{P\star} \right).
\label{eq:mapping}$$ The three methods (shown in Fig. \[fig:pushingMethods\]) are:
1. **Maximum:** Given the sampled contours ${\mathbf{x}}^{P}_k$ and ${\mathbf{x}}^{P\star}$, the pushing action starts and ends at the pair of matched pixels that are the farthest (have highest Euclidean distance, among all matched pairs), on the two contours. In a nutshell, the maximum method pushes in the location where the current and desired sampled contours are the farthest. More formally, the push action is defined as $$\begin{array}{l}
{\mathbf{p}}_k \left( {\mathbf{x}}^P_k, {\mathbf{x}}^{P\star} \right) = \left[ \begin{array}{l}
u_{j,k} \\
v_{j,k} \\
u_{j,\star} \\
v_{j,\star} \\
\end{array}
\right] \text{ such that }
\\
j = \underset{i = 1 ... N}{\mathrm{argmax}}
\sqrt{\left(u_{i,k}-u_{i,\star}\right)^2+\left(v_{i,k}-v_{i,\star}\right)^2}
\end{array}
\label{eq:toolActionPushMax}$$ If $\sqrt{\left(u_{i,k}-u_{i,\star}\right)^2+\left(v_{i,k}-v_{i,\star}\right)^2}$ has multiple maxima, we randomly choose one.
2. **Average:** While the previous method is greedy and aims at acting on the farthest contour points, with the Average method the robot pushes along the contours’ centroids. Thus, the push action is defined as
$${\mathbf{p}}_k \left( {\mathbf{x}}^P_k, {\mathbf{x}}^{P\star} \right) =
\frac{1}{N} \sum^N_{i=1} \left[
\begin{array}{c}
u_{i,k} \\
v_{i,k} \\
u_{i,\star} \\
v_{i,\star} \\
\end{array}
\right].
\label{eq:toolActionPushAvg}$$
3. **Learning-based:** We trained an Artificial Neural Network (ANN) using the 3,539 triplets in ${\cal{T}}$ (see Sect. \[sect:gettingTriplets\]). This ANN learns the mapping in (\[eq:mapping\]) from current and desired states to the pushing action to perform: $${\mathbf{p}}_k = {\mathbf{p}}_{\mbox{ANN}}(\mathbf{x}^P_k, \mathbf{x}^{P\star}).
\label{eq:toolActionPushANN}$$
The input to the network is the current contour ${\mathbf{x}}^{P}_k$ and the target contour ${\mathbf{x}}^{P\star}$. With $N=10$ contour samples, the input consists of 40 scalars (the image coordinates of the sample pixels on the two contours). The output of the network is the action $\mathbf{p}_k$, which consists of 4 scalars, *i.e.*, the pixel coordinates of the start and end pixels, see (\[eq:toolActionPushing\]).
The network architecture consists of 3 fully connected layers with 100 hidden nodes each. Early experiments conducted to choose the network architecture showed small differences. However, we found that on our particular dataset, deeper networks quickly resulted in diminishing returns (and eventually overfitting). Thus, we chose a network with sufficient capacity to perform well, but not overfit. The standard ReLu activation is used for all layers, since it is computationally efficient while modeling non-linearities (see [@glorot2011deep]). We randomly split the set ${\cal{T}}$ of 3539 sample triplets into training ($\%80$), test ($\%10$) and validation ($\%10$) sets. The network is trained for 25,000 episodes where loss has converged. The training takes about fifteen minutes, and the mean absolute testing errors on the output variables are shown in Table \[table:neural\_network\_errors\].
As shown in the table, the testing error for all output variables is within reasonable bounds, considering that the tool has a size of $30 \times 40$ pixels. Note that the trained network predicts the tool end position much better than the tool start position. This is a very interesting result. Clearly, the effect of a push on the contour is much more dependent on the tool end position (when it is in contact with the contour) than on its start position. The results show that the ANN has inferred this characteristic of the pushing action from the human dataset.
Output Mean Error (in pixels)
-------- ------------------------
$u_S$ $15.4$
$v_S$ $12.5$
$u_E$ $3.1$
$v_E$ $1.3$
: Mean prediction error of the neural network trained to output pushing actions ${\mathbf{p}} = \left[ u_S \; v_S \; u_E \; v_E \right]^\top$ given current and desired contours.[]{data-label="table:neural_network_errors"}
Experimental Setup {#sect:setup}
==================
Objectives
----------
We have run a series of experiments on a robotic manipulator to validate our methods. The robot should mold the kinetic sand into desired shapes. At each iteration $k$, the molding action is determined by comparing current and desired images of the kinetic sand shape.
The experiments rely on the hypotheses defined in Sect. \[Sect:problem\]. ** has been used to design the whole methodology (presented in the previous sections) that the robot uses to mold the kinetic sand. The objective of the experiments is to verify this Hypothesis, i.e., to check whether designing the actions according to the kinetic sand state ((\[eq:toolActionTappingMax\]) for tapping and either (\[eq:toolActionPushMax\]), (\[eq:toolActionPushAvg\]) or (\[eq:toolActionPushANN\]) for pushing) is effective in shaping the kinetic sand. To deal with **, we use as desired shapes only those feasible by the robot: matter cannot be added nor pulled, and the kinetic sand must be in the accessible robot workspace. As for **, a human operator selects at each iteration $k$ the action to be executed (either pushing and tapping).
To evaluate the quality of molding, we use mutual information error (\[eq:mutInfoError\]), which we calculate via the MATLAB Toolbox by [@MImatlab:07]. The software used for the experiments is available online at: <https://github.com/acosgun/sand_manipulation>. The experiments are shown in the videos available here: <https://www.lirmm.fr/recherche/equipes/idh/flexbot>.
Workspace and material
----------------------
The experimental setup is shown in Fig. \[fig:expSetup\] and detailed here. In all experiments, we use a Kinova Mico arm[^3] with 6 degrees of freedom. The tool mounted on the robot end-effector has been custom-designed and 3D-printed, so that it can be used by the robot for both pushing and tapping. We have designed the tool so that it is as similar as possible – considering the 3D printer constraints – to the tools used by the humans in the pilot study (see Fig. \[Fig:setup\], left). Its final shape is that of a poker with a connection to the robot. The Intel RealSense camera – which points downward at the sandbox – is the same as in the user study. We position the camera so that it can view the whole sandbox as well as the tool. The walls and bottom of the sandbox are rigid (cardboard). We place a foam support under the sandbox; this was needed to avoid breaking the sandbox in the preliminary experiments, when the software had some instabilities. Considering its kinematics and workspace, the robot can access and manipulate only a portion of the kinetic sand (highlighted in Fig. \[fig:expSetup\] as the hatched area). Thus, we define the desired shapes only in this area of the sandbox.
We perform all computations (image processing, control and machine learning) on the CPU (Intel i7) of a Linux based computer, and use ROS Indigo along with the official Kinova-ROS packages for robot communication and control. We use OpenCV 3.0 for image processing and PyTorch to design and run the neural network.
Robot control for pushing and tapping {#sec:robControl}
-------------------------------------
We decided to control only the translational components of the tool operational space velocity. With reference to the robot frame shown in Fig. \[fig:expSetup\], $\mathbf{v}_x$ and $\mathbf{v}_y$ (the velocity components that are parallel to the sandbox plane) are controlled using visual servoing, while $\mathbf{v}_z$ is controlled to regulate the tool height along the $z$-axis. Since the tool can be easily detected as the centroid of a blue blob, IBVS provides a robust and elegant way of regulating it in the image. Furthermore, since the image plane is parallel to the $xy$ plane, the formulation of the interaction matrix is simple and the scene depth and camera focal length can be included directly in the control gain. In a nutshell, to drive the tool centroid from the current pixel $\left(u, v\right)$ to the desired one $\left(u^*, v^*\right)$ we can simply apply: $$\left[
\begin{array}{c}
\mathbf{v}_x \\
\mathbf{v}_y
\end{array}
\right] = \overline{\mathbf{v}}_{xy} \widehat{
\left[
\begin{array}{c}
- u^* + u\\
v^* - v
\end{array}
\right]
}
\label{eq:visServo}$$ where the minus sign on the first component is due to the orientation of image and robot frame axes (see Fig. \[fig:expSetup\]), and the $\hat{}$ symbol indicates error vector normalization. We introduce this to avoid asymptotic convergence (in which case the velocities become very small as the error diminishes) and to maintain the velocity norm in the $xy$ plane constantly equal to pre-tuned value $\overline{\mathbf{v}}_{xy}>0$. Simultaneously, we regulate the tool height to $z^*$ using: $$\mathbf{v}_z = \overline{\mathbf{v}}_z \operatorname{sign}{\left( z^* - z \right)}.
\label{eq:posZ}$$ As for $\mathbf{v}_x$ and $\mathbf{v}_y$, the $\operatorname{sign}$ function is introduced to avoid asymptotic convergence and to maintain the velocity norm constant and equal to pre-tuned value $\overline{\mathbf{v}}_{z}>0$.
We can apply control laws (\[eq:visServo\]) and (\[eq:posZ\]) on each acquired image, until the visual error is below some threshold, at which point the task is deemed finished.
![Steps of the IBVS control scheme for pushing the kinetic sand, here using action $\mathbf{p}$ (red). The robot tool is blue and the waypoint pixels (H, B, S, E and U) are white.[]{data-label="fig:wayPoints"}](wayPoints.pdf){width="\columnwidth"}
For tapping, the desired pixel $\left(u^*, v^*\right)$ is set to ${\mathbf{t}} = \left( u_T \; v_T \right)$ derived from (\[eq:toolActionTappingMax\]). For pushing, the tool must move sequentially first to the start pixel $\left( u_S \; v_S \right)$, then to the end pixel $\left( u_E \; v_E \right)$, which are defined according to either (\[eq:toolActionPushMax\]), (\[eq:toolActionPushAvg\]) or (\[eq:toolActionPushANN\]).
Yet, for both actions, waypoints are needed to guarantee that the tool does not touch the kinetic sand nor occlude the camera view when it is not intended to. Both the pushing and tapping motions start and end at a constant home pixel $H$, placed on the sandbox far from the kinetic sand. It is only when the motion is finished and the tool has returned at $H$, that the acquired image is processed to determine the next action and the iteration index $k$ is increased. Since the tool is at $H$, there is no risk of camera occlusion and the kinetic sand has stopped moving. On the other hand, the images acquired by the camera while the tool moves between the waypoints are only used to drive the IBVS control scheme according to (\[eq:visServo\])-(\[eq:posZ\]), not to determine the next $\mathbf{p}$ or $\mathbf{t}$.
For pushing, these waypoints are shown in Fig. \[fig:wayPoints\]. The second waypoint $B$ is placed on the same pixel row as the start pixel $S$, but on the side opposite to the kinetic sand. This waypoint is indispensable to avoid accidentally hitting the kinetic sand while moving from $H$ to $S$. After $B$, the tool moves to $S$ and then $E$ (this is the actual *push action* $\mathbf{p}$). We move the tool along these waypoints, using only control law (\[eq:visServo\]), and setting $\mathbf{v}_z = 0$. Then, the tool is raised to a waypoint $U$ placed at higher $z$; for this, we only apply control law (\[eq:posZ\]) on $\mathbf{v}_z$, and set $\mathbf{v}_x = \mathbf{v}_y = 0$. Finally, the tool is brought back to the home pixel, using both (\[eq:visServo\]) and (\[eq:posZ\]). Similarly, to tap the kinetic sand, the tool must first rise from $H$ to a given height (above kinetic sand level), then translate to a waypoint above $T$ (defined by the tapping action $\mathbf{t}$), lower to $T$, rise again, and finally return to $H$.
[0.9]{} {width="\textwidth"}
[0.9]{} {width="\textwidth"}
[0.9]{} {width="\textwidth"}
{width="\textwidth"}
Experimental Results {#sect:experiments}
====================
We have run a number of experiments to test the two actions first separately, and then together. In all cases, the human operator must press a key to launch the action at each iteration $k$. He also terminates the experiment when he considers that the performance cannot further improve. This is a subjective choice, that we would like to avoid, by finding an objective termination condition – related to the image error $e_k$ – for the robot to stop autonomously. In the experiments with both pushing and tapping together, the human also selects – via the keyboard – which action to perform at each iteration $k$, depending on the current kinetic sand state. In all the experiments, the desired shapes $\mathbf{I}^*$ are images acquired after a human has shaped the kinetic sand by hand, trying at best to guarantee ** (feasible shape).
Pushing experiments {#sect:pushXP}
-------------------
The results of the pushing experiments are shown in Figures \[fig:CESexperiments\] and \[fig:mutInfoPush\]. We used three desired shapes (reminiscent of letters C, E and $\Sigma$) framed in green in Fig. \[fig:CESexperiments\]. For each shape, the robot has to mold the kinetic sand using the three strategies: maximum (\[eq:toolActionPushMax\]), average (\[eq:toolActionPushAvg\]) and learning-based (\[eq:toolActionPushANN\]). In all nine experiments, we start from an initial shape with straight contour (red framed in Fig. \[fig:CESexperiments\]).
Figure \[fig:CESexperiments\] shows the sequence of images obtained after each pushing action, using the maximum strategy. The final shapes obtained by the robot are shown in the image preceding the green framed ones. The differences are mainly due to the tool resolution and to the fact that we decided not to control the tool orientation. In Fig. \[fig:mutInfoPush\] we plot the mutual information error $e_k$ from (\[eq:mutInfoError\]) at each iteration $k$, for each desired shape (left to right: C, E and $\Sigma$) and each strategy (maximum in blue, average in red and learning-based in black). This experiment is also useful to see if it is possible to identify a termination condition for the molding. Inspired by (\[eq:errorDecrease\]), we test the following condition: the robot must stop when the mutual information error increases, *i.e.*, at iteration $k+1$ such that $e_{k+1} > e_k$. The solid curves show the values of $e_k$ until this condition is verified, whereas the dashed plots represent the error values until manual termination by the operator[^4].
Let us first comment the solid curves. Note that even for the same desired shape, since the initial image is not identical in the three strategies, the initial values are slightly different. Nevertheless, for all nine plots the general trend is decreasing. We also note that the strongest slope – at least in the initial iterations – is obtained with the learning-based approach (black). The second best is maximum (blue), followed by average (red). However, for shapes E and $\Sigma$, the learning-based strategy is terminated earlier than the maximum one, which ends up with the lowest overall error (0.42 for E shape, and 0.27 for $\Sigma$). The reason is most probably due to the data distribution: the neural network has not been trained on small contour variations and is therefore less efficient in such situations. Among the three methods, average (red) is the worse. This could be expected, since this heuristic is easily driven into local minima (particularly for very convex shapes, as E and $\Sigma$).
Comparing the dashed and solid curves, it is noteworthy that most strategies continue to reduce error $e_k$ even after having reached the termination condition. This is probably due to the complexity of the task, which can be seen as a non-convex optimization problem. In brief, an action may occasionally increase $e_k$ (for an iteration), but on a longer time horizon the error can still diminish. This leads to questioning the convergence condition (\[eq:errorDecrease\]), which may be too strict for the shape servoing task.
Tapping experiments {#Sect.tapExp}
-------------------
To assess the tapping strategy we have run three experiments, depicted in Fig. \[fig:tapExperiments\] from left to right. The tool size is $w_{TCP}\times h_{TCP} = 30 \times 40$ pixels. The experiments are characterized by different initial (upper row of images) and desired (middle row) conditions on the kinetic sand height, while the contours were kept unchanged. On the lower row of the figure we show the images obtained by the robot after having applied tapping strategy (\[eq:toolActionTappingMax\]) for a few iterations. The termination condition is given by the human operator when he esteems that there will be no further improvements (*i.e.*, after respectively 30, 19 and 24 images).
Experiments 1 and 3 have similar desired images (both require flattening the whole accessible workspace) but differ in the initial image. Experiment 1 starts with a pile of kinetic sand in the upper part of the image, whereas Experiment 3 starts with two smaller piles on each side of the workspace. Experiment 2 starts with loose kinetic sand and the desired task consists in flattening a square in the upper part of the image. As the final images show, in Experiments 1 and 3 the robot manages to flatten the piles, whereas in Experiment 2 it acts over the whole workspace and not only on the specified square. The mutual information error does not improve during any of the three experiments. In our opinion, this is due to two reasons indicated below.
- Even more than for pushing, because of the tool size and of the robot characteristics, the robot cannot tap on the kinetic sand as accurately as a human. Typically, since the constant tool height reference $z^*$ is not related to the – varying but unmeasurable – kinetic sand height, the tool often penetrates the kinetic sand, and leaves a footprint with a rectangular shade that appears in the robot images, but not in the human ones (compare bottom and middle row of images). We also tried replacing position control with force control along the $z$ axis, but the Kinova embedded force sensing is not accurate enough for this.
- Feature $\tilde{\mathbf{I}}$ is not the best feedback for this action. Indeed, since the goal of tapping is to modify the kinetic sand shape along directions perpendicular to the image plane, the best feedback would rather be a point cloud from a depth image. Yet, as we mentioned, the RealSense depth image is not exploitable. The RGB image cannot characterize directly the kinetic sand depth, because of effects such as the material granularity and shades. This confirms the formidable capacity of the human sensorimotor system, which through stereovision and experience alone, can mold very precisely along the depth axis. Despite the questionnaire results, probably haptic feedback also plays a more important role here, than it does for pushing.
Although $\tilde{\mathbf{I}}$ and generally RGB data is inappropriate for tapping, the designed action does properly level the kinetic sand. This is visible in Fig. \[fig:tapSide\], where we show a side view of Experiment 1, with the desired shape (top) and a sequence of nine consecutive snapshots during the experiment (bottom). The figure clearly shows that the kinetic sand level is gradually reduced where required, and that a side view such as this one would be much more useful – as feedback signal for tapping – than the top view used in our work.
Also note that tapping on the kinetic sand boundaries may cause its expansion and alter the shape contours (see the bottom row of images in Fig. \[fig:tapExperiments\]). Nevertheless, our framework can overcome this issue by alternating pushing and tapping actions until convergence of the overall error. In the next Section, we present the results of experiments where we alternate between pushing and tapping.
Experiments requiring both pushing and tapping {#Sect.pushtapExp}
----------------------------------------------
Finally, we have run three experiments requiring both pushing and tapping actions, and depicted in Fig. \[fig:pushtapExperiments\] from left to right. Since our framework is not yet capable of autonomously selecting the best action type at each iteration (**), the human operator chooses it using the keyboard. Among the pushing strategies, he can also choose between *maximum* and *learning-based*, since these performed better than *average* in the experiments of Sect. \[sect:pushXP\]. In short, at each iteration of these three experiments, the operator can select from the keyboard between: tap action, push action using maximum strategy and push action using learning-based strategy.
The experiments are characterized by different initial (upper row of images) and desired (middle row) conditions on both the kinetic sand height and contours. On all initial images the contour is a straight line and on all desired images all the workspace kinetic sand has been flattened. The experiments differ in the desired contour and in the initial height: a wide and low pile of kinetic sand in Exp. PT1, loose kinetic sand everywhere in Exp. PT2, and a thin and high pile in Exp. PT3.
On the lower row of the figure we show the images obtained by the robot after having applied pushing and tapping actions for a few iterations. As the final images show, the three experiments confirm that pushing is more effective than tapping. In fact, we can see qualitatively that all three final contours resemble the desired ones: by pushing, the robot has even corrected the contour expansion effect of tapping, mentioned in Sect. \[Sect.tapExp\]. Instead, in Exp. PT2 the robot has not tapped the whole workspace as required. In Fig. \[fig:mutInfoPushTap\], we have plotted $e_k$ at each iteration for the three experiments. Since the termination condition used in the pushing experiments of Sect. \[sect:pushXP\] seemed too conservative, we have decided to relax it and use $e_{k+1} - e_k > 0.005$. In practice, we tolerate a maximum increase of $0.005$ in mutual information, from one iteration to the next. We plot the curves until this condition met. The curves confirm the results seen in Fig. \[fig:pushtapExperiments\]: in all three cases, using the proposed termination condition, the error is reduced by more than 0.2. Furthermore, the choice of this condition is appropriate, because using $e_{k+1} > e_k$ as in Sect. \[sect:pushXP\] would have interrupted the robot too early on experiments PT1 and PT2, because of the weakness of the tapping action. Using $e_{k+1} > e_k$ instead of $e_{k+1} - e_k > 0.005$, the robot would have stopped: for PT1 after 7 iterations at $e_7 = 0.33$ instead of after 15 iterations at $e_{15} = 0.29$, and for PT2 after 3 iterations at $e_3 = 0.48$ instead of after 8 iterations at $e_{8} = 0.34$.
![Evolution of the mutual information error $e_k$ at each iteration $k$ in experiments with both pushing and tapping, to obtain three desired shapes.[]{data-label="fig:mutInfoPushTap"}](miPlotsPushTap.pdf){width="\columnwidth"}
Discussion {#sect:discussion}
----------
The experiments show two important limitations of our framework: first, it cannot regulate the tool orientation during neither pushing nor tapping; second, it is not capable of autonomous selection of the action type to be applied at each iteration. In the following paragraphs we propose solutions to these two problems. We also comment on the robustness of our framework with respect to the variability of experimental conditions.
### Controlling the Tool Orientation {#sec:toolOrient}
As mentioned in Sect 5.1, if the contact between tool and material can be approximated by a point or by a sphere, the effect of pushing and tapping on the kinetic sand will be invariant to the tool orientation. Since the tool we use in our setup does not fulfill such hypothesis, its orientation will affect the kinetic sand shape. We hereby explain how one could add tool orientation control to our framework.
Among the three tool orientations in space, let us shortly discuss the one around the camera optical axis, which is the easiest to visually measure and control in our setup. For tapping, the definition of the controlled feature $\tilde{\mathbf{I}}$ explicitly accounts for the tool resolution and tolerates that changes in the image $\tilde{\mathbf{I}}$ cannot be smaller than the tool size. Hence, rotating the tool will not enhance the performances. For pushing, orienting the tool is more interesting since it could avoid the differences between obtained and desired images visible, for example, in Figures 9 and 13. The orientation can be included in ${\mathbf{p}}$: $${\mathbf{p}}_k = \left[ u_k \; v_k \; \theta_k \; u_\star \; v_\star \; \theta_\star \right]^\top.
\label{eq:toolActionPushing}$$ One way of designing $\theta_k$ and $\theta_\star$ is by aligning the tool with the translational direction, i.e., setting: $$\theta_k = \theta_{\star} = \operatorname{atan2}\left( u_\star - u_k, v_\star - v_k \right).$$ Alternatively, one can design $\theta_k$ and $\theta_{\star}$ independently from the action start and end pixels. The design will differ for each of the three pushing strategies of Sect. \[sect:strategies\]. For the *maximum* strategy, the most intuitive solution is to take as start and end orientations those of the normals to the contours ${\mathbf{x}}^P_k$ and ${\mathbf{x}}^{P\star}$, at the points where the contours are the farthest. Naming $$n \left( u, v, {\mathbf{x}}^P \right) \in \left] -\pi, \pi \right]$$ the orientation of the normal to contour ${\mathbf{x}}^P$ at pixel $\left( u, v \right)$, the *maximum* strategy would yield: $$\theta_k = n \left( u_{j,k}, v_{j,k}, {\mathbf{x}}^P_k \right) \quad \quad \theta_{\star} = n \left( u_{j,\star}, v_{j,\star}, {\mathbf{x}}^{P\star} \right)$$ such that: $$j = \underset{i = 1 ... N}{\mathrm{argmax}}
\sqrt{\left(u_{i,k}-u_{i,\star}\right)^2+\left(v_{i,k}-v_{i,\star}\right)^2}.
\label{eq:toolActionPushMax}$$ For the *average* strategy, one could take the average orientation of all $N$ normals to contours ${\mathbf{x}}^P_k$ and ${\mathbf{x}}^{P\star}$: $$\begin{array}{l}
\theta_k = \frac{1}{N} \sum^N_{i=1} n \left( u_{i,k}, v_{i,k}, {\mathbf{x}}^P_k \right) \\
\theta_\star = \frac{1}{N} \sum^N_{i=1} n \left( u_{i,\star}, v_{i,\star}, {\mathbf{x}}^{P\star} \right)
\end{array}
\label{eq:toolActionPushAvg}$$ Finally, for the *learning-based* strategy, the start and end orientations should be learned by the Artificial Neural Network using the dataset. This requires extracting the tool orientation from each image in the dataset, i.e., adding such feature to the output of the image processing pipeline in Sect. \[sect:gettingTriplets\].
To make the robot rotate the tool while pushing, the visual servoing controller in Sect. \[sec:robControl\] must also be adapted. Since the current and desired tool orientations are directly measurable in the image, this can be done, as for the components of $\mathbf{v}$, through a feedback controller on the tool angular velocity around the optical axis, $\mathbf{\omega}_z$: $$\mathbf{\omega}_z = \bar{\mathbf{\omega}}_z \operatorname{sign}\left( \theta^* - \theta_k \right).$$ with $\bar{\mathbf{\omega}}_z$ a pre-tuned positive scalar.
### Action selection
It could be possible to automatically choose the action to be realized at each iteration $k$. One way of doing this is by verifying on the current image $\mathbf{I}_k$ which feature “requires the most change” to look as it does in desired image $\mathbf{I}^*$. This can be done by comparing the Euclidean distances between current and desired push- and tap-controlled features. Since these features (respectively contour $\mathbf{x}^P$ and image $\tilde{\mathbf{I}}$) are defined in different sets and expressed in different measurement units, we must include some positive weight $\alpha > 0$ to make the comparison consistent. Note that the tuning of scalar weight $\alpha$ is crucial here. Choosing a high (respectively, low) value will make tapping (respectively, pushing) prevail more often. The value of $\alpha$ could also be learned from the dataset. The described algorithm would look as follows.
**Algorithm:** Automatic selection of action type $\mathbf{a}_k$ at iteration $k$
------------------------------------------------------------------------
Desired image $\mathbf{I}^*$ and current image $\mathbf{I}_k$. Action type $\mathbf{a}_k$ (either push $\mathbf{p}$ or tap $\mathbf{t}$).
------------------------------------------------------------------------
Extract $\mathbf{x}_k^P$ from $\mathbf{I}_k$ and $\mathbf{x}^{P*}$ from $\mathbf{I}^*$ Resize $\mathbf{I}_k$ to $\tilde{\mathbf{I}}_k$ and $\mathbf{I}^*$ to $\tilde{\mathbf{I}}^*$ $\mathbf{a}_k \leftarrow \mathbf{p} \left( \mathbf{x}_k^P, \mathbf{x}^{P*} \right)$ $\mathbf{a}_k \leftarrow \mathbf{t} \left( \tilde{\mathbf{I}}_k, \tilde{\mathbf{I}}^* \right)$ $\mathbf{a}_k$
------------------------------------------------------------------------
### Robustness to variability of the experimental conditions
Since part of the action selection process is randomized, and since the initial and desired images vary from one setup the other, it is difficult to objectively assess the performance of our framework over multiple experiments. To this end, we have processed the results obtained in all $9$ setups (shown in Figures \[fig:CESexperiments\], \[fig:tapExperiments\] and \[fig:pushtapExperiments\]). First, since the three setups in Fig. \[fig:CESexperiments\] have each been tackled with all three pushing strategies (see Fig. \[fig:mutInfoPush\]), for each setup we have averaged the three values of $e_k$ (one per strategy) at each iteration $k$. Now, we have $9$ trends of $e_k$ corresponding to the $9$ setups: $3$ for pushing obtained by averaging as mentioned just above, $3$ for tapping and $3$ for pushing+tapping obtained via the experiments of Sections \[Sect.tapExp\] and \[Sect.pushtapExp\], respectively. Then, we compute the mean and standard deviation of $e_k$ at each iteration $k$ of these $9$ trends. We do so on the first $15$ iterations, since only $5$ of the $9$ experiments have lasted longer, due to the termination conditions. The results are plotted as error bars in Fig. \[fig:barErrorAll\]. As the reader can see, the trend is decreasing, showing that our framework is capable of reducing the mutual information error despite the variability of actions and setups.
![Evolution of the mean and standard deviation of $e_k$ at each iteration $k$ for all $9$ setups and experiments.[]{data-label="fig:barErrorAll"}](barError.pdf){width="0.9\columnwidth"}
Conclusions and future work {#sect:conclusion}
===========================
In this paper, we have addressed the problem of non-prehensile shaping of plastic materials. Inspired by our human study, we have designed two actions, pushing and tapping. Both are realized using image-based visual servoing (neither force nor tactile feedback) to control a tool held by a robot manipulator. We assume that these two actions are specialized: pushing alters the external shape contours, whereas tapping modifies the image.
The key issue is how to relate the parameters of these two actions to the current and next states. While for tapping this seems simple, since the effect is local and has constant size (equal to the tool contact surface dimensions), pushing requires a deeper reflection, which constitutes in our opinion one of the main contributions of the paper. We draw inspiration from the user dataset to derive the main parameters – contour and action size – that humans use when operating. Based on these parameters, we break the global problem of regulating the current image towards the desired one into smaller local problems – in the state space – which can be solved more easily.
We also propose three strategies: two heuristics (maximum and average) and a neural network trained with the dataset images. To compare the approaches, we use mutual information – an objective metric of image similarity. The results show that while slightly outperforming the other strategies on the short run, the neural network quickly settles when the image error is small. Paradoxically, the much simpler maximum strategy is more efficient afterwards (when the error is small). This result is particularly interesting in the light of the current euphoria that surrounds machine learning worldwide. It turns out that for our problem, while requiring a huge pre-processing effort (many images are unusable and deriving push-action triplets is non-trivial, see Sect. \[sect:gettingTriplets\]) the outcome of learning is barely better than that of a simple heuristic, because of the curse of dimensionality. This result per se may discourage the user from going through the data acquisition and pre-processing steps. Nevertheless, these steps are also necessary to infer the human parameters mentioned in the paragraph above, which make the global task separable into smaller local ones. Once data acquisition and pre-processing are done, training the neural network is straightforward, so why not use it afterwards?
All in all, the results show that our framework succeeds in making the robot realize numerous shapes. Nevertheless, we acknowledge the many limitations of our exploratory work. These could be the object of future work of researchers interested by this fascinating topic. A non-exhaustive list is given below to conclude the paper.
- The pushing actions clearly outperform the tapping ones. Possible reasons have been mentioned in Sect. \[Sect.tapExp\]. The main one seems to be the choice of the feature: an image, rather than a point cloud. Yet, with an accurate depth image, tapping could be formulated as pushing in any plane perpendicular to the image. Then, our approach for pushing could be directly applied in any depth plane and would likely succeed in generalization.
- Many limitations are due to the hardware constraints: tool size, design, sensed data. The use of a soft tool or of multiple tools (e.g., fingers) could improve the performance, while raising other interesting research problems.
- In our approach, inspired by model-based control, we map current and desired state to action. Alternatively, one could map current state and action to next state. This second paradigm is more relevant if planning or reinforcement learning were to be applied to this problem.
- It would be useful to integrate haptic feedback (measured by force or tactile sensors) to vision. This would be valuable, e.g., to control the force required to overcome the kinetic sand resistance.
- We have trained the neural network on the user dataset. One could speed up the learning process via self-learning (*i.e.*, the robot acquires new data while it molds the material).
- A larger set of actions could be studied. For instance, the analysis of the user dataset shows that often humans perform hybrid actions (e.g., simultaneous “push-and-tap”), which alter multiple features at once.
- Our user analysis is quite limited (with only nine partakers). While being a world premiere, our public dataset should be enriched by other researchers, particularly cognitive scientists, who could provide their expertise on human studies.
- Our framework is not autonomous in selecting the best action type, given the system state. This problem is of interest not only for deformable object manipulation, but also for other applications that need heterogeneous action sequencing.
- The current version of our framework does not regulate the tool orientation. This feature could be added in future work, by drawing inspiration from the suggestions given in Sect. \[sec:toolOrient\].
This work has been partly funded by the PHC FASIC project FlexBot. Ortenzi, Cosgun and Corke are supported by the Australian Research Council Centre of Excellence for Robotic Vision (project number CE140100016). We would like to thank Steven Martin for the help with the custom 3D printed tool, and El Mustapha Mouaddib, André Crosnier and Juxi Leitner for the constructive discussions.
[^1]: <https://en.m.wikipedia.org/wiki/Kinetic_Sand>
[^2]: <https://cloudstor.aarnet.edu.au/plus/s/Vii90T72WFM8Qwp> (password: sandman)
[^3]: <https://www.kinovarobotics.com/en>
[^4]: Coincidentally, the two termination conditions are identical in the average E-shape experiment (red curve in the center).
|
---
abstract: 'The electronic structure and magnetic moments of free Mn$_2^+$ and Mn$_3^+$ are characterized by $2p$ x-ray absorption and x-ray magnetic circular dichroism spectroscopy in a cryogenic ion trap that is coupled to a synchrotron radiation beamline. Our results show directly that localized magnetic moments of 5 $\mu_B$ are created by $3d^5 (^6\mathrm{S})$ states at each ionic core, which are coupled in parallel to form molecular high-spin states via indirect exchange that is mediated in both cases by a delocalized valence electron in a singly-occupied $4s$ derived orbital with an unpaired spin. This leads to total magnetic moments of 11 $\mu_B$ for Mn$_2^+$ and 16 $\mu_B$ for Mn$_3^+$, with no contribution of orbital angular momentum.'
author:
- 'V. Zamudio-Bayer'
- 'K. Hirsch'
- 'A. Langenberg'
- 'M. Kossick'
- 'A. Ławicki'
- 'A. Terasaki'
- 'B. v. Issendorff'
- 'J. T. Lau'
title: 'Direct Observation of High-Spin States in Manganese Dimer and Trimer Cations by X-ray Magnetic Circular Dichroism Spectroscopy in an Ion Trap'
---
Introduction {#introduction .unnumbered}
============
Manganese is an element with peculiar electronic and magnetic properties. Of all $3d$ transition elements, the manganese atom carries the second largest magnetic moment of 5$\mu_B$ because of the high-spin $3d^5\ (^6\mathrm{S})$ subshell configuration, while bulk manganese has an unusual 58-atom unit cell with noncollinear antiferromagnetic order [@Hobbs03] below the Néel temperature. A similar complex behavior can also be found in manganese molecules and clusters [@Morse86; @Alonso00]. Stern-Gerlach deflection studies of Mn$_n\ (n = 5 - 99)$ cluster beams [@Knickelbein01; @Knickelbein04a] showed superparamagnetism or ferrimagnetism with average magnetic moments that oscillate between $0.4 - 1.7\ \mu_B$ per atom. Heisenberg behavior [@Negodaev08] was postulated for Mn$_2$ and a transition from ferro- to antiferromagnetism [@BobadovaParvanova05; @Kabir06] with increasing cluster size, as well as noncollinear spin structure [@Pederson98; @Longo08; @Zeleny09], was predicted. Combined photoelectron spectroscopy and density functional theory studies [@Gutsev08] found indications [@Jellinek06] for half-metallic Mn$_n$ clusters. Turning to the smallest clusters and molecules, the electronic ground states of molecular manganese cations have been studied by matrix-isolation electron spin resonance spectroscopy [@VanZee81; @Baumann83; @VanZee88; @Cheeseman90] and photodissociation spectroscopy [@Terasaki01; @Terasaki03]. In this size range, Mn$_n^+$ cations are characterized by low dissociation energies [@Ervin83; @Jarrold85; @Terasaki03; @Tono05] of 1.39 eV for Mn$_2^+$ and $0.83 \pm 0.05$ eV for Mn$_3^+$ that increase only slightly to $1.06 \pm 0.03$ eV for Mn$_7^+$. For Mn$_2^+$, the experimental studies agree on a $^{12}\Sigma_g^+$ ground state as do most theoretical studies [@Bauschlicher89; @Nayak98; @Desmarais00; @Wang05] even though a $^{10}\Pi_u$ ground state [@Gutsev03a] is also considered. For Mn$_3^+$ a $^5B_2$ ground state [@Gutsev06] is predicted by theory but photodissociation spectroscopy [@Terasaki03] favors a $^{17}B_2$ high-spin ground state. Local high-spin states [@Lau09b; @Hirsch12b] of Mn$_2^+$ and Mn$_3^+$ are also inferred from x-ray absorption spectroscopy. This illustrates the need for a direct experimental probe for the spin states of size-selected free molecular ions [@ZamudioBayer15]. One interesting aspect of the interplay of magnetism and chemical bonding in manganese is that large magnetic moments could in principle be obtained if the atomic Mn $3d^5\ (^6\mathrm{S})$ high-spin state could be preserved in larger manganese entities and if parallel spin alignment could be achieved by long range ferromagnetic interaction. Here we show experimentally that Mn$_2^+$ and Mn$_3^+$ are characterized by fully occupied majority spin states and local $3d^5\ (^6\mathrm{S})$ high spin terms that couple to 11 $\mu_B$ and 16 $\mu_B$, respectively. Non-collinear spin arrangements can be ruled out for the smallest molecular cations.
Experimental and Computational Details {#experimental-and-computational-details .unnumbered}
======================================
Sample Preparation {#sample-preparation .unnumbered}
------------------
Mn$_2^+$ and Mn$_3^+$ were prepared in situ in a cluster ion beam apparatus [@Lau08; @Hirsch09; @Niemeyer12] by direct-current magnetron sputtering of a high-purity manganese target (99.95 %, Lesker) in a mixed (approx. 5:1 volume flow ratio) helium-argon (99.9999 %) atmosphere of 0.1 - 1 mbar at liquid nitrogen temperature. The magnetron discharge creates neutral and ionic species that grow by gas aggregation. A distribution of Mn$_n^+$ ions was extracted from the ion source and guided through differential pumping stages into a radio-frequency quadrupole mass filter (Extrel) to select either Mn$_2^+$ or Mn$_3^+$ parent ions, which were then stored in a liquid-helium-cooled linear quadrupole ion trap [@Niemeyer12; @ZamudioBayer13; @Langenberg14; @ZamudioBayer15; @Hirsch15a] filled with $10^{-4} - 10^{-3}$ mbar high purity ($> 99.9999$ %) helium buffer gas. The number density of helium atoms in the ion trap is $\approx 7 \cdot 10^{13} - 9 \cdot 10^{14}$ cm$^{-3}$ at our experimental parameters. Under these conditions, vibrations and rotations are thermalized to equilibrium on a time scale of micro to milliseconds [@Gerlich95; @Gerlich09; @Otto13; @Hansen14; @Boyarkin14]. The ion trap was continuously filled with parent ions to the space charge limit. Typical storage times of the parent ions in the ion trap were $1 - 10$ s. This excludes the possibility of trapping metastable configurations [@Hirsch12a]. The purity of the parent ions in the ion trap was verified by reflectron time-of-flight mass spectrometry. The homogeneous static magnetic field of a superconducting solenoid [@Terasaki07] (JASTEC) that surrounds the ion trap vacuum chamber was used to magnetize the Mn$_2^+$ and Mn$_3^+$ samples. The inhomogeneity of the applied magnetic field is $\le 1$ % over the entire ion trap volume.
Spectroscopic Technique {#spectroscopic-technique .unnumbered}
-----------------------
X-ray absorption and x-ray magnetic circular dichroism (XMCD) spectroscopy at the manganese $L_{2,3}$ edges of the Mn$_2^+$ and Mn$_3^+$ parent ions was performed inside the ion trap in ion yield mode by monitoring the intensity of Mn$^{2+}$ product ions in both cases. These product ions are generated by dissociation of highly excited intermediates that result from x-ray absorption that is followed by Auger decay of the $2p$ core-excited state of the parent ion. Parent and product ion bunches (a small fraction of the trap filling) were extracted at a rate of $\approx 0.3$ kHz from the ion trap and guided into the acceleration region of the reflectron time-of-flight mass spectrometer for detection. The incident photon energy was scanned across the manganese $L_{2,3}$ absorption edges from $610 - 690$ eV in $250 - 500$ meV steps with 625 meV photon energy resolution. At every photon energy step, the sample was irradiated with monochromatized x-rays for 8 s and the product ion intensity was recorded in a photoionization mass spectrum. For XMCD spectroscopy, a static magnetic field with $\mu_0$H = 5 T was applied along the ion trap axis, with parallel or antiparallel orientation to the photon helicity of the incoming elliptically-polarized soft-x-ray beam. The difference of the spectra that are recorded for negative and positive helicity of the x-ray photons gives the XMCD spectrum, and the average is the isotropic x-ray absorption spectrum. All spectra were normalized to incoming photon flux, detected by a GaAsP photo diode, and were corrected for the 90 % polarization degree of the elliptically polarized soft x-ray photons. The experiments were carried out at beamlines UE52-SGM and UE52-PGM of the BESSY II synchrotron radiation facility at Helmholtz-Zentrum Berlin.
Atomic Hartree-Fock Calculations {#atomic-hartree-fock-calculations .unnumbered}
--------------------------------
To analyze the experimental data, we have calculated the x-ray absorption and XMCD spectra of Mn$^+$ for the \[Ar\]$3d^5\,4s^1$ configuration in the $^7$S ground state and $^5$S excited state term. Dipole accessible $2p^5\,3d^6\,4s^1$ and $2p^5\,3d^5\,4s^2$ final state configurations were taken into account. These Hartree-Fock calculations were performed with the [@Cowan68] as implemented in [@MISSING]. In our calculations the usual scaling-down of the Coulomb and exchange interaction parameters to 85 % of the *ab initio* values was applied in order to account for intra-atomic relaxation effects [@Cowan68]. The calculated spectra were convoluted with a lifetime (Lorentz) broadening of 0.1 eV at the $L_3$ and 0.2 eV at the $L_2$ resonance, and an instrument (Gaussian) broadening with 0.25 eV full width at half maximum is applied to match the experimental photon energy resolution. The increased broadening at the $L_2$ resonance is due to the reduced lifetime of the $2p_{\nicefrac{1}{2}}$ core hole [@vanderLaan91]. The calculated spectra were redshifted by 2.57 eV in order to match the experimental excitation energy, and the *ab initio* $2p$ spin-orbit splitting parameter $\zeta$ is 6.74 eV. Direct $2p$ photoionization was not included in the Hartree-Fock calculation. This causes the offset of the experimental x-ray absorption spectrum (cf. Fig. \[fig:XSpectra\]) at higher excitation energies, but has no effect on the XMCD spectrum.
Results {#results .unnumbered}
=======
Atomic Localization of 3d Electrons in Mn$_2^+$ and Mn$_3^+$ {#atomic-localization-of-3d-electrons-in-mn_2-and-mn_3 .unnumbered}
------------------------------------------------------------
In Fig. \[fig:XSpectra\] the experimental $L_{2,3}$ x-ray absorption and XMCD spectra of Mn$^+_2$ and Mn$_3^+$ are shown along with the corresponding theoretical spectra of Mn$^+$ in its \[Ar\]$3d^5\,4s^1$ $^7$S ground state configuration. All spectra were normalized to the integrated signal of resonant $2p \rightarrow 3d$ transitions, i.e., to the number of unoccupied $3d$ states. This normalization and the good agreement of the spectral fingerprints allows us to directly compare theoretical and experimental XMCD signals in order to obtain information on the electronic ground states of Mn$_2^+$ and Mn$_3^+$. For the Mn$_2^+$ and Mn$_3^+$ molecular ions, the respective XAS and XMCD spectra in Fig. \[fig:XSpectra\] are identical in shape to the calculated spectrum [@Lau09b; @Hirsch12a; @Hirsch12b] of atomic Mn$^+$ in its ground state configuration. This immediately shows that the experimental spectra originate from an unperturbed atomic $3d^5$ $(^6\mathrm{S})$ electronic configuration of the $3d$ subshell, i.e., the $3d$ electrons form local high-spin states but do not or only very weakly participate in bonding [@Lau09b; @Hirsch12a; @Hirsch12b; @ZamudioBayer15] in Mn$_2^+$ and Mn$_3^+$. The fact that the $3d$ orbitals remain atomically localized [@Lau09b; @Hirsch12a; @Hirsch12b; @ZamudioBayer15] has implications for the geometric structure. For unperturbed $3d$ orbitals, the overlap of $3d$ electrons at different nuclei must be very weak. A rough estimate of the interatomic distance [@ZamudioBayer15] in Mn$_2^+$ and Mn$_3^+$ can therefore be made with the atomic manganese $3d$ radial distribution function, from which it can be seen that the radial $3d$ electron density decreases to less than 1 % of its maximum at $r \geq 1.3$ [Å]{}, leading to a corresponding equilibrium distance for pure $4s\sigma$ bonding $r_e(\text{Mn}^+_2) \geq 2.6$ [Å]{}. This estimate is in good agreement with calculated values [@Nesbet64; @Bauschlicher89; @Terasaki01; @Wang05] of $r_e \approx 2.9 - 3.0$ [Å]{} for Mn$_2^+$ and Mn$_3^+$ high-spin states. Thus the equilibrium distance in Mn$_2^+$ is significantly larger than the nearest neighbor distance in bulk manganese [@Hobbs03] of 2.24 [Å]{} but also larger than the typical bond distances of diatomic transition metal cations, e.g., the experimental values [@Asher94; @Yang00] of $r_e(\text{V}^+_2) = 1.73$ [Å]{} and $r_e(\text{Ni}^+_2) = 2.22$ [Å]{}, where $3d$ orbitals participate in molecular bonding.
High-Spin Ground State of Mn$_2^+$ {#high-spin-ground-state-of-mn_2 .unnumbered}
----------------------------------
For the lowest energy dissociation limit of Mn$_2^+$ into Mn $3d^5\,4s^2\ ^6$S and Mn$^+$ $3d^5\,4s^1\ ^7$S, the localized $3d^5\ (^6\mathrm{S})$ high-spin states at the manganese cores and the delocalized single $4s$ derived spin could couple to states with a total spin [@Wigner28] $S$ of $11 \ge 2S \ge 1$ in Mn$_2^+$. In the following treatment we will assume for simplicity that the localized $3d^5\ (^6\mathrm{S})$ states first couple to a total $3d$ spin $S_{3d}$ and then with the single unpaired $4s$ derived spin to give a total spin $S$. This should be a good approximation of the real angular momentum coupling.
![\[fig:XSpectra\] Experimental $2p$ x-ray absorption and XMCD spectra (bullets) of Mn$_2^+$ and Mn$_3^+$ along with x-ray absorption (broken line) and XMCD (solid line) from an atomic Mn$^+$ $3d^5\,4s^1\ ^7$S Hartree-Fock calculation [@Cowan68; @MISSING]. Calculated XMCD spectra are scaled to match the experimental amplitude to give the experimental magnetization. Inset: detailed pre-edge region of experimental and theoretical XMCD spectra. The strong overshoot at 639.5 eV that is observed in the experimental spectra agrees better with the overshoot that is obtained in the calculated spectrum for the initial \[Ar\]$3d^5\, 4s^1$ configuration in the $^7$S term (solid line) than in the $^5$S term (dashed line) of Mn$^+$. This indicates parallel $3d - 4s$ coupling.](MnHighSpinFig1.eps)
Applied to the experimental spectrum, the orbital angular momentum sum rule of XMCD [@Thole92; @Carra93] yields a molecular orbital magnetic moment of $\mu_L = (0.1 \pm 0.4)$ $\mu_B$ for Mn$_2^+$ and clearly indicates a $\Sigma$ ground state. The total angular momentum $J$ of Mn$_2^+$ thus is exclusively spin. Because of the identical atomic $3d^5$ signature in the x-ray absorption and XMCD spectra, the experimental $3d$ spin magnetization can be obtained by fitting the calculated Mn$^+$ XMCD signal to the experimental Mn$_2^+$ spectrum [@ZamudioBayer15; @Hirsch15a] after normalization. This circumvents the empiric correction [@OBrien94b; @Piamonteze09] that would be required to the XMCD spin sum rule [@Carra93] for manganese. From the fit of the calculated XMCD of Mn$^+$ to the experimental spectrum of Mn$_2^+$ in Fig. \[fig:XSpectra\], a total $3d$ spin magnetization of $(0.53 \pm 0.04) \cdot 2\cdot 5\ \mu_B\ = (5.3 \pm 0.4)\ \mu_B$ is determined. This rules out all states with $2S_{3d} \le 4$ and leaves only six possible states with $2S_{3d} = 6\ (2S = 5, 7)$, $2S_{3d} = 8\ (2S = 7, 9)$, and $2S_{3d} = 10\ (2S = 9, 11)$. The temperature at which $S_{3d}$ of these states would reach a magnetization of $(5.3 \pm 0.4)\ \mu_B$ at $\mu_0H = 5$ T is given by the Brillouin function for the total spin $S$. An ion temperature of $\le 7$ K that would be necessary for the $2S_{3d} = 6$ states can be ruled out because the experiment was performed at a temperature of the ion trap of $8 \pm 1$ K and radio-frequency heating of the ions is inevitable at our conditions [@Niemeyer12; @Langenberg14; @ZamudioBayer15]. For a $3d$ magnetization of $(5.3 \pm 0.4)\ \mu_B$ in the two $2S_{3d} = 8\ (2S = 7, 9)$ states, an ion temperature of $11 \pm 2$ K and $14 \pm 2$ K would be required, which would correspond to a radio-frequency heating of $3 \pm 2$ K and $6 \pm 2$ K, respectively. As will be shown below, radio-frequency heating at the conditions of our experiment is $\ge 10 \pm 2$ K for Mn$_2^+$, which rules out both states. The remaining $2S_{3d} = 10\ (2S = 9, 11)$ states with fully parallel alignment of the $3d$ states but antiparallel $(2S = 9)$ or parallel $(2S = 11)$ alignment of the $4s$ spin would correspond to ion temperatures of $20 \pm 2$ K and $23 \pm 2$ K, respectively, and cannot be distinguished by consideration of the very similar radio frequency heating of $12 \pm 2$ K and $15 \pm 2$ K. However, the $2S_{3d} = 10\ (2S = 9)$ state with antiparallel coupling of the $4s$ derived electron spin should be about 1 eV higher in energy than the $2S_{3d} = 10\ (2S = 11)$ state because of the strong intra-atomic $3d - 4s$ exchange coupling in manganese that leads to parallel spin coupling and favors the $3d^5\,4s^1\ ^7$S term over the $3d^5\,(^6\mathrm{S})\,4s^1\ ^5$S term in the ground state of free Mn$^+$ ions [@Sugar85] by 1.17 eV. This argument is very similar to the case of the maximum spin ground state [@ZamudioBayer15] of Cr$_2^+$. Parallel $3d - 4s \sigma$ spin alignment is also indicated at the onset of the $L_3$ line of the experimental XMCD spectrum as shown in detail as an inset to Fig. \[fig:XSpectra\]. Here the experimental spectra of Mn$_2^+$ and Mn$_3^+$ are compared to calculated spectra for $^7$S and $^5$S terms, i.e., for parallel and antiparallel alignment of the $4s$ derived spin. As can be seen, the calculated XMCD spectra differ significantly in the intensity of the overshoot at 639.25 eV, and by the sign of the signal at 638.5 eV. The strong overshoot at 639.5 eV and the dip at 638.75 eV in the experiment agree better with the calculated spectrum for parallel than for antiparallel alignment. In addition to the energy consideration above, this is an experimental indication of parallel alignment of the $4s$ derived spin and leaves only $2S_{3d} = 10\ (2S = 11)$ for the spin in the ground state of Mn$_2^+$. A similar signature of parallel $3d - 4s \sigma$ spin coupling has also been observed in the case of Cr$_2^+$ where it appears not only in XMCD but also in the x-ray absorption spectrum as a well separated line at photon energies below the $L_3$ main line [@Lau09b; @ZamudioBayer15]. This prepeak at the onset of $2p$ excitation of the free molecular ion is a sensitive measure of parallel $3d - 4s$ spin alignment that vanishes upon cluster deposition [@Lau09b; @ZamudioBayer15; @Lau00b; @Lau05; @Reif05]. In conclusion, our results from XMCD spectroscopy provide the first direct experimental evidence for the $^{12}\Sigma^+_g$ ground state of free $\mathrm{Mn}_2^+$ ions, in agreement with electron spin resonance spectroscopy of $\mathrm{Mn}_2^+$ isolated in inert gas matrices [@VanZee88; @Cheeseman90] and in agreement with indirect evidence from photodissociation spectroscopy of free $\mathrm{Mn}_2^+$ ions [@Terasaki01].
High-Spin Ground State of Mn$_3^+$ {#high-spin-ground-state-of-mn_3 .unnumbered}
----------------------------------
The analysis of the spin and orbital angular momentum in the ground state of Mn$_3^+$ follows the same reasoning as for Mn$_2^+$ above. In Mn$_3^+$, the localized $3d^5\ (^6\mathrm{S})$ configurations of the $3d$ subshells can couple with the $4s$ derived spin to a total spin $S$ of $16 \ge 2S \ge 0$ as given by the 2 Mn ($3d^5\,4s^2\ ^6$S) + Mn$^+$ ($3d^5\,4s^1\ ^7$S) lowest energy dissociation limit. We will again assume for simplicity that the localized $3d^5\ (^6\mathrm{S})$ states in Mn$_3^+$ first couple to a total $3d$ spin $S_{3d}$ and then with the single unpaired $4s$ derived spin to give a total spin $S$. Again, the orbital angular momentum sum rule of XMCD [@Thole92; @Carra93] returns a molecular orbital magnetic moment of $\mu_L = (0.06 \pm 0.48)$ $\mu_B$ for Mn$_3^+$, i.e., no orbit contribution to the total angular momentum $J$ in agreement with half-filled $3d^5$ subshells localized at each of the three manganese cores. As for Mn$_2^+$, the total magnetization of the $3d$ spins in Mn$_3^+$ is determined from a fit of the calculated Mn$^+$ XMCD signal to the experimental XMCD spectrum as shown in Fig. \[fig:XSpectra\]. This procedure gives $\left(0.70 \pm 0.04 \right) \cdot 3 \cdot 5\ \mu_B = (10.5 \pm 0.6) \ \mu_B$ as the total $3d$ magnetization and excludes all states with $2S_{3d} \le 9$. Out of the remaining states, those with $2S_{3d} = 11\ (2S = 10, 12)$ and $2S_{3d} = 13\ (2S = 12, 14)$ can be ruled out because these would reach a $3d$ magnetization of $(10.5 \pm 0.6) \ \mu_B$ at $\mu_0H = 5$ T only at ion temperatures of $\le 7$ K and $12 \pm 2$ K, respectively, according to the Brillouin function. These ion temperatures are incompatible with an ion trap temperature of $10 \pm 1$ K and radio-frequency heating that is inevitably present in the experiment. This result indicates fully parallel spin coupling of the localized $3d^5$ subshells to the maximum possible value of $2S_{3d} = 15$ in Mn$_3^+$. In principle, two states with a total spin of $2S = 14$ and $2S = 16$ could be formed by antiparallel or parallel coupling of the $4s$ derived spin with the $3d$ spin $S_{3d}$. Analogous to the case of Mn$_2^+$, antiparallel coupling can be ruled out from energy considerations and from the experimental XMCD spectrum, shown in detail as an inset to Fig. \[fig:XSpectra\]. Again, the state with antiparallel coupling of the $4s$ derived spin to $S_{3d}$ should be $\approx 1.17$ eV higher in energy [@Sugar85] because of the strong intra-atomic $3d - 4s$ exchange interaction that the $4s$ derived electron experiences with the localized $3d$ electrons. Furthermore, the calculated spectrum for the $^7$S ground state of Mn$^+$ with parallel coupling fits the details of the experimental spectrum at the onset of $2p$ excitation better than the $^5$S excited state with antiparallel $4s - 3d$ coupling. The case is is even clearer than for Mn$_2^+$ because of the better signal-to-noise ratio. In summary, the total magnetic moment of Mn$_3^+$ is equal to $16\ \mu_B$ and is purely determined by the electron spin of the molecule. This is in agreement with results from photodissociation spectroscopy [@Terasaki03] of Mn$_3^+$. A total $3d$ magnetization of $(10.5 \pm 0.6) \ \mu_B$ at $\mu_0H = 5$ T for the $2S = 16\ (2S_{3d} = 15)$ state of Mn$_3^+$ corresponds to an ion temperature of $20 \pm 2$ K and thus to a radio-frequency heating of $10 \pm 2$ K at our experimental conditions. Since radio-frequency heating is more pronounced for Mn$_2^+$ than for Mn$_3^+$ because of the lighter mass at otherwise identical conditions, this result serves as an *ex post* justification of the anticipated strong radio frequency heating of Mn$_2^+$ that was made above.
Discussion {#discussion .unnumbered}
==========
Our experimental results show that Mn$_2^+$ and Mn$_3^+$, similar to Cr$_2^+$ (Ref. ), possess electronic ground states with maximum $3d$ spin magnetic moments of 5 $\mu_B$ per atom and fully occupied $4s$ and $3d$ majority spin states. This confirms previous experimental results [@VanZee88; @Cheeseman90; @Terasaki01; @Terasaki03; @Lau09b; @Hirsch12b] on free and matrix-isolated Mn$_2^+$ and Mn$_3^+$ but contradicts the theoretical predictions of a $^5B_2$ ground state [@Gutsev06] for Mn$_3^+$ and a $^{10}\Pi_u$ ground state [@Gutsev03a] for Mn$_2^+$. The mechanism that mediates parallel spin alignment of the $3d^5\ (^6\mathrm{S})$ high spin configurations located at the individual nuclei in Mn$_2^+$ and Mn$_3^+$ is indirect (double) exchange coupling [@Zener51a; @Anderson55] via the spin of a single unpaired $4s$ derived electron. This is identical in both molecular ions. The reason for this indirect exchange coupling is the strong intra-atomic $3d - 4s$ exchange interaction in manganese that leads to an energy difference of 1.17 eV between the $3d^5\,(^6\mathrm{S})\,4s^1\ ^7$S ground state of Mn$^+$ with parallel alignment of the unpaired $3d$ and $4s$ spins, and the $3d^5\,(^6\mathrm{S})\,4s^1\ ^5$S first excited state where the alignment is antiparallel [@Sugar85]. This intra-atomic exchange interaction favors a ferromagnetic coupling of the localized $3d$ shells in Mn$_2^+$ and Mn$_3^+$ because the unpaired $4s$ derived electron can only delocalize over the entire molecular ion with its spin aligned parallel to the $3d$ spins at the atomic sites. This is analogous to spin coupling [@ZamudioBayer15] in the ground state of Cr$_2^+$. Indirect exchange coupling [@Zener51a] was first proposed for Mn$_2^+$ by @Bauschlicher89 who discussed how competing effects of intra-atomic $3d - 4s$ and interatomic $3d - 3d$ exchange coupling in Mn$^+_2$ would result in parallel coupling of the spins of the unpaired $4s \sigma$ electron and both localized $3d$ subshells in the ground state. A similar approach by @Wang05 regards Mn$_2^+$ as a mixed valence system and leads to identical results. One important requirement for indirect or double exchange [@Zener51a] is atomic-like spin correlation in the open $3d$ shell, which is not shown easily in general. Here, our experimental results demonstrate that this atomic-like spin correlation is clearly valid for Mn$_2^+$ and Mn$_3^+$ as for the case of Cr$_2^+$ from the atomic-like x-ray absorption and XMCD spectra [@Lau09b; @Hirsch12a; @Hirsch12b; @ZamudioBayer15] that are very sensitive to spin correlations in the initial and final states. Finally, it is known that the XMCD spin sum rule [@Carra93] cannot be applied without empiric correction to manganese [@Duerr97; @Piamonteze09] because $L_3$ and $L_2$ edges are not strictly separable as required. This leads to an underestimation of the spin magnetization. A comparison of the magnetization of Mn$_2^+$ and Mn$_3^+$ as obtained above to the result that would be given by application of the XMCD spin sum rule to the experimental spectra yields experimental correction factors to the spin sum rule of $1.7\pm0.3$ for Mn$_2^+$ and $1.5\pm0.1$ for Mn$_3^+$. These values agree well with results of 1.5 and 1.47 that are obtained by Hartree-Fock or charge-transfer multiplet calculations [@Duerr97; @Piamonteze09].
Conclusion {#conclusion .unnumbered}
==========
In conclusion, x-ray absorption and x-ray magnetic circular dichroism spectroscopy of free Mn$_2^+$ and Mn$_3^+$ give direct experimental evidence of molecular high spin states. For both molecular ions we find a maximum $3d$ spin ground state with atomically localized $3d$ electrons and fully parallel alignment of all $3d$ electron spins, which leads to the largest possible magnetic spin moments of 11 $\mu_B$ and 16 $\mu_B$, respectively. We do not see any evidence for noncollinear or canted spin arrangements in Mn$_2^+$ and Mn$_3^+$. We find that the same mechanism is responsible for the electronic structure and chemical bonding in Mn$_2^+$ and Mn$_3^+$. These molecular ions are characterized by localized $3d^5\ (^6\mathrm{S})$ high-spin configurations of half-filled $3d$ subshells, and a single delocalized $4s$ derived electron that mediates a strong and ferromagnetic indirect exchange coupling, analogous to the case of Cr$_2^+$. From the observed localized atomic magnetic moment and $3d^5$ configuration we deduce an experimental lower limit on the interatomic distances of $r_e \geq 2.6$ [Å]{} in agreement with theoretical predictions for the high spin states. There is no orbital contribution to the total angular momentum and thus no coupling of the spin to the molecular axis or frame to first order.
Beam time for this project was granted at BESSY II beamlines UE52-SGM and UE52-PGM, operated by Helmholtz-Zentrum Berlin. Skillful technical assistance by Thomas Blume is gratefully acknowledged. This project was partially funded by the German Federal Ministry of Education and Research (BMBF) through grant BMBF-05K13Vf2. The superconducting solenoid was kindly provided by Toyota Technological Institute. AT acknowledges financial support by Genesis Research Institute, Inc. BvI acknowledges travel support by Helmholtz-Zentrum Berlin.
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|
---
abstract: |
We prove the conjecture that any Grothendieck -topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to -toposes, just as higher-order logic is used for 1-toposes.
As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2-monad theory.
author:
- Michael Shulman
bibliography:
- 'all.bib'
title: 'All $(\infty,1)$-toposes have strict univalent universes'
---
\[5\]
|
---
abstract: 'We exhibit a family of graphs that develop turning singularities (i.e. their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem where the permeability is given by a nonnegative step function. We study the influence of different choices of the permeability and different boundary conditions (both at infinity and considering finite/infinite depth) in the development or prevention of singularities for short time. In the general case (inhomogeneous, confined) we prove a bifurcation diagram concerning the appearance or not of singularities when the depth of the medium and the permeabilities change. The proofs are carried out using a combination of classical analysis techniques and computer-assisted verification.'
author:
- 'Javier Gómez-Serrano$^{\mbox{{\footnotesize 1}}}$ and Rafael Granero-Belinchón$^{\mbox{{\footnotesize 2}}}$'
bibliography:
- 'bibliografia.bib'
- 'references.bib'
title: 'On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof'
---
**Keywords**: Darcy’s law, inhomogeneous Muskat problem, blow-up, computer-assisted, singularity, turning, water waves.
**MSC (2010)**: 35R35, 65G30, 76B03, 35Q35.
**Acknowledgments**: The authors are supported by the Grant MTM2011-26696 from Ministerio de Ciencia e Innovación (MICINN) and MINECO: ICMAT Severo Ochoa project SEV-2011-0087. Javier Gómez-Serrano is supported by StG-203138CDSIF of the ERC. Rafael Granero-Belinchón is grateful to Luigi Berselli and Rafael Orive for productive comments in an early version of these results. Javier Gómez-Serrano thanks Rafael de la Llave for fruitful discussions. We thank Diego Córdoba for his guidance and useful suggestions. We wish to thank the Instituto de Ciencias Matemáticas (Madrid) for computing facilities.
Introduction {#IIIsec1}
============
In this paper we study the evolution of the interface between two different incompressible fluids with the same viscosity in a two-dimensional porous medium. This problem is worthwhile studying since it is a model of an aquifer or an oil well (see [@muskat1937flow] and the references therein) or a model of a geothermal reservoir (see [@CF] and the references therein). We address the differences between the dynamics of the singularity of turning waves when the assumptions of the model change. In this context, we will refer to a turning singularity whenever we speak about curves such that initially have a point with vertical tangent, backwards in time can be parametrized as graphs and forward in time they can not, as seen in Figure \[FigTurning\].
![Turning singularity: graph, vertical tangent and turning of the interface.[]{data-label="FigTurning"}](turningqualitTime.eps)
We notice that, according to our definition, a point with vertical tangent, by itself, is not a turning singularity, since the curve can recoil and move into the stable regime where it can be parametrized as a graph. In order to be considered as a singularity, the curve necessarily has to turn over. Parametrized as a curve, the interface remains analytic while parametrized as a graph has a singularity. In other words, the parametrization blows up. In this framework, the singularity is equivalent to the fact that $\partial_\alpha z_1<0$ at some point for short time.
We consider two incompressible fluids with the same viscosity but different densities, $\rho^1$ and $\rho^2$, evolving in a two dimensional porous medium with permeability $\kappa(x)$. The velocity field obeys Darcy’s law: $$\label{IIIdarcy}
\mu\frac{v}{\kappa}=-\nabla p-g(0,\rho)^t,$$ where $\mu$ is the viscosity and $g$ is the acceleration due to gravity, and the incompressibility condition $$\label{IIIincom}
\nabla\cdot v=0.$$ We take $\mu=g=1$. The fluids also satisfy the conservation of mass equation $$\label{IIIconser}
{\partial_t}\rho+v\cdot\nabla\rho=0.$$ Given $l>0$, the spatial domains considered are $\Omega={\mathbb R}\times(-l,l),{\mathbb R}^2$ and ${\mathbb T}\times{\mathbb R}.$ We denote by $S^1$ the volume occupied by the fluid with density $\rho^1$ and by $S^2$ the volume occupied by the fluid with density $\rho^2$. The interface between both fluids is the curve $z(\alpha,t)$. Given $0<h_2<l$, we consider that the permeability is $$\label{IIIperm}
\kappa(x)=\kappa^1\textbf{1}_{\{(x,y)\in\Omega, y>-h_2\}}+\kappa^2\textbf{1}_{\{(x,y)\in\Omega, y\leq-h_2\}},$$ *i.e.* the curve $h(\alpha)=(\alpha,-h_2)$ separates the regions with different permeabilities. We assume that the initial curve $z(\alpha,0)$ does not touch the curve $h(\alpha)$. Moreover we consider that $z(\alpha,0)$ is in the region with permeability equal to $\kappa^1$. See Figure \[FigInhomogeneous\] for an illustration of the previous domains.
![Situation of the different fluids and permeabilities.[]{data-label="FigInhomogeneous"}](inhomogeneousv4.eps)
We define the Rayleigh-Taylor condition $$RT(\alpha,t)=-(\nabla p^2(z(\alpha,t))-\nabla p^1(z(\alpha,t)))\cdot\partial_\alpha^\bot z(\alpha,t).$$ Fix $t>0$. If $RT(\alpha,t)>0 , \;\forall\alpha\in{\mathbb R}$ we will say that the curve is in the Rayleigh-Taylor stable regime and if $RT(\alpha,t)<0$ for some $\alpha$, we will say that the curve is in the Rayleigh-Taylor unstable regime. We note that when the interface is a graph and is parametrized as $(\alpha,f(\alpha))$, this function reduces to $$RT=g(\rho^2-\rho^1),$$ and the curve is in the RT stable regime whenever $\rho^1<\rho^2$.
The Muskat problem where the permeability is constant and the depth is infinite has been studied in many works. A proof of local existence of classical solutions in the Rayleigh-Taylor stable regime and ill-posedness in the unstable regime can be encountered in [@c-g07]. A maximum principle for $\|f(t)\|_{L^\infty}$ can be found in [@c-g09]. Moreover, the authors showed in [@c-g09] that if $\|{\partial_x}f_0\|_{L^\infty}<1$, then $\|{\partial_x}f(t)\|_{L^\infty}<\|{\partial_x}f_0\|_{L^\infty}$. In [@ccfgl], the authors determined that the initial curve becomes analytic for every positive time and they also proved the existence of turning singularities. For other results see [@ambrose2004well; @castro2012breakdown; @Castro-Cordoba-Gancedo:recent-results-muskat; @ccgs-10; @c-g-o08; @e-m10; @KK; @SCH].
The case with finite depth (equivalently, when the permeability is supported in the strip ${\mathbb R}\times(-l,l)$) has been addressed in [@Cordoba-GraneroBelinchon-Orive:confined-muskat]. In this work the authors found the existence and uniqueness of solutions in the RT stable regime, a smoothing effect, ill-posedness in the RT unstable regime, a maximum principle and a decay estimate for $\|f(t)\|_{L^\infty}$ which is slower than in the case where the depth is infinity. The authors also proved that if the initial datum has small amplitude and slope (in a very precise sense depending on the depth), $\|{\partial_x}f(t)\|_{L^\infty}$ verifies a uniform bound and under more restrictive conditions for the initial amplitude and slope, the derivative obeys a maximum principle. We remark that the condition is not only on the size of the slope. Moreover, in this region there are global weak solutions (see [@G]).
The Muskat problem where the permeability is given by has been treated in [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat]. For this model the authors proved well-posedness and the existence of turning singularities when the physical parameters are in a precise range. One of our main contributions is to extend the range of physical parameters where the waves turn by means of a computer-assisted proof. In [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat] there is numerical evidence that supports the aforementioned results.
In the present paper we compare different models. First, we show the existence of waves such that when the depth is $l=\pi/2$ the wave turns and if $l=\infty$ then the slope of the wave decreases for a short enough time. In [@Cordoba-GraneroBelinchon-Orive:confined-muskat] there is numerical evidence of this result. The same result is true when the Muskat equation is replaced with the water waves equations (or free boundary incompressible Euler equations, see [@Castro-Cordoba-Fefferman-Gancedo-GomezSerrano:finite-time-singularities-water-waves-surface-tension; @Castro-Cordoba-Fefferman-Gancedo-GomezSerrano:splash-water-waves; @Castro-Cordoba-Fefferman-Gancedo-GomezSerrano:finite-time-singularities-free-boundary-euler] and the references therein), which are given by $$\label{ww}
\left\{\begin{array}{lll}
{\partial_t}z(\alpha,t) & = & BR(\varpi,z)z(\alpha,t)+c({\alpha},t){\partial_\alpha}z({\alpha},t) \\
{\partial_t}\varpi(\alpha,t) & = & -2{\partial_t}BR(\varpi,z)z(\alpha,t)\cdot{\partial_\alpha}z(\alpha,t)-{\partial_\alpha}\left(\frac{|\varpi|^2}{4|{\partial_\alpha}z|}\right)+{\partial_\alpha}\left(c\varpi\right)\\
&&+2c({\alpha},t){\partial_\alpha}BR(\varpi,z)z(\alpha,t)\cdot{\partial_\alpha}z({\alpha},t)-2g{\partial_\alpha}z_2({\alpha},t),
\end{array}\right.$$ where $z$ is the interface, $\varpi$ is the amplitude of the vorticity, $c$ accounts for the reparametrization freedom of the curve and $BR$ denotes the Birkhoff-Rott kernel (see in Section \[IIIsec2\] below). Notice that this kernel depends on the domain.
Second, we study a model where the permeability is given by a nonnegative step function. In this case, we are interested in the effect of this inhomogeneity in the interface. For this model we obtain that with different permeabilities there is no global in time solution in the Rayleigh-Taylor stable regime corresponding to an arbitrary, large (in $C^1$) initial data which is a graph. Moreover, if the permeabilities verify some conditions we get that they can help or prevent the formation of turning singularities for some families of initial data. These results are true for both the periodic and the flat at infinity cases.
Finally, we consider the most general model where, in addition to the change of permeabilities, the medium is bounded by impervious walls. For this case, we define a family of curves depending on the height $h_2$ where the permeability jump is located in a way that the curves are located above $h_2$. For this family, we perform rigorous computations of a bifurcation diagram in which the parameters are $h_2$ and the permeability values and the outcomes are $\{\text{turning, not turning, unknown}\}$. We obtain that the family exhibits different behaviours depending on $h_2$: for some of them the outcome is independent of the permeability values and for some it is not. Moreover, we see that the property of turning/not turning is persistent, *i.e.* small variation of the parameters give rise to the same outcome and we prove the existence of a smooth curve in parameter space that delimits turning from not turning. The role of the permeability is rather subtle. Assuming that the initial data can be parametrized as a graph, in [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat Section 4], the numerics show that if $\kappa^1-\kappa^2<0$ the evolution for $\|f(t)\|_{L^\infty}$ is smoother than in the case with only one permeability ($\kappa^1-\kappa^2=0$) in the sense that the decay of this quantity is faster. In the same way, if $\kappa^1-\kappa^2>0$ the decay of the $L^\infty$ norm is slower than in the homogeneous case. However, when the evolution of $\|{\partial_x}f(t)\|_{L^\infty}$ is addressed, the same numerics show that the situation is reversed. For the Lipschitz seminorm, the decay is faster in the case $\kappa^1-\kappa^2>0$ than in the homogeneous case.
The use of computers to perform floating-point arithmetic can lead to numerical errors. To overcome this difficulty and prove rigorous results, we use the so-called *interval arithmetics*, in which instead of working with arbitrary real numbers, we perform computations over intervals which have representable numbers as endpoints. On these objects, an arithmetic is defined in such a way that we are guaranteed that for every $x \in X, y \in Y$ $$\begin{aligned}
x \star y \in X \star Y,\end{aligned}$$
for any operation $\star$. For example, $$\begin{aligned}
[\underline{x},\overline{x}] + [\underline{y},\overline{y}] & = [\underline{x} + \underline{y}, \overline{x} + \overline{y}] \nonumber \\
[\underline{x},\overline{x}] \times [\underline{y},\overline{y}] & = [\min\{\underline{x}\underline{y},\underline{x}\overline{y},\overline{x}\underline{y},\overline{x}\overline{y}\},\max\{\underline{x}\underline{y},\underline{x}\overline{y},\overline{x}\underline{y},\overline{x}\overline{y}\}] \nonumber \\
\max\{[\underline{x},\overline{x}],[\underline{y},\overline{y}]\} & = [\max\{\underline{x},\underline{y}\},\max\{\overline{x},\overline{y}\}].
\label{defMAX}\end{aligned}$$ We can also define the interval version of a function $f(X)$ as an interval $I$ such that for every $x \in X$, $f(x) \in I$. Rigorous computation of integrals has been theoretically developed since the seminal work of Moore and many others [@Berz-Makino:high-dimensional-quadrature; @Kramer-Wedner:adaptive-gauss-legendre-verified-computation; @Lang:multidimensional-verified-gaussian-quadrature; @Moore-Bierbaum:methods-applications-interval-analysis], and has had applications in physics [@Holzmann-Lang-Schutt:gravitation-verified-quadrature]. An important ingredient of our proofs will be the rigorous computation of some integrals. Having a tight enclosure of the result is crucial for the sake of determining if an initial condition will develop a turning singularity or not for short time. In order to perform the rigorous computations we used the C-XSC library [@CXSC].
The organization of this paper is as follows: the contour equations are obtained in Section \[IIIsec2\], a precise statement of the theorems is given in Section \[MainResultsMuskat\], their proofs can be found in Section \[SectionProofs\] and the codes in the supplementary material. The codes are intended to be read in order. Some of the strategies of the Theorems are built upon the ones used for the previous ones. Moreover, we have sacrificed performance for readability in the first 3 Theorems where the computation was less intensive than in the last one. In any case, we have tried to achieve the optimal asymptotic complexity but without optimizing in a very deep low level.
**Notation:** we denote $(a,b)^\perp=(-b,a)$ and define $$\mathcal{K}=\frac{\kappa^1-\kappa^2}{\kappa^1+\kappa^2}\text{ and } \bar{\rho}=\frac{\kappa^1(\rho^2-\rho^1)}{4\pi}.$$ We notice that $\mathcal{K}$ is an dimensionless number and satisfies $-1 < \mathcal{K} < 1$. From now on, we also drop the dependence in $t$.
For readability purposes, instead of writing the intervals as for example $[123456,123789]$ we will refer to them as $123^{456}_{789}$.
The contour equation {#IIIsec2}
====================
In this section we obtain the contour equation. Now we consider the bounded porous medium ${\mathbb R}\times (-l,l)$. This regime is equivalent to the case with more than two $\kappa^i$ because the boundaries can be understood as regions with $\kappa=0$. Given a scalar function $a$ and curves $f=(f_1,f_2),g=(g_1,g_2)$, we denote the Birkhoff-Rott integral by $$\label{IIIeq2}
BR(a,g)f(\alpha)=\text{P.V.}\int_{\mathbb R}a(\beta) BS(f(\alpha),g(\beta))d\beta,$$ where $BS$ denotes the Biot-Savart law in ${\mathbb R}\times(-l,l)$, which is given by the kernel (see [@Cordoba-GraneroBelinchon-Orive:confined-muskat]) $$\begin{gathered}
\label{IIIBSconf}
BS(x,y,\mu,\nu)=\frac{1}{8l}\left(\frac{-\sin\left(\frac{\pi}{2l}(y-\nu)\right)}{\cosh\left(\frac{\pi}{2l}(x-\mu)\right)-\cos\left(\frac{\pi}{2l}(y-\nu)\right)}+\frac{\sin\left(\frac{\pi}{2l}(y+\nu)\right)}{\cosh\left(\frac{\pi}{2l}(x-\mu)\right)+\cos\left(\frac{\pi}{2l}(y+\nu)\right)},\right.\\
\left.\frac{\sinh\left(\frac{\pi}{2l}(x-\mu)\right)}{\cosh\left(\frac{\pi}{2l}(x-\mu)\right)-\cos\left(\frac{\pi}{2l}(y-\nu)\right)}-\frac{\sinh\left(\frac{\pi}{2l}(x-\mu)\right)}{\cosh\left(\frac{\pi}{2l}(x-\mu)\right)+\cos\left(\frac{\pi}{2l}(y+\nu)\right)}\right).\end{gathered}$$ To simplify notation we take the depth to be $l=\pi/2$. Notice that if $z({\alpha})$ is a solution of the Muskat problem - with depth $l=\pi/2$, then $z^\lambda({\alpha},t)=\lambda z(\lambda{\alpha},t/\lambda)$ is the interface corresponding to a solution of the Muskat problem with depth equal to $l=\pi\lambda/2$.
Due to ,, and the vorticity concentrates on the two interfaces as long as a weak solution exists for the full system considered. Thus, we can write it as $$\label{IIIvort}
\omega(\alpha,t)=\varpi_1(\alpha,t)\delta((x,y)-z(\alpha,t)) + \varpi_2(\alpha,t)\delta((x,y)-h(\alpha)),$$ where $\varpi_1$ and $\varpi_2$ stand for the different vorticity amplitudes. Computing the limits of the velocity towards the two interfaces we see that $$\label{IIIeq3}
v^\pm(z(\alpha))=\lim_{\epsilon\rightarrow0}v(z(\alpha)\pm\epsilon{\partial_\alpha}^\perp z(\alpha))=BR(\varpi_1,z)z({\alpha})+BR(\varpi_2,h)z({\alpha})\mp\frac{1}{2}\frac{\varpi_1(\alpha)}{|{\partial_\alpha}z(\alpha)|^2}{\partial_\alpha}z(\alpha),$$ and $$\label{IIIeq4}
v^\pm(h(\alpha))=\lim_{\epsilon\rightarrow0}v(h(\alpha)\pm\epsilon{\partial_\alpha}^\perp h(\alpha))=BR(\varpi_1,z)h({\alpha})+BR(\varpi_2,h)h({\alpha})\mp\frac{1}{2}\frac{\varpi_2(\alpha)}{|{\partial_\alpha}h(\alpha)|^2}{\partial_\alpha}h(\alpha).$$ We observe that $v^+(z(\alpha))$ is the limit inside $S^1$ (the upper subdomain) and $v^-(z(\alpha))$ is the limit inside $S^2$ (the lower subdomain). The curve $z(\alpha)$ does not touch the curve $h(\alpha)$, therefore the limits for the curve $h$ are in the same subdomain $S^2$.
Using Darcy’s Law, we have $$\begin{aligned}
(v^-(z(\alpha))-v^+(z(\alpha)))\cdot{\partial_\alpha}z(\alpha)&=\kappa^1\left(-{\partial_\alpha}(p^-(z(\alpha))-p^+(z(\alpha)))\right)-\kappa^1(\rho^2-\rho^1){\partial_\alpha}z_1(\alpha)\\
&=0-\kappa^1(\rho^2-\rho^1){\partial_\alpha}z_2(\alpha),\end{aligned}$$ since the pressure is continuous along the interface (see [@c-c-g10 Section 2]). Using we conclude $$\label{IIIeq5}
\varpi_1(\alpha)=-\kappa^1(\rho^2-\rho^1){\partial_\alpha}z_2(\alpha).$$ We need to determine $\varpi_2$. We consider $$\begin{aligned}
\left[\frac{v}{\kappa}\right]&\equiv &\left(\frac{v^-(h(\alpha))}{\kappa^2}-\frac{v^+(h(\alpha))}{\kappa^1}\right)\cdot{\partial_\alpha}h(\alpha)\\
&=&-{\partial_\alpha}(p^-(h(\alpha))-p^+(h(\alpha)))\\
&=&0,\end{aligned}$$ where the first equality is obtained by Darcy’s Law. Expression leads us to $$\left[\frac{v}{\kappa}\right]=\left(\frac{1}{\kappa^2}-\frac{1}{\kappa^1}\right)\left(BR(\varpi_1,z)h({\alpha})+BR(\varpi_2,h)h({\alpha})\right)\cdot{\partial_\alpha}h(\alpha)+\left(\frac{1}{2\kappa^2}+\frac{1}{2\kappa^1}\right)\varpi_2.$$ We have a Fredholm integral equation of the second kind: $$\label{IIIw2def}
\varpi_2(\alpha)+\frac{\mathcal{K}}{2\pi}\;\text{P.V.}\int_{\mathbb R}\frac{\varpi_2(\beta)\sin(2h_2)}{\cosh(\alpha-\beta)+\cos(2h_2)}d\beta=-2\mathcal{K}BR(\varpi_1,z)h({\alpha})\cdot(1,0).$$
We define the Fourier transform as $$\mathcal{F}(f)(\zeta)=\frac{1}{\sqrt{2\pi}}\int_{{\mathbb R}}e^{-ix\zeta}f(x)dx,$$ and using some of its basic properties, we obtain $$\mathcal{F}(\varpi_2)(\zeta)\left(1+\frac{\mathcal{K}}{\sqrt{2\pi}}\mathcal{F}\left(\frac{\sin(2h_2)}{\cosh(x)+\cos(2h_2)}\right)(\zeta)\right)=-2\mathcal{K}\mathcal{F}(BR(\varpi_1,z)h\cdot(1,0))(\zeta).$$ In [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat] the equation for $\varpi_2$ is solved for every $|\mathcal{K}|<\delta(h_2)$ with $$\label{IIIdelta}
\delta(h_2)=\min\left\{1,\frac{\sqrt{2\pi}}{\displaystyle \max_{\zeta}\left|\mathcal{F}\left(\frac{\sin(2h_2)}{\cosh(x)+\cos(2h_2)}\right)(\zeta)\right|}\right\}.$$ We have the following result concerning the range of correct parameters:
\[lemacitado\] Let $0<h_2<\pi/2$ be a constant, then $\delta(h_2)=1$. Thus, there exists a solution to for every $-1<\mathcal{K}<1$.
We prove the result by computing explicitly $$J = \mathcal{F}\left(\frac{\sin(2h_2)}{\cosh(x)+\cos(2h_2)}\right)(\zeta)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb R}e^{-ix\zeta}\frac{\sin(2h_2)}{\cosh(x)+\cos(2h_2)}dx.$$ Take $\zeta\in{\mathbb R}$, $\zeta<0$. We consider the complex extension $$I^{j}=\int_{\partial\Gamma^{j}} e^{-iz\zeta}\frac{\sin(2h_2)}{\cosh(z)+\cos(2h_2)}dz,$$ where $\Gamma^{j}=(-\pi-2j\pi,\pi+2j\pi)\times(0,2j \pi)\in{\mathbb C}, \; j \in \mathbb{N}$. The poles of the function (see Figure \[FigPoles\]) are $$\gamma_k^-=(\pi-2h_2+2k\pi)i\text{ and }\gamma_k^+=(\pi+2h_2+2k\pi)i, \quad k \in \mathbb{Z}.$$ Given that $$\cosh(z)+\cos(2h_2)=2\cosh\left((z+2h_2i)/2\right)\cosh\left((z-2h_2i)/2\right),$$ $\gamma_k^\pm$ are simple poles. We split the contour integral in $$I^{j}=I_{1}^{j} + I_{2}^{j} + I_{3}^{j} + I_{4}^{j},$$ with $$\begin{aligned}
I_{1}^{j}& =\int^{\pi+2j\pi}_{-(\pi+2j\pi)}\frac{e^{-ix\zeta}\sin(2h_2)}{\cosh(x)+\cos(2h_2)}dx, \\
I_{2}^{j}& =\int_{\pi+2j\pi}^{-(\pi+2j\pi)}\frac{e^{-i(x+2 \pi j i)\zeta}\sin(2h_2)}{\cosh(x+2 \pi j i)+\cos(2h_2)}dx, \\
I_{3}^{j}& =\int_0^{2j\pi}\frac{e^{-i(\pi+2j\pi+iy)\zeta}\sin(2h_2)}{\cosh(\pi+2j\pi+iy)+\cos(2h_2)}dy, \\
I_{4}^{j}& =\int_{2j\pi}^0\frac{e^{-i(-\pi-2j\pi+iy)\zeta}\sin(2h_2)}{\cosh(-\pi-2j\pi+iy)+\cos(2h_2)}dy.\end{aligned}$$
![Situation of the poles: $\gamma_{i}^{+}$ in black, $\gamma_{i}^{-}$ in grey.[]{data-label="FigPoles"}](PolosIJ.eps)
Using classical trigonometric identities, we get $$|I_{2}^{j}|\leq\int_{-\infty}^{\infty}\frac{e^{2j\pi\zeta}\sin(2h_2)}{\cosh(x)+\cos(2h_2)}dx\leq c_{h_2}e^{2j\pi\zeta},$$ which tends to zero when $j$ tends to infinity since $\zeta < 0$. We can bound the third integral as $$\begin{gathered}
|I_{3}^{j}|\leq\int_0^{2j\pi}\frac{e^{y\zeta}}{|\cosh(\pi+2j\pi)\cos(y)+\sinh(\pi+2j\pi)\sin(y)+\cos(2h_2)|}dy\\
\leq\int_0^{\infty}\frac{e^{y\zeta}}{(\cosh(\pi+2j\pi)-1)^2+\cos(2h_2)-2}dy.\end{gathered}$$ The same remains valid for $I_{4}^{j}$. Then, taking the limit $j\rightarrow\infty$: $$J = \lim_{j \to \infty}\frac{1}{\sqrt{2\pi}}I^{j}=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}\frac{e^{-ix\zeta}\sin(2h_2)}{\cosh(x)+\cos(2h_2)}dx.$$ By the Residue Theorem, this implies $$\begin{aligned}
I^{j}& =2\pi i\sum_{k=0}^{j}\text{Res}\left(\frac{e^{-iz\zeta}\sin(2h_2)}{\cosh(z)+\cos(2h_2)},\gamma_k^\pm\right) \\
J & = \lim_{j \to \infty}\frac{1}{\sqrt{2\pi}}I^{j} = 2\pi i \sum_{k\geq0}\frac{1}{\sqrt{2\pi}}ie^{\pi\zeta}(e^{2\pi\zeta})^k2\sinh(2h_2\zeta)=
\sqrt{2\pi}\frac{\sinh(2h_2\zeta)}{\sinh(\pi\zeta)}.\end{aligned}$$ The result can be easily extended to every $\zeta > 0$ by the evenness of $\frac{\sin(2h_2)}{\cosh(x)+\cos(2h_2)}$ and to $\zeta = 0$ by the the continuity of the Fourier transform. Finally we obtain $$\delta(h_2)=\min\left\{1,\frac{\pi}{2h_2}\right\},$$ and we conclude that $\delta(h_2)=1$ for every $0<h_2<\pi/2$. Moreover, this extends the result in [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat Remark 2], where numerical evidence of its validity was found.
Thus, for every $|\mathcal{K}|<1$, we can write the expression of $\varpi_2$ as $$\begin{aligned}
\varpi_2({\alpha})=&-2\mathcal{K}BR(\varpi_1,z)h({\alpha})\cdot(1,0)+\frac{2\mathcal{K}^2}{2\pi}BR(\varpi_1,z)h({\alpha})\cdot(1,0)*G_{h_2,\mathcal{K}}\nonumber\\
=&\, 2\mathcal{K}\bar{\rho}\left[\text{P.V.}\int_{\mathbb R}{\partial_\alpha}z_2(\beta)\frac{\sin(h_2+z_2(\beta))}{\cosh(\alpha-z_1(\beta))-\cos(h_2+z_2(\beta))}d\beta\nonumber\right.\\
&-\text{P.V.}\int_{\mathbb R}{\partial_\alpha}z_2(\beta)\frac{\sin(-h_2+z_2(\beta))}{\cosh(\alpha-z_1(\beta))+\cos(-h_2+z_2(\beta))}d\beta\nonumber\\
&-\frac{\mathcal{K}}{2\pi}G_{h_2,\mathcal{K}}*\text{P.V.}\int_{\mathbb R}\frac{{\partial_\alpha}z_2(\beta)\sin(h_2+z_2(\beta))}{\cosh(\alpha-z_1(\beta))-\cos(h_2+z_2(\beta))}d\beta\nonumber\\
&+\left.\frac{\mathcal{K}}{2\pi}G_{h_2,\mathcal{K}}*\text{P.V.}\int_{\mathbb R}\frac{{\partial_\alpha}z_2(\beta)\sin(-h_2+z_2(\beta))}{\cosh(\alpha-z_1(\beta))+\cos(-h_2+z_2(\beta))}d\beta\right], \label{IIIw2defb}\end{aligned}$$ where $$G_{h_2,\mathcal{K}}(\xi)=\mathcal{F}^{-1}\left(\frac{\mathcal{F}\left(\frac{\sin(2h_2)}{\cosh(x)+\cos(2h_2)}\right)}{1+\frac{\mathcal{K}}{\sqrt{2\pi}}\mathcal{F}\left(\frac{\sin(2h_2)}{\cosh(x)+\cos(2h_2)}\right)}\right)=\int_{\mathbb R}\frac{\cos(y\xi)\sinh(2h_2 y)}{\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y)}dy.$$
We observe that $G_{h_2,\mathcal{K}}$ is a function in the Schwartz class. Using $$\int_{\mathbb R}\partial_\beta\log\left(\cosh({\alpha}-z_1(\beta))\pm\cos(y\pm z_2(\beta))\right)d\beta=0,$$ and adding the correct tangential term (see [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat; @c-c-g10; @Cordoba-GraneroBelinchon-Orive:confined-muskat]), we obtain $$\begin{aligned}
{\partial_t}z(\alpha)=&\bar{\rho}\text{P.V.}\int_{\mathbb R}\frac{({\partial_\alpha}z(\alpha)-{\partial_\alpha}z(\beta))\sinh(z_1(\alpha)-z_1(\beta))}{\cosh(z_1(\alpha)-z_1(\beta))-\cos(z_2(\alpha)-z_2(\beta))}d\beta\nonumber\\
&+\bar{\rho}\text{P.V.}\int_{\mathbb R}\frac{({\partial_\alpha}z_1(\alpha)-{\partial_\alpha}z_1(\beta),{\partial_\alpha}z_2(\alpha)+{\partial_\alpha}z_2(\beta))\sinh(z_1(\alpha)-z_1(\beta))}{\cosh(z_1(\alpha)-z_1(\beta))+\cos(z_2(\alpha)+z_2(\beta))}d\beta\nonumber\\
&+\frac{1}{4\pi}\text{P.V.}\int_{\mathbb R}\varpi_2(\beta)BS(z_1(\alpha),z_2(\alpha),\beta,-h_2)d\beta\nonumber\\
&+\frac{{\partial_\alpha}z(\alpha)}{4\pi}\text{P.V.}\int_{\mathbb R}\varpi_2(\beta)\frac{\sin(z_2(\alpha)+h_2)}{\cosh(z_1(\alpha)-\beta)-\cos(z_2(\alpha)+h_2)}d\beta\nonumber\\
&+\frac{{\partial_\alpha}z(\alpha)}{4\pi}\text{P.V.}\int_{\mathbb R}\varpi_2(\beta)\frac{\sin(z_2(\alpha)-h_2)}{\cosh(z_1(\alpha)-\beta)+\cos(z_2(\alpha)-h_2)}d\beta\label{IIIeqv2}.\end{aligned}$$ In the case where the fluids fill the whole plane, we can take the limit $l\rightarrow\infty$ in and write $$\begin{gathered}
\label{IIIeq9}
{\partial_t}z(\alpha)=2\bar{\rho}\text{P.V.}\int_{\mathbb R}\frac{z_1(\alpha)-z_1(\beta)}{|z(\alpha)-z(\beta)|^2}({\partial_\alpha}z(\alpha)-{\partial_\alpha}z(\beta))d\beta\\
+\frac{1}{2\pi}\text{P.V.}\int_{\mathbb R}\varpi_2(\beta)\frac{(z(\alpha)-h(\beta))^\perp}{|z(\alpha)-h(\beta)|^2}d\beta\\
+{\partial_\alpha}z(\alpha)\frac{1}{2\pi}\text{P.V.}\int_{\mathbb R}\varpi_2(\beta)\frac{z_2(\alpha)+h_2}{|z(\alpha)-h(\beta)|^2}d\beta,\end{gathered}$$ with $$\varpi_2(\alpha)=4\mathcal{K}\bar{\rho}\text{P.V.}\int_{\mathbb R}{\partial_\alpha}z_2(\beta)\frac{h_2+z_2(\beta)}{|h(\alpha)-z(\beta)|^2}d\beta,\label{IIIeq10A}$$ and, if the initial curve is periodic in the horizontal variable, using complex variables notation for the curve $z=(z_1,z_2)=z_1+iz_2$ and the identity $$\frac{1}{z}+\sum_{k\geq1}\frac{2z}{z^2-(2k\pi)^2}=\frac{1}{2\tan(z/2)},\;\;\forall z\in{\mathbb C},$$ we get $$\begin{gathered}
\label{IIIeq13}
{\partial_t}z(\alpha)=\bar{\rho}\text{P.V.}\int_{\mathbb T}\frac{\sin(z_1(\alpha)-z_1(\beta))({\partial_\alpha}z(\alpha)-{\partial_\alpha}z(\beta))d\beta}{\cosh(z_2(\alpha)-z_2(\beta))-\cos(z_1(\alpha)-z_1(\beta))}\\
+\frac{{\partial_\alpha}z_1(\alpha)-1}{4\pi}\text{P.V.}\int_{\mathbb T}\frac{\sinh(z_2(\alpha)+h_2)\varpi_2(\beta)d\beta}{\cosh(z_2(\alpha)+h_2)-\cos(z_1(\alpha)-\beta)}\\
+\frac{i}{4\pi} \text{P.V.}\int_{\mathbb T}\frac{({\partial_\alpha}z_2(\alpha)\sinh(z_2(\alpha)+h_2)+\sin(z_1(\alpha)-h_1(\beta)))\varpi_2(\beta)d\beta}{\cosh(z_2(\alpha)+h_2)-\cos(z_1(\alpha)-\beta)},\end{gathered}$$ where the second vorticity amplitude can be written as $$\varpi_2(\alpha)=2\bar{\rho}\mathcal{K} \text{P.V.}\int_{\mathbb T}\frac{\sinh(h_2+z_2(\beta)){\partial_\alpha}z_2(\beta)d\beta}{\cosh(-h_2-z_2(\beta))-\cos(\alpha-z_1(\beta))}.\label{IIIeq14A}$$
Statement of the results {#MainResultsMuskat}
========================
In this section we will state the theorems that will be proved in the next one. We show that the fact of having a confined medium plays a role in the mechanism for achieving turning singularities. Moreover, we also show that there are cases for which the jump in the permeabilities can lead to either prevent or promote these singularities, and cases in which the heterogeneity of the medium has no impact on whether the wave turns or not.
Notice that the confined (and homogeneous) Muskat problem corresponds to $\varpi_2=0$ in , while the unconfined (and homogeneous) satisfies $\varpi_2=0$ in . For these cases we have the next theorem:
There exists a family of analytic curves $z({\alpha}) = (z_1({\alpha}),z_2({\alpha}))$, flat at infinity, for which there exists a finite time $T$ such that the solution to the confined Muskat problem develops a turning singularity before $t=T$ and the non confined does not. \[ThmConfTurnsNoConfNoTurns\]
This theorem also implies the following result:
\[water\] There exists a family of analytic curves $z({\alpha}) = (z_1({\alpha}),z_2({\alpha}))$, flat at infinity, for which there exists a finite time $T$ such that the solution to the confined water waves problem develops a turning singularity before $t=T$ and the non confined does not.
The shallowness parameter (see [@Bona-Lannes-Saut:asymptotic-internal-waves]) is defined as $$\mu\equiv \left(\frac{\text{\emph{typical} depth}}{\text{\emph{ typical } wavelength}}\right)^2.$$ Waves with $\mu \ll 1$ are in the shallow water regime. In this example we have $\mu=\frac{1}{16}$, thus the shallow water regime is reached.
We will say that an analytic initial condition $z({\alpha}) = (z_1({\alpha}),z_2({\alpha}))$ ***turns unconditionally*** if there exists a finite time $T > 0$ for which the solution to the Muskat problem with initial condition equal to $z({\alpha})$ develops a turning singularity before time $T$ independently of the permeability parameter $\mathcal{K}$. Analogously, we will say that it ***recoils unconditionally*** if there exists a finite time $T > 0$ for which the solution to the Muskat problem with initial condition equal to $z({\alpha})$ does not develop a turning singularity before time $T$ independently of the permeability parameter $\mathcal{K}$. If the curve $z({\alpha},t) = (z_1({\alpha}),z_2({\alpha}))$ does not satisfy any of the two conditions mentioned before, we will say that the initial condition ***turns conditionally***.
These behaviours are *local in time*, thus, they refer to times $0\leq t \leq T$. In other words, for some initial data that *turns unconditionally*, there may exist some $T_2>T$ such that for $t>T_2$ the interface can be parametrized as a graph. The converse is also true: there may exist initial data such that they become smooth graphs for $0<t<T$ and $T_2>T$ such that $$\limsup_{t\rightarrow T_2}\|{\partial_x}f(t)\|_{L^\infty}=\infty.$$
We will say that for a given analytic initial condition $z({\alpha}) = (z_1({\alpha}),z_2({\alpha}))$, the permeabilities ***help*** the formation of singularities if the curve turns conditionally and for $\mathcal{K} = 0$ there exists a time $T > 0$ such that it does not develop a turning singularity before time $T$. Analogously, we will say that they ***prevent*** the formation of singularities if the curve turns conditionally and for $\mathcal{K} = 0$ there exists a time $T > 0$ such that it develops a turning singularity before time $T$.
For the unconfined, inhomogeneous Muskat problem we have
There exist 3 different families of analytic curves $z({\alpha}) = (z_1({\alpha}),z_2({\alpha}))$, periodic in the horizontal variable such that the corresponding solution to the unconfined, inhomogeneous Muskat :
- They turn unconditionally.
- The permeabilities help the formation of singularities.
- The permeabilities prevent the formation of singularities.
\[ThmDepKappaIndepKappaPeriodic\]
There exist 3 different families of analytic curves $z({\alpha}) = (z_1({\alpha}),z_2({\alpha}))$, flat at infinity such that the corresponding solution to the unconfined, inhomogeneous Muskat :
- They turn unconditionally.
- The permeabilities help the formation of singularities.
- The permeabilities prevent the formation of singularities.
\[ThmDepKappaIndepKappaFlat\]
Moreover, for both Theorems, in cases $(b)$ and $(c)$, there exists a unique parameter $K^{*}$ such that for all $\mathcal{K} < K^{*}$ the curve exhibits one behaviour and for all $\mathcal{K} > K^{*}$ it exhibits the other.
We should remark that Theorems \[ThmDepKappaIndepKappaPeriodic\] and \[ThmDepKappaIndepKappaFlat\] are more general than the ones in [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat Theorems 3 and 4] since we are suppressing any smallness assumption in $|\mathcal{K}|$ or largeness in $h_2$.
Finally, regarding the confined, inhomogeneous problem, we prove:
\[ThmConfinedInhomogeneous\] There exists a family of analytic initial data $z({\alpha},h_2) = (z_1({\alpha},h_2),z_2({\alpha},h_2))$, depending on the height at which the permeability jump is located, such that the corresponding solution to the confined, inhomogeneous Muskat :
(a) 1. For all $0.25 < h_2 < h_2^{ntu} = 0.648$, the curve recoils unconditionally.
2. For all $ 0.676 < h_2 < 0.686$, the permeabilities help the formation of singularities.
3. For all $ 0.715 < h_2 < 0.738$, the permeabilities prevent the formation of singularities.
4. For all $ 0.77 = h_2^{tu} < h_2 < 1.25$, the curve turns unconditionally.
(b) There exists a $C^1$ curve $(h_2,\mathcal{K}(h_2))$, located in $[0.648,0.77] \times (-1,1)$, such that for every $h_2$ for which the curve is defined, for every $\mathcal{K}<\mathcal{K}(h_2)$ the curve does not turn and for every $\mathcal{K}>\mathcal{K}(h_2)$ the curve turns.
Proof of the Theorems {#SectionProofs}
=====================
The idea of these proofs is to transform the problem on the turning or not into finding a sign of a given quantity (${\partial_\alpha}v_1(0,0)$). This sign will be validated using interval arithmetics. First, we consider curves such that ${\partial_\alpha}z_1(0,0)=0$ and define $\displaystyle m(t)=\min_\alpha {\partial_\alpha}z_1(\alpha,t)$. We will assume that $m(0)={\partial_\alpha}z_1(0,0)=0$ holds, and this minimum is only attained at ${\alpha}= 0$. Now, if ${\partial_\alpha}v_1(0,0)=\partial_\alpha{\partial_t}z_1(0,0)>0$ then we get $\frac{d}{dt}m(t)>0$ for $t>0$ small enough. This implies $m(\delta)>0$ for a small enough $\delta>0$ and the curve can be parametrized as a graph. Indeed, we compute $$m(\delta)=m(0)+\int_0^ \delta \frac{d}{dt}m(s)ds=\int_0^ \delta {\partial_t}{\partial_\alpha}z_1(\alpha_s,s)ds>0,$$ where ${\alpha}_s$ is a point where the minimum is attained. If ${\partial_\alpha}v_1(0,0)=\partial_\alpha{\partial_t}z_1(0,0)<0$, then $m(t)<0$ if $t$ is small enough and the curve can not be parametrized as a graph. After this goal is achieved, we approximate our initial data with analytic curves with the same properties (for instance by convolving it with the heat kernel). All these analytic curves that approximate our explicit constructed example satisfy the same symmetry hypotheses (see below). For these approximating curves, we apply the local existence forward and backward in time theorems proved in [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat; @ccfgl; @Cordoba-GraneroBelinchon-Orive:confined-muskat].
The homogeneous case {#SubsectionProofThm1}
--------------------
In this section we prove that the boundaries make the Muskat problem more singular from the point of view of singularity formation. Equivalently, the boundaries decrease the diffusion rate (see [@Cordoba-GraneroBelinchon-Orive:confined-muskat]).
We take $l=\pi/2$, $\bar{\rho} = 1$.We consider curves $z(\alpha)=(z_1(\alpha),z_2(\alpha))$ such that:
1. $z_i$ are analytic, odd functions.
2. $\partial_\alpha z_1(\alpha)>0, \forall \alpha\neq 0$, $\partial_\alpha z_1(0)=0$, and $\partial_\alpha z_2(0)>0$.
We want to show that $\partial_\alpha v_1(0,0)=\partial_\alpha{\partial_t}z_1(0,0)<0$. The equation for this regime is $$\begin{gathered}
{\partial_t}z(\alpha)=\text{P.V.}\int_{\mathbb R}\frac{({\partial_\alpha}z(\alpha)-{\partial_\alpha}z(\alpha-\beta))\sinh(z_1(\alpha)-z_1(\alpha-\beta))}{\cosh(z_1(\alpha)-z_1(\alpha-\beta))-\cos(z_2(\alpha)-z_2(\alpha-\beta))}d\beta\\
+\text{P.V.}\int_{\mathbb R}\frac{({\partial_\alpha}z_1(\alpha)-{\partial_\alpha}z_1(\alpha-\beta),{\partial_\alpha}z_2(\alpha)+{\partial_\alpha}z_2(\alpha-\beta))\sinh(z_1(\alpha)-z_1(\alpha-\beta))}{\cosh(z_1(\alpha)-z_1(\alpha-\beta))+\cos(z_2(\alpha)+z_2(\alpha-\beta))}d\beta.\end{gathered}$$ Taking one derivative we get $${\partial_\alpha}{\partial_t}z_1(\alpha)=I_1(\alpha)+I_2(\alpha)+I_3(\alpha),$$ where $$I_1(0)=\text{P.V.}\int_{\mathbb R}\left(\frac{-{\partial_\alpha}^2 z_1(-\beta)\sinh(-z_1(-\beta))}{\cosh(-z_1(-\beta))-\cos(-z_2(-\beta))}+\frac{-{\partial_\alpha}^2 z_1(-\beta)\sinh(-z_1(-\beta))}{\cosh(-z_1(-\beta))+\cos(z_2(-\beta))}\right)d\beta,$$ $$I_2(0)=\text{P.V.}\int_{\mathbb R}\left(\frac{\cosh(-z_1(-\beta))\left(-{\partial_\alpha}z_1(-\beta)\right)^2}{\cosh(-z_1(-\beta))-\cos(-z_2(-\beta))}+\frac{\left(-{\partial_\alpha}z_1(-\beta)\right)^2\cosh(-z_1(-\beta))}{\cosh(-z_1(-\beta))+\cos(z_2(-\beta))}\right)d\beta,$$ and $$\begin{gathered}
I_3(0)=-\text{P.V.}\int_{\mathbb R}\frac{\left[\sinh(-z_1(-\beta))\left(-{\partial_\alpha}z_1(-\beta)\right)\right]^2}{\left(\cosh(-z_1(-\beta))-\cos(-z_2(-\beta))\right)^2}d\beta\\
-\text{P.V.}\int_{\mathbb R}\frac{(-{\partial_\alpha}z(-\beta))\sinh(-z_1(-\beta))\left[\sin(-z_2(-\beta))\left(-{\partial_\alpha}z_2(-\beta)\right)\right]}{\left(\cosh(-z_1(-\beta))-\cos(-z_2(-\beta))\right)^2}d\beta\\
+\text{P.V.}\int_{\mathbb R}\frac{\left[-{\partial_\alpha}z_1(-\beta)\sinh(-z_1(-\beta))\right]^2}{\left(\cosh(-z_1(-\beta))+\cos(z_2(-\beta))\right)^2}d\beta\\
+\text{P.V.}\int_{\mathbb R}\frac{-{\partial_\alpha}z_1(-\beta)\sinh(-z_1(-\beta))\left[-\sin(z_2(-\beta)){\partial_\alpha}z_2(-\beta)\right]}{\left(\cosh(-z_1(-\beta))+\cos(z_2(-\beta))\right)^2}d\beta.\end{gathered}$$ Then, after some integration by parts and using the properties of $z_i$, we get the following expression for the derivative of the velocity in the confined case: $$\begin{gathered}
I^A_{neg}\equiv\frac{\partial_\alpha v_1(0)}{2}=\partial_{\alpha} z_2(0)\int_0^\infty \partial_\alpha z_1(\eta)\sinh(z_1(\eta))\sin(z_2(\eta))\bigg{(}\frac{1}{(\cosh(z_1(\eta))-\cos(z_2(\eta)))^2}\\
+\frac{1}{(\cosh(z_1(\eta))+\cos(z_2(\eta)))^2}\bigg{)}d\eta.\end{gathered}$$ With the same approach, for the unconfined case the expression is $$I^A_{pos}\equiv\frac{\partial_\alpha v_1(0)}{8}=\partial_{\alpha} z_2(0)\int_0^\infty \frac{\partial_\alpha z_1(\eta)z_1(\eta)z_2(\eta)}{(z_1(\eta))^2+(z_2(\eta))^2)^2}d\eta.$$
Thus, we are left to validate the following signs: $$\begin{aligned}
\label{signconfnoconf}
I^A_{neg}<0,\quad I^A_{pos}>0.\end{aligned}$$
![The curve in Theorem \[ThmConfTurnsNoConfNoTurns\]. Inset: Close caption around zero, solid: initial condition, dotted: normal component of the velocity for the infinitely deep case, squared: normal component of the velocity for the finitely deep case. The normal components have been scaled by a factor $1/100$.[]{data-label="figteo1"}](CAPTurningThmA.eps)
We rigorously validate them for the following data (see Figure \[figteo1\]):
$$\begin{aligned}
z_1({\alpha}) & = {\alpha}- \sin({\alpha})e^{-K{\alpha}^{2}}, \quad K = 10^{-4} \\
z_2({\alpha}) & = \left\{
\begin{array}{lr}
\displaystyle \frac{\sin(3{\alpha})}{3} & \displaystyle \text{ if } 0 \leq {\alpha}\leq \frac{\pi}{3} \vspace{0.2cm} \\
\displaystyle - {\alpha}+ \frac{\pi}{3} & \displaystyle \text{ if } \frac{\pi}{3} \leq {\alpha}\leq \frac{\pi}{2} \vspace{0.2cm} \\
\displaystyle {\alpha}- \frac{2\pi}{3} & \displaystyle \text{ if } \frac{\pi}{2} \leq {\alpha}\leq \frac{2\pi}{3}\vspace{0.2cm} \\
\displaystyle 0 & \displaystyle \text{ if } \frac{2\pi}{3} \leq {\alpha}, \\
\end{array}
\right.\end{aligned}$$
where $z_2$ is extended such that it is an odd function. This corresponds to the numerical scenario given in [@Cordoba-GraneroBelinchon-Orive:confined-muskat]. A first attempt is to compute the normal velocity of the curve in a nonrigorous way using the integral representations in and and the trapezoidal rule with an equispaced grid for several points around the point with vertical tangent. In Figure \[figteo1\] (inset), we plot it for the two scenarios (confined and non confined), both scaled by a factor $1/100$. We can observe that the velocity denoted by squares, which corresponds to the confined case, will make the curve develop a turning singularity, where the dotted one (non-confined case) will force the curve to stay in the stable regime.
In order to validate the sign, we split each of $I_{pos}^{A}$ and $I_{neg}^{A}$ into three pieces, each corresponding to a different piece of the piecewise defined $z_2$ in which $z_2({\alpha})$ is not identically 0.
In the second and third pieces, the integrand is analytic and we can apply Simpson’s rule on a uniform (equispaced) mesh $\eta_{0} < \eta_{1} < \ldots < \eta_{N+1}$ for the computation of the integrals:
$$\begin{aligned}
\int_{a}^{b} f(\eta) d\eta \in \sum_{i=0}^{N} \int_{\eta_{i}}^{\eta_{i+1}}f(\eta)d\eta & = \sum_{i=0}^{N} \frac{(\eta_{i+1}-\eta_{i})}{6}\left(f(\eta_{i})+f(\eta_{i+1})+4f\left(\frac{\eta_{i}+\eta_{i+1}}{2}\right)\right)\\
& -\frac{1}{2880}(\eta_{i+1}-\eta_{i})^{5}f^{4}([\eta_{i},\eta_{i+1}]).\end{aligned}$$
The first piece needs special care since the integrand is of type $\frac{0}{0}$ when $\alpha$ goes to zero. We should remark that the function is integrable: the numerator is $O(\alpha^{6})$ and the denominator is $O(\alpha^{4})$ when expanded both around $\alpha = 0$ in the two problematic cases, namely $I^{A}_{pos}$ and the first summand of $I^{A}_{neg}$. We further split the integral into two pieces, one ranging from 0 to ${\varepsilon}$ and another from ${\varepsilon}$ to $\frac{\pi}{3}$. In the validation of the theorem, the choice of the constant ${\varepsilon}$ equal to $\frac{1}{128}$ was enough. The integrand of the second piece is analytic and is calculated as before, while for the first piece we expand both the numerator and the denominator and cancel out the extra factors $\alpha$. In our case this means (for $I_{pos}^{A}$):
$$\begin{aligned}
\int_{0}^{{\varepsilon}}\frac{{\partial_\alpha}z_1(\alpha)\sin(z_1(\alpha))\sinh(z_2(\alpha))}{(\cosh(z_2(\alpha))-\cos(z_1(\alpha)))^2}d\alpha
\equiv \int_{0}^{{\varepsilon}}\frac{\mathcal{N}({\alpha})}{\mathcal{D}({\alpha})}d\alpha \in \int_{0}^{{\varepsilon}} \frac{\sum_{i=0}^{5}a_{i}\alpha^{i} + \frac{1}{6!}{\partial_\alpha}^{6} \mathcal{N}([0,{\varepsilon}])\alpha^{6}}{\sum_{j=0}^{3}b_j\alpha^{j} + \frac{1}{4!}{\partial_\alpha}^{4}\mathcal{D}([0,{\varepsilon}])\alpha^{4}}d\alpha.\end{aligned}$$
Since $a_0, \ldots, a_5, b_0, \ldots, b_3$ are zero, we get
$$\begin{aligned}
\label{pax4D}
\int_{0}^{{\varepsilon}}\frac{\mathcal{N}({\alpha})}{\mathcal{D}({\alpha})}d\alpha \in \int_{0}^{{\varepsilon}} \frac{4!}{6!}\frac{{\alpha}^2{\partial_\alpha}^{6} \mathcal{N}([0,{\varepsilon}])}{ {\partial_\alpha}^{4}\mathcal{D}([0,{\varepsilon}])}d\alpha
\subset \frac{{\varepsilon}^{3}}{3}\frac{1}{30}\frac{{\partial_\alpha}^{6} \mathcal{N}([0,{\varepsilon}])}{ {\partial_\alpha}^{4}\mathcal{D}([0,{\varepsilon}])}.\end{aligned}$$
The code is flexible so that $N$ can be specified by the user of the program. One can see that for small values of $N$, the intervals in which the value of $I^{A}_{pos}, I^{A}_{neg}$ are enclosed are not small enough such that 0 does not belong to them, needing further precision. However, for $N = 8192$ the grid is fine enough to check conditions . The calculations for $N = 8192$ can be found in Table \[tabla1\].
Quantity Enclosure
--------------- ----------------------------
$I^{A}_{pos}$ $ 0.0212172^{1922}_{7301}$
$I^{A}_{neg}$ $-0.0136819^{7345}_{1981}$
: Results of Theorem \[ThmConfTurnsNoConfNoTurns\].[]{data-label="tabla1"}
The computation took 2.96 seconds on an Intel i5 processor with 4 GB of RAM. Choosing as initial data a sufficiently close analytic perturbation of $z_2$ finishes the theorem.
Corollary \[water\] follows:
Take the same curve as before and define the initial amplitude for the vorticity as ${\partial_\alpha}z_2(\alpha,0)$. With these initial data we have a solution $(z,\varpi)$ of the water waves problem and we obtain the result (see [@ccfgl] for more details).
The inhomogeneous, unconfined case {#SubsectionProofsThm23}
----------------------------------
In this section we prove the existence of turning waves for a physical parameter region bigger than the one in [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat]. In both proofs, we consider curves $z(\alpha)=(z_1(\alpha),z_2(\alpha))$ such that:
1. $z_i$ are analytic, odd functions.
2. $\partial_\alpha z_1(\alpha)>0, \forall \alpha\neq 0$, $\partial_\alpha z_1(0)=0$, and $\partial_\alpha z_2(0)>0$.
3. $|z_2(\alpha)|<h_2$.
In this case, the question whether the interface turns over or not is reduced to find a negative sign (resp. positive) of $$\begin{aligned}
{\partial_\alpha}z_2(0)& \left(\int_0^\pi\frac{{\partial_\alpha}z_1(\beta)\sin(z_1(\beta))\sinh(z_2(\beta))}{(\cosh(z_2(\beta))-\cos(z_1(\beta)))^2}d\beta\right.\nonumber \\
& \left.+\frac{1}{4\pi}\int_0^\pi\frac{(\omega^{B}(\beta)+\omega^{B}(-\beta))(-1+\cosh(h_2)\cos(\beta))}{(\cosh(h_2)
-\cos(\beta))^2}d\beta\right),
\label{velocityperiodic}\end{aligned}$$ where $\omega^{B}$ is $$\begin{aligned}
\label{omegaperiodic}
\omega^{B}(\beta)=\mathcal{K}\int_{-\pi}^{\pi}\frac{\sin(\beta-z_1(\gamma)){\partial_\alpha}z_1(\gamma)}{\cosh(h_2+z_2(\gamma))-\cos(\beta-z_1(\gamma))}d\gamma\end{aligned}$$ and we assume $\bar{\rho} = \frac{1}{2}$. We refer to [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat] for the computations leading to these expressions. Plugging into we have to compute
$$\begin{aligned}
\displaystyle I^{B} & \displaystyle \equiv {\partial_\alpha}z_2(0)\left(\int_0^\pi\frac{{\partial_\alpha}z_1(\beta)\sin(z_1(\beta))\sinh(z_2(\beta))}{(\cosh(z_2(\beta))-\cos(z_1(\beta)))^2}d\beta\right. \nonumber \\
& \displaystyle +\frac{\mathcal{K}}{4\pi}\int_{0}^{\pi}\int_{-\pi}^{\pi}\frac{\sin(\beta-z_1(\gamma)){\partial_\alpha}z_1(\gamma)(-1+\cosh(h_2)\cos(\beta))}{(\cosh(h_2)-\cos(\beta))^2} \nonumber\\
&\times \displaystyle \left.\left(\frac{1}{\cosh(h_2+z_2(\gamma))-\cos(\beta-z_1(\gamma))}+\frac{1}{\cosh(h_2+z_2(\gamma))-\cos(-\beta-z_1(\gamma))}\right)d\beta d\gamma\right) \nonumber \\
& \displaystyle \equiv I^{B}_{1} + I^{B}_{2}.
\label{velocityflatbis}\end{aligned}$$
We remark that the integrand of the 2D integral above is regular (does not even have an indetermination such as the 1D one) since we are assuming that $|z_2({\alpha})| < h_2$. We calculate $I^{B}_{1}$ as in the first case. However, the choice of a uniform grid in $I^{B}_{2}$ leads to high execution times or low precision. In order to ameliorate the performance of the algorithm, we perform the integration using an adaptive algorithm. We will start with the full domain $[0,\pi] \times [-\pi,\pi]$ and in each iteration we will use a 2D Simpson’s rule.
$$\begin{aligned}
\int_{a}^{b}\int_{c}^{d} f(x,y) dx dy & \in \frac{(b-a)(d-c)}{36}\left(16f\left(\frac{a+b}{2},\frac{c+d}{2}\right)\right. \\
& +4\left(f\left(a,\frac{c+d}{2}\right)+f\left(b,\frac{c+d}{2}\right)+f\left(\frac{a+b}{2},c\right)+f\left(\frac{a+b}{2},d\right)\right) \\
& +\left.\frac{}{}\left(f\left(b,c\right)+f\left(b,d\right)+f\left(a,c\right)+f\left(a,d\right)\right)\right) \\
& -\frac{(b-a)(d-c)}{2880} \left((b-a)^4\partial^{4}_xf\left([a,b],[c,d]\right) + (d-c)^4\partial^{4}_{y}f\left([a,b],[c,d]\right)\right).\end{aligned}$$
If the result meets some tolerance requirements in the form of having absolute or relative (with respect to the volume of the integration region - see Tables \[tabla2\],\[tabla3\] for the values used) width smaller than two constants (AbsTol and RelTol) we will save it and add it to the total. Otherwise, we bisect our domain by the midpoint in each of the two directions and call the integrator again with the new 4 subdomains recursively. We also keep track of the number of calls to the integrator and limit the depth of the levels of splitting in order to prevent infinite loops or stack overflows because of too stringent tolerances, but this was not necessary for the parameters specified below.
In order to prove the theorem we will take the following curves defined for $\alpha \in [-\pi,\pi]$ and extended periodically in the horizontal variable.
$$\begin{aligned}
z_1({\alpha}) & = {\alpha}- \sin({\alpha}), \\
z_2({\alpha}) & = \left\{
\begin{array}{lr}
\displaystyle \frac{\sin(3{\alpha})}{3}- \sin({\alpha})\left(e^{-({\alpha}+2)^2}+e^{-({\alpha}-2)^2}\right)& \displaystyle \text{ in case }(a).\\
\displaystyle \frac{\sin(2{\alpha})}{2}- \frac{2}{3}\sin({\alpha})\left(e^{-({\alpha}+2)^2}+e^{-({\alpha}-2)^2}\right) & \displaystyle \text{ in case } (b).\\
\displaystyle \frac{\sin(2{\alpha})}{1.4}- 0.5\sin({\alpha})\left(e^{-({\alpha}+2)^2}+e^{-({\alpha}-2)^2}\right) & \displaystyle \text{ in case } (c).\\
\end{array}
\right.\end{aligned}$$
After running the program with the previous data we get the results summarized in Table \[tabla2\]. This shows the theorem.
Quantity (a) (b) (c)
---------------------- --------------------------- -------------------------- ---------------------------
$I^{B}_{1}$ $-0.7910^{7003}_{6993}$ $0.124312^{5192}_{6103}$ $-0.180519^{6014}_{5579}$
$I^{B}_{2}$ $-0.12^{703437}_{699367}$ $-0.1414^{5494}_{1422}$ $-0.2127^{8188}_{1946}$
$I^{B}$ $-0.918^{1044}_{0636}$ $-0.0171^{4242}_{0161}$ $-0.393^{3015}_{2390}$
Runtime (sec) 6.10 4.98 6.25
Number of calls 7305 5677 6405
$N$
(RelTol,AbsTol)
$(\mathcal{K}, h_2)$
: Results of Theorem \[ThmDepKappaIndepKappaPeriodic\].[]{data-label="tabla2"}
Notice that since $I^B$ is linear in $\mathcal{K}$, it will change sign at most once. Together with the values at $\mathcal{K} = \{-1,0,1\}$, it guarantees existence and uniqueness of $K^{*}$. We should remark that although $\mathcal{K}=\pm 1$ are not physical, they are meaningful by understanding them in the sense of the appropriate limit.
Let us assume $\bar{\rho} = \frac{1}{2}$. The turning or not (for a short enough time) for the flat at infinity case can be shown to be equivalent [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat] to finding a sign of
$$\begin{aligned}
\label{velocityflat}
I^{C} \equiv {\partial_\alpha}z_2(0)\left(\text{P.V.}\int_0^\infty\frac{4{\partial_\alpha}z_1(\beta)z_1(\beta)z_2(\beta)}{((z_1(\beta))^2+(z_2(\beta))^2)^2}-\frac{1}{2\pi}\frac{(\omega^{C}(\beta)+\omega^{C}(-\beta))\beta^2}{(\beta^2+h_2^2)^2}d\beta\right),\end{aligned}$$
where $\omega^{C}$ is defined by $$\begin{aligned}
\label{omegaflat}
\omega^{C}(\beta)=2\mathcal{K}\text{P.V.}\int_{-\infty}^{\infty}\frac{(h_2+z_2(\gamma)){\partial_\alpha}z_2(\gamma)}{(h_2+z_2(\gamma))^2+(\beta-z_1(\gamma))^2}d\gamma.\end{aligned}$$
Plugging into we have to compute [ $$\begin{aligned}
\begin{array}{rl}
\displaystyle I^{C} & \displaystyle \equiv {\partial_\alpha}z_2(0)\left( 4\text{P.V.}\int_0^\infty\frac{{\partial_\alpha}z_1(\beta)z_1(\beta)z_2(\beta)}{((z_1(\beta))^2+(z_2(\beta))^2)^2}d\beta\right. -\frac{\mathcal{K}}{\pi}\int_0^\infty\int_{-\infty}^{\infty}\frac{(h_2+z_2(\gamma)){\partial_\alpha}z_2(\gamma)\beta^2}{(\beta^2+h_2^2)^2} \\
&\displaystyle \times \left(\frac{1}{(h_2+z_2(\gamma))^2+(\beta-z_1(\gamma))^2}\right.
\left.\left.+\frac{1}{(h_2+z_2(\gamma))^2+(-\beta-z_1(\gamma))^2}\right)d\beta d\gamma\right) \\
& \displaystyle \equiv I^{C}_{1} + I^{C}_{2}.
\end{array}
\label{velocityflatbis2}\end{aligned}$$ ]{} Again, we compute $I^{C}_{1}$ as in Theorem \[ThmConfTurnsNoConfNoTurns\]. It is important to notice that we are now integrating $I^{C}_{2}$ in an unbounded region. Even in the case that $z_2$ has compact support and the integral in $\gamma$ is different than zero in a compact set, the integral in $\beta$ cannot be reduced to integrate in a bounded region. Therefore, we split $I^{C}_{2}$ into a bounded part and an unbounded one. We now explain how to deal with the latter since the former is computed as in the previous Theorem.
We want to bound [ $$\begin{aligned}
I^C_{2,ub} \equiv \displaystyle -\frac{\mathcal{K}}{\pi} {\partial_\alpha}z_2(0) \int_M^\infty\int_{-\pi}^{\pi}\frac{(h_2+z_2(\gamma)){\partial_\alpha}z_2(\gamma)\beta^2}{(\beta^2+h_2^2)^2}
&\displaystyle \left(\frac{1}{(h_2+z_2(\gamma))^2+(\beta-z_1(\gamma))^2}\right. \\
&\displaystyle \left.+\frac{1}{(h_2+z_2(\gamma))^2+(-\beta-z_1(\gamma))^2}\right) d\gamma d\beta\end{aligned}$$ ]{}and we will take the following curves:
$$\begin{aligned}
z_1({\alpha}) & = {\alpha}- \sin({\alpha})e^{-K{\alpha}^{2}}, \quad K = 10^{-2}, \\
z_2({\alpha}) & = \left\{
\begin{array}{lr}
\displaystyle \left(\frac{\sin(3{\alpha})}{3}- \sin({\alpha})\left(e^{-({\alpha}+2)^2}+e^{-({\alpha}-2)^2}\right)\right)1_{\{|{\alpha}| \leq \pi\}} & \displaystyle \text{ in case }(a).\\
\displaystyle \left(\frac{\sin(2{\alpha})}{2}- 0.85\sin({\alpha})\left(e^{-({\alpha}+2)^2}+e^{-({\alpha}-2)^2}\right)\right)1_{\{|{\alpha}| \leq \pi\}} & \displaystyle \text{ in case } (b).\\
\displaystyle \left(\frac{\sin(2{\alpha})}{1.8}- 0.7\sin({\alpha})\left(e^{-({\alpha}+2)^2}+e^{-({\alpha}-2)^2}\right)\right)1_{\{|{\alpha}| \leq \pi\}} & \displaystyle \text{ in case } (c).\\
\end{array}
\right.\end{aligned}$$
We will provide bounds for $I^C_{2,ub}$ in this way:
[ $$\begin{aligned}
\label{eqBd2D}
|I^C_{2,ub}| \leq \displaystyle \frac{|\mathcal{K}|}{\pi} |{\partial_\alpha}z_2(0)| \int_M^\infty \frac{\beta^2}{(\beta^2+h_2^2)^2} d\beta \int_{-\pi}^{\pi}
&\displaystyle \left(\frac{|h_2+z_2(\gamma)||{\partial_\alpha}z_2(\gamma)|}{(h_2+z_2(\gamma))^2+(\beta-z_1(\gamma))^2}\right. \nonumber \\
&\displaystyle \left.+\frac{|h_2+z_2(\gamma)||{\partial_\alpha}z_2(\gamma)|}{(h_2+z_2(\gamma))^2+(-\beta-z_1(\gamma))^2}\right)d\gamma\end{aligned}$$ ]{}and let $$\begin{aligned}
G(\beta) \equiv \frac{|\mathcal{K}|}{\pi} \int_{-\pi}^{\pi}
\left(\frac{|h_2+z_2(\gamma)||{\partial_\alpha}z_2(\gamma)|}{(h_2+z_2(\gamma))^2+(\beta-z_1(\gamma))^2}\right.
\left.+\frac{|h_2+z_2(\gamma)||{\partial_\alpha}z_2(\gamma)|}{(h_2+z_2(\gamma))^2+(-\beta-z_1(\gamma))^2}\right)d\gamma.\end{aligned}$$
It is easy to check that $G(\beta)$ is monotone in $\beta$ for $\beta$ larger than $\|z_1\|_{L^{\infty}(-\pi,\pi)}$. Indeed,
$$G(\beta) \leq G(M), \quad \text{ if we take } M = 14 \pi,$$ which is our choice of $M$ for the computer verification. Plugging this relation into we obtain
$$\begin{aligned}
|I^C_{2,ub}| & \leq |{\partial_\alpha}z_2(0)| G(M) \int_M^\infty \frac{\beta^2}{(\beta^2+h_2^2)^2} d\beta \nonumber \\
& = |{\partial_\alpha}z_2(0)| G(M) \left(\frac{\pi}{4h_2} - \frac{1}{h_2}\arctan\left(\frac{M}{h_2}\right) + \frac{M}{2(h_2^2+M^2)}\right).\end{aligned}$$
Thus, we are left to compute rigorous bounds for $G$. Let us denote by $$\begin{aligned}
\label{defIG}
IG(\beta,\gamma) = \frac{|\mathcal{K}|}{\pi}
\left(\frac{|h_2+z_2(\gamma)||{\partial_\alpha}z_2(\gamma)|}{(h_2+z_2(\gamma))^2+(\beta-z_1(\gamma))^2}\right.
\left.+\frac{|h_2+z_2(\gamma)||{\partial_\alpha}z_2(\gamma)|}{(h_2+z_2(\gamma))^2+(-\beta-z_1(\gamma))^2}\right)
\end{aligned}$$ the integrand of $G$. That means $$\begin{aligned}
G(\beta) = \int_{-\pi}^{\pi} IG(\beta,\gamma)d\gamma.\end{aligned}$$ We perform the following integration scheme: $$\begin{aligned}
\int_{\gamma_i}^{\gamma_{i+1}} IG(\beta,\gamma)d\gamma =
\left\{
\begin{array}{cc}
IG(\beta,[\gamma_{i},\gamma_{i+1}])(\gamma_{i+1}-\gamma_{i}) & \text{ if } 0 \in IG(\beta,[\gamma_{i},\gamma_{i+1}]) \\
\frac{(\gamma_{i+1}-\gamma_{i})}{6}\left(IG(\beta,\gamma_{i})+IG(\beta,\gamma_{i+1})+4IG\left(\beta,\frac{\gamma_{i}+\gamma_{i+1}}{2}\right)\right) & \\
-\frac{1}{2880}(\gamma_{i+1}-\gamma_{i})^{5}\partial^{4}_{\gamma}IG(\beta,[\gamma_{i},\gamma_{i+1}]) & \text{ otherwise}
\end{array}
\right.\end{aligned}$$ in which we apply a Simpson rule for the case where the integrand is smooth, otherwise we take the full interval that results in evaluating the integrand in the whole integration interval. We perform the integration in $\gamma$ over a uniform mesh $-\pi = \gamma_0 < \gamma_1 < \ldots < \gamma_{N_2} = \pi,$ $\gamma_i = -\pi + \frac{2\pi}{N_2}i$.
Therefore, adding all the contributions $$\begin{aligned}
G(M) = \sum_{i=0}^{N_2-1} \int_{\gamma_{i}}^{\gamma_{i+1}}IG(M,\gamma)d\gamma,\end{aligned}$$ we get the desired bound on $T$. The variable $N_2$ is user-specified in our program. The results are summarized in Table \[tabla3\]. These prove the Theorem.
Quantity (a) (b) (c)
-------------------------- --------------------------- ------------------------- ----------------------------
$I^{C}_{1}$ $-0.745640^{1337}_{0299}$ $0.00147^{1972}_{2074}$ $-0.0087191^{9854}_{1782}$
$|I^C_{2,ub}|$ $0.0000^{0000}_{2668}$ $0.0000^{0000}_{2697}$ $0.0000^{0000}_{3183}$
$I^{C}_{2} - I^C_{2,ub}$ $-0.020^{6841}_{3465}$ $-0.011^{6887}_{3785}$ $-0.009^{9556}_{5855}$
$I^{C}$ $-0.76^{63509}_{59599}$ $-0.0^{10244}_{09879}$ $-0.018^{7067}_{2728}$
Runtime (sec) 6.96 8.30 8.11
Number of calls 9205 9177 8805
$(N,N_2)$
(RelTol,AbsTol)
$(\mathcal{K}, h_2)$
: Results of Theorem \[ThmDepKappaIndepKappaFlat\][]{data-label="tabla3"}
Again, as in the previous Theorem, $I^{C}$ is linear in $\mathcal{K}$ and by the same reasoning, we have existence and uniqueness of $K^{*}$.
Notice our choice of the numerical parameters $N, N_2, M$ is not optimal. Smaller parameters might also work, however, as the time required to compute the intervals is not very long, we didn’t try to optimize in terms of choosing different values of $N, N_2, M$. In Theorem \[ThmConfinedInhomogeneous\], where the computational costs are higher, we integrate in an adaptive way without fixing the number of points.
The inhomogeneous, confined case {#SubsectionProofThm4}
--------------------------------
In this subsection we will detail the refinements and technical details that led to the bifurcation diagram shown in Figure \[FigBifurcacion\], which illustrates Theorem \[ThmConfinedInhomogeneous\].
### Dimension reduction by complex integration
In [@Berselli-Cordoba-GraneroBelinchon:local-solvability-singularities-muskat] the existence of turning singularities is proved by a continuity argument for the full problem . Here we obtain these turning waves for the full range $|\mathcal{K}|<1$. We will write the equation for the velocity in a more suitable way by calculating explicitly some of the integrals using complex integration. We remark that we are transforming an a priori 4-dimensional problem into a 2-dimensional one, dramatically reducing the resources needed for its computation. We will denote the complex argument function, *i.e.* the function that given a complex number returns its phase, by $\arg(z)$ and consider the branch that takes values in $[-\pi,\pi)$. We start with some useful Lemmas whose proof (similar to the proof of Lemma \[lemacitado\]) we omit for the sake of brevity:
\[IIIlem1\] We have, for $-1 < d < 1$, $c, y \in \mathbb{R}$: $$\int_{\mathbb R}\frac{\cos(y\xi)d\xi}{\cosh(c-\xi)+d}=-\frac{2\pi}{\sqrt{1-d^2}}\frac{\cos(yc)\sinh(y \arg(-d+\sqrt{1-d^2}i)-y\pi)}{\sinh(\pi y)}.$$
\[IIIlem2\] We have, for $-1 < d < 1$, $b, c, y \in \mathbb{R}$: $$\begin{gathered}
\int_{\mathbb R}\frac{\cos(y(\xi+c))(-\cosh(\xi)\cos(h_2)+b)d\xi}{(\cosh(\xi)+d)^ 2}\\
=\frac{2\pi}{\sinh(\pi y)}\left(\frac{y\cosh(y\cdot \emph{arg}(-d+i\sqrt{1-d^2})-\pi y)\cos(cy)(d\cos(h_2)+b)}{1-d^2}\right.\\
\left.+\frac{\sinh\left(y\cdot \emph{arg}(-d+i\sqrt{1-d^2})-\pi y\right)\cos(cy)\left(\cos(h_2)+db\right)}{(\sqrt{1-d^2})^3}\right).\end{gathered}$$
\[IIIlem3\] We have, for $0<a<\pi\leq c$, $b\in{\mathbb R}$ $$\int_{\mathbb R}\frac{\cos(yb)\sinh(ay)}{\sinh(cy)}dy=\frac{\pi}{c}\frac{\sin\left(\frac{\pi}{c}a\right)}{\cos\left(\frac{\pi}{c}a\right)+\cosh\left(\frac{\pi}{c}b\right)}.$$
Using classical trigonometric identities, we have, $$\cos(yb)\sinh(ay)=\frac{1}{2}\left[\sinh(ay-iby)+\sinh(ay+iby)\right],$$ so, we need to compute the integral $$I_{ss}=\int_{{\mathbb R}}\frac{\sinh(wy)}{\sinh(cy)}dy,$$ for the appropriate $w$.
![Contour of integration.[]{data-label="poloslema4"}](PolosLema4.eps)
We use complex integration and the Residue Theorem. We consider the contour given by $\displaystyle\mathcal{C}_1=\cup_i\Gamma_i$ and by $R_1$ the interior region delimited by $\mathcal{C}_1$, where $$\Gamma_1=\left\{z=x+\frac{3}{2}\frac{\pi}{c}i, x\in (-R,R)\right\}, \text{ }\Gamma_2=\{z=x, x\in(-R,-\delta)\cup(\delta,R)\},$$ $$\Gamma_3=\{z=\delta e^{i\theta}, \theta\in(0,\pi)\}, \Gamma_{4}=\left\{z=\pm R+iy,y\in(0,\frac{3}{2}\frac{\pi}{c})\right\}.$$ Since $c>a$ and $b$ only deals with oscillations with bounded amplitude, we have $$\lim_{R\rightarrow\infty}\int_{\Gamma_4}\frac{\sinh(wz)}{\sinh(cz)}dz=0,$$ $$\lim_{\delta\rightarrow 0}\int_{\Gamma_3}\frac{\sinh(wz)}{\sinh(cz)}dz\leq \lim_{\delta\rightarrow 0}C\delta=0.$$ Thus, we are left with $$\lim_{\delta\rightarrow 0, R\rightarrow\infty}\int_{\mathcal{C}_1}\frac{\sinh(wz)}{\sinh(cz)}dz=\int_{-\infty}^\infty\frac{\sinh(wy)}{\sinh(cy)}dy+\int_{-\infty}^\infty\frac{\sinh\left(wy+w\frac{3}{2}\frac{\pi}{c}i\right)}{i\cosh(c y)}dy=2\pi i\text{Res}_{ss},$$ where $$\text{Res}_{ss}=\sum_{\xi\in R_1, \xi \text{ poles}} \text{Res}\left(\frac{\sinh(wz)}{\sinh(cz)},\xi\right)=-\frac{\sinh\left(\frac{w\pi}{c}i\right)}{c}.$$ We need to compute a helpful integral $$I_{cc}=\int_{-\infty}^\infty \frac{\cosh(wy)}{\cosh(cy)}dy.$$ We define the contour $\mathcal{C}_2$ as the boundary of the rectangle $R_2=[-R,R]\times [0,\frac{\pi}{c}]$. We get $$\int_{\mathcal{C}_2}\frac{\cosh(wz)}{\cosh(cz)}dz=\int_{-\infty}^\infty\frac{\cosh(wy)}{\cosh(cy)}dy+\int_{-\infty}^\infty\frac{\cosh\left(wy+w\frac{\pi}{c}i\right)}{\cosh(c y)}dy=2\pi i\text{Res}_{cc},$$ where $$\text{Res}_{cc}=\sum_{\xi\in R_2, \xi \text{ poles}} \text{Res}\left(\frac{\cosh(wz)}{\cosh(cz)},\xi\right)=-\frac{\cosh\left(\frac{w\pi}{2c}i\right)}{ci}.$$ Using trigonometric identities, we obtain $$\left(1+\cos\left(w\frac{\pi}{c}\right)\right)I_{cc}+i\sin\left(w\frac{\pi}{c}\right)\int_{-\infty}^\infty\frac{\sinh\left(wy\right)}{\cosh(c y)}dy=2\pi i\text{Res}_{cc}.$$ Therefore, using the oddness of the second integrand, $$I_{cc}=\frac{\frac{2\pi}{c}\cosh\left(\frac{w\pi i}{2c}\right)}{1+\cos\left(w\frac{\pi}{c}\right)}=\frac{\pi}{c}\frac{1}{\cos\left(w\frac{\pi}{2c}\right)}.$$ Inserting this value in the previous expression for $I_{ss}$, we obtain $$I_{ss}+\sin\left(w\frac{3\pi}{2c}\right)I_{cc}=2\pi i \frac{\sinh\left(w\frac{\pi i}{c}\right)}{-c}=\frac{2\pi\sin\left(w\frac{\pi}{c}\right)}{c},$$ thus, $$I_{ss}=\frac{\pi}{c}\tan\left(w\frac{\pi}{2c}\right).$$ Finally, $$\int_{\mathbb R}\frac{\cos(yb)\sinh(ay)}{\sinh(cy)}dy=\frac{\pi}{2c}\left(\tan\left((a-ib)\frac{\pi}{2c}\right)+\tan\left((a+ib)\frac{\pi}{2c}\right)\right)
=\frac{\pi}{c}\frac{\sin\left(\frac{\pi}{c}a\right)}{\cos\left(\frac{\pi}{c}a\right)+\cosh\left(\frac{\pi}{c}b\right)}.$$
Then, according to Lemmas \[IIIlem1\] and \[IIIlem2\], we have $$\begin{gathered}
\label{IIIa}
\int_{\mathbb R}\frac{\cos(y\xi)d\xi}{\cosh(-\beta-z_1(\gamma)-\xi)-\cos(h_2+z_2(\gamma))}\\
=\frac{2\pi}{\sin(h_2+z_2(\gamma))}\frac{\cos(y(\beta+z_1(\gamma)))\sinh(y(\pi-h_2-z_2(\gamma)))}{\sinh(\pi y)}\end{gathered}$$ and $$\begin{gathered}
\label{IIIb}
\int_{\mathbb R}\frac{\cos(y\xi)d\xi}{\cosh(-\beta-z_1(\gamma)-\xi)+\cos(h_2-z_2(\gamma))}\\
=\frac{2\pi}{\sin(h_2-z_2(\gamma))}\frac{\cos(y(\beta+z_1(\gamma)))\sinh(y(h_2-z_2(\gamma)))}{\sinh(\pi y)},\end{gathered}$$ $$\label{IIIc}
\int_{\mathbb R}\frac{\cos(y(\beta+z_1(\gamma)))(-\cosh(\beta)\cos(h_2)+1)d\beta}{(\cosh(\beta)-\cos(h_2))^ 2}=2\pi\frac{y\cosh(y (\pi- h_2))\cos(z_1(\gamma)y)}{\sinh(\pi y)}$$ and $$\begin{gathered}
\label{IIId}
\int_{\mathbb R}\frac{\cos(y(\beta+z_1(\gamma)))(-\cosh(\beta)\cos(h_2)-\cos^ 2(h_2)+\sin^2(h_2))d\beta}{(\cosh(\beta)+\cos(h_2))^ 2}\\
=\frac{2\pi \cos(z_1(\gamma)y)}{\sinh(\pi y)}\left(y\cosh(y h_2)-\frac{2 \sinh\left(y h_2\right)}{\tan(h_2)}\right).\end{gathered}$$
In particular, using Lemma \[IIIlem3\], $$\begin{gathered}
\label{eqaux}
\int_{\mathbb R}\frac{\cos(y(\beta+z_1(\gamma)))\sinh((\pi-h_2-z_2(\gamma))y)}{\sinh(\pi y)}dy\\
=\frac{\sin\left(\pi-h_2-z_2(\gamma)\right)}{\cos\left(\pi-h_2-z_2(\gamma)\right)+\cosh\left(\beta+z_1(\gamma)\right)}\\
=\frac{\sin\left(h_2+z_2(\gamma)\right)}{-\cos\left(h_2+z_2(\gamma)\right)+\cosh\left(\beta+z_1(\gamma)\right)},\end{gathered}$$ and $$\begin{gathered}
\label{eqaux2}
\int_{\mathbb R}\frac{\cos(y(\beta+z_1(\gamma)))\sinh((-h_2+z_2(\gamma))y)}{\sinh(\pi y)}dy=\frac{\sin\left(-h_2+z_2(\gamma)\right)}{\cos\left(-h_2+z_2(\gamma)\right)+\cosh\left(\beta+z_1(\gamma)\right)}.\end{gathered}$$
We proceed now to calculate ${\partial_\alpha}v_1(0)$. We fix $\bar{\rho} = 1$. Then, the appropriate expression is $${\partial_\alpha}v_1(0)={\partial_t}{\partial_\alpha}z_1(0)=I_1+I_2,$$ where $$I_1=2{\partial_\alpha}z_2(0)\int_0^\infty \frac{{\partial_\alpha}z_1(\beta)\sinh(z_1(\beta))\sin(z_2(\beta))}{\left(\cosh(z_1(\beta))-\cos(z_2(\beta))\right)^2}+\frac{{\partial_\alpha}z_1(\beta)\sinh(z_1(\beta))\sin(z_2(\beta))}{\left(\cosh(z_1(\beta))+\cos(z_2(\beta))\right)^2}d\beta,$$ and $$\begin{gathered}
I_2=\frac{{\partial_\alpha}z_2(0)}{4\pi}\text{P.V.}\int_{\mathbb R}\frac{\varpi_2(-\beta)(-\cosh(\beta)\cos(h_2)+1)}{(\cosh(\beta)-\cos(h_2))^2}d\beta\\
+\frac{{\partial_\alpha}z_2(0)}{4\pi}\text{P.V.}\int_{\mathbb R}\frac{\varpi_2(-\beta)(-\cosh(\beta)\cos(h_2)-\cos^2(h_2)+\sin^2(h_2))}{(\cosh(\beta)+\cos(h_2))^2}d\beta,\end{gathered}$$ where $\varpi_2$ is given in . Now we use Lemmas \[IIIlem1\] and \[IIIlem2\] to compute explicitly some of the integrals in $I_2$. Notice that the space is $\sigma-$finite and, taking the absolute value, we can apply Tonelli-Fubini Theorem. First, we integrate in $\xi$ using and , and by Lemma \[IIIlem3\] and equations and , we obtain $$\begin{aligned}
I_2=&\frac{{\partial_\alpha}z_2(0)\mathcal{K}}{2\pi}\bigg{[}\text{P.V.}\int_{\mathbb R}\int_{\mathbb R}\text{P.V.}\int_{\mathbb R}\frac{
\frac{{\partial_\alpha}z_2(\gamma)\sinh(y(\pi-h_2-z_2(\gamma)))}{\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y)}\cos(y(\beta+z_1(\gamma))) }{(\cosh(\beta)-\cos(h_2))^2 (-\cosh(\beta)\cos(h_2)+1)^{-1}}dy d\gamma d\beta\\
&+\text{P.V.}\int_{\mathbb R}\int_{\mathbb R}\text{P.V.}\int_{\mathbb R}\frac{\frac{{\partial_\alpha}z_2(\gamma)\sinh(y(\pi-h_2-z_2(\gamma)))}{\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y)}\cos(y(\beta+z_1(\gamma)))}{(\cosh(\beta)+\cos(h_2))^2(-\cosh(\beta)\cos(h_2)-\cos(2h_2))^{-1}}dy d\gamma d\beta\\
&+\text{P.V.}\int_{\mathbb R}\int_{\mathbb R}\text{P.V.}\int_{\mathbb R}\frac{\frac{{\partial_\alpha}z_2(\gamma)\sinh(y(h_2-z_2(\gamma)))}{\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y)}\cos(y(\beta+z_1(\gamma)))}{(\cosh(\beta)-\cos(h_2))^2(-\cosh(\beta)\cos(h_2)+1)^{-1}} dy d\gamma d\beta\\
&+\text{P.V.}\int_{\mathbb R}\int_{\mathbb R}\text{P.V.}\int_{\mathbb R}\frac{\frac{{\partial_\alpha}z_2(\gamma)\sinh(y(h_2-z_2(\gamma)))}{\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y)}\cos(y(\beta+z_1(\gamma)))}{(\cosh(\beta)+\cos(h_2))^2(-\cosh(\beta)\cos(h_2)-\cos(2h_2))^{-1}}dy d\gamma d\beta\bigg{]}.\end{aligned}$$
Now we integrate in $\beta$ using and : $$\begin{gathered}
I_2={\partial_\alpha}z_2(0)\mathcal{K}\int_{\mathbb R}\text{P.V.}\int_{\mathbb R}\frac{{\partial_\alpha}z_2(\gamma)\cos(yz_1(\gamma))}{(\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y))\sinh(\pi y)}\\
\times\left(\sinh(y(\pi-h_2-z_2(\gamma)))+\sinh(y(h_2-z_2(\gamma)))\right)\\
\times \left(y\cosh(y(\pi-h_2))+y\cosh(yh_2)-\frac{2\sinh(yh_2)}{\tan(h_2)}\right) d\gamma dy.\end{gathered}$$ Using the oddness of $z_i$, we obtain $$\begin{gathered}
I_2=4{\partial_\alpha}z_2(0)\mathcal{K}\int_0^\infty\int_0^\infty
\frac{{\partial_\alpha}z_2(\gamma)\cos(yz_1(\gamma))}{(\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y))\sinh(\pi y)}\\
\times\left(2\sinh\left(y\frac{\pi}{2}\right)\cosh\left(y z_2(\gamma)\right)\cosh\left(y\left(\frac{\pi}{2}-h_2\right)\right)\right)\\
\times \left(y\cosh(y(\pi-h_2))+y\cosh(yh_2)-\frac{2\sinh(yh_2)}{\tan(h_2)}\right) d\gamma dy,\end{gathered}$$ and, using trigonometrical identities, we get the final expression $$\begin{gathered}
I_2=4{\partial_\alpha}z_2(0)\mathcal{K}\int_0^\infty\int_0^\infty
\frac{{\partial_\alpha}z_2(\gamma)\cos(z_1(\gamma)y)}{(\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y))\cosh\left(y\frac{\pi}{2}\right)}\\
\times\left(2y\cosh\left(\frac{y\pi}{2}- y h_2\right)\cosh\left(\frac{y \pi}{2}\right)-\frac{2\sinh\left(y h_2\right)}{\tan(h_2)}\right)\\
\times \cosh\left(y z_2(\gamma)\right)\cosh\left(y\left(\frac{\pi}{2}-h_2\right)\right) d\gamma dy.\end{gathered}$$
### Technical details concerning Theorem \[ThmConfinedInhomogeneous\](a)
The first four statements of Theorem \[ThmConfinedInhomogeneous\] can be deduced from Figure \[FigBifurcacion\]. In this section, we explain the algorithms and the technical details that led us to the rigorous computation of the bifurcation diagram.
We have been more careful in the optimization of the codes concerning the diagram since we were expecting a higher computation time. However, the possibility of parallelization did not force us to optimize up to a very low level, just to simply maintain the correct complexity of the code. The implementation is now split into several files, and many of the headers of the functions (such as the integration methods) contain pointers to functions (the integrands) so that they can be reused for an arbitrary amount of integrals with minimal changes and easy and safe debugging.
The initial condition family we used for the bifurcation diagram was $$\begin{aligned}
z_1({\alpha}) & = {\alpha}- \sin({\alpha})e^{-K {\alpha}^{2}}, \quad K = 10^{-4} \nonumber \\
z_2({\alpha}) & = h_2\frac{3}{\pi}\left(\frac{\sin(3{\alpha})}{3}- \frac{\sin({\alpha})}{2.5}\left(e^{-({\alpha}+2)^2}+e^{-({\alpha}-2)^2}\right)\right)1_{\{|{\alpha}| \leq \pi\}}. \label{initialConditionBif}\end{aligned}$$
It is easy to check that $z$ is odd, $z_1({\alpha})$ is strictly monotone except at ${\alpha}= 0$, ${\partial_\alpha}z_2(0) > 0$ and $|z_2| < h_2$. A more precise bound is given in Lemma \[LemmaBoundZ2\]. We will compute the bifurcation diagram in the region $(h_2,\mathcal{K}) = \left[\frac{1}{4},\frac{5}{4}\right] \times \left[-1,1\right]$.
The algorithm for the computation of the bifurcation diagram is as follows: we define a structure called `ParameterSet`, which encapsulates all the necessary information about the parameters and the information needed by the integration procedures in order to compute ${\partial_\alpha}v_1(0,0)$ for those parameters. More precisely, a ParameterSet contains:
- Two intervals, `Left` and `Right`, which set the limits for the bounded, singularity and unbounded regions (i.e. singularity $ = [0,\text{Left}]$, bounded $= [\text{Left,Right}]$, unbounded $ = [\text{Right},\infty)$, whenever they make sense). In our proof, Left $=0.125$, Right = $16.125$.
- Two doubles, `AbsTol` and `RelTol`, which limit the precision up to which the integrals are computed. In our proof, AbsTol $=$ RelTol $= 10^{-5}$.
- Two intervals, `h_2` and `Kappa`, which are the rectangle in the parameter space we are calculating.
We mantain a queue (implemented using the Standard Template Library (STL) `Queue`), in which we store all the ParameterSets to be computed. While the queue is not empty, we take the top element, pop it and give an enclosure of ${\partial_\alpha}v_1(0,0)$ for this region. Three different cases arise:
- The enclosure is positive.
- The enclosure is negative.
- We can not say anything about its positivity.
In both the first two cases, the result is output to its corresponding file (one for the regions for which there is a turning singularity, another for the ones for which there is not). If, on the contrary, we are in the third case, the ParameterSet is split into other narrower ParameterSets which are pushed in the queue. This splitting is only done if the dimensions (both in $h_2$ and $\mathcal{K}$) are bigger than a given limit, which in our case was set to $5 \cdot 10^{-3}$ for the 2 parameters. Moreover, the splitting is not done in a uniform way. We found heuristically that a splitting that cut in 4 in the $h_2$-dimension and in 2 in the $\mathcal{K}$-dimension balanced the width of $I_1$ and $I_2$. If the parameter interval is too narrow, we output the result to a third file, which accounts for the unknown regions.
$I_1$ is split into two parts as in the discussion from Subsection \[SubsectionProofsThm23\]: a bounded one and a singularity one. The bounded part is calculated using a Gauss-Legendre quadrature of order 2, given by $$\begin{gathered}
\int_{a}^{b} f(\eta) d\eta \in \frac{b-a}{2}\left(f\left(\frac{b-a}{2}\frac{\sqrt{3}}{3} + \frac{b+a}{2}\right)\right.\\\left.+f\left(-\frac{b-a}{2}\frac{\sqrt{3}}{3}+ \frac{b+a}{2}\right)\right)
+\frac{1}{4320}(b-a)^{5}f^{4}([a,b]).\end{gathered}$$
Other quadratures of several orders (Gauss-Legendre, Newton-Cotes) were tested and they resulted either in worse results or similar results but worse runtime performance. Moreover, the integration was done in an adaptive way. For each region, we accepted or rejected the result depending on the width in an absolute and a relative way. It is important to notice that because of the uncertainty of the parameters, division by zero is easy to find, even in small integration intervals. In such cases, bisection in the parameter space is needed. We developed extra mechanisms to take care of these cases and discard a ParameterSet once a division by zero is found.
The number of levels of subdivision was also limited, since the uncertainty of the parameters might yield wide enclosures of the integral even with infinite precision. In our case, the maximum number of subdivisions for a non-singular one-dimensional integral was 18, totaling a maximum number of subintervals equal to $2^{18}$, which can be carried out roughly under 90 seconds. Another feature of the integration method is that instead of subdividing the integration intervals by the midpoint (in other words, by the arithmetic mean of the endpoints), we subdivided by the *geometric* mean of the endpoints. While the arithmetic division minimizes the length of the longest piece after the division, the geometric one minimizes the piece with the biggest ratio between its endpoints. This can be particularly useful in many cases: for example in order to avoid divisions by zero for integrands of the type $\frac{1}{\sinh(ay)-\sinh(by)}$, which is the case of $I_2$. However, the geometric division also performs better for $I_1$ and we bisect using that method.
The singular part of $I_1$ was also integrated and not bounded as in the previous sections. In this case, the algorithm works as follows: we perform Taylor series of order 6 and 4 respectively of the numerator and the denominator and integrate as in . Potentially this could fail because the uncertainty in $h_2$ (and therefore in $z_2$) could yield a Taylor series in which $0$ belongs to ${\partial_\alpha}^{4} \mathcal{D}([0,\varepsilon])$. Whenever this happens, we try to integrate using a Gauss-Legendre quadrature of order 2. The integration division is in this case arithmetic since 0 belongs to our integration domain. The maximum subdivision level was set to 12 ($2^{12}$ intervals). If even the Gauss-Legendre quadrature fails, then we return an error and bisect in the space of parameters.
We note that both the singular and the bounded part of $I_1$ are independent of $\mathcal{K}$. For performance purposes, we kept two STL `Map<ParameterSet,Interval,ComparisonFunction>`, with a ComparisonFunction that only sorts by $h_2$ so that for a ParameterSet, we check if we have calculated the values of $I_1$ for that $h_2$ before. If not, once we calculate them we store them in the map.
Regarding $I_2$, we divide it into three regions: singularity, bounded and unbounded. The bounded region is calculated using a 2 dimensional Gauss-Legendre quadrature and geometric division of the subintervals. Here it is clear the need of this subdivision because we want to avoid division by zero and $$\begin{aligned}
0 \in \sinh(\pi Y) - \sinh(2 h_2 Y) \Leftrightarrow \frac{\sup(Y)}{\inf(Y)} \geq \frac{\pi}{2h_2},\end{aligned}$$ so the objective is to keep the quotient $\frac{\sup(I)}{\inf(I)}$ as small as possible for every integration interval $I$. The maximum number of subdivision levels was set to 8 ($2^{16}$ rectangles) and the computation time of the bounded part of $I_2$ was well under the 2 minute mark.
For the singularity part, we took the intersection between the interval computed by Gauss-Legendre integration and Taylor expansions (in this case, the expansion is of order 1 in both the numerator and the denominator). The maximum number of subdivision levels was 7 ($2^{14}$ rectangles).
We end the discussion with the estimations of the unbounded region $$\begin{gathered}
I_2^{ub}=4{\partial_\alpha}z_2(0)|\mathcal{K}|\int_M^\infty\bigg{(}\int_0^\pi\bigg{|}
\frac{{\partial_\alpha}z_2(\gamma)\cos(z_1(\gamma)y)}{(\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y))\cosh\left(y\frac{\pi}{2}\right)}\\
\times\left(2y\cosh\left(\frac{y\pi}{2}- y h_2\right)\cosh\left(\frac{y \pi}{2}\right)-\frac{2\sinh\left(y h_2\right)}{\tan(h_2)}\right)\\
\times \cosh\left(y z_2(\gamma)\right)\cosh\left(y\left(\frac{\pi}{2}-h_2\right)\right)\bigg{|} d\gamma \bigg{)} dy.\end{gathered}$$
We will bound the tails using the following inequalities, which are very easy to check,
$$\begin{aligned}
\frac{1}{2} e^{x} \leq \cosh(x) \leq e^{x} \\
\frac{1}{4} e^{x} \leq \sinh(x) \leq \frac{1}{2} e^{x}, \quad x \geq \log(2).\end{aligned}$$
Now we can show the following naive bounds
$$\begin{aligned}
4\cosh\left(y z_2(\gamma)\right)\cosh\left(y\left(\frac{\pi}{2}-h_2\right)\right)
& \leq 4 e^{-y\left(h_2-\frac{\pi}{2}-\|z_2\|_{L^{\infty}}\right)} \\
\left|\frac{2y\cosh\left(\frac{y\pi}{2}- y h_2\right)\cosh\left(\frac{y \pi}{2}\right)-\frac{2\sinh\left(y h_2\right)}{\tan(h_2)}}{\cosh\left(y\frac{\pi}{2}\right)}\right|
& \leq 2e^{y\left(\frac{\pi}{2}-h_2\right)}y+2\frac{e^{y\left(h_2-\frac{\pi}{2}\right)}}{\tan\left(h_2\right)}.\end{aligned}$$
For the last factor, we distinguish two cases:
$$\begin{gathered}
\label{bichaco}
\frac{1}{(\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y))} \leq
\left\{
\begin{array}{cc}
\displaystyle\frac{2e^{-\pi y}}{1-e^{-2\pi M}}, & \text{ if } \mathcal{K} \geq 0 \\
\displaystyle e^{-\pi y} \frac{2}{1-e^{-2\pi M} - |\mathcal{K}|e^{-M(\pi - 2h_2)}}, & \text{ if } \mathcal{K} < 0 \\
\end{array}
\right\}\\
\equiv e^{-\pi y}C(M,\mathcal{K},h_2),\end{gathered}$$
where we have used $$\begin{aligned}
\frac{1}{\sinh(\pi y)} = \frac{2}{e^{\pi y} - e^{-\pi y}} = e^{-\pi y}\frac{2}{1 - e^{-2\pi y}}
\leq e^{-\pi y}\frac{2}{1 - e^{-2\pi M}}.\end{aligned}$$ Putting all the estimates together, we need to integrate in $y$ and we get
$$\begin{aligned}
& 8 \max_{\mathcal{K}, h_2}\{C(M,\mathcal{K},h_2)\}\int_{M}^{\infty}\left(\frac{e^{-y(\pi - \|z_2\|_{L^{\infty}})}}{\tan(h_2)}+e^{-y(2h_2-\|z_2\|_{L^{\infty}})}y\right)dy \\
= & 8 \max_{\mathcal{K},h_2}\{C(M,\mathcal{K},h_2)\}\frac{e^{-M(\pi - \|z_2\|_{L^{\infty}})}}{\tan(h_2)\left(\pi - \|z_2\|_{L^{\infty}}\right)} \\
+ & 8 \max_{\mathcal{K},h_2}\{C(M,\mathcal{K},h_2)\}\frac{e^{-M(2h_2 - \|z_2\|_{L^{\infty}})}}{2h_2 - \|z_2\|_{L^{\infty}}}\left(M + \frac{1}{2h_2-\|z_2\|_{L^{\infty}}}\right)\end{aligned}$$
Finally, we can bound $\|z_2\|_{L^{\infty}}$ in terms of $h_2$ in the following way:
\[LemmaBoundZ2\] Let $z_2({\alpha})$ be $$\begin{aligned}
z_2({\alpha}) & = \frac{3}{\pi}\left(\frac{\sin(3{\alpha})}{3}- \frac{\sin({\alpha})}{2.5}\left(e^{-({\alpha}+2)^2}+e^{-({\alpha}-2)^2}\right)\right)1_{\{|{\alpha}| \leq \pi\}}\end{aligned}$$ Then $\|z_2\|_{L^{\infty}} < 0.65$.
The proof is computer-assisted and the code can be found in the supplementary material. The algorithm is the classical branch and bound [@Neumaier:branch-and-bound]: given an interval $I$ we first compute an enclosure $z_2(I)$. If the diameter is not small enough (smaller than a given tolerance), we split $I$ into $I^{L}, I^{R}$ such that $I \subset I^{L} \cup I^{R}$ and call the same function to get their $L^{\infty}$ norms recursively. We merge the results using that $$\begin{aligned}
\|z_2\|_{L^{\infty}(I)} \subset \max\left\{\|z_2\|_{L^{\infty}(I^{L})}, \|z_2\|_{L^{\infty}(I^{R})}\right\},\end{aligned}$$ where the $\max$ operation between intervals was defined in . For a tolerance equal to $2 \cdot10^{-6}$, our program outputs the following bound: $$\begin{aligned}
\|z_2\|_{L^{\infty}} \in 0.64627^{3239}_{4666}.\end{aligned}$$ This proves the Lemma.
Thus, we can bound the contribution of the unbounded part $I_{2}^{ub}$ by $$\begin{gathered}
{\partial_\alpha}z_2(0)|\mathcal{K}| \int_{0}^\pi |\partial_{\gamma} z_2(\gamma)| d\gamma\bigg{(}
8 \max_{\mathcal{K},h_2}\{C(M,\mathcal{K},h_2)\}\frac{e^{-M(\pi - 0.65 h_2})}{\tan(h_2)\left(\pi - 0.65h_2\right)} \\
+ 8 \max_{\mathcal{K},h_2}\{C(M,\mathcal{K},h_2)\}\frac{e^{-M(1.35h_2})}{1.35h_2}\left(M + \frac{1}{1.35h_2}\right)\bigg{)}.\end{gathered}$$
For the computation of the integral of $|\partial_\gamma z_2|$, we note that this integral is linear in $h_2$ (since it is linear in $z_2$) and we use an unnormalized version of $z_2$, namely $\frac{z_2}{h_2}$ and multiply by $h_2$ at the end. This narrows the resulting interval.
We computed the bifurcation diagram depicted in Figure \[FigBifurcacion\]. We could give an answer regarding the question of turning or not to $97.14\%$ of the parameter space. $53.23\%$ of the space turned (red) and $43.91\%$ did not turn (yellow). The remaining $2.86\%$ is painted in white. The computation was done in parallel (every core was allocated an initial region) over 8 cores. The division along the cores was made in such a way that core $i = 1,\ldots,8$ started to compute the region $\left[\frac{1}{4},\frac{5}{4}\right] \times \left[-1 + \frac{i-1}{4}, -1 + \frac{i}{4}\right]$. The average runtime was about 30 hours per core, and a total of 5960 rectangles were calculated (an average of 2.5 minutes per rectangle): 8 of the first generation, 64 of the second, 512 of the third, 1880 of the fourth and 3496 of the fifth, out of which 1871 gave a positive result (not turning), 2407 gave a negative (turning) and the rest did not give an answer to the sign and were subdivided or output to a file depending on their width.
![Bifurcation diagram corresponding to the phenomenon of turning/not turning for the initial condition given by the family of curves . Yellow (lighter color): not turning, red (darker color): turning.[]{data-label="FigBifurcacion"}](BifurcacionOKOKRY.eps)
### Technical details concerning Theorem \[ThmConfinedInhomogeneous\](b)
We want to invoke the Implicit Function Theorem. Thus, we have to check that $$\frac{d}{d\mathcal{K}}{\partial_t}{\partial_\alpha}z_1(0,0)\neq0\text{ for points }(h_2,\mathcal{K}) \text{ such that }{\partial_t}{\partial_\alpha}z_1(0,0)=0.$$ In particular, we have to check the previous condition in an open set containing the white region in Figure \[FigBifurcacion\]. We compute $$\begin{gathered}
DI_2 \equiv \frac{d}{d\mathcal{K}}{\partial_t}{\partial_\alpha}z_1(0,0)=4{\partial_\alpha}z_2(0)\int_0^\infty\int_0^\infty
\frac{\sinh(\pi y){\partial_\alpha}z_2(\gamma)\cos(z_1(\gamma)y)}{(\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y))^2\cosh\left(y\frac{\pi}{2}\right)}\\
\times\left(2y\cosh\left(\frac{y\pi}{2}- y h_2\right)\cosh\left(\frac{y \pi}{2}\right)-\frac{2\sinh\left(y h_2\right)}{\tan(h_2)}\right)\\
\times \cosh\left(y z_2(\gamma)\right)\cosh\left(y\left(\frac{\pi}{2}-h_2\right)\right) d\gamma dy.\end{gathered}$$
As in Theorem \[ThmConfinedInhomogeneous\](a), we divide the integral into three diferent regions: singularity, bounded and unbounded, which are calculated in the same way as for the previous Theorem. All what is left is to estimate the tails.
Using , we have $$\frac{\sinh(\pi y)}{\sinh(\pi y)+\mathcal{K}\sinh(2h_2 y)}\leq \sinh(\pi y)e^{-\pi y}C(M,\mathcal{K},h_2)\leq \frac{C(M,\mathcal{K},h_2)}{2}.$$ With the previous estimates we have that the tail contribution can be bounded by $$\begin{gathered}
{\partial_\alpha}z_2(0) \int_{0}^\pi |\partial_{\gamma} z_2(\gamma)| d\gamma\bigg{(}
4 \max_{\mathcal{K},h_2}\{C(M,\mathcal{K},h_2)\}^2\frac{e^{-M(\pi - 0.65 h_2})}{\tan(h_2)\left(\pi - 0.65h_2\right)} \\
+ 4 \max_{\mathcal{K},h_2}\{C(M,\mathcal{K},h_2)\}^2\frac{e^{-M(1.35h_2})}{1.35h_2}\left(M + \frac{1}{1.35h_2}\right)\bigg{)}.\end{gathered}$$
Again, the computation was split among 8 cores, which took as input the intervals output as “unknown” in Theorem \[ThmConfinedInhomogeneous\](a) and ran for about 4 hours. All of them verified a negative sign for $DI_2$ in those intervals, without needing to split them into further subintervals.
|
---
abstract: 'We address the problem of finding the minimum decomposition of a permutation in terms of transpositions with non-uniform cost. For arbitrary non-negative cost functions, we describe polynomial-time, constant-approximation decomposition algorithms. For metric-path costs, we describe exact polynomial-time decomposition algorithms. Our algorithms represent a combination of Viterbi-type algorithms and graph-search techniques for minimizing the cost of individual transpositions, and dynamic programing algorithms for finding minimum cost cycle decompositions. The presented algorithms have applications in information theory, bioinformatics, and algebra.'
author:
- 'Farzad Farnoud (Hassanzadeh) and Olgica Milenkovic, *IEEE Member*'
bibliography:
- 'bib.bib'
title: 'Sorting of Permutations by Cost-Constrained Transpositions'
---
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Introduction
============
Permutations are ubiquitous combinatorial objects encountered in areas as diverse as mathematics, computer science, communication theory, and bioinformatics. The set of all permutations of $n$ elements – the symmetric group of order $n$, $\mathbb{S}_{n}$ – plays an important role in algebra, representation theory, and analysis of algorithms [@goulden_combinatorial_2004; @van_lint_course_2001; @chung_algebraic_????; @hofri_analysis_1995]. As a consequence, the properties of permutations and the symmetric group have been studied extensively.
One of the simplest ways to generate an arbitrary permutation is to apply a sequence of transpositions - swaps of two elements - on a given permutation, usually the identity permutation. The sequence of swaps can be reversed in order to recover the identity permutation from the original permutation. This process is referred to as sorting by transpositions.
A simple result, established by Cayley in the 1860’s, asserts that the minimum number of transpositions needed to sort a permutation so as to obtain the identity permutation is the difference of the size of the permutation and the number of cycles formed by the elements of the permutation. Cayley’s result is based on a simple constructive argument, which reduces to a linear-complexity procedure for breaking cycles into sub-cycles. Sorting a permutation is equivalent to finding the transposition distance between the permutation and the identity permutation. Since permutations form a group, the transposition distance between two arbitrary permutations equals the transposition distance between the identity permutation and the composition of the inverse of one permutation and the other permutation.
We address the substantially more challenging question: assuming that each transposition has a non-negative, but otherwise arbitrary cost, is it possible to find the minimum sorting cost and the sequence of transpositions used for this sorting in polynomial time? In other words, can one compute the cost-constrained transposition distance between two permutations in polynomial time? Although at this point it is not known if the problem is NP hard, at first glance, it appears to be computationally difficult, due to the fact that it is related to finding minimum generators of groups and the subset-sum problem[@jerrum_complexity_1985]. Nevertheless, we show that large families of cost functions – such as costs based on metric-paths – have exact polynomial-time decomposition algorithms. Furthermore, we devise algorithms for approximating the minimum sorting cost for any non-negative cost function, with an approximation constant that does not exceed four.
Our investigation is motivated by three different applications.
The first application pertains to sorting of genomic sequences, while the second application is related to a generalization of the notion of a chemical channel (also known as trapdoor channel [@permuter_capacity_2008]). The third application is in the area of coding for storage devices.
Genomic sequences – such as DNA sequences – evolved from one common ancestor, and therefore frequently contain conserved subsequences. During evolution or during the onset of a genomic disease, these subsequences are subject to mutations, and they may exchange their locations. As an example, genomes of cancer cells tend to contain the same sequence of blocks as normal cells, but in a reshuffled (permuted) order. This finding motivated a large body of work on developing efficient algorithms for reverse-engineering the sequence of shuffling steps performed on conserved subsequences. With a few exceptions, most of the methods for sorting use reversals rather than transpositions, they follow the uniform cost model (each change in the ordering of the blocks is equally likely) and the most parsimonious sorting scenario (the sorting scenario with smallest number of changes is the most likely explanation for the observed order). Several approaches that do not fit into this framework were described in [@pinter_genomic_2002; @pevzner_transforming_2003]. Sorting by cost-constrained transpositions can be seen as a special instant of the general subsequence sorting problem,where the sequence is allowed to break at three or four points. Unfortunately, the case of two sequence breakpoints, corresponding to so called reversals, cannot be treated within this framework.
The second application arises in the study of chemical channels. The chemical channel is a channel model in which symbols are used to describe molecules, and where the channel permutes the molecules in a queue using adjacent transpositions[^1]. In information theory, the standard chemical channel model assumes that there are only two molecules, and that the channel has only two states - hence the use of adjacent transpositions. If all the molecules are different, and the channel is allowed to output molecules with time-varying probabilities, one arrives at a channel model for which the output is a cost-constrained permutation of the input. Finding the minimum cost sequence decomposition therefore represents an important step in the maximum likelihood decoding algorithm fort the channel.
The third application is concerned with flash memories and rank permutation coding (see [@Bruck] and [@5485013]). In this case, one is also interested in sorting permutations using adjacent transpositions and computing the Kendall distance between permutations [@chadwick69]. If one considers more precise charge leakage models for memory cells, the costs of adjacent transpositions become non-uniform. This can easily be captured by a transposition cost model in which non-adjacent transpositions have unbounded cost, while the costs of adjacent transpositions are unrestricted. Hence, the proposed decomposition algorithms can be used as part of general soft-information rank modulation decoders.
Our findings are organized as follows. Section II introduces the notation followed in the remainder of the paper, as well as relevant definitions. Sections III and IV contain the main results of our study: a three-stage polynomial-time approximation algorithm for general cost-constrained sorting of permutations, an exact polynomial-time algorithm for sorting with metric-path costs, as well as a complexity analysis of the described techniques. Section V contains the concluding remarks.
Notation and Definitions\[sec:Notation-and-Definitions\]
========================================================
A permutation $\pi$ of $\{1,2,\cdots,n\}$ is a bijection from $\{1,2,\cdots,n\}$ to itself. The set of permutations of $\{1,2,\cdots,n\}$ is denoted by $\mathbb{S}_{n}$, and is called the symmetric group on $\{1,2,\cdots,n\}$. A permutation can be represented in several ways. In the two-line notation, the domain is written on top, and its image below. The one-line representation is the second row of the two-line representation. A permutation may also be represented as the set of elements and their images.
For example, one can write a permutation $\pi$ as $\pi\left(1\right)=3,\pi\left(2\right)=1,\pi\left(3\right)=2,\pi\left(4\right)=5,\pi\left(5\right)=4,$ or more succinctly as $\pi=31254$, or in the two-line notation as $$\pi=\left(\begin{array}{ccccc}
1 & 2 & 3 & 4 & 5\\
3 & 1 & 2 & 5 & 4\end{array}\right).$$ Yet another way of writing a permutation is via a set of mappings, for example $\pi=\left\{ 1\rightarrow3,2\rightarrow1,3\rightarrow2,4\rightarrow5,5\rightarrow4\right\} .$ It will be helpful to think of a permutation as a mapping from *positions* to *objects*. For example, $\pi\left(1\right)=3$ means object 3 occupies position 1. Alternatively, we can also say that element 1 is a predecessor of element 3. If not otherwise stated, the word *predecessor* will be henceforth used in this context.
The product $\pi_{2}\pi_{1}$ of two permutations $\pi_{1}$ and $\pi_{2}$ is the permutation obtained by first applying $\pi_{1}$ and then $\pi_{2}$ to $\{1,2,\cdots,n\}$, i.e., the product represents the composition of $\pi_{1}$ and $\pi_{2}$.
The *functional digraph* **** of a function $f:\{1,2,\cdots,n\}\rightarrow\{1,2,\cdots,n\}$, denoted by $\mathcal{G}\left(f\right)$, is a directed graph with vertex set $\{1,2,\cdots,n\}$ and an edge from $i$ to $f\left(i\right)$ for each $i\in\{1,2,\cdots,n\}$. We use the words vertex and element interchangeably. For a permutation $\pi$ of $\{1,2,\cdots,n\}$, $\mathcal{G}\left(\pi\right)$ is a collection of disjoint *cycles,* since the in-degree and out-degree of each vertex is exactly one. Each cycle can be written as a $k-$tuple $\sigma=\left(a_{1}a_{2}\cdots a_{k}\right)$, where $k$ is the length of the cycle and $a_{i+1}=\sigma\left(a_{i}\right)$. For each cycle of length $k$, the indices are evaluated modulo $k$, so that $a_{k+1}$ equals $a_{1}$. A planar embedding of $\mathcal{G}\left(\pi\right)$ can be obtained by placing vertices of each of the disjoint cycles on disjoint circles*.* We hence reserve the symbol $\sigma$ for single cycles, and $\pi$ for multiple cycle permutations.
We use $\mathcal{G}\left(\pi\right)$ to refer to the planar embedding of the functional digraph of $\pi$ on circles, as well as the functional digraph of $\pi$. As a convention, we do not explicitly indicate the direction of edges on the circle. Instead, we assume a clockwise direction and treat $\mathcal{G}\left(\pi\right)$ as a non-directional graph, unless otherwise stated.
A cycle of length two is called a *transposition*. A *transposition decomposition* **$\tau$** (or simply a decomposition) of a permutation $\pi$ is a sequence $t_{m}\cdots t_{1}$ of transpositions $t_{i}$ whose product is $\pi$. Note that the transpositions are applied from right to left. A *sorting* **** $s$ of a permutation $\pi$ is a sequence of transpositions that transform $\pi$ into $\imath$, where $\imath$ denotes the identity element of $\mathbb{S}_{n}$. In other words, $s\pi=\imath$. Note that a decomposition $\tau$ in reverse order equals a sorting $s$ of the same permutation.
The cycle representation of a permutation is the list of its cycles. For example, the cycle representation of $31254$ is $\left(132\right)\left(45\right)$. Cycles of length one are usually omitted. The product of non-disjoint cycles is interpreted as a product of permutations. As an illustration, $\left(124\right)\left(213\right)=\left(\left(124\right)\left(3\right)\right)\left(\left(213\right)\left(4\right)\right)=\left(2\right)\left(134\right)$.
A permutation $\pi$ is said to be odd (even) if the number of pairs $a,b\in\{1,2,\cdots,n\}$ such that $a<b$ and $\pi\left(a\right)>\pi\left(b\right)$ is odd (even). If a permutation is odd (even), then the number of transpositions in any of its decompositions is also odd (even).
The following definitions regarding graphs $G=\left(V,E\right)$ will be ** used throughout the paper. An edge with endpoints $u$ and $v$ is denoted by $\left(uv\right)\in E$. A graph is said to be planar if it can be embedded in the plane without intersecting edges. The subgraph of $G$ induced by the vertices in the set $S\subset V$ is denoted by $G\left[S\right]$. The degree of a vertex $v$ in $G$ is denoted by $\deg_{G}\left(v\right)$ or, if there is no ambiguity, by $\deg\left(v\right)$. Deletion of an edge $e$ from a graph $G$ is denoted by $G-e$ and deletion of a vertex $v$ and its adjacent edges from $G$ is denoted by $G-v$. The same notions can be defined for multigraphs - graphs in which there may exist multiple edges between two vertices.
We say that an edge $e$ in $G$ is a cut edge for two vertices $a$ and $b$, denoted by $\cutedge ab$, if in $G-e$ there exists no path between $a$ and $b$. The well known Menger’s theorem[@doug_west_comb_math] asserts that the minimum number of edges one needs to delete from $G$ to disconnect $a$ from $b$ is also the maximum number of pairwise edge-disjoint paths between $a$ and $b.$ This theorem holds for multigraphs as well.
Let $\mathcal{T}\left(\tau\right)$ be a (multi)graph with vertex set $\{1,2,\cdots,n\}$ and edges $(a_{i}b_{i})$ for each transposition $t_{i}=\left(a_{i}b_{i}\right)$ of $\tau$. We use the words transposition and edge interchangeably. The embedding of $\mathcal{T}\left(\tau\right)$ with vertex set $\{1,2,\cdots,n\}$ into $\mathcal{G}\left(\pi\right)$ is also denoted by $\mathcal{T}\left(\tau\right)$. In the derivations to follow, we make frequent use of the spanning trees of the (multi)graphs $\mathcal{T}\left(\tau\right),\mathcal{G}\left(\pi\right)$ and $\mathcal{G}\left(\pi\right)\cup\mathcal{T}\left(\tau\right)$. A spanning tree is a standard notion in graph theory: it is a tree that contains all vertices of the underlying (multi)graph.
We are concerned with the following problem: given a non-negative cost function $\varphi$ on the set of transpositions, the cost of a transposition decomposition is defined as the sum of costs of its transpositions. The task is to find an efficient algorithm for generating the Minimum Cost Transposition Decomposition (MCD) of a permutation $\pi\in\mathbb{S}_{n}$ . The cost of the MCD of a permutation $\pi$ under cost function $\varphi$ is denoted by $M_{\varphi}\left(\pi\right)$. If there is no ambiguity, the subscript is omitted.
For a non-negative cost function $\varphi$, let $\mathcal{K}\left(\varphi\right)$ be the undirected complete graph in which the cost of each edge $(ab)$ equals $\varphi\left(a,b\right)$. The cost of a graph $ $$G\subseteq\mathcal{K}\left(\varphi\right)$ is the sum of the costs of its edges, $$\cost\left(G\right)=\sum_{\left(ab\right)\in G}\varphi\left(a,b\right).$$ The shortest path, i.e., the path with minimum cost, between $i$ and $j$ in $\mathcal{K}\left(\varphi\right)$ is denoted by $p^{*}\left(i,j\right)$.
The following definitions pertaining to cost functions are useful in our analysis. A cost function $\varphi$ is a metric if for $a,b,c\in\{1,2,\cdots,n\}$$$\varphi\left(a,c\right)\le\varphi\left(a,b\right)+\varphi\left(b,c\right).$$ A cost function $\varphi$ is a **** *metric-path* **** cost if it is defined in terms of a weighted path, denoted by $\Theta_{s}$. The weights of edges $\left(uv\right)$ in $\Theta_{s}$ are equal to $\varphi\left(u,v\right)$, and the cost of any transposition $\left(ij\right)$ equals $$\varphi\left(i,j\right)=\sum_{t=1}^{l}\varphi\left(c_{t},c_{t+1}\right),$$ where $c_{1}\cdots c_{l}c_{l+1}$, $c_{1}=i$, $c_{l+1}=j$, represents the unique path between $i$ and $j$ in $\Theta_{s}$. The path $\Theta_{s}$ is called the *defining path* **** of $\varphi$. A cost function $\varphi$ is an **** *extended-metric-path* **** cost function if for a defining path $\Theta_{s}$, $\varphi\left(i,j\right)$ is finite only for the edges $\left(ij\right)$ of the defining path, and unbounded otherwise.
\#1\#2\#3[(\#3,(\#1\#2))]{}
Applying a transposition $\left(ab\right)$ to a permutation $\pi$ is equivalent to exchanging the predecessors of $a$ and $b$ in $\mathcal{G}\left(\pi\right)$. We define a generalization of the notion of a transposition, termed *h-transposition*, where the predecessor of $a$ can be changed independently of the predecessor of $b$. For example, let $a,b,c,d\in\{1,2,\cdots,n\}$ and let $\pi\left(c\right)=a$ and $\pi\left(d\right)=b$. Let $\pi'=\htrans abc\pi$, where we used $\htrans abc$ to denote an h-transposition. This h-transposition takes $c$, the predecessor of $a$, to $b$, without modifying the predecessor of $b$. That is, we have a mapping in which $\pi'\left(c\right)=\pi'\left(d\right)=b$, and $a$ has no predecessor. Note that $\pi'$ is no longer a bijection, and several elements may be mapped to one element. A transposition represents the product of a pair of h-transpositions, as in $$\left(ab\right)\pi=\htrans ab{\pi^{-1}\left(a\right)}\htrans ba{\pi^{-1}\left(b\right)}\pi.$$
An *h-decomposition* **** $h$ of a permutation $\pi$ is a sequence of h-transpositions such that $h\imath=\pi$. Similar to transpositions, a cost $\psi\left(a,b\right)\ge0$ can be assigned to h-transpositions $\htrans abc$, where $c$ is the predecessor of $a$. Note that the cost $\psi$ is not dependent on $c$. We say that the transposition cost $\varphi$ and the h-transposition cost $\psi$ are *consistent* if for all transpositions $\left(ab\right)$ it holds that $\varphi\left(a,b\right)=\psi\left(a,b\right)+\psi\left(b,a\right)$.
For a permutation $\pi$ and a transposition $\left(ab\right)$, it can be easily verified that $\left(ab\right)\pi$ consist of one more (or one less) cycle than $\pi$ if and only if $a$ and $b$ are in the same cycle (in different cycles). Since the identity permutation has $n$ cycles, a *Minimum Length Transposition Decomposition (MLD)* of $\pi$ has length $n-\ell$, where $\ell$ denotes the number of cycles in $\pi$. The minimum cost of an MLD of $\pi$, with respect to cost function $\varphi$, is denoted by $L_{\varphi}\left(\pi\right)$. For example, $\left(132\right)\left(45\right)=\left(45\right)\left(23\right)\left(12\right)$ is decomposed into three transpositions. In particular, if $\pi$ is a single cycle, then the MLD of the cycle has length $n-1$. A cycle of length $k$ has $k^{k-2}$ MLDs [@NilsMeier03011976]. An MCD is not necessarily an MLD, as illustrated by the following example.
\[exa:navin\]Consider the cycle $\sigma=(1\cdots5)$ with $\varphi\left(i,i+1\right)=3$ and $\varphi\left(i,i+2\right)=1$. It is easy to verify that $\left(14\right)(13)(35)(24)(14)(13)$ is an MCD of $\sigma$ with cost six, i.e., $M\left(\sigma\right)=6$. However, as we shall see later, the cost of a minimum cost MLD is eight, i.e., $L\left(\sigma\right)=8$. One such MLD is $\left(14\right)\left(23\right)\left(13\right)\left(45\right)$ [@navin_personal_2010].
Our approach to finding the minimum cost decomposition of a permutation consists of three stages:
1. First, we find the minimum cost decomposition for each individual transposition. In particular, we show that the minimum cost decomposition of a transposition can be obtained by recursively substituting transpositions with triples of transpositions. This step is superfluous for the case when the cost function is a metric.
2. In the second step, we consider cycles only and assume that each transposition cost is optimized. Cycles have the simplest structure among all permutations, and furthermore, each permutation is a collection of cycles. Hence, several approximation algorithms operate on individual cycles and combine their decompositions. As part of this line of results, we describe how to find the minimum cost MLD and show that its cost is not more than a constant factor higher than that of the corresponding MCD. We also present a particularly simple-to-implement class of decompositions whose costs lie between the cost of a minimum MLD and a constant multiple of the cost of an MCD.
3. We generalize the results obtained for single cycles to permutations with multiple cycles.
Optimizing Individual Transposition\[sec:Optimizing-Individual-Transposition\]
==============================================================================
Let $\tau$ be a transposition decomposition and let $\left(ab\right)$ be a transposition in $\tau$. Since a transposition is an odd permutation, it may only be written as the composition of an odd number of transpositions. For example, $$\left(ab\right)=\left(ac\right)\left(bc\right)\left(ac\right),\label{eq:trans_decompose}$$ where $c\in\{1,2,\cdots,n\}$ and $c\neq a,b$. It is straightforward to see that any decomposition of a transposition of length three must be of the form , with a possible reversal of the roles of the elements $a$ and $b$.
If $\varphi\left(ab\right)>2\varphi\left(ac\right)+\varphi\left(bc\right),$ then replacing $\left(ab\right)$ by $\left(ac\right)\left(bc\right)\left(ac\right)$ reduces the overall cost of $\left(ab\right)$. Thus, the first step of our decomposition algorithm is to find the optimal cost of each transposition. As will be shown, it is straightforward to develop an algorithm for finding *minimum cost decompositions of transpositions of the form* . One such algorithm – Alg. \[alg:opt-trans\] – performs a simple search on the ordered set of transpositions in order to check if their product, of the form of , yields a decomposition of lower cost for some transposition. It then updates the costs of transpositions and performs a new search for decompositions of length three that may reduce some transposition cost.
The optimized costs produced by the algorithm are denoted by $\varphi^{*}$. Note that $\varphi^{*}(a,b)\leq2\varphi^{*}(a,x)+\varphi^{*}(b,x)$, for any $x\neq a,b.$ Although an optimal decomposition of the form produced by Alg. \[alg:opt-trans\]is not guaranteed to produce the overall minimum cost decomposition of any transposition, we show that this is indeed the case after the expositions associated with Alg. \[alg:opt-trans\].
Observe that if the cost function is such that $$\varphi\left(b,c\right)+2\varphi\left(a,c\right)\geq\varphi\left(a,b\right),\quad a,b,c\in\{1,2,\cdots,n\},\label{eq:V-ineq}$$ as in Example 1, Alg. \[alg:opt-trans\] is redundant and can be omitted when computing the MCD. In particular, if the cost function is a metric, then Alg. \[alg:opt-trans\] is not needed.
The input to the algorithm Alg. \[alg:opt-trans\] is an ordered list $\Omega$ of transpositions and their costs. Each row of $\Omega$ corresponds to one transposition and is of the form $\left[\left(ab\right)|\varphi\left(a,b\right)\right]$. Sorting of $\Omega$ means reordering its rows so that transpositions are sorted in increasing order of their costs. The output of the algorithm is a list with the same format, but with minimized costs for each transposition.
Input: $\Omega$ (the list of transpositions and their cost) Sort $\Omega$ $(a_1 b_1)\leftarrow \Omega(i)$ $\phi_1 \leftarrow \varphi {\left(a_1,b_1\right)}$ $(a_2 b_2)\leftarrow \Omega(j)$ $\phi_2 \leftarrow \varphi {\left(a_2,b_2\right)}$ $a_{com} \leftarrow \{a_1,b_1\}\cap\{a_2,b_2\}$ $\{a_3,b_3\}\leftarrow \{a_1,a_2,b_1,b_2\}-\{a_{com}\}$ update $\varphi {\left(a_3,b_3\right)}$ in $\Omega$ Sort $\Omega$
\[lem:opt-trans\] Alg. \[alg:opt-trans\] optimizes the costs $\varphi$ of all transpositions with respect to the triple transposition decomposition.
Let $\Omega_{i}$ be the list $\Omega$ at the beginning of iteration $i$, obtained immediately before executing line 4 of Alg. \[alg:opt-trans\]. We prove, by induction, that transpositions in $\Omega_{i}(1\upto i)$ have minimum triple decomposition costs that do not change in subsequent iterations of the algorithm, and that the transpositions in $\Omega_{i}(i+1\upto\left|\Omega\right|)$ cannot be written as a product of transpositions exclusively in $\Omega_{i}(1:i-1)$ that have smaller overall cost.
The claim is obviously true for $i=2$.
Assume that the claim holds for $i$. Let $t_{1}=\Omega_{i}(i)$ and consider $s\in\Omega_{i}(i+1:\left|\Omega\right|)$. By the induction assumption, $s$ cannot be written as a product of transpositions exclusively in $\Omega_{i}(1:i-1)$ having smaller overall cost. Thus, the cost of $s$ may be reduced only if one can write $s$ as $t_{2}t_{1}t_{2}$, where $t_{2}\in\Omega(1\upto i-1)$. The list $\Omega_{i+1}$ is obtained after considering all such transpositions, updating $\varphi$ and sorting $\Omega_{i}$. The transposition of minimum cost in $\Omega_{i+1}(i+1\upto\left|\Omega\right|)$ is $\Omega_{i+1}(i+1)$. Now $\Omega_{i+1}(i+1\upto\left|\Omega\right|)$ cannot be written in terms of transpositions in $\Omega_{i+1}(1:i)$ only, and hence the cost of any transposition in $\Omega_{i+1}(i+1:\left|\Omega\right|)$ cannot be reduced below the cost of $\Omega_{i+1}(i+1)$. Hence, the cost of $\Omega_{i+1}(i+1)$ is minimized.
The left-most list in represents the input $\Omega$ to the algorithm, with transpositions in increasing order of their costs. The two lists that follow represent updates of $\Omega$ produced by Alg. \[alg:opt-trans\]. In the first step, the algorithm considers the transposition $(13)$, for $i=2$, and the transposition $\left(34\right)$, for $j=1$. Using these transpositions we may write $\left(34\right)\left(13\right)\left(34\right)=\left(14\right)$. The initial cost of $\left(14\right)$ is 12 which exceeds $2\varphi\left(3,4\right)+\varphi\left(1,3\right)=8$. Hence, the list representing $\Omega$ is updated to form the second list in . Next, for $i=3$ and $j=1$, the algorithm considers $\left(24\right)$ and $\left(34\right)$. Since $\left(34\right)\left(24\right)\left(34\right)=(23)$, we update the cost of $\left(23\right)$ from $23$ to $11$ as shown in the third list in . Additional iterations of the algorithm introduce no further changes in the costs.
$$\left[\begin{array}{c|c}
\left(34\right) & 2\\
\left(13\right) & 4\\
\left(24\right) & 7\\
\left(14\right) & 12\\
\left(12\right) & 15\\
\left(23\right) & 23\end{array}\right]\rightarrow\left[\begin{array}{c|c}
\left(34\right) & 2\\
\left(13\right) & 4\\
\left(24\right) & 7\\
\mathit{\left(14\right)} & \mathit{8}\\
\left(12\right) & 15\\
\left(23\right) & 23\end{array}\right]\rightarrow\left[\begin{array}{c|c}
\left(34\right) & 2\\
\left(13\right) & 4\\
\left(24\right) & 7\\
\left(14\right) & 8\\
\mathit{\left(23\right)} & \mathit{11}\\
\left(12\right) & 15\end{array}\right]\label{eq:trans_opt_ex}$$
Upon executing the algorithm, the cost of each transposition is set to its minimal value. Only after the last stage of the MCD approximation algorithm is completed will each transposition be replaced by its minimal cost decomposition. For each index $i$ the number of operations performed in the algorithm is $O(\left|\Omega\right|)$. Thus, the total complexity of the algorithm is $O(\left|\Omega\right|^{2})$. Since $\left|\Omega\right|$ is at most equal to the number of transpositions, we have $\left|\Omega\right|={n \choose 2}$. Hence, the complexity of Alg. \[alg:opt-trans\] equals $O(n^{4})$.
In the analysis that follows, denote the optimized transposition costs by the superscript $*$, as in $\varphi^{*}$.
Since the transposition costs are arbitrary non-negative values, it is not clear that the minimum cost decomposition of a transposition is necessarily of the form generated by Alg. \[alg:opt-trans\]. This algorithm only guarantees that one can identify the *optimal sequence of consecutive replacements of transpositions by triples of transpositions*. Hence, the minimum cost of a transposition $(ab)$ may be smaller than $\varphi^{*}(a,b)$, i.e. there may be decompositions of length five, seven, or longer, which allow for an even smaller decomposition cost of a transposition.
Fortunately, this is not the case: we first prove this claim for decompositions of length five via exhaustive enumeration and then proceed to prove the general case via the use of Mengers’s theorem for multigraphs[@goulden_combinatorial_2004]. We choose to provide the example of length-five decompositions since it illustrates the difficulty of proving statements about non-minimal decompositions of permutations using exhaustive enumeration techniques. Graphical representations, on the other hand, allow for much more general and simpler proofs pertaining to non-minimal decompositions of transpositions.
We start by considering all possible transposition decompositions of length five, for which the transposition costs are first optimized via Alg. \[alg:opt-trans\]. In other words, we investigate if there exist decompositions of $(ab)$ of length five that have cost smaller than $\varphi^{*}(a,b)$. Once again, observe that the costs of all transpositions used in such decompositions are first optimized via a sequence of triple-transposition decompositions. To reduce the number of cases, we present the following lemma restricting the possible configurations in a multigraph corresponding to the decomposition of a transposition $\left(ab\right)$.
Let $\tau$ be a decomposition of a transposition $\left(ab\right)$. The multigraph $\mathcal{M}=\mathcal{T}\left(\tau\right)$, where $\tau$ does not contain $\left(ab\right)$, has the following properties:\[lem:properties-of-T\]
1. Both $a$ and $b$ have degree at least one.\[enu:deg-a-b-1\]
2. The degree of at least one of the vertices $a$ and $b$ is at least two.\[enu:deg-a-b-2\]
3. Every vertex of $\mathcal{M}=\mathcal{T}\left(\tau\right)$, other than $a$ and $b$, appears in a closed path (cycle) with no repeated edges in $\mathcal{M}$. \[enu:Every-vertex-cycle\]
The proof follows from the simple observations that :
1. In order to swap $a$ and $b$, both $a$ and $b$ must be moved.
2. If both vertices $a$ and $b$ have degree one, then $a$ and $b$ are moved exactly once. This is only possible only if $\left(ab\right)\in\tau.$
3. Let $\tau=t_{m}\cdots t_{2}t_{1}$. Let $t_{i}$ be the transposition with the smallest index $i$ that includes $x\in V(\mathcal{M})$. In the permutation $\tau_{1}=t_{i}\cdots t_{2}t_{1}$, $x$ is not in its original location but rather occupies the position of another element, say, $y$. As we shall see in the proof of Lemma and Example , this means that there is a path from $x$ to $y$ in $\mathcal{T}(\tau_{1})$. Similarly, there must exist a path from $y$ to $x$ in $\mathcal{T}\left(t_{m}\cdots t_{i+2}t_{i+1}\right)$. Thus there is a closed path with no repeated edges from $x$ to itself in $\mathcal{M}$.
Let $x_{1},x_{2},\cdots,x_{N}$ be vertices included in the decomposition $\tau$ other than $a$ and $b$. If $E(\mathcal{M})$ denotes the number of edges in the multigraph $\mathcal{M}=\mathcal{T}(\tau)$, then $$2\left|E\left(\mathcal{M}\right)\right|=\sum_{i=1}^{N}\deg\left(x_{i}\right)+\deg\left(a\right)+\deg\left(b\right).$$ From parts \[enu:deg-a-b-1\] and \[enu:deg-a-b-2\] of Lemma \[lem:properties-of-T\], note that $\deg\left(a\right)+\deg\left(b\right)\ge3$ and, from part \[enu:Every-vertex-cycle\], it holds that $\deg\left(x_{i}\right)\ge2$. Hence, $2\left|E\left(\mathcal{M}\right)\right|\ge2N+3$, and since $N$ has to be an integer, $$N\le\lfloor\left|E\left(\mathcal{M}\right)\right|-\frac{3}{2}\rfloor=\left|E\left(\mathcal{M}\right)\right|-2.\label{eq:vertex-bound}$$ Suppose that $\tau=t_{5}t_{4}t_{3}t_{2}t_{1}$ is the minimum cost decomposition of $\left(ab\right)$ with cost $\phi$, and that the cost of the optimal decomposition produced by Alg. \[alg:opt-trans\] exceeds $\phi$. Then there is no vertex $x$ such that $$G_{1}=\left\{ \left(ax\right),\left(ax\right),\left(bx\right)\right\}$$ is a subset of edges in the multigraph $\mathcal{M}$ since, in that case, $$\varphi^{*}\left(a,b\right)\le2\varphi^{*}\left(a,x\right)+\varphi^{*}\left(b,x\right)\le\phi.$$ Also, there exists no pair of vertices $x,y$ such that $$G_{2}=\left\{ \left(ax\right),\left(bx\right),\left(ay\right),\left(by\right)\right\}$$ is a subset of edges in the multigraph $\mathcal{M}$. To prove this claim, suppose that $G_{2}\subseteq E(\mathcal{M})$. Without loss of generality, assume that $$\varphi^{*}\left(a,x\right)+\varphi^{*}\left(b,x\right)\le\varphi^{*}\left(a,y\right)+\varphi^{*}\left(b,y\right).$$ Then,$$\begin{aligned}
\varphi^{*}\left(a,b\right) & \le2\varphi^{*}\left(a,x\right)+\varphi^{*}\left(b,x\right)\\
& \le2\varphi^{*}\left(a,x\right)+2\varphi^{*}\left(b,x\right)\\
& \le\cost\left(G_{2}\right)\\
& \le\phi.\end{aligned}$$
Hence, any decomposition of length five that contains $G_{2}$ must have cost at least $\varphi^{*}\left(a,b\right)$.
For any five-decomposition $\tau$, we have $\left|E\left(\mathcal{M}\right)\right|=5$ and, thus, $N\leq3$. We consider all five-decompositions of $\left(ab\right)$ such that $\mathcal{M}$ is $G_{1}-$free and $G_{2}-$free, and which contain at most five vertices in $\mathcal{M}$. Assume that the three extra vertices, in addition to $a$ and $b$, are $c$, $d$, and $e$. We now show that for each decomposition of length five, there exists a decomposition obtained via Alg. \[alg:opt-trans\] with cost at most $\phi$, denoted by either $\mu$ or $\mu'$. The following scenarios are possible.
1. Suppose that $\deg\left(a\right)=2$ and $\deg\left(b\right)=1$. Furthermore, suppose that there exist a vertex that is adjacent to both $a$ and $b$ in $\mathcal{M}$. Without loss of generality, assume that $\left(ad\right)\in\mathcal{M},\left(bd\right)\in\mathcal{M}$ (Figure \[fig:ab21-1\]). We consider two cases, depending on the existence of the edge $\left(cd\right)$ in $\mathcal{M}$.\
First, assume that $\left(cd\right)\in\mathcal{M}$ (Figure \[fig:ab21-cd\]). If $\varphi^{*}\left(a,c\right)+\varphi^{*}\left(c,d\right)\le\varphi^{*}\left(a,d\right)$, then the decomposition $$\begin{aligned}
\mu & =\left(ac\right)\left(cd\right)\left(bd\right)\left(cd\right)\left(ac\right)\end{aligned}$$ has cost at most $\phi$. Note that $\mu$ can be obtained from Alg. \[alg:opt-trans\], since $$\begin{aligned}
\mu & =\left(ac\right)\left(bc\right)\left(ac\right)=\left(ab\right).\end{aligned}$$ On the other hand, if $\varphi^{*}\left(a,c\right)+\varphi^{*}\left(c,d\right)>\varphi^{*}\left(a,d\right)$, then the decomposition $\mu'=\left(ad\right)\left(bd\right)\left(ad\right)$ has cost at most $\phi$.\
Next assume that $\left(cd\right)\notin\mathcal{M}$. Since both $c$ and $d$ each must lie on a cycle, the only possible decompositions of $(ab)$ are shown in Figure \[fig:ab21-notcd\]. Now, if $\varphi^{*}\left(ad\right)\le\varphi^{*}\left(d,e\right)+\varphi^{*}\left(e,c\right)+\varphi^{*}\left(a,c\right)$, then the decomposition $$\mu=\left(ad\right)\left(bd\right)\left(ad\right)$$ has cost at most $\phi$. On the other hand, if $\varphi^{*}\left(ad\right)>\varphi^{*}\left(d,e\right)+\varphi^{*}\left(e,c\right)+\varphi^{*}\left(a,c\right)$, then the decomposition $$\mu'=\left(ac\right)\left(ec\right)\left(ed\right)\left(bd\right)\left(ed\right)\left(ce\right)\left(ac\right)\label{eq:mu-ac-ec-be}$$ has cost at most $\phi$. Note that $\mu'$ can be obtained from Alg. \[alg:opt-trans\], since $$\begin{aligned}
\mu' & =\left(ac\right)\left(ec\right)\left(be\right)\left(ec\right)\left(ac\right)\\
& =\left(ac\right)\left(cb\right)\left(ac\right)\\
& =\left(ab\right).\end{aligned}$$
2. Suppose that $\deg\left(a\right)=2$ and $\deg\left(b\right)=1$, but that there is no vertex adjacent to both $a$ and $b$. Without loss of generality, assume $c$ and $d$ are adjacent to $a$ and $e$ is adjacent to $b$ (Figure \[fig:ab21-be\]). Since $c,d,$ and $e$ each must lie on a cycle, one must include two more edges in the graph, as shown in Figure \[fig:ab21-adec\]. Since $d$ and $c$ have a symmetric role in the decomposition, we may without loss of generality, assume that $\varphi^{*}\left(a,c\right)+\varphi^{*}\left(c,e\right)\le\varphi^{*}\left(a,d\right)+\varphi^{*}\left(d,e\right)$. Let $\mu$ be equal to $$\mu=\left(ac\right)\left(ce\right)\left(eb\right)\left(ce\right)\left(ac\right).$$ Similarly to , it is easy to see that the cost of $\mu$ is at most $\phi$ and that it can be obtained from Alg. .
3. Assume that $\deg\left(a\right)=\deg\left(b\right)=2$ (Figure \[fig:ab22\]). Since $e$ and $c$ must lie on a cycle, the fifth transposition in the decomposition must be $\left(ec\right)$ (Figure \[fig:ab22ec\]). If $\varphi^{*}\left(a,d\right)+\varphi^{*}\left(b,d\right)\le\varphi^{*}\left(b,e\right)+\varphi^{*}\left(e,c\right)+\varphi^{*}\left(c,a\right)$, then the decomposition $$\mu=\left(ad\right)\left(bd\right)\left(ad\right)$$ has cost at most $\phi$. Otherwise, if $\varphi^{*}\left(a,d\right)+\varphi^{*}\left(b,d\right)>\varphi^{*}\left(b,e\right)+\varphi^{*}\left(e,c\right)+\varphi^{*}\left(c,a\right)$, the decomposition$$\mu'=\left(ac\right)\left(ec\right)\left(be\right)\left(ce\right)\left(ac\right)$$ has cost at most $\phi$. Note that both $\mu$ and $\mu'$ represent decompositions of a form optimized over by Alg. \[alg:opt-trans\].
4. Suppose that $\deg\left(a\right)=3$, $\deg\left(b\right)=1$, and that all edges adjacent to $a$ and $b$ are simple (not repeated). Without loss of generality, assume that $e$ is adjacent to both $a$ and $b$ (Figure \[fig:ab31e\]). One edge must complete cycles that include $c,d,$ and $e$. Since creating such cycles with one edge is impossible, this configuration is impossible.
5. Suppose that $\deg\left(a\right)=3$, $\deg\left(b\right)=1$, one edge adjacent to $a$ appears twice, and there is a vertex adjacent to both $a$ and $b$. Without loss of generality, assume that this vertex is $d$ (Figure \[fig:ab31d\]). Since $d$ must be in a cycle, it must be adjacent to the “last edge”, i.e., the fifth transposition. If the last edge is $\left(ed\right)$, then one more edge is needed to create a cycle passing through $e$. Thus, the last edge cannot be $\left(ed\right)$. The only other choice is $\left(cd\right)$ (Figure \[fig:ab31ddc\]). Now, if $\varphi^{*}\left(a,d\right)\ge\varphi^{*}\left(c,d\right)$, then the decomposition $$\mu=\left(ac\right)\left(cd\right)\left(bd\right)\left(cd\right)\left(ac\right)$$ has cost at most $\phi$. Otherwise, if $\varphi^{*}\left(a,d\right)<\varphi^{*}\left(c,d\right)$, the decomposition $$\mu'=\left(ad\right)\left(bd\right)\left(ad\right)$$ has cost at most $\phi$.
6. Suppose that $\deg\left(a\right)=3,$ $\deg\left(b\right)=1$, and no vertex is adjacent to both $a$ and $b$. The two possible cases are shown in Figures \[fig:ab31cc\] and \[fig:ab31ccc\]. Since one edge cannot create all the necessary cycles, both configurations are impossible.
Next, we state a general theorem pertaining to the optimality of Alg. \[alg:opt-trans\].
\[thm:correctness\_of\_alg1\]The minimum cost decompositions of all transpositions are generated by Alg. \[alg:opt-trans\].
The proof proceeds in two steps. First, we show that the multigraph $\mathcal{M}$ for a transposition $(ab)$ cannot have more than one $\cutedge ab$. If $\mathcal{M}$ has no $\cutedge ab$, then there exist at least two edge-disjoint paths between $a$ and $b$ in $\mathcal{M}$. This claim follows by invoking Menger’s theorem. The costs of the paths can be combined, leading to a cost of the form induced by a transposition decomposition optimized via . If the multigraph has exactly one $\cutedge ab$, this case can be reduced to the case of no $\cutedge ab$. This completes the proof.
Before proving the impossibility of the existence of more than one $\cutedge ab$, we explain how a $\cutedge ab$ imposes a certain structure in the decomposition of $\left(ab\right)$. Consider the decomposition $t_{m}t_{m-1}\cdots t_{i}\cdots t_{1}$ of $\left(ab\right)$ and suppose that $t_{i}=\left(x_{1}y_{1}\right)$ is an $\cutedge ab$, as shown in Figure \[fig:one-cut-edge\]. Let $\pi_{j}=t_{j}\cdots t_{1}$. Since there exists a path between $a$ to $b$, there also exists a path between $a$ and $x_{1}$ that does not use the edge $\left(x_{1}y_{1}\right)$. Thus, in $\mathcal{M}-\left(x_{1}y_{1}\right)$, $a$ and $x_{1}$ are in the same “component”. Denote this component by $B_{1}$. Similarly, a component, denoted by $B_{2}$, must contain both the vertices $b$ and $y_{1}$. Since there is no transposition in $\pi_{i-1}$ with endpoints in both **$B_{1}$** and $B_{2}$, there is no element $z\in B_{1}$ such that $\pi_{i-1}\left(z\right)\in B_{2}$. Similarly, there is no element $z\in B_{2}$ such that $\pi_{i-1}\left(z\right)\in B_{1}$. This implies that $\pi_{i-1}\left(a\right)\in B_{1}$ and $\pi_{i-1}\left(b\right)\in B_{2}$. Since $\left(x_{1}y_{1}\right)$ is the only edge connecting $B_{1}$ and $B_{2}$, we must have $$\begin{aligned}
\pi_{i-1}^{-1}\left(x_{1}\right) & =a,\\
\pi_{i-1}^{-1}\left(y_{1}\right) & =b,\end{aligned}$$ and$$\begin{aligned}
\pi_{i}^{-1}\left(x_{1}\right) & =b,\\
\pi_{i}^{-1}\left(y_{1}\right) & =a.\end{aligned}$$
Now suppose there are at least two $\cutedge ab$s in $T$ as shown in Figure \[fig:two-cut-edge\]. Let the decomposition of $(ab)$ be $t_{m}\cdots t_{l}\cdots t_{i}\cdots t_{1}$, where $t_{l}=\left(x_{2}y_{2}\right)$ and $t_{i}=\left(x_{1}y_{1}\right)$, for some $i<l$. Define $B_{1}$, $B_{2}$, and $B_{3}$ to be the components containing $a$, $y_{1}$, and $y_{2}$, respectively, in $\mathcal{M}-\left(x_{1}y_{1}\right)-\left(x_{2}y_{2}\right)$. By the same reasoning as above we must have $$\begin{aligned}
\pi_{l-1}^{-1}\left(x_{2}\right) & =a,\\
\pi_{l-1}^{-1}\left(y_{2}\right) & =b.\end{aligned}$$ However, this cannot be true: after applying $\left(x_{1}y_{1}\right)$, the successor of $b$ belongs to $B_{1}$, and there are no other edges connecting the two components of the multigraph. Hence, the successor of $b$ before transposing $\left(x_{2}y_{2}\right)$ (that is, the successor of $b$ in $\pi_{l-1}$) cannot be $y_{2}$.
Since $\mathcal{M}$ cannot contain more than one $\cutedge ab$, it must contain either one $\cutedge ab$ or it must contain no $\cutedge ab$s.
Consider next the case when there is no $\cutedge ab$ in $\mathcal{M}$. In this case, based on Menger’s theorem, there must exist at least two pairwise edge disjoint paths between $a$ and $b$. The cost of one of these paths has to be less than or equal to the cost of the other path. Refer to this path as the *minimum path*. Clearly, the cost of the decomposition of $(ab)$ described by $\mathcal{M}$ is greater than or equal to twice the cost of the minimum path.
Let the edges of the minimum path be $(az_{1})(z_{1}z_{2})...(z_{m-1}z_{m})(z_{m}b)$, for some integer $m$. The cost of $(ab)$ is greater than or equal to
$$\begin{split}
\text{\small $2\varphi^{*}(a,z_{1})+2\varphi^{*}(z_{1},z_{2})+\cdots+2\varphi^{*}(z_{m-1}z_{m})+2\varphi^{*}(z_{m},b)\ge$}\\
\text{\small $\varphi^{*}(a,z_{1})+2\varphi^{*}(z_{1},z_{2})+...+2\varphi^{*}(z_{m-1}z_{m})+2\varphi^{*}(z_{m},b)\geq$}\\
\text{\small $\varphi^{*}(a,z_{2})+2\varphi^{*}(z_{2},z_{3})+...+2\varphi^{*}(z_{m},b)\geq$}\\
\text{\small $\cdots\ge\varphi^{*}(a,z_{m})+2\varphi^{*}(z_{m},b)\geq\varphi^{*}(a,b),$}
\end{split}$$
and the cost of the decomposition associated with $\mathcal{M}$ cannot be smaller than the cost of the optimal decomposition produced by Alg. \[alg:opt-trans\].
Next, consider the case when there is one $\cutedge ab$ in $\mathcal{M}$. In this case, we distinguish two scenarios: when $x_{1}=a$, and when $x_{1}\neq a$.
In the former case, the transposition $(ay_{1})$ plays the role of the transposition $(az_{1})$ and the remaining transpositions used in the decomposition lie in the graph $\mathcal{M}-(ay_{1})$. Since $\mathcal{M}-(ay_{1})$ has no $\cutedge ab$, continuing with line two of (6) proves that the cost of the decomposition associated with $\mathcal{M}$ cannot be smaller than $\varphi^{*}(a,b).$
In the later case, the procedure we described for the case $x_{1}=a$ is first applied to the multigraph containing the edge $(x_{1}y_{1})$ and the sub-multigraph containing the vertex $a$. As a result, the edge $(x_{1}y_{1})$ is replaced by $(ay_{1}),$ with cost greater than or equal to $\varphi^{*}(a,y_{1})$. Applying the same procedure again, now for the case $x_{1}=a$, proves the claimed result.
As an illustration, one can see in Figures 1a-1l that the multigraphs corresponding to decompositions of length five have no more than one $\cutedge ab$.
A quick inspection of Alg. 1 reveals that it has the structure of a Viterbi-type search for finding a minimum cost path in a transposition graph. An equivalent search procedure can be devised to operate on the graph $\mathcal{K}\left(\varphi\right)$, rather than on a trellis. The underlying search algorithm is described in the Appendix, and is based on a modification of the well known Bellman-Ford procedure [@cormen24introduction].
For an arbitrary path $p=c_{1}c_{2}\cdots c_{m}$ in $\mathcal{K}\left(\varphi\right)$, the *transposition path cost* is defined as $$\bar{\varphi}\left(p\right)=2\sum_{i=1}^{m}\varphi\left(c_{i},c_{i+1}\right)-\max_{i}\varphi\left(c_{i},c_{i+1}\right).$$ Let $\hat{p}\left(a,b\right)$ be a path with minimum transposition path cost among paths between $a$ and $b$. That is, $$\bar{\varphi}(\hat{p}\left(a,b\right))=\min_{p}\bar{\varphi}\left(p\right),$$ where the minimum is taken over all paths $p$ in $\mathcal{K}\left(\varphi\right)$ between $a$ and $b$. Furthermore, let $p^{*}\left(a,b\right)$ be the standard shortest path between $a$ and $b$ in the cost graph $\mathcal{K}\left(\varphi\right)$.
\[lem:star-and-bar\]The minimum cost of a transpositions $\left(ab\right)$ is at most $\bar{\varphi}\left(\hat{p}\left(a,b\right)\right)$. That is, $$\varphi^{*}\left(a,b\right)\le\bar{\varphi}\left(\hat{p}\left(a,b\right)\right).$$
Suppose that $\hat{p}=c_{0}c_{1}\cdots c_{m}c_{m+1}$ where $a=c_{0}$ and $b=c_{m+1}$. Note that, for any $0\le i\le m$,
$$\begin{aligned}
(ab) & =\left(c_{0}c_{1}\cdots c_{i-1}c_{i}\right)\left(c_{m+1}c_{m}\cdots c_{i+2}c_{i+1}\right)\nonumber \\
& \left(c_{i}c_{i+1}\right)\left(c_{i+1}c_{i+2}\cdots c_{m}c_{m+1}\right)\left(c_{i}c_{i-1}\cdots c_{1}c_{0}\right),\label{eq:basic-path}\end{aligned}$$
Choose $i=\arg\max_{j}\varphi\left(c_{j},c_{j+1}\right)$ so that $\left(c_{i}c_{i+1}\right)$ is the most costly edge in $\hat{p}\left(a,b\right)$.
Each of the cycles in can be decomposed using the edges of $p$ as$$\begin{aligned}
\left(c_{0}c_{1}\cdots c_{i-1}c_{i}\right) & =\left(c_{i-1}c_{i}\right)\cdots\left(c_{2}c_{1}\right)\left(c_{1}c_{0}\right),\nonumber \\
\left(c_{i}c_{i-1}\cdots c_{1}c_{0}\right) & =\left(c_{0}c_{1}\right)\left(c_{1}c_{2}\right)\cdots\left(c_{i-1}c_{i}\right),\nonumber \\
\left(c_{i+1}c_{i+2}\cdots c_{m}c_{m+1}\right) & =\left(c_{m}c_{m+1}\right)\left(c_{m-1}c_{m}\right)\cdots\left(c_{i+2}c_{i+1}\right),\nonumber \\
\left(c_{m+1}c_{m}\cdots c_{i+2}c_{i+1}\right) & =\left(c_{i+1}c_{i+2}\right)\cdots\left(c_{m-1}c_{m}\right)\left(c_{m}c_{m+1}\right).\label{eq:basic-subs}\end{aligned}$$ Thus, the minimum cost of $\left(ab\right)$ does not exceed $$2\sum_{j=0}^{m}\varphi\left(c_{j},c_{j+1}\right)-\varphi\left(c_{i},c_{i+1}\right).$$
\[lem:Transpositional-path\]The minimum cost of a transposition $\left(ab\right)$ equals the minimum transposition path cost $\varphi(\hat{p}\left(a,b\right))$. That is,$$\varphi^{*}\left(a,b\right)=\bar{\varphi}\left(\hat{p}\left(a,b\right)\right).$$
Suppose $\tau$ is the minimum cost decomposition of $\left(ab\right)$. Let $\mathcal{M}=\mathcal{T}\left(\tau\right)$, and note that $\cost\left(\mathcal{M}\right)=\varphi^{*}\left(a,b\right)$.
In the proof of Theorem \[thm:correctness\_of\_alg1\], we showed that $\mathcal{M}$ has at most one $\cutedge ab$. Suppose that $\mathcal{M}$ has no $\cutedge ab$. Then there are two edge-disjoint paths between $a$ and $b$ in $\mathcal{M}$. Define the minimum path as in Theorem \[thm:correctness\_of\_alg1\]. Suppose the minimum path is $p=c_{0}c_{1}\cdots c_{m}c_{m+1}$. It is easy to see that $$\bar{\varphi}\left(\hat{p}\left(a,b\right)\right)\le\bar{\varphi}\left(p\right)\le\cost\left(\mathcal{M}\right)=\varphi^{*}\left(a,b\right).$$ From Lemma , we have $\varphi^{*}\left(a,b\right)\le\bar{\varphi}\left(\hat{p}\left(a,b\right)\right)$. Hence, $ $in this case, we conclude that $\varphi^{*}\left(a,b\right)=\bar{\varphi}\left(\hat{p}\left(a,b\right)\right)$.
Next, suppose that $\mathcal{M}$ has one $\cutedge ab$, as shown in Figure \[fig:one-cut-edge\]. Menger’s theorem implies that there are two edge-disjoint paths between $a$ and $x_{1}$ and two edge-disjoint paths between $b$ and $y_{1}$. Let $p_{1}$ be the path with smaller cost among the pair of paths between $a$ and $x_{1}$, and similarly, let $p_{2}$ be the path with smaller cost between the pair of paths between $b$ and $y_{1}$. Let $p$ be the path obtained by concatenating $p_{1}$, the edge $\left(x_{1}y_{1}\right)$, and $p_{2}$. Note that $\bar{\varphi}\left(\hat{p}\left(a,b\right)\right)\le\bar{\varphi}\left(p\right)\le\cost\left(\mathcal{M}\right)\le\varphi^{*}\left(a,b\right)$. Since $\varphi^{*}\left(a,b\right)\le\bar{\varphi}\left(\hat{p}\left(a,b\right)\right)$, we have $\varphi^{*}\left(a,b\right)=\bar{\varphi}\left(\hat{p}\left(a,b\right)\right)$.
It is easy to see that $\varphi^{*}\left(a,b\right)\le2\cost\left(p^{*}\left(a,b\right)\right)$ since we have $$\begin{aligned}
\varphi^{*}\left(a,b\right) & =\bar{\varphi}\left(\hat{p}\left(a,b\right)\right)\nonumber \\
& \le\bar{\varphi}\left(p^{*}\left(a,b\right)\right)\nonumber \\
& \le2\cost\left(p^{*}\left(a,b\right)\right).\label{eq:p-hat-p}\end{aligned}$$
Note that the Bellman-Ford Alg. \[alg:SSBF\], presented in the Appendix, finds the paths $\hat{p}$ in $\mathcal{K}(\varphi)$ between a given vertex $s$ and all other vertices in the graph.
Lemma \[lem:Transpositional-path\] provides an easy method for computing $\varphi^{*}\left(i,j\right)$ when there is only one path with finite cost between $i$ and $j$ in $\mathcal{K}\left(\varphi\right)$. For example, for an extended-metric cost function $\varphi$, we have $$\varphi^{*}\left(i,j\right)=2\sum_{t=1}^{l}\varphi\left(c_{t},c_{t+1}\right)-\max_{1\le t\le l}\varphi\left(c_{t},c_{t+1}\right),\label{eq:def-navin}$$ where $\hat{p}\left(i,j\right)=c_{1}\cdots c_{l+1}$, $c_{1}=i$, $c_{l+1}=j$, is the unique path between $i$ and $j$ in $\Theta_{s}$.
Optimizing Individual Cycles\[sec:Minimum-Cost-MLDs\]
=====================================================
We consider next the cost optimization problem over single cycles. First, we find the minimum cost MLD via a dynamic programming algorithm. The minimum cost MLD is obtained with respect to the optimized cost function $\varphi^{*}$ of the previous section. For simplicity, we henceforth omit the superscript in the cost whenever there is no ambiguity in terms which cost function is used.
We also present a second algorithm to find decompositions whose cost, along with the cost of the minimum cost MLD, is not more than a constant factor higher than the cost of the MCD. Both algorithms are presented for completeness.
The results in this section apply to any cycle $\sigma$. However, for clarity of presentation, and without loss of generality, we consider the cycle $\sigma=\left(12\cdots k\right)$.
Minimum Cost, Minimum Length Transposition Decomposition
--------------------------------------------------------
Recall that the vertices of $\mathcal{G}\left(\sigma\right)$ are placed on a circle. For an MLD $\tau$ of a permutation $\pi$ with $\ell$ cycles, $\mathcal{T}\left(\tau\right)$ is a forest with $\ell$ components; each tree in the forest is the decomposition of one cycle of $\pi$. This can be easily seen by observing that each cycle corresponds to a tree. The following lemma provides a rigorous proof for this statement.
The graph $\mathcal{T}\left(\tau\right)$ of an MLD $\tau$ of a cycle $\sigma$ is a tree.\[lem:MLD-graph\]
First, we show that $\mathcal{T}\left(\tau\right)$ is connected. The decomposition $\tau$ transform the identity permutation $\imath$ to $\sigma$ by transposing pairs of elements. Note that every transposition exchanges the predecessors of two elements. In $\imath$, each element $i$ is a fixed point (i.e., it is its own predecessor) and in $\sigma$, $i$ is the predecessor of $\sigma\left(i\right)$. Thus there exists a path, formed by a sequence of transpositions, between $i$ and $\sigma\left(i\right)$. An instance of such a path is described in Example \[exa:path-from-a-to-b-(ab)\].
To complete the proof, observe that $\mathcal{T}\left(\tau\right)$ has $k$ vertices and $k-1$ edges, since an MLD of a cycle of length $k$ contains $k-1$ transpositions. Hence, $\mathcal{T}\left(\tau\right)$ is a tree.
As already pointed out, we provide an example that illustrates the existence of a path from $i$ to $\sigma\left(i\right)$ in the decomposition $\tau$ of $\sigma$, for the special case when $\sigma$ is a cycle of length two.
\[exa:path-from-a-to-b-(ab)\]Consider the cycle $\sigma=\left(14\right)$. It is easy to see that $\tau=\left(34\right)\left(12\right)\left(23\right)\left(34\right)\left(12\right)$ is a decomposition of $\sigma$. Figure \[fig:path-a-b-(ab)\] illustrates a path from vertex 1 to vertex 4 in $\mathcal{T}\left(\tau\right)$. For instance, the transposition $\left(12\right)$ in $\tau$ corresponds to the edge $\left(12\right)$ in $\mathcal{T}\left(\tau\right)$, as shown in Figure \[fig:1b\], and the transposition $\left(23\right)$ corresponds to the edge $\left(23\right)$ etc. The cycle $\sigma$ is a cycle in $\pi_{e}$ in Figure \[fig:1e\]. The path from $1$ to $4$ in $\mathcal{T}\left(\tau\right)$ is $1\rightarrow2\rightarrow3\rightarrow4$.
For related ideas regarding permutation decompositions and graphical structures, the interested reader is referred to [@goulden_tree-like_2002].
The following definitions will be used in the proof of a lemma which states that $\mathcal{G}\left(\pi\right)\cup\mathcal{T}\left(\tau\right)$ is planar, provided that $\tau$ is an MLD of $\sigma$.
Let $R$ be the region enclosed by edges of $\mathcal{G}\left(\sigma\right)$. Let $T$ be a tree with vertex set $\{1,2,\cdots,k\}$, such that $\mathcal{G}\left(\sigma\right)\cup T$ is planar. Since $T$ is a tree with edges contained in $R$, the edges of $T$ partition $R$ into smaller regions; each of these parts is the enclosure of a subset of edges of $\mathcal{G}\left(\sigma\right)\cup T$ and includes the vertices of these edges. These vertices can be divided into *corner vertices,* lying at the intersection of at least two regions, and *inner vertices*, belonging only to one region. In Figure \[fig:regions\], $\mathcal{G}\left(\sigma\right)$ with vertices $V=\left\{ 1,2,3,4,5,6\right\} $ is partitioned into four regions, $R_{1},R_{2},R_{3}$ and $R_{4}$. In $R_{2}$, vertices $1$ and $3$ are corner vertices, while vertex $2$ is an inner vertex.
\[lem:MLD-planar\] For an MLD $\tau=t_{1}\cdots t_{k-1}$ of $\sigma$, $\mathcal{T}\left(\tau\right)\cup\mathcal{G}\left(\sigma\right)$ is planar. That is, for $t_{i}=\left(a_{1}a_{2}\right)$, where $a_{1}<a_{2}$, and $t_{j}=\left(b_{1}b_{2}\right)$, where $b_{1}<b_{2}$, if $a_{1}<b_{1}<a_{2}$, then $a_{1}<b_{2}<a_{2}$.
Note that $\tau^{-1}\sigma=\imath$. Let $\tau_{i}=t_{i-1}\cdots t_{1}$. Since $\tau$ is an MLD of $\sigma$, $\tau_{i}\sigma$ has $i$ cycles. The proof proceeds by showing that for all $1\le i\le k$, the following two claims are true:
\(I) $\mathcal{G}\left(\sigma\right)\cup\mathcal{T}\left(\tau_{i}\right)$ is planar.
\(II) Each cycle of $\tau_{i}$ corresponds to a subregion $R$ of $\mathcal{G}\left(\sigma\right)\cup\mathcal{T}\left(\tau_{i}\right)$. The cycle corresponding to $R$ contains all of its inner vertices and some of its corner vertices but no other vertex.
Both claims (I) and (II) are obvious for $i=1$. We show that if (I) and (II) are true for $\tau_{i}$, then they are also true for $\tau_{i+1}$.
Let $t_{i}=\left(ab\right)$. Clearly, $\tau_{i+1}\sigma=t_{i}\tau_{i}\sigma$ has one more cycle than $\tau_{i}\sigma$, and by assumption, $\mathcal{G}\left(\sigma\right)\cup\mathcal{T}\left(\tau_{i}\right)$ is planar and partitioned into a set of subregions. Note that $a$ and $b$ are in the same cycle, and thus are inner or corner veritices of some subregion $R^{*}$ of $\mathcal{G}\left(\sigma\right)\cup\mathcal{T}\left(\tau_{i}\right)$. The edge $\left(ab\right)$ divides $R^{*}$ into two subregions, $R_{a}$ and $R_{b}$ (without crossing any edge in $\mathcal{G}\left(\sigma\right)\cup\mathcal{T}\left(\tau_{i}\right)$). This proves (I).
Let the cycle corresponding to $R^{*}$ be $$\mu=\left(aa_{1}\cdots a_{l}bb_{1}\cdots b_{l'}\right)$$ as seen in Figure \[fig:ab-cycle-b\]. Then, $$\left(ab\right)\mu=\left(aa_{1}\cdots a_{l}\right)\left(bb_{1}\cdots b_{l'}\right).$$ Now the cycles $\left(aa_{1}\cdots a_{l}\right)$ and $\left(bb_{1}\cdots b_{l'}\right)$ in $\tau_{i+1}$ correspond to subregions $R_{a}$ and $R_{b}$, respectively, as see in Figure \[fig:ab-cycle-c\]. This proves claim (II) since the cycle corresponding to each subregion contains all of its inner vertices and some of its corner vertices but no other vertex.
The following lemma establishes a partial converse to the previous lemma.
\[lem:Planar-MLD\] For a cycle $\sigma$ and a spanning tree $T$ over the vertices $\left\{ 1,2,\cdots,k\right\} $, for which $\mathcal{G}\left(\sigma\right)\cup T$ is planar, there exists at least one MLD $\tau$ of $\sigma$ such that $T=\mathcal{T}\left(\tau\right)$.
We prove the lemma by recursively constructing an MLD corresponding to $T$. If $k=2$, then $T$ has exactly one edge and the MLD is the transposition corresponding to that edge. For $k>2$, some vertex has degree larger than one. Without loss of generality, assume that $\deg\left(1\right)>1$. Let$$r=\max\left\{ u|\left(1u\right)\in T\right\} .$$ Since $T$ is a tree, $T-\left(1r\right)$ has two components. These two components have vertex sets $\left\{ 1,\cdots,s\right\} $ and $\left\{ s+1,\cdots,k\right\} $, for some $s$. It is easy to see that $$\left(1\,\cdots\, k\right)=\left(s+1\,\cdots\, k\,1\right)\left(1\,\cdots\, s\right).\label{eq:two-cycle}$$ Let$$\begin{aligned}
T' & = & T\left[\left\{ 1,\cdots,s\right\} \right],\\
T'' & = & T\left[\left\{ s+1,\cdots,k,1\right\} \right].\end{aligned}$$ Note that $T'$ and $T''$ have fewer than $k$ vertices. Furthermore, $T'\cup\mathcal{G}\left(\left(1\cdots\, s\right)\right)$ and $T''\cup\mathcal{G}\left(\left(s+1\cdots k\,1\right)\right)$ are planar. Thus, from the induction hypothesis, $\left(1\,\cdots\, s\right)$ and $\left(s+1\,\cdots\, k\,1\right)$ have decompositions $\tau'$ and $\tau''$ of length $s-1$ and $k-s$, respectively. By , $\tau''\tau'$ is an MLD for $\sigma$.
In Figure \[fig:no-last-edge-1\], we have $r=10$ and $s=8$. The cycle $\left(1\,\cdots\,12\right)$ can be decomposed into two cycles,$$\left(1\,\cdots\,12\right)=\left(9\,10\,11\,12\,1\right)\left(1\,2\,\cdots\,8\right).$$ Now, each of these cycles is decomposed into smaller cycles, for example $$\begin{aligned}
\left(9\,10\,11\,12\,1\right) & =\left(9\,10\right)\left(10\,11\,12\,1\right),\\
\left(1\cdots8\right) & =\left(8\,1\right)\left(1\cdots7\right).\end{aligned}$$ The same type of decomposition can be performed on the cycles $\left(9\,10\right),\cdots,\left(17\right)$.
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Since any MLD of a cycle can be represented by a tree that is planar on the circle, the search for an MLD of minimum cost only needs to be performed over the set of planar trees. This search can be performed using a dynamic program, outlined in Alg. \[alg:opt-cycle\]. Lemma \[lem:opt-cycle\] establishes that Alg. \[alg:opt-cycle\] produces a minimum cost MLD.
Input: Optimized transposition cost function $\Phi^*$ where $\Phi^*_{i,j}=\varphi^*(i,j)$ (Output of Alg. \[alg:opt-trans\]) $C(i,j)\leftarrow\infty $ for $i,j\in[k]$ $C\left(i,i\right)\leftarrow 0$ for $i\in \left[k\right]$ $C\left(i,i+1\right)\leftarrow \varphi^*\left(i,i+1\right)$ for $i\in \left[k\right]$ $j \leftarrow i+l $ $A\leftarrow C(i,s)+C(s+1,r)+C(r,j )+\varphi^*(i,r) $ $C(i,j)\leftarrow A$
\[lem:opt-cycle\]The output cost of Alg. , $C\left(1,k\right)$, equals $L\left(\sigma\right)$.
The algorithm finds the minimum cost MLD of $\left(1\cdots k\right)$ by first finding the minimum cost of MLDs of shorter cycles of the form $\left(i\cdots j\right)$, where $1\le i<j\le k$. We look at the computations performed in the algorithm from a top-down point of view.
Let $C_{T}\left(i,j\right)$ be the cost of the decomposition of the cycle $\sigma^{i,j}=\left(i\cdots j\right)$, using edges of $T\left[\left\{ i,\cdots,j\right\} \right]$, where $T$ is an arbitrary planar spanning tree over the vertices $\left\{ 1,\cdots,k\right\} $ arranged on a circle. For a fixed $T$, let $r$ and $s$ be defined as in the proof of Lemma \[lem:Planar-MLD\]. We may write$$\left(i\cdots j\right)=\left(s+1\cdots r\right)\left(ir\right)\left(r\cdots j\right)\left(i\cdots s\right)\label{eq:CT0}$$ where $i\le s<r\le j$. Thus $$\begin{aligned}
C_{T}\left(i,j\right) & =C_{T}\left(s+1,r\right)+\varphi^{*}\left(i,r\right)+C_{T}\left(r,j\right)+C_{T}\left(i,s\right).\label{eq:CT-1}\end{aligned}$$ Define $C\left(i,j\right)=C_{T^{*}}\left(i,j\right)$, where $$T^{*}=\arg\min_{T}C_{T}\left(i,j\right)$$ denotes a tree that minimizes the cost of the decomposition of $\left(i\cdots j\right)$. Then, we have$$C\left(i,j\right)=C\left(s^{*}+1,r^{*}\right)+\varphi^{*}\left(i,r^{*}\right)+C\left(r^{*},j\right)+C\left(i,s^{*}\right),\label{eq:CT1}$$ where $s^{*}$ and $r^{*}$ are the values that minimize the right-hand-side of under the constraint $1\le i\le s<r\le j$. Since the cost of each cycle can be computed from the cost of shorter cycles, $C\left(i,j\right)$ can be obtained recursively, with initialization $$C\left(i,i+1\right)=\varphi^{*}\left(i,i+1\right).\label{eq:CT2}$$ The algorithm searches over $s$ and $r$ and computes $C\left(1,k\right)$ using and .
Although these formulas are written in a recursive form, Alg. \[alg:opt-cycle\] is written as a dynamic program. The algorithm first computes $C\left(i,j\right)$ for small values of $i$ and $j$, and then finds the cost of longer cycles. That is, for each $2\le l\le k-1$ in increasing order, $C\left(i,i+l\right)$ is computed by choosing its optimal decomposition in terms of costs of smaller cycles.
As an example, let us find the minimum cost decomposition of the cycle $\sigma=(1234)$ using the above algorithm. Let $\Phi$ be the matrix of transposition costs, with $\Phi_{ij}=\varphi\left(i,j\right)$: $$\Phi=\left[\begin{array}{cccc}
0 & 5 & 10 & 3\\
- & 0 & 2 & 3\\
- & - & 0 & 9\\
- & - & - & 0\end{array}\right],\label{eq:cost_raw}$$ $$\Phi^{*}=\left[\begin{array}{cccc}
0 & 5 & 9 & 3\\
- & 0 & 2 & 3\\
- & - & 0 & 7\\
- & - & - & 0\end{array}\right]$$ After optimizing the transposition costs in $\Phi$ via Alg. \[alg:opt-trans\], we obtain $\Phi^{*}$, shown beneath $\Phi$. From Alg. \[alg:opt-cycle\], we obtain$$\begin{aligned}
C\left(1,3\right) & =C\left(2,3\right)+\varphi^{*}\left(1,2\right)=7,\left(s,r\right)=\left(1,2\right),\\
C\left(2,4\right) & =C\left(2,3\right)+\varphi^{*}\left(2,4\right)=5,\left(s,r\right)=\left(3,4\right).\end{aligned}$$ Consider the cycle $\left(1234\right)$, where $i=1$ and $j=4$. The algorithm compares $\binom{4}{2}=6$ ways to represent the cost of this cycle using the cost of shorter cycles. The minimum cost is obtained by choosing $s=2$ and $r=4$, so that$$C\left(1,4\right)=C\left(2,4\right)+\varphi^{*}\left(1,4\right)=8.$$ Writing $C$ as a matrix, where $C\left(i,j\right)=C_{ij}$, we have: $$C=\left[\begin{array}{cccc}
0 & 5 & 7 & 8\\
- & 0 & 2 & 5\\
- & - & 0 & 7\\
- & - & - & 0\end{array}\right]$$
Note that we can modify the above algorithm to also find the underlying MLD by using to write the decomposition of every cycle with respect to $r$ and $s$ that minimize the cost of the cycle. For example, from , by substituting the appropriate values of $r$ and $s$, we obtain $$\begin{aligned}
\left(1234\right) & =\left(234\right)\left(14\right)\\
& =\left(34\right)\left(24\right)\left(14\right).\end{aligned}$$
The initialization steps are performed in $O\left(k\right)$ time. The algorithm performs a constant number of steps for each $i,j,r$, and $s$ such that $1\le i\le s<r\le j\le k$. Hence, the computational cost of the algorithm is $O\left(k^{4}\right)$.
Note that Alg. \[alg:opt-cycle\] operates on the optimized cost function $\varphi^{*}$, obtained as the output of Alg. \[alg:opt-trans\]. Figure \[fig:f1\] illustrates the importance of first reducing individual transposition costs using Alg. \[alg:opt-trans\] before applying the dynamic program. Since the dynamic program can only use $k-1$ transpositions of minimum cost, it cannot optimize the individual costs of transpositions and strongly relies on the reduction of Alg. \[alg:opt-trans\] for producing low cost solutions. In Figure \[fig:f1\], the transposition costs were chosen independently from a uniform distribution over $\left[0,1\right]$.
![The average minimum MLD cost vs the length of the cycle. Transposition costs are chosen independently and uniformly in [\[]{}0,1[\]]{}.[]{data-label="fig:f1"}](compare_MST_OUR_0_2){width="3.4in"}
Constant-factor approximation for cost of MCD
---------------------------------------------
For the cycle $\sigma=\left(12\cdots k\right)$ and $1\le j\le k$, consider the decomposition $$\begin{gathered}
\left(j+1\ j+2\right)\left(j+2\ j+3\right)\cdots\\
\left(k-1\ k\right)\left(k1\right)\left(12\right)\left(23\right)\cdots\left(j-1\ j\right).\end{gathered}$$ The cost of this decomposition equals $$\sum_{i\in\sigma}\varphi^{*}\left(i,\sigma\left(i\right)\right)-\varphi^{*}\left(j,\sigma\left(j\right)\right).$$ To minimize the cost of the decomposition, we choose $j$ such that the transpositions $\left(j\ j+1\right)$ has maximum cost. This choice leads to the decomposition$$\begin{gathered}
\left(j^{*}+1\ j^{*}+2\right)\left(j^{*}+2\ j^{*}+3\right)\cdots\\
\left(k-1\ k\right)\left(k1\right)\left(12\right)\left(23\right)\cdots\left(j^{*}-1\ j^{*}\right)\label{eq:j*}\end{gathered}$$ where $$j^{*}=\arg\max_{j\in\sigma}\varphi^{*}\left(j,\sigma\left(j\right)\right).$$ The decomposition in is termed the Simple Transposition Decomposition (STD) of $\sigma$. The cost of the STD of $\sigma$, denoted by $S\left(\sigma\right)$, equals $$\begin{aligned}
S\left(\sigma\right) & =\sum_{i\in\sigma}\varphi^{*}\left(i,\sigma\left(i\right)\right)-\varphi^{*}\left(j^{*},\sigma\left(j^{*}\right)\right).\end{aligned}$$
\[thm:MLS4M-1\] For a cycle $\sigma$, $M\left(\sigma\right)\le L\left(\sigma\right)\le S\left(\sigma\right)\le4M\left(\sigma\right).$
Clearly, $M\left(\sigma\right)\le L\left(\sigma\right)$. It is easy to see that the STD is itself an MLD and, thus, $L\left(\sigma\right)\le S\left(\sigma\right)$. For $S\left(\sigma\right)$, we have$$\begin{aligned}
S\left(\sigma\right) & =\sum_{i\in\sigma}\varphi^{*}\left(i,\sigma\left(i\right)\right)-\varphi^{*}\left(j^{*},\sigma\left(j^{*}\right)\right)\nonumber \\
& \le\sum_{i\in\sigma}\varphi^{*}\left(i,\sigma\left(i\right)\right)\nonumber \\
& \le2\sum_{i\in\sigma}\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right)\label{eq:STD-Cost}\end{aligned}$$ where the last inequality follows from . To complete the proof, we need to show that $M\left(\sigma\right)\ge\frac{1}{2}\sum_{i}\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right)$. Since this result is of independent importance in our subsequent derivations, we state it and prove it in Lemma \[lem:M-best-path-1\].
In order to prove Lemma \[lem:M-best-path-1\], we first prove Lemma \[lem:consistent-costs-1\] and a corollary.
Consider a transposition cost function $\varphi$ and a h-transposition cost function $\psi$. Recall from Section \[sec:Notation-and-Definitions\] that$$\htrans ab{\sigma^{-1}\left(a\right)}\htrans ba{\sigma^{-1}\left(b\right)}\sigma=\left(ab\right)\sigma.$$
\[lem:consistent-costs-1\]The minimum cost of an h-decomposition of $\sigma$ is upper-bounded by the cost of the MCD of $\sigma$, provided that $\varphi$ and $\psi$ are consistent.
We prove the lemma by showing that there exists an h-decomposition of $\sigma$ with cost $M\left(\sigma\right)$. Suppose that the MCD of $\sigma$ is $\tau=t_{m}t_{m-1}\cdots t_{1}$, where $t_{i}=\left(a_{i}b_{i}\right)$, $1\le i\le m$, and where $m$ is the length of the MCD. Let the permutation $t_{i}t_{i-1}\cdots t_{1}$ be denoted by $\sigma_{i}$. The cost of the MCD is $$M_{\varphi}\left(\sigma\right)=\sum_{i=1}^{m}\varphi\left(a_{i},b_{i}\right).$$ By replacing each transposition $t_{i}=\left(a_{i}b_{i}\right)$ in $\tau$ by a corresponding pair of h-transpositions $\htrans{a_{i}}{b_{i}}{\sigma_{i-1}^{-1}\left(a_{i}\right)}\htrans{b_{i}}{a_{i}}{\sigma_{i-1}^{-1}\left(b_{i}\right)}$, one can see that $$\begin{aligned}
M\left(\sigma\right) & = & \sum_{i=1}^{m}\left(\psi\left(a_{i},b_{i}\right)+\psi\left(b_{i},a_{i}\right)\right),\end{aligned}$$ since $\psi$ and $\varphi$ are consistent. Hence, the h-decomposition $$\begin{gathered}
\htrans{b_{m}}{a_{m}}{\sigma_{m-1}^{-1}\left(b_{m}\right)}\htrans{a_{m}}{b_{m}}{\sigma_{m-1}^{-1}\left(a_{m}\right)}\cdots\\
\htrans{b_{1}}{a_{1}}{\sigma_{0}^{-1}\left(b_{1}\right)}\htrans{a_{1}}{b_{1}}{\sigma_{0}^{-1}\left(a_{1}\right)}\end{gathered}$$ has cost $M\left(\sigma\right)$. In other words, decomposing each transposition in an MCD into h-transpositions establishes the claimed result.
\[cor:maximin-1\]For a fixed $\varphi$ and a cycle $\sigma$, one has $$M\left(\sigma\right)\ge\max_{\psi}\min_{H}C_{\psi}\left(H\right)$$ where the maximum is taken over all h-transposition costs $\psi$ consistent with $\varphi$, and the minimum is taken over all h-decompositions $H$ of $\sigma$ with cost $C_{\psi}\left(H\right)$.
\[lem:M-best-path-1\]It holds that $M\left(\sigma\right)\ge\frac{1}{2}\sum_{i}\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right)$.
Define $\psi_{1/2}$ as$$\psi_{1/2}\left(a,b\right)=\psi_{1/2}\left(b,a\right)=\varphi\left(a,b\right)/2.$$ It is clear that $\psi_{1/2}$ is consistent with $\varphi$. Hence, we have
$$\begin{aligned}
M\left(\sigma\right) & \ge & \max_{\psi}\min_{H}C_{\psi}\left(H\right)\\
& \ge & \min_{H}C_{\psi_{1/2}}\left(H\right)\\
& \stackrel{\mathtt{(\star)}}{=} & \sum_{i}\sum_{\left(ab\right)\in p^{*}\left(i,\sigma\left(i\right)\right)}\psi_{1/2}\left(a,b\right)\\
& = & \frac{1}{2}\sum_{i}\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right)\end{aligned}$$
where $\mathtt{(\star)}$ follows from the fact that the minimum cost h-decomposition uses the shortest path $p^{*}\left(i,\sigma\left(i\right)\right)$ between $i$ and $\sigma\left(i\right)$. In this case, $i$ becomes the predecessor of $\sigma\left(i\right)$ through the following sequence of h-transpositions:$$\htrans{v_{m}}{\sigma\left(i\right)}i\cdots\htrans{v_{1}}{v_{2}}i\htrans i{v_{1}}i$$ where $p^{*}\left(i,\sigma\left(i\right)\right)=iv_{1}v_{2}\cdots v_{m}\sigma\left(i\right)$ is the shortest path between $i$ and $\sigma\left(i\right)$.
Observe that Theorem \[thm:MLS4M-1\] asserts that *a minimum cost MLD never exceeds the cost of the corresponding MCD by more than a factor of four*. Hence, a minimum cost MLD represents a good approximation for an MCD, independent of the choice of the cost function. On the other hand, STDs and their corresponding path search algorithms are attractive alternatives to MLDs and dynamic programs, due to the fact that they are particularly simple to implement.
\[exa:13-1\]Consider the cycle $\sigma=\left(12345\right)$ and the cost function $\varphi$, with $\varphi\left(2,4\right)=\varphi\left(2,5\right)=\varphi\left(3,5\right)=1$, and $\varphi\left(i,j\right)=100$ for all remaining transposition. First, observe that the costs are not reduced according to Alg. \[alg:opt-trans\]. Nevertheless, one can use the upper-bound for the transposition cost in terms of the shortest paths defined in the proof of Lemma \[lem:M-best-path-1\]. In this case, one obtains $$M\left(\sigma\right)\ge\frac{1}{2}\left(100+2+3+2+100\right)=103.5.$$ For example, the second term in the sum corresponds to a path going from $2$ to $5$ and then from $5$ to $3$. The cost of this path is two.
Since $M\left(\sigma\right)$ has to be an integer, it follows that $M\left(\sigma\right)\ge104$.
The optimized cost function, $\varphi^{*}$, obtained from Alg. \[alg:opt-trans\] gives $$\varphi^{*}\left(i,j\right)=\begin{cases}
1, & \quad\left(ij\right)\in\left\{ \left(25\right),\left(35\right),\left(24\right)\right\} \\
3, & \quad\left(ij\right)\in\left\{ \left(23\right),\left(45\right)\right\} \\
5, & \quad\left(ij\right)=\left(34\right)\\
100, & \quad\text{otherwise }\end{cases}$$
A minimum cost MLD can be computed using the dynamic program of Alg. \[alg:opt-cycle\]. One minimum cost MLD equals $\tau_{L}=\left(45\right)\left(35\right)\left(12\right)\left(25\right)$, and has cost $L\left(\sigma\right)=105$. By substituting each of the transposition in $\tau_{L}$ with their minimum cost transposition decomposition, we obtain $\left(24\right)\left(25\right)\left(24\right)\left(35\right)\left(12\right)\left(25\right)$.
It is easy to see that $$\tau_{s}=\left(12\right)\left(23\right)\left(34\right)\left(45\right)$$ is the STD of $\sigma$ with cost $S\left(\sigma\right)=100+3+5+3=111$.
Hence, the inequality $M\left(\sigma\right)\le L\left(\sigma\right)\le S\left(\sigma\right)\le4M\left(\sigma\right)$ holds. Furthermore, note that $\sigma$ is an even cycle, and hence must have an even number of transpositions in any of its decompositions. This shows that $M\left(\sigma\right)=L\left(\sigma\right)=105$.
Metric-Path and Extended-Metric-Path Cost Functions
---------------------------------------------------
We show next that for two non-trivial families of cost functions, one can improve upon the bounds of Theorem \[thm:MLS4M-1\]. For metric-path cost functions, a minimum cost MLD is actually an MCD, i.e., $L\left(\sigma\right)=M\left(\sigma\right)$. For extended-metric-path costs, it holds that $L\left(\sigma\right)\le2M\left(\sigma\right)$.
Note that metric-path costs are not the only cost functions which admit MCDs of the form of MLDs – another example includes star transposition costs. For such costs, one has $\varphi\left(i,j\right)=\infty$ for all $i,j$ except for one index $i$. The remaining costs are arbitrary, but non-negative. The proof for this special case is straightforward and hence omitted.
\[lem:MLD-metric-tree\]For a cycle $\sigma$ and a metric-path cost function $\varphi$, $L\left(\sigma\right)\le\frac{1}{2}\sum_{i}\varphi\left(i,\sigma\left(i\right)\right)=\frac{1}{2}\sum_{i}\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right)$.
The equality in the lemma follows from the definition of metric-path cost functions.
We recursively construct a spanning tree $T\left(\sigma\right)$ of cost $B\left(\sigma\right)=\frac{1}{2}\sum_{i}\varphi\left(i,\sigma\left(i\right)\right)$, such that $\mathcal{G}\left(\sigma\right)\cup T\left(\sigma\right)$ is planar. Since $T\left(\sigma\right)$ corresponds to an MLD, $L\left(\sigma\right)\le B\left(\sigma\right)$.
The validity of the recursive construction can be proved by induction. For $k=2$, $T\left(\sigma\right)$ is the edge $\left(12\right)$. Assume next that the cost of $T\left(\sigma\right)$ for any cycle of length $\le k-1$ equals $B\left(\sigma\right)$.
For a cycle of length $k$, without loss of generality, assume that the vertex labeled $1$ is a leaf in $\Theta_{s}$, the defining path of $\varphi$, and that $t$ is its parent. We construct $T\left(\sigma\right)$ from smaller trees by letting $$T\left(\sigma\right)=\left(1t\right)\cup T\left(\left(2\cdots t\right)\right)\cup T\left(\left(t\cdots k\right)\right).$$
See Figure \[fig:MLD-metric-tree\] for an illustration. The cost of $T\left(\sigma\right)$ is equal to $B\left(\left(2\cdots t\right)\right)+B\left(\left(t\cdots k\right)\right)+\varphi\left(1,t\right)$. Note that we can write$$\begin{aligned}
B\left(\left(2\cdots t\right)\right) & =\frac{1}{2}\sum_{i=2}^{t-1}\varphi\left(i,\sigma\left(i\right)\right)+\frac{1}{2}\varphi\left(2,t\right)\\
& =\frac{1}{2}\sum_{i=1}^{t-1}\varphi\left(i,\sigma\left(i\right)\right)+\frac{1}{2}\varphi\left(2,t\right)-\frac{1}{2}\varphi\left(1,2\right),\end{aligned}$$ $$\begin{aligned}
B\left(\left(t\cdots k\right)\right) & =\frac{1}{2}\sum_{i=t}^{k-1}\varphi\left(i,\sigma\left(i\right)\right)+\frac{1}{2}\varphi\left(t,k\right)\\
& =\frac{1}{2}\sum_{i=t}^{k}\varphi\left(i,\sigma\left(i\right)\right)+\frac{1}{2}\varphi\left(t,k\right)-\frac{1}{2}\varphi\left(1,k\right).\end{aligned}$$ Since $\varphi\left(1,2\right)=\varphi\left(1,t\right)+\varphi\left(t,2\right)$ and $\varphi\left(1,k\right)=\varphi\left(1,t\right)+\varphi\left(t,k\right)$, it follows that$$B\left(\left(2\cdots t\right)\right)+B\left(\left(t\cdots k\right)\right)=B\left(\left(1\cdots k\right)\right)-\varphi\left(1,t\right).$$ This completes the proof of the Lemma.
\[lem:MLD-construction-metric-tree\]For a cycle $\sigma$ and a metric-path cost function, one has$$L\left(\sigma\right)=M\left(\sigma\right)=\frac{1}{2}\sum_{i}\varphi\left(i,\sigma\left(i\right)\right).$$
Since $L\left(\sigma\right)\ge M\left(\sigma\right)$, it suffices to show that $L\left(\sigma\right)\le\frac{1}{2}\sum_{i}\varphi\left(i,\sigma\left(i\right)\right)$ and $M\left(\sigma\right)\ge\frac{1}{2}\sum_{i}\varphi\left(i,\sigma\left(i\right)\right)$. Lemma \[lem:MLD-metric-tree\] establishes that $L\left(\sigma\right)\le\frac{1}{2}\sum_{i}\varphi\left(i,\sigma\left(i\right)\right)$. From Lemma \[lem:M-best-path-1\], it also follows that $$M\left(\sigma\right)\ge\frac{1}{2}\sum_{i}\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right).$$ Since $\varphi$ is a metric-path cost function, we have $\varphi\left(i,\sigma\left(i\right)\right)=\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right)$. This proves the claimed result.
\[thm:L<=00003D2M-ext-metric-tree\]For extended-metric-path cost functions $\varphi_{e}$, $L_{\varphi_{e}}\left(\sigma\right)\le2M_{\varphi_{e}}\left(\sigma\right)$.
We prove the theorem by establishing that $$L_{\varphi_{e}}\left(\sigma\right)\stackrel{\mathsf{\left(a\right)}}{\le}\sum_{i}\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right)\stackrel{\mathsf{\left(b\right)}}{\le}2M_{\varphi_{e}}\left(\sigma\right)$$ where $p^{*}\left(i,\sigma\left(i\right)\right)$ is the shortest path between $i$ and $\sigma\left(i\right)$ in $\mathcal{K}\left(\varphi_{e}\right)$ and is calculated with respect to the cost function $\varphi_{e}$.
Let $\Theta_{s}$ be the defining path of an extended-metric-path cost $\varphi_{e}$. Consider the *metric-path cost* function, $\varphi_{m}$, with defining path $\Theta_{s}$, and with costs of all edges $\left(ij\right)\in\Theta_{s}$ doubled. If the edge $\left(ij\right)\notin\Theta_{s}$, and if $c_{1}c_{2}\cdots c_{l+1}$ is the unique path from $c_{1}=i$ to $c_{l+1}=j$ in $\Theta_{s}$, then $$\varphi_{m}\left(i,j\right)=\sum_{t=1}^{l}\varphi_{m}\left(c_{t},c_{t+1}\right)=2\sum_{t=1}^{l}\varphi_{e}\left(c_{t},c_{t+1}\right).$$ By , $\varphi_{e}\left(i,j\right)\le\varphi_{m}\left(i,j\right)$, for all $i,j$. Hence, $L_{\varphi_{e}}\left(\sigma\right)\le L_{\varphi_{m}}\left(\sigma\right)$. Now, following along the same lines of the proof of Lemma \[lem:MLD-metric-tree\], it can be shown that $$\begin{aligned}
L_{\varphi_{e}}\left(\sigma\right) & \le L_{\varphi_{m}}\left(\sigma\right)\label{eq:lower-bound-non-metric}\\
& =\frac{1}{2}\sum_{i}\varphi_{m}\left(i,\sigma\left(i\right)\right)\nonumber \\
& =\sum_{i}\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right),\nonumber \end{aligned}$$ which proves $\mathsf{\left(a\right)}$.
Note that Lemma \[lem:M-best-path-1\] holds for all non-negative cost functions, including extended-metric-path cost functions. Thus, $$M_{\varphi_{e}}\left(\sigma\right)\ge\frac{1}{2}\sum_{i}\cost\left(p^{*}\left(i,\sigma\left(i\right)\right)\right),$$ which proves $\mathsf{\left(b\right)}$.
Consider the cycle $\sigma=\left(12345\right)$ and the extended-metric-path cost function of Example \[exa:navin\].
By inspection, one can see that an MCD of $\sigma$ is $\left(14\right)\left(13\right)\left(35\right)\left(24\right)\left(14\right)\left(13\right)$, with cost $M\left(\sigma\right)=6$. A minimum cost MLD of $\sigma$ is $\left(14\right)\left(23\right)\left(13\right)\left(45\right)$, with cost $L\left(\sigma\right)=8$. The STD is $\left(12\right)\left(23\right)\left(34\right)\left(45\right)$, with cost $S\left(\sigma\right)=12$. Thus, we observe that the inequality $L\left(\sigma\right)\le S\left(\sigma\right)\le2M\left(\sigma\right)$ is satisfied.
Optimizing Permutations with Multiple Cycles
============================================
Most of the results in the previous section generalize to permutations with multiple cycles without much difficulty. We present next the generalization of those results.
Let $\pi$ be a permutation in $\mathbb{S}_{n}$, with cycle decomposition $\sigma_{1}\sigma_{2}\cdots\sigma_{\ell}$. A decomposition of $\pi$ with minimum number of transpositions is the product of MLDs of individual cycles $\sigma_{i}$. Thus, the minimum cost MLD of $\pi$ equals $$L\left(\pi\right)=\sum_{t=1}^{\ell}L\left(\sigma_{t}\right).$$ The STD of $\pi$ is the product of the STDs of individual cycles $\sigma_{i}$.
The following theorem generalizes the results presented for single cycle permutations to permutations with multiple cycles.
Consider a permutation $\pi$ with cycle decomposition $\sigma_{1}\sigma_{2}\cdots\sigma_{\ell}$, and cost function $\varphi$. The following claims hold.
1. $S\left(\pi\right)\le2\sum_{i}\cost\left(p^{*}\left(i,\pi\left(i\right)\right)\right).$
2. $M\left(\pi\right)\ge\frac{1}{2}\sum_{i}\cost\left(p^{*}\left(i,\pi\left(i\right)\right)\right).$
3. 4. If $\varphi$ is a metric-path cost function, then $$M\left(\pi\right)=L\left(\pi\right).$$
5. If $\varphi$ is an extended-metric-path cost function, then $$L\left(\pi\right)\le2M\left(\pi\right).$$
1. For each cycle it holds that $$S\left(\sigma_{t}\right)\le2\sum_{i\in\sigma_{t}}\cost\left(p^{*}\left(i,\sigma_{t}\left(i\right)\right)\right),$$ which can be seen by referring to in the proof of Theorem \[thm:MLS4M-1\]. Thus, $$\begin{aligned}
S\left(\pi\right) & =\sum_{t=1}^{\ell}S\left(\sigma_{t}\right)\\
& \le\sum_{t=1}^{\ell}2\sum_{i\in\sigma_{t}}\cost\left(p^{*}\left(i,\pi\left(i\right)\right)\right)\\
& =2\sum_{i=1}^{n}\cost\left(p^{*}\left(i,\pi\left(i\right)\right)\right).\end{aligned}$$
2. The same argument as in Lemma \[lem:M-best-path-1\] applies without modifications.
3. For each $1\le t\le\ell$, from the proof of Theorem \[thm:MLS4M-1\], we have $L\left(\sigma_{t}\right)\le S\left(\sigma_{t}\right)$. Consequently,$$L\left(\pi\right)=\sum_{t=1}^{\ell}L\left(\sigma_{t}\right)\le\sum_{t=1}^{\ell}S\left(\sigma_{t}\right)=S\left(\pi\right).$$ Furthermore, from parts 1 and 2 of this theorem, it follows that $S\left(\pi\right)\le4M\left(\pi\right)$. Therefore $L\left(\pi\right)\le S\left(\pi\right)\le4M\left(\pi\right)$.
4. From Lemma \[lem:MLD-metric-tree\], for each $\sigma_{t}$, it holds that $$L\left(\sigma_{t}\right)\le\frac{1}{2}\sum_{i\in\sigma_{t}}\cost\left(p^{*}\left(i,\pi\left(i\right)\right)\right).$$ By summing over all cycles, we obtain $$L\left(\pi\right)\le\frac{1}{2}\sum_{i=1}^{n}\cost\left(p^{*}\left(i,\pi\left(i\right)\right)\right).$$ The claimed result follows from part 2 and the fact that $M\left(\pi\right)\le L\left(\pi\right)$.
5. From the proof of Theorem \[thm:L<=00003D2M-ext-metric-tree\], we have $L\left(\sigma_{t}\right)\le\sum_{i\in\sigma_{t}}\cost\left(p^{*}\left(i,\pi\left(i\right)\right)\right)$. By summing over all cycles, we obtain$$L\left(\pi\right)\le\sum_{i=1}^{n}\cost\left(p^{*}\left(i,\pi\left(i\right)\right)\right)\le2M\left(\pi\right),$$ where the last inequality follows from part 2 of this theorem.
Merging cycles
--------------
In Section \[sec:Minimum-Cost-MLDs\], we demonstrated that the minimum cost of an MLD for an arbitrary permutation represents a constant approximation for an MCD. The MLD of a permutation represents the product of the MLDs of individual cycles of the permutation. Clearly, optimization of individual cycle costs may not lead to the minimum cost decomposition of a permutation. For example, it may happen that the cost of transpositions within a cycle are much higher than the costs of transpositions between elements in different cycles. It is therefore useful to analyze how merging of cycles may affect the overall cost of a decomposition.
We propose a simple merging method that consists of two steps:
1. Find a sequence of transpositions $$\tau'=t_{k-1}\cdots t_{1}$$ so that $\sigma'=\tau'\pi$ is a single cycle. Ideally, this sequence should have minimum cost, although this is not required in the proofs to follow.
2. Find the minimum cost MLD $\tau$ of $\sigma'$.
The resulting decomposition is of the form $\tau'^{-1}\tau$.
Suppose that $\pi$ has $k$ cycles. Joining $k$ cycles requires $k-1$ transpositions. Hence, each $t_{i}$ is a transposition joining two cycles of $\pi$. The cost of $\tau'$ equals $\sum_{i=1}^{k-1}\varphi\left(a_{i},b_{i}\right)$, where $t_{i}=\left(a_{i}b_{i}\right)$. The cost of the resulting decomposition using $\tau'$ and the single cycle MLD equals $$C=\sum_{i=1}^{k-1}\varphi\left(a_{i},b_{i}\right)+L(\sigma').\label{eq:app1-1}$$ Since $\pi=t_{1}\cdots t_{k}\sigma'$, we also have $$L(\sigma')\le4M\left(\sigma'\right)\le4\left(\sum_{i=1}^{k-1}\varphi\left(a_{i},b_{i}\right)+M(\pi)\right).\label{eq:app2-1}$$ Hence, from and , $C$ is upper bounded by $$\begin{split}C & \le\sum_{i=1}^{k-1}\varphi\left(a_{i},b_{i}\right)+4\left(\sum_{i=1}^{k-1}\varphi\left(a_{i},b_{i}\right)+M(\pi)\right)\\
& \le5k\varphi_{max}+4M(\pi),\end{split}$$ where $\varphi_{max}$ is the highest cost in $\varphi$. The approximation ratio, defined as $C/M(\pi)$, is upper bounded by $$\alpha\le4+\frac{5k}{n-k}\frac{\varphi_{max}}{\varphi_{min}}=4+\frac{5k/n}{1-k/n}\frac{\varphi_{max}}{\varphi_{min}},\label{eq:alpha}$$ which follows from the fact $M(\pi)\ge\left(n-k\right)\varphi_{min}$, where $\varphi_{min}$ is the smallest cost in $\varphi$, assumed to be nonzero.
Although $\alpha$ is bounded by a value strictly larger than four, according to the expression above, this does not necessarily imply that merging cycles is sub-optimal compared to running the MLD algorithm on individual cycles. Furthermore, if the MCDs of single cycles can be computed correctly, one can show that $$\begin{aligned}
C & = & \sum_{i=1}^{k-1}\varphi\left(a_{i},b_{i}\right)+M(\sigma').\label{eq:app1-1-1}\\
& \le2 & \sum_{i=1}^{k-1}\varphi\left(a_{i},b_{i}\right)+M(\pi)\\
& \le & 2k\varphi_{max}+M\left(\pi\right)\end{aligned}$$ The approximation ratio in this case is upper bounded by $$\alpha\le1+\frac{2k}{n-k}\frac{\varphi_{max}}{\varphi_{min}}=1+\frac{2k/n}{1-k/n}\frac{\varphi_{max}}{\varphi_{min}}\cdot$$
\[lem:perm-approx-1\]Let $\pi$ be a randomly chosen permutation from $\mathbb{S}_{n}$. Given that the MCDs of single cycles can be computed correctly, and provided that $\varphi_{max}=o\left(n/\log n\right)$, $\alpha$ goes to one in probability as $n\rightarrow\infty$.
Let $X_{n}$ be the random variable denoting the number of cycles in a random permutation $\pi_{n}\in\mathbb{S}_{n}$. It is well known that $EX_{n}=\sum_{j=1}^{n}\frac{1}{j}=H\left(n\right)$ and that $EX_{n}\left(X_{n}-1\right)=\left(EX_{n}\right)^{2}-\sum_{j=1}^{n}\frac{1}{j^{2}}$ [@knuth2005art]. Here, $H\left(n\right)$ denotes the $n$th Harmonic number. Thus $$\begin{split}EX_{n}^{2} & =O\left(\left(\ln n\right)^{2}\right)\end{split}
,$$ which shows that $X_{n}/n\to0$ in quadratic mean as $n\to\infty$. Hence $X_{n}/n\to0$ in probability. By Slutsky’s theorem [@slutsky], $\alpha\to1$ in probability as $n\to\infty$.
In the following example, all operations are performed modulo 10, with zero replaced by 10.
Consider the permutation $\pi=\sigma_{1}\sigma_{2}$, where $\sigma_{1}=\left(1\,7\,3\,9\,5\right)$ and $\sigma_{2}=\left(2\,8\,4\,10\,6\right)$, and the cost function $\varphi$,$$\varphi\left(i,j\right)=\begin{cases}
1, & \quad d\left(i,j\right)=1\\
\infty, & \quad\text{otherwise}\end{cases}$$ where $d\left(i,j\right)=\min\left\{ \left|i-j\right|,10-\left|i-j\right|\right\} $. Note that $$\cost\left(p^{*}\left(i,j\right)\right)=d\left(i,j\right).$$ where $p^{*}$ is the shortest path from $i$ to $j$. We make the following observations regarding the decompositions of $\pi$.
1. MCD: We cannot find the MCD of $\pi$, but we can easily obtain the following bound: $$M\left(\pi\right)\ge\lceil\frac{1}{2}\sum_{i=1}^{10}\cost\left(p^{*}\left(i,\pi\left(i\right)\right)\right)\rceil=\frac{2\cdot5\cdot4}{2}=20.$$
2. MLD: As before, let the output of Alg. \[alg:opt-trans\] be denoted by $\varphi^{*}$. We have $\varphi^{*}\left(i,j\right)=2d\left(i,j\right)-1$. The minimum cost MLDs for the cycles are$$\begin{aligned}
\left(1\,7\,3\,9\,5\right) & = & \left(1\,9\right)\left(3\,7\right)\left(1\,3\right)\left(9\,5\right),\\
\left(2\,8\,4\,10\,6\right) & = & \left(2\,10\right)\left(4\,8\right)\left(2\,4\right)\left(6\,10\right),\end{aligned}$$ each of cost 20. The MLD of $\pi$ is the concatenation of the MLDs of $\sigma_{1}$ and $\sigma_{2}$: $$\pi=\left(2\,10\right)\left(4\,8\right)\left(2\,4\right)\left(6\,10\right)\left(1\,9\right)\left(3\,7\right)\left(1\,3\right)\left(9\,5\right),$$ with overall cost equal to 40.
3. STD: It can be shown that the STD of $\sigma_{1}$ is$$\begin{aligned}
\sigma_{1} & = & \left(1\ 7\right)\left(7\ 3\right)\left(3\ 9\right)\left(9\ 5\right),\\
\sigma_{2} & = & \left(2\ 8\right)\left(8\ 4\right)\left(4\ 10\right)\left(10\ 6\right),\end{aligned}$$ each with cost 28. The total cost of the STD is $S\left(\pi\right)=56$.
4. Merging cycles: Instead of finding the minimum cost MLD of each cycle separately, we may join the cycles and find the MLD of a larger cycle. Here, we find the MLD of $\sigma'=\left(1\,2\right)\pi=\left(1\,7\,3\,9\,5\,2\,8\,4\,10\,6\right)$. The cost of the minimum MLD of $\sigma'$ can be shown to be 37. Since the cost of the transposition $\left(1\,2\right)$ must also be accounted for, the total cost is 38. Observe that this cost is smaller than the MLD cost of part 2, and hence merging cycles may provide better solutions than the ones indicated by the bound or as obtained through optimization of individual cycles.
Conclusions\[sec:Conclusions\]
==============================
We introduced the problem of minimum cost transposition sorting and presented an algorithm for computing a transposition decomposition of an arbitrary permutation, with cost at most four times the minimum cost. We also described an algorithm that finds the minimum cost of each transposition in terms of a product of other transpositions, as well as an algorithm that computes the minimum cost/minimum length decomposition using dynamic programing methods.
We also showed that more accurate solutions are possible for two particular families of cost functions: for metric-path costs, we derived optimal decomposition algorithms, while for extended-metric-path costs, we described a 2-approximation method.
The algorithms presented in this paper are of polynomial complexity. Finding the minimum cost of a transposition has complexity $O(n^{4})$. Given the optimized cost transpositions, the minimum length decomposition can also be constructed in $O(n^{4})$ steps. Computing a decomposition whose cost does not exceed the minimum cost by more than a factor of four requires $O(n^{4})$ steps as well.
The authors gratefully acknowledge useful discussions with Chien-Yu Chen, Chandra Chekuri, and Alon Orlitsky. They would also like to thank Navin Kashyap for describing Example 1. This work was funded by the NSF grants NSF CCF 08-21910 and NSF CCF 08-09895.
The Bellman-Ford Algorithm
==========================
We describe an algorithm for finding $\bar{\varphi}\left(\hat{p}\left(a,b\right)\right)$. The algorithm represents a variant of the Bellman-Ford procedure, described in detail in [@cormen24introduction].
Recall that for each path $p=v_{1}\cdots v_{m+1}$ in $\mathcal{K}\left(\varphi\right)$, we define two types of costs: the standard cost of the path, $$\cost\left(p\right)=\sum_{i=1}^{m}\varphi\left(v_{i},v_{i+1}\right),\label{eq:cost}$$ and the transposition path cost, $$\bar{\varphi}\left(p\right)=2\cost\left(p\right)-\max_{i}\varphi\left(v_{i},v_{i+1}\right).\label{eq:barvarphi}$$ The goal is to find the path that minimizes the transposition path cost in .
Before describing our algorithm, we briefly review the standard Single-Source Bellman-Ford shortest path algorithm, and its relaxation techniques.
Given a fixed source $s$, for each vertex $v\neq s$, the algorithm maintains an upper bound on the distance between $s$ and $v$, denoted by $D\left(v\right)$. Initially, for each vertex $v$, we have $D\left(v\right)=\varphi(s,v)$.
“Relaxing” an edge $\left(uv\right)$ means testing that the upper-bounds $D\left(u\right)$ and $D\left(v\right)$ satisfy the conditions,$$\begin{aligned}
D\left(u\right) & \le D\left(v\right)+w,\nonumber \\
D\left(v\right) & \le D\left(u\right)+w,\label{eq:relaxation}\end{aligned}$$ where $w$ denotes the cost of the edge $(uv)$. If the above conditions are not satisfied, then one of the two upper-bounds can be improved, since one can reach $u$ by passing through $v$, and vice versa.
In our algorithm, we maintain the upper-bound for two types of costs. The source $s$ is an arbitrary vertex in $\mathcal{K}(\varphi)$. For a path between $s$ and a vertex $v$, we use $D_{1}\left(v\right)$ to denote the bound on the minimum transposition path cost, and we use $D_{2}\left(v\right)$ to denote the bound on twice the minimum cost of the path. From the definitions of these costs, it is clear that$$\begin{aligned}
D_{2}\left(u\right) & \le2w+D_{2}\left(v\right),\label{eq:confusing}\\
D_{2}\left(v\right) & \le2w+D_{2}\left(u\right),\nonumber \\
D_{1}\left(u\right) & \le\min\left\{ w+D_{2}\left(v\right),2w+D_{1}\left(v\right)\right\} ,\nonumber \\
D_{1}\left(v\right) & \le\min\left\{ w+D_{2}\left(u\right),2w+D_{1}\left(u\right)\right\} .\nonumber \end{aligned}$$ The relaxation algorithm for these inequalities, Alg. \[alg:relax\], is straightforward to implement. To describe the properties of the output of the Bellman-Ford algorithm, we briefly comment on a simple property of the algorithm, termed the path-relaxation property.
Suppose $p=v_{1}\cdots v_{m+1}$ is the shortest path (in terms of or ) from $s=v_{1}$ to $u=v_{m+1}$. After relaxing the edges $\left(v_{1}v_{2}\right),\left(v_{2}v_{3}\right),\cdots\left(v_{m}v_{m+1}\right)$, in that given order, the upper-bound $D_{i}\left(u\right)$ (for $i=1,2$) equals the optimal cost of the corresponding path. Note that the property still holds even if the relaxations of the edges $\left(v_{1}v_{2}\right),\left(v_{2}v_{3}\right),\cdots,\left(v_{m}v_{m+1}\right)$ are interleaved by relaxations of some other edges. In other words, it suffices to identify only a subsequence of relaxations of the edges $\left(v_{1}v_{2}\right),\left(v_{2}v_{3}\right),\cdots\left(v_{m}v_{m+1}\right)$.
In the algorithm below, we use $pred_{i}(v)$ to denote the predecessor of node $v$ used for tracking the updates of the cost $D_{i}(v)$, $i=1,2$, and $(u,i),i=1,2$, to indicate from which of the two costs, minimized over in , $u$ originated. Note that this notion of predecessor is not to be confused with the predecessor element in a two-line permutation representation.
$w\leftarrow \varphi(u,v)$ $D_2(v)\leftarrow D_2(u)+2w$ $pred_2(v)\leftarrow (u,2)$ $D_2(u)\leftarrow D_2(v)+2w$ $pred_2(u)\leftarrow (v,2)$ $D_1(v)\leftarrow D_2(u)+w$ $pred_1(v)\leftarrow (u,2)$ $D_1(u)\leftarrow D_2(v)+w$ $pred(u,1)\leftarrow (v,2)$ $D_1(v)\leftarrow D_1(u)+2w$ $pred_1(v)\leftarrow (u,1)$ $D_1(u)\leftarrow D_1(v)+2w$ $pred_1(u)\leftarrow (v,1)$
The Bellman-Ford algorithm performs $n-1$ rounds of relaxation on the edges of the graph $\mathcal{K}(\varphi)$. Lemma \[lem:BF\] proves the correctness of the algorithm.
Input: vertex $s$ Output: $\hat{p}(s,u)$ for $1\le u\le n$ $D_1(u)\leftarrow \varphi(s,u)$ $pred_1(u)\leftarrow s$ $D_2(u)\leftarrow 2\varphi(s,u)$ $pred_2(u)\leftarrow s$ $(u,v)$ initialize path at $u$ backtrack min cost alg to recover path to $s$ output $\hat{p}(s,u)$
An example of the steps of Alg. \[alg:SSBF\] is given in Figure \[fig:BFEx \]. Initially, only edges between $a$, $b$, and $c$ and $a$ have finite costs, as a result of steps 4-8 of the algorithm. Next, $n-1$ passes are executed and in each of them all edges of the graph are relaxed. Edge $\left(bd\right)$ is relaxed first, as seen in Figure \[fig:bd\]. Next, the relaxation of edge $\left(cd\right)$ reduces the cost of $D_{1}\left(d\right)$ from 12 to 10. Continuing with the algorithm, we obtain the final result in Figure \[fig:fin\]. Note that in this example, the result obtained after the first pass is the final result. In general, however, the final costs may be obtained only after all $n-1$ passes are performed.
\[lem:BF\]Given $n$, a cost function $\varphi$, and a source $s$, after the execution of Alg. , one has $D_{1}\left(u\right)=\varphi^{*}\left(s,u\right)$ and $D_{2}\left(u\right)=2\cost\left(p^{*}\left(s,u\right)\right)$.
Let $\hat{p}\left(s,u\right)=v_{1}v_{2}\cdots v_{m+1}$ be the path that minimizes $\bar{\varphi}\left(p\right)$ among all paths $p$ between $v_{1}=s$ and $v_{m+1}=u$. Since any path $p$ has at most $n$ vertices, we have $m\le n-1$. The algorithm makes $n-1$ passes and in each pass relaxes all edges of the graph. Thus, there exist a subsequence of relaxations that relax $\left(v_{1}v_{2}\right),\left(v_{2}v_{3}\right),\cdots,\left(v_{m}v_{m+1}\right)$, in that order. The proof for the claim regarding $D_{1}\left(u\right)$ follows by invoking the path-relaxation property and the fact that $\varphi^{*}\left(s,u\right)=\bar{\varphi}\left(\hat{p}\left(s,u\right)\right)$. The proof for the claim regarding $D_{2}\left(u\right)$ is similar.
[^1]: Usually, the channel is initialized by a molecule that may appear in the queue as well.
|
---
abstract: 'We propose to measure the weak cosmic shear using the spatial derivatives of the galaxy surface brightness field. The measurement should be carried out in Fourier space, in which the point spread function (PSF) can be transformed to a desired form with multiplications, and the spatial derivatives can be easily measured. This method is mathematically well defined regardless of the galaxy morphology and the form of the PSF, and involves simple procedures of image processing. Furthermore, with high resolution galaxy images, this approach allows one to probe the shape distortions of galaxy substructures, which can potentially provide much more independent shear measurements than the ellipticities of the whole galaxy. We demonstrate the efficiency of this method using computer-generated mock galaxy images.'
author:
- |
Jun Zhang[^1]\
\
Department of Astronomy, University of California, Berkeley, CA 94720, USA\
title: Measuring the Cosmic Shear in Fourier Space
---
\[firstpage\]
cosmology: theory - weak lensing
Introduction {#intro}
============
The coherent distortions of background galaxy images by the intervening metric perturbations provide us a direct probe of the large scale mass distribution (see reviews by @bs01 [@wittman02; @refregier03]). Recently, several groups have claimed positive detections of the weak lensing effect and obtained useful constraints on the cosmological model (@bre00 [@kwl00; @vw00; @wittman00; @maoli01; @rhodes01; @vw01; @hyg02; @rrg02; @bmre03; @brown03; @hamana03; @jarvis03; @rhodes04; @heymans05; @mbre05; @vmh05; @dahle06; @hetterscheidt06; @hoekstra06; @jjbd06; @schrabback06; @semboloni06]). In future weak lensing observations (VST-KIDS, DES[^2], VISTA darkCAM, Pan-STARRS[^3], LSST[^4], DUNE, SNAP[^5], JDEM[^6]), if the photometric redshift can be well calibrated, we will be able to study the dark energy properties (its abundance and equation of state) using the redshift dependence of the shear fields (@hu02 [@abazajian03; @jain03; @bernstein04; @hu04; @song04; @takada04; @takadawhite04; @ishak05; @simpson05; @zhang05; @hannestad06; @schimd06; @taylor06; @zhan06]). By constraining the growth factor of the mass perturbation and the geometrical distance as functions of redshift separately, weak lensing provides a consistency check of the cosmological model (@kratochvil04 [@simpson05; @zhang05; @knox06]), and opens a window for testing alternative gravity theories (@acquaviva04 [@song05; @ishak06]).
An important and challenging job in weak lensing is to measure the weak cosmic shear (of order a few percent) from the shapes (or ellipticities) of the background galaxy images, which have large intrinsic variations. The existing methods are all based on convoluting the galaxy images with some weighting functions, and are called the INTEGRAL methods hereafter (see @tyson90 [@bonnet95; @kaiser95; @luppino97; @hoekstra98; @rhodes00; @kaiser00; @bridle01; @bernstein02; @refregierbacon03; @massey05; @kuijken06; @nakajima06]). The INTEGRAL methods typically have disadvantages in three aspects: [**1.**]{} since the galaxy images are smeared by the PSF (either instrumental or environmental), the INTEGRAL methods involve at least two folds of convolutions, the math of which is complicated; [**2.**]{} the details of the methods are often sensitive to the galaxy morphology and the form of the PSF; [**3.**]{} the shear information from the shape distortions of galaxy substructures is not considered. Strictly speaking, the shapelets method (see, ,Refregier 2003) may not be called an INTEGRAL method, because the galaxy weighting functions form a complete set of orthonormal shapelets which have very convenient mathematical properties. It also has the potential of measuring the cosmic shears on galaxy substructures. However, since this method requires calibrations of the intrinsic distributions of the shapelet coefficients, it has strong dependence on the galaxy morphology.
In this paper, we propose to use the spatial derivatives of the galaxy surface brightness field to measure the cosmic shear. This method was first used by Seljak and Zaldarriaga (1999) on CMB lensing. We generalize their analysis by including the PSF and carrying out the measurement in Fourier space. This approach is well defined regardless of the galaxy morphology and the form of the PSF, and involves simple image processing procedures. Given a high image resolution, the method can potentially probe the cosmic shear from galaxy substructures, greatly suppressing the shape noise.
We begin by introducing the method in §\[method\]. In §\[test\], this approach is shown to work well on different types of computer-generated mock galaxy images with general forms of PSF. A brief summary is given in §\[summary\].
The Method {#method}
==========
We derive the relation between the cosmic shear and the spatial derivatives of the galaxy surface brightness field without a PSF in §\[withoutPSF\]. In the presence of an isotropic Gaussian PSF, the relation is modified and shown in §\[withisopsf\]. In §\[Fourier\], Fourier transformation is introduced not only to simplify the measurement of the spatial derivatives, but also to deal with general forms of PSF.
Without the PSF {#withoutPSF}
---------------
The surface brightness on the image plane $f_I(\vti)$ and on the source plane $f_S(\vts)$ ($\vti$ and $\vts$ are the position angles on the image and source plane respectively) are related through a simple relation: $$\begin{aligned}
\label{fifstits}
&&f_I(\vti)=f_S(\vts)\\ \nonumber
&&\vti=\mathbf{A}\vts\end{aligned}$$ where $\mathbf{A}_{ij}=\delta_{ij}+\Phi_{ij}$, and $\Phi_{ij}=\partial\delta\theta^I_i/\partial\theta^S_j$ are the spatial derivatives of the lensing deflection angle, which can be expressed in terms of the convergence $\kappa=(\Phi_{xx}+\Phi_{yy})/2$ and the two shear components $\gamma_1=(\Phi_{xx}-\Phi_{yy})/2$ and $\gamma_2=\Phi_{xy}$. Using eq.\[\[fifstits\]\], we get: $$\begin{aligned}
\label{dfidfs}
\frac{\partial f_I}{\partial \theta^I_i}&=&\frac{\partial\theta^S_j}{\partial\theta^I_i}\frac{\partial f_S}{\partial\theta^S_j}\\ \nonumber
&=&(\delta_{ij}-\Phi_{ij})\frac{\partial f_S}{\partial\theta^S_j}\end{aligned}$$ where we have implicitly assumed that $\Phi_{ij}$ is small, which is true for weak lensing. Assuming the original surface brightness field $f_S$ is isotropic on the source plane, the quadratic combinations of the derivatives of the lensed image provide a direct measure of the shear components (@seljak99): $$\begin{aligned}
\label{shear12}
&&\frac{1}{2}\frac{\langle (\partial_xf_I)^2-(\partial_yf_I)^2\rangle}{\langle (\partial_xf_I)^2+(\partial_yf_I)^2\rangle}=-\gamma_1 \\ \nonumber
&&\frac{\langle\partial_xf_I\partial_yf_I\rangle}{\langle (\partial_xf_I)^2+(\partial_yf_I)^2\rangle}=-\gamma_2\end{aligned}$$ where the averages are taken over the whole galaxy.
With an Isotropic Gaussian PSF {#withisopsf}
------------------------------
The presence of PSF brings both advantages and disadvantages. On the positive side, the PSF smooths out the galaxy surface brightness field, which is originally not differentiable due to structures on arbitrarily small scales. On the other hand, the convolution of the galaxy image with the PSF leads to a nontrivial modification to eq.\[\[shear12\]\], the form of which is calculated in this section. For simplicity, we assume the PSF is isotropic and Gaussian. General forms of PSF will be discussed in §\[Fourier\].
The observed galaxy surface brightness distribution $f_O$ is related to $f_I$ via: $$\label{fofi}
f_O(\vt)=\int d^2\vti W_{\beta}(\vt -\vti)f_I(\vti)$$ where $W_{\beta}$ is the Gaussian PSF with scale length $\beta$: $$\label{wbeta}
W_{\beta}(\vt)=\frac{1}{2\pi\beta^2}\exp\left(-\frac{\vert\vt\vert^2}{2\beta^2}\right)$$ Using eq.\[\[fifstits\]\] to replace $f_I$ with $f_S$ and $\vti$ with $\vts$ in eq.\[\[fofi\]\], we get: $$\label{fofs}
f_O(\vt)=\vert{\rm det}(\mathbf{A})\vert\int d^2\vts W_{\beta}(\vt -\mathbf{A}\vts)f_S(\vts)$$ or equivalently: $$\begin{aligned}
\label{fofs2}
f_O(\mathbf{A}\vt)&=&\vert{\rm det}(\mathbf{A})\vert\int d^2\vts W_{\beta}[\mathbf{A}(\vt-\vts)]f_S(\vts)\\ \nonumber
&\doteq&\vert{\rm det}(\mathbf{A})\vert\int d^2\vts f_S(\vts)W_{\beta}(\vt-\vts)\\ \nonumber
&\times&\left[1-(\vt-\vts)\cdot(\mathbf{A}-\mathbf{I})\cdot(\vt-\vts)/\beta^2\right]\end{aligned}$$ where $\mathbf{I}$ is the $2\times 2$ unitary matrix. Note that the second part of eq.\[\[fofs2\]\] is a result of Taylor expansion of the term $W_{\beta}[\mathbf{A}(\vt-\vts)]$ due to the small amplitudes of the lensing components $\Phi_{ij}$. For convenience, let us define: $$\label{F_S}
F_S(\vt)=\int d^2\vts f_S(\vts)W_{\beta}(\vt-\vts)$$ which is the surface brightness field we would observe in absence of lensing. Eq.\[\[fofs2\]\] can then be re-written as: $$\begin{aligned}
\label{fofs3}
\frac{f_O(\mathbf{A}\vt)}{\vert{\rm det}(\mathbf{A})\vert}&=&(1-\Phi_{xx}-\Phi_{yy})F_S(\vt)\\ \nonumber
&-&\beta^2\left(\Phi_{xx}\frac{\partial^2F_S}{\partial\theta_x^2}+2\Phi_{xy}\frac{\partial^2F_S}{\partial\theta_x\partial\theta_y}+\Phi_{yy}\frac{\partial^2F_S}{\partial\theta_y^2}\right)\end{aligned}$$ Let $\vto=\mathbf{A}\vt$, then: $$\label{fvto}
\frac{\partial f_O(\vto)}{\partial\theta^O_i}=(\mathbf{A}^{-1})_{ij}\frac{\partial f_O(\mathbf{A}\vt)}{\partial\theta_j}$$ Using eq.\[\[fofs3\]\] and eq.\[\[fvto\]\], it is not hard to express the derivatives of $f_O$ in terms of the derivatives of $F_S$: $$\begin{aligned}
\label{dfodFs}
\frac{\partial_xf_O}{\vert{\rm det}(\mathbf{A})\vert}&=&(1-2\Phi_{xx}-\Phi_{yy})F_x-\Phi_{xy}F_y\\ \nonumber
&-&\beta^2(\Phi_{xx}F_{xxx}+2\Phi_{xy}F_{xxy}+\Phi_{yy}F_{xyy})\\ \nonumber
\frac{\partial_yf_O}{\vert{\rm det}(\mathbf{A})\vert}&=&(1-\Phi_{xx}-2\Phi_{yy})F_y-\Phi_{xy}F_x\\ \nonumber
&-&\beta^2(\Phi_{yy}F_{yyy}+2\Phi_{xy}F_{xyy}+\Phi_{xx}F_{xxy})\end{aligned}$$ where $$\label{defineDF}
F_{i_1\cdot\cdot\cdot i_n}=\frac{\partial^nF_S}{\partial\theta_{i_1}\cdot\cdot\cdot\partial\theta_{i_n}}$$ Note that we have implicitly assumed that the spatial fluctuation of the cosmic shear is negligible on galactic scales. Assuming the distribution of $F_S$ is isotropic, we obtain the following relation between the shear components and the spatial derivatives of the surface brightness field: $$\begin{aligned}
\label{shear12PSF}
&&\frac{1}{2}\frac{\langle (\partial_xf_O)^2-(\partial_yf_O)^2\rangle}{\langle (\partial_xf_O)^2+(\partial_yf_O)^2\rangle+\Delta}=-\gamma_1 \\ \nonumber
&&\frac{\langle\partial_xf_O\partial_yf_O\rangle}{\langle (\partial_xf_O)^2+(\partial_yf_O)^2\rangle+\Delta}=-\gamma_2\end{aligned}$$ where $$\label{Delta}
\Delta=\frac{\beta^2}{2}\langle\vec{\nabla}f_O\cdot\vec{\nabla}(\nabla^2f_O)\rangle$$
The derivation of eq.\[\[shear12PSF\]\] and eq.\[\[Delta\]\] is shown in the appendix. Note that in the limit when the galaxy image is very smooth over the scale length $\beta$, the correction $\Delta$ approaches zero, eq.\[\[shear12PSF\]\] then reduces to eq.\[\[shear12\]\].
Fourier Transform and General PSF {#Fourier}
---------------------------------
For the method to become useful, there are at least two remaining issues to be addressed: [**1.**]{} how to measure the spatial derivatives of the surface brightness field; [**2.**]{} how to deal with other forms of the PSF. It turns out that Fourier transformation provides a solution to both problems.
Since convolutions in real space correspond to multiplications in Fourier space, one can easily transform the PSF to a desired form (an isotropic Gaussian form in our case) by multiplying the Fourier modes of the observed image with the ratios between the Fourier modes of the desired PSF and those of the original PSF (known from calibrations with stars). This operation is usually well defined if the scale length of the desired PSF is larger than that of the original PSF. Moreover, it turns out that for the purpose of measuring the cosmic shear, one does not need to transform the new image back to real space, because the derivatives of the surface brightness field can be more easily measured in Fourier space.
As an example, we show how to measure quantities such as $\langle\vert\vec{\nabla}f\vert^2\rangle$, where $f$ is the surface brightness field of interest. First of all, the distribution $f$ in real space should be sampled with an interval $\Delta\theta$ which is a few times less than the size of the PSF to avoid translating high frequency power into the frequency range determined by the sampling resolution through discrete Fourier transform (@press92). In other words, the galaxy image should be “oversampled” to avoid aliasing power from small scales in the discrete Fourier transform. For undersampled images, one can smooth the images with an additional large enough PSF, which can be treated as a part of the PSF from the instrumentation, and therefore does not affect our discussion below. Similarly, to avoid such aliasing power at low frequency, the box size for the Fourier transform should be a few times larger than the image size. Given this setup, the Fourier transform of the image is defined as: $$\label{fourierf}
\tilde{f}(l_i,l_j)=\Delta\theta^2\sum_{m=0}^{N-1}\sum_{n=0}^{N-1}f(\theta_m,\theta_n)\exp\left[i(\theta_ml_i+\theta_nl_j)\right]$$ where $$\begin{aligned}
\label{thetal}
&&\theta_{m(n)}=m(n)\times\Delta\theta, \mbox{ } m(n)=0,1,...,N-1\\ \nonumber
&&l_{i(j)}=i(j)\times\Delta l, \mbox{ } i(j)=-N/2,..,N/2 \\ \nonumber
&&\Delta l=2\pi/(N\Delta\theta)\end{aligned}$$ N is the box size, chosen to be a power of $2$ for the Fast Fourier Transform. It is now straightforward to show that $\langle\vert\vec{\nabla}f\vert^2\rangle$ can be expressed as the sum over the Fourier modes weighted by the wave numbers: $$\begin{aligned}
\label{fourierfinverse}
&&\sum_{m=0}^{N-1}\sum_{n=0}^{N-1}\vert\vec{\nabla}f(\theta_m,\theta_n)\vert^2\\ \nonumber
&=&\frac{1}{N^2\Delta\theta^4}\sum_{i=-N/2}^{N/2}\sum_{j=-N/2}^{N/2}\vert\tilde{f}(l_i,l_j)\vert^2(l_i^2+l_j^2)\end{aligned}$$ Eq.\[\[fourierfinverse\]\] gives exactly the quantity $\langle\vert\vec{\nabla}f\vert^2\rangle$ multiplied by the number of bright pixels covered by the galaxy, because the dark pixels have no contributions[^7]. Similarly, we can calculate the other terms in eq.\[\[shear12PSF\]\] in Fourier space. Note that for the purpose of obtaining $\gamma_1$ and $\gamma_2$, it is not necessary to calculate the number of bright pixels because it appears in both the nominator and the denominator in eq.\[\[shear12PSF\]\].
The Test {#test}
========
This section is organized as follows: in §\[diskgalaxy\], we test the method using mock regular galaxies smeared by different forms of PSF; in §\[irregulargalaxy\], using mock irregular galaxies generated by 2-D random walks, we further demonstrate the usefulness of this approach on galaxies with a different morphology, and explore the possibility of suppressing the shape noise in the shear measurements by including the information from galaxy substructures.
With Mock Regular Galaxies {#diskgalaxy}
--------------------------
Each regular galaxy in our simulation contains a thin circular disk with an exponential profile and a co-axial de Vaucouleurs-type spheroidal component (@vaucouleurs91). When viewed face-on, the surface brightness distribution (before lensing and smearing by the PSF) of the galaxy can be parameterized as: $$\label{diskprofile}
f(r)=\exp(-r/r_d)+f_{s/d}\exp\left[-(r/r_s)^{\frac{1}{4}}\right]$$ where $r$ is the distance to the galaxy center, $r_d$($r_s$) is the scale length of the disk(spheroid), and $f_{s/d}$ determines the relative importance of the spheroid. The overall luminosity of the galaxy is only important in the presence of noise, which will be discussed in a future paper.
Our simulation box is $128\times128$. We choose $r_d$ to be $1/32$ of the box size of the simulation, $r_s=r_d/2$, and $f_{s/d}=1$. Note that changing these particular numbers does not affect our main conclusions. Once the galaxy’s face-on image is generated, it is projected onto the source plane with a random inclination angle along a random direction perpendicular to the line of sight[^8].
The projected galaxy image is subsequently distorted by a constant cosmic shear and smeared by the PSF in real space. We consider two PSF models given by the following forms rotated by certain angles (shown in Fig.\[PSFs\]): $$\begin{aligned}
\label{PSFforms}
&&W_r^{(1)}(x,y)\propto\exp\left[-(\vert x-y\vert +\vert x+y\vert)^2/(8r^2)\right]\\ \nonumber
&&W_r^{(2)}(x,y)\propto\exp\left[-(x^2+0.8y^2)/(2r^2)\right]\\ \nonumber\end{aligned}$$ where $r$ is the scale length, which is equal to six times the grid size, comparable to the galaxy size. The shear components $(\gamma_1,\gamma_2)$ are chosen to be $(-0.012,0.035)$, $(-0.032,-0.005)$, $(0.01,0.02)$ for $W_r^{(1)}$, and $(0.015,-0.024)$, $(0.05,0.01)$, $(-0.04,-0.04)$ for $W_r^{(2)}$.
To measure the cosmic shear, we follow the procedures described in §\[Fourier\]. The desired PSF has an isotropic Gaussian form with a scale length about $4/3$ times that of the original PSF. The results are plotted in Fig.\[shear\_disk\]. The results are consistent within $1\sigma$ error regardless of the form of the PSF.
With Mock Irregular Galaxies {#irregulargalaxy}
----------------------------
Our irregular galaxies are generated using 2-D random walks. The random walk starts from the center of the simulation box for 20000 steps, each of which is equal to the grid size of the simulation box (which is now $1024\times1024$). Once the distance from the center is more than $1/6$ of the box size, the walk starts from the center again to finish the rest of the steps. The surface brightness of the galaxy is equal to the density of the trajectories. Note that these galaxies naturally have abundant substructures, which are useful not only for further testing the method, but also for illustrating how much lensing information may be contained in the substructures. We caution that our random-walk-type galaxies are not based on any physical models, therefore they do not necessarily mimic observed irregular galaxies. In a future paper, more realistic galaxy models will be adopted to study this topic.
For the purpose of this section, we smooth the galaxies directly with the isotropic Gaussian PSF of different scale lengths, which correspond to different angular resolutions. The scale length $\beta$ (defined in eq.\[\[wbeta\]\]) is chosen to be $1/256$, $1/128$, $1/64$, and $1/32$ of the box size (roughly corresponding to $1/85$, $1/43$, $1/21$, and $1/10$ of the galaxy size). Fig.\[rwgalaxy\] shows typical images of our irregular galaxy under these four different angular resolutions. For convenience, we plot the minimum $\beta$ as unity in the figures of this section.
The shear component $\gamma_2$ is set to zero, and $\gamma_1$ is fixed at $0.03$. After averaging over $10000$ irregular galaxies, we find that the measured $\gamma_1$ is $0.0324\pm0.0029$ for $\beta=8$, $0.0301\pm0.0022$ for $\beta=4$, $0.0291\pm0.0015$ for $\beta=2$, and $0.0293\pm0.0010$ for $\beta=1$. More interestingly, as shown in Fig.\[results\], the statistical error bar is found to decrease significantly when the angular resolution is increased. This is further illustrated in Fig.\[sigma\], which shows an approximate power-law relation between the measured variance of $\gamma_1$ and $\beta$, the exponent of which is close to one. Note that as the angular resolution increases, one gets additional information on the cosmic shears from the galaxy substructures. If we naively assume that each bright pixel on the galaxy map provides an independent measurement of the cosmic shear, we expect the variance of the measured cosmic shear to scale as the inverse of the number of the bright pixels, or $\beta^d$, where $d$ is the Hausdorff dimension of the galaxy image (@hausdorff1919). Since the Hausdorff dimension of our random-walk-generated irregular galaxies is $2$ (@falconer86), the variance of $\gamma_1$ should scale as $\beta^{2}$, which is not too far from what we have observed in our numerical experiments. In reality, substructures generated by the 2-D random walks are correlated at some unknown level, therefore, the observed exponent indicated in Fig.\[sigma\] is less than the Hausdorff dimension.
Summary
=======
We have presented a simple approach of measuring the weak cosmic shear using the spatial derivatives of the galaxy surface brightness field. The measurement should be carried out in Fourier space, in which it is easy to evaluate the spatial derivatives and to transform the PSF to a desired form. The accuracy of the method is demonstrated using computer-generated mock regular and irregular galaxies. We find no systematic errors on the measured shear components in the numerical experiments.
Given high image resolutions, this new method may reduce the shape noise in the shear measurement significantly, because it takes into account the shape information on the galaxy substructures. Using the mock irregular galaxies generated by 2-D random walks, we have shown that the variance of the measured shear is indeed suppressed by a large factor when the image resolution is increased. This example encourages us to test this method on real galaxies of a wide range of morphology classes in a future paper by joining the Shear TEsting programme (@heymans06 [@massey06]), the results of which may be useful for optimizing the signal to noise ratio in shear measurements and planning future weak lensing survey.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Chung-Pei Ma, Tony Tyson, Martin White, David Wittman for useful discussions, Gary Bernstein, Lam Hui, Bhuvnesh Jain, Nick Kaiser, and the anonymous referee for comments on an earlier version of this manuscript. Research for this work is supported by NASA, and by the TAC Fellowship of UC Berkeley.
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Appendix – Relating the Cosmic Shears with the Spatial Derivatives of the Surface Brightness field {#appendix .unnumbered}
==================================================================================================
From eq.\[\[dfodFs\]\], we have: $$\begin{aligned}
\label{appendix1}
&&\frac{1}{\vert{\rm det}(\mathbf{A})\vert^2}\left[(\partial_xf_O)^2-(\partial_yf_O)^2\right]\\ \nonumber
&=&(1-6\kappa)(F_x^2-F_y^2)-2\gamma_1(F_x^2+F_y^2)\\ \nonumber
&-&\beta^2\left[2\kappa\Pi_1+\gamma_1(\Lambda+\Upsilon_1)+\gamma_2(\Upsilon_2-\tilde{\Lambda})\right]\end{aligned}$$ and $$\begin{aligned}
\label{appendix2}
&&\frac{2}{\vert{\rm det}(\mathbf{A})\vert^2}\partial_xf_O\partial_yf_O\\ \nonumber
&=&2(1-6\kappa)F_xF_y - 2\gamma_2(F_x^2+F_y^2)\\ \nonumber
&-&\beta^2\left[2\kappa\Pi_2+\gamma_1(\tilde{\Lambda}+\Upsilon_2)+\gamma_2(\Lambda-\Upsilon_1)\right]\end{aligned}$$ where $$\begin{aligned}
\label{appendix3}
&&\Lambda=F_xF_{xxx}+F_xF_{xyy}+F_yF_{xxy}+F_yF_{yyy}\\ \nonumber
&&\tilde{\Lambda}=F_yF_{xxx}+F_yF_{xyy}-F_xF_{xxy}-F_xF_{yyy}\\ \nonumber
&&\Pi_1=F_xF_{xxx}+F_xF_{xyy}-F_yF_{xxy}-F_yF_{yyy}\\ \nonumber
&&\Pi_2=F_xF_{xxy}+F_xF_{yyy}+F_yF_{xyy}+F_yF_{xxx}\\ \nonumber
&&\Upsilon_1=F_xF_{xxx}-3F_xF_{xyy}-3F_yF_{xxy}+F_yF_{yyy}\\ \nonumber
&&\Upsilon_2=F_yF_{xxx}-3F_yF_{xyy}+3F_xF_{xxy}-F_xF_{yyy}\end{aligned}$$
Note that according to the definitions in eq.\[\[appendix3\]\], $\Lambda$ is a scalar, $\tilde{\Lambda}$ is a pseudo scalar, $\Pi_1+i\Pi_2$ is a spin-2 field, and $\Upsilon_1+i\Upsilon_2$ is a spin-4 field. If the intrinsic surface brightness distribution is isotropic, the spatial averages of $\tilde{\Lambda}$, $\Pi_1$, $\Pi_2$, $\Upsilon_1$, and $\Upsilon_2$ must vanish. As a result of this, we have: $$\begin{aligned}
\label{appendix4}
&&\frac{1}{2}\langle(\partial_xf_O)^2-(\partial_yf_O)^2\rangle\\ \nonumber
&=&-\gamma_1\left(\langle F_x^2+F_y^2\rangle+\frac{\beta^2}{2}\langle\Lambda\rangle\right)\\ \nonumber
&&\langle\partial_xf_O\partial_yf_O\rangle\\ \nonumber
&=&-\gamma_2\left(\langle F_x^2+F_y^2\rangle+\frac{\beta^2}{2}\langle\Lambda\rangle\right)\end{aligned}$$ We have neglected the factor $\vert{\rm det}(\mathbf{A})\vert$ which is equal to unity to the 0th order. Using the fact that $\Lambda=\vec{\nabla}F_S\cdot\vec{\nabla}(\nabla^2F_S)$, and $F_S=f_O$ to the 0th order, it is now straightforward to prove eq.\[\[shear12PSF\]\].
\[lastpage\]
[^1]: E-mail:[email protected]
[^2]: see www.darkenergysurvey.org
[^3]: see pan-starrs.ifa.hawaii.edu
[^4]: see www.lsst.org
[^5]: see snap.lbl.gov
[^6]: see destiny.asu.edu
[^7]: In the presence of noise, extra procedures may be required to clean the galaxy map before the Fourier transform. We shall discuss this in a future paper.
[^8]: The intrinsic flattening parameter $q$ of the spheroid part is set to one for simplicity.
|
---
abstract: 'We investigate the difficulties of training sparse neural networks and make new observations about optimization dynamics and the energy landscape within the sparse regime. Recent work of @Gale2019 [@Liu2018] has shown that sparse ResNet-50 architectures trained on ImageNet-2012 dataset converge to solutions that are significantly worse than those found by pruning. We show that, despite the failure of optimizers, there is a linear path with a monotonically decreasing objective from the initialization to the “good” solution. Additionally, our attempts to find a decreasing objective path from “bad” solutions to the “good” ones in the sparse subspace fail. However, if we allow the path to traverse the dense subspace, then we consistently find a path between two solutions. These findings suggest that traversing extra dimensions may be needed to escape stationary points found in the sparse subspace.'
bibliography:
- 'index.bib'
---
Introduction {#intro}
============
Reducing parameter footprint and inference latency of machine learning models is an active area of research, fostered by diverse applications like mobile vision and on-device intelligence. Sparse networks, that is, neural networks in which a large subset of the model parameters are zero, have emerged as one of the leading approaches for reducing model parameter count. It has been shown empirically that deep neural networks can achieve state-of-the-art results under high levels of sparsity [@han2015learning; @louizos2017bayesian; @Gale2019] , and this property has been leveraged to significantly reduce the parameter footprint and inference complexity [@kalchbrenner2018] of densely connected neural networks. However, pruning-based sparse solutions require to train densely connected networks and uses the same, or even greater, computational resources compared to fully dense training, which imposes an upper limit on the size of sparse networks we can train.
![[**Test accuracy of ResNet-50 networks trained on ImageNet-2012 dataset at different sparsity levels**]{}. We observe a large gap in generalization accuracy between approaches based on pruning and other approaches. See text for details. \[fig:val-acc\]](img/training_eval_accuracy){width="0.9\columnwidth"}
-0.2in
[**Training Sparse Networks.**]{} In the context of sparse networks, state-of-the-art results have been obtained through training densely connected networks and modifying their topology during training through a technique known as pruning [@gupta2018; @baiduexploringsparsity; @han2015learning]. A different approach is to *reuse* the sparsity pattern found through pruning and train a sparse network from scratch. This can be done with a random initialization ([**“scratch”**]{}) or the same initialization as the original training ([**“lottery”**]{}). Previous work [@Gale2019; @Liu2018] demonstrated that both approaches achieve similar final accuracies, but lower than pruning[^1]. The difference between pruning and both approaches to training while sparse can be seen in Figure \[fig:val-acc\]. Despite being in the same energy landscape, “scratch” and “lottery” solutions fail to match the performance of the solutions found by pruning. Given the utility of being able to train sparse from scratch, it is critical to understand the reasons behind the failure of current techniques at training sparse neural networks. There exists a line of work on training sparse networks [@Bellec2017; @Mocanu2018; @Pieterse2019; @Mostafa2019] which allows the connectivity pattern to change over time. These techniques generally achieve higher accuracy compared with fixed sparse connectivity, but generally worse than pruning. In this work we focus on *fixed* sparsity patterns, although our results (Section \[section32\]) also give insight into the success of these other approaches.
Motivated by the disparity in accuracy observed in Figure \[fig:val-acc\], we perform a series of experiments to improve our understanding of the difficulties present in training sparse neural networks, and identify possible directions for future work.
More precisely, our [**main contributions**]{} are:
- A set of experiments showing that the objective function is monotonically decreasing along the straight lines that interpolate from:
- the original dense initialization,
- the original dense initialization projected into the sparse subspace ,
- a random initialization in the sparse subspace,
to the solution obtained by pruning[^2]. This demonstrates that even when the optimization process fails, there was a monotonically decreasing path to the “good” solution.
- In contrast, the linear path between the *scratch* and the *pruned* solutions depicts a high energy barrier between the two solutions. Our attempts to find quadratic and cubic Bézier curves [@Garipov2018] with a decreasing objective between the two sparse solutions fails suggesting that the optimization process gets attracted into a “bad” local minima.
- Finally, by removing the sparsity constraint from the path, we are consistently able to find decreasing objective Bézier curves between the two sparse solutions. This result suggests that allowing for dense connectivity might be necessary and sufficient to escape the stationary point converged in the sparse subspace.
The rest of the paper is organized as follows: In §\[methods\], we describe the experimental setup. In §\[results\] we present the results from these experiments, followed by a discussion in §\[scs:conclusion\].
Methods
=======
**Training methods.** The different training strategies considered in this paper are summarized in Figure \[fig:diagram\]. In *dense training*, we train the densely connected model and apply model pruning [@gupta2018] to find the **Pruned Solution (P-S)**. The other strategies start instead with a sparse connectivity pattern (represented as a binary mask) obtained from the pruned solution. The solution obtained from the same random initialization as the pruned solution [@frankle2018] is denoted **Lottery Solution (L-S)**, while the solution obtained from another random initialization [@Liu2018] is named **Scratch Solution (S-S)**. All of our experiments are based on the Resnet-50 architecture [@resnet] and the Imagenet-2012 dataset [@imagenet]. Abbreviations defined in Figure \[fig:diagram\] below the boxes are re-used to in the remaining of the text to indicate start and end points of the interpolation experiments.
![[**Experimental setup**]{}. In this paper we consider three different methods for obtaining sparse solutions: pruned, lottery and scratch. The pruned solution is obtained by starting with a densely connected network and gradually removing connections during training, whereas the other two solutions are obtained by training sparse networks from start. \[fig:diagram\]](img/diagram){width="0.9\columnwidth"}
-0.3in
**Pruning strategy.** We use magnitude based model pruning [@gupta2018] in our experiments. This has been shown [@Gale2019] to perform as well as the more complex and computationally demanding variational dropout [@Molchanov2017] and $\ell_0$ regularization approaches [@Louizos2018]. In our experiments, we choose the 3 top performing pruning schedules for each sparsity level using the code and checkpoints provided by @Gale2019. The hyper-parameters involved in the pruning algorithm were found by grid search separately for each sparsity level. The 80% sparse model loses almost no accuracy over the baseline, while the 98% sparse model drops to 69% top-1 accuracy (see Figure \[fig:val-acc\]-*pruned*). Training details and the exact pruning schedules used in our experiments are detailed in Appendix \[apx:expdetail\].
{width="1.8\columnwidth"}
-0.4in
**Interpolation in parameter space.** Visualizing the energy landscape of neural network training is an active area of research. @Goodfellow2014 measured the training loss on MNIST [@mnist] along the line segment between the initial point $\mathbf{\theta_s}$ and the solution $\mathbf{\theta_e}$, observing a monotonically decreasing curve. Motivated by this, we were curious if this was still true (a) for Resnet-50 on Imagenet-2012 dataset and (b) if it was still true in the sparse subspace. We hypothesized that if (a) was true but (b) was not, then this could help explain some of the training difficulties encountered with sparse networks.
In our linear interpolation experiments, we generate networks along the segment $\mathbf{\theta}=t\mathbf{\theta_e}+(1-t)\mathbf{\theta_s}$ for $t \in [-0.2, 1.2]$ with increments of $0.01$ and evaluate them on the training set of 500k images. Interpolated parameters include the weights, biases and trainable batch normalization parameters. We enable training mode for batch normalization layers so that the batch statistics are used during the evaluation. The objective is identical to the objective used during training, which includes a weight decay term scaled by $10^{-4}$.
We now seek to find a non-linear path between the initial point and solution using parametric Bézier curves of order $n = 2$ and $3$. These are curves given by the expression -0.2in $$B_n(t)=\sum_{i=0}^n \binom{n}{i} (1-t)^{n-i}t^i\theta_i~,$$ where $\theta_0=\theta_e$ and $\theta_n=\theta_s~$. We optimize the following, -0.2in $$\min_{\theta_1,\cdots,\theta_{n-1}} \int_0^1 L(B_n(t))dt~,$$ using the stochastic method as proposed by @Garipov2018 with a batch size of 2048 where $L(\theta)$ denotes the training loss as a function of trainable parameters. Mirroring our original training settings, we set the weight decay coefficient to $10^{-4}$. We performed a hyper-parameter search over base learning rates ($1,10^{-1},10^{-2},10^{-3}$) and momentum coefficients ($0.9, 0.95, 0.99, 0.995$), obtaining similar learning curves for most of the combinations. We choose $0.01$ as the base learning rate and $0.95$ as the momentum coefficient for these path finding experiments.
Results and Discussion {#results}
======================
{width="1.85\columnwidth"}
-0.3in
Our experiments highlight a gap in our understanding of energy landscape of sparse deep networks. Why does training a sparse network from scratch gets stuck at a neighborhood of a stationary point with a significantly higher objective? This is in contrast with recent work that has proven that such a gap does not exist for certain kinds of over-parameterized dense networks [@exploringhighdimensionallandscapes2014; @lossurfaces2014]. Since during pruning dimensions are slowly removed, we conjecture that this prevents the optimizer from getting stuck into “bad” local minima. The failure of the optimizer is even more surprising in the light of the linear interpolation experiments of Section \[section31\], which show that sparse initial points are connected to the pruning solutions through a path in which the training loss is monotonically decreasing.
In high dimensional energy landscapes, it is difficult to assess whether the training converges to a local minimum or to a higher order saddle point. @sagun2017 shows that the Hessian of a convolutional network trained on MNIST is degenerate and most of its eigenvalues are very close to zero indicating an extremely flat landscape at solution. @dauphin2014 comments on @Bray2007’s results and argues that critical points that are far from the global minima in Gaussian fields are most likely to be saddle points. In Section \[section32\], we examine the linear interpolation between solutions and attempt to find a parametric curve between them with decreasing loss. This is because finding a decreasing path from the high loss solution (“scratch”) to the low loss solution(“pruned”) would demonstrate that the former solution is at a saddle point.
Path between start and end {#section31}
--------------------------
Linear interpolations from Initial-Point-1 (Dense), Initial-Point-1 (Sparse) and Initial-Point-2 (Sparse) to Pruned Solution at different sparsity levels are shown in Figure \[fig:interpolation-linear\] respectively; all cases show monotonically decreasing curves. The training loss represented in the y-axis consists of a cross entropy loss and an $\ell_2$ regularization term. While in Figure \[fig:interpolation-linear\] the $y$ axis represents the full training loss, the two terms composing this loss are shown separately in Appendix \[apx:loss-decomp\]. There we observe that the sum is dominated by the cross entropy loss.
In Figure \[fig:interpolation-linear\]-left, we observe a long flat plateau followed by a sharp drop: this is unlike typical learning curves, which are steepest at the beginning and then level off. Model pruning allows the optimizer to take the path of steepest descent while still allowing it to find a good solution as dimensions are slowly removed.
Finally, the linear interpolation from a random point sampled from the original initialization distribution (“scratch”) also depicts a decreasing curve (Figure \[fig:interpolation-linear\]-right), almost identical to the interpolations that originates from the lottery initialization (Figure \[fig:interpolation-linear\]-middle). This brings further evidence against the “lottery ticket” initialization being special relative to other initializations.
Path between two solutions {#section32}
--------------------------
The training loss along the linear segment and the parametric Bézier curve connecting the scratch and the pruned solutions are shown in Figure \[fig:interpolation-solution\]. As observed by @Keskar2016, linear interpolation (Figure \[fig:interpolation-solution\]-left) depicts a barrier between solutions, as high as the values observed by randomly initialized networks. The sparse parametric curve (Figure \[fig:interpolation-solution\]-middle) found through optimization also fails at connecting the two solutions with a monotonically decreasing path (although it has much smaller loss value than the straight line). Using a third order Bézier curve also fails to decrease the maximum loss value over the second order curve (Appendix \[apx:loss-bezier3\]). The failure of the third order curve does not prove that a path cannot be found. However, as a second order curve was sufficient to connect solutions in dense networks [@Garipov2018], it does show that if such a path exists, then it must be significantly more complex than those necessary in dense networks.
We continue our experiments by removing the sparsity constraint from the quadratic Bézier curve and optimize over the full parameter space (Figure \[fig:interpolation-solution\]-right). With all dimensions unmasked, our algorithm consistently finds paths along which the objective is significantly smaller[^3]. This result suggests enabling extra connections might be necessary and sufficient to escape bad critical points converged. It also gives insight why allowing the sparsity pattern to change over training to be beneficial for training sparse networks.
Conclusion and Future Work {#scs:conclusion}
==========================
In this work we have provided some insights into the dynamics of optimization in the sparse regime which we hope will guide progress towards better regularization techniques, initialization schema, and/or optimization algorithms for training sparse networks.
Training of sparse neural networks is still not fully understood from an [optimization]{} perspective. In the sparse regime, we show that optimizers converge to stationary points with a sub-optimal generalization accuracy. This is despite monotonically decreasing paths existing from the initial point to the pruned solution. And despite nearly monotonically decreasing paths in the dense subspace from the “bad” local minimum to the pruned one.
Optimizers that avoid “bad” local minima in the sparse regime are yet to be found. We believe that understanding why popular optimizers used in deep learning fail in the sparse regime will yield important insights leading us towards more robust optimizers.
### Acknowledgments {#acknowledgments .unnumbered}
The authors would like to thank; Trevor Gale, Hugo Ponte, Saurabh Kumar, Vincent Dumoulin, Danny Tarlow, Ross Goroshin, Pablo Castro, Nicolas Le Roux, Levent Sagun and Kevin Swersky for helpful discussions and feedback.
Experimental details {#apx:expdetail}
====================
Our experiments use the code made publicly available by [^4]. The pruning algorithm uses magnitude based iterative strategy to reach to a predefined final sparsity goal over the coarse of the training. We use a batch size of 4096 and train the network with 48000 steps (around 153 epochs). Our learning rate starts from 0 and increases linearly towards 0.1 in 5 epoch and stays there until 50th epoch. The learning rate is dropped by a factor of 10 afterwards at 50th, 120th and 140th epochs [@Goyal2017].
0.15in
Sparsity Start End Frequency
---------- ------- ------- -----------
12500 40000 2000
12500 40000 500
12500 36000 1000
12500 36000 2000
10000 40000 4000
7500 36000 1000
7500 36000 1000
10000 32000 500
7500 40000 2000
12500 36000 500
7500 32000 500
12500 32000 500
: Pruning strategies used in various settings. []{data-label="table:1"}
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Due to high sensitivity observed, we don’t prune the first convolutional layer and cap the maximum sparsity for the final fully connected layer with 80%. Top 3 performing pruning schedules selected for each sparsity level are shared in Table \[table:1\]. We calculate the average $\ell_1$ norm of the gradient for the first setting in the table and obtain 4e-6. As is the case for most iterative (especially stochastic) methods, our solutions do not qualify as stationary points since the gradient is never exactly zero. By slight abuse of language we refer to stationary point as any point where the gradient is below of $10^{-5}$.
![At the beginning of the training we randomly set a fraction of weights to zero and train the network with default parameters for 32k steps. We observe a sudden drop only if more than 99% of the parameters are set to zero. \[fig:sparse-init\]](img/sparse_init){width=".5\linewidth"}
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Sparse Initialization {#apx:sparse-init}
=====================
Initialization methods that control variance of activations during the first forward and backward-pass is known to be crucial for training deep neural networks [@glorot2010; @he2015]. However, with batch normalization [@Ioffe2015] and skip connections the importance of initialization is expected to be less pronounced. *sparse-init* experiments shared in Figure \[fig:val-acc\] can be seen as a demonstration of such tolerance. In sparse-init experiments we train a dense ResNet-50 but use the sparse binary mask found by pruning to set a fraction of initial weights to zero. At all sparsity levels considered (0.8, 0.91, 0.96, 0.98), we observe that the training succeeds and reaches to final accuracy around 76% matching the performance of the original training. Thus the initialization point alone cannot be the reason for the failure of sparse training.
To understand the extent which we are able to train sparsely initialized networks without compromising performance, we perform experiments where we randomly set a given fraction of weights to zero. Figure \[fig:sparse-init\] shows the results. We observe no significant difference until 99.5% after which we observe a sharp drop in performance. The initialization requires a very small number of non-zero weights to succeed.
Loss Decomposition {#apx:loss-decomp}
==================
Figure \[fig:interpolation-reg\_loss\] depicts the value of the $\ell_2$ regularization term over the linear interpolations described in Section \[section31\]. The curve demonstrates that the solutions found are consistently of lower weight magnitude than their initialization, and they also demonstrate that the regularization terms are a factor of ten smaller than the objective function.
{width="\linewidth"}
Figure \[fig:interpolation-cross\_loss\] depicts the value of the cross entropy loss over the linear interpolation described above. Figure \[fig:interpolation-cross\_loss\]-(left) the sparse to sparse and random sparse to sparse interpolation maintain the monotonic decreasing pattern observed in the interpolations plots of the total loss (Figure \[fig:interpolation-linear\]), whereas dense to sparse interpolation shows a slight increase in the objective before a rapid descent.
{width="\linewidth"}
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Dense-sparse cross entropy is increasing. Time to drop is much less with original initializations, higher with random and least with dense.
Path Finding Experiments with Cubic Bézier Curve {#apx:loss-bezier3}
================================================
In Section \[section32\] our experiments fail to find paths along which the loss is decreasing between the “pruned” and “scratch” solutions. Would optimizing more complex parametric curves change the result? Figure \[fig:bezier3\_loss\] depicts the objective along the third order Bézier curves. Though the integral of the loss over the segment between solutions seems less than Figure \[fig:interpolation-solution\]-middle, we still observe a small barrier between solutions.
{width="\linewidth"}
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[^1]: @frankle2019 closes this gap by using trained parameters from 5th epoch to initialize the network. However they don’t quantify how different these values are from the solution.
[^2]: sparse subspace refers to the sparsity pattern found by pruning and same in all settings.
[^3]: While this path is not strictly monotonically decreasing, this is not unexpected, given that our algorithm minimizes the integral of the objective over the interpolation segment and so monotonicity is not enforced.
[^4]: <https://github.com/google-research/google-research/tree/master/state_of_sparsity>
|
---
author:
- 'Jing Li[^1]'
- 'Yan Song[^2]'
- 'Zhongyu Wei[^3]'
- 'Kam-Fai Wong[^4]'
bibliography:
- 'compling/compling\_style.bib'
title: A Joint Model of Conversational Discourse and Latent Topics on Microblogs
---
[^1]: Shenzhen, China. Email: `[email protected]`. Jing Li is the corresponding author. This work was partially conducted when Jing Li was at Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, HKSAR, China.
[^2]: Shenzhen, China. Email: `[email protected]`.
[^3]: School of Data Science, Fudan University, Shanghai, China. Email: `[email protected]`
[^4]: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, HKSAR, China. Email: `[email protected]`
|
---
abstract: 'We show that boundary states in the generic on-shell background satisfy a universal nonlinear equation of closed string field theory. It generalizes our previous claim for the flat background. The origin of the equation is factorization relation of boundary conformal field theory which is always true as an axiom. The equation necessarily incorporates the information of open string sector through a regularization, which implies the equivalence with Cardy condition. We also give a more direct proof by oscillator representations for some nontrivial backgrounds (torus and orbifolds). Finally we discuss some properties of the closed string star product for non-vanishing $B$ field and find that a commutative and non-associative product (Strachan product) appears naturally in Seiberg-Witten limit.'
---
hep-th/0409069\
UT-04-23September, 2004\
1.5 cm
[Cardy states, factorization and]{}\
[idempotency in closed string field theory]{}
\
[*[ ]{}\
Department of Physics, Faculty of Science, University of Tokyo\
Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan\
[ ]{}\
*]{}
Introduction
============
Since its discovery, D-brane has been one of the central objects of interest in string theory. It represents fundamental nonperturbative features and is an analog of the soliton excitation in string theory.
In conformal field theory, the D-brane is described by boundary state. It belongs to the [*closed*]{} string Hilbert space and is implemented by the boundary conditions such as, $$\partial_\tau X^\mu|_{\tau=0} |B\rangle_{\mathrm{Neumann}} = 0\,,\quad
\mbox{or }\quad \partial_\sigma X^\mu|_{\tau=0}
|B\rangle_{\mathrm{Dirichlet}} = 0\,.$$ These equations determine the state $|B\rangle$ up to normalization constant. The information of the open strings which live on D-brane can be extracted from $|B\rangle$ after modular transformation, $$\langle B|q^{{1\over 2}\left(L_0+\tilde L_0-\frac{c}{12}\right)}
|B\rangle =
\mbox{Tr}_{\mathcal{H}_{\mathrm{open}}} \tilde q^{L_0-\frac{c}{24}}\,,\quad
\tilde q\equiv e^{4\pi^2/\log q}\,.$$
For more generic (conformal invariant) background where we can not use the free field oscillators as above, the boundary condition can be implemented only through generators of Virasoro algebra, $$\label{e_boundary}
(L_n-\tilde L_{-n})|B\rangle =0\,.$$ This condition is [*universal*]{} in a sense that it does not depend on a particular representation of Virasoro algebra which corresponds to the background.
This linear equation, however, is not enough to characterize the D-brane completely. We need further constraints that the open string sectors derived from them should be well-defined. More explicitly, take two states $|B_i\rangle$ ($i=1,2$) which satisfy eq. (\[e\_boundary\]). The open string sector appears in the annulus amplitude after the modular transformation can be written as, $$\label{e_Cardy}
\chi_{12}(q) = \langle B_1 |
q^{{1\over 2}\left(L_0+\tilde L_0-\frac{c}{12}\right)} |B_2\rangle
= \sum_i \mathcal{N}_{12}^i\,
\chi_i(\tilde q)\,,\quad
$$ where $\chi_i$ are characters of the irreducible representations in the open string channel. In order to have well-defined open string sector, the coefficients $\mathcal{N}_{12}^i$ must be non-negative integers. This is called [*Cardy condition*]{} [@Cardy:ir]. We note that these are non-linear (quadratic) constraints in terms of the boundary states.
These conditions (\[e\_boundary\], \[e\_Cardy\]) are written in terms of the boundary conformal field theory. In this sense, it is at the level of the first quantization. In order to consider the off-shell process, such as tachyon condensation, we need to use the second quantized description. One of the strong candidates of the off-shell descriptions of string theory is string field theory. Therefore, it is natural to ask whether one may derive conditions which are equivalent to (\[e\_boundary\], \[e\_Cardy\]) in that language.
There are a few species of string field theories. The best-established one is Witten’s open bosonic string field theory [@Witten:1985cc]. In terms of this formulation, the annihilation process of unstable D-branes was first studied extensively and it established the idea of “tachyon vacuum” by computation of the D-brane tension numerically [@Taylor:2003gn].
In this paper, however, we do not use this formulation since the dynamical variables of open string field theory depend essentially on the D-brane where the open string is attached. Since our goal is to find the characterization of generic consistent boundaries, this variable is not particularly natural because of this particular reference to the specific D-brane.
Since the boundary state belongs to the closed string Hilbert space, we will instead take the closed string field theory as the basic language. With closed string variable, the description of the linear constraint (\[e\_boundary\]) is trivial: $(L_n-\tilde L_{-n})\Phi=0$. On the other hand, the description of the Cardy condition (\[e\_Cardy\]) is much more nontrivial since it is the requirement to the [*open*]{} string channel which appears only after the modular transformation. Furthermore, it is a nonlinear relation. If it is possible to represent it in string field theory, one needs to use the closed string star product to express nonlinear relations.
There are two types of closed string star products which have been studied in the literature. The first one is Zwiebach’s star product which is a closed string version of Witten’s open string star product [@Zwiebach; @Saadi] and the other one is a covariant version of the light-cone string field theory (HIKKO’s vertex) [@HIKKO2]. They are defined through the overlap of three strings as depicted in Fig. \[fig:ovelapping\].
\[0.5\][![ Closed string vertices []{data-label="fig:ovelapping"}](overlapping.eps "fig:")]{}
These vertices are constructed to define the closed string field theories proposed by these authors. In this paper, however, our main focus is the algebraic structure between boundary states. Indeed, the nonlinear relation which we are going to study holds for both of these two vertices. In a sense, our relation does not seem to be a consequence of their proposed action at least at this moment but rather comes directly from the basic properties of the boundary conformal field theory.
The nonlinear equation has been proposed and studied in our previous papers [@kmw1; @kmw2; @km1]. It can be written as an idempotency relation[^1] among boundary states, $$\label{e_idempotency}
\Phi \star \Phi = \mathcal{C}\,T_B^{-1}c_0^+ \Phi,\quad
(\Phi\equiv c_0^-b_0^+|B\rangle\,,\,T_B\equiv \langle 0|
c_{-1}\tilde{c}_{-1}c_0^-|B\rangle)\,,$$ where $T_B$ is the tension of the D-brane associated with the boundary state $|B\rangle$. In the first paper [@kmw1], we proved it for the usual D$p$-brane boundary states for $\Phi$ and HIKKO vertex for $\star$. A surprise was that this equation is universal for any boundary states which we considered including the coefficient $\mathcal{C}$ [@kmw2]. The proof is based on an explicit calculation with the oscillator representation [@HIKKO2]. In the second paper [@kmw2], we gave an outline of the proof of the same equation for Zwiebach’s vertex. The equation takes the same form for these two vertices except for the overall constant $\mathcal{C}$. From these observations, we conjectured that eq. (\[e\_idempotency\]) is a background independent characterization of the boundary state.
Toward that direction, in [@kmw2], we used the path integral definition of the string field theory in terms of the conformal mapping [@LPP1] and tried to prove the relation in the general background. After some efforts, we have arrived at a weaker statement: suppose $|B_i\rangle$ ($i=1,2$) satisfy the linear constraint (\[e\_boundary\]), the state $|B_1\rangle \star |B_2\rangle $ also satisfies the same constraint. It proves that product of any boundary state in weak sense becomes again boundary state in weak sense. This does not, of course, imply the Cardy constraint (\[e\_Cardy\]) and, in particular, we could not understand the role of the open string sector.
One of the purposes of this paper is to discuss the link with open string which was missed in our previous studies and establish more explicit relation between (\[e\_idempotency\]) and the Cardy condition. A crucial hint to this problem is that the coefficient $\mathcal{C}$ in the relation (\[e\_idempotency\]) is actually divergent and it is necessary to introduce some sort of regularization in the computation. In [@kmw1], we cut-off the rank of Neumann coefficient by $K$. Then the divergent coefficient behaves as $\mathcal{C}\sim K^3$. In LPP approach, on the other hand, another type of the regularization can be introduced by slightly shifting the interaction point on the world sheet. Such a shift gives a small strip which interpolates between two holes associated with the boundary. In the limit of turning-off the regulator, the moduli parameter which describes the shape of the strip becomes zero. This is the usual factorization process where the world sheet becomes degenerate [@r:Factorization]. An essential point is that such a factorization process occurs in the [*open string channel*]{} between the two holes. In this way, the equation for the string field (\[e\_idempotency\]) can be related with the consistency of the dual open string channel. The leading singularity comes from the propagation of the open string tachyon which is universal for any boundary states in arbitrary background and it explains our claim that the divergent factor $\mathcal{C}$ is also universal for any Cardy states.
This outline will be explained in detail in section 2. We will also repeat our previous discussion [@kmw2] that only the boundary states can satisfy the equation (\[e\_boundary\]). By combining these ideas, it will be obvious that the nonlinear equation (\[e\_boundary\]) plays an essential rôle to understand D-branes in the context of string field theory.
At this point, it may be worth while to mention that our claims are remarkably similar to the scenario conjectured in vacuum string field theory [@r:VSFT]. The form of the equation is exactly the same except that the dynamical variables and the star product are totally different. There is no nontrivial solution in the vicinity of $\Phi=0$ and the non-vanishing solutions correspond to D-branes. In a sense, our equation is an explicit realization of VSFT scenario in the dual closed string channel. While the dynamical variables is closed string field, the physical excitations around the boundary state are on-shell open string mode [@kmw1; @kmw2].
Our discussion in section 2 is based on the path integral and the argument becomes necessarily formal to some extent. In this sense, it is desirable to check the consistency of the argument in the oscillator representation for some non-trivial backgrounds. Fortunately, there are explicit forms of the three string vertex for (1) toroidal $T^d$ and (2) orbifold $T^d/Z_2$ compactification [@HIKKO_torus; @Itoh_Kunitomo].
In both cases, the three string vertex has some modifications compared to that on the flat background ${\bf R}^d$ [@HIKKO2]. One needs to include cocycle factor due to the existence of winding mode and take into account the twisted sector in orbifold case. The cocycle factor is needed to keep Jacobi identity for the $\star$ product of the closed string field theory. In section 3, we perform explicit computation of $\star$ product among the boundary (Ishibashi) states. For torus case, there appear extra cocycle factors in the algebra of Ishibashi states and some care is needed to construct Cardy states as idempotents of the algebra. For the orbifold case, mixing between untwisted sector and twisted sector is needed to describe the fractional D-branes. The ratio of the coefficients of the two sectors is given as the ratio of the determinants of the Neumann matrices for the (un)twisted sectors. We use various regularization methods to calculate them explicitly. This result is consistent with our previous arguments [@km1].
Finally, the third issue which will be discussed in section 4 is to incorpolate the noncommutativity on the D-brane. We have already seen in [@kmw1] that non-commutativity on the world volume of the D-brane forces us to use the open string metric to write down the on-shell conditions. This is possible since the explicit form of the boundary state is known for such cases. On the other hand, for the noncommutativity in the transverse directions, it is difficult to express in the language of the boundary conformal field theory. This is, however, an important set-up for matrix models or noncommutative Yang-Mills theory.
A motivation toward this direction is our previous study [@kmw2] where we have seen that an analog of the noncommutative soliton arises in the commutative limit. We note that the idempotency relation for the D$p$-brane takes the following form in the matter sector, $$|B,x^\perp\rangle \star |B, y^\perp\rangle
=\mathcal{C}_d\,\delta^{d-p-1}(x^\perp-y^\perp)\,|B,y^\perp\rangle\,\,,$$ where $x^\perp,\,y^\perp$ are the coordinates in the transverse directions. In order to recover the universal relation (\[e\_idempotency\]), we need to take a linear combination, $$|B\rangle_f\equiv \int d^{d-p-1} x^{\perp}\, f(x^\perp) |B,x^\perp\rangle
\,\,.$$ Eq. (\[e\_idempotency\]) then implies $f^2(x^\perp)=f(x^\perp)$ which is the same as (\[e\_projector\]) in the commutative limit. It is, therefore, tempting to study what kind of modification will be necessary in the presence of $B$ field.
In order to study it, we consider a particular deformation of Ishibashi states which seems to be relevant to describe the noncommutativity along the transverse directions. We take the star product between them and take Seiberg-Witten limit. The constraint for $f$ is deformed to the following form, $$(f\diamond f)(x^\perp) = f(x^\perp)\,\,,\qquad
\diamond\equiv \frac{\sin(\Lambda)}{\Lambda}\,,\quad
\Lambda=\frac{1}{2}\overleftarrow{\partial_i}\theta^{ij}
\overrightarrow{\partial_j}\,.$$ This $\diamond$ product is commutative but breaks associativity. It appeared in mathematical literature [@Strachan] and is related to the loop corrections in noncommutative super Yang-Mills theory [@start]. The appearance of such deformation seems to be natural since the star product of two boundaries is topologically equivalent to one loop from the open string viewpoint.
Idempotency relation in generic background
==========================================
In this section, we prove the relation (\[e\_idempotency\]) in the generic background by using a sequence of conformal maps. Our proof depends only on a generic property of the boundary conformal field theory — factorization — which should be satisfied axiomatically in any BCFT. Our discussion also shows a clear link between the Cardy condition and the idempotency relation.
$\star$ product and factorization
---------------------------------
The factorization is a general behavior of the correlation functions of the conformal field theory defined on a pinched Riemann surface. The relevant process for us is the degeneration of a strip between two holes where the correlation function behaves as $$\langle \mathcal{O}\cdots \rangle \rightarrow
\sum_i \langle \mathcal{O}\cdots A_i(z_1) A_i(z_2)\rangle q^{\Delta_i}$$ where $i$ is the label of orthonormal basis $\left\{A_i(z)\right\}$ of the open string Hilbert space between two holes and $\Delta_i$ is the conformal dimension of $A_i$. The open string channel depends on the boundary conditions at two holes. $q$ is a real parameter which describes the degeneration of the strip and $z_{1,2}$ are the coordinates of the two points along the boundary where the two ends of the strip are attached (Fig. \[fig:degenerate\]).
\[0.55\][![ Factorization associated with merging two holes. []{data-label="fig:degenerate"}](degenerate.eps "fig:")]{}
In the star product, such a degeneration of a strip appears as we explained in our previous paper [@kmw2]. It comes from combining the geometrical nature of the boundary state as a surface state and three string vertex. In order to explain the former, we consider an inner product between a closed string state $|\chi\rangle$ and a boundary state: $\langle B|\chi\rangle$. On the world-sheet, it is equivalent to the one-point function on a disk $\langle \chi(0) \rangle$ with the boundary condition at $|z|=1$ specified by the boundary state. If we map from the sphere to a cylinder by a conformal transformation, $w=\log z=\tau+i\sigma$, the geometrical role of the boundary state can be summarized as follows: (Fig. \[fig:boundary\])
\[0.5\][![ Geometrical interpretation of boundary state as a surface state. []{data-label="fig:boundary"}](boundary.eps "fig:")]{}
1. cut the infinite cylinder at $\tau=0$ and strip off the region $\tau>0$,
2. set the boundary condition specified by $|B\rangle$ at the boundary.
We combine this property of the boundary state with that of the three string vertex, which is represented by Mandelstam diagram (Fig. \[fig:vertex\]-a) for HIKKO type vertex.
\[0.9\][![ (a) Putting boundary states at two legs of trousers that is associated with three string vertex. (b) Stripping two legs at the origin. (c) Shifting the interaction point as a regularization. []{data-label="fig:vertex"}](vertex.eps "fig:")]{}
The matrix element $\Phi_1\cdot(\Phi_2\star\Phi_3)$ corresponds to putting three local operators $\Phi_i$ ($i=1,2,3$) at the ends of three half cylinders.
As we see previously, the boundary states are not described by local operators but should be interpreted as the surface state. To take the product of two boundary states $|B\rangle \star |B\rangle$ is then geometrically represented as the Mandelstam diagram whose two legs are stripped at the interaction time $\tau=0$ (see a Fig. \[fig:vertex\]-b).
This configuration is, however, singular since two boundaries are attached at one point (interaction point) and we need a regularization to obtain a smooth surface. A natural regularization is to shift the location of the boundary slightly, for example at $\tau=\tau_1/2>0$ (Fig. \[fig:vertex\]-c). As we see later, this is equivalent to a cut-off of the Neumann matrix with finite size $K$ which was used in our previous paper [@kmw1]. The correspondence of the regulator turns out to be $K\sim \tau_1^{-1}$ . With this regularization, the world-sheet becomes a cylinder with one vertex operator insertion. The limit $\tau_1\rightarrow 0$ is equivalent to shrinking a strip of this diagram and reducing it to a disk. We can use the discussion of factorization as, $$|B\rangle \star_{\tau_1} |B\rangle =
\sum_i q^{\Delta_i} A_i(\sigma_1) A_i(\sigma_2)|B\rangle\,,$$ where again $A_i(\sigma_i)$ belongs to a set of orthonormal operators in the [*open*]{} string Hilbert space with both end attached to a brane specified by $|B\rangle$ and $\sigma_{1,2}$ are the coordinates along the boundary.
For the consistency of boundary states of the bosonic string, the lowest dimensional operator in the Hilbert space is always tachyon state which is written as, $|0\rangle^{\mathrm{m}}\otimes c_1|0\rangle^{\mathrm{gh}}$ where $|0\rangle^{\mathrm{m,gh}}$ are $SL(2,R)$ invariant vacuum for the matter and ghost. The conformal dimension of this state is $-1$. Other terms depend on the detail of the boundary state but they always give less singular terms as $q\rightarrow 0$. Similarly, if we $\star$-multiply two different boundary states $|B_1\rangle \star |B_2\rangle$, the open string sector is described by the Hilbert space of the [*mixed*]{} boundary condition and the lowest dimensional operator always has a dimension $\Delta$ greater than $0$. This simple argument then implies, $$|B_\alpha\rangle \star |B_\beta\rangle \sim q^{-1} c(\sigma_1)
(c\partial c)(\sigma_2) \delta_{\alpha\beta}|B_\beta\rangle + \
\mbox{less singular terms in $K$}\,.$$ Although the less singular terms do depend on the background and boundary state, the first term is universal. As we see, the precise structure for ghost and singularity is more involved due to the ghost insertions in the three string vertex and the singular behaviour themselves should be modified in order to obtain the precise agreement with the oscillator computation.
Computation by conformal mappings \[sec:compC\]
------------------------------------------------
In order to see the degeneration in detail, we consider three surfaces which can be related with each other by conformal mappings (Fig. \[fig:mapping\]-a,b,c).
\[0.8\][![ (a) (reguralized) string vertex with two boundary states; (b) a cylinder diagram; (c) a disk diagram with operator insertions. []{data-label="fig:mapping"}](mapping2.eps "fig:")]{}
The first one is the regularized version of the Mandelstam diagram for the star product of two boundary states (Fig. \[fig:vertex\]-a). A natural coordinate for this diagram is $\rho=\tau+i\sigma$ ($\tau>0$, $-\pi\alpha\leq \sigma\leq \pi\alpha$). The interaction point is $\rho=\tau_1/2\pm i\pi\alpha_1$, ($0\leq \alpha_1\leq \alpha$) and the parameter $\tau_1>0$ is introduced for the regularization.
This diagram has two holes together with one vertex operator insertion at infinity. Since it is topologically annulus, it can be mapped to the standard annulus diagram (Fig. \[fig:mapping\]-b). A complex parameter $u$ ($|\mbox{Re}(u)|\leq 1/2$, $-t/2\leq\mbox{Im}(u)\leq 0$, $\tilde\tau:=it$) is a flat complex coordinate and $\tilde\tau$ is the moduli parameter. These two diagrams are related with each other by a generalized Mandelstam mapping [@r:Mandel][@r:AKT], $$\label{eq:Mandel1loop}
\rho(u) = \alpha \ln \frac{\vartheta_1(u-Z_1|\tilde\tau)}{\vartheta_1(u-Z_2|
\tilde\tau)}
- 2\pi i \alpha_1 u\,.$$ We note that it can be extended as a mapping between the doubles of above diagrams, i.e., a torus $-\frac{1}{2}\leq \mbox{Re}(u)\leq \frac{1}{2}$, $-t/2\leq \mbox{Im}(u)\leq t/2$ and corresponding Mandelstam diagram. The parameters $Z_{1,2}:=\mp \beta\tilde\tau/2$ are mapped to the infinities $\mbox{Re}(\rho)=\mp \infty$. The interaction point $\tau_1/2 \mp i\pi\alpha_1$ is mapped to $\hat{z}_{\pm}=\pm \frac{1}{2}-\tilde{\tau}y$. There is a set of relations among parameters [@r:AKT], $$\begin{aligned}
&& \beta=-\frac{\alpha_1}{\alpha}\,,\\
&& \frac{\tau_1}{\alpha} =2 \ln
\frac{\vartheta_2(\tilde\tau(-\beta/2+y)|\tilde\tau)}
{\vartheta_2(\tilde\tau(-\beta/2-y)|\tilde\tau)}
-4\pi i \beta\tilde\tau y\,,\\
&& g_2(\tilde\tau(-\beta/2+y)|\tilde\tau)+
g_2(\tilde\tau(-\beta/2-y)|\tilde\tau)=2\pi i \beta\,,\end{aligned}$$ where $ g_2(\nu|\tilde\tau)=\partial_\nu \ln\vartheta_2(\nu|\tilde\tau)$. In the degenerate limit $\tau_1\rightarrow 0$, these are reduced to, $$\begin{aligned}
&& y\sim\frac{1}{4}\,,\qquad
e^{-\frac{i\pi}{\tilde\tau}}
\equiv q^{1/2}\sim\frac{\tau_1}{8\alpha\sin(-\pi\beta)}\,.
\label{eq:yq}\end{aligned}$$
The third diagram (Fig. \[fig:mapping\]-c) is disk-like diagram with two short slits. It is parametrized by a complex coordinate $w$ with $|w|\leq 1$. The relation with the Mandelstam diagram is very simple, $$ w=f(\rho)\equiv \exp(-\rho/\alpha)\,.$$ Two slits are located at $x_\pm=\exp(\pm \pi i\alpha_1/\alpha)
=\exp(\mp \pi i \beta)$.
Each diagram has its own role in the computation of $|B\rangle \star |B\rangle$. Firstly, the Mandelstam diagram gives the definition of the star product. The expression,[^2] $$\mathcal{F}\equiv \left(
\langle B_1|_{\tau_1\over 2\alpha_1}b^+_0c_0^-\star
\langle B_2|_{\tau_1\over 2\alpha_2}b^+_0c_0^-
\right)|\phi\rangle
\quad (\langle B|_T\equiv
\langle B|e^{-T(L_0+\bar L_0)}) \,,$$ can be evaluated as the one point function of $\phi$ inserted at $\tau=\infty$ in $\rho$-plane with two boundaries defined by $| B_{1,2}\rangle$. $b_0^+$ insertion is used to cancel $c_0^+$ factor contained in the boundary state and to set the ghost number to be two. The ket vector $|\phi\rangle =\phi(0)|0\rangle$ has ghost number two as usual. If we map it to the standard annulus diagram (Fig. \[fig:mapping\]-b), it can be rewritten as,[^3] $$\begin{aligned}
\label{eq:calF}
\mathcal{F}&=&
\alpha_1\alpha_2
\langle B_1 | \tilde{q}^{{1\over 2}(L_0+\tilde{L}_0)}\,b_\rho^+\,
(\rho^{-1}\!\circ\phi)(Z_2)\, b_\rho^+ |B_2\rangle\,,\\
&&\tilde{q}:=e^{2\pi i\tilde{\tau}}\,,~~~~
b_\rho^+=\oint \frac{du}{2\pi i} b(u)
\left(\frac{d\rho}{du}\right)^{-1}+c.c.\,.\end{aligned}$$
We need to evaluate this expression in the limit $\tilde\tau\rightarrow
0$ and take the conformal transformation to disk diagram (Fig. \[fig:mapping\]-c).
As we have argued, taking the limit corresponds to taking the lowest dimensional operator in the [*open*]{} string channel. Therefore, one can rewrite $\mathcal{F}$ in this limit as, (after the conformal map to $w$ plane), $$\mathcal{F}\sim \delta_{12}\langle B|b_{w}^{1+}b_{w}^{2+}
(g\circ c)(x_+) (g\circ c\partial c)(x_-)|\phi\rangle\,,$$ where $x_{\pm}=e^{\mp i\pi\beta}$ are the locations of tachyon insertions and $g=f\circ \rho$. $b^{1+}_w$ and $b^{2+}_w$ are the conformal transformations of $b_\rho^+$ along two boundaries. The ghost insertions become very complicated but we have already proved in the oscillator formulation [@kmw1] that $$\label{eq:Bb1b2}
\langle B|b_{w}^{1+}b_{w}^{2+}
(g\circ c)(x_+) (g\circ c\partial c)(x_-)\sim\langle B|c_0^-\,.$$ In order to fix the coefficient, we take the simplest example, $|\phi\rangle=c_{1}\tilde{c}_{1}|0\rangle$ and calculate both sides of the equation.
It is convenient to divide the computation into matter and ghost sectors. Let us first consider the matter part. The vertex operator for $|\phi\rangle$ is simply 1. The inner product between two boundary states is $$\langle B^{\mathrm{m}}_1|
{\tilde{q}}^{{1\over 2}\left(L_0+\tilde{L}_0-{c\over 12}\right)}
|B^\mathrm{m}_2\rangle =
q^{-{c\over 24}}\,\delta_{12}+\mbox{(higher order in $q$)}\,,
\label{eq:B1B2m}$$ where we have supposed that these boundary states $|B_1^{\rm m}\rangle,|B_2^{\rm m}\rangle $ satisfy Cardy condition and used the leading behavior of the character of identity operator $\chi_1(q)\simeq q^{-{c\over 24}}$ ($q\rightarrow 0$). On the other hand, the right hand side becomes, $$\langle B^\mathrm{m}_2|0\rangle =T_{B_2}\,.
\label{eq:TB2}$$ Namely the tension for the D-brane [@Harvey:1999gq].
For the ghost part, we compute ${\cal F}$ (\[eq:calF\]) with $\phi=c\tilde{c}$ and take the limit of ${\tilde{\tau}}\rightarrow +i0$ after modular transformation. In order to compute explicitly, we map the $u$-plane to $\tilde{\rho}=2\pi i u$ such that $\tilde{\rho}$-plane becomes closed string strip with period $2\pi$ in the ${\rm Im}\tilde{\rho}$ direction and then we expand the ghosts as $$\begin{aligned}
&&b(\tilde{\rho})=\sum_{n=-\infty}^{\infty}b_ne^{-n\tilde{\rho}}\,,~~~
c(\tilde{\rho})=\sum_{n=-\infty}^{\infty}c_ne^{-n\tilde{\rho}}\,,~~~
\{b_n,c_m\}=\delta_{n+m,0}\,,~~~~~~\end{aligned}$$ and similar ones for $\tilde{b}(\bar{\tilde{\rho}})$ and $\tilde{c}(\bar{\tilde{\rho}})$. Using the property of the boundary states in the ghost sector: $(b_n-\tilde{b}_{-n})|B\rangle=0$, we calculate ghost contribution for ${\cal F}$ (\[eq:calF\]) as $$\begin{aligned}
{\cal F}_{c\tilde{c}}
&=&4\alpha_1\alpha_2(2\pi)^2
\int_{C_1} {du_1\over 2\pi i}
{du_1\over d\rho}
\int_{C_2} {du_2\over 2\pi i}{du_2\over d\rho}
\left[\left.{du\over dw_3}\right|_{w_3=0}
\left.{d\bar{u}\over d\bar{w}_3}\right|_{\bar{w}_3=0}
\right]^{-1}\nonumber\\
&&\times\,
\langle B|\tilde{q}^{{1\over 2}\left(L_0+\tilde{L}_0+{13\over 6}\right)}
b(2\pi i u_1)c(2\pi i Z_2)\tilde{c}(-2\pi i \bar{Z}_2)b(2\pi i u_2)
|B\rangle\,.~~~~~
\label{eq:Fcc}\end{aligned}$$ ($C_1$ and $C_2$ are given in Fig. \[fig:mapping\]-b.) Here the conformal factor $C_{c\tilde{c}}$ for $c\tilde{c}$ (given by $[\cdots]^{-1}$ in the integrand) can be evaluated using the Mandelstam map (\[eq:Mandel1loop\]) with $\rho(u)=\alpha_3 \log w_3$ ($\alpha_3=-\alpha_1-\alpha_2=-\alpha$) for string $3$ region where $\phi$ is inserted: $$\begin{aligned}
\label{eq:confC}
C_{c\tilde{c}}=
\left|{\vartheta_1(Z_2-Z_1|\tilde{\tau})\over
\vartheta_1^{\prime}(0|\tilde{\tau})}e^{i\pi\tilde{\tau}\beta^2}
\right|^{-2}
\sim \pi^2|\tilde{\tau}|^{-2}(\sin\pi\beta)^{-2}\,,
~~~(\tilde{\tau}\rightarrow +i0)\,.\end{aligned}$$ We have used modular transformation for $\vartheta$-function in order to obtain the last expression. The above inner product $\langle B|\cdots |B\rangle$ is computed straightforwardly: $$\begin{aligned}
&& \langle B|\tilde{q}^{{1\over 2}\left(L_0+\tilde{L}_0+{13\over 6}\right)}
b(2\pi i u_1)c(2\pi i Z_2)\tilde{c}(-2\pi i \bar{Z}_2)b(2\pi i u_2)
|B\rangle\nonumber\\
&=&{-i\over 8\pi}\,e^{i\pi\tilde{\tau}\over 6}
\prod_{n=1}^{\infty}(1-e^{2\pi i\tilde{\tau}n})^2\,
\bigl[g_1(u_1-Z_1|\tilde{\tau})
-g_1(u_1-Z_2|\tilde{\tau})-g_1(u_2-Z_1|\tilde{\tau})
+g_1(u_2-Z_2|\tilde{\tau})
\bigr]\nonumber\\
&=&{-i\over 8\pi \alpha}
\,\eta(\tilde{\tau})^2\left[
{d\rho\over du}(u_1)-{d\rho\over du}(u_2)
\right],
\label{eq:BqB}\end{aligned}$$ where we have used $$\begin{aligned}
&&g_1(\nu|\tau):={\vartheta_1^{\prime}(\nu|\tau)\over \vartheta_1(\nu|\tau)}
=\pi \cot \pi \nu+4\pi \sum_{n=1}^{\infty}{e^{2\pi i n\tau}\over
1-e^{2\pi i n\tau}}\sin(2\pi n \nu)\,,\\
&&L_0:=\sum_{n=1}^{\infty}n(c_{-n}b_n+b_{-n}c_n)-1\,,~~
\tilde{L}_0:=
\sum_{n=1}^{\infty}n(\tilde{c}_{-n}\tilde{b}_n+\tilde{b}_{-n}\tilde{c}_n)-1\,,
~~~~~~~~\end{aligned}$$ and adopted the normalization as: $$\begin{aligned}
&&|B\rangle = e^{\sum_{n=1}^{\infty}(c_{-n}\tilde{b}_{-n}
+\tilde{c}_{-n}b_{-n})}c_0^+c_1\tilde{c}_1|0\rangle\,,~~~~~
\langle 0|c_{-1}\tilde{c}_{-1}c_0\tilde{c}_0c_1\tilde{c}_1|0\rangle
=1\,.\end{aligned}$$ We note that (\[eq:BqB\]) corresponds to (5.16) in [@r:AKT] which was calculated as the correlation function at the 1-loop of [*open*]{} string, as expected. We perform contour integration for $b$-ghost in (\[eq:Fcc\]) and reduce it to evaluation of the residue at the interaction point $\hat{z}_{\pm}$ on $u$-plane, where ${d\rho\over du}=0$, by deforming the contour $C_1+C_2$: $$\begin{aligned}
{\cal F}_{c\tilde{c}}&=&{2\pi\alpha_1\alpha_2\over \alpha}\,
C_{c\tilde{c}}\,\eta(\tilde{\tau})^2\,{\cal R}\,,\\
{\cal R}&=&\left(\left.{d^2\rho\over du^2}\right|_{u=\hat{z}_{\pm}}\right)^{-1}
={1\over \alpha}
\left(g^{\prime}_1(\hat{z}_{\pm}-Z_1|\tilde{\tau})
-g^{\prime}_1(\hat{z}_{\pm}-Z_2|\tilde{\tau})
\right)^{-1}.~~~~\end{aligned}$$ In the degenerate limit, using (\[eq:yq\]), the above residue ${\cal R}$ behaves as $$\begin{aligned}
{\cal R}&\sim&{1\over 4\alpha}\,
(\log q)^{-2}\,{q^{-1/2}\over \sin\pi\beta}\,.\end{aligned}$$ Then, from $\eta(\tilde{\tau})^2=i\tilde{\tau}^{-1}\eta(-1/\tilde{\tau})^2
\sim |\tilde{\tau}|^{-1}e^{-{\pi\over 6|\tilde{\tau}|}}$ and (\[eq:confC\]), the ghost contribution to ${\cal F}$ with $\phi=c\tilde{c}$ is evaluated as $$\begin{aligned}
{\cal F}_{c\tilde{c}}\sim
{-\alpha_1\alpha_2\over 16\alpha^2}\,(\log q)\,
q^{13\over 12}
\left(q^{-1/2}\over \sin\pi\beta\right)^3\end{aligned}$$ in the degenerating limit. On the other hand, the inner product of the right hand side of (\[eq:Bb1b2\]) and $\phi=c\tilde{c}$ gives $$\begin{aligned}
\langle B|c_0^-c_1\tilde{c}_1|0\rangle =1\,,
\label{eq:TBgh}\end{aligned}$$ in the ghost sector.
After combining contributions from the matter and the ghost sector, we note that there is a constraint $\alpha=2p^+$ in order to get physical amplitudes [@KZ]. Namely, the string length parameter $\alpha$ should be identified with a light cone momentum $p^+$. It gives extra factor $(\log q)^{-1}$ compared to the right hand side in (\[eq:B1B2m\]). (See, (5.41) in [@r:AKT].) Taking into account of it in open string description, ${\cal F}$ (\[eq:calF\]) is evaluated as $$\begin{aligned}
{\cal F}_{c\tilde{c}}&\sim&\delta_{12}\,
{-\alpha_1\alpha_2\over 16\alpha^2}\,
q^{26-c\over 24}
\left(q^{-1/2}\over \sin\pi\beta\right)^3\nonumber\\
&\sim&32\,\delta_{12}\,\alpha_1\alpha_2(\alpha_1+\alpha_2)\tau_1^{-3},
\label{eq:Ftotal}\end{aligned}$$ where we have substituted $c=26$ as total central charge in the matter sector and used (\[eq:yq\]) in the second line.
After all, using the above results: (\[eq:Ftotal\]),(\[eq:TB2\]) and (\[eq:TBgh\]) for $\phi=c\tilde{c}$, we can evaluate the proportional constant of $B_1\star B_2\propto
\delta_{12} B_2$ for Cardy states: $$\begin{aligned}
\left(
\langle B_1|_{\tau_1\over 2\alpha_1}b^+_0c_0^-\star
\langle B_2|_{\tau_1\over 2\alpha_2}b^+_0c_0^-
\right)|\phi\rangle/\langle B_2|b_0^+ c_0^- c_0^+|\phi\rangle&\sim& 32\,\delta_{12}\,\alpha_1\alpha_2(\alpha_1+\alpha_2)
\tau_1^{-3}\,T_{B_2}^{-1}\,,\end{aligned}$$ with regularization parameter $\tau_1$. This implies ${\cal C}\sim 32\,\delta_{12}\,
\alpha_1\alpha_2(\alpha_1+\alpha_2)\tau_1^{-3}$ in (\[e\_idempotency\]) and is consistent with the result in [@kmw2] by identifying a regularization parameter $\tau_1$ with $K^{-1}$ [@kmw2].
Algebra of Ishibashi states and fusion ring \[sec:km1\]
--------------------------------------------------------
Before we proceed, we point out that the idempotency relation implies that Ishibashi state satisfies a simple algebra with the $\star$ product. We have discussed such relation in our previous paper [@km1] by assuming the relation for the generic background. Since it is proved in this paper, it is worth mentioning the result again with a slight generalization.
We focus on the matter part of the idempotency relation (\[e\_idempotency\]) which may be written as, $$|a\rangle\star |b\rangle =
q^{-{c\over 24}}\,\delta_{ab}\,T_b^{-1}|b\rangle\,.
\label{eq:m_idempotency}$$ Here $q\,(\rightarrow 0)$ is a regularization parameter which was introduced in the previous subsection. The factor $q^{-\frac{c}{24}}$ will contribute, when we combine it with ghost fields with other matter sector, to a universal divergent factor. We will therefore drop it in the following discussion to illuminate the nature of the algebra.
For the rational conformal field theory, the relation between Cardy state with Ishibashi states, (with slight generalization after [@Z] eq. (2.10)), $$|a\rangle = \sum_{j}\frac{\psi_{a}^j}{\sqrt{S_{j1}}}|j{\rangle\!\rangle}\,.
\label{eq:Cs}$$ The coefficient $\psi_a^j$ should satisfy the orthogonality (eqs. (2.18) (2.19) of [@Z]), $$\sum_a \psi_a^i (\psi_a^j)^*=\delta_{ij}\,,\quad
\sum_i \psi_a^i (\psi_b^i)^*=\delta_{ab}\,,\quad$$ and also generalized Verlinde formula (2.16): $$n_{ia}{}^b=\sum_j\frac{S_{ij}}{S_{1j}}\psi^j_a (\psi^j_b)^*\,,$$ where $n_{ia}{}^b$ are non-negative integers. With this combination, the tension $T_a$ can be written as $$T_a=\frac{\psi^1_a}{\sqrt{S_{11}}}\,.$$ The idempotency relation between Cardy states (in matter sector) can be rewritten as the algebra between the Ishibashi states $|i{\rangle\!\rangle}$, $$\label{fr}
|i{\rangle\!\rangle}'\star |j{\rangle\!\rangle}' = \sum_k \mathcal{N}_{ij}{}^k |k{\rangle\!\rangle}'\,,$$ where we changed the normalization of Ishibashi states as, $$|i{\rangle\!\rangle}' \equiv (S_{i1}S_{11})^{-1/2}|i{\rangle\!\rangle}\,,$$ and ${\cal N}_{ij}^{~~k}$ is given by $$\begin{aligned}
\mathcal{N}_{ij}^{~~k}&=&
\sum_{a}{(\psi^i_a)^*(\psi^j_a)^*\psi^k_a\over \psi^1_a}\,.\end{aligned}$$ Eq. (\[fr\]) is a natural generalization of the fusion ring for the generic BCFT, namely the coefficient $\mathcal{N}_{ij}{}^k$ is also known to be non-negative integers [@Z]. This relation looks natural since (generalized) fusion ring describes the number of channels in OPE of primary fields and Ishibashi states are directly related with the irreducible representation.
One may summarize the observation as, [*Cardy states are projectors of (generalized) fusion ring*]{}. We believe that this nonlinear relation is a natural replacement of Cardy condition in the first quantized language.
As a preparation of the next section, we present an application of this result to the orbifold CFT [@km1]. We consider an orbifold $\mathbf{R}^d/\Gamma$ where $\Gamma$ is a finite group which may be nonabelian in general. At the orbifold singularity, there exist fractional D-branes which are given as combinations of various twisted sector. We apply the above idea to these fractional D-branes.
In this setup, there is a boundary state which belongs to the twisted sector specified by $h\in \Gamma$, $$(X(\sigma+2\pi)-h\cdot X(\sigma))|h\rangle\!\rangle=0\,.$$ When $\Gamma$ is nonabelian, however, such a state is not invariant under conjugation. Ishibashi state is, therefore, given as a linear combination of such boundary state which belongs to a conjugacy class $C_j$ of $\Gamma$: $$|j\rangle\!\rangle:={1\over \sqrt{r_j}}\sum_{h_j\in C_j}
|h_j\rangle\!\rangle,$$ where $r_j$ is the number of elements in $C_j$.
In this case, Cardy state $|a\rangle$ is given by eq. (\[eq:Cs\]) where the coefficients $\psi^j_a$, $S_{j1}$ are [@Billo] $$\begin{aligned}
&&\psi_a^j=\sqrt{r_j\over |\Gamma|}\,\zeta^{(a)}_j,~~~~~
S_{j1}={1\over \sigma(e,h_j)},~~(h_j\in C_j)\,,
$$ $\zeta^{(a)}_j$ is the character of an irreducible representation $a$ for $g\in C_j$, $e$ is the identity element of $\Gamma$ and $\sigma(e,h)$ is determined by the modular transformation of the character $\chi_g^h(q)\equiv {\rm Tr}_{{\cal H}_g}(h\,q^{L_0-{c\over 24}})$: $$\begin{aligned}
\chi_e^h(q)=\sigma(e,h)\chi_h^e(\tilde{q})\,,~~~~
(q=e^{2\pi i\tau},~\tilde{q}=e^{-{2\pi i\over \tau}})\,.
\label{eq:Modular}\end{aligned}$$ The normalization of Ishibashi state $|h{\rangle\!\rangle}$ is specified as $\langle\!\langle h|\tilde{q}^{{1\over
2}\left(L_0+\tilde{L}_0-{c\over 12}\right)}|h\rangle\!\rangle
= \chi_h^e(\tilde{q})\,.$ In this case, eq. (\[fr\]) is equivalent to $$\begin{aligned}
&& e_i\star e_j = \sum_k{\cal N}_{ij}^{~~k}\,e_k\,,~~~~
{\cal N}_{ij}^{~~k}=\sum_{a}{r_ir_j\zeta_i^{(a)}\zeta_j^{(a)}\zeta_k^{(a)*}
\over \zeta_1^{(a)}}\,,
\label{eq:orb_alg}\\
&&e_i:=|\Gamma|\sqrt{r_i\,\sigma(e,h_i)}\,|i\rangle\!\rangle\,.
\label{eq:rescal}\end{aligned}$$ Namely the (generalized) fusion ring is equivalent to the group ring ${\bf C}^{[\Gamma]}$ [@group].
The example in the next section is a simple example of this general algebra. The orbifold group $\Gamma$ is ${\bf Z}_2$ and we have only two Ishibashi states in untwisted and twisted sector. We write them as $|+{\rangle\!\rangle}$ and $|-{\rangle\!\rangle}$. The above algebra (\[eq:orb\_alg\]) is simply, $$\begin{aligned}
&&e_{\pm}\star e_{\pm}=e_{+}\,,\quad
e_{\pm}\star e_{\mp}=e_{-}\,,\\
&&e_{+}:=2|+\rangle\!\rangle\,,\quad
e_{-}:=2\sqrt{\sigma(e,g)}\,|-\rangle\!\rangle\,,
~~{\bf Z}_2=\{e,g\}\,,\end{aligned}$$ (using $\sigma(e,e)=1$) and its idempotents $P_{\pm}$ are easily obtained: $$\begin{aligned}
P_{\pm}={1\over 2}(e_+\pm e_-)
=|+\rangle\!\rangle
\pm \sqrt{\sigma(e,g)}\,|-\rangle\!\rangle\,,
\label{eq:orb_Cardy}\end{aligned}$$ which is the same as the Cardy states (\[eq:Cs\]) up to overall factor.
Explicit computation: toroidal and $Z_2$ orbifold compactifications
===================================================================
As nontrivial examples of general arguments in the previous section, we calculate the $\star$ product between Ishibashi states on torus and $Z_2$ orbifold. We use explicit oscillator representations of three string vertices which were formulated in [@HIKKO_torus] and [@Itoh_Kunitomo], respectively. These simple examples contain nontrivial ingredients such as winding modes, twisted sector, cocycle factor, etc. which make the explicit computation more interesting compared to ${\bf R}^d$ case in [@kmw1].
We use ${\bf R}^d\times T^D$ and ${\bf R}^d\times T^D/{\bf Z}_2$ as a background spacetime and consider the HIKKO $\star$ product on them. For the torus $T^D$, we identify its coordinates as $X^i\sim X^i+2\pi\sqrt{\alpha'}~(i=1,\cdots,D)$ and introduce constant background metric $G_{ij}$ and antisymmetric tensor $B_{ij}$.[^4] In the case of $Z_2$ orbifold $T^D/{\bf Z}_2$, the action of ${\bf Z}_2$ is defined by $X^i\rightarrow
-X^i~(i=1,\cdots,D)$. The ghost sector and ${\bf R}^d$ sector of the star product are the same as the original HIKKO’s construction [@HIKKO2]. We will compute the star product of string fields of the form $|\Phi(\alpha)\rangle=|B_D\rangle\otimes |\Phi_B(x^{\perp},\alpha)
\rangle$, where $|B_D\rangle$ is boundary states in $D$-dimensional sector: $T^D$ or $T^D/{\bf Z}_2$ and $|\Phi_B(x^{\perp},\alpha)\rangle=c_0^-b_0^+\,|B\rangle_{\rm
matter}\otimes|B\rangle_{\rm ghost}\otimes|\alpha\rangle$ represents a boundary state for D-brane at $x^{\perp}(\in {\bf R}^{d-p-1})$ including ghost and $\alpha$-sector. For the ${\bf R}^d$ sector, conventional boundary states for D$p$-brane were proved to be idempotent in [@kmw1; @kmw2]: $$\begin{aligned}
&&|\Phi_B(x^{\perp},\alpha_1)\rangle\star|\Phi_B(y^{\perp},\alpha_2)\rangle
\nonumber\\
&&~=~\delta^{d-p-1}(x^{\perp}-y^{\perp})\,\mu^2\,
{\det}^{-{d-2\over2}}(1-(\tilde{N}^{33})^2)\,
c_0^+|\Phi_B(x^{\perp},\alpha_1+\alpha_2)\rangle,~~~~
\label{eq:starRd}\\
&& \mu=e^{-\tau_0\sum_{r=1}^3\alpha_r^{-1}},~~~
\tau_0=\sum_{r=1}^3\alpha_r\log|\alpha_r|,~~~
(\alpha_3\equiv -\alpha_2-\alpha_1)\,.\end{aligned}$$ In this section, we will focus on the matter $T^D$ or $T^D/{\bf Z}_2$ sector and prove a similar relation for Cardy states on those backgrounds.
By toroidal compactification, winding mode is introduced in addition to momentum; the zero mode sector $|p\rangle$ changes to $|p,w\rangle$ with $p_i,w^i\in {\bf Z}$. Due to this mode, the boundary states and the 3-string vertex should be modified. The definition of the boundary state will be given in (\[eq:Ishi\_T\]). The 3-string vertex should be modified to include “cocycle factor” such as $e^{-i\pi(p_3w_2-p_1w_1)}$. (See, Appendix \[sec:starproduct\] for detail.) It is necessary to guarantee “Jacobi identity” with respect to closed string fields : $$\begin{aligned}
(\Phi_1\star\Phi_2)\star\Phi_3
+(-1)^{|\Phi_1|(|\Phi_2|+|\Phi_3|)}(\Phi_2\star \Phi_3)\star\Phi_1
+(-1)^{|\Phi_3|(\Phi_1|+|\phi_2|)}(\Phi_3\star\Phi_1)\star\Phi_2&=&0\,,~~\end{aligned}$$ which plays an important role to prove gauge invariance of the action of closed string field theory [@HIKKO_torus]. It can be also derived by careful treatment of the connection condition of light-cone type in [@Maeno_Takano]. When the boundary state has non-vanishing winding number, this cocycle factor becomes relevant.
$T^D/{\bf Z}_2$ is one of the simplest examples of orbifold background on which we gave a general argument in [@km1] and previous subsection (§\[sec:km1\]). Cardy state $|a_{\pm} \rangle$ (\[eq:Cs\], \[eq:orb\_Cardy\]), which represents fractional D-brane, is given by: $$\begin{aligned}
|a_{\pm} \rangle={1\over \sqrt{2}}\left(|\iota\rangle\!\rangle_u
\pm 2^{D\over 4}|\iota\rangle\!\rangle_t\right),\end{aligned}$$ where $|\iota\rangle\!\rangle_u$ or $|\iota\rangle\!\rangle_t$ is a linear combination of Ishibashi states in the untwisted or twisted sector, respectively. The ratio of coefficients $2^{D\over 4}$ comes from the factor $\sqrt{\sigma(e,g)}$ for $T^D/{\bf Z}_2$. We will demonstrate that string fields given in (\[eq:fractional\_reg\]) which are of the above form satisfy idempotency relations (\[eq:idem1\], \[eq:idem2\]). It provides a consistency check for the previous general arguments. The oscillator computation, however, has a limitation in determining coefficients of Ishibashi state. They are given by determinants of infinite rank Neumann matrices and are divergent in general. As a regularization, we slightly shift the interaction time which is specified by overlapping of three strings as we discussed in §\[sec:compC\]. We reduce the ratio of determinants to the degenerating limit of the ratio of 1-loop amplitudes in the sense of (\[eq:ct\_derive\]).
We will also comment on compatibility of idempotency relations on $T^D$ and $T^D/{\bf Z}_2$ with T-duality transformation in string field theory which was investigated in [@KZ] for $T^D$.
Star product between Ishibashi states \[sec:Ishi\_star\]
---------------------------------------------------------
In this subsection, we first introduce Ishibashi states $|\iota\rangle\!\rangle$ for the backgrounds $T^D$ and $T^D/{\bf Z}_2$ and then compute the star product between them. In Appendix \[sec:starproduct\], we give some definitions and our convention of free oscillators.
#### Ishibashi states
The Ishibashi states $|\iota\rangle\!\rangle$ for the torus $T^D$ are obtained by solving $(\alpha^i_{n}+{\cal
O}^i_{~j}\tilde{\alpha}_{-n}^j)|\iota\rangle\!\rangle=0$. ${\cal O}^i_{~j}$ is an orthogonal matrix in the sense ${\cal O}^TG{\cal O}=G$. Explicitly it is written as $$\begin{aligned}
\label{eq:Ishi_T}
&&|\iota({\cal O},p,w)\rangle\!\rangle=
e^{-\sum_{n=1}^{\infty}{1\over n}
\alpha_{-n}^iG_{ij}{\cal O}^j_{~k}\tilde{\alpha}^k_{-n}}|p,w\rangle\,,
~~p_i=-2\pi\alpha'F_{ij}w^j\,,\end{aligned}$$ with labels of momentum $p_i$ and winding number $w^j$ . The antisymmetric matrix $F_{ij}$ is given by ${\cal O}=(E^T-2\pi\alpha'F)^{-1}(E+2\pi\alpha'F)$ (where $E_{ij}:=G_{ij}+2\pi\alpha'B_{ij}$). It must be quantized in order to keep $p_i$ and $w^j$: integers with a relation $(1+{\cal O}^{T-1})p-(E-{\cal O}^{T-1}E^T)w=0$, which corresponds to $(\alpha_0^i+{\cal O}^i_{~j}\tilde{\alpha}_0^j)|\iota\rangle\!\rangle=0$. In particular, for Dirichlet type boundary condition, we should set $w^i=0$ since ${\cal O}=-1$.
For $T^D/{\bf Z}_2$, there are Ishibashi states in untwisted and twisted sectors. For the untwisted sector, they can be obtained by multiplying ${\bf Z}_2$-projection to the ones for the torus: $$\begin{aligned}
\label{eq:Ishi_u}
{\cal P}_u^{Z_2}|\iota({\cal O},p,w)\rangle\!\rangle_u
={1\over 2}(|\iota({\cal O},p,w)\rangle\!\rangle_u+
|\iota({\cal O},-p,-w)\rangle\!\rangle_u)\,,\end{aligned}$$ where we add a subscript $u$ to make a distinction from the Ishibashi states for the torus. For the twisted sector, we have Ishibashi states of the form: $$\begin{aligned}
\label{eq:Ishi_tw}
|\iota({\cal O},n^f)\rangle\!\rangle_t&=&
e^{-\sum_{r=1/2}^{\infty}{1\over r}
\alpha_{-r}^iG_{ij}{\cal O}^j_{~k}\tilde{\alpha}^k_{-r}}|n^f\rangle\,.\end{aligned}$$ The label $(n^f)^i$ takes value $0$ or $1$ and specifies a fixed point. This state has ${\bf Z}_2$ invariance: ${\cal P}^{Z_2}_t|\iota({\cal O},n^f)\rangle\!\rangle_t=|\iota({\cal
O},n^f)\rangle\!\rangle_t$.
#### $\star$ product
For $T^D$ case, the $\star$ product of the states (\[eq:Ishi\_T\]) becomes: $$\begin{aligned}
\label{eq:Ishi_star_T}
&&|\iota({\cal O},p_1,w_1)\rangle\!\rangle_{\alpha_1}\star
|\iota({\cal O},p_2,w_2)\rangle\!\rangle_{\alpha_2}\nonumber\\
&&=~{\det}^{-{D\over 2}}(1-(\tilde{N}^{33})^2)
\,(-1)^{p_1 w_2}|\iota({\cal O},p_1+p_2,w_1+w_2)
\rangle\!\rangle_{\alpha_1+\alpha_2}\,,\end{aligned}$$ where we have assigned $\alpha_r$ for each string (we consider the case of $\alpha_1\alpha_2>0$ here and following) and omitted the ghost and the matter ${\bf R}^d$ sector. Differences from ${\bf R}^d$ case [@kmw1] are limited to the existence of winding mode and the cocycle factor and the proof is similar; we use eqs. (\[eq:Ruu\]) and (\[eq:Vuuu\]) without ${\cal P}^{Z_2}_{u1}{\cal P}^{Z_2}_{u2}{\cal
P}^{Z_2}_{u3}$. The cocycle factor appeared as an extra sign factor $(-1)^{p_1 w_2}
=(-1)^{w_1 (2\pi\alpha'F) w_2}$. We note that this factor is irrelevant for the Dirichlet type boundary state since we need to set $w=0$.
For $T^D/{\bf Z}_2$, we have to compute three combinations of Ishibashi states: $({\rm untwisted})\star({\rm untwisted})$, $({\rm twisted})\star({\rm twisted})$ and $({\rm untwisted})\star({\rm twisted})$. The first one can be obtained by ${\bf Z}_2$-projection of the torus case (\[eq:Ishi\_star\_T\]): $$\begin{aligned}
\label{eq:u_star_u}
&&{\cal P}^{Z_2}_u|\iota({\cal O},p_1,w_1)\rangle\!\rangle_{u,\alpha_1}\star
{\cal P}^{Z_2}_u|\iota({\cal O},p_2,w_2)\rangle\!\rangle_{u,\alpha_2}\nonumber\\
&&={\det}^{-{D\over 2}}(1-(\tilde{N}^{33})^2)
\,{(-1)^{p_1 w_2}\over 2}\,{\cal P}^{Z_2}_u\\
&&~~~\times
\left[
|\iota({\cal O},p_1+p_2,w_1+w_2)
\rangle\!\rangle_{u,\alpha_1+\alpha_2}
+|\iota({\cal O},p_1-p_2,w_1-w_2)
\rangle\!\rangle_{u,\alpha_1+\alpha_2}
\right].\nonumber\end{aligned}$$ The star product for two Ishibashi states (\[eq:Ishi\_tw\]) in the twisted sector can be computed by the vertex operators (\[eq:Rtt\], \[eq:Vutt\]) (with appropriate permutation such that string $3$ is in the untwisted sector). Using the identities among Neumann coefficients given in (\[eq:Yoneya\]), a direct computation which is similar to that in [@kmw1] yields $$\begin{aligned}
\label{eq:tt_pre}
&&|\iota({\cal O},n_1^f)\rangle\!\rangle_{t,\alpha_1}
\star |\iota({\cal O},n_2^f)\rangle\!\rangle_{t,\alpha_2}\\
&&=~e^{{D\over 8}\tau_0(\alpha_1^{-1}+\alpha_2^{-1})}
{\det}^{-{D\over 2}}(1-(\tilde{T}^{3_u3_u})^2)\wp{\cal P}^{Z_2}_u \sum_{p,w}
\gamma({\bf p};n_1^f,n_2^f)\,e^{\Delta E}
\,e^{-\sum_{n>0}{1\over n}\alpha_{-n}{\cal O}\tilde{\alpha}_{-n}}
|p,w\rangle_{\alpha_1+\alpha_2}\,,\nonumber\end{aligned}$$ where $$\begin{aligned}
\Delta E &=& -\sum_{n>0}n^{-{1\over 2}}
(\alpha_{-n}+\tilde{\alpha}_{-n}{\cal O}^T)\sum_{r,s=1,2}\,\sum_{m_r,l_s>0}
\tilde{T}^{3_ur}_{nm_r}
[(\tilde{T}^{\,\cdot\, 3_u}\tilde{T}^{3_u\cdot})^{-1}]^{rs}_{m_rl_s}
\tilde{T}^{s 3_u}_{l_s0}{\bf p}_+\nonumber\\
&&+{1\over 4}\left(
T^{3_u3_u}_{00}-\sum_{r,s=1,2}\,\sum_{n_r,m_s>0}
\tilde{T}^{3_ur}_{0n_r}[(1+\tilde{T})^{-1}]^{rs}_{n_rm_s}
\tilde{T}^{s 3_u}_{m_s0}
\right){\bf p}_+G^{-1}{\bf p}_+\,,\\
({\bf p}_+)_i&=&
{1\over \sqrt{2}}[(1+{\cal O}^{T-1})p-(E-{\cal O}^{T-1}E^T)w]_i\,.
$$ The above peculiar exponent $\Delta E$ can be ignored because the coefficient of positive definite factor ${\bf p}_+G^{-1}{\bf p}_+$ can be evaluated by using various formulae in Appendix \[sec:TNeumann\] as $$\begin{aligned}
T^{3_u3_u}_{00}-\sum_{r,s=1,2}\,\sum_{n_r,m_s>0}
\tilde{T}^{3_ur}_{0n_r}[(1+\tilde{T})^{-1}]^{rs}_{n_rm_s}
\tilde{T}^{s 3_u}_{m_s0}
&=&-\sum_{n=1}^{\infty}{2\cos^2\left({\alpha_1\over \alpha_3}n\pi\right)
\over n}\,.~~~~~~~~
$$ Since it gives $-\infty$, the ${\bf p}_+\ne 0$ terms in the summation in (\[eq:tt\_pre\]) is suppressed. The constraint ${\bf p}_+=0$ implies $p_i=-2\pi\alpha'F_{ij}w^j$ in (\[eq:Ishi\_T\]), which is consistent with $(L_n-\tilde{L}_{-n})\left(|\iota({\cal O},n_1^f)\rangle\!\rangle_{t,\alpha_1}
\star |\iota({\cal O},n_2^f)\rangle\!\rangle_{t,\alpha_2}\right)=0
$. This is an example of our general claim in Ref. [@kmw2] that the star product between the conformal invariant states is again conformal invariant; $(L_n-\tilde{L}_{-n})|B_i\rangle=0,i=1,2,\forall n\in {\bf Z}$ $\rightarrow$ $(L_n-\tilde{L}_{-n})(|B_1\rangle\star |B_2\rangle)=0$. The final form of the $\star$ product becomes, $$\begin{aligned}
\label{eq:t_star_t}
&&|\iota({\cal O},n_1^f)\rangle\!\rangle_{t,\alpha_1}
\star |\iota({\cal O},n_2^f)\rangle\!\rangle_{t,\alpha_2}\\
&&=~e^{{D\over 8}\tau_0(\alpha_1^{-1}+\alpha_2^{-1})}
{\det}^{-{D\over 2}}(1-(\tilde{T}^{3_u3_u})^2)\sum_{p,w,{\bf p}_+=0}(-1)^{p\,n_2^f}
\sum_m\delta^D_{n_2^f-n_1^f+w+2m,0}
|\iota({\cal O},p,w)\rangle\!\rangle_{u,\alpha_1+\alpha_2}\,.
\nonumber\end{aligned}$$
Finally the $\star$ product between the Ishibashi states in untwisted and twisted sectors can be computed similarly by using the formulae in (\[eq:Yoneya\]): $$\begin{aligned}
&& {\cal P}_u^{Z_2}|\iota({\cal O},p_1,w_1)\rangle\!\rangle_{u,\alpha_1}
\star |\iota({\cal O},n_2^f)\rangle\!\rangle_{t,\alpha_2}
\nonumber\\
&&=~e^{{D\over 8}\tau_0(\alpha_2^{-1}-(\alpha_1+\alpha_2)^{-1})}
\,{\det}^{-{D\over 2}}(1-(\tilde{T}^{3_t3_t})^2)
\,(-1)^{p_1n_2^f}
|\iota({\cal O},[n_2^f-w_1]_{{\rm mod}\, 2})
\rangle\!\rangle_{t,\alpha_1+\alpha_2}\,.
\label{eq:u_star_t}\end{aligned}$$ We have similar formula for \[twisted(\[eq:Ishi\_tw\])\] $\star$ \[untwisted(\[eq:Ishi\_u\])\] by appropriate replacement in the above.
We have confirmed that Ishibashi states on ${\bf Z}_2$ orbifold (\[eq:Ishi\_u\]) and (\[eq:Ishi\_tw\]) (resp., on torus (\[eq:Ishi\_T\])) form a closed algebra with respect to the $\star$ product as eqs. (\[eq:u\_star\_u\]),(\[eq:t\_star\_t\]) and (\[eq:u\_star\_t\]) (resp., eq. (\[eq:Ishi\_star\_T\])).
Cardy states as idempotents
---------------------------
We proceed to compare the Cardy state and idempotent of $\star$ product algebra (fusion ring) for Ishibashi state that we have just computed. We note that the algebra for the Dirichlet type boundary states are simpler since there is no winding number and consequently the cocycle factor in the vertex operator vanishes. Because of this simplicity we divide our discussion into Dirichlet and Neumann type boundary states.
#### Dirichlet type
We start our consideration from Dirichlet type states, namely ${\cal O}^i_{~j}=-\delta^i_j$ for the torus. The Cardy state which describes the Dirichlet boundary condition is given by a Fourier transformation of Ishibashi states (\[eq:Ishi\_T\]) with respect to momentum $p_i$: $$\begin{aligned}
\label{eq:B(x)}
|B(x)\rangle&=&\left(\det(2G_{ij})\right)^{-{1\over 4}}
\sum_{p\in {\bf Z}^D}e^{-ix^ip_i}|\iota(-1,p,0)\rangle\!\rangle\,.\end{aligned}$$ One can check that it satisfies $[\alpha^{\prime-{1\over
2}}X^i(\sigma)-x^i]_{\rm mod\, 2\pi}|B(x)\rangle=0$. We have chosen its normalization by $$\begin{aligned}
\langle B(x)|q^{{1\over 2}(L_0+\tilde{L}_0-{D\over 12})}|B(x')\rangle
&=&\eta(\tau)^{-D}\left[{\det}^{-{1\over 2}}(2G_{ij})
\sum_{p\in {\bf Z}^D}e^{-i(x-x')p}q^{{1\over 4}pG^{-1}p}\right]
\nonumber\\
&=&
\eta(-1/\tau)^{-D}\sum_{m\in {\bf Z}^D}
e^{-{i\over 2\pi\tau}(x-x'+2\pi m)G
(x-x'+2\pi m)}\,,
\label{eq:normD}\end{aligned}$$ where $q=e^{2\pi i\tau}$ and $\eta(\tau)=q^{1\over 24}\prod_{n=1}^{\infty}(1-q^n)$. The last representation implies that it gives 1-loop amplitude of open string whose boundaries are on D-branes at $x$ and $x'$ on the torus $T^D$.
On the other hand, from (\[eq:Ishi\_star\_T\]), the star product between them becomes $$\begin{aligned}
&&|B(x)\rangle_{\alpha_1}\star |B(x')\rangle_{\alpha_2}\\
&&=~\delta^D([x-x^{\prime}])(2\pi)^D\left(\det(2G_{ij})\right)^{-{1\over 4}}
{\det}^{-{D\over 2}}(1-(\tilde{N}^{33})^2)\,
|B(x)\rangle_{\alpha_1+\alpha_2}\,,\nonumber\end{aligned}$$ where $\delta^D([x-x^{\prime}]):=\sum_{m\in{\bf Z}^D}\delta^D(x-x'+2\pi
m)=(2\pi)^{-D}\sum_{p\in{\bf Z}^D}e^{-i(x-x')p}$. This is the idempotency relation in [@kmw1] for the toroidal compactification. For $T^D/{\bf Z}_2$, the boundary state with ${\bf Z}_2$ projection, ${{\cal P}^{Z_2}_u}|B(x)\rangle_{u}={1\over 2}(|B(x)\rangle_{u}
+|B(-x)\rangle_{u})$ gives idempotents in the sense: $$\begin{aligned}
&&{{\cal P}^{Z_2}_u}|B(x)\rangle_{u,\alpha_1}\star {{\cal P}^{Z_2}_u}
|B(x')\rangle_{u,\alpha_2}\\
&&=\,{1\over 2}\,(\delta^D([x-x^{\prime}])+\delta^D([x+x^{\prime}]))
(2\pi)^D\left(\det(2G_{ij})\right)^{-{1\over 4}}
{\det}^{-{D\over 2}}(1-(\tilde{N}^{33})^2)\,
{{\cal P}^{Z_2}_u}|B(x)\rangle_{u,\alpha_1+\alpha_2}\,.\nonumber\end{aligned}$$ It is clear that the combination of delta functions is well-defined on $T^D/{\bf Z}_2$.
At the fixed point, there are fractional D-branes. To see them, we consider a restriction of $x$ to a fixed point $\pi n^f$, $$\begin{aligned}
\label{eq:Bnf}
|B_{n^f}\rangle_u
&=&\left(\det(2G_{ij})\right)^{-{1\over 4}}
\sum_{p\in {\bf Z}^D}(-1)^{p\,n^f}|\iota(-1,p,0)\rangle\!\rangle_u,\end{aligned}$$ it is ${\bf Z}_2$ invariant by itself ${\cal P}^{Z_2}_u|B_{n^f}\rangle_u=|B_{n^f}\rangle_u$ and is idempotent: $$\begin{aligned}
&&|B_{n^f_1}\rangle_{u,\alpha_1}\star|B_{n^f_2}\rangle_{u,\alpha_2}\nonumber
\\
&&~=(2\pi\delta(0))^D
\delta^D_{n_1^f,n_2^f}\left(\det(2G_{ij})\right)^{-{1\over 4}}
\,{\det}^{-{D\over 2}}(1-(\tilde{N}^{33})^2)\,
|B_{n^f_1}\rangle_{u,\alpha_1+\alpha_2}.~~~~~~~
\label{eq:uu_D}\end{aligned}$$ For the twisted sector, we can derive from eqs. (\[eq:t\_star\_t\]), (\[eq:u\_star\_t\]) and (\[eq:Bnf\]), $$\begin{aligned}
&&|B_{n_1^f}\rangle_{t,\alpha_1}\star|B_{n_2^f}\rangle_{t,\alpha_2}
\nonumber\\
&&=\delta^D_{n_1^f,n_2^f}\left(\det(2G_{ij})\right)^{1\over 4}
e^{{D\over 8}\tau_0(\alpha_1^{-1}+\alpha_2^{-1})}
{\det}^{-{D\over 2}}(1-(\tilde{T}^{3_u3_u})^2)\,
|B_{n^f_1}\rangle_{u,\alpha_1+\alpha_2}\,,\label{eq:tt_D}\\
&&|B_{n_1^f}\rangle_{u,\alpha_1}\star|B_{n_2^f}\rangle_{t,\alpha_2}
\nonumber\\
&&=\delta^D_{n_1^f,n_2^f}\left(\det(2G_{ij})\right)^{-{1\over 4}}
(2\pi\delta(0))^D\,
e^{{D\over 8}\tau_0(\alpha_2^{-1}-(\alpha_1+\alpha_2)^{-1})}
{\det}^{-{D\over 2}}(1-(\tilde{T}^{3_t3_t})^2)\,
|B_{n^f_1}\rangle_{t,\alpha_1+\alpha_2}\,,\label{eq:ut_D}~~~~~~~\end{aligned}$$ where $|B_{n^f}\rangle_t:=
|\iota(-1,n^f)\rangle\!\rangle_t$. These eqs. (\[eq:uu\_D\]),(\[eq:tt\_D\]) and (\[eq:ut\_D\]) show that the Dirichlet boundary states at fixed points, $|B_{n^f}\rangle_u$ and $|B_{n^f}\rangle_t$, form a closed algebra with respect to the $\star$ product. It can be diagonalized by taking a linear combination of the untwisted and twisted sectors: $$\begin{aligned}
\label{eq:fractional_D}
&&|\Phi_B(n^f,x^{\perp},\alpha)\rangle_{\pm}\\
&&={1\over 2}(2\pi\delta(0))^{-D}\left(
{\det}^{1\over 4}(2G_{ij})
|B_{n^f}\rangle_u\pm c_t(2\pi\delta(0))^{D\over 2}
|B_{n^f}\rangle_t\right)\otimes|\Phi_B(x^{\perp},\alpha)\rangle,
\nonumber\end{aligned}$$ where we have included a string field $|\Phi_B(x^{\perp},\alpha)\rangle$, which is a contribution from the other part of matter sector ${\bf R}^d$ and ghost sector. It is essentially a boundary state for D$p$-brane. The coefficient of the boundary states in the twisted sector is given by a ratio of the determinants of Neumann matrices: $$\begin{aligned}
c_t&:=&\sqrt{{\cal C}\over {\cal C}'}
=\left(
e^{-{\tau_0\over 4}(\alpha_1^{-1}+\alpha_2^{-1})}\,{
\det(1-(\tilde{T}^{3_u 3_u})^2)
\over \det(1-(\tilde{N}^{3 3})^2)}
\right)^{D\over 4},
\label{eq:ct}
\\
{\cal C}&=&\mu^2\,{\det}^{-{d+D-2\over 2}}(1-(\tilde{N}^{33})^2)\,,~~~~
\mu=e^{-\tau_0(\alpha_1^{-1}+\alpha_2^{-1}-(\alpha_1+\alpha_2)^{-1})},
~~~\\
{\cal C}'&=&\mu^2\,e^{{D\over 8}\tau_0(\alpha_1^{-1}+\alpha_2^{-1})}
{\det}^{-{D\over 2}}(1-(\tilde{T}^{3_u 3_u})^2)\,
{\det}^{-{d-2\over 2}}(1-(\tilde{N}^{33})^2)\,.\end{aligned}$$ They satisfy idempotency relations of the following form: $$\begin{aligned}
&&|\Phi_B(n_1^f,x^{\perp},\alpha_1)\rangle_{\pm}
\star |\Phi_B(n_2^f,y^{\perp},\alpha_2)\rangle_{\pm}\nonumber\\
&&~~~=~\delta^D_{n_1^f,\,n_2^f}\,\delta^{d-p-1}(x^{\perp}-y^{\perp})\,
{\cal C}\,c_0^+|\Phi_B(n_1^f,x^{\perp},\alpha_1+\alpha_2)\rangle_{\pm}\,,
\label{eq:idem1}
\\
\label{eq:idem2}
&&|\Phi_B(n_1^f,x^{\perp},\alpha_1)\rangle_{\pm}
\star |\Phi_B(n_2^f,y^{\perp},\alpha_2)\rangle_{\mp}=0\,,\end{aligned}$$ where ${\cal C}$ was computed in [@kmw2] and is proportional to $K^3\alpha_1\alpha_2(\alpha_1+\alpha_2)$ for $d+D=26$ with cutoff parameter $K$. In the above computation, we have used the relation of determinants of Neumann matrices: $$\begin{aligned}
\label{eq:CG_T}
{\det}^{-{1\over 2}}(1-(\tilde{N}^{33})^2)
=e^{{1\over 8}\tau_0(\alpha_2^{-1}-(\alpha_1+\alpha_2)^{-1})}
{\det}^{-{1\over 2}}(1-(\tilde{T}^{3_t 3_t})^2)\,,\end{aligned}$$ which can be proved analytically by using Cremmer-Gervais identity as in Ref. [@kmw2]. Outline of the proof is given in Appendix \[sec:CG\]. It can be also checked numerically by truncating the size of Neumann matrices to $L\times L$. ($L\sim K$) As for the coefficient $c_t$ (\[eq:ct\]) in front of the twisted sector, it can be evaluated by another regularization as §\[sec:compC\] (See, Appendix \[sec:1-loop\] for detail.) The result is given in (\[eq:ct\_derive\]): $$\begin{aligned}
\label{eq:ct_value}
c_t(2\pi\delta(0))^{D\over 2}
=2^{D\over 4}\left(\det(2G)\right)^{1\over 4}
=\sqrt{\sigma(e,g)}\left(\det(2G)\right)^{1\over 4}\,,\end{aligned}$$ where $\sigma(e,g)=2^{D\over 2}$ is the Modular transformation matrix defined in (\[eq:Modular\]) and is given in [@Billo]. This implies that the idempotents (\[eq:fractional\_D\]) is proportional to the Cardy state for the fractional D-branes, $$\begin{aligned}
\label{eq:fractional_reg}
|\Phi_B(n^f,x^{\perp},\alpha)\rangle_{\pm}
={1\over 2}\,{{\det}^{1\over 4}(2G)\over (2\pi\delta(0))^D}
\left(|B_{n^f}\rangle_u\pm 2^{D\over 4}
|B_{n^f}\rangle_t\right)\otimes|\Phi_B(x^{\perp},\alpha)\rangle,
~~~~~~\end{aligned}$$ after a proper regularization.
#### Neumann type
We call the boundary states with ${\cal O}^i_{~j}\ne -\delta^i_j$ as Neumann type while they may have mixed boundary condition in general. As we wrote, the derivation of idempotent for such states is slightly more nontrivial because of the cocycle factor in the vertex. We start again from the toroidal compactification and consider a particular linear combination of Ishibashi states (\[eq:Ishi\_T\]) of the form: $$\begin{aligned}
\label{eq:Bq}
|B(q),F\rangle:={\det}^{-{1\over 4}}(2G_O^{-1})
\sum_{w}e^{-iqw+i\pi wF_uw}
|\iota({\cal O},-2\pi\alpha'Fw,w)\rangle\!\rangle\,,\end{aligned}$$ where we denote $(F_u)_{ij}= 2\pi\alpha'F_{ij}$ for $i<j$ and $(F_u)_{ij}=0$ for $i\ge j$. As we explained, $2 \pi \alpha' F_{ij}$ should be quantized for the consistency with the momentum quantization. We have chosen the normalization factor ${\det}^{-{1\over 4}}(2G_O^{-1})$ by $$\begin{aligned}
\langle B(q'),F|\,e^{\pi i\tau\left(L_0+\tilde{L}_0-{D\over 12}\right)}
|B(q),F\rangle&=&{\det}^{-{1\over 2}}(2G_O^{-1})\,\eta(\tau)^{-D}\sum_{w^i}
e^{-i(q-q')_iw^i}e^{{\pi i\tau\over 2}w^iG_{Oij}w^j}\nonumber\\
&=&\eta(-1/\tau)^{-D}\sum_{m}
e^{-{i\over 2\pi \tau}(q-q'+2\pi m)_iG_O^{ij}(q-q'+2\pi m)_j}\,,
\label{eq:normN}\end{aligned}$$ as (\[eq:normD\]). Here $q_i\equiv q_i+2\pi$ corresponds to Wilson line on the D-brane and ${G_O}_{ij}:=G_{ij}-(E+2\pi\alpha' F)_{ik}G^{kl}(E^T-2\pi\alpha' F)_{lj}
$ is the open string metric.
We compute the $\star$ product of (\[eq:Bq\]) using (\[eq:Ishi\_star\_T\]): $$\begin{aligned}
&&|B(q_1),F\rangle_{\alpha_1}\star|B(q_2),F\rangle_{\alpha_2}
\\
&&~=\delta^D([q_1-q_2])\,
(2\pi)^D{\det}^{-{1\over 4}}(2G_O^{-1})
\,{\det}^{-{D\over 2}}(1-(\tilde{N}^{33})^2)\,
|B(q_1),F\rangle_{\alpha_1+\alpha_2},\nonumber\end{aligned}$$ which is the idempotency relation for $T^D$. We note that due the phase factor $e^{i\pi wF_u w}$ in (\[eq:Bq\]), Cardy state is not the Fourier transform of the Ishibashi state. It is necessary to cancel the cocycle factor in the 3-string vertex (\[eq:Vuuu\]). It is also necessary to keep T-duality symmetry in closed string field theory on the torus $T^D$, (see, eq. (\[eq:upsilon\]) in particular).
For the orbifold, we can check that ${\cal P}^{Z_2}_u|B(q),F\rangle_{u}={1\over 2}(
|B(q),F\rangle_{u}+|B(-q),F\rangle_{u})$ is idempotent in the untwisted sector on $T^D/{\bf Z}_2$: $$\begin{aligned}
&&{{\cal P}^{Z_2}_u}|B(q),F\rangle_{u,\alpha_1}\star {{\cal P}^{Z_2}_u}
|B(q'),F\rangle_{u,\alpha_2}\\
&&~~=~{1\over 2}(\delta^D([q-q^{\prime}])+\delta^D([q+q^{\prime}]))
(2\pi)^D{\det}^{-{1\over 4}}(2G_O^{-1}){\det}^{-{D\over 2}}(1-(\tilde{N}^{33})^2)\,
{{\cal P}^{Z_2}_u}|B(q),F\rangle_{u,\alpha_1+\alpha_2}\,.\nonumber\end{aligned}$$
Mixing with the twisted sector occurs when the Wilson line takes special values, $q_i=\pi m^f_i$ ($m_i^f=0,1$) for the untwisted sector: $$\begin{aligned}
\label{eq:Bmf}
&&|B_{m^f},F\rangle_u
={\det}^{-{1\over 4}}(2G_O^{-1})
\sum_{w}(-1)^{m^f w+wF_uw}
|\iota({\cal O},-2\pi\alpha'Fw,w)\rangle\!\rangle_u\,.~~~~~~~~\end{aligned}$$ These states are by themselves ${\bf Z}_2$ invariant: ${\cal P}^{Z_2}_u|B_{m^f},F\rangle_u=|B_{m^f},F\rangle_u$. The star product between them is, $$\begin{aligned}
&&|B_{m^f_1},F\rangle_{u,\alpha_1}\star |B_{m^f_2},F\rangle_{u,\alpha_2}
\label{eq:uu_N}\\
&&~=\delta^D_{m^f_1,m^f_2}{\det}^{-{1\over 4}}(2G_O^{-1})(2\pi\delta(0))^D
{\det}^{-{D\over 2}}(1-(\tilde{N}^{33})^2)\,
|B_{m^f_1},F\rangle_{u,\alpha_1+\alpha_2}\,.\nonumber\end{aligned}$$ In the twisted sector, we consider a particular linear combination of Ishibashi states (\[eq:Ishi\_tw\]) such as $$\begin{aligned}
\label{eq:Bmft}
|B_{m^f},F\rangle_t:=2^{-{D\over 2}}\sum_{n^f_i=0,1}
(-1)^{m^fn^f+n^fF_un^f}
|\iota({\cal O},n^f)\rangle\!\rangle_t\,,\end{aligned}$$ which is a generalization of the twisted Neumann boundary state in [@oshi]. Here, we have also multiplied the phase factor $(-1)^{n^fF_un^f}$ as in the untwisted sector (\[eq:Bmf\]) for the idempotency. We can derive the $\star$ product formulae $$\begin{aligned}
&&|B_{m_1^f},F\rangle_{t,\alpha_1}
\star |B_{m^f_2},F\rangle_{t,\alpha_2}\\
&&=\delta^D_{m^f_1,\,m^f_2}
{\det}^{1\over 4}(2G_O^{-1})\,
e^{{D\over 8}\tau_0(\alpha_1^{-1}+\alpha_2^{-1})}
{\det}^{-{D\over 2}}(1-(\tilde{T}^{3_u3_u})^2)
|B_{m^f_1},F\rangle_{u,\alpha_1+\alpha_2},\nonumber\\
&&|B_{m_1^f},F\rangle_{u,\alpha_1}
\star |B_{m_2^f},F\rangle_{t,\alpha_2}\\
&&=\delta_{m_1^f,\,m_2^f}{\det}^{-{1\over 4}}(2G_O^{-1})
(2\pi\delta(0))^De^{{D\over 8}\tau_0(\alpha_2^{-1}-(\alpha_1+\alpha_2)^{-1})}
{\det}^{-{D\over 2}}(1-(\tilde{T}^{3_t3_t})^2)\,
|B_{m_1^f},F\rangle_{t,\alpha_1+\alpha_2}
,\nonumber\end{aligned}$$ from eqs. (\[eq:t\_star\_t\]),(\[eq:u\_star\_t\]),(\[eq:Bmf\]) and (\[eq:Bmft\]). Using the above formulae, noting eq. (\[eq:CG\_T\]), we obtain idempotents which include the twisted sector: $$\begin{aligned}
\label{eq:fractional_DN}
&&|\Phi_B(m^f,F,x^{\perp},\alpha)\rangle_{\pm}={1\over 2}\,{{\det}^{1\over 4}(2G_O^{-1})\over (2\pi\delta(0))^D}
\left(|B_{m^f},F\rangle_u\pm 2^{D\over 4}\,
|B_{m^f},F\rangle_t\right)\otimes|\Phi_B(x^{\perp},\alpha)\rangle\,.~~~\end{aligned}$$ Here we again include the extra matter fields on ${\bf R}^d$ and ghost sector: $|\Phi_B(x^{\perp},\alpha)\rangle$. We evaluate the ratio of determinants $c_t$ (\[eq:ct\]) using the regularization given by (\[eq:BBuN\]) instead of (\[eq:BBu\]) because we are treating Neumann type boundary states. Their star product becomes idempotent as expected, $$\begin{aligned}
&&|\Phi_B(m_1^f,F,x^{\perp},\alpha_1)\rangle_{\pm}
\star |\Phi_B(m_2^f,F,y^{\perp},\alpha_2)\rangle_{\pm}\nonumber\\
&&~~~=~\delta^D_{m_1^f,\,m_2^f}\,\delta^{d-p-1}(x^{\perp}-y^{\perp})\,
{\cal C}\,c_0^+|\Phi_B(m_1^f,x^{\perp},\alpha_1+\alpha_2)\rangle_{\pm}\,,
\label{eq:idem1N}
\\
\label{eq:idem2N}
&&|\Phi_B(m_1^f,x^{\perp},\alpha_1)\rangle_{\pm}
\star |\Phi_B(m_2^f,y^{\perp},\alpha_2)\rangle_{\mp}=0\,.\end{aligned}$$
Comments on T-duality
---------------------
We have seen that the Dirichlet type idempotent and the Neumann type one are constructed in slightly different manner due to the cocycle factor. They are related, however, by T-duality transformation and we would like to see explicitly how the difference can be absorbed. In this subsection we follow the argument of [@KZ].
A key ingredient is the existence of the following operator $\mathcal{U}_g^\dagger$, $$\begin{aligned}
\label{eq:T-dualU}
{\cal U}_g^{\dagger}|A\star B\rangle_E
&=&|({\cal U}_g^{\dagger}A)\star({\cal U}_g^{\dagger}B)\rangle_{g(E)}\,.\end{aligned}$$ Here the subscripts of the ket: $E$ and $g(E)$ specify the constant background $E=G+2\pi \alpha' B$ and its T-duality transformation specified by $g\in O(D,D;{\bf Z})$: $$\begin{aligned}
&&g(E):=(aE+b)(cE+d)^{-1},\\
&&g=\left(
\begin{array}[tb]{cc}
a & b \\
c & d
\end{array}
\right),
~~~~g^T J g=g\,,~J:=\left(
\begin{array}[tb]{cc}
0&1_D \\
1_D&0
\end{array}
\right).\end{aligned}$$ The operator ${\cal U}_g$ is defined by [@KZ]: $$\begin{aligned}
&& {\cal U}_g=U_g\,\Upsilon(g,\hat{p},\hat{w})\,,~~
U_g^{\dagger}|p,w\rangle_E=|ap+bw,cp+dw\rangle_{g(E)},\\
&&U_g^{\dagger}\alpha_n(E) U_g=(d-cE^T)^{-1}\alpha_n(g(E))\,,\\
&&U_g^{\dagger}\tilde{\alpha}_n(E) U_g=(d+cE)^{-1}
\tilde{\alpha}_n(g(E))\,,~~~~~\\
&&\Upsilon(g,\hat{p},\hat{w})=\exp(i\pi(\hat{p}(dc^T)_u\hat{p}
+\hat{w}(ba^T)_u\hat{w}+\hat{w}bc^T\hat{p}))\,,
\label{eq:upsilon}\end{aligned}$$ where matrices in the exponent of $\Upsilon$ with subscript $u$ are defined by $(A_u)_{ij}= A_{ij}\,(i<j)$ and $(A_u)_{ij}=0\,(i\ge j)$. In particular, we consider a class of $O(D,D;{\bf Z})$-transformation of the form: $$\begin{aligned}
\label{eq:gDN}
g_{\rm DN}&=&\left(
\begin{array}[tb]{cc}
-2\pi\alpha'F& 1\\
1&0
\end{array}
\right)\,,~~~~~~(2\pi\alpha'F_{ij}=-2\pi\alpha'F_{ji}\in\mathbf{Z}).\end{aligned}$$ They give T-duality transformations between the idempotents $$\begin{aligned}
{\cal U}_{g_{\rm DN}}^{\dagger}|B(x^i)\rangle_{E}=
|B(q_i=x^i),F\rangle_{g_{\rm DN}(E)}\,,\end{aligned}$$ on the torus. Note that the original metric $G$ is mapped to the inverse open string metric $G^{-1}_O$ by the transformation $g_{\rm DN}$: $$\begin{aligned}
G=(E^{\prime T}-2\pi\alpha'F)^{-1}G'(E'+2\pi\alpha' F)^{-1}=G^{\prime-1}_O
\,,~~~~~E'=g_{\rm DN}(E)\,.\end{aligned}$$ Indeed, this is consistent with general property of the $\star$ product (\[eq:T-dualU\]).
We can extend such an analysis to $T^D/{\bf Z}_2$ case. We define a unitary operator ${\cal U}_{g_{\rm DN}}$ which represent the action of $g_{\rm DN}$ (\[eq:gDN\]) to the twisted sector: $$\begin{aligned}
{\cal U}_{g_{\rm DN}}^{\dagger}\alpha_r(E) {\cal U}_{g_{\rm DN}}&=&-E^{T-1}
\alpha_r(g_{\rm DN}(E)),\\
{\cal U}_{g_{\rm DN}}^{\dagger}\tilde{\alpha}_r(E) {\cal U}_{g_{\rm DN}}
&=&E^{-1}\tilde{\alpha}_r(g_{\rm DN}(E)),\end{aligned}$$ where $\alpha_r(E),~(r\in {\bf Z}+{1/2})$ is the oscillator on the background $E$. For the oscillator vacuum $|n^f\rangle_E$, we define $$\begin{aligned}
{\cal U}^{\dagger}_{g_{\rm DN}}|n^f\rangle_{E}
&=&2^{-{D\over 2}}\sum_{m^f_i=0,1}(-1)^{n^f_im^f_i+m^f_i(F_u)_{ij}m^f_j}
|n^f\rangle_{g_{\rm DN}(E)}\,.\end{aligned}$$ Then, with ${\bf Z}_2$ projection, we can prove $$\begin{aligned}
{\cal U}^{\dagger}_{g_{\rm DN}}|A\star B\rangle_E
=|( {\cal U}^{\dagger}_{g_{\rm DN}}A)
\star({\cal U}^{\dagger}_{g_{\rm DN}}B)\rangle_{g_{\rm DN}(E)},\end{aligned}$$ not only in the untwisted sector but also in the twisted sector by investigating reflectors (\[eq:Ruu\]),(\[eq:Rtt\]) and 3-string vertices (\[eq:Vuuu\]),(\[eq:Vutt\]). This implies that we obtain Neumann type idempotents (\[eq:fractional\_DN\]) from Dirichlet type (\[eq:fractional\_D\]) by $ {\cal U}^{\dagger}_{g_{\rm DN}}$: $$\begin{aligned}
{\cal U}^{\dagger}_{g_{\rm DN}}|\Phi_B(n^f_i,x^{\perp},\alpha)\rangle_{\pm,E}
=|\Phi_B(m^f_i=n^f_i,F,x^{\perp},\alpha)\rangle_{\pm,\,g_{\rm DN}(E)}\,.\end{aligned}$$
Deformation of the algebra by $B$ field
=======================================
In this section, we consider a deformation along the transverse directions by the introduction of $B$ field. In Seiberg-Witten limit, it induces noncommutativity to the ring of functions on these directions. Since our equation, $\Phi\star \Phi=\Phi$ formally resembles GMS soliton equation, it is curious how our star product is modified in such limit.
In particular, the algebra of Ishibashi state in transverse dimension was, $$|p{\rangle\!\rangle}\star | q{\rangle\!\rangle}= a( p,
q)|{p+q}{\rangle\!\rangle}\,,\quad
(a(p,q)=1)$$ when there is no $B$ field. In order to obtain a projector for this algebra, we perform a Fourier transformation $|x^\perp\rangle=\int dk \,e^{ikx^\perp}|k{\rangle\!\rangle}$, which combines Ishibashi states to Cardy state, and this is identical to the the boundary state for the transverse direction.
A naive guess is that the product becomes Moyal product, namely $a(p,q)$ becomes $\exp\left(-\frac{i}{2} p_i\theta^{ij} q_j\right)$. This can not, however, be the case since the closed string star product is commutative. We will see that in a [*specific*]{} setup which we are going to consider, the factor becomes $$a(p,q)=\frac{\sin(-\beta\lambda)}{-\beta\lambda}
\frac{\sin((1+\beta)\lambda)}{(1+\beta)\lambda}\,,
\quad \lambda=-{1\over 2}\,p_i\theta^{ij}q_j\,,$$ for HIKKO type star product in the Seiberg-Witten limit. If we expand in terms of $\lambda$, it is easy to see that this expression reduces to $1$ when $\theta\rightarrow 0$. It is commutative and non-associative which are the basic properties of closed string star product.
If we know the boundary state in the presence of $B$ field in the transverse dimensions, our computation would be straightforward since the definition of the star product itself remains the same. Actually, however, the boundary state which corresponds to GMS soliton is not known. Namely, the treatment of the massive particles is difficult. Such modes can be decoupled from zero-mode only when Seiberg-Witten limit is taken.
Therefore, we are going to take the following path to obtain the deformation of the algebra,
1. define an operator $V_\theta$ (\[eq:KTop\]) which describes the deformation by $B$ field and apply that operator to Ishibashi states, $|p{\rangle\!\rangle}'=V_\theta |p{\rangle\!\rangle}$,
2. calculate $\star$ product between these states $|p{\rangle\!\rangle}'\star |q{\rangle\!\rangle}'$,
3. and take Seiberg-Witten limit.
Actually the state obtained in the step 1 does not satisfy the conformal invariance $(L_n-\tilde L_{-n})|B\rangle=0$. It means that they are not, precisely speaking, the boundary states. Instead, we will see that the deformed Ishibashi state is equivalent to Neumann type boundary state with tachyon vertex insertion (\[eq:VD=VBF\]). It may imply that our computation in the following should be related to the loop correction factor in noncommutative Yang-Mills theory.
A deformation of boundary state in the presence of $B$ field
------------------------------------------------------------
Let us first introduce “KT-operator” $V_{\theta}=e^M$ [@Kawano_Takahashi; @KT2], which defines the deformation associated with the noncommutativity for the constant $B$-field background in Witten’s open string field theory and HIKKO open-closed string field theory. In that context, it was demonstrated that this operator $V_{\theta}$ transforms open string fields on $B=0$ background to that on $B\ne 0$. The KT operator $V_{\theta}$ on a constant metric $g_{ij}$ background is given by $$\begin{aligned}
\label{eq:KTop}
V_{\theta}&=&\exp\left(-{i\over 4}\oint d\sigma\oint d\sigma'
P_i(\sigma)\theta^{ij}
\epsilon(\sigma-\sigma')
P_j(\sigma')\right)\end{aligned}$$ where $P_i(\sigma)={1\over 2\pi}\left[
\hat{p}_i+{1\over\sqrt{2\alpha'}}\sum_{n\ne 0,\,n\in {\bf Z}}g_{ij}
\left(\alpha_n^j
e^{in\sigma}+\tilde{\alpha}_n^je^{-in\sigma}\right)
\right]$, and $\epsilon(x)$ is the sign function. Formally, we get $$V_{\theta}\partial_{\sigma}X^i(\sigma)
V^{-1}_{\theta}=\partial_{\sigma}X^i(\sigma)-\theta^{ij}P_j(\sigma)$$ by canonical commutation relation, and therefore, we can expect that the operator $V_{\theta}$ induces a map from Dirichlet boundary state to Neumann one with constant flux.[^5]
A subtlety in (\[eq:KTop\]) is how to define $\epsilon(\sigma-\sigma')$ or $\oint
d\sigma\oint d\sigma'$ since we need to impose the periodicity of [*closed*]{} strings $P_i(\sigma+2\pi)=P_i(\sigma)$. Here, we introduce a cut $\sigma_c$ and set the integration region to $\sigma\in [\sigma_c,2\pi+\sigma_c]$. Then, by taking normal ordering using a formula given in (\[eq:KP\]), an explicit oscillator representation of KT operator $V_{\theta}$ (\[eq:KTop\]) becomes, $$\begin{aligned}
V_{\theta,\sigma_c}&:=&\exp\left(-{i\over 4}
\int_{\sigma_c}^{2\pi+\sigma_c} d\sigma
\int_{\sigma_c}^{2\pi+\sigma_c} d\sigma'
P_i(\sigma)\theta^{ij}
\epsilon(\sigma-\sigma')
P_j(\sigma')\right)\nonumber\\
&=&\left(\det\left(1-C\right)\right)^{-{1\over 2}}
e^{{1\over 2}DN(1-C)^{-1}D^T}
e^{-{1\over 2}a^{\dagger}NC(1+C)^{-1}a^{\dagger}+D(1+C)^{-1}a^{\dagger}}
\nonumber\\
&&\times\,
e^{-a^{\dagger}\log(1-C)a}
e^{{1\over 2}aNC(1-C)^{-1}a+DN(1-C)^{-1}a},\label{eq:KTop2}\end{aligned}$$ where $$\begin{aligned}
&&a=\left(
\begin{array}[tb]{c}
{(e\alpha_n)^a\over \sqrt{n}}\\
{(e\tilde{\alpha}_n)^a\over \sqrt{n}}
\end{array}
\right),~
a^{\dagger}=\left(
\begin{array}[tb]{c}
{(e\alpha_{-n})^a\over \sqrt{n}}\\
{(e\tilde{\alpha}_{-n})^a\over \sqrt{n}}
\end{array}
\right),~(n\ge 1);
~~g_{ij}=e_i^a\eta_{ab}e_j^b\,,
~g^{ij}=\tilde{e}^i_a\eta^{ab}\tilde{e}^j_b\,,
~e_i^a\tilde{e}_a^j=\delta_i^j,~~~~~\\
&&C=-C^T=-{1\over 4\pi\alpha'}\left(
\begin{array}[tb]{cc}
(e\theta e)^{ab}\delta_{n,m}& 0\\
0&-(e\theta e)^{ab}\delta_{n,m}
\end{array}
\right),\\
&&N=N^T=\left(
\begin{array}[tb]{cc}
0& \eta_{ab}\delta_{n,m}\\
\eta_{ab}\delta_{n,m}&0
\end{array}
\right),
~~~~
D=-{1\over 2\pi\sqrt{2\alpha'}}\,\hat{p}_i\,\theta^{ij}\left(e_j^a\,
{e^{-im\sigma_c}\over \sqrt{m}},
-e_j^a\,{e^{im\sigma_c}\over \sqrt{m}}\right).\end{aligned}$$ By multiplying $V_{\theta,\sigma_c}$ (\[eq:KTop2\]) to the Dirichlet type Ishibashi state with momentum $p$: $|p\rangle\!\rangle_D:=e^{\sum_{n\ge 1}{1\over n}\alpha^i_{-n}
g_{ij}\tilde{\alpha}^j_{-n}}|p\rangle$, we obtain $$\begin{aligned}
&&V_{\theta,\sigma_c}|p\rangle\!\rangle_D=
\left[\det\left(1-(2\pi\alpha')^{-1}g\theta\right)\right]^{-\sum_{n\ge 1}1}\,
e^{-\alpha'pG^{-1}_{\theta}p\,\sum_{n\ge 1}{1\over n}}\\
&&~~~~~~~~~~~~~~~~
\times\exp\left(-\sum_{n=1}^{\infty}{1\over n}
\alpha_{-n}g{\cal O}_{\theta}\tilde{\alpha}_{-n}
+\sum_{n=1}^{\infty}(\lambda_n\alpha_{-n}+\tilde{\lambda}_n
\tilde{\alpha}_{-n})\right)|p\rangle,\nonumber\\
&&{\cal O}_{\theta}=(g+2\pi\alpha'\theta^{-1})^{-1}
(g-2\pi\alpha'\theta^{-1})\,,~~~
G^{-1}_{\theta}=(g-2\pi\alpha'\theta^{-1})^{-1}
g(g+2\pi\alpha'\theta^{-1})^{-1},\\
&&(\lambda_m,\tilde{\lambda}_m)=\sqrt{2\alpha'}\,p
\left(
(g-2\pi\alpha'\theta^{-1})^{-1}g\,{e^{-im\sigma_c}\over m}
,(g+2\pi\alpha'\theta^{-1})^{-1}g\,{e^{im\sigma_c}\over m}
\right).$$ We redefine the normalization of this state as $$\begin{aligned}
\label{eq:KT_D}
\hat{V}_{\theta,\sigma_c}|p\rangle\!\rangle_D:=
\left[\det\left(1-(2\pi\alpha')^{-1}g\theta\right)\right]^{\sum_{n\ge 1}1}
e^{\alpha'pG^{-1}_{\theta}p\,\sum_{n\ge 1}{1\over n}}\,
V_{\theta,\sigma_c}|p\rangle\!\rangle_D,\end{aligned}$$ so that $\langle p'|\hat{V}_{\theta,\sigma_c}|p\rangle\!\rangle_D=
(2\pi)^d\delta^d(p'-p)$. Then, we find an identity $$\begin{aligned}
\label{eq:VD=VBF}
\hat{V}_{\theta,\sigma_c}|p\rangle\!\rangle_D
&=& V_p(\sigma_c)|B(F_{ij}=-(\theta^{-1})_{ij})\rangle\,,\end{aligned}$$ where $$\begin{aligned}
|B(F)\rangle=e^{-
\sum_{n=1}^{\infty}{1\over n}\alpha_{-n}g{\cal O}\tilde{\alpha}_{-n}}
|p=0\rangle\end{aligned}$$ with ${\cal O}=(g-2\pi\alpha'F)^{-1}(g+2\pi\alpha'F)$, is the Neumann boundary state with constant flux $F$ and $$\begin{aligned}
&&V_k(\sigma)={\cal N}_k:e^{ikX(\sigma)}:\nonumber\\
&&=
e^{{1\over 2}\alpha'kG^{-1}_Ok\sum_{n\ge 1}{1\over n}}\,
e^{k\sum_{n=1}^{\infty}\left(\alpha'\over 2\right)^{1/2}{1\over n}
\left(
\alpha_{-n}e^{-in\sigma}
+\tilde{\alpha}_{-n}e^{in\sigma}
\right)}e^{ik\hat{x}}\,
e^{-k\sum_{n=1}^{\infty}\left(\alpha'\over 2\right)^{1/2}{1\over n}
\left(\alpha_ne^{in\sigma}+\tilde{\alpha}_ne^{-in\sigma}\right)
},~~~~~~~
\label{eq:tach}\end{aligned}$$ where $G_O^{-1}=(g+2\pi\alpha'F)^{-1}
g(g-2\pi\alpha'F)^{-1}$ is the open string metric, represents the tachyon vertex operator at $\sigma$ with momentum $k_i$.
The above identity (\[eq:VD=VBF\]) implies that the KT operator (\[eq:KTop2\]) maps the Dirichlet type Ishibashi state of momentum $p$ to Neumann boundary states [*with*]{} tachyon vertex with momentum $p$, where the position of the tachyon insertion $\sigma_c$ corresponds to the cut in the definition of the exponent of (\[eq:KTop2\]). This combination was investigated as a fluctuation around boundary states in [@kmw1; @kmw2] and can be used to calculate their star product in the following.
$\star$ product of deformed Ishibashi state
-------------------------------------------
Let us proceed to the step 2, namely the computation of the $\star$ product of $\hat{V}_{\theta,\sigma_c}|p\rangle\!\rangle_D$ (\[eq:KT\_D\]). We use eqs. (4.6) and (4.7) in [@kmw1] to give $$\begin{aligned}
&&\hat{V}_{\theta,\sigma_c}|p_1\rangle\!\rangle_{D,\alpha_1}
\star\hat{V}_{\theta,\sigma_c}|p_2\rangle\!\rangle_{D,\alpha_2}\nonumber\\
&&={\cal N}_{12}\,{\det}^{-{d\over 2}}(1-(\tilde{N}^{33})^2)
\oint {d\sigma_1\over 2\pi}
\oint {d\sigma_2\over 2\pi}|e^{i\sigma^{(1)}(\sigma_1)}
-e^{i\sigma^{(2)}(\sigma_2)}|^{2\alpha'p_1G^{-1}_Op_2}\,
e^{i\Theta_{12}}\nonumber\\
&&~~~~\times\,\wp\, e^{\sum_{n\ge 1}(\lambda_n^{(12)}\alpha_{-n}
+\tilde{\lambda}_n^{(12)}\tilde{\alpha}_{-n})-{\sum_{n\ge 1}{1\over n}
\alpha_{-n}G{\cal O}\tilde{\alpha}_{-n}}}|p_1+p_2\rangle_{\alpha_1+\alpha_2}\,,
\label{eq:VDstar}\end{aligned}$$ where we have assigned $\alpha_1,\alpha_2$ ($\alpha_1\alpha_2>0$) and omitted ghost sector. Here, the coordinates $\sigma^{(1)}(\sigma_1)$ and $\sigma^{(2)}(\sigma_2)$ are given by $$\begin{aligned}
\sigma^{(1)}(\sigma_1)&=&{\alpha_1\over \alpha_1+\alpha_2}(\sigma_c+\sigma_1)
-\pi\,{\rm sgn}(\sigma_c+\sigma_1)\,,\label{eq:sigma1}\\
\sigma^{(2)}(\sigma_2)&=&{\alpha_2\over \alpha_1+\alpha_2}
\left(\sigma_c+\sigma_2
-\pi\,{\rm sgn}(\sigma_c+\sigma_2)\right),\label{eq:sigma2}\end{aligned}$$ for $|\sigma_c+\sigma_r|<\pi,~r=1,2$, which represent the positions of tachyon vertices on the boundary of the joined string $3$ specified by the overlapping condition for the 3-string vertex. Note that the phase factor $e^{i\Theta_{12}}$ appears as a result of the $\star$ product of closed string field theory which is computed from the last term in eq. (4.7) in [@kmw1] using (\[eq:formulae\]) as $$\begin{aligned}
\label{eq:Theta12}
\Theta_{12}&=&-{1\over 2\pi}\,p_{1i}\vartheta^{ij}p_{2j}
(\sigma^{(1)}(\sigma_1)-\sigma^{(2)}(\sigma_2))+{1\over 2}p_{1i}\vartheta^{ij}p_{2j}\,
\epsilon(\sigma^{(1)}(\sigma_1)-\sigma^{(2)}(\sigma_2))\,,\end{aligned}$$ where $$\begin{aligned}
{\vartheta}^{ij}=(2\pi\alpha')^2\left[(g-2\pi\alpha'\theta^{-1})^{-1}
{\theta}^{-1}(g+2\pi\alpha'\theta^{-1})^{-1}\right]^{ij}\end{aligned}$$ corresponds to the noncommutativity parameter. In the exponent, linear terms with respect to oscillators are given by $$\begin{aligned}
\lambda_n^{(12)}&=&{\sqrt{2\alpha'}\over n}
\left(p_1\,e^{-in\sigma^{(1)}(\sigma_1)}
+p_2\,e^{-in\sigma^{(2)}(\sigma_2)}\right)(g-2\pi\alpha'\theta^{-1})^{-1}g\,,
~~\\
\tilde{\lambda}_n^{(12)}&=&{\sqrt{2\alpha'}\over n}
\left(p_1\,e^{in\sigma^{(1)}(\sigma_1)}
+p_2\,e^{in\sigma^{(2)}(\sigma_2)}\right)(g+2\pi\alpha'\theta^{-1})^{-1}g\,.\end{aligned}$$ The factor ${\cal N}_{12}$ is evaluated as $$\begin{aligned}
{\cal N}_{12}&=&\lim_{L\rightarrow\infty}\left[
e^{\alpha'p_1G^{-1}_Op_1\left(
\sum_{n=1}^{|\alpha_1|L}{1\over n}-
\sum_{p=1}^{|\alpha_3|L}{1\over p}\right)}
e^{\alpha'p_2G^{-1}_Op_2\left(
\sum_{n=1}^{|\alpha_2|L}{1\over n}-
\sum_{p=1}^{|\alpha_3|L}{1\over p}\right)}
\right]\nonumber\\
&=&(-\beta)^{\alpha'p_1G^{-1}_Op_1}(1+\beta)^{\alpha'p_2G^{-1}_Op_2},
~~~(\beta={\alpha_1/\alpha_3},~\alpha_3=-\alpha_1-\alpha_2),\end{aligned}$$ where we take cutoffs for the mode number of strings such that they are proportional to each string length parameter $|\alpha_r|$. This prescription was used in [@kmw1]v4 in order to investigate the on-shell condition from idempotency and is consistent with conformal factor of the open string tachyon vertex [@kmw2]. We can also rewrite (\[eq:VDstar\]) as $$\begin{aligned}
&& \hat{V}_{\theta,\sigma_c}|p_1\rangle\!\rangle_{D,\alpha_1}
\star\hat{V}_{\theta,\sigma_c}|p_2\rangle\!\rangle_{D,\alpha_2}\nonumber\\
&&=~(-\beta)^{\alpha'p_1G^{-1}_Op_1}(1+\beta)^{\alpha'p_2G^{-1}_Op_2}
\,{\det}^{-{d\over 2}}(1-(\tilde{N}^{33})^2)\\
&&~~~~\times\oint {d\sigma_1\over 2\pi}
\oint {d\sigma_2\over 2\pi}\wp\,
V_{p_1}(\sigma^{(1)}(\sigma_1))
V_{p_1}(\sigma^{(2)}(\sigma_2))
|B(F=-\theta^{-1})\rangle_{\alpha_1+\alpha_2}\,,\nonumber\end{aligned}$$ using tachyon vertex given in (\[eq:tach\]). This implies that the $\star$ product of $\hat{V}_{\theta,\sigma_c}|p\rangle\!\rangle_{D}$ induces conventional operator product of tachyon vertices on the Neumann boundary state.
Seiberg-Witten limit
--------------------
Next, we proceed the third step to take Seiberg-Witten limit [@SW] of (\[eq:VDstar\]) in order to obtain the deformed algebra. In the limit $ \alpha'\sim \varepsilon^{1\over 2}\rightarrow 0\,,~
g_{ij}\sim\varepsilon \rightarrow 0$, the $\star$ product formula (\[eq:VDstar\]) is simplified as $$\begin{aligned}
\label{eq:SWstar}
&& \hat{V}_{\theta,\sigma_c}|p_1\rangle\!\rangle_{D,\alpha_1}
\star\hat{V}_{\theta,\sigma_c}|p_2\rangle\!\rangle_{D,\alpha_2}
\sim a(p_1,p_2) \hat{V}_{\theta,\sigma_c}
|p_1+p_2\rangle\!\rangle_{D,\alpha_1+\alpha_2}\,,
\\
&&a(p_1,p_2)\equiv {\det}^{-{d\over 2}}(1-(\tilde{N}^{33})^2)
\oint {d\sigma_1\over 2\pi}
\oint {d\sigma_2\over 2\pi}\,e^{i\Theta_{12}}\,,\end{aligned}$$ where we have estimated using $\alpha^i_{-n}=\sqrt{n}\,
\tilde{e}^i_aa^{\dagger a}_n \sim
\varepsilon^{-{1\over 2}}$ and ignored linear terms in the exponent. We can interprete that, in this limit, the deformed Ishibashi states: $\hat{V}_{\theta,\sigma_c}|p\rangle\!\rangle_{D}$ form a closed algebra with respect to the $\star$ product of closed string field theory. After we drop the determinant factor, the coefficient $a(p_1,p_2)$ can be evaluated as $$\begin{aligned}
a(p_1,p_2)=
{\sin (-\beta p_{1i}\theta^{ij}p_{2j})\over
-\beta p_{1i}\theta^{ij}p_{2j}}
{\sin ((1+\beta) p_{1i}\theta^{ij}p_{2j})\over
(1+\beta) p_{1i}\theta^{ij}p_{2j}}\,,\end{aligned}$$ where we introduce a parameter $\beta={-\alpha_1\over \alpha_1+\alpha_2}$ ($-1<\beta<0$) which comes from the assigned $\alpha$-parameters for string fields in the $\star$ product. The integration intervals for $\sigma_1,\sigma_2$ are taken as $-\pi<\sigma_c+\sigma_1<\pi$, $-\pi<\sigma_c+\sigma_2<\pi$ and we have used eqs. (\[eq:sigma1\]) and (\[eq:sigma2\]). We note that the last expression does not depend on the cut $\sigma_c$ in the KT operator. This independence is caused by the level matching projections $\wp_1,\wp_2$ in the 3-string vertex. By taking Fourier transformation, the induced product is represented in the coordinate space as, $$\begin{aligned}
f_{\alpha_1}(x)\diamond_{\beta} g_{\alpha_2}(x)&=&f_{\alpha_1}(x){\sin(-\beta\lambda)\sin((1+\beta)\lambda)\over
(-\beta)(1+\beta)\lambda^2}g_{\alpha_2}(x)
\label{eq:closed_star}\\
&=&
f_{\alpha_1}(x)\,
\sum_{k=0}^{\infty}{(-1)^k\lambda^{2k}\over (2k+1)!}
\sum_{l=0}^k{(1+2\beta)^{2l}\over k+1}\,g_{\alpha_2}(x)\,,~~~~
\left(
\lambda={1\over 2}
{\overleftarrow{\partial}\over \partial x^i}
\theta^{ij}
{\overrightarrow{\partial}\over \partial x^j}
\right),\nonumber\end{aligned}$$ where we have specified $\alpha_1,\alpha_2$ for coefficient functions because the parameter $\beta$ in the above $\diamond_{\beta}$ is given by their ratio. In fact, for the string fields of the form $$\begin{aligned}
\label{eq:PhiNC}
|\hat{\Phi}_{f_{\alpha}}(\alpha)\rangle&=&\int d^dx\,f_{\alpha}(x)
\hat{V}_{\theta,\sigma_c}|\Phi_B(x,\alpha)\rangle\,,\end{aligned}$$ where we have included the ghost and $\alpha$ sector: $|\Phi_B(x,\alpha)\rangle=c_0^-b_0^+|B(x)\rangle\otimes
|B\rangle_{\rm ghost}\otimes|\alpha\rangle$ and $\alpha$ dependence in the coefficient function, we can express the above $\diamond_{\beta}$ product in terms of the $\star$ product: $$\begin{aligned}
&&\langle x, \alpha_1+\alpha_2|c_{-1}\tilde{c}_{-1}
|\hat{\Phi}_{f_{\alpha_1}}(\alpha_1)\rangle\star
|\hat{\Phi}_{g_{\alpha_2}}(\alpha_2)\rangle
\nonumber\\
&&=~[\mu^2\,{\det}^{-{d-2\over 2}}(1-(\tilde{N}^{33})^2)\,2\pi\delta(0)]\,
f_{\alpha_1}(x)\diamond_{\beta} g_{\alpha_2}(x)\label{eq:star12SW}\end{aligned}$$ in the Seiberg-Witten limit. Here, we give some comments on this $\diamond_{\beta}$ product (\[eq:closed\_star\]). It is commutative in the sense: $$\begin{aligned}
f_{\alpha_1}(x)\diamond_{\beta} g_{\alpha_2}(x)
=g_{\alpha_2}(x)\diamond_{\beta} f_{\alpha_1}(x)
=f_{\alpha_1}(x)\diamond_{-1-\beta} g_{\alpha_2}(x)\,.\end{aligned}$$ (Note that exchange of $\alpha_1\leftrightarrow \alpha_2$ corresponds to $\beta\leftrightarrow -1-\beta$.) We can take the “commutative” background limit $\theta^{ij}\rightarrow 0$ : $$\begin{aligned}
\lim_{\theta^{ij}\rightarrow 0}
f_{\alpha_1}(x)\diamond_{\beta} g_{\alpha_2}(x)
=f_{\alpha_1}(x)\,g_{\alpha_2}(x)\end{aligned}$$ where the right hand side is ordinary product. In the case that one of the string length parameter $\alpha_1,\alpha_2$ equals to zero, our product (\[eq:closed\_star\]) is reduced to the Strachan product [@Strachan]: $$\begin{aligned}
&&\lim_{\alpha_1\rightarrow 0}
f_{\alpha_1}(x)\diamond_{\beta} g_{\alpha_2}(x)
=f_0(x)\diamond g_{\alpha_2}(x)\,,\nonumber\\
&&\lim_{\alpha_2\rightarrow 0}
f_{\alpha_1}(x)\diamond_{\beta} g_{\alpha_2}(x)
=f_{\alpha_1}(x)\diamond g_0(x)\,,\nonumber\\
&&{\rm where}~~~
f(x)\diamond g(x):=f(x){\sin\lambda\over \lambda}g(x)\,,~~~
\left(
\lambda={1\over 2}
{\overleftarrow{\partial}\over \partial x^i}
\theta^{ij}
{\overrightarrow{\partial}\over \partial x^j}
\right),~~~~
\label{eq:strachan}\end{aligned}$$ which is also one of the generalized star product: $*_2$ [@start].
In the literature [@start], Strachan product appeared in one-loop correction to the non-commutative Yang-Mills theory. The appearance of the similar product here may be interpreted naturally. As we have seen in section 2, taking the closed string star product of boundary states is equivalent to the degeneration limit of the open string one loop correction. In this interpretation, the star product we considered can be mapped to one-loop open string diagram with one open string external lines attached to each of the two boundaries. It reduces to a diagram which is similar to the one in [@start] in the Seiberg-Witten limit.
It will be very interesting to obtain the explicit form of the projector to the Strachan product, $$f\diamond f=f,$$ since it may describe the zero-mode part of the Cardy state that corresponds to GMS soliton. One important task before proceeding that direction may be, however, to construct the argument which is valid without taking the Seiberg-Witten limit.
We comment that the observation made here is parallel to the situation in [*open*]{} string field theory. In the limit of a large $B$-field, Witten’s star product factorizes into that of zero mode and nonzero modes. The star product is then reduced to Moyal product on the zero mode sector. The noncommutativity appears as the coefficient functions on the lump solution $|{\cal S}\rangle$ in the context of vacuum string field theory [@openGMS]. The correspondence is: $$\begin{aligned}
|{\cal S}\rangle~&\leftrightarrow &~\hat V_{\theta,\sigma_c}|B(x)\rangle
\nonumber\\
\mbox{Moyal product}& \leftrightarrow & \mbox{Strachan product}
\nonumber\\
\mbox{Open string field theory}&\leftrightarrow &
\mbox{Closed string field theory}\,.\nonumber\end{aligned}$$ This may be a natural extension of open vs. closed “VSFT” correspondence, as we suggested in [@kmw1; @kmw2], for a constant $B$-field background in the transverse directions.
Conclusion and Discussion
=========================
A main observation in this article is that the nonlinear relation (\[e\_idempotency\]) is satisfied by arbitrary consistent boundary states in the sense of Cardy for any conformal invariant background. The origin of such a simple relation is the factorization property of the boundary conformal field theory. Since this should be true for any background as an axiom, our nonlinear equation should be true for any consistent closed string field theory. In fact, we have checked this relation for torus and $Z_2$ orbifold by direct calculation in terms of explicit oscillator formulation of the HIKKO closed string field theory. Although the relation (\[e\_idempotency\]) looks exactly like a VSFT equation, it is not a consequence of a particular proposal of the closed string field theory. Usually it is believed that such an equation for the vacuum theory can be obtained from the re-expansion around the tachyon vacuum of some consistent string field theory. However, our equation is not, at least at present, obtained in that way. It is rather a direct consequence of an axiom of the boundary conformal field theory. It is very interesting that a universal nonlinear equation can be obtained in this way. In a sense, it is more like loop equation.
A weak point of our equation may be that it contains the regularization parameter $K$ explicitly and divergent while it is milder for the HIKKO type vertex than Zwiebach’s one. This can be overcome by the generalization to superstring field theory. The factorization property of two holes attached to a BPS D-brane is regular since the open string channel does not contain tachyon. In this sense, it will be possible to write down a regular nonlinear equation which characterize the BPS D-branes. A complication arises when we consider a product of non-BPS D-brane or different type of BPS D-branes. In such a situation, there appears the open string tachyon and their star product will be divergent. This will be very different from the bosonic case where eq. (\[e\_idempotency\]) is universally true for any D-brane. We will come back to this question in our future study.
Acknowledgements {#acknowledgements .unnumbered}
================
We are obliged to H. Isono and E. Watanabe for helpful discussions. I. K. would like to thank H. Kunitomo for valuable comments. I. K. would also like to thank the Yukawa Institute for Theoretical Physics at Kyoto University. Discussions during the YITP workshop YITP-W-04-03 on “Quantum Field Theory 2004” were useful to complete this work. Y. M would like to thank G. Semenoff for the invitation to a workshop “String Field Theory Camp” at BIRS where very useful discussions with the participants were possible. He would like to thank, especially, S. Minwalla and W. Taylor for their comments and interests. I. K. is supported in part by JSPS Research Fellowships for Young Scientists. Y. M. is supported in part by Grant-in-Aid (\# 16540232) from the Ministry of Education, Science, Sports and Culture of Japan.
Star product on $Z_2$ orbifold \[sec:starproduct\]
==================================================
In this section, we briefly review the star product on $T^D/{\bf Z}_2$ orbifold [@Itoh_Kunitomo] and fix our convention which is mainly based on [@KZ]. By restricting to the untwisted sector and removing ${\bf Z}_2$ projection, we obtain the star product on a torus $T^D$.
We define the $\star$ product for the string fields $|A\rangle,|B\rangle$ by: $$\begin{aligned}
&&|A\rangle \star |B\rangle\equiv |A\star B\rangle_3
\equiv~{}_1\langle A|~ {}_2\langle B|V(1,2,3)\rangle\,,\\
&&~~{\rm where}~~~
{}_2\langle \Phi|\equiv \langle R(1,2)|\Phi\rangle_1\,,\end{aligned}$$ which gives cubic interaction term in an action of closed string filed theory. In order to define the above concretely, we should specify the reflector $\langle
R(1,2)|$ and the 3-string vertex $|V(1,2,3)\rangle$ in $T^D/{\bf Z}_2$ sector. We expand the coordinates $X^i(\sigma)$ and their canonical conjugate momentum $P_i(\sigma)$ to express them in terms of oscillators as follows. In the untwisted sector, $X^i(\sigma+2\pi)\equiv X^i(\sigma)~({\rm
mod}~2\pi\sqrt{\alpha'})$: $$\begin{aligned}
\label{eq:mode_st}
X^i(\sigma)&=&\sqrt{\alpha'}[x^i+w^i\sigma]
+i\sqrt{\alpha'\over 2}\sum_{n\ne 0,\,n\in {\bf Z}}{1\over n}
\left[\alpha_n^ie^{in\sigma}
+\tilde{\alpha}_n^ie^{-in\sigma}\right],~~~~\\
P_i(\sigma)&=&{1\over 2\pi\sqrt{\alpha'}}\left[
p_i+{1\over\sqrt{2}}\sum_{n\ne 0,\,n\in {\bf Z}}\left(E^T_{ij}\alpha_n^j
e^{in\sigma}+E_{ij}\tilde{\alpha}_n^je^{-in\sigma}\right)
\right],\end{aligned}$$ where $E_{ij}=G_{ij}+2\pi\alpha' B_{ij},E^T_{ij}=G_{ij}-2\pi\alpha'
B_{ij}$. The commutation relations are given by $
[x^i,p_j]=i\delta^i_j,[\alpha^i_n,\alpha^j_m]=nG^{ij}\delta_{n+m,0},
[\tilde{\alpha}^i_n,\tilde{\alpha}^j_m]=nG^{ij}\delta_{n+m,0}.
$ In our compactification, we should identify as $x^i\equiv x^i+2\pi$ and then the zero mode momentum $p_i$ takes integer eigenvalue. In the twisted sector, $X^i(\sigma+2\pi)\equiv -X^i(\sigma)~({\rm
mod}~2\pi\sqrt{\alpha'})$: $$\begin{aligned}
\label{eq:t_mode_st}
X^i(\sigma)&=&\sqrt{\alpha'}\,x^i
+i\sqrt{\alpha'\over 2}\sum_{r\in {\bf Z}+{1\over 2}}{1\over r}
\left[\alpha_r^ie^{ir\sigma}
+\tilde{\alpha}_r^i e^{-ir\sigma}\right],\\
P_i(\sigma)&=&{1\over 2\pi\sqrt{2\alpha'}}
\sum_{r\in {\bf Z}+{1\over 2}}\left(E^T_{ij}\alpha_r^j
e^{ir\sigma}+E_{ij}\tilde{\alpha}_r^je^{-ir\sigma}\right).\end{aligned}$$ The commutation relations of nonzero modes are given by $[\alpha^i_r,\alpha^j_s]=rG^{ij}\delta_{r+s,0}$, $[\tilde{\alpha}^i_r,\tilde{\alpha}^j_s]=rG^{ij}\delta_{r+s,0}$ and the zero mode $x^i$ takes eigenvalue corresponding to fixed points of ${\bf Z}_2$ action: $x^i=\pi (n^f)^i$ where $(n^f)^i=0$ or $1$.
#### Reflector
We use reflector to obtain a bra $\langle \Phi|$ from a ket $|\Phi\rangle$. There are two types of reflector according to the twisted/untwisted sector. For the untwisted sector,[^6] $$\begin{aligned}
\label{eq:Ruu}
\langle R_u(1,2)|&=&
\sum_{p_r,w_r}\delta^D_{p_1+p_2,0}\delta^D_{w_1+w_2,0}
\langle p_1,w_1|\langle p_2,w_2|\,
e^{E_u(1,2)}e^{-i\pi p_1w_1}\wp_{12},~~~~\\
E_u(1,2)&=&-\sum_{n\ge 1} {(-1)^n\over n}
G_{ij}\left(\alpha_n^{(1)i} \alpha_n^{(2)j}+
\tilde{\alpha}_n^{(1)i} \tilde{\alpha}_n^{(2)j} \right)\,,~\end{aligned}$$ where the the prefactor $e^{-i\pi p_1w_1}$ comes from the connection condition $X^{(1)}(\sigma)-X^{(2)}(\pi-\sigma)=0$ without projector $\wp_{12}$ [@KZ][^7] and the oscillator vacuum with zero mode eigen value $(p_i,w^i)$: $\langle p,w|$ is normalized as $\langle p,w|p',w'\rangle=\delta^D_{p,p'}\delta^D_{w,w'}$. For the twisted sector, the reflector is given by $$\begin{aligned}
\label{eq:Rtt}
\langle R_t(1,2)|&=&
\sum_{n_1^f,\,n_2^f}\delta^D_{n_1^f,\,n_2^f}
\,\langle n_1^f|\langle n_2^f|\,e^{-\sum_{r\ge {1\over 2}}{1\over r}
G_{ij}\left(\alpha_r^{(1)i} \alpha_r^{(2)j}+
\tilde{\alpha}_r^{(1)i} \tilde{\alpha}_r^{(2)j} \right)
}\wp_{12},~~~\end{aligned}$$ which represents $X^{(1)}(\sigma)-X^{(2)}(-\sigma)=0$ without $\wp_{12}$ and we take the normalization of the oscillator vacuum for the fixed point $\pi n^f$ as $\langle
n^f|n^{f\prime}\rangle=\delta^D_{n^f,n^{f\prime}}$.
#### 3-string vertex
We have two types of 3-string interaction: (uuu) all strings are in the untwisted sector; (utt) one is in the untwisted sector and the other two are in the twisted sector. Correspondingly, there are two types of 3-string vertex. They are constructed by a connection condition based on HIKKO type interaction, i.e., joining/splitting of closed strings at one interaction point. (Odd number of twisted sectors such as (ttt), (uut) are not contained in 3-string interaction terms to be consistent with ${\bf Z}_2$ action.)
For (uuu)-type 3-string vertex, by assigning $\alpha_r$ for each string, we have $$\begin{aligned}
\label{eq:Vuuu}
|V(1_u,2_u,3_u)\rangle
&=&\wp_{123}{\cal P}_{u1}^{Z_2}{\cal P}_{u2}^{Z_2}{\cal P}_{u3}^{Z_2}
\sum_{p_r,w_r}\delta_{p_1+p_2+p_3,0}\delta_{w_1+w_2+w_3,0}
\nonumber\\
&&\times\, e^{-i\pi(p_3w_2-p_1w_1)}e^{E_u(1,2,3)}
|p_1,w_1\rangle|p_2,w_2\rangle|p_3,w_3\rangle,~~~~~\end{aligned}$$ where the exponent is given by $$\begin{aligned}
&&E_u(1,2,3)
={1\over 2}\sum_{r,s=1}^3\sum_{n,m\ge 0}
\bar{N}^{rs}_{nm}G_{ij}\left(\alpha_{-n}^{i(r)}
\alpha_{-m}^{j(s)}+\tilde{\alpha}_{-n}^{i(r)}
\tilde{\alpha}_{-m}^{j(s)}\right)\,.\end{aligned}$$ Here $\bar{N}^{rs}_{nm}$ is the same as the Neumann coefficient on ${\bf R}^d$ (we also use the notation: $\tilde{N}^{rs}_{nm}:=\sqrt{nm}\bar{N}^{rs}_{nm}\,~(n,m>0)$) [@HIKKO2] and we define zero modes as: $\alpha_0^i=G^{ij}(p_j-E_{jk}w^k)/\sqrt{2}$, $\tilde{\alpha}_0^i=G^{ij}(p_j+E_{jk}^Tw^k)/\sqrt{2}$. The prefactor ${\cal P}_{u}^{Z_2}$ is ${\bf Z}_2$-projection for the untwisted sector and is given by ${\cal P}_u^{Z_2}={1\over 2}(1+RO_u)$ with $R|p,w\rangle=|-p,-w\rangle\,,~O_u\alpha_n^iO^{-1}_u=-\alpha_n^i\,,
~O_u\tilde{\alpha}_n^iO^{-1}_u=-\tilde{\alpha}_n^i$. The phase factor $e^{-i\pi(p_3w_2-p_1w_1)}$ is necessary to satisfy Jacobi identity [@HIKKO_torus; @Maeno_Takano]. The above vertex $|V(1_u,2_u,3_u)\rangle$ is also obtained by multiplying ${\bf Z}_2$-projection ${\cal P}_{u1}^{Z_2}{\cal P}_{u2}^{Z_2}{\cal P}_{u3}^{Z_2}$ to the 3-string vertex on the torus $T^D$ [@HIKKO_torus].
For (utt)-type 3-string vertex, by assigning $\alpha_r$ for each string, we have $$\begin{aligned}
\label{eq:Vutt}
|V(1_u,2_t,3_t)\rangle&=&
e^{{D\over 8}\tau_0\left(\alpha_2^{-1}+\alpha_3^{-1}\right)}\wp_{123}
{\cal P}_{u1}^{Z_2}{\cal P}_{t2}^{Z_2}{\cal P}_{t3}^{Z_2}\\
&&\times\,
\sum_{p_1,w_1}\sum_{n_2^f,\,n_3^f}
\gamma({\bf p}_1;n_2^f,n_3^f)
e^{E_t(1_t,2_u,3_u)}|p_1,w_1\rangle|n_2^f\rangle|n_3^f\rangle\,,\nonumber\\
E_t(1_t,2_u,3_u)&=&{1\over 2}\sum_{r,s=1}^3\sum_{n_r,m_s\ge 0}
T^{rs}_{n_r m_s}G_{ij}\left(\alpha_{-n_r}^{i(r)}
\alpha_{-m_s}^{j(s)}+\tilde{\alpha}_{-n_r}^{i(r)}
\tilde{\alpha}_{-m_s}^{j(s)}\right),~~~~~~\end{aligned}$$ where Neumann coefficients $T^{rs}_{n_rm_s}$ are given explicitly in Appendix \[sec:TNeumann\] and $$\begin{aligned}
\gamma({\bf p}_1;n_2^f,n_3^f) &=&(-1)^{p_1n_3^f}\,\sum_{m^i\in {\bf Z}}\delta^D_{n_3^f-n_2^f+w_1+2m,0}\end{aligned}$$ is the cocycle factor [@Itoh_Kunitomo; @IKKS] and ${\cal P}_{t}^{Z_2}={1\over 2}(1+O_t),$ which is given by $O_t\alpha_r^i O^{-1}_t=-\alpha_r^i,
O_t\tilde{\alpha}_r^i O^{-1}_t=-\tilde{\alpha}_r^i$, is the ${\bf Z}_2$-projection. The extra factor $e^{{D\over
8}\tau_0\left(\alpha_2^{-1}+\alpha_3^{-1}\right)}$, ($\tau_0=\sum_{r=1}^3\alpha_r\log|\alpha_r|$, $\alpha_1+\alpha_2+\alpha_3=0$), can be identified with the conformal factor of twist fields in CFT language.
Note that the complete 3-string vertex is given by including ghost, matter ${\bf R}^d$ and $\alpha$ sector in the above expression (\[eq:Vuuu\]) or (\[eq:Vutt\]).
Neumann coefficients for the twisted sector on ${\bf Z}_2$ orbifold \[sec:TNeumann\]
====================================================================================
The Neumann coefficients $T^{rs}_{n_rm_s}$ in (\[eq:Vutt\]) are given by $T^{11}_{00}=-2\log 2+{\tau_0\over \alpha_1}$ and integration form in [@Itoh_Kunitomo]. We can demonstrate that there is a relation: $$\label{eq:Trs}
T^{rs}_{n_r m_s}={\alpha_1n_rm_s\over \alpha_rm_s+\alpha_sn_r}
\,T^{r1}_{n_r0}\,T^{s1}_{m_s0}\,,~~~(n_r,m_r>0)\,,$$ and $T^{r1}_{n_r0}$ are explicitly obtained: $$\begin{aligned}
T^{11}_{n0}&=&{e^{n{\tau_0\over \alpha_1}}\over n}
{\Gamma\left({1\over 2}-{\alpha_2\over \alpha_1}n\right)\over
n!\,\Gamma\left({1\over 2}+{\alpha_3\over\alpha_1}n\right)}
\,,~~~n=1,2,\cdots,\\
T^{21}_{r0}&=&{e^{r{\tau_0\over \alpha_2}}\over r}
{(-1)^{r+{1\over 2}}\,\Gamma\left(-{\alpha_1\over \alpha_2}r\right)
\over \left(r-{1\over 2}\right)!\,
\Gamma\left({1\over 2}+{\alpha_3\over \alpha_2}r\right)}\,,~~~
r={1\over 2},{3\over 2},\cdots,\\
T^{31}_{r0}&=&{e^{r{\tau_0\over \alpha_3}}\over r}
{(-1)^{r+{1\over 2}}\,\Gamma\left(-{\alpha_1\over \alpha_3}r\right)
\over \left(r-{1\over 2}\right)!\,
\Gamma\left({1\over 2}+{\alpha_2\over \alpha_3}r\right)}\,,~~~
r={1\over 2},{3\over 2},\cdots.\end{aligned}$$ Note that only string 1 is in the untwisted sector which includes zero mode $(p,w)$ in the (utt) type 3-string vertex (\[eq:Vutt\]). However, this structure of the Neumann coefficients $T^{rs}_{n_rm_s}$ is similar to that of $\bar{N}^{rs}_{nm}$ [@Mandel64] in the untwisted 3-string vertex (\[eq:Vuuu\]) in which all 3 strings have zero mode $(p,w)$.
Using continuity of Neumann function $T(\rho,\tilde{\rho})$ which is given in [@Itoh_Kunitomo] with the method in Appendix B in [@Kawano_Takahashi], namely, from the identity $\sum_{t=1}^3\int_{-\pi}^{\pi}d\sigma_t^{\prime}
T(\rho_r,\rho_t^{\prime})
{\partial\over \partial\xi_t^{\prime}}T(\rho_t^{\prime},\rho_s^{\prime\prime})
=0$ (where ${\rm Re}\,\rho'_t=\tau_0$), we have obtained the relations: $$\begin{aligned}
&& \sum_{t=1}^3\sum_{l_t> 0}T^{rt}_{n_r l_t}\, l_t\, T^{ts}_{l_t m_s}
=\delta_{r,s}\delta_{n_r,m_s}{1\over n_r}\,,\nonumber\\
&& \sum_{t=1}^3\sum_{l_t> 0}T^{1t}_{0 l_t}\, l_t\, T^{ts}_{l_t m_s}
=-T^{1s}_{0 m_s}\,,~~~~~~~~~
\sum_{t=1}^3\sum_{l_t> 0}T^{1t}_{0 l_t}\, l_t\, T^{t1}_{l_t 0}
=-2T^{11}_{00}\,,~~~
\label{eq:Yoneya}\end{aligned}$$ which correspond to Yoneya formulae for the untwisted sector [@Yoneya]. These are essential to simplify some expressions in terms of Neumann coefficients which appear in computation of the $\star$ product.
Furthermore, in the case of $\alpha_1>0,\alpha_2<0,\alpha_3<0$, we can derive following formulae using the method in [@GS]: $$\begin{aligned}
\tilde{T}_{n_rn_s}^{rs}&:=&\sqrt{n_rm_s}\,T_{n_rn_s}^{rs}=
(\delta_{r,s}1-2A^{\prime(r)T}\Gamma^{\prime-1}A^{\prime(s)})_{n_rn_s}
~~~(n_r,n_s>0)\nonumber\\
&=&{\alpha_1n_rn_s\over n_s\alpha_r+n_r\alpha_s}
(A^{\prime(r)T}\Gamma^{\prime-1}B^{\prime})_{n_r}
(A^{\prime(s)T}\Gamma^{\prime-1}B^{\prime})_{n_s}\,,\\
\tilde{T}^{r1}_{n_r0}&:=&\sqrt{n_r}\,T^{r1}_{n_r0}=
(A^{\prime(r)T}\Gamma^{\prime-1}B^{\prime})_{n_r}\,,
~~~~~~~~~~~~~~~~~~(n_r>0)\\
T^{11}_{00}&=&-{1\over 2}B^{\prime T}\Gamma^{\prime-1}B^{\prime}\,,\end{aligned}$$ where the infinite matrices $A^{\prime(r)}_{nm_r},\Gamma^{\prime}_{nm}$ and the infinite vector $B^{\prime}_n$ are given by $$\begin{aligned}
&&A^{\prime(r)}_{nm_r}=(-1)^{n+m_r-{1\over 2}}
{2n^{3\over 2}
\left(\alpha_r\over \alpha_1\right)^2
\cos \left({\alpha_r\over \alpha_1}n\pi\right)
\over \pi m_r^{1\over 2}
\left[m_r^2-n^2\left(\alpha_r\over \alpha_1\right)^2\right]}
\,,~~~r=2,3;~~m_r\ge {1\over 2}\,,~~~~~~\\
&&A^{\prime(1)}_{nm}=\delta_{n,m}\,,~~~~
B^{\prime}_n={2(-1)^n\cos n\pi\beta\over \sqrt{n}}\,,~~~
~~~n,m\ge 1\,,\\
&&\Gamma^{\prime}_{nm}=\sum_{r=1}^3\sum_{l_r>0}
A^{\prime(r)}_{nl_r}A^{\prime(r)}_{ml_r}\,,~~~~~
n,m\ge 1\,.\end{aligned}$$ Using these formulae, we can prove various identities, including (\[eq:Yoneya\]), which correspond to those in [@GS] such as $$\begin{aligned}
&&\sum_{l\ge 1}A^{\prime(r)}_{ln_r}{1\over l}A^{\prime(s)}_{ln_s}
=-{\alpha_r\over n_r\alpha_1}\delta_{r,s}\,,~~~~~r,s=2,3\,,\\
&&\sum_{r=2,3}\sum_{l_r\ge{1\over 2}}A^{\prime(r)}_{ml_r}
{l_r\over \alpha_r}
A^{\prime(r)}_{nl_r}=-{m\over \alpha_1}\delta_{m,n}\,,~~~~~~
m,n\ge 1\,,\\
&&\sum_{r=1}^3\sum_{l_r>0}{\alpha_r\over \alpha_1}A^{\prime(r)}_{nl_r}
{1\over l_r}A^{\prime(r)}_{ml_r}
=-{1\over 2}B^{\prime}_nB^{\prime}_m\,,~~~~~~
m,n\ge 1\,.\end{aligned}$$
Cremmer-Gervais identity for $T^{rs}_{n_rm_s}$ \[sec:CG\]
=========================================================
We demonstrate the relation (\[eq:CG\_T\]) by using an analogue of Cremmer-Gervais identity [@CG]. Let us consider matrices such as $$\begin{aligned}
\label{eq:AB}
&& {\tilde{{\cal N}}^{66}}_{nm}={nm\over n+m}A_{n}A_{m}\,,~~~~
{\tilde{{\cal N}}^{55}}_{t\,nm}={nm\over n+m}B_{n}B_{m}
e^{-(n+m)t}\,,~~~~~~\end{aligned}$$ which are the same form as the Neumann matrix $\tilde{N}^{rr}$ for 3-string vertex in the untwisted sector and its $T=|\alpha_5| t$ evolved one. We can derive a differential equation: $$\begin{aligned}
\label{eq:CG_diffeq}
&&{\partial^2\over \partial t^2}
\log \det(1-{\tilde{\cal{N}}^{66}}{\tilde{\cal N}_t}^{55})=
-{1\over 4}\left(\partial_t^2a_{00}\over \partial_t b_{00}\right)^2\,,\end{aligned}$$ by direct computation, where $$\begin{aligned}
a_{00}&=&\sum_{n,m}A_n\left(\tilde{{\cal N}}^{55}_t
(1-\tilde{{\cal N}}^{66}\tilde{\cal N}^{55}_t)^{-1}\right)_{nm}A_m\,,\\
b_{00}&=&\sum_{n,m}B_ne^{-nt}\left(
(1-\tilde{{\cal N}}^{66}\tilde{\cal N}^{55}_t)^{-1}\right)_{nm}A_m\,.\end{aligned}$$ The counterpart of (\[eq:CG\_diffeq\]) was integrated by identifying $a_{00},b_{00}$ with Neumann coefficients for 4-string vertex [@CG]. As we have noted in Appendix \[sec:TNeumann\], the Neumann matrices for the twisted sector $\tilde{T}^{rs}$ also has the same structure. Therefore, we consider the replacement in (\[eq:AB\]): $$\begin{aligned}
\label{eq:repl}
A_n,~~B_n,~~t~~~\longrightarrow~~~
\left(\alpha_2/\alpha_6\right)^{1\over 2}\tilde{T}^{62}_{n0},~~
\left(\alpha_3/\alpha_5\right)^{1\over 2}\tilde{T}^{53}_{n0},~~
T/\alpha_5\,,\end{aligned}$$ respectively, to evaluate the determinant $\det(1-\tilde{T}^{6_t6_t}\tilde{T}^{5_t5_t}_t)$. We have depicted this situation in Fig. \[fig:CG(P)\].
\[0.6\][![ 4-string configuration in $\rho$-plane. We have drawn ${\rm Im}\,\rho\ge 0$ only. We take strings 1,4 (2,3) in the twisted (untwisted) sector. The intermediate strings 6,5 are in the twisted sector. (There is a ${\bf Z}_2$ cut at ${\rm Im}\,\rho=0$.)[]{data-label="fig:CG(P)"}](cgp.eps "fig:")]{}
In particular strings 2 and 3 are in the untwisted sector. The Neumann coefficients for this 4-string amplitude can be obtained by expanding the Neumann function $$\begin{aligned}
&&{\cal T}(\rho,\tilde{\rho})=\log\!\left[\!\sqrt{z-Z_4\over z-Z_1}
-\sqrt{\tilde{z}-Z_4\over \tilde{z}-Z_1}\right]\!
-
\log\!\left[\!\sqrt{z-Z_4\over z-Z_1}\!
+\!\sqrt{\tilde{z}-Z_4\over \tilde{z}-Z_1}\right],~~~~~~~\end{aligned}$$ with the Mandelstam mapping: $ \rho(z)=\sum_{r=1}^4\alpha_r\log(z-Z_r)
=\alpha_r\zeta_r+\tau^{(r)}_0+i\beta_r$, where $\tau_0^{(1)}=\tau_0^{(2)}={\rm Re}\,\rho(z_-),\,
\tau_0^{(3)}=\tau_0^{(4)}={\rm Re}\,\rho(z_+)$ are interaction time: $\left.{d\rho\over dz}\right|_{z_{\pm}}=0$ (Fig. \[fig:CG(P)\]). This procedure is parallel to that for constructing 3-string vertex (\[eq:Vutt\]) in [@Itoh_Kunitomo]. In particular, the coefficient for zero modes are obtained as $$\begin{aligned}
T^{(4)rs}_{~~00}&=&\log\!\left[\!\sqrt{Z_r-Z_4\over Z_r-Z_1}
\!-\sqrt{Z_s-Z_4\over Z_s-Z_1}\right]\!
-
\log\!\left[\!\sqrt{Z_r-Z_4\over Z_r-Z_1}\!
+\!\sqrt{Z_s-Z_4\over Z_s-Z_1}\right],~~\nonumber\\
&&~~~~~~~~(r,s=2,3,~r\ne s),~~\\
T^{(4)rr}_{~~00}&=&-2\log 2+{\tau_0^{(r)}+i\beta_r\over \alpha_r}
-\log(Z_r-Z_1)-\log(Z_r-Z_4)\nonumber\\
&&+\log(Z_4-Z_1)
-\sum_{l\ne r,l=1,\cdots,4}{\alpha_l\over \alpha_r}\log(Z_r-Z_l),
~~(r=2,3)\,.\end{aligned}$$ By comparing them with zero mode dependence in the exponent of $$\begin{aligned}
\langle R(5_t,6_t)|e^{-{T\over \alpha_5}(L_0^{(5)}+{\tilde{L}_0^{(5)}})}
|V_0(1_t,2_u,6_t)\rangle|V_{0}(5_t,3_u,4_t)\rangle\,,\end{aligned}$$ (where $\langle R(5_t,6_t)|$ and $|V_0(r,s,t)\rangle$ are reflector (\[eq:Rtt\]) and 3-string vertex (\[eq:Vutt\]) respectively, with appropriate replacement and without projections) which represents Fig. \[fig:CG(P)\], we can make an identification: $$\begin{aligned}
\label{eq:T4ab}
T^{(4)22}_{~~00}
=-{\alpha_5\over \alpha_2}a_{00}+T^{22}_{00}\,,~~~~
T^{(4)23}_{~~00}
=-{\alpha_5\over \sqrt{-\alpha_2\alpha_3}}\,b_{00}\,,\end{aligned}$$ up to pure imaginary constant where $
\alpha_1+\alpha_2+\alpha_3+\alpha_4=0\,,~\alpha_3+\alpha_4+\alpha_5=0\,
$. By fixing as $Z_1=\infty,Z_2=1,Z_4=0$, we have some relations: $$\begin{aligned}
z_{\pm}&=&-(2\alpha_1)^{-1}(\alpha_{34}+\alpha_{24}Z_3\pm
\Delta^{1\over 2})\,,~~~~(\alpha_{ij}:=\alpha_i+\alpha_j)\,,\\
\Delta&=&\alpha_{24}^2Z_3^2+2(\alpha_2\alpha_3+\alpha_4\alpha_1)Z_3
+\alpha_{34}^2\,,\\
{\partial T\over \partial Z_3}&=&-{\Delta^{1\over 2}\over Z_3(1-Z_3)}
\,,~~~
T:=\tau_0^{(3)}-\tau_0^{(1)}\,,\end{aligned}$$ which are the same convention as in [@HIKKO2] Appendix C, and then we obtain a differential equation for determinant of Neumann coefficients with regularization parameter $T$: $$\begin{aligned}
\label{eq:det(P)}
&& {\partial^2\over \partial T^2}\log
\det(1-\tilde{T}^{66}\tilde{T}^{55}_t)=
{\alpha_2\alpha_3(\alpha_5-(\alpha_2+\alpha_4)Z_3)^2Z_3(1-Z_3)^2\over
4\Delta^3},~~~~~~~~\end{aligned}$$ using (\[eq:CG\_diffeq\]),(\[eq:repl\]) and (\[eq:T4ab\]). This can be rewritten by subtracting the counterpart in the untwisted sector as: $$\begin{aligned}
&& {\partial^2\over \partial T^2}\log\left[
\det(1-\tilde{T}^{66}\tilde{T}^{55}_t)\over
\det(1-\tilde{N}^{66}\tilde{N}^{55}_t)
\right]
={\partial^2\over \partial T^2}\left[
-{1\over 4}\,a_{00}+{\alpha_2\alpha_5(\alpha_1-\alpha_4)\over 8\alpha_4}\,a
\right]\!,~~~~~~~~\end{aligned}$$ where $a$ is given in (C.18) [@HIKKO2]. Around $Z_3\sim 0$, we can estimate these determinants: $
\log\det(1-\tilde{T}^{66}\tilde{T}^{55}_t)={\cal O}(Z_3),
\log\det(1-\tilde{N}^{66}\tilde{N}^{55}_t)={\cal O}(Z_3^2)
$ by definition. Therefore, we have obtained: $$\begin{aligned}
&&\log\left[
\det(1-\tilde{T}^{66}\tilde{T}^{55}_t)\over
\det(1-\tilde{N}^{66}\tilde{N}^{55}_t)
\right]\\
&&={\alpha_4\alpha_{34}-\alpha_1(\alpha_3-\alpha_4)
\over 16\alpha_1\alpha_4}\!\left[
{\tau^{534}_0-\tau^{126}_0-T\over \alpha_5}-\log Z_3\right]\!
+\!{\alpha_{14}^2\over 16\alpha_1\alpha_4}\log(1-Z_3),\nonumber\end{aligned}$$ up to pure imaginary constant, where $\tau_0^{ijk}:=\sum_{r=i,j,k}\alpha_r\log|\alpha_r|$. In order to evaluate the ratio of left and right hand side in (\[eq:CG\_T\]) by regularizing the Neumann matrices with $T$ such as Appendix B in [@kmw2], we take $\alpha_3=-\alpha_2,\alpha_4=-\alpha_1$ in particular, and we get $$\begin{aligned}
&&\log\left|
{e^{{1\over 8}\tau_0(\alpha_2^{-1}-(\alpha_1+\alpha_2)^{-1})}
(\det(1-\tilde{T}^{66}\tilde{T}^{55}_t))^{-{1\over 2}}\over
(\det(1-\tilde{N}^{66}\tilde{N}^{55}_t))^{-{1\over 2}}}
\right|
=
-{\alpha_2\over 16\alpha_1}\left({T\over \alpha_1+\alpha_2}+
\log Z_3\right)={\cal O}(T)\,.~~~~~~~~\end{aligned}$$ (Note that $T\sim 0$ corresponds to $Z_3\sim 1$.) This implies the relation (\[eq:CG\_T\]) for $T\rightarrow +0$.
Evaluation of the coefficient $c_t$ \[sec:1-loop\]
==================================================
Using the similar method in Appendix \[sec:CG\], we cannot evaluate $c_t$ (\[eq:ct\]) because the counterpart in Fig. \[fig:CG(P)\] is 4-twisted string and we cannot refer to $T^{(4)rs}_{~~00}$ in order to solve a differential equation such as (\[eq:CG\_diffeq\]). Therefore, we consider a different regularization such as §\[sec:compC\]. Using (\[eq:tt\_D\]),(\[eq:Bnf\]) and (\[eq:Vutt\]), the determinant of Neumann coefficients is represented as: $$\begin{aligned}
\label{eq:CDp}
{\cal C}'_D&:=&e^{{D\over 8}\tau_0(\alpha_1^{-1}+\alpha_2^{-1})}
{\det}^{-{D\over 2}}(1-(\tilde{T}^{3_u3_u})^2)\nonumber\\
&=&
{}_{\alpha_1+\alpha_2}\langle p=0,w=0 |\,(\,|B_{n^f}
\rangle_{t,\alpha_1}\star|B_{n^f}\rangle_{t,\alpha_2})\,.\end{aligned}$$ We regularize ${\cal C}'_D$ by inserting $e^{-{\tau_1\over
2\alpha_r}(L_0+\tilde{L}_0-2a_t)}$ in front of $|B_{n^f}\rangle_{t,\alpha_r}$ ($r=1,2$) where $L_0-a_t=\sum_{r\ge 1/2}\alpha_{-r}^iG_{ij}\alpha_r^j+{D\over 48}$ and $\tilde{L}_0-a_t=\sum_{r\ge 1/2}\tilde{\alpha}_{-r}^iG_{ij}
\tilde{\alpha}_r^j+{D\over 48}$. In order to evaluate ${\cal C}^{\prime}_D$ using the method in §\[sec:compC\], we should take degenerate limit of $$\begin{aligned}
\label{eq:BBt}
{}_t\langle B_{n^f}|
\tilde{q}^{{1\over 2}(L_0+\tilde{L}_0-2a_t)}
|B_{n^f}\rangle_t
=\left(\eta(\tilde{\tau})\over \vartheta_0(0|\tilde{\tau})\right)^{D\over 2}
=\left(\eta(-1/\tilde{\tau})\over
\vartheta_2(0|-1/\tilde{\tau})\right)^{D\over 2}\,,\end{aligned}$$ which comes from evaluation of the amplitude in Fig. \[fig:mapping\]-b with ${\bf Z}_2$ cut along ${\rm Re}\,u=-1/2$. Similarly, we regularize $$\begin{aligned}
\label{eq:CDdef}
{\cal C}_D&:=&
{\det}^{-{D\over 2}}(1-(\tilde{N}^{33})^2)\\
&=&
{}_{\alpha_1+\alpha_2}\langle p=0,w=0 |(|B_{n^f}
\rangle_{u,\alpha_1}\star|B_{n^f}\rangle_{u,\alpha_2})(2\pi\delta(0))^{-D}
{\det}^{1\over 2}(2G)\,,\nonumber\end{aligned}$$ (which follows from (\[eq:uu\_D\])) and evaluate it by taking degenerate limit of $$\begin{aligned}
\label{eq:BBu}
&& (2\pi\delta(0))^{-D}{\det}^{1\over 2}(2G)\,{}_u\langle B_{n^f}|
\tilde{q}^{{1\over 2}(L_0+\tilde{L}_0-{D\over 12})}
|B_{n^f}\rangle_u\nonumber\\
&&~=(2\pi\delta(0))^{-D}(\det(2G))^{1\over 2}\,
\eta(-1/\tilde{\tau})^{-D}
\sum_m e^{-{2\pi i\over\tilde{\tau}}mGm}\,,\end{aligned}$$ where we have used eqs. (\[eq:Bnf\]) and (\[eq:normD\]). From (\[eq:BBt\]) and (\[eq:BBu\]), the coefficient $c_t$ (\[eq:ct\]) is evaluated as $$\begin{aligned}
\label{eq:ct_derive}
c_t&=&\sqrt{{\cal C}_D\over {\cal C}_D'}
=\lim_{\tilde{\tau}\rightarrow +i0}\left[
{(\det(2G))^{1\over 2}\over (2\pi\delta(0))^{D}}\,
{\vartheta_2(0|-1/\tilde{\tau})^{D\over 2}\over
\eta(-1/\tilde{\tau})^{3D\over 2}}
\sum_m e^{-{2\pi i\over\tilde{\tau}}mGm}
\right]^{1\over 2}\nonumber\\
&=&2^{D\over 4}(\det(2G))^{1\over 4}(2\pi\delta(0))^{-{D\over 2}}\,.\end{aligned}$$ We have used eqs. (\[eq:BBt\]) and (\[eq:BBu\]) instead of ${\cal
C}_D',{\cal C}_D$, respectively. Although this replacement itself is valid up to factor, their ratio ${\cal C}_D'/{\cal C}_D$ is invariant because they are related by the same conformal mapping (\[eq:Mandel1loop\]).
In the case of Neumann type boundary states, we evaluate $c_t$ in the same way as above. In the twisted sector $({\cal C}_D')$, we can use the same value as the Dirichlet type (\[eq:BBt\]) because of the identity: $$\begin{aligned}
\label{eq:BBtN}
{}_t\langle B_{m^f},F|\tilde{q}^{{1\over 2}(L_0+\tilde{L}_0-2a_t)}
|B_{m^f},F\rangle_t
={}_t\langle B_{n^f}|\tilde{q}^{{1\over 2}(L_0+\tilde{L}_0-2a_t)}
|B_{n^f}\rangle_t\,,\end{aligned}$$ which follows from (\[eq:Bmft\]). On the other hand, for untwisted sector, we replace (\[eq:BBu\]) with $$\begin{aligned}
\label{eq:BBuN}
&& (2\pi\delta(0))^{-D}{\det}^{1\over 2}(2G^{-1}_O)\,{}_u\langle B_{m^f},F|
\tilde{q}^{{1\over 2}(L_0+\tilde{L}_0-{D\over 12})}
|B_{m^f},F\rangle_u\nonumber\\
&&~=(2\pi\delta(0))^{-D}(\det(2G_O^{-1}))^{1\over 2}\,
\eta(-1/\tilde{\tau})^{-D}
\sum_{m}
e^{-{2\pi i\over \tilde{\tau}}m_iG_O^{ij}m_j},\end{aligned}$$ to evaluate ${\cal C}_D$. Note (\[eq:uu\_N\]) and (\[eq:Bmf\]) comparing to (\[eq:CDdef\]) for the prefactor. We have used the modular transformation in (\[eq:normN\]). This gives the ratio of the determinant $c_t=
2^{D\over 4}(\det(2G^{-1}_O))^{1\over 4}(2\pi\delta(0))^{-{D\over 2}}$ and the coefficient of the twisted term of (\[eq:fractional\_DN\]), which is consistent with T-duality transformation: $G\rightarrow G^{-1}_O$ compared to Dirichlet type idempotents (\[eq:fractional\_D\]).
Some formulae \[sec:formulae\]
===============================
For the operators $a_i,a^{\dagger}_j$ such as $[a_i,a^{\dagger}_j]=\delta_{ij}$ and $[a_i,a_j]=[a^{\dagger}_i,a^{\dagger}_j]=0$, we have a normal ordering formula: $$\begin{aligned}
\label{eq:KP}
&&e^{a^{\dagger}Aa^{\dagger}+aBa+{1\over 2}(a^{\dagger}Ca
+aC^Ta^{\dagger})+Da^{\dagger}+Ea}\nonumber \\
&&=~\det{}^{-{1\over 2}}(1-C)\,e^{E{4-C\over 12(1-C)}AE^T
+DB{4-C\over 12(1-C)}D^T+E{6-4C+C^2\over 12(1-C)}D^T}\\
&&~~~~\times\, e^{a^{\dagger}(1-C)^{-1}Aa^{\dagger}
+\left(EA+D\left(1-{C^T\over 2}\right)\right)(1-C^T)^{-1}a^{\dagger}}
e^{-a^{\dagger}\log (1-C) a}e^{aB(1-C)^{-1}a
+\left(E\left(1-{C\over 2}\right)+DB\right)(1-C)^{-1}a}~~~~
\nonumber\end{aligned}$$ for matrices $A,B,C$, which satisfy the relations $$\label{eqn:ABC}
A^T=A,\ \ B^T=B,\ \ C^2=4AB,\ \ AC^T=CA,\ \ C^TB=BC,$$ and vectors $D,E$. This formula is obtained, for example, by using similar technique in [@KP] Appendix A.
We use following formulae in order to compute (\[eq:VDstar\]) explicitly: $$\begin{aligned}
\sum_{n=1}^{\infty}
{\sin nx\over n}&=&-{1\over 2}x+{1\over 2}\pi\epsilon(x)\,,
~~~~~(|x|\le 2\pi)\,,\nonumber\\
\sum_{n=1}^{\infty}{\sin nx\sin ny\over n^2}&=&
{x(\pi-y)\over 2}-{\pi(x-y)\over 2}\theta(x-y)\,,~~~~(-y\le x\le 2\pi-y)\,,
\label{eq:formulae}\end{aligned}$$ where $\epsilon(x),\theta(x)$ are sign and step function respectively.
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[^1]: This equation takes the form of the projector equation of $C^*$ algebra. It reminds us of the fact that the topological charge carried by the D-brane is given by the K-theory [@r:K-theory]. In the context of the noncommutative geometry, an element which represents a K-theory class is given by the projection equation [@NCK], $$\label{e_projector}
\phi \star \phi =\phi\,,$$ where $\star$ is the product of the (noncommutative) background geometry. Solutions of this equation are called noncommutative soliton [@Gopakumar:2000zd]. Later, even in the bosonic string which does not have RR charge, it was argued [@Harvey:2000jt] that the noncommutative solitons still represent unstable D-branes. Topological charges of D-branes are represented in terms of the projector $\phi$. For instance, the D-brane number is related to the rank of $\phi$.
[^2]: We use the notation $b_0^+=b_0+\tilde{b}_0,c_0^-=c_0-\tilde{c}_0$ and $b_0^-={1\over 2}(b_0-\tilde{b}_0),c_0^+={1\over
2}(c_0+\tilde{c}_0)$. The extra ghost zero mode $c_0^-$ is needed for our convention of the HIKKO $\star$ product. Here, we assign the string length parameter $\alpha_1,\,\alpha_2=\alpha-\alpha_1(>0)$ to each string $1,2$ in order to use the HIKKO 3-string vertex.
[^3]: The prefactor $\alpha_1\alpha_2$ comes from the conformal factor $(dw_r/d\rho)^{-1}$ ($r=1,2$) in gluing the local disks (in $w_r$-plane) which represents strings 1 and 2 to $\rho$-plane in Fig. \[fig:mapping\]-a.
[^4]: We mainly use the convention in [@KZ] although we introduce $\sqrt{\alpha'}$ to specify a unit length. By taking $\alpha'=1$ and replacing $2\pi\alpha' B_{ij}\rightarrow B_{ij}$, we recover some formulae in [@KZ] for torus.
[^5]: This operator was also obtained using path integral formulation [@Oku] in the process of constructing boundary state for D$p$-brane from that for D-instanton.
[^6]: We often denote $\wp_1\cdots \wp_N$ as $\wp_{1\cdots N}$ where $\wp_r$ is a projector which imposes the level matching condition $L_0^{(r)}-\tilde{L}_0^{(r)}=0$ on each string field.
[^7]: This factor should be removed if we remove $(-1)^n$ in $E_u(1,2)$ and this implies a different connection condition $X^{(1)}(\sigma)-X^{(2)}(-\sigma)=0$ without $\wp_{12}$. By multiplying $\wp_{12}$, these two conventions become equivalent for the reflector $ \langle R_u(1,2)|$.
|
---
abstract: 'We study estimates of the Green’s function in ${\mathbb{R}}^d$ with $d \ge 2$, for the linear second order elliptic equation in divergence form with variable uniformly elliptic coefficients. In the case $d \ge 3$, we obtain estimates on the Green’s function, its gradient, and the second mixed derivatives which scale optimally in space, in terms of the “minimal radius” $r_*$ introduced in \[Gloria, Neukamm, and Otto: A regularity theory for random elliptic operators; ArXiv e-prints (2014)\]. As an application, our result implies optimal stochastic Gaussian bounds in the realm of homogenization of equations with random coefficient fields with finite range of dependence. In two dimensions, where in general the Green’s function does not exist, we construct its gradient and show the corresponding estimates on the gradient and mixed second derivatives. Since we do not use any scalar methods in the argument, the result holds in the case of uniformly elliptic systems as well.'
author:
- Peter Bella
- Arianna Giunti
title: 'Green’s function for elliptic systems: moment bounds'
---
Max Planck Institute for Mathematics in the Sciences
Inselstrasse 22, 04103 Leipzig, Germany
Introduction
============
This paper is a contribution to recently very active area of quantitative stochastic homogenization of second order uniformly elliptic operators, the main goal of which is to quantify how close is the large scale behavior of the heterogeneous operator $(-\nabla \cdot A(x) \nabla)^{-1}$ to the behavior of the constant-coefficient solution operator $(-\nabla \cdot {A_{\textrm{hom}}}\nabla)^{-1}$. Here $A(x)$ stands for the non-constant (random) coefficient field defined on ${\mathbb{R}}^d$ and ${A_{\textrm{hom}}}$ is called the matrix of homogenized coefficients.
As originally realized in their seminal papers by Papanicolaou and Varadhan [@papvar] and, independently, by Kozlov [@kozlov], the central object in the homogenization of elliptic operators with random coefficients is the *corrector* $\phi_\xi$, defined for each direction $\xi \in {\mathbb{R}}^d$ as a solution of the following elliptic problem $$\nonumber
-\nabla_x \cdot (A(x) \nabla_x (x \cdot \xi + \phi_\xi(A,x))) = 0$$ in the whole space ${\mathbb{R}}^d$. The function $\phi$ is called corrector since it corrects the linear function $x \cdot \xi$, which is clearly solution to the constant-coefficient equation, to be a solution of the equation with heterogeneous coefficients. Since $\phi$ serves as a correction of a linear function, it should naturally be smaller, i.e., sublinear. Assuming the distribution of random coefficient fields $A$ is stationary (meaning the joint distribution of $A$ at any two points in ${\mathbb{R}}^d$ is the same) and ergodic (meaning any shift-invariant random variable is almost surely constant, a property encoding decorrelation of coefficient fields over large scales), they showed that correctors are almost surely sublinear and can be used to define the homogenized coefficient $$\nonumber {A_{\textrm{hom}}}e_i := {\left< A(e_i + \nabla \phi_{e_i}) \right>}.$$ Since the problem is linear, it clearly suffices to study the $d$ correctors $\phi_i := \phi_{e_i}$ for $i=1,\ldots,d$. Second, borrowing notation from the statistical physics, ${\left< \cdot \right>}$ stands for the ensemble average (expected value) with respect to a probability distribution on the space of coefficient fields $A$. Here and also later, we will often drop the argument $A$ in random quantities like the corrector as well as the argument $x$ related to the spatial dependence of quantities like the coefficient field $A$ or the corrector $\phi$.
Both mentioned works [@kozlov; @papvar] were purely qualitative in the sense that they showed the sublinearity of the corrector in the limit of large scales without any rate. Assuming the correlation of the coefficient fields decays with a specific rate (either encoded by some functional inequality like the Spectral Gap estimate or the Logarithmic Sobolev Inequality, or by some mixing conditions or even assuming finite range of dependence), one goal of quantitative theory is to *quantify* the sublinearity (smallness) of the corrector and consequences thereof.
Though the present result is purely deterministic in the sense that it translates the fact that the energy of any $A$-harmonic function satisfies “mean value property” from some scale on (a fact that follows from the sublinearity of the corrector, see [@GNO4]) into estimates on Green’s function and its derivatives, we will first mention some recent results related to sublinearity of the corrector without going too much into details.
In [@GNO4], Gloria, Neukamm, and Otto introduced the random variable $r_* = r_*(A)$ called *minimal radius*, which for given fixed $\delta = \delta(d,\lambda) > 0$ (here $\lambda$ denotes the ellipticity contrast) is defined as $$\label{defrstar}
r_* := \inf \biggl\{ r \ge 1, \textrm{ dyadic } : \forall R \ge r, \textrm{ dyadic }: \frac{1}{R^2} \fint_{B_R} \biggl|(\phi,\sigma) - \fint_{B_R} (\phi,\sigma)\biggr|^2 \le \delta \biggr\}.$$ Here $(\phi,\sigma)$ stands for the augmented corrector, where the new element $\sigma$ (called vector potential) can be used for obtaining good error estimates and which was originally introduced for the periodic homogenization (see, e.g., [@AL1]). Here and in what follows $B_R$ stands for a ball of radius $R$ centered at the *origin* and $\fint$ denotes the average integral. In [@GNO4] they showed that for small enough $\delta=\delta(d,\lambda)$ the sublinearity of the corrector implies the mean value property, meaning that for $R \ge r \ge r_*$ and for any $A$-harmonic function $u$ on $B_R$ (i.e., a solution of $-\nabla \cdot A \nabla u = 0$ in $B_R$) one has $$\nonumber
\fint_{B_r} |\nabla u|^2 \le C(d,\lambda) \fint_{B_R} |\nabla u|^2.$$ Moreover, assuming that the ensemble on the coefficient fields satisfies a coarsened version of the Logarithmic Sobolev Inequality, they showed that the minimal radius $r_*$ has stretched exponential moments $$\nonumber
{\left< \exp {\left( \tfrac{1}{C} r_*^{d(1-\beta)} \right)} \right>} \le C,$$ where $0 \leq \beta < 1$ appearing in the exponent is related to the coarsening rate in the Logarithmic Sobolev Inequality. Observe that in the case $\beta = 0$, i.e., the case when we consider the Logarithmic Sobolev Inequality without coarsening, this is the optimal Gaussian bound.
Recently, reviving the parabolic approach used in the discrete setting [@GNO2short], which has a benefit of conveniently disintegrating contributions to the corrector from different scales, Gloria and Otto [@GloriaOttoNearOptimal] obtained a similar results assuming the coefficient fields have finite range of dependence (it is a known fact that the assumption of the finite range of dependence does not imply any of the functional inequalities we have mentioned above). As a by-product, assuming finite range of dependence Gloria and Otto get the estimates for the minimal radius $r_*$ with optimal stochastic integrability of the form $$\nonumber
{\left< \exp {\left( \tfrac{1}{C} r_*^{d(1-\epsilon)} \right)} \right>} < \infty,\quad \forall \epsilon > 0.$$
Finally, on the other side of the spectrum, Fischer and Otto [@FischerOtto2] combined Meyer’s estimate together with sensitivity analysis to show that for strongly correlated coefficient fields (more precisely, they consider coefficient fields which are $1$-Lipschitz images of a stationary Gaussian field with correlations bounded by $|x|^{-\beta}$, where $0 < \beta \ll 1$ is coming from Meyer’s estimates) it holds $$\nonumber
{\left< \exp {\left( \tfrac{1}{C} r_*^\beta \right)} \right>} \le C.$$
For the sake of completeness, without discussing any details, let us also mention the work of Armstrong and Smart [@armstrongsmart2014] which predates the previously mentioned works of Otto and coauthors (see also subsequent works [@ArmstrongKuusiMourrat; @ArmstrongMourrat] for more general results) and which contains estimates with the optimal stochastic integrability for some quantity related but different from $r_*$.
In the present paper we will obtain deterministic estimates for the Green’s function based on the minimal radii $r_*$ at different points. More precisely, fixing two points $x_0, y_0 \in {\mathbb{R}}^d$, we take as the input the coefficient field $A$ and the corresponding minimal radii $r_*(x_0), r_*(y_0)$ (here and in what follows $r_*(x_0)$ stays for the minimal radius $r_*$ of the shifted coefficient field $A(\cdot - x_0)$ ), and produce estimates on the Green’s function $G$ and its derivatives $\nabla_xG, \nabla_yG, \nabla_x\nabla_yG$, averaged over small scale around the points $x_0$ and $y_0$. This averaging is necessary since we do not assume any smoothness of the coefficient fields.
Our only goal in this paper is to obtain bounds, and not to show existence (or other properties) of the Green’s function. In fact, a well known counterexample of De Giorgi [@DeGiorgiCounterexample] shows that there are uniformly elliptic coefficient fields for which the Green’s function does not exist. Nevertheless, as recently shown in [@ConlonGiuntiOtto] by Conlon, Otto, and the second author, this is not a generic behavior. More precisely, in [@ConlonGiuntiOtto] they show that for *any* uniformly elliptic coefficient field $A$ the Green’s function $G=G(A;x,y)$ exists at *almost every* point $y \in {\mathbb{R}}^d$, provided the dimension $d \ge 3$. Therefore, in the case $d \ge 3$, we will assume that the Green’s function $G(A;\cdot,y) \in L^1_{\textrm{loc}}({\mathbb{R}}^d)$ exists, at least in the almost everywhere sense (i.e., for a.e. $y \in {\mathbb{R}}^d$), and focus solely on the estimates. Since in ${\mathbb{R}}^2$ the Green’s function does not have to exist, but its “gradient” can possibly exists, using a reduction from $3D$ (where the Green’s function exists) in Section \[sec2d\] we construct and estimate $\nabla G$.
There are several works studying estimates on the Green’s function in the context of uniformly elliptic equations with random coefficients. Using De Giorgi-Nash-Moser approach for a parabolic equation (which is naturally restricted to the scalar case), Delmotte and Deuschel [@delmottedeuschel] obtained annealed estimates on the first and second gradient of the Green’s function, in $L^2$ and $L^1$ in probability respectively, under mere assumption of stationarity of the ensemble (see also [@MO2] for a different approach). Using different methods, Conlon, Otto, and the second author [@ConlonGiuntiOtto] recently obtained similar estimates, together with other properties of the Green’s function, but without the restriction to the scalar case. For a single equation and in the discrete case, assuming that the spatial correlation of the coefficient fields decays sufficiently fast to the effect that the Logarithmic Sobolev Inequality is satisfied, Marahrens and Otto [@MO] upgraded the Delmotte-Deuschel bounds to any stochastic moments. Recently, for a single equation this work was extended by Gloria and Marahrens [@gloriamarahrens] into the continuum setting.
Before we state the main result, let us mention other works relating the smallness of the corrector and the properties of solutions to the heterogeneous equation. Together with Otto [@BellaGiuntiOttoPCMI], we compare the finite energy solution $u$ of $$\nonumber
- \nabla \cdot A \nabla u = \nabla \cdot g,$$ with $g \in L^2({\mathbb{R}}^d;{\mathbb{R}}^d)$ being supported in a unit ball around the origin, with *twice* corrected solution $u_\textrm{hom}$ of the homogenized equation $$\nonumber
- \nabla \cdot {A_{\textrm{hom}}}\nabla u_\textrm{hom} = \nabla \cdot \tilde g.$$ Here by twice corrected we mean that first the right-hand side $g$ from the heterogeneous equation is replaced by $\tilde g = g(\textrm{Id} + \nabla \phi)$ in the constant-coefficient equation, and second, we compare $u$ with $(\textrm{Id} + \phi_i \partial_i)v$ at the level of gradients. Using duality argument together with a compactness lemma, this gives an estimate of the difference between $\nabla_x \nabla_y G(x,y)$ and $\partial_i \partial_j G_\textrm{hom} (e_i + \nabla \phi_i(x)) \otimes (e_j + \nabla \phi_j(y))$ (averaged over small balls both in $x$ and $y$). In order to get such estimates, it is not enough to assume that the corrector is at most linear with small slope (as in ), but rather we need to assume that for some $\beta \in (0,1)$, it grows in the $L^2$-sense at most like $|x|^{1-\beta}$: $$\label{growthbeta}
\frac{1}{R^2} \fint_{B_R} \biggl|(\phi,\sigma) - \fint_{B_R} (\phi,\sigma)\biggr|^2 \le C R^{-2\beta}, \quad \forall R \ge r_*.$$ Hence, in comparison with the present work, in [@BellaGiuntiOttoPCMI] we get a stronger statement (since we estimate the difference between the heterogeneous Green’s function and corrected constant-coefficient Green’s function while in the present paper we only control the heterogeneous Green’s function alone), at the expense of stronger assumption on the smallness of the corrector and more involved proof. More precisely, here we show that the second mixed derivative of the Green’s function $\nabla_x \nabla_y G$ behaves like $C|x-y|^{-d}$ (clearly this estimate is sharp in scaling since it agrees with the behavior of the constant-coefficient Green’s function), while in [@BellaGiuntiOttoPCMI] we show that the homogenization error, i.e., the difference between $\nabla_x \nabla_y G$ and twice corrected mixed second derivative of the constant-coefficient Green’s function, is estimated by $C|x-y|^{-(d+\beta)}$ - that means we gain a factor of $|x-y|^{-\beta}$, where the exponent $\beta \in (0,1)$ is the one appearing in .
Last, let us mention the work of Otto and the authors [@BellaGiuntiOtto2nd], where we push farther the results of [@BellaGiuntiOttoPCMI] using higher order correctors. The second and higher order correctors were introduced into the stochastic homogenization setup by Fischer and Otto [@FischerOtto], in order to extend the $C^{1,\alpha}$ regularity estimates on large scales [@GNO4] to $C^{2,\alpha}$ estimates and $C^{k,\alpha}$ estimates respectively. In [@BellaGiuntiOtto2nd], under the assumption of smallness of the corrector, we obtain two results about $A$-harmonic function in exterior domains: first, for any integer $k$ we construct a finite dimensional space of functions such that the distance between any $A$-harmonic function in the exterior domain and this space is bounded by $C|x-y|^{-(d+k)}$ (this statement can be seen as an analogue of Liouville statements for finite energy solutions in the exterior domain). Second, assuming smallness of both the first and second order augmented correctors (i.e., also including the second order vector potential, which we had to introduce), compared with [@BellaGiuntiOttoPCMI] we improve by $1$ the exponent in the estimate between the solution of the heterogeneous equation in the exterior domain and of some corrected solution of the constant-coefficient equation.
The paper is organized as follows: In the next section we will state our assumptions together with the main result, Theorem \[thm1\], and its corollaries, Corollary \[cor1\], Corollary 2, and Corollary 3. In Section \[sec3d\] we prove Theorem \[thm1\] and in Section \[sec2d\] we give the argument for Corollary \[cor1\], which is the only corollary which does not immediately follow from the theorem.
[**Notation.**]{} Throughout the article, we denote by $C$ a positive generic constant which is allowed to depend on the dimension $d$ and the ellipticity contrast $\lambda$, and which may be different from line to line of the same estimate. By $\lesssim$ we will mean $\le C$. Finally, the integrals without specified domain of integration are meant as integrals over the whole space ${\mathbb{R}}^d$.
The main result
===============
We fix a coefficient field $A \in L^\infty({\mathbb{R}}^d;{\mathbb{R}}^{d\times d})$, which we assume to be uniformly elliptic in the sense that $$\label{coer}
\begin{aligned}
\int_{{\mathbb{R}}^d} \nabla \varphi \cdot A(x) \nabla \varphi {{\, \mathrm{d} x}}&\ge \lambda \int_{{\mathbb{R}}^d} {\left| \nabla \varphi \right|}^2,& \quad &\forall \varphi \in \mathcal{C}^{\infty}_c({\mathbb{R}}^d),\\
|A(x)\xi|
&\le |\xi|,& \quad &\forall \textrm{a.e. }x \in {\mathbb{R}}^d, \forall \xi \in {\mathbb{R}}^d,
\end{aligned}$$ where $\lambda \in (0,1)$ is fixed throughout the paper. Then we have the following result:
\[thm1\] Let $d \ge 3$, let $A$ be a uniformly elliptic coefficient field on ${\mathbb{R}}^d$ in the sense of , and let $x_0,y_0 \in {\mathbb{R}}^d$ with ${\left| x_0 - y_0 \right|} \ge 10$. For a point $x \in {\mathbb{R}}^d$, let $r_*(x)=r_*(A, x)$ denote a radius such that for $R \ge r \ge r_*(x)$ and any $A$-harmonic function $u$ in $B_R(x)$ we have $$\label{E37}
\fint_{B_r(x)} {\left| \nabla u \right|}^2 \le C(d,\lambda) \fint_{B_R(x)} {\left| \nabla u \right|}^2.$$ Let $G = G(A;x,y)$ be the Green’s function defined through $$\nonumber
-\nabla_x \cdot A \nabla_x G(A;\cdot,y) = \delta(\cdot-y),$$ assuming it exists for a.e. $y \in {\mathbb{R}}^d$. Then we have $$\begin{aligned}
\label{E47}
\int_{B_1(x_0)} \int_{B_1(y_0)} {\left| \nabla_x \nabla_y G(A; x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}&\le C(d,\lambda) {\left( \frac{r_*(x_0)r_*'(y_0)}{|x_0-y_0|^2}\right)}^{d},
\\ \label{E52}
\int_{B_1(x_0)} \int_{B_1(y_0)} {\left| \nabla_y G(A; x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}&\le C(d,\lambda) |x_0-y_0|^2 {\left( \frac{r_*(x_0)r_*'(y_0)}{|x_0-y_0|^2}\right)}^{d},
\\ \label{E53}
\int_{B_1(x_0)} \int_{B_1(y_0)} {\left| \nabla_x G(A; x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}&\le C(d,\lambda) |x_0-y_0|^2 {\left( \frac{r_*'(x_0)r_*(y_0)}{|x_0-y_0|^2}\right)}^{d},
\\ \label{E54}
\int_{B_1(x_0)} \int_{B_1(y_0)} {\left| G(A; x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}&\le C(d,\lambda) |x_0-y_0|^4 {\left( \frac{r_*(x_0)r_*'(y_0) + r_*'(x_0)r_*(y_0)}{|x_0-y_0|^2}\right)}^{d}.\end{aligned}$$ where $r_*'(y)=r_*(A^t, y)$ denotes the minimal radius for the adjoint coefficient field $A^t$ at a point $y$.
Though the Green’s function does not have to exist in $2D$, with the help of the Green’s function in $3D$ we can at least define and estimate “its gradient & second mixed derivatives”:
\[cor1\] Let $d=2$. Let $A$ be a uniformly elliptic coefficient field on ${\mathbb{R}}^2$ in the sense of , such that for its extension into ${\mathbb{R}}^3$ of the form $$\label{2d1}
\bar A(x,x_3) := \begin{pmatrix} A(x) & 0\ \\ 0 & 1\ \end{pmatrix}$$ there exists two points $\bar X, \bar Y \in {\mathbb{R}}^3$ so that the minimal radii for $\bar A^t$ and $\bar A$ at those points are finite, respectively (i.e., $r_*(\bar A^t,\bar X) < \infty$, $r_*(\bar A,\bar Y) < \infty$).
Then for a.e. $y \in {\mathbb{R}}^2$ there exists a function on ${\mathbb{R}}^2$, which we denote $\nabla G(A;\cdot,y)$, so that it satisfies in a weak sense $$\nonumber
-\nabla_x \cdot A \nabla G(A;\cdot ,y) = \delta(\cdot-y).$$ Moreover, given $x_0,y_0 \in {\mathbb{R}}^2$ with ${\left| x_0 - y_0 \right|} \ge 10$, we have estimates for $\nabla G$ as well as for $\nabla_y \nabla G$: $$\begin{aligned}
\int_{B_1(x_0)}\int_{B_1(y_0)} |\nabla_y\nabla{G}(A; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}&\le C(\lambda) \frac{{\left(r_*(A,x_0)r_*(A^t,y_0)\right)}^2}{|x_0-y_0|^4},\label{2d5b}\\
\int_{B_1(x_0)}\int_{B_1(y_0)} |\nabla{G}(A; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}&\le C(\lambda) \frac{{\left(r_*(\bar A^t,(x_0,0))r_*(\bar A,(y_0,0))\right)}^2}{|x_0-y_0|^2}.\label{2d5}
$$
Assuming that the coefficient field $A$ in the statement of Corollary \[cor1\] is chosen at random with respect to a stationary and ergodic ensemble, by the standard ergodic argument (see, e.g., [@GNO4]), applied in $3D$ to the ensemble obtained as a push-forward of the $2D$ ensemble through , the assumption on the finiteness of the minimal radii is almost surely satisfied (say with $\bar X=\bar Y=(0,0,0)$).
\[rmk1\] It is clear from the proof of Theorem \[thm1\] that all the above estimates, i.e. -, are true also if the domains of integration $B_1(x_0)$ and $B_1(y_0)$ are replaced by larger balls with the corresponding radii $r_*$. Moreover, the radii of the balls could be even larger than the minimal radii (as long as these new radii are not larger than one third of a distance between centers of those balls), in which case we need to replace the minimal radii on the right-hand sides of those estimates with the actual radii of the balls.
The appearance of different minimal radii in and (in the minimal radii are related to the equation in $2D$, while in they are the minimal radii for the equation in $3D$) is not a typo. The reason is that while is proved directly in $2D$, the proof of passes through $3D$ - hence the need to consider the minimal radii in $3D$. In view of the relation $r_*(A,x) \le r_*(\bar A,\bar x)$, which easily follows from the fact that any $A$-harmonic function in $B_R \subset {\mathbb{R}}^2$ can be trivially extended to an $\bar A$-harmonic function in $\bar B_R \subset {\mathbb{R}}^3$, the estimate seems to be less optimal.
For notational convenience we state the result for a single equation. Since in the proof of Theorem \[thm1\] we do not use any *scalar* methods (like for example De Giorgi-Nash-Moser iteration), the result holds also in the case of elliptic systems - for that one just considers that $u$ has values in some finite-dimensional Hilbert space. Naturally, in that case all the constants will depend on the dimension of this Hilbert space.
Using the Gaussian bounds on $r_*$ for the case of coefficient fields with finite range of dependence, which were obtained recently in [@GloriaOttoNearOptimal], Theorem \[thm1\] implies the following bounds:
Suppose ${\left< \cdot \right>}$ is an ensemble of $\lambda$-uniformly elliptic coefficient fields which is stationary and of unity range of dependence, and let $d \ge 2$. Then there exist $C(d,\lambda)$ such that for every two points $x_0,y_0 \in {\mathbb{R}}^d$, $|x_0 - y_0| \ge 10$, and every $\epsilon > 0$ we have $$\begin{aligned}
{\left< \exp \biggl( \biggl( C|x_0 - y_0|^{2d} \int_{B_1(x_0)} \int_{B_1(y_0)} |\nabla_x \nabla_y G( A ; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\biggr)^{d(1-\epsilon)} \biggr) \right>} &< \infty,
\\
{\left< \exp \biggl( \biggl( C|x_0 - y_0|^{2d-2} \int_{B_1(x_0)} \int_{B_1(y_0)} |(\nabla_x,\nabla_y) G( A ; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\biggr)^{d(1-\epsilon)} \biggr) \right>} &< \infty,\end{aligned}$$ and in $d \ge 3$ also $$\nonumber
{\left< \exp \biggl( \biggl( C|x_0 - y_0|^{2d-4} \int_{B_1(x_0)} \int_{B_1(y_0)} |G( A ; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\biggr)^{d(1-\epsilon)} \biggr) \right>} < \infty.$$
In the case of coefficient fields with stronger correlations we can use the result from [@GNO4]:
Suppose $d \ge 2$, and that the ensemble ${\left< \cdot \right>}$ is stationary and satisfies a logarithmic Sobolev inequality of the following type: There exists a partition $\{ D \}$ of ${\mathbb{R}}^d$ not too coarse in the sense that for some $0 \le \beta < 1$ it satisfies $$\textrm{diam} (D) \le (\textrm{dist}(D) + 1)^\beta \le C(d) \textrm{diam}(D).$$ Moreover, let us assume that there is $0 < \rho \le 1$ such that for all random variables $F$ $${\left< F^2 \log F^2 \right>} - {\left< F^2 \right>} \log {\left< F^2 \right>} \le \frac{1}{\rho} \biggl< \left\| \frac{\partial F}{\partial A}\right\|^2 \biggr>,$$ where the carré-du-champ of the Malliavin derivative is defined as $$\biggl\| \frac{\partial F}{\partial A}\biggr\|^2 := \sum_D \biggl( \int_D \biggl|\frac{\partial F}{\partial A}\biggr| ^2 \biggr).$$ Then there exists a constant $0 < C < \infty$, depending only on $d, \lambda, \rho, \beta$, such that $$\begin{aligned}
{\left< \exp \biggl( \biggl( C|x_0 - y_0|^{2d} \int_{B_1(x_0)} \int_{B_1(y_0)} |\nabla_x \nabla_y G( A ; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\biggr)^{d(1-\beta)} \biggr) \right>} &< \infty,
\\
{\left< \exp \biggl( \biggl( C|x_0 - y_0|^{2d-2} \int_{B_1(x_0)} \int_{B_1(y_0)} |(\nabla_x,\nabla_y) G( A ; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\biggr)^{d(1-\beta)} \biggr) \right>} &< \infty,\end{aligned}$$ and in $d \ge 3$ also $${\left< \exp \biggl( \biggl( C|x_0 - y_0|^{2d-4} \int_{B_1(x_0)} \int_{B_1(y_0)} |G( A ; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\biggr)^{d(1-\beta)} \biggr) \right>} < \infty.$$
Proof of Theorem \[thm1\] {#sec3d}
=========================
The proof is inspired by a duality argument of Avellaneda and Lin [@AL1 Theorem 13], which they used to obtain Green’s function estimates in the periodic homogenization. After stating and proving two auxiliary lemmas, we first prove the estimate on the second mixed derivative . Then, will follow from using Poincaré inequality and one additional estimate. Next we observe that can be obtained from by replacing the role of $x$ and $y$, which can be done by considering the adjoint $A^t$ instead of $A$. Finally, will follow from in a similar way as follows from .
We thus start with the following two auxiliary lemmas. The first one is very standard:
\[lmcac\] Let $\rho > 0$, $\delta > 0$, and let $u$ be a solution of a uniformly elliptic equation $-\nabla \cdot A\nabla u = 0$ in $B_{(1+\delta)\rho}$. Then $$\label{E99}
\int_{B_\rho} {\left| \nabla u \right|}^2 \le \frac{C(d)}{\lambda\rho^2\delta^2} \int_{B_{(1+\delta)\rho}} {\left| u - c \right|}^2$$ for any $c \in {\mathbb{R}}$.
By considering $u-c$ instead of $u$, it is enough to show estimate (\[E99\]) with $c = 0$. We test the equation for $u$ with $\eta^2 u$, where $\eta$ is a smooth cut-off function for $B_\rho$ in $B_{(1+\delta)\rho}$ with ${\left| \nabla \eta \right|} \lesssim (\delta\rho)^{-1}$, use and Young’s inequality to get $$\nonumber
\int_{{\mathbb{R}}^d} {\left| \nabla (\eta u) \right|}^2 \le \frac{C(d)}{\lambda} \int {\left| \nabla \eta \right|}^2 u^2.$$ Since ${\left| \nabla \eta \right|} \le \frac{C}{\rho \delta}$, immediately follows.
\[lm1\] Let $R_0 \ge r_*(0)$, and let $u$ be an $A$-harmonic function in $B_{R_0}$. Then we have $$\label{E130}
\fint_{B_{r_*(0)}} {\left| u \right|}^2 \le C(d,\lambda) \fint_{B_{R_0}} {\left| u \right|}^2.$$
Throughout the proof we write $r_*$ instead of $r_*(0)$. We assume that $2r_* < R_0$, since otherwise is trivial. For $r \in [r_*,R_0]$ we denote $u_r := \fint_{B_r} u$. We have $$\begin{aligned}
\fint_{B_{r_*}} {\left| u - u_{r_*} \right|}^2 &\overset{\textrm{Poincar\'e}}{\lesssim} r_*^2 \fint_{B_{r_*}} {\left| \nabla u \right|}^2 \overset{\eqref{E37}}{\lesssim} r_*^2 \fint_{B_{{R_0/2}}} {\left| \nabla u \right|}^2
\\ &{\underset{\phantom{\textrm{Poincar\'e}}}{\overset{\eqref{E99}}{\lesssim}}}
{\left( \frac{r_*}{R_0} \right)}^{2} \fint_{B_{R_0}} {\left| u \right|}^2 \le \fint_{B_{R_0}} {\left| u \right|}^2.
\end{aligned}$$ Hence, to prove it is enough to show $$\label{E198}
{\left| u_{r_*} \right|}^2 = {\left| \fint_{B_{r_*}} u \right|}^2 \lesssim \fint_{B_{R_0}} {\left| u \right|}^2.$$ To prove it, we use the following estimate $$\label{E199}
{\left| u_r - u_{2r} \right|} \lesssim r {\left( \fint_{B_{2r}} {\left| \nabla u \right|}^2 \right)}^{{\textstyle\frac{1}{2}}},$$ which in fact holds for any function $u \in W^{1,2}(B_{2r})$.
We first argue how to obtain (\[E198\]) the proof thanks to estimate (\[E199\]): Let $n \ge 0$ be the largest integer that satisfies $2^n r_* \le R_0/2$; using multiple times we get $$\begin{aligned}
{\left| u_{r_*} - u_{2^n r_*} \right|} &{\underset{\phantom{\eqref{E37}}}{\overset{}{\le}}} \sum_{k=0}^{n-1} {\left| u_{2^k r_*} - u_{2^{k+1} r_*} \right|} \overset{\eqref{E199}} \lesssim \sum_{k=0}^{n-1} 2^k r_* {\left( \fint_{B_{2^{k+1}r_*}} {\left| \nabla u \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}\\
&\overset{\eqref{E37}}\lesssim {\left( \fint_{B_{R_0/2}} {\left| \nabla u \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}\sum_{k=0}^{n-1} 2^k r_* \overset{\eqref{E99}} \lesssim R_0 {\left( \frac{1}{R_0^2} \fint_{B_{R_0}} {\left| u \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}= {\left( \fint_{B_{R_0}} {\left| u \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}.\end{aligned}$$ Using Jensen’s inequality and the fact that $R_0 \le 2^{n+2}r_*$ we get $$\nonumber
{\left| u_{2^n r_*} \right|} = {\left| \fint_{B_{2^n r_*}} u \right|} \le {\left( \fint_{B_{2^n r_*}} {\left| u \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}\lesssim {\left( \fint_{B_{R_0}} {\left| u \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}.$$ Combination of the two previous estimates then gives .
It remains to prove . Using Jensen’s and Poincaré’s inequalities we get $$\begin{aligned}
{\left| u_r - u_{2r} \right|} &= {\left| \fint_{B_r} (u-u_r) - (u-u_{2r}) \right|} \lesssim \fint_{B_r} {\left| u - u_r \right|} + \fint_{B_{2r}} {\left| u - u_{2r} \right|}
\\
&\lesssim {\left( \fint_{B_r} {\left| u - u_r \right|}^2\right)}^{{\textstyle\frac{1}{2}}}+ {\left(\fint_{B_{2r}} {\left| u - u_{2r} \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}\lesssim r {\left( \fint_{B_{2r}} {\left| \nabla u \right|}^2\right)}^{{\textstyle\frac{1}{2}}}.\end{aligned}$$
Proof of
---------
We denote $R_0 := |x_0 - y_0|/3$. We split the proof of into 4 steps. In the first step we show that $$\label{E174}
\int_{B_1(y_0)} {\left| {F_\rho}{\left(\nabla_x \nabla_y G(\cdot,y)\right)} \right|}^2 {{\, \mathrm{d} y}}\lesssim {\left( \frac{r_*(x_0)r_*'(y_0)}{R_0^2}\right)}^{d}$$ for any $\rho \in [1,2]$ and any functional $F_\rho$ on $L^2(B_\rho(x_0))$ which satisfies $$\label{E62}
{\left| {F_\rho}(\nabla v) \right|}^2 \le \int_{B_\rho(x_0)} {\left| \nabla v \right|}^2,$$ for any $v \in W^{1,2}(B_\rho(x_0))$. In the second step, using Neumann eigenfunctions on a ball, we define a family of functionals $F_k$ satisfying , which will play the role of Fourier coefficients. Using these $F_k$ we then estimate $\int {\left| v \right|}^2$ with a sum of $N$ terms ${\left| F_k(v) \right|}^2$ plus a residuum in the form $\frac{1}{\lambda_N} \int {\left| \nabla v \right|}^2$. Here, $\lambda_N$ denotes the $N$-th Neumann eigenvalue of Laplacian on a ball. Using together with the second step we get an estimate on $\int {\left| \nabla_x \nabla_y G \right|}^2$ in terms of a (good) term and a small prefactor times $\int {\left| \nabla_x \nabla_y G \right|}^2$, integrated over a slightly larger ball. In the last step, we apply iteratively this estimate.
[**Step 1.**]{} Proof of (inspired by the duality argument of Avellaneda and Lin [@AL1]).
Let $f \in L^2(B_{R_0}(y_0);{\mathbb{R}}^d)$, and let $u$ be the finite energy solution of $$\nonumber
-\nabla \cdot A \nabla u = -\nabla \cdot f$$ in ${\mathbb{R}}^d$, for which holds the energy estimate $$\label{energy1}
\int_{{\mathbb{R}}^d} {\left| \nabla u \right|}^2 \lesssim \int_{{\mathbb{R}}^d} {\left| f \right|}^2.$$ Then on the one hand, the Green’s function representation formula yields $$\label{greenrep}
\nabla u(x) = \int_{B_{R_0}(y_0)} \nabla_x \nabla_y G(x,y) f(y) {{\, \mathrm{d} y}}.$$ If $r_*(x_0) \le R_0$ (w. l. o. g. we assume $r_*(x_0) \ge \rho$), we use this in to get $$\begin{gathered}
\nonumber
{\left| {F_\rho}(\nabla u) \right|}^2 \le \int_{B_\rho(x_0)} {\left| \nabla u \right|}^2 {{\, \mathrm{d} x}}\le \int_{B_{r_*(x_0)}(x_0)} {\left| \nabla u \right|}^2 \lesssim {\left( \frac{r_*(x_0)}{R_0}\right)}^d \int_{B_{R_0}(x_0)} {\left| \nabla u \right|}^2
\\
\overset{\eqref{energy1}}{\lesssim} {\left( \frac{r_*(x_0)}{R_0}\right)}^d \int_{{\mathbb{R}}^d} {\left| f \right|}^2. \end{gathered}$$ If $r_*(x_0) \ge R_0$, we simply have $$\nonumber
{\left| {F_\rho}(\nabla u) \right|}^2 \le \int_{B_\rho(x_0)} {\left| \nabla u \right|}^2 {{\, \mathrm{d} x}}\overset{\eqref{energy1}}\lesssim \int_{{\mathbb{R}}^d} {\left| f \right|}^2 {{\, \mathrm{d} x}}\le {\left( \frac{r_*(x_0)}{R_0}\right)}^d \int_{{\mathbb{R}}^d} {\left| f \right|}^2.$$
Since ${F_\rho}$ is linear, using we have $$\nonumber
{F_\rho}(\nabla u) = \int_{B_{R_0}(y_0)} {F_\rho}{\left( \nabla_x \nabla_y G(\cdot,y)\right)} f(y) {{\, \mathrm{d} y}},$$ where the dot means that ${F_\rho}$ acts on the first variable. The previous relations then give $$\nonumber
{\left| \int_{B_{R_0}(y_0)} {F_\rho}{\left( \nabla_x \nabla_y G(\cdot,y)\right)} f(y) {{\, \mathrm{d} y}}\right|}^2 \lesssim {\left( \frac{r_*(x_0)}{R_0}\right)}^d \int_{B_{R_0}(x_0)} {\left| f \right|}^2.$$ Using definition of the norm $L^2(B_{R_0}(y_0))$ by duality we get $$\label{E98}
\int_{B_{R_0}(y_0)} {\left| {F_\rho}{\left( \nabla_x \nabla_y G(\cdot,y)\right)} \right|}^2 {{\, \mathrm{d} y}}\lesssim {\left( \frac{r_*(x_0)}{R_0}\right)}^{d}.$$
Let $r_*'(y_0)$ play the same role as $r_*(x_0)$ but for the adjoint equation. In the case $r_*'(y_0) \le R_0$, since $y \mapsto {F_\rho}(\nabla_x G(\cdot,y))$ solves the adjoint equation $-\nabla \cdot A^t \nabla{F_\rho}(\nabla_x G(\cdot,y))=0$ in $B_{R_0}(y_0)$, an analogue of implies $$\begin{gathered}
\label{E88}
\int_{B_1(y_0)} {\left| {F_\rho}(\nabla_x \nabla_y G(\cdot,y)) \right|}^2 {{\, \mathrm{d} y}}\le \int_{B_{r_*'(y_0)}(y_0)} {\left| {F_\rho}(\nabla_x \nabla_y G(\cdot,y)) \right|}^2 {{\, \mathrm{d} y}}\\
\lesssim {\left( \frac{r_*'(y_0)}{R_0}\right)}^{d} \int_{B_{R_0}(y_0)} {\left| {F_\rho}(\nabla_x \nabla_y G(\cdot,y)) \right|}^2 {{\, \mathrm{d} y}}\overset{\eqref{E98}}{\lesssim} {\left( \frac{r_*(x_0)r_*'(y_0)}{R_0^2}\right)}^{d}.\end{gathered}$$ If $r_*'(y_0) \ge R_0$, we get the same conclusion for free: $$\begin{aligned}
\int_{B_1(y_0)} {\left| {F_\rho}(\nabla_x \nabla_y G(\cdot,y)) \right|}^2 {{\, \mathrm{d} y}}&{\underset{\phantom{\eqref{E98}}}{\overset{}{\le}}} \int_{B_{R_0}(y_0)} {\left| {F_\rho}(\nabla_x \nabla_y G(\cdot,y)) \right|}^2 {{\, \mathrm{d} y}}\\
&\overset{\eqref{E98}}\lesssim {\left( \frac{r_*(x_0)}{R_0}\right)}^{d} \le {\left( \frac{r_*(x_0)r_*'(y_0)}{R_0^2}\right)}^{d}.\end{aligned}$$
[**Step 2.**]{} Let $\rho \in [1,2)$ and $\delta > 0$ be fixed such that $(1+\delta)\rho \le 2$. For $n \ge 0$, let $F_n$ denote the functional on $L^2(B_{(1+\delta)\rho})$, defined as an inner product of the $n$-th eigenfunction of Neumann Laplacian with $v$, and let $\lambda_n$ be the associated eigenvalue. By writing any $v \in W^{1,2}(B_{(1+\delta)\rho})$ in terms of Neumann eigenfunctions (which can be done since they form an orthonormal basis in $L^2$) we get $$\label{E110}
\begin{aligned}
\int_{B_{(1+\delta)\rho}} {\left| v \right|}^2 &= \sum_{k=0}^\infty {\left| F_k(v) \right|}^2 =
\sum_{k=0}^{N-1} {\left| F_k(v) \right|}^2 + \sum_{k=N}^\infty \frac{1}{\lambda_k} {\left| F_k(\nabla v) \right|}^2 \\
&\le \sum_{k=0}^{N-1} {\left| F_k(\nabla v) \right|}^2 + \frac{1}{\lambda_N} \int_{B_{(1+\delta)\rho}} {\left| \nabla v \right|}^2.
\end{aligned}$$
[**Step 3.**]{} Combination of Step 1 and Step 2 (applied to $\nabla_y G(\cdot,y)$) and use of yields $$\begin{aligned}\label{43}
&\int_{B_1(y_0)} \int_{B_\rho(x_0)} {\left| \nabla_x \nabla_y G(x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\overset{\eqref{E99}}\lesssim
\frac{1}{\delta^2} \int_{B_1(y_0)} \int_{B_{(1+\delta)\rho}(x_0)} {\left| \nabla_y G(x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\le
\\
&\overset{\eqref{E110}}\lesssim \frac{1}{\delta^2} {\left( \sum_{k=0}^{N-1} \int_{B_1(y_0)} {\left| F_k(\nabla_x \nabla_y G(\cdot,y)) \right|}^2 + \frac{1}{\lambda_N} \int_{B_1(y_0)} \int_{B_{(1+\delta)\rho}(x_0)} {\left| \nabla_x \nabla_y G(x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\right)}
\\
&\overset{\eqref{E88}}\lesssim \frac{1}{\delta^2} {\left( N {\left( \frac{r_*(x_0)r_*'(y_0)}{R_0^2}\right)}^{d} + \frac{1}{\lambda_N} \int_{B_1(y_0)} \int_{B_{(1+\delta)\rho}(x_0)} {\left| \nabla_x \nabla_y G(x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\right)}.
\end{aligned}$$
[**Step 4.**]{} For a given sequence $\delta_k > 0$ such that $\rho \Pi_{k=1}^\infty (1+\delta_k) \le 2$ we consider the following iteration procedure. Let $\rho_0 := 1$, and for $k\ge 1$ set $\rho_k := (1+\delta_k)\rho_{k-1}$. We denote $$\nonumber
M_k := {\left( \frac{r_*(x_0)r_*'(y_0)}{R_0^2}\right)}^{-d} \int_{B_1(y_0)} \int_{B_{\rho_k}(x_0)} {\left| \nabla_x \nabla_y G(x,y)) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}.$$ For any $N_k \ge 1$, estimate in Step 3 yields $$\label{E121}
M_k \le \frac{C}{\delta_k^2} N_k + \frac{C}{\delta_k^2} \frac{1}{\lambda_N} M_{k+1},$$ where the values of $\delta_k$ and $N_k$ are at our disposal. We choose $\delta_k := (2k)^{-2}$ and $N_k := \alpha k^{2d} 2^d$. Since $\Pi_{k=1}^\infty (1+\frac{1}{4k^2}) \sim 1.46 \le 2$, for this choice of $\delta_k$ for all $k\ge 1$ we have $\rho_k \in [1,2]$. Using lower bound on the Neumann eigenvalues for the ball in the form $\lambda_k \ge Ck^{\frac{2}{d}}$ (in the case of a cube one can use trigonometric functions to explicitly write down the formula for eigenfunctions and eigenvalues; for the ball, one uses the monotonicity of the eigenvalues with respect to the domain, which follows from the variational formulation of the eigenvalues), we can find large enough $\alpha$ such that the prefactor in front of $M_{k+1}$ above satisfies $$\nonumber
\frac{C}{\delta_k^2} \frac{1}{\lambda_N} \le C' k^4 (\alpha k^{2d} 2^d)^{-\frac{2}{d}} = \frac{C'}{\alpha^{\frac{2}{d}}} \frac{1}{4} \le \frac{1}{4}.$$ For this choice turns into $$\nonumber
M_k \le C\alpha k^4 k^{2d} 2^d + \frac{1}{4} M_{k+1}.$$ Iterating this we get $$\nonumber
M_1 \le C\alpha \sum_{k=1}^K 4^{-k} k^4 k^{2d} 2^d + {\left(\frac{1}{4}\right)}^{K} M_{K+1}.$$ Assuming we have $\sup_k M_k < \infty$, we send $K \to \infty$ to get $$\nonumber
M_1 \le C\alpha2^d \sum_{k=1}^\infty 4^{-k} k^{4+2d}.$$ Since the sum on the right-hand side is summable, we get that $M_1 \lesssim 1$.
It remains to justify the assumption $\sup_k M_k < \infty$. For any $\Lambda \ge 1$, let $\chi_\Lambda(y)$ be the characteristic function of the set $\left\{ y \in B_1(y_0) : \int_{B_2(x_0)} |\nabla_x \nabla_y G(x,y)|^2 \le \Lambda \right\}$. Using the previous arguments, applied to $\nabla_x\nabla_y G(x,y) \chi_\Lambda(y)$, we get that $$\nonumber
{\left( \frac{r_*(x_0)r_*'(y_0)}{R_0^2}\right)}^{-d} \int_{B_1(y_0)} \biggl( \int_{B_{1}(x_0)} {\left| \nabla_x \nabla_y G(x,y)) \right|}^2 {{\, \mathrm{d} x}}\biggr) \chi_\Lambda(y) {{\, \mathrm{d} y}}\le C,$$ where the right-hand side does not depend on $\Lambda$. Now we send $\Lambda \to \infty$, and get $$\nonumber
M_1 = {\left( \frac{r_*(x_0)r_*'(y_0)}{R_0^2}\right)}^{-d} \int_{B_1(y_0)} \int_{B_{1}(x_0)} {\left| \nabla_x \nabla_y G(x,y)) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\le C$$ by the Monotone Convergence Theorem. This completes the proof of .
Proof of
---------
We first observe that using Poincaré’s inequality we can control the difference between $\nabla_y G$ and its averages over $B_1(x_0)$ by the $L^2$-norm of $\nabla_x \nabla_y G$, which we already control by . Hence, to obtain it is enough to estimate averages $\fint_{B_1(x_0)} \nabla_y G(x,y) {{\, \mathrm{d} x}}$. Such estimate will follow from an analogue of applied to one particular functional $F$. Since in this setting we need to work with $\int {\left| u \right|}^2$ and not with previously used $\int {\left| \nabla u \right|}^2$, we will need to use Lemma \[lm1\].
[**Step 1.**]{} By Poincaré inequality in the $x$-variable we have $$\begin{aligned}
\label{E288}
&\int_{B_1(y_0)} {\left( \int_{B_1(x_0)} {\left| \nabla_y G(x,y) - {\left( \fint_{B_1(x_0)} \nabla_y G(x',y) {\, \mathrm{d} x}' \right)} \right|}^2 {{\, \mathrm{d} x}}\right)} {{\, \mathrm{d} y}}\\ \nonumber
&{\underset{\phantom{\eqref{E47}}}{\overset{}{\lesssim}}} \int_{B_1(y_0)} \int_{B_1(x_0)} {\left| \nabla_x \nabla_y G(x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\\ \nonumber
&\overset{\eqref{E47}}\lesssim {\left( \frac{r_*(x_0)r_*'(y_0)}{R_0^2}\right)}^{d}.\end{aligned}$$ By the triangle inequality we have $$\begin{aligned}
&\int_{B_1(y_0)} \int_{B_1(x_0)} {\left| \nabla_y G(x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\\
&\lesssim \int_{B_1(y_0)} {\left( \int_{B_1(x_0)} {\left| \nabla_y G(x,y) - {\left( \fint_{B_1(x_0)} \nabla_y G(x',y) {\, \mathrm{d} x}' \right)} \right|}^2 {{\, \mathrm{d} x}}\right)} {{\, \mathrm{d} y}}\\
&\quad +{\left| B_1 \right|} \int_{B_1(y_0)} {\left( \fint_{B_1(x_0)} \nabla_y G(x,y) {{\, \mathrm{d} x}}\right)}^2 {{\, \mathrm{d} y}},\end{aligned}$$ and so follows from provided we show $$\begin{aligned}
\label{E312}
\int_{B_1(y_0)} {\left( \fint_{B_1(x_0)} \nabla_y G(x,y) {{\, \mathrm{d} x}}\right)}^2 {{\, \mathrm{d} y}}\lesssim \frac{{\left(r_*(x_0)r_*'(y_0)\right)}^d}{R_0^{2d-2}}.\end{aligned}$$ [**Step 2.**]{} Proof of . Similarly as for , consider arbitrary $f \in L^2(B_{R_0}(y_0);{\mathbb{R}}^d)$ and the finite energy solution $u$ of $$\nonumber
- \nabla \cdot A \nabla u = -\nabla \cdot f$$ in ${\mathbb{R}}^d$, which satisfies the energy estimate $$\label{E246}
\int_{{\mathbb{R}}^d} {\left| \nabla u \right|}^2 \lesssim \int_{{\mathbb{R}}^d} {\left| f \right|}^2.$$ Let $F$ be a linear functional on $L^2(B_1(x_0))$ such that ${\left| F(v) \right|}^2 \le \int_{B_1(x_0)} {\left| v \right|}^2$ for any $v \in L^2(B_1(x_0))$. Then, if $r_*(x_0) \leq R_0$ $$\begin{aligned}
{\left| F(u) \right|}^2 &{\underset{\phantom{\textrm{Sobolev}}}{\overset{}{\le}}} \int_{B_1(x_0)} {\left| u \right|}^2 \le \int_{B_{r_*(x_0)}(x_0)} {\left| u \right|}^2 \overset{\textrm{Lemma \ref{lm1}}}\lesssim r_*^d(x_0) \fint_{B_{R_0}(x_0)} {\left| u \right|}^2
\\
&{\underset{\phantom{\textrm{Sobolev}}}{\overset{\textrm{Jensen}}{\le}}} r_*^d(x_0) {\left( \fint_{B_{R_0}(x_0)} {\left| u \right|}^{\frac{2d}{d-2}} \right)}^{\frac{d-2}{d}}
\lesssim \frac{r_*^d(x_0)}{R_0^{d-2}} {\left( \int_{{\mathbb{R}}^d} {\left| u \right|}^{\frac{2d}{d-2}} \right)}^{\frac{d-2}{d}}
\\
&\overset{\textrm{Sobolev}}\lesssim \frac{r_*^d(x_0)}{R_0^{d-2}} \int_{{\mathbb{R}}^d} {\left| \nabla u \right|}^2 \overset{\eqref{E246}}\lesssim \frac{r_*^d(x_0)}{R_0^{d-2}} \int_{{\mathbb{R}}^d} {\left| f \right|}^2. \end{aligned}$$ If otherwise $r_*(x_0) > R_0$, we do not need anymore to appeal to Lemma \[lm1\] and may directly bound $$\begin{aligned}
{\left| F(u) \right|}^2 &{\underset{\phantom{\textrm{Sobolev}}}{\overset{}{\le}}} \int_{B_1(x_0)} {\left| u \right|}^2 \le \int_{B_{R_0}(x_0)} {\left| u \right|}^2 \lesssim r_*^d(x_0) \fint_{B_{R_0}(x_0)} {\left| u \right|}^2 \end{aligned}$$ and proceed as in the previous inequality. As before, we use linearity of $F$ and write $$\nonumber
{\left| F(u) \right|} = {\left| \int_{B_{R_0}(y_0)} F( \nabla_y G(\cdot,y) ) f(y) {{\, \mathrm{d} y}}\right|}.$$ Since $f \in L^2(B_{R_0}(y_0);{\mathbb{R}}^d)$ was arbitrary, combination of the two previous estimates yields $$\label{E260}
\int_{B_{R_0}(y_0)} {\left| F(\nabla_y G(\cdot,y)) \right|}^2 {{\, \mathrm{d} y}}\lesssim \frac{r_*^d(x_0)}{R_0^{d-2}}.$$ As before, it remains to argue that by going from $\int_{B_{R_0}(y_0)}$ to $\int_{B_1(y_0)}$ we gain a factor $R_0^{-d}$. We define $v(y) := F(G(\cdot,y))$, and observe that $-\nabla A^t \nabla v = 0$ in $B_{R_0}(y_0)$, where $A^t$ denotes the adjoint coefficient field. Then by definition of $v$ estimate implies $$\label{E268}
\begin{aligned}
\int_{B_{1}(y_0)} {\left| F(\nabla_y G(\cdot,y)) \right|}^2 {{\, \mathrm{d} y}}&= \int_{B_{1}(y_0)} {\left| \nabla v \right|}^2 {{\, \mathrm{d} y}}\le \int_{B_{r_*'(y_0)}(y_0)} {\left| \nabla v \right|}^2 {{\, \mathrm{d} y}}\\
&\lesssim {\left(\frac{r_*'(y_0)}{R_0}\right)}^d \int_{B_{R_0}(y_0)} {\left| \nabla v \right|}^2 \lesssim \frac{{\left(r_*(x_0)r_*'(y_0)\right)}^d}{R_0^{2d-2}}.
\end{aligned}$$ For the choice $F(v) = \fint_{B_1(x_0)} v$ is exactly .
Proof of
---------
Similarly to the proof of , we use Poincaré’s inequality (Step 1) to show that follows from provided we control averages of $G$ (Step 2).
[**Step 1.**]{} By Poincaré’s inequality in the $x$-variable we have $$\begin{aligned}
\nonumber &\int_{B_1(y_0)} {\left( \int_{B_1(x_0)} {\left| G(x,y) - {\left( \fint_{B_1(x_0)} G(x',y) {\, \mathrm{d} x}' \right)} \right|}^2 {{\, \mathrm{d} x}}\right)} {{\, \mathrm{d} y}}\\ \nonumber
&{\underset{\phantom{\eqref{E53}}}{\overset{}{\lesssim}}} \int_{B_1(y_0)} \int_{B_1(x_0)} {\left| \nabla_x G(x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\\ \nonumber
&\overset{\eqref{E53}}\lesssim R_0^2 {\left( \frac{r_*'(x_0)r_*(y_0)}{R_0^2}\right)}^{d}.\end{aligned}$$ Then by the triangle inequality we have $$\begin{aligned}
\nonumber
&\int_{B_1(y_0)} \int_{B_1(x_0)} {\left| G(x,y) \right|}^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\\ \nonumber
&\lesssim \int_{B_1(y_0)} {\left( \int_{B_1(x_0)} {\left| G(x,y) - {\left( \fint_{B_1(x_0)} G(x',y) {\, \mathrm{d} x}' \right)} \right|}^2 {{\, \mathrm{d} x}}\right)} {{\, \mathrm{d} y}}\\ \nonumber
&\quad + {\left| B_1 \right|} \int_{B_1(y_0)} {\left( \fint_{B_1(x_0)} G(x,y) {{\, \mathrm{d} x}}\right)}^2 {{\, \mathrm{d} y}},\end{aligned}$$ and so follows provided we show $$\begin{aligned}
\label{E402}
\int_{B_1(y_0)} {\left( \fint_{B_1(x_0)} G(x,y) {{\, \mathrm{d} x}}\right)}^2 {{\, \mathrm{d} y}}\lesssim \frac{{\left(r_*(x_0)r_*'(y_0)\right)}^d}{R_0^{2d-4}}.\end{aligned}$$
[**Step 2.**]{} Proof of . Similarly as for , consider arbitrary $f \in L^2(B_{R_0}(y_0))$, but this time $u$ being a finite energy solution of $$\nonumber
- \nabla \cdot A \nabla u = f$$ in ${\mathbb{R}}^d$. In order to get the energy estimate, we test the equation with $u$ to obtain: $$\begin{aligned}
\lambda \int_{{\mathbb{R}}^d} {\left| \nabla u \right|}^2 &{\underset{\phantom{\textrm{Jensen},d\ge 3}}{\overset{}{\le}}} \int_{B_{R_0}(y_0)} f u \le R_0^\frac{d}{2}{\left( \int_{B_{R_0}(y_0)} {\left| f \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}{\left( \fint_{B_{R_0}(y_0)} {\left| u \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}\\
&\overset{\textrm{Jensen},d\ge 3}\le
R_0^\frac{d}{2}{\left( \int_{B_{R_0}(y_0)} {\left| f \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}{\left( \fint_{B_{R_0}(y_0)} {\left| u \right|}^\frac{2d}{d-2} \right)}^\frac{d-2}{2d}
\\
&{\underset{\phantom{\textrm{Jensen},d\ge 3}}{\overset{}{=}}}
R_0 {\left( \int_{B_{R_0}(y_0)} {\left| f \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}{\left( \int_{B_{R_0}(y_0)} {\left| u \right|}^\frac{2d}{d-2} \right)}^\frac{d-2}{2d}
\\
&{\underset{\phantom{\textrm{Jensen},d\ge 3}}{\overset{\textrm{Sobolev}}{\lesssim}}}
R_0 {\left( \int_{B_{R_0}(y_0)} {\left| f \right|}^2 \right)}^{{\textstyle\frac{1}{2}}}{\left( \int_{{\mathbb{R}}^d} {\left| \nabla u \right|}^2 \right)}^{{\textstyle\frac{1}{2}}},\end{aligned}$$ and so $$\label{E424}
\int_{{\mathbb{R}}^d} {\left| \nabla u \right|}^2 \lesssim R_0^2 \int_{B_{R_0}(y_0)} {\left| f \right|}^2.$$ We point out that compared to the proof of or , we got additional $R_0^2$ due to the right-hand side of the equation being $f$ and not $\nabla \cdot f$.
Let $F$ be a linear functional on $L^2(B_1(x_0))$ such that ${\left| F(v) \right|}^2 \le \int_{B_1(x_0)} {\left| v \right|}^2$. If $r_*(x_0) \leq R_0$, then $$\begin{aligned}
{\left| F(u) \right|}^2 &{\underset{\phantom{\textrm{Jensen},d\ge 3}}{\overset{}{\le}}} \int_{B_1(x_0)} {\left| u \right|}^2 \le \int_{B_{r_*(x_0)}(x_0)} {\left| u \right|}^2 \overset{\textrm{Lemma \ref{lm1}}}\lesssim r_*^d(x_0) \fint_{B_{R_0}(x_0)} {\left| u \right|}^2
\\
&\overset{\textrm{Jensen},d\ge 3}\le r_*^d(x_0) {\left( \fint_{B_{R_0}(x_0)} {\left| u \right|}^{\frac{2d}{d-2}} \right)}^{\frac{d-2}{d}}
\lesssim \frac{r_*^d(x_0)}{R_0^{d-2}} {\left( \int_{{\mathbb{R}}^d} {\left| u \right|}^{\frac{2d}{d-2}} \right)}^{\frac{d-2}{d}}
\\
&{\underset{\phantom{\textrm{Jensen},d\ge 3}}{\overset{\textrm{Sobolev}}{\lesssim}}} \frac{r_*^d(x_0)}{R_0^{d-2}} \int_{{\mathbb{R}}^d} {\left| \nabla u \right|}^2 \overset{\eqref{E424}}\lesssim \frac{r_*^d(x_0)}{R_0^{d-4}} \int_{{\mathbb{R}}^d} {\left| f \right|}^2. \end{aligned}$$ If otherwise $r_*(x_0) > R_0$, then we directly bound $$\begin{aligned}
{\left| F(u) \right|}^2 &{\underset{\phantom{\textrm{Jensen},d\ge 3}}{\overset{}{\le}}} \int_{B_1(x_0)} {\left| u \right|}^2 \le \int_{B_{R_0}(x_0)} {\left| u \right|}^2 \lesssim r_*^d(x_0) \fint_{B_{R_0}(x_0)} {\left| u \right|}^2\end{aligned}$$ and proceed analogously to the other case. Using the Green’s function representation formula we have $u(x) = \int_{B_{R_0}(y_0)} G(x,y) f(y) {{\, \mathrm{d} y}}$, and thus the linearity of $F$ yields $$\nonumber
{\left| F(u) \right|} = {\left| \int_{B_{R_0}(y_0)} F( G(\cdot,y) ) f(y) {{\, \mathrm{d} y}}\right|}.$$ Since $f \in L^2(B_{R_0}(y_0))$ was arbitrary, we may combine the two previous estimates and conclude $$\label{E445}
\int_{B_{R_0}(y_0)} {\left| F(G(\cdot,y)) \right|}^2 {{\, \mathrm{d} y}}\lesssim \frac{r_*^d(x_0)}{R_0^{d-4}}.$$ As before, it remains to argue that by going from $\int_{B_{R_0}(y_0)}$ to $\int_{B_1(y_0)}$ we gain a factor $R_0^{-d}$. We define $v(y) := F(G(\cdot,y))$, and observe that $-\nabla A^t \nabla v = 0$ in $B_{R_0}(y_0)$. Now we use Lemma \[lm1\] with $v$ to get $$\label{E451}
\int_{B_1(x_0)} {\left| v \right|}^2 \le \int_{B_{r_*'(y_0)}} {\left| v \right|}^2 \overset{\textrm{Lemma \ref{lm1}}}\lesssim {\left(\frac{r_*'(y_0)}{R_0}\right)}^d \int_{B_{R_0}(y_0)} {\left| v \right|}^2 \overset{\eqref{E445}}\lesssim \frac{{\left(r_*(x_0)r_*'(y_0)\right)}^d}{R_0^{2d-4}}.$$ For the choice $F(v) = \fint_{B_1(x_0)} v$, relation is exactly .
Proof of Corollary \[cor1\] {#sec2d}
===========================
We provide a generalization of (\[E52\])-(\[E53\]) in the two-dimensional case. When $d=2$, the Green’s function for the whole space $\mathbb{R}^2$ does not have to exist; nevertheless, we may give a definition for $\nabla G$ via the Green’s function on $\mathbb{R}^3$. To this purpose we introduce the following notation: If $\bar x \in \mathbb{R}^3$, we write $\bar x= (x, x_3) \in \mathbb{R}^2 \times \mathbb{R}$ and denote by $\bar B_r \subset {\mathbb{R}}^3$ and $B_r \subset {\mathbb{R}}^2$ the balls of radius $r$ and centered at the origin. For a given bounded and uniformly elliptic coefficient field $A$ in $\mathbb{R}^2$, recall that its trivial extension $\bar A$ to $\mathbb{R}^3$ was defined in by $$\bar A(x,x_3) := \begin{pmatrix} A(x) & 0\ \\ 0 & 1\ \end{pmatrix},$$ and the three-dimensional Green’s function $\bar{G}=\bar{G}(\bar{A};\bar{x},\bar{y})$ is defined as a solution of $$-\nabla_{\bar x} \cdot \bar{A}\nabla_{\bar x} \bar{G}(\bar A; \cdot , \bar y)=\delta (\cdot -\bar y).$$ It will become clear below that the argument for the representation formula for $\nabla G$ through $\nabla_x \bar G$ calls for the notion of pointwise existence in $\bar y \in \mathbb{R}^3$ of the Green’s function $\bar G (\bar A; \cdot , \bar y)$. As mentioned in Section 1, in the case of systems we may only rely on a definition of the Green’s function for almost every singularity point $\bar y$. Therefore, differently from the previous sections, we need to bear in mind this weaker notion of existence of $\bar G$.
[**Step 1.**]{} We argue that for almost every $y \in \mathbb{R}^2$ the function $\nabla G$ (since $G$ does not exist, $\nabla G$ should be understood as a symbol for a function and not as a gradient of some function $G$), defined through $$\label{2d4}
\nabla{G(A;\cdot ,y)}:=\int_{\mathbb{R}}\nabla_x\bar{G}(\bar{A};(\cdot,x_3),(y,y_3)) {{\, \mathrm{d} x}}_3,$$ satisfies for every $\zeta \in C^\infty_0(\mathbb{R}^2)$ $$\begin{aligned}
\label{2d10}
\int \nabla_x \zeta(x) \cdot A(x) \nabla G(A; x, y) {{\, \mathrm{d} x}}= \zeta(y),\end{aligned}$$ i.e., in a weak sense it solves $-\nabla_x \cdot A \nabla G(A; \cdot, y)= \delta(\cdot -y)$.\
By definition of $\bar G(\bar A; \cdot, \cdot)$, we have for almost every $\bar y\in \mathbb{R}^3$ and every $\bar \zeta \in C^\infty_0( \mathbb{R}^3)$ $$\begin{aligned}
\nonumber
\int \nabla_{\bar x} \bar \zeta(\bar x) \cdot \bar A \nabla_{\bar x}\bar G(\bar A; \bar x, \bar y) {{\, \mathrm{d} \bar x}}= \bar \zeta(\bar y).\end{aligned}$$ Thus, for any $\bar \rho \in C^\infty_0(\mathbb{R}^3)$ this yields $$\begin{aligned}
\nonumber
\int \bar \rho(\bar y) \int \nabla_{\bar x}\bar \zeta(\bar x) \cdot \bar A \nabla_{\bar x}\bar G(A; \bar x, \bar y) {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}= \int \bar \rho(y) \bar \zeta(\bar y) {{\, \mathrm{d} \bar y}}.\end{aligned}$$ We now choose a sequence $\{\bar \zeta_n \}_{n \in \mathbb{N}}$ of test functions $\bar \zeta_n = \eta_n \zeta$, with $\zeta= \zeta(x) \in C^\infty_0( \mathbb{R}^2)$ and $\eta_n= \eta_n(x_3)$ smooth cut-off function for $\{|x_3|< n\}$ in $\{|x_3| < n+1 \}$: From the previous identity and definition (\[2d1\]) it follows $$\begin{aligned}
\int \bar \rho(\bar y) \int &\zeta(x) \eta_n'(x_3) \partial_{x_3}\bar G(\bar A; \bar x, \bar y) {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}\\
&+ \int \bar \rho(\bar y) \int \eta_n(x_3) \nabla\zeta(x) \cdot A \nabla \bar G(\bar A; \bar x, \bar y) {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}= \int \bar \rho(y) \zeta(y) {{\, \mathrm{d} \bar y}}.\end{aligned}$$ We now want to send $n \rightarrow +\infty$ in the previous identity : By our assumptions on $\bar \rho$ and $\bar \zeta_n$, if we show that $$\begin{aligned}
\label{2d9}
\int_{\textrm{supp}(\bar\rho)}\int_{\textrm{supp}(\zeta) \times \mathbb{R}} |\nabla_{\bar x} \bar G( \bar A; \bar x, \bar y)| {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}< +\infty,\end{aligned}$$ then by the Dominated Convergence Theorem we may conclude that $$\begin{aligned}
\nonumber
\int\bar \rho(\bar y) \int \nabla\zeta(x) \cdot A \bigl( \int_{\mathbb{R}}\nabla \bar G(\bar A; \bar x, \bar y) {{\, \mathrm{d} x}}_3 \bigr) {{\, \mathrm{d} x}}{{\, \mathrm{d} \bar y}}= \int\bar \rho(\bar y) \zeta(y) {{\, \mathrm{d} \bar y}},\end{aligned}$$ and thus (\[2d10\]) by the arbitrariness of the test function $\bar \rho$ and the separability of $C^\infty_0(\mathbb{R}^2)$.
To argue inequality (\[2d9\]) we proceed as follows: We define a finite radius $M$ such that $$\nonumber
M \ge \max(r_*(\bar A^t,\bar X),r_*(A,\bar Y)) \quad \textrm{and} \quad \textrm{supp}(\bar \rho) \subset \bar B_{M}(\bar Y), \ \textrm{supp}(\zeta) \subset B_{M/2}(X),$$ and observe that inequality (\[2d9\]) is implied by $$\begin{aligned}
\label{2d11}
\int_{\bar B_{M}(\bar Y)} \int_{B_{M/2}(X) \times \mathbb{R}} |\nabla_{\bar x} \bar G| {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}< +\infty.\end{aligned}$$ Since $\bar A$ is translational invariant in $x_3$, the minimal radius $r_*(\bar A^t,\cdot)$ is independent of $x_3$. Then, by the definition of $M$ and Remark \[rmk1\] we have $$\begin{aligned}
\label{2d14}
\int_{\bar B_{M}(\bar Y)}\int_{\bar B_{M}((X ,X_3))} |\bar\nabla_x \bar G(\bar A; \bar x, \bar y)|^2 {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}\lesssim \frac{ M^6}{|Y - (X ,X_3)|^4} \le \frac{ M^6}{|Y_3 - X_3|^4}\end{aligned}$$ provided $|X_3 - Y_3| \ge 3M$.
We now cover the cylinder $B_{M/2}(X) \times \mathbb{R}$ with countably many balls of radius $M$ centered at the points $(X, \pm Mn) \in \mathbb{R}^3$. By translational invariance we can w. l. o. g. assume that $Y_3=0$. We thus bound the integral in by $$\begin{aligned}
\nonumber
\int_{\bar B_M(\bar Y)} &\int_{B_{M/2}(X) \times \mathbb{R}} |\nabla_{\bar x} \bar G| {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}\leq \sum_{n=0}^{+\infty}\int_{\bar B_M(\bar Y)} \int_{\bar B_{M}(X,\pm Mn)} |\nabla_{\bar x} \bar G| {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}\\ \label{p102}
&\lesssim \int_{\bar B_M(\bar Y)} \int_{\bar B_{4M}((X,0))} |\nabla_{\bar x} \bar G| {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}+ \sum_{n > 4}\int_{\bar B_M(\bar Y)} \int_{\bar B_{M}(X,\pm Mn)} |\nabla_{\bar x} \bar G| {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}.\end{aligned}$$ We claim that $\nabla_{\bar x} \bar G( \bar A; \cdot, \cdot) \in L^1_{loc}( \mathbb{R}^3 \times \mathbb{R}^3)$, and so the first integral on the r. h. s. of the previous identity is finite.
Here we only sketch the idea why $\bar \nabla_{\bar x} \bar G \in L^1_{loc}({\mathbb{R}}^3 \times {\mathbb{R}}^3)$; for the proof with all the details we refer to the proof of [@ConlonGiuntiOtto Theorem 1]. To show that $\bar \nabla_{\bar x} \bar G \in L^1_{loc}$ it suffices to show that $\int_{\bar B_R(0)} \int_{\bar B_R(0)} |\bar \nabla_{\bar x}\bar G| < \infty$. In order to do that we observe that for given two distinct points $\bar x, \bar y \in {\mathbb{R}}^3$, the proof of Theorem \[thm1\] (without the use of $r_*$ to go to smaller scales; see also Remark \[rmk1\]) implies in $3D$ $$\biggl( \int_{\bar B_r(\bar x)} \int_{\bar B_r(\bar y)} |\bar \nabla_{\bar x}\bar G|^2 \biggr)^{\frac{1}{2}} \lesssim \frac{|\bar B_r|}{r^2},$$ where $r = |\bar x-\bar y|/3$, which by Hölder’s inequality turns into $$\int_{\bar B_r(\bar x)} \int_{\bar B_r(\bar y)} |\bar \nabla_{\bar x}\bar G| \lesssim \frac{|\bar B_r|^2}{r^2}.$$ Using a simple covering argument, the above estimate holds also in the case when the balls are replaced by cubes. Since $\bar B_R(0) \times \bar B_R(0)$ can be written as a null-set plus a countable union of pairs of open cubes $\bar Q_{r_n}(\bar x_n) \times \bar Q_{r_n}(\bar y_n)$, each with size $r_n := |\bar x_n - \bar y_n|/3$ and such that each pair of points $(\bar x,\bar y) \in \bar B_R(0) \times \bar B_R(0)$ belongs to at most one such pair of cubes, we conclude $$\nonumber
\int_{\bar B_R(0)} \int_{\bar B_R(0)} |\bar \nabla_{\bar x}\bar G| \lesssim \int_{\bar B_{2R}(0)} \int_{\bar B_{2R}(0)} |\bar x- \bar y|^{-2} {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}< \infty,$$ where we used that for $(\bar x,\bar y) \in \bar Q_{r_n}(\bar x_n) \times \bar Q_{r_n}(\bar y_n)$ we have $|\bar x - \bar y| \sim r_n$.
Going back to the second term on the right-hand side of , an application of Hölder’s inequality in both variables $\bar x $ and $\bar y$ yields for the the sum over $n$ $$\begin{aligned}
\sum_{n > 4}\int_{\bar B_M(\bar Y)} \int_{\bar B_{M}(X,\pm Mn)} |\nabla_{\bar x} \bar G|& \lesssim M^3 \sum_{n > 4} \biggl(\int_{\bar B_{M}(\bar Y)} \int_{\bar B_{M}(X,\pm M n)} |\bar \nabla_{\bar x} \bar G|^2 \biggr)^{\frac 1 2}.\end{aligned}$$ We now may apply to the r.h.s. the bound (\[2d14\]) and thus obtain $$\begin{aligned}
\sum_{n > 4}\int_{\bar B_M(\bar Y)} \int_{\bar B_{M}(X,\pm Mn)} |\nabla_{\bar x} \bar G|& \lesssim M^6 \sum_{n > 4} (Mn)^{-2}
\lesssim M^4 < \infty.$$ We have established (\[2d9\]).
Before concluding Step 1, we show that the representation formula (\[2d4\]) does not depend on the choice of the coordinate $y_3 \in \mathbb{R}$, namely that for almost every two values $y_{0,3},y_{1,3} \in \mathbb{R}$, for almost every $y_0, x_0 \in \mathbb{R}^2$ $$\begin{aligned}
\label{2d14b}
\int_{\mathbb{R}}\nabla_x \bar{G}(\bar{A};(x_0, x_3),(y_0, y_{0,3})) {{\, \mathrm{d} x}}_3& = \int_{\mathbb{R}} \nabla_x\bar{G}(\bar{A};(x_0, x_3),(y_0, y_{1,3}) ) {{\, \mathrm{d} x}}_3.\end{aligned}$$ Without loss of generality we assume $y_{0,3}=0$: Since by the uniqueness of $\bar G(\bar A; \cdot , \cdot)$, for every $\bar z \in \mathbb{R}^3$ and almost every $\bar x, \bar y \in \mathbb{R}^3$ $$\bar G(\bar A;\bar x+\bar z , \bar y+\bar z) = \bar G(\bar A( \cdot + \bar z); \bar x, \bar y),$$ by choosing $\bar z=(0, z_3)$ and using definition (\[2d1\]) for $\bar A$, we get $$\begin{aligned}
\label{2d13}
\bar G(\bar A;\bar x+\bar z , \bar y+\bar z) = \bar G(\bar A; \bar x, \bar y).\end{aligned}$$ Let $x_0, y_0\in \mathbb{R}^2$ and $y_{1,3} \in \mathbb{R}^3$ be fixed: For every $\delta > 0$ we may write $$\begin{aligned}
\fint_{B_\delta (x_0)}&\fint_{\bar B_\delta((y_0, y_{1,3}))}\int_{\mathbb{R}}\nabla_x \bar{G}(\bar{A};\bar{x},\bar{y}) {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}\\
&=\fint_{B_\delta (x_0)}\fint_{\bar B_\delta((y_0, y_{1,3}))}\int_{\mathbb{R}}\nabla_x \bar{G}(\bar{A};(x,x_3-y_{1,3}+y_{1,3}),(y, y_3-y_{1,3}+y_{1,3})) {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}},\end{aligned}$$ and use (\[2d13\]) with $\bar z= (0, y_{1,3})$ to get $$\begin{aligned}
\fint_{B_\delta (x_0)}&\fint_{\bar B_\delta((y_0, y_{1,3}))}\int_{\mathbb{R}}\nabla_x \bar{G}(\bar{A};\bar{x},\bar{y}) {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}=\fint_{B_\delta (x_0)}\fint_{\bar B_\delta((y_0, 0))}\int_{\mathbb{R}}\nabla_x \bar{G}(\bar{A};\bar{x},\bar y) {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}.\end{aligned}$$ We now appeal to Lebesgue’s theorem and conclude (\[2d14\]).
[**Step 2.**]{} Proof of (\[2d5\]). For this part we denote $r_x := r_*(\bar A^t,(x_0,0))$ and $r_y := r_*(\bar A, (y_0,0))$. By translational invariance of $\bar A$ and $\bar A^t$ we have $r_x = r_*(\bar A^t,(x_0,x_3))$ and $r_y = r_*(\bar A,(y_0,y_3))$ for any $x_3, y_3 \in {\mathbb{R}}$. Denoting ${\mathcal{B}}:= B_1(y_0) \times (-r_y/2,r_y/2)$, the independence of (\[2d4\]) from $y_3$ yields $$\begin{aligned}
\int_{{\mathcal{B}}} \int_{B_1(x_0)}|\int_{\mathbb{R}}\nabla_{\bar{x}} \bar{G}(\bar{x},\bar{y}){{\, \mathrm{d} x}}_3|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} \bar y}}&{\underset{\phantom{\eqref{2d4}}}{\overset{}{=}}} r_y \int_{B_1(y_0)} \int_{B_1(x_0)}|\int_{\mathbb{R}}\nabla_{\bar{x}} \bar{G}(\bar{x},(y, 0 ) )dx_3|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\\
&\stackrel{(\ref{2d4})}{=} r_y \int_{B_1(y_0)} \int_{B_1(x_0)} | \nabla G(A; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}.\end{aligned}$$ Since ${\mathcal{B}}\subset \bar B_{r_y}((y_0,0))$, the previous identity implies $$\nonumber \begin{aligned}
r_y \int_{B_1(x_0)}&\int_{B_1(y_0)}|\nabla_x G(A; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}= \int_{{\mathcal{B}}} \int_{B_1(x_0)}|\int_{\mathbb{R}}\nabla_x \bar{G}(\bar A;\bar{x},\bar{y}) {{\, \mathrm{d} x}}_3|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} \bar y}}\\
&\lesssim \int_{\bar B_{r_y}((y_0,0))} \int_{B_1(x_0)}|\int_{\mathbb{R}}\nabla_x \bar{G}(\bar A; \bar{x},\bar{y}) {{\, \mathrm{d} x}}_3|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} \bar y}}\\
&\le \int_{\bar B_{r_y}((y_0,0))} \int_{B_1(x_0)} \biggl( \sum_{n=-\infty}^{\infty} \int_{nr_x}^{(n+1)r_x} |\nabla_x \bar{G}(\bar A; \bar{x},\bar{y})| {{\, \mathrm{d} x}}_3 \biggr)^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} \bar y}}.
\end{aligned}$$ We define a sequence $$a_n := \frac{(r_x r_y)^{\frac{3}{4}}}{( |x_0 - y_0|^2 + n^2 (r_x)^2)^{\frac{1}{2}}}$$ and observe that $$\begin{aligned}
\biggl( \sum_{n=-\infty}^\infty & \int_{nr_x}^{(n+1)r_x} |\nabla_x \bar G(\bar A;\bar x,\bar y)| {{\, \mathrm{d} x}}_3 \biggr)^2
= \biggl( \sum_{n=-\infty}^\infty a_n \frac{r_x}{a_n} \fint_{nr_x}^{(n+1)r_x} |\nabla_x \bar G(\bar A;\bar x,\bar y)|{{\, \mathrm{d} x}}_3 \biggr)^2
\\
&\overset{\textrm{H\"older}}{\le} \biggl( \sum_{n=-\infty}^\infty a_n^2 \biggr) \biggl( \sum_{n=-\infty}^\infty \frac{(r_x)^2}{a_n^2} \biggl( \fint_{nr_x}^{(n+1)r_x} |\nabla_x \bar G(\bar A;\bar x,\bar y)|{{\, \mathrm{d} x}}_3 \biggr)^2 \biggr)
\\
&\overset{\textrm{Jensen}}{\le} \biggl( \sum_{n=-\infty}^\infty a_n^2 \biggr) \biggl( \sum_{n=-\infty}^\infty \frac{r_x}{a_n^2} \int_{nr_x}^{(n+1)r_x} |\nabla_x \bar G(\bar A;\bar x,\bar y)|^2{{\, \mathrm{d} x}}_3 \biggr).\end{aligned}$$ Since $$\label{p101}
\sum_{n=-\infty}^\infty a_n^2 \lesssim \frac{(r_x r_y)^{\frac{3}{2}}}{|x_0-y_0| r_x},$$ where for simplicity we assumed $|x_0-y_0| \ge r_x$, we combine the three above relations to infer $$\begin{aligned}
r_y \int_{B_1(x_0)} &\int_{B_1(y_0)}|\nabla_x G(A; x,y)|^2 {{\, \mathrm{d} x}}{{\, \mathrm{d} y}}\\
&{\underset{\phantom{\eqref{E53},d=3}}{\overset{}{\lesssim}}} \frac{(r_x r_y)^{\frac{3}{2}}}{|x_0-y_0| r_x} \sum_{n} \frac{r_x}{a_n^2}
\\
&\qquad \qquad \quad \times \int_{\bar B_{r_y}((y_0,0))} \int_{\bar B_{r_x}{(x_0,(n+1/2)r_x)}} |\nabla_x \bar G(\bar A;\bar x,\bar y)|^2 {{\, \mathrm{d} \bar x}}{{\, \mathrm{d} \bar y}}\\
&\overset{\eqref{E53},d=3}{\lesssim} \frac{(r_x r_y)^{\frac{3}{2}}}{|x_0-y_0| r_x)} \sum_{n} \frac{r_x}{a_n^2} a_n^4 \overset{\eqref{p101}}{\lesssim} \frac{(r_x r_y)^3}{|x_0-y_0|^2 r_x},\end{aligned}$$ which is exactly .
Concerning (\[2d5b\]), there are two possible ways how to proceed. For the first we observe that implies for every test function $\phi \in C^\infty_c({\mathbb{R}}^2)$ $$\nonumber
\int \nabla \phi(x) \cdot A(x) \biggl( \int \nabla_y \nabla G(x,y) \cdot f(y) {{\, \mathrm{d} y}}\biggr) {{\, \mathrm{d} x}}= \int \nabla \phi \cdot f = \int \nabla\phi \cdot A\nabla u,$$ where $f \in L^2({\mathbb{R}}^2;{\mathbb{R}}^2)$ and $u$ is a solution of $-\nabla \cdot A \nabla u = -\nabla \cdot f$. Therefore we have that $$\nabla u(x) = \int \nabla_y \nabla G(x,y) \cdot f(y) {{\, \mathrm{d} y}},$$ and the proof of applies verbatim. A different way would be to mimic the argument for , i.e., to define $\nabla_y \nabla G$ as an integral of the second mixed derivative of the Green’s function in three dimension. Unfortunatelly, this way we would obtain the estimate where the minimal radii in $2D$ appearing on the right-hand side of would need to be replaced (with possibly larger) minimal radii for $3D$.
Acknowledgment {#acknowledgment .unnumbered}
==============
We warmly thank Felix Otto for introducing us into the world of stochastic homogenization and also for valuable discussions of this particular problem.
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|
---
author:
- 'Miguel A. de Avillez'
- Dieter Breitschwerdt
bibliography:
- 'bibliography.bib'
date: 'Received March 16, 2015; accepted May 11, 2015'
title: 'Temperature-averaged and total free-free Gaunt factors for $\kappa$ and Maxwellian distributions of electrons'
---
[Optically thin plasmas may deviate from thermal equilibrium and thus, electrons (and ions) are no longer described by the Maxwellian distribution. Instead they can be described by $\kappa-$distributions. The free-free spectrum and radiative losses depend on the temperature-averaged (over the electrons distribution) and total Gaunt factors, respectively. Thus, there is a need to calculate and make available these factors to be used by any software that deals with plasma emission.]{} [We recalculated the free-free Gaunt factor for a wide range of energies and frequencies using hypergeometric functions of complex arguments and the Clenshaw recurrence formula technique combined with approximations whenever the difference between the initial and final electron energies is smaller than $10^{-10}$ in units of $z^{2}Ry$. We used double and quadruple precisions. The temperature-averaged and total Gaunt factors calculations make use of the Gauss-Laguerre integration with 128 nodes.]{} [The temperature-averaged and total Gaunt factors depend on the $\kappa$ parameter, which shows increasing deviations (with respect to the results obtained with the use of the Maxwellian distribution) with decreasing $\kappa$. Tables of these Gaunt factors are provided.]{}
Introduction
============
Astrophysical plasma emission codes are a powerful tool for calculating spectra and energy losses in a plasma such as the interstellar or intergalactic medium and for comparing the results with observations. However, such plasmas are complex systems in which frequently made assumptions, like establishing the Maxwell-Boltzmann distribution (hereafter referred as Maxwellian distribution) of the electrons and ions, are not always fulfilled and may lead to erroneous interpretations of the plasma properties. We have therefore reexamined the nonrelativistic free-free Gaunt factor, which is the quantum correction for the semiclassical cross section of @kramers1923. The factor has been the subject of many papers over the years, comprising analytical approximations [e.g., @menzel1935; @elwert1954; @grant1958; @brussaard1962; @hummer1988; @beckert2000] to the exact quantum mechanical expressions (in terms of hypergeometric functions) derived, for example, by @menzel1935, @sommerfeld1953 [using nonrelativistic dipole approximation], @kulsrud1954, and @biedenharn1956, and detailed numerical computations first discussed in the seminal work of @karzas1961 [hereafter KL61] and followed by a series of publications during the past five decades.
Karzas & Latter computed the Gaunt factor using hypergeometric functions of complex variables and presented their results in graphical form. The problem reduces to calculating the solution of differential equations by means of power series of the real variable $x$ (which is negative) at two regimes ($\left|x\right|>1$ and $\left|x\right|\leq 1$) with a very slow convergence when $\left|x\right|\to 1$. Since KL61, several authors recalculated the Gaunt factor with increasing precision and size of parameter space [e.g., @obrien1971; @carson1988; @hummer1988; @nicholson1989; @janicki1990; @sutherland1998; @vanhoof2014]. Special care was taken to overcome the slow convergence of the solution by (i) redefining the regimes through a change in variables [@obrien1971], which has been adopted, in combination with KL61 formulae, in most of the calculations that followed, (ii) increase in precision, and (iii) by using the approximation of @menzel1935 [see discussion in @vanhoof2014, who corrected a term in their formulae].
The Gaunt factor can be averaged over a distribution of electrons (also known as the temperature-averaged Gaunt factor) and then integrated over the full range in frequency (the total Gaunt factor). These in turn are used in the determination of the emission spectra and radiative losses by a plasma as a result of this process. In general, the averaging is made over a Maxwellian distribution of electrons, thus relying on the assumption that thermal equilibrium is the rule [see, e.g., @karzas1961; @gayet1970; @armstrong1971; @feng1983; @carson1988; @hummer1988; @nicholson1989; @janicki1990; @sutherland1998; @vanhoof2014].
However, this condition may not be attained if high-energy electrons are injected into the system on timescales shorter than that needed to achieve thermalization (Livadiotis & McComas 2009) or when long-range forces are present in the plasma [@collier2004]. Deviations may also occur in the presence of strong temperature and/or density gradients [@collier1993; @collieretal2004 see discussion in, e.g., @dudik2011 and references therein]. An example of nonthermal distributions is the so-called $\kappa$-distribution, which was first used by @vasyliunas1968 to match the observed electron distribution in the Earth’s magnetosphere. $\kappa-$ distributions have also been used to explain the discrepancies observed in the abundances and temperatures in [[H[<span style="font-variant:small-caps;">ii</span>]{}]{}]{} regions and planetary nebulae when derived using collisional excitation lines and optical recombination lines [see @binette2012; @nicholls2012; @nicholls2013]. The $\kappa$-distribution is characterized by a high-energy power-law tail and has the form $$f_{\kappa}(E)dE=\frac{2E^{1/2}}{\pi^{1/2}(k_{B}T)^{3/2}} \displaystyle A_{\kappa} \left[ 1+\frac{E}{(\kappa-3/2)k_{B}T}\right]^{-\kappa-1}dE,$$ with $$A_{\kappa}=\frac{\Gamma(\kappa+1)}{\Gamma(\kappa-1/2)(\kappa-3/2)^{3/2}},$$ and $\Gamma(x)$ denoting the gamma function of the variable $x$. When $\kappa\to \infty,$ the Maxwellian distribution is recovered. As $\kappa$ decreases, deviations from the Maxwellian distribution increase, reaching maximum when $\kappa$ approaches 3/2 (Fig. \[kappadist\], which displays the $\kappa$ and Maxwellian distributions at different temperatures for electrons with energies varying between $10^-2$ and $10^4$ eV). Similarly to the Maxwellian distribution, the mean energy of the $\kappa$-distribution is independent of $\kappa$ and is given by $\langle E\rangle=3/2k_{B}T$. Hence, $T$ can be defined as the thermodynamic temperature for these distributions. For a review of the $\kappa$-distribution and its applications see @pierrard2010 and @livadiotis2009.
@dudik2011 [@dudik2012] calculated the free-free contribution to the emission spectra and to the radiative losses of a plasma with the nonthermal distributions ($\kappa$ and $n$ distributions) and evolving under collisional ionization equilibrium conditions (CIE), that is, the number of recombinations equals the number of ionizations by electron impact. The authors calculated the temperature-averaged Gaunt factors for nonthermal distributions using the free-free Gaunt factors of @sutherland1998.
![Not normalized Maxwellian and $\kappa$-distributions (with different $\kappa$) at $10^{4}$ K (top) and $10^{6}$ K (bottom panel). The largest deviation of the Maxwellian distribution occurs for the lowest $\kappa,$ showing an increase in electrons at the low- and high-energy ranges when compared to the Maxwellian distribution.[]{data-label="kappadist"}](figures/maxwell+k_distributions-logT=4.pdf){width="0.85\hsize"}
![Not normalized Maxwellian and $\kappa$-distributions (with different $\kappa$) at $10^{4}$ K (top) and $10^{6}$ K (bottom panel). The largest deviation of the Maxwellian distribution occurs for the lowest $\kappa,$ showing an increase in electrons at the low- and high-energy ranges when compared to the Maxwellian distribution.[]{data-label="kappadist"}](figures/maxwell+k_distributions-logT=6.pdf){width="0.85\hsize"}
The temperature-averaged and total Gaunt factors are needed for detailed simulations involving the coupling of the dynamical and thermal evolutions of the interstellar medium, bubbles and superbubbles (including the Local Bubble), and formation of galaxies, to name just a few. Hence, we carried out detailed calculations of the free-free Gaunt factor and of the temperature-averaged and total Gaunt factors for Maxwellian and $\kappa-$distributions (considering a large grid of $\kappa$ parameters). We present these results in tabulated form to be used in any plasma emission software through convenient one- and two-dimensional interpolations.
The structure of this paper is as follows: Section 2 describes the calculation of the free-free Gaunt factors. Section 3 deals with the temperature-averaged and total Gaunt factor for Maxwellian and $\kappa$ distributions of electrons. Section 4 describes the tabulated data, and we conclude in Sect. 5 with some final remarks.
Calculations of the Gaunt factor
================================
The calculation of the free-free Gaunt factor in double and quadruple precision follows the prescription of @janicki1990 with some adaptations taken from @carson1988 and @vanhoof2014. It is assumed that an electron with initial energy $E_{i}$ absorbs a photon of energy $h\nu$. Thus, the electron transits to a higher state with an energy $E_{f}$=$E_{i}+h\nu$. The free-free Gaunt factor is then given by (KL61) $$g_{_{\rm ff}}=
\frac{2\sqrt{3}}{\pi}I_{0}\left[I_{0}\left(\frac{\eta_{i}}{\eta_{f}}+\frac{\eta_{f}}{\eta_{i}}+2\eta_{i}\eta_{f}\right)-2I_{1}\sqrt{1+\eta_{i}^{2}}\sqrt{1+\eta_{f}^{2}}\right],$$ where $\eta_{i,f}^{2}=1/\epsilon_{i,f}$, $\epsilon_{i}=E_{i}/Z^{2}Ry$ and $\epsilon_{i}=E_{f}/Z^{2}Ry$ are the electron scaled initial and final energies, respectively; $Ry$ is the infinite-mass Rydberg unit of energy, and $Z$ the atomic number of the ion. The functional $I_{l}$ (with $l=0.1$) is defined as $$I_{l}=\frac{1}{4}\left(\frac{4\eta_{i}\eta_{f}}{(\eta_{i}-\eta_{f})^{2}}\right)^{l+1} e^{\pi\left|\eta_{i}-\eta_{f}\right|/2}\frac{\left|\Gamma(l+1+i\eta_{i})\Gamma(l+1+i\eta_{f})\right|}{\Gamma(2l+2)}G_{l},$$ where $G_{l}$ is a real function given by $$G_{l}=(1-x)^{-i\eta_{i}-i\eta_{f}} {}_{2}F_{1}(a,b;c;x)$$ with $x=-4 \eta_{i}\eta_{f}/(\eta_{i}-\eta_{f})^2$ (which is always negative), $a=l+1-i\eta_{f}$, $b=l+1-i\eta_{i}$, and $c=2l+2$; ${}_{2}F_{1}(a,b;c;x)$ is the hypergeometric function, which satisfies the equation $$x(1-x)F^{\prime\prime}+\left[c-(a+b+1)xF^{\prime}\right]-abF=0.$$ $G_{l}$ satisfies the equation (Janicki 1990) $$x(1-x^{2})G^{\prime\prime}+(1-x)f_{1}\,G^{\prime}+f_{2}\, G=0,$$ with $f_{1}=c+x(2d-a-b-1)$ and $f_{2}=x[d^{2}+ab-d(a+b)]-ab+dc]$.
![Free-free Gaunt factor variation with the normalized initial electron energy (top panel) for specific photon energies running from $\log h\nu/z^{2}\mbox{Ry}=8$ through $-4$ with steps of dex=1, and with the normalized photon energy (bottom panel) for initial electron energies of $\log E_{i}/z^{2}\mbox{Ry}=-5$ through $8$ with steps of dex=1. []{data-label="gff"}](figures/gaunt-ff-ei_hnu.pdf){width="0.9\hsize"}
Thus, the free-free Gaunt factor determination reduces to the calculation of solutions for $G_{l}$, and therefore for $I_{l}$, for $l=0,\,1$ in terms of a power series in $x$ for $\left|x\right|<1$ and in $y=-1/x$ for $\left|x\right|>1$ (KL61). These solutions converge very slowly near 1, but a change in variables to [@janicki1990] $$y=\left\{ \begin{array}{lll}
x/(x-1) & if&\left|x\right|\leq 1.61804 \\
& \\
-1/x & if & \left|x\right|> 1.61804
\end{array}\right.$$ facilitates the calculation. The threshold results from equating $\left|x\right|/(\left|x\right|-1)$ to $-1/\left|x\right|$ and taking into account that $x<0$ (see text above). Another step comprises the usage of the Clenshaw recurrence formula to calculate $G_{l}$ at each regime by means of a series of terms. The number of terms $n$ to be considered in the sum to within a certain precision, for example, $10^{-\delta}$, is given by $-\delta/\log \left|x/(x-1)\right|$ (for $\left|x\right|\leq 1.61804$) and $-\delta/\log \left|-1/x\right|$ ($\left|x\right|> 1.61804$). For further details see @janicki1990.
In the range $\epsilon_{i,f}^{3/2}/w\leq 10^{-4}$, with $w=\epsilon_{f}-\epsilon_{i}=h\nu/Z^{2}Ry$, the exact solution fails [see @carson1988; @hummer1988; @vanhoof2014]. Hence, the approximation of @menzel1935 with the correction by @vanhoof2014 is used: $$g_{_{\rm ff}}=1+c_{1}\frac{1+k^{2}}{\chi^{2}}+c_{2}\frac{1-4/3k^{2}+k^{4}}{\chi^{4}}+c_{3}\frac{1-1/3k^{2}-1/3k^{4}+k^{6}}{\chi^{6}}
,$$ with $k=\eta_{f}/\eta_{i}$, $\chi=\left[(1-k^{2})\eta_{f}\right]^{1/3}$, $c_{1}=0.1728260369...$, $c_{2}=-0.04959570168...$, and $c_{3}=-0.01714285714...$. The maximum error in this approximation is smaller than $5.5\times 10^{-10}$ [@vanhoof2014].
Figure \[gff\] displays the variation of the free-free Gaunt factor with the normalized initial electron energy (top panel) for specific photon energies running from $\log h\nu/z^{2}\mbox{Ry}=8$ through $-4$ with steps of dex=1, and with the normalized photon energy (bottom panel) for initial electron energies of $\log E_{i}/z^{2}\mbox{Ry}=-5$ through $8$ with steps of dex=1. These Gaunt factors overlap with those reported by @vanhoof2014 and @sutherland1998.
Temperature-averaged and total Gaunt factor
===========================================
The energy spectrum by free-free emission from electrons with an energy distribution $f(E)$ is given by [see, e.g., @kwok2007] $$\label{spectrum}
\frac{dP_{_{\rm ff}}}{d \nu}=\frac{8 \pi^{2}}{c^{3}}\left(\frac{2}{3 m_{_{e}}}\right)^{3/2} e^{6} z^{2}n_{e}
n_{Z,z}\displaystyle \int_{h\nu}^{+\infty}\frac{1}{E^{1/2}}f(E) g_{_{\rm ff}}(E,\nu) dE
,$$ where $\nu$ is the frequency of the emitted photon, $T$ is the temperature, $k_{B}$ is the Boltzmann constant, $n_{e}$ is the electron density, $n_{_{Z,z}}$ is the number density of ion with atomic number $Z$ and ionization stage $z$. For electrons with a $\kappa$ or Maxwellian ($\kappa\to\infty$) distribution, and after suitable change of variables, Eq. (\[spectrum\]) becomes $$\begin{aligned}
\label{power}
\frac{dP_{_{\rm ff}}}{du} &=& C_{_{\rm ff}} z^{2} n_{e} n_{_{Z,z}} T^{1/2} \times \\
& \times & \left\{ \begin{array}{lllc}
\displaystyle \int_{0}^{+\infty}g_{_{\rm ff}}(\gamma^{2},u)\frac{A_{\kappa}}{\left[1+\frac{x+u}{\kappa-3/2}\right]^{\kappa+1}}dx & if & \kappa> 3/2 \\
&& \nonumber\\
\displaystyle e^{-u} \int_{0}^{+\infty}g_{_{\rm ff}}(\gamma^{2},u)e^{-x}dx & if & \kappa\to \infty,
\end{array}\right.\end{aligned}$$ where $\displaystyle C_{_{\rm ff}}=16 \left(\frac{2 \pi}{3 m_{_{e}}}\right)^{3/2} \frac{e^{6} k_{B}^{1/2}}{hc^{3}}=1.4256\times 10^{-27}$, and the parameters $x$, $u$ and $\gamma$ have the forms $$x=\frac{E}{k_{B}T}-\frac{h\nu}{k_{B}T},\, u=\frac{h\nu}{k_{B}T} \mbox{~~and~~}
\gamma^{2}=\frac{z^{2}Ry}{k_{B}T}=z^{2}\frac{1.579\times 10^{5} \mbox{K}}{T}.$$ The integral on the right-hand side of Eq. (\[power\]) is the temperature-averaged Gaunt factor (KL61). Integration of Eq. (\[power\]) over the photon frequency spectrum gives the total free-free power associated with an ion $(Z,z)$ $$P_{_{\rm ff}}(T)=C_{_{\rm ff}} z^{2} n_{e}n_{_{Z,z}}T^{1/2} \int_{0}^{+\infty} \langle g_{_{\rm
ff}}(\gamma^{2},u)\rangle f(u) du
\label{cooling}
,$$ with $f(u)=e^{-u}$ (for $\kappa \to \infty$; Maxwellian distribution) and $f(u)=1$ for $\kappa>3/2$ ($\kappa$ distribution). The total free-free Gaunt factor is defined as (see, e.g., KL61) $$g_{_{\rm ff}}(T)=\int_{0}^{+\infty}\langle g_{_{\rm ff}}(\gamma^{2},u) \rangle e^{-u} du
,$$ and, thus, can be calculated for both distributions.
![Temperature-averaged Gaunt factors calculated for $\kappa=2$, 5, 10, 15, and 25) and Maxwellian distributions of electrons for the range $10^{-4}\leq \gamma^{2}\leq10^{4}$ (top panel) and zoomed-in to the region $\gamma^{2}\in[10^{2},10^{4}]$ and $\langle g_{_{ff}}(\gamma^{2},u)\rangle \in[0.8,2]$ (bottom panel).[]{data-label="temperature_averaged"}](figures/gaunt-ff-temp-averaged-kappa-vs-gam2-20150510.pdf "fig:"){width="0.85\hsize"} ![Temperature-averaged Gaunt factors calculated for $\kappa=2$, 5, 10, 15, and 25) and Maxwellian distributions of electrons for the range $10^{-4}\leq \gamma^{2}\leq10^{4}$ (top panel) and zoomed-in to the region $\gamma^{2}\in[10^{2},10^{4}]$ and $\langle g_{_{ff}}(\gamma^{2},u)\rangle \in[0.8,2]$ (bottom panel).[]{data-label="temperature_averaged"}](figures/gaunt-ff-temp-averaged-kappa-vs-gam2-zoom2-20150510.pdf "fig:"){width="0.85\hsize"}
Figures \[temperature\_averaged\] and \[total\_ff\] display the variation with $\gamma^{2}$ of the temperature-averaged, $\langle g_{_{\rm ff}}(\gamma^{2},u)\rangle$, and total free-free, $g_{_{ff}}(T)$ Gaunt factor. The results correspond to $\kappa=2$, 5, 10, 15, 25 and $\infty$ (the latter value corresponds to the Maxwellian distribution).
The temperature-averaged Gaunt factors depend on the $\kappa$ parameter - with the decrease of $\kappa,$ the deviations of $\langle g_{_{\rm ff}}(\gamma^{2},u)\rangle$ increase with respect to the Maxwellian-averaged value (Fig. \[temperature\_averaged\]). The strongest deviation always occurs for $\kappa\to 3/2$. These deviations depend strongly on the $\gamma^{2}$ (i.e., on the temperature) and mildly on the $u$ (i.e., on the frequency) parameters. As $\gamma^{2}$ tends to $-\infty,$ the $\langle g_{_{\rm ff}}(\gamma^{2},u)\rangle$ for different $\kappa$ approximate the Maxwellian-averaged value (black dashed line; top panel of Fig. \[temperature\_averaged\]), depending on the value of $u$. With the decrease in $u,$ the blending occurs for lower $\gamma^{2}$ than for $u=5$. With an increase in $\gamma^{2}$ (decrease in temperature), the deviations become larger and more pronounced for $\gamma^{2}>10$ (bottom panel of Fig. \[temperature\_averaged\]). However, regardless of the value of $\gamma^{2}$ , the $\langle g_{_{\rm ff}}(\gamma^{2},u)\rangle$ shows a small variation for all $\kappa\geq 5$ even for $\gamma^{2}>10$.
The total Gaunt factor shows a similar dependence on $\kappa$ for the different $\gamma^{2}$ as $\langle g_{_{\rm
ff}}(\gamma^{2},u)\rangle$. That is, the deviations increase with increasing $\gamma^{2}$ (Fig. \[total\_ff\]). The total Gaunt factor maximum decreases with increasing $\kappa$ from 2.038 to 1.441 for $\kappa=2$ and $\kappa\to \infty$, respectively. When $\gamma^{2}\to \infty,$ the total Gaunt factor tends to 1.596 for $\kappa=2,$ decreasing toward 1.0 with $\kappa\to \infty$ (the Maxwellian distribution). With $\gamma^{2}\to -\infty$ $g_{_{\rm
ff}}(T)\to 1.224$ and 1.1 for $\kappa=2$ and $\kappa \to \infty $, respectively.
![Total free-free Gaunt factor calculated for $\kappa$ (2, 3, 5, 10, 15, and 25) and Maxwellian distributions. The maximum of $g_{\rm ff}(T)$ moves to the left with the increase in $\kappa$.\[total\_ff\]](figures/Total_free-free-gaunt-kappa-20150510.pdf){width="0.9\hsize"}
As the total Gaunt factor decreases with increase in $\kappa,$ the losses of energy due to free-free emission follow the same path as a result of Eq. (\[cooling\]).
Tables
======
Temperature-averaged and total free-free Gaunt factors calculated for $\kappa$ (2, 3, 5, 10, 15, 25, and 50) and Maxwellian distributions are displayed in Tables A.1-A.8 ($\langle g_{_{\rm ff}}(\gamma^{2},u)\rangle$ vs. $\gamma^{2}$ for different $u$) and B.1 (appendices A and B, respectively). The parameter space in display comprises $\gamma^{2}\in[10^{-8},10^{10}]$ and $u\in [10^{-4},10^{4}]$, but our calculations, and the publically available data in http://www.lca.uevora.pt, cover a wider range in these parameters. More data can be provided by the authors upon request.
Final remarks
=============
Optically thin plasmas in the interstellar medium may deviate from thermal equilibrium and thus, electrons are no longer described by the Maxwellian distribution. Instead they can be described by $\kappa-$distributions. These have been used to explain the deviations between derived abundances and temperatures in [[H[<span style="font-variant:small-caps;">ii</span>]{}]{}]{}regions and planetary nebulae. Free-free emission dominates the cooling function of optically thin plasmas at temperatures greater than $10^7$ K. The free-free spectrum and radiative losses depend on the temperature-averaged (over the electron energy) and total Gaunt factors, respectively. Thus, there is a need to calculate and make available these factors to be used by any software that deals with plasma emission.
Notable astrophysical plasmas, which are dominated by free-free emission, apart from supernova remnants and superbubbles in the interstellar medium, are the intracluster and intergalactic media in clusters of galaxies, for instance, where the hot medium dominates the baryonic matter. In particular, merger events in which smaller clusters and groups of galaxies fall into larger ones are accompanied by shocks, and hence deviations from thermal equilibrium are expected. On a larger scale still, structure formation shocks can develop as a consequence of gas infall onto dark matter halos, thereby converting gravitational into thermal energy [see, e.g., @pfrommer2006]. The missing-baryon problem and its possible solution by the existence of a widespread warm hot intergalactic medium [see @cen1999; @cen2006] is another example for the importance of free-free emission at high temperatures. In many of these contexts, $\kappa$-distributions are therefore expected to provide a better description than assuming a Maxwellian.
Here we have recalculated the nonrelativistic free-free Gaunt factor and its temperature averaged over a large spectrum of $\kappa$ parameters (including the Maxwellian distribution) and integrated it over the frequency to obtain the total Gaunt factor. We found that the $\kappa$ parameter most affects the temperature-averaged and total Gaunt factor at lower temperatures.
The authors thank the anonymous referee for the comments improving the paper. This research was supported by the project “Hybrid computing using accelerators & coprocessors-modelling nature with a novell approach” (PI: M.A.) funded by the InAlentejo program, CCDRA, Portugal. Partial support to M.A. and D.B. was provided by the *Deutsche Forschungsgemeinschaft*, DFG project ISM-SPP 1573. The computations made use of the ISM Xeon Phi Cluster of the Computational Astrophysics Group, University of Évora.
Tables of the temperature-averaged free-free Gaunt factor for different $\kappa$
================================================================================
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
log $\gamma^{2}$
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00
-8.00 5.528E+00 4.259E+00 3.002E+00 1.806E+00 8.424E-01 3.035E-01 9.801E-02 3.107E-02 9.826E-03
-7.80 5.528E+00 4.259E+00 3.002E+00 1.806E+00 8.425E-01 3.035E-01 9.802E-02 3.107E-02 9.827E-03
-7.60 5.528E+00 4.259E+00 3.002E+00 1.806E+00 8.425E-01 3.035E-01 9.803E-02 3.107E-02 9.828E-03
-7.40 5.528E+00 4.259E+00 3.002E+00 1.807E+00 8.426E-01 3.035E-01 9.804E-02 3.108E-02 9.830E-03
-7.20 5.528E+00 4.259E+00 3.002E+00 1.807E+00 8.426E-01 3.036E-01 9.806E-02 3.108E-02 9.831E-03
-7.00 5.528E+00 4.259E+00 3.002E+00 1.807E+00 8.427E-01 3.036E-01 9.808E-02 3.109E-02 9.834E-03
-6.80 5.528E+00 4.259E+00 3.002E+00 1.807E+00 8.428E-01 3.037E-01 9.810E-02 3.110E-02 9.836E-03
-6.60 5.528E+00 4.259E+00 3.002E+00 1.807E+00 8.429E-01 3.038E-01 9.814E-02 3.111E-02 9.840E-03
-6.40 5.528E+00 4.259E+00 3.002E+00 1.807E+00 8.431E-01 3.039E-01 9.818E-02 3.112E-02 9.845E-03
-6.20 5.528E+00 4.260E+00 3.002E+00 1.807E+00 8.433E-01 3.040E-01 9.823E-02 3.114E-02 9.850E-03
-6.00 5.528E+00 4.260E+00 3.002E+00 1.807E+00 8.436E-01 3.042E-01 9.829E-02 3.116E-02 9.857E-03
-5.80 5.528E+00 4.260E+00 3.002E+00 1.808E+00 8.439E-01 3.044E-01 9.838E-02 3.119E-02 9.866E-03
-5.60 5.528E+00 4.260E+00 3.002E+00 1.808E+00 8.443E-01 3.046E-01 9.848E-02 3.122E-02 9.877E-03
-5.40 5.528E+00 4.260E+00 3.002E+00 1.808E+00 8.449E-01 3.050E-01 9.861E-02 3.127E-02 9.892E-03
-5.20 5.528E+00 4.260E+00 3.003E+00 1.809E+00 8.455E-01 3.054E-01 9.877E-02 3.132E-02 9.909E-03
-5.00 5.528E+00 4.260E+00 3.003E+00 1.810E+00 8.464E-01 3.059E-01 9.898E-02 3.139E-02 9.932E-03
-4.80 5.528E+00 4.260E+00 3.003E+00 1.810E+00 8.474E-01 3.066E-01 9.924E-02 3.148E-02 9.960E-03
-4.60 5.528E+00 4.260E+00 3.004E+00 1.812E+00 8.487E-01 3.074E-01 9.957E-02 3.159E-02 9.996E-03
-4.40 5.528E+00 4.260E+00 3.004E+00 1.813E+00 8.504E-01 3.084E-01 9.999E-02 3.173E-02 1.004E-02
-4.20 5.528E+00 4.260E+00 3.005E+00 1.815E+00 8.525E-01 3.098E-01 1.005E-01 3.191E-02 1.010E-02
-4.00 5.527E+00 4.260E+00 3.006E+00 1.817E+00 8.552E-01 3.114E-01 1.012E-01 3.214E-02 1.017E-02
-3.80 5.527E+00 4.260E+00 3.007E+00 1.819E+00 8.586E-01 3.136E-01 1.020E-01 3.242E-02 1.026E-02
-3.60 5.527E+00 4.260E+00 3.008E+00 1.823E+00 8.628E-01 3.163E-01 1.031E-01 3.279E-02 1.038E-02
-3.40 5.527E+00 4.260E+00 3.010E+00 1.827E+00 8.682E-01 3.197E-01 1.045E-01 3.325E-02 1.053E-02
-3.20 5.526E+00 4.260E+00 3.011E+00 1.832E+00 8.750E-01 3.240E-01 1.062E-01 3.384E-02 1.072E-02
-3.00 5.525E+00 4.259E+00 3.013E+00 1.838E+00 8.835E-01 3.296E-01 1.084E-01 3.460E-02 1.097E-02
-2.80 5.523E+00 4.258E+00 3.015E+00 1.846E+00 8.944E-01 3.367E-01 1.113E-01 3.558E-02 1.128E-02
-2.60 5.520E+00 4.257E+00 3.016E+00 1.855E+00 9.079E-01 3.457E-01 1.150E-01 3.684E-02 1.169E-02
-2.40 5.516E+00 4.254E+00 3.018E+00 1.865E+00 9.250E-01 3.573E-01 1.198E-01 3.847E-02 1.222E-02
-2.20 5.510E+00 4.249E+00 3.018E+00 1.877E+00 9.461E-01 3.722E-01 1.260E-01 4.060E-02 1.291E-02
-2.00 5.501E+00 4.241E+00 3.016E+00 1.890E+00 9.719E-01 3.913E-01 1.340E-01 4.337E-02 1.381E-02
-1.80 5.489E+00 4.231E+00 3.012E+00 1.903E+00 1.003E+00 4.156E-01 1.445E-01 4.702E-02 1.500E-02
-1.60 5.472E+00 4.216E+00 3.004E+00 1.914E+00 1.040E+00 4.464E-01 1.582E-01 5.181E-02 1.656E-02
-1.40 5.449E+00 4.195E+00 2.991E+00 1.923E+00 1.081E+00 4.850E-01 1.759E-01 5.812E-02 1.863E-02
-1.20 5.419E+00 4.167E+00 2.971E+00 1.928E+00 1.126E+00 5.328E-01 1.990E-01 6.644E-02 2.137E-02
-1.00 5.379E+00 4.131E+00 2.944E+00 1.926E+00 1.172E+00 5.905E-01 2.286E-01 7.735E-02 2.499E-02
-0.80 5.329E+00 4.083E+00 2.907E+00 1.917E+00 1.216E+00 6.581E-01 2.662E-01 9.158E-02 2.975E-02
-0.60 5.267E+00 4.025E+00 2.860E+00 1.898E+00 1.253E+00 7.343E-01 3.132E-01 1.100E-01 3.597E-02
-0.40 5.193E+00 3.955E+00 2.803E+00 1.869E+00 1.279E+00 8.157E-01 3.706E-01 1.335E-01 4.404E-02
-0.20 5.107E+00 3.873E+00 2.735E+00 1.831E+00 1.294E+00 8.975E-01 4.388E-01 1.631E-01 5.439E-02
0.00 5.010E+00 3.782E+00 2.658E+00 1.785E+00 1.296E+00 9.731E-01 5.173E-01 1.998E-01 6.753E-02
0.20 4.904E+00 3.682E+00 2.574E+00 1.733E+00 1.287E+00 1.037E+00 6.044E-01 2.445E-01 8.403E-02
0.40 4.792E+00 3.577E+00 2.486E+00 1.678E+00 1.270E+00 1.084E+00 6.967E-01 2.981E-01 1.046E-01
0.60 4.676E+00 3.468E+00 2.396E+00 1.622E+00 1.248E+00 1.113E+00 7.898E-01 3.610E-01 1.299E-01
0.80 4.557E+00 3.357E+00 2.306E+00 1.566E+00 1.224E+00 1.127E+00 8.782E-01 4.334E-01 1.610E-01
1.00 4.436E+00 3.245E+00 2.216E+00 1.512E+00 1.200E+00 1.129E+00 9.561E-01 5.146E-01 1.987E-01
1.20 4.315E+00 3.134E+00 2.129E+00 1.461E+00 1.178E+00 1.124E+00 1.019E+00 6.031E-01 2.440E-01
1.40 4.194E+00 3.024E+00 2.045E+00 1.413E+00 1.157E+00 1.115E+00 1.064E+00 6.960E-01 2.979E-01
1.60 4.073E+00 2.915E+00 1.963E+00 1.369E+00 1.137E+00 1.104E+00 1.091E+00 7.892E-01 3.609E-01
1.80 3.953E+00 2.808E+00 1.885E+00 1.328E+00 1.120E+00 1.093E+00 1.104E+00 8.773E-01 4.334E-01
2.00 3.833E+00 2.703E+00 1.811E+00 1.291E+00 1.104E+00 1.083E+00 1.107E+00 9.548E-01 5.146E-01
2.20 3.714E+00 2.601E+00 1.740E+00 1.257E+00 1.091E+00 1.073E+00 1.103E+00 1.017E+00 6.031E-01
2.40 3.596E+00 2.500E+00 1.673E+00 1.227E+00 1.079E+00 1.064E+00 1.096E+00 1.062E+00 6.960E-01
2.60 3.479E+00 2.403E+00 1.610E+00 1.199E+00 1.068E+00 1.056E+00 1.087E+00 1.089E+00 7.891E-01
2.80 3.363E+00 2.308E+00 1.552E+00 1.175E+00 1.059E+00 1.049E+00 1.078E+00 1.102E+00 8.772E-01
3.00 3.249E+00 2.216E+00 1.497E+00 1.153E+00 1.051E+00 1.042E+00 1.069E+00 1.104E+00 9.547E-01
3.20 3.136E+00 2.127E+00 1.446E+00 1.133E+00 1.044E+00 1.036E+00 1.061E+00 1.101E+00 1.017E+00
3.40 3.025E+00 2.042E+00 1.399E+00 1.116E+00 1.038E+00 1.032E+00 1.054E+00 1.094E+00 1.061E+00
3.60 2.916E+00 1.960E+00 1.355E+00 1.101E+00 1.032E+00 1.027E+00 1.047E+00 1.085E+00 1.089E+00
3.80 2.808E+00 1.882E+00 1.316E+00 1.087E+00 1.028E+00 1.023E+00 1.041E+00 1.076E+00 1.101E+00
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[rccccccccc]{}\
\
\[-2ex\] log $\gamma^{2}$ &\
& -4.00 & -3.00 & -2.00 & -1.00 & 0.00 & 1.00 & 2.00 & 3.00 & 4.00\
4.00 & 2.703E+00 & 1.808E+00 & 1.280E+00 & 1.076E+00 & 1.024E+00 & 1.020E+00 & 1.036E+00 & 1.068E+00 & 1.104E+00\
4.20 & 2.600E+00 & 1.737E+00 & 1.247E+00 & 1.065E+00 & 1.021E+00 & 1.017E+00 & 1.031E+00 & 1.060E+00 & 1.100E+00\
4.40 & 2.500E+00 & 1.671E+00 & 1.218E+00 & 1.056E+00 & 1.018E+00 & 1.015E+00 & 1.027E+00 & 1.053E+00 & 1.093E+00\
4.60 & 2.402E+00 & 1.608E+00 & 1.191E+00 & 1.049E+00 & 1.015E+00 & 1.013E+00 & 1.023E+00 & 1.046E+00 & 1.085E+00\
4.80 & 2.307E+00 & 1.549E+00 & 1.167E+00 & 1.042E+00 & 1.013E+00 & 1.011E+00 & 1.020E+00 & 1.040E+00 & 1.076E+00\
5.00 & 2.215E+00 & 1.495E+00 & 1.146E+00 & 1.036E+00 & 1.011E+00 & 1.010E+00 & 1.017E+00 & 1.035E+00 & 1.068E+00\
5.20 & 2.127E+00 & 1.444E+00 & 1.128E+00 & 1.031E+00 & 1.010E+00 & 1.008E+00 & 1.015E+00 & 1.030E+00 & 1.060E+00\
5.40 & 2.042E+00 & 1.397E+00 & 1.111E+00 & 1.027E+00 & 1.008E+00 & 1.007E+00 & 1.013E+00 & 1.026E+00 & 1.053E+00\
5.60 & 1.960E+00 & 1.354E+00 & 1.096E+00 & 1.023E+00 & 1.007E+00 & 1.006E+00 & 1.011E+00 & 1.023E+00 & 1.046E+00\
5.80 & 1.882E+00 & 1.314E+00 & 1.084E+00 & 1.020E+00 & 1.006E+00 & 1.005E+00 & 1.009E+00 & 1.019E+00 & 1.040E+00\
6.00 & 1.807E+00 & 1.279E+00 & 1.072E+00 & 1.017E+00 & 1.005E+00 & 1.004E+00 & 1.008E+00 & 1.017E+00 & 1.035E+00\
6.20 & 1.737E+00 & 1.246E+00 & 1.063E+00 & 1.015E+00 & 1.004E+00 & 1.004E+00 & 1.007E+00 & 1.014E+00 & 1.030E+00\
6.40 & 1.670E+00 & 1.217E+00 & 1.054E+00 & 1.013E+00 & 1.004E+00 & 1.003E+00 & 1.006E+00 & 1.012E+00 & 1.026E+00\
6.60 & 1.608E+00 & 1.190E+00 & 1.047E+00 & 1.011E+00 & 1.003E+00 & 1.003E+00 & 1.005E+00 & 1.011E+00 & 1.023E+00\
6.80 & 1.549E+00 & 1.167E+00 & 1.040E+00 & 1.009E+00 & 1.003E+00 & 1.002E+00 & 1.004E+00 & 1.009E+00 & 1.019E+00\
7.00 & 1.494E+00 & 1.146E+00 & 1.035E+00 & 1.008E+00 & 1.002E+00 & 1.002E+00 & 1.004E+00 & 1.008E+00 & 1.017E+00\
7.20 & 1.444E+00 & 1.127E+00 & 1.030E+00 & 1.007E+00 & 1.002E+00 & 1.002E+00 & 1.003E+00 & 1.007E+00 & 1.014E+00\
7.40 & 1.397E+00 & 1.110E+00 & 1.026E+00 & 1.006E+00 & 1.002E+00 & 1.002E+00 & 1.003E+00 & 1.006E+00 & 1.012E+00\
7.60 & 1.354E+00 & 1.096E+00 & 1.022E+00 & 1.005E+00 & 1.002E+00 & 1.001E+00 & 1.002E+00 & 1.005E+00 & 1.011E+00\
7.80 & 1.314E+00 & 1.083E+00 & 1.019E+00 & 1.004E+00 & 1.001E+00 & 1.001E+00 & 1.002E+00 & 1.004E+00 & 1.009E+00\
8.00 & 1.278E+00 & 1.072E+00 & 1.016E+00 & 1.004E+00 & 1.001E+00 & 1.001E+00 & 1.002E+00 & 1.004E+00 & 1.008E+00\
8.20 & 1.246E+00 & 1.062E+00 & 1.014E+00 & 1.003E+00 & 1.001E+00 & 1.001E+00 & 1.002E+00 & 1.003E+00 & 1.007E+00\
8.40 & 1.217E+00 & 1.054E+00 & 1.012E+00 & 1.003E+00 & 1.001E+00 & 1.001E+00 & 1.001E+00 & 1.003E+00 & 1.006E+00\
8.60 & 1.190E+00 & 1.046E+00 & 1.010E+00 & 1.002E+00 & 1.001E+00 & 1.001E+00 & 1.001E+00 & 1.002E+00 & 1.005E+00\
8.80 & 1.167E+00 & 1.040E+00 & 1.009E+00 & 1.002E+00 & 1.001E+00 & 1.001E+00 & 1.001E+00 & 1.002E+00 & 1.004E+00\
9.00 & 1.146E+00 & 1.034E+00 & 1.008E+00 & 1.002E+00 & 1.001E+00 & 1.000E+00 & 1.001E+00 & 1.002E+00 & 1.004E+00\
9.20 & 1.127E+00 & 1.030E+00 & 1.007E+00 & 1.001E+00 & 1.000E+00 & 1.000E+00 & 1.001E+00 & 1.001E+00 & 1.003E+00\
9.40 & 1.110E+00 & 1.026E+00 & 1.006E+00 & 1.001E+00 & 1.000E+00 & 1.000E+00 & 1.001E+00 & 1.001E+00 & 1.003E+00\
9.60 & 1.096E+00 & 1.022E+00 & 1.005E+00 & 1.001E+00 & 1.000E+00 & 1.000E+00 & 1.001E+00 & 1.001E+00 & 1.002E+00\
9.80 & 1.083E+00 & 1.019E+00 & 1.004E+00 & 1.001E+00 & 1.000E+00 & 1.000E+00 & 1.000E+00 & 1.001E+00 & 1.002E+00\
10.00 & 1.072E+00 & 1.016E+00 & 1.004E+00 & 1.001E+00 & 1.000E+00 & 1.000E+00 & 1.000E+00 & 1.001E+00 & 1.002E+00\
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
log $\gamma^{2}$
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00
-8.00 7.857E+00 5.836E+00 3.859E+00 2.101E+00 8.838E-01 3.054E-01 9.814E-02 3.110E-02 9.836E-03
-7.80 7.857E+00 5.836E+00 3.860E+00 2.101E+00 8.839E-01 3.055E-01 9.816E-02 3.111E-02 9.839E-03
-7.60 7.857E+00 5.836E+00 3.860E+00 2.101E+00 8.840E-01 3.055E-01 9.818E-02 3.111E-02 9.841E-03
-7.40 7.857E+00 5.836E+00 3.860E+00 2.101E+00 8.841E-01 3.056E-01 9.820E-02 3.112E-02 9.843E-03
-7.20 7.857E+00 5.837E+00 3.860E+00 2.101E+00 8.843E-01 3.056E-01 9.823E-02 3.113E-02 9.846E-03
-7.00 7.857E+00 5.837E+00 3.860E+00 2.101E+00 8.845E-01 3.057E-01 9.826E-02 3.114E-02 9.849E-03
-6.80 7.857E+00 5.837E+00 3.860E+00 2.102E+00 8.847E-01 3.059E-01 9.831E-02 3.115E-02 9.854E-03
-6.60 7.857E+00 5.837E+00 3.860E+00 2.102E+00 8.850E-01 3.060E-01 9.836E-02 3.117E-02 9.860E-03
-6.40 7.857E+00 5.837E+00 3.861E+00 2.102E+00 8.854E-01 3.062E-01 9.843E-02 3.119E-02 9.867E-03
-6.20 7.857E+00 5.837E+00 3.861E+00 2.103E+00 8.859E-01 3.064E-01 9.851E-02 3.122E-02 9.876E-03
-6.00 7.857E+00 5.837E+00 3.861E+00 2.104E+00 8.865E-01 3.067E-01 9.862E-02 3.126E-02 9.887E-03
-5.80 7.857E+00 5.837E+00 3.862E+00 2.105E+00 8.873E-01 3.071E-01 9.876E-02 3.130E-02 9.901E-03
-5.60 7.857E+00 5.837E+00 3.863E+00 2.106E+00 8.883E-01 3.076E-01 9.893E-02 3.136E-02 9.919E-03
-5.40 7.857E+00 5.837E+00 3.864E+00 2.108E+00 8.896E-01 3.082E-01 9.915E-02 3.143E-02 9.942E-03
-5.20 7.857E+00 5.838E+00 3.865E+00 2.109E+00 8.912E-01 3.089E-01 9.942E-02 3.152E-02 9.971E-03
-5.00 7.857E+00 5.838E+00 3.866E+00 2.112E+00 8.932E-01 3.099E-01 9.976E-02 3.163E-02 1.001E-02
-4.80 7.857E+00 5.838E+00 3.868E+00 2.115E+00 8.957E-01 3.111E-01 1.002E-01 3.178E-02 1.005E-02
-4.60 7.857E+00 5.839E+00 3.870E+00 2.119E+00 8.989E-01 3.126E-01 1.008E-01 3.196E-02 1.011E-02
-4.40 7.856E+00 5.839E+00 3.873E+00 2.124E+00 9.030E-01 3.146E-01 1.015E-01 3.219E-02 1.019E-02
-4.20 7.856E+00 5.840E+00 3.876E+00 2.130E+00 9.081E-01 3.170E-01 1.023E-01 3.248E-02 1.028E-02
-4.00 7.856E+00 5.840E+00 3.880E+00 2.137E+00 9.146E-01 3.201E-01 1.035E-01 3.285E-02 1.040E-02
-3.80 7.855E+00 5.841E+00 3.885E+00 2.147E+00 9.228E-01 3.241E-01 1.049E-01 3.333E-02 1.055E-02
-3.60 7.854E+00 5.841E+00 3.891E+00 2.159E+00 9.332E-01 3.292E-01 1.068E-01 3.394E-02 1.075E-02
-3.40 7.852E+00 5.841E+00 3.898E+00 2.174E+00 9.465E-01 3.356E-01 1.091E-01 3.472E-02 1.100E-02
-3.20 7.849E+00 5.841E+00 3.905E+00 2.192E+00 9.634E-01 3.440E-01 1.122E-01 3.573E-02 1.132E-02
-3.00 7.845E+00 5.839E+00 3.914E+00 2.215E+00 9.850E-01 3.547E-01 1.161E-01 3.703E-02 1.174E-02
-2.80 7.837E+00 5.835E+00 3.923E+00 2.243E+00 1.012E+00 3.686E-01 1.212E-01 3.873E-02 1.228E-02
-2.60 7.826E+00 5.828E+00 3.932E+00 2.276E+00 1.047E+00 3.866E-01 1.279E-01 4.096E-02 1.300E-02
-2.40 7.809E+00 5.817E+00 3.939E+00 2.314E+00 1.091E+00 4.098E-01 1.367E-01 4.387E-02 1.394E-02
-2.20 7.785E+00 5.799E+00 3.941E+00 2.356E+00 1.145E+00 4.399E-01 1.481E-01 4.771E-02 1.517E-02
-2.00 7.752E+00 5.772E+00 3.938E+00 2.400E+00 1.211E+00 4.786E-01 1.631E-01 5.277E-02 1.680E-02
-1.80 7.708E+00 5.736E+00 3.927E+00 2.443E+00 1.289E+00 5.280E-01 1.828E-01 5.944E-02 1.896E-02
-1.60 7.651E+00 5.688E+00 3.906E+00 2.481E+00 1.379E+00 5.902E-01 2.083E-01 6.821E-02 2.180E-02
-1.40 7.579E+00 5.625E+00 3.872E+00 2.509E+00 1.478E+00 6.675E-01 2.414E-01 7.970E-02 2.555E-02
-1.20 7.490E+00 5.548E+00 3.824E+00 2.525E+00 1.582E+00 7.613E-01 2.837E-01 9.468E-02 3.046E-02
-1.00 7.384E+00 5.454E+00 3.762E+00 2.525E+00 1.682E+00 8.720E-01 3.372E-01 1.140E-01 3.685E-02
-0.80 7.260E+00 5.344E+00 3.686E+00 2.509E+00 1.770E+00 9.980E-01 4.038E-01 1.388E-01 4.510E-02
-0.60 7.120E+00 5.220E+00 3.597E+00 2.479E+00 1.840E+00 1.135E+00 4.851E-01 1.703E-01 5.569E-02
-0.40 6.966E+00 5.083E+00 3.497E+00 2.436E+00 1.887E+00 1.277E+00 5.821E-01 2.096E-01 6.915E-02
-0.20 6.801E+00 4.937E+00 3.390E+00 2.383E+00 1.910E+00 1.414E+00 6.952E-01 2.583E-01 8.614E-02
0.00 6.628E+00 4.784E+00 3.278E+00 2.326E+00 1.913E+00 1.538E+00 8.233E-01 3.179E-01 1.074E-01
0.20 6.448E+00 4.627E+00 3.165E+00 2.265E+00 1.902E+00 1.638E+00 9.636E-01 3.899E-01 1.340E-01
0.40 6.265E+00 4.469E+00 3.051E+00 2.205E+00 1.882E+00 1.711E+00 1.111E+00 4.756E-01 1.669E-01
0.60 6.081E+00 4.311E+00 2.941E+00 2.146E+00 1.858E+00 1.756E+00 1.260E+00 5.761E-01 2.074E-01
0.80 5.896E+00 4.155E+00 2.834E+00 2.090E+00 1.832E+00 1.777E+00 1.401E+00 6.916E-01 2.569E-01
1.00 5.711E+00 4.001E+00 2.731E+00 2.038E+00 1.806E+00 1.781E+00 1.525E+00 8.212E-01 3.171E-01
1.20 5.528E+00 3.851E+00 2.634E+00 1.990E+00 1.782E+00 1.774E+00 1.624E+00 9.625E-01 3.894E-01
1.40 5.347E+00 3.705E+00 2.542E+00 1.946E+00 1.760E+00 1.761E+00 1.696E+00 1.111E+00 4.754E-01
1.60 5.168E+00 3.564E+00 2.455E+00 1.906E+00 1.739E+00 1.746E+00 1.739E+00 1.259E+00 5.760E-01
1.80 4.991E+00 3.427E+00 2.374E+00 1.870E+00 1.721E+00 1.731E+00 1.760E+00 1.400E+00 6.916E-01
2.00 4.818E+00 3.296E+00 2.299E+00 1.837E+00 1.705E+00 1.716E+00 1.764E+00 1.524E+00 8.212E-01
2.20 4.647E+00 3.170E+00 2.229E+00 1.808E+00 1.691E+00 1.702E+00 1.758E+00 1.623E+00 9.624E-01
2.40 4.480E+00 3.049E+00 2.165E+00 1.782E+00 1.678E+00 1.689E+00 1.746E+00 1.694E+00 1.111E+00
2.60 4.317E+00 2.934E+00 2.105E+00 1.758E+00 1.667E+00 1.677E+00 1.733E+00 1.737E+00 1.259E+00
2.80 4.157E+00 2.825E+00 2.051E+00 1.738E+00 1.657E+00 1.666E+00 1.719E+00 1.758E+00 1.400E+00
3.00 4.002E+00 2.721E+00 2.001E+00 1.720E+00 1.649E+00 1.657E+00 1.705E+00 1.762E+00 1.523E+00
3.20 3.850E+00 2.624E+00 1.956E+00 1.704E+00 1.642E+00 1.649E+00 1.692E+00 1.756E+00 1.623E+00
3.40 3.704E+00 2.532E+00 1.916E+00 1.690E+00 1.635E+00 1.642E+00 1.681E+00 1.745E+00 1.694E+00
3.60 3.562E+00 2.446E+00 1.879E+00 1.677E+00 1.630E+00 1.635E+00 1.670E+00 1.732E+00 1.737E+00
3.80 3.426E+00 2.366E+00 1.845E+00 1.666E+00 1.625E+00 1.630E+00 1.660E+00 1.718E+00 1.758E+00
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[rccccccccc]{}\
\
\[-2ex\] log $\gamma^{2}$ &\
& -4.00 & -3.00 & -2.00 & -1.00 & 0.00 & 1.00 & 2.00 & 3.00 & 4.00\
4.00 & 3.294E+00 & 2.291E+00 & 1.816E+00 & 1.657E+00 & 1.621E+00 & 1.625E+00 & 1.652E+00 & 1.704E+00 & 1.762E+00\
4.20 & 3.168E+00 & 2.222E+00 & 1.789E+00 & 1.649E+00 & 1.618E+00 & 1.621E+00 & 1.644E+00 & 1.691E+00 & 1.756E+00\
4.40 & 3.047E+00 & 2.158E+00 & 1.765E+00 & 1.642E+00 & 1.614E+00 & 1.618E+00 & 1.638E+00 & 1.680E+00 & 1.745E+00\
4.60 & 2.933E+00 & 2.100E+00 & 1.744E+00 & 1.635E+00 & 1.612E+00 & 1.615E+00 & 1.632E+00 & 1.669E+00 & 1.731E+00\
4.80 & 2.823E+00 & 2.046E+00 & 1.725E+00 & 1.630E+00 & 1.610E+00 & 1.612E+00 & 1.627E+00 & 1.660E+00 & 1.718E+00\
5.00 & 2.720E+00 & 1.997E+00 & 1.709E+00 & 1.625E+00 & 1.608E+00 & 1.610E+00 & 1.623E+00 & 1.651E+00 & 1.704E+00\
5.20 & 2.623E+00 & 1.953E+00 & 1.694E+00 & 1.621E+00 & 1.606E+00 & 1.608E+00 & 1.619E+00 & 1.644E+00 & 1.691E+00\
5.40 & 2.531E+00 & 1.912E+00 & 1.682E+00 & 1.618E+00 & 1.605E+00 & 1.606E+00 & 1.616E+00 & 1.637E+00 & 1.680E+00\
5.60 & 2.445E+00 & 1.876E+00 & 1.670E+00 & 1.615E+00 & 1.603E+00 & 1.605E+00 & 1.613E+00 & 1.632E+00 & 1.669E+00\
5.80 & 2.365E+00 & 1.843E+00 & 1.660E+00 & 1.612E+00 & 1.602E+00 & 1.603E+00 & 1.611E+00 & 1.627E+00 & 1.660E+00\
6.00 & 2.291E+00 & 1.813E+00 & 1.652E+00 & 1.610E+00 & 1.601E+00 & 1.602E+00 & 1.609E+00 & 1.623E+00 & 1.651E+00\
6.20 & 2.222E+00 & 1.787E+00 & 1.644E+00 & 1.608E+00 & 1.601E+00 & 1.601E+00 & 1.607E+00 & 1.619E+00 & 1.644E+00\
6.40 & 2.158E+00 & 1.764E+00 & 1.638E+00 & 1.606E+00 & 1.600E+00 & 1.601E+00 & 1.605E+00 & 1.616E+00 & 1.637E+00\
6.60 & 2.099E+00 & 1.743E+00 & 1.632E+00 & 1.605E+00 & 1.599E+00 & 1.600E+00 & 1.604E+00 & 1.613E+00 & 1.632E+00\
6.80 & 2.046E+00 & 1.724E+00 & 1.627E+00 & 1.603E+00 & 1.599E+00 & 1.599E+00 & 1.603E+00 & 1.610E+00 & 1.627E+00\
7.00 & 1.997E+00 & 1.708E+00 & 1.623E+00 & 1.602E+00 & 1.598E+00 & 1.599E+00 & 1.602E+00 & 1.608E+00 & 1.623E+00\
7.20 & 1.952E+00 & 1.693E+00 & 1.619E+00 & 1.601E+00 & 1.598E+00 & 1.598E+00 & 1.601E+00 & 1.607E+00 & 1.619E+00\
7.40 & 1.912E+00 & 1.681E+00 & 1.616E+00 & 1.601E+00 & 1.598E+00 & 1.598E+00 & 1.600E+00 & 1.605E+00 & 1.616E+00\
7.60 & 1.875E+00 & 1.670E+00 & 1.613E+00 & 1.600E+00 & 1.597E+00 & 1.598E+00 & 1.600E+00 & 1.604E+00 & 1.613E+00\
7.80 & 1.843E+00 & 1.660E+00 & 1.611E+00 & 1.599E+00 & 1.597E+00 & 1.597E+00 & 1.599E+00 & 1.603E+00 & 1.610E+00\
8.00 & 1.813E+00 & 1.651E+00 & 1.609E+00 & 1.599E+00 & 1.597E+00 & 1.597E+00 & 1.599E+00 & 1.602E+00 & 1.608E+00\
8.20 & 1.787E+00 & 1.644E+00 & 1.607E+00 & 1.598E+00 & 1.597E+00 & 1.597E+00 & 1.598E+00 & 1.601E+00 & 1.607E+00\
8.40 & 1.763E+00 & 1.637E+00 & 1.605E+00 & 1.598E+00 & 1.597E+00 & 1.597E+00 & 1.598E+00 & 1.600E+00 & 1.605E+00\
8.60 & 1.743E+00 & 1.632E+00 & 1.604E+00 & 1.598E+00 & 1.597E+00 & 1.597E+00 & 1.598E+00 & 1.600E+00 & 1.604E+00\
8.80 & 1.724E+00 & 1.627E+00 & 1.603E+00 & 1.597E+00 & 1.596E+00 & 1.597E+00 & 1.597E+00 & 1.599E+00 & 1.603E+00\
9.00 & 1.708E+00 & 1.623E+00 & 1.602E+00 & 1.597E+00 & 1.596E+00 & 1.596E+00 & 1.597E+00 & 1.599E+00 & 1.602E+00\
9.20 & 1.693E+00 & 1.619E+00 & 1.601E+00 & 1.597E+00 & 1.596E+00 & 1.596E+00 & 1.597E+00 & 1.598E+00 & 1.601E+00\
9.40 & 1.681E+00 & 1.616E+00 & 1.600E+00 & 1.597E+00 & 1.596E+00 & 1.596E+00 & 1.597E+00 & 1.598E+00 & 1.600E+00\
9.60 & 1.670E+00 & 1.613E+00 & 1.600E+00 & 1.597E+00 & 1.596E+00 & 1.596E+00 & 1.597E+00 & 1.598E+00 & 1.600E+00\
9.80 & 1.660E+00 & 1.611E+00 & 1.599E+00 & 1.597E+00 & 1.596E+00 & 1.596E+00 & 1.596E+00 & 1.597E+00 & 1.599E+00\
10.00 & 1.651E+00 & 1.608E+00 & 1.599E+00 & 1.596E+00 & 1.596E+00 & 1.596E+00 & 1.596E+00 & 1.597E+00 & 1.599E+00\
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
log $\gamma^{2}$
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00
-8.00 6.444E+00 4.888E+00 3.351E+00 1.930E+00 8.594E-01 3.041E-01 9.806E-02 3.108E-02 9.830E-03
-7.80 6.444E+00 4.888E+00 3.351E+00 1.930E+00 8.594E-01 3.041E-01 9.807E-02 3.108E-02 9.831E-03
-7.60 6.444E+00 4.888E+00 3.351E+00 1.930E+00 8.595E-01 3.041E-01 9.808E-02 3.109E-02 9.833E-03
-7.40 6.444E+00 4.888E+00 3.352E+00 1.931E+00 8.596E-01 3.042E-01 9.809E-02 3.109E-02 9.835E-03
-7.20 6.444E+00 4.888E+00 3.352E+00 1.931E+00 8.597E-01 3.042E-01 9.812E-02 3.110E-02 9.837E-03
-7.00 6.444E+00 4.888E+00 3.352E+00 1.931E+00 8.598E-01 3.043E-01 9.814E-02 3.111E-02 9.840E-03
-6.80 6.444E+00 4.888E+00 3.352E+00 1.931E+00 8.599E-01 3.044E-01 9.817E-02 3.112E-02 9.843E-03
-6.60 6.444E+00 4.888E+00 3.352E+00 1.931E+00 8.601E-01 3.045E-01 9.821E-02 3.113E-02 9.847E-03
-6.40 6.444E+00 4.888E+00 3.352E+00 1.931E+00 8.604E-01 3.046E-01 9.827E-02 3.115E-02 9.853E-03
-6.20 6.444E+00 4.888E+00 3.352E+00 1.932E+00 8.607E-01 3.048E-01 9.833E-02 3.117E-02 9.860E-03
-6.00 6.444E+00 4.888E+00 3.352E+00 1.932E+00 8.611E-01 3.050E-01 9.841E-02 3.120E-02 9.868E-03
-5.80 6.444E+00 4.888E+00 3.353E+00 1.933E+00 8.616E-01 3.053E-01 9.851E-02 3.123E-02 9.879E-03
-5.60 6.444E+00 4.888E+00 3.353E+00 1.933E+00 8.623E-01 3.056E-01 9.864E-02 3.127E-02 9.893E-03
-5.40 6.444E+00 4.888E+00 3.353E+00 1.934E+00 8.631E-01 3.061E-01 9.881E-02 3.133E-02 9.911E-03
-5.20 6.444E+00 4.888E+00 3.354E+00 1.935E+00 8.641E-01 3.066E-01 9.901E-02 3.140E-02 9.933E-03
-5.00 6.444E+00 4.888E+00 3.354E+00 1.936E+00 8.653E-01 3.073E-01 9.927E-02 3.148E-02 9.961E-03
-4.80 6.444E+00 4.888E+00 3.355E+00 1.938E+00 8.669E-01 3.082E-01 9.960E-02 3.159E-02 9.996E-03
-4.60 6.444E+00 4.889E+00 3.356E+00 1.940E+00 8.690E-01 3.093E-01 1.000E-01 3.173E-02 1.004E-02
-4.40 6.444E+00 4.889E+00 3.357E+00 1.942E+00 8.715E-01 3.107E-01 1.005E-01 3.191E-02 1.010E-02
-4.20 6.444E+00 4.889E+00 3.359E+00 1.946E+00 8.748E-01 3.124E-01 1.012E-01 3.213E-02 1.017E-02
-4.00 6.444E+00 4.889E+00 3.361E+00 1.950E+00 8.788E-01 3.146E-01 1.020E-01 3.241E-02 1.026E-02
-3.80 6.444E+00 4.889E+00 3.363E+00 1.955E+00 8.840E-01 3.175E-01 1.031E-01 3.277E-02 1.037E-02
-3.60 6.443E+00 4.890E+00 3.366E+00 1.961E+00 8.906E-01 3.210E-01 1.045E-01 3.323E-02 1.052E-02
-3.40 6.442E+00 4.890E+00 3.369E+00 1.968E+00 8.989E-01 3.256E-01 1.062E-01 3.381E-02 1.071E-02
-3.20 6.441E+00 4.889E+00 3.372E+00 1.978E+00 9.094E-01 3.315E-01 1.085E-01 3.456E-02 1.095E-02
-3.00 6.439E+00 4.888E+00 3.376E+00 1.990E+00 9.227E-01 3.390E-01 1.113E-01 3.552E-02 1.126E-02
-2.80 6.435E+00 4.886E+00 3.380E+00 2.004E+00 9.396E-01 3.486E-01 1.150E-01 3.677E-02 1.166E-02
-2.60 6.430E+00 4.883E+00 3.384E+00 2.021E+00 9.609E-01 3.610E-01 1.199E-01 3.838E-02 1.218E-02
-2.40 6.422E+00 4.877E+00 3.387E+00 2.041E+00 9.876E-01 3.769E-01 1.261E-01 4.049E-02 1.286E-02
-2.20 6.410E+00 4.868E+00 3.388E+00 2.063E+00 1.021E+00 3.975E-01 1.343E-01 4.326E-02 1.376E-02
-2.00 6.393E+00 4.854E+00 3.385E+00 2.087E+00 1.061E+00 4.238E-01 1.449E-01 4.689E-02 1.493E-02
-1.80 6.370E+00 4.835E+00 3.378E+00 2.110E+00 1.110E+00 4.575E-01 1.588E-01 5.166E-02 1.648E-02
-1.60 6.340E+00 4.808E+00 3.365E+00 2.131E+00 1.166E+00 5.002E-01 1.769E-01 5.795E-02 1.853E-02
-1.40 6.299E+00 4.773E+00 3.345E+00 2.147E+00 1.229E+00 5.535E-01 2.005E-01 6.623E-02 2.123E-02
-1.20 6.248E+00 4.727E+00 3.314E+00 2.156E+00 1.297E+00 6.189E-01 2.309E-01 7.708E-02 2.480E-02
-1.00 6.184E+00 4.668E+00 3.273E+00 2.155E+00 1.364E+00 6.970E-01 2.697E-01 9.124E-02 2.948E-02
-0.80 6.106E+00 4.597E+00 3.220E+00 2.143E+00 1.425E+00 7.873E-01 3.185E-01 1.096E-01 3.559E-02
-0.60 6.014E+00 4.512E+00 3.156E+00 2.119E+00 1.475E+00 8.873E-01 3.788E-01 1.330E-01 4.350E-02
-0.40 5.908E+00 4.415E+00 3.080E+00 2.084E+00 1.510E+00 9.923E-01 4.516E-01 1.626E-01 5.366E-02
-0.20 5.791E+00 4.307E+00 2.995E+00 2.040E+00 1.528E+00 1.096E+00 5.372E-01 1.996E-01 6.657E-02
0.00 5.663E+00 4.190E+00 2.902E+00 1.988E+00 1.530E+00 1.190E+00 6.348E-01 2.451E-01 8.285E-02
0.20 5.528E+00 4.066E+00 2.805E+00 1.932E+00 1.520E+00 1.267E+00 7.423E-01 3.003E-01 1.032E-01
0.40 5.388E+00 3.939E+00 2.706E+00 1.875E+00 1.502E+00 1.324E+00 8.558E-01 3.662E-01 1.285E-01
0.60 5.245E+00 3.809E+00 2.606E+00 1.817E+00 1.480E+00 1.360E+00 9.702E-01 4.435E-01 1.596E-01
0.80 5.100E+00 3.679E+00 2.508E+00 1.762E+00 1.456E+00 1.376E+00 1.079E+00 5.324E-01 1.978E-01
1.00 4.954E+00 3.550E+00 2.413E+00 1.709E+00 1.431E+00 1.379E+00 1.174E+00 6.322E-01 2.441E-01
1.20 4.808E+00 3.422E+00 2.320E+00 1.659E+00 1.408E+00 1.373E+00 1.251E+00 7.409E-01 2.998E-01
1.40 4.663E+00 3.297E+00 2.232E+00 1.613E+00 1.387E+00 1.363E+00 1.306E+00 8.550E-01 3.660E-01
1.60 4.518E+00 3.174E+00 2.148E+00 1.571E+00 1.367E+00 1.350E+00 1.340E+00 9.694E-01 4.434E-01
1.80 4.375E+00 3.054E+00 2.068E+00 1.532E+00 1.350E+00 1.338E+00 1.355E+00 1.078E+00 5.324E-01
2.00 4.233E+00 2.937E+00 1.993E+00 1.497E+00 1.334E+00 1.325E+00 1.359E+00 1.173E+00 6.322E-01
2.20 4.093E+00 2.823E+00 1.922E+00 1.466E+00 1.320E+00 1.314E+00 1.354E+00 1.249E+00 7.409E-01
2.40 3.955E+00 2.714E+00 1.856E+00 1.437E+00 1.308E+00 1.303E+00 1.345E+00 1.304E+00 8.550E-01
2.60 3.818E+00 2.608E+00 1.795E+00 1.412E+00 1.298E+00 1.294E+00 1.335E+00 1.338E+00 9.694E-01
2.80 3.684E+00 2.506E+00 1.738E+00 1.389E+00 1.288E+00 1.285E+00 1.324E+00 1.353E+00 1.078E+00
3.00 3.552E+00 2.409E+00 1.685E+00 1.369E+00 1.280E+00 1.278E+00 1.313E+00 1.356E+00 1.173E+00
3.20 3.423E+00 2.315E+00 1.637E+00 1.351E+00 1.273E+00 1.271E+00 1.303E+00 1.352E+00 1.249E+00
3.40 3.297E+00 2.227E+00 1.592E+00 1.335E+00 1.267E+00 1.265E+00 1.294E+00 1.343E+00 1.304E+00
3.60 3.173E+00 2.143E+00 1.552E+00 1.321E+00 1.261E+00 1.260E+00 1.286E+00 1.333E+00 1.337E+00
3.80 3.053E+00 2.063E+00 1.515E+00 1.309E+00 1.257E+00 1.256E+00 1.278E+00 1.322E+00 1.353E+00
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[rccccccccc]{}\
\
\[-2ex\] log $\gamma^{2}$ &\
& -4.00 & -3.00 & -2.00 & -1.00 & 0.00 & 1.00 & 2.00 & 3.00 & 4.00\
4.00 & 2.936E+00 & 1.988E+00 & 1.482E+00 & 1.298E+00 & 1.253E+00 & 1.252E+00 & 1.272E+00 & 1.312E+00 & 1.356E+00\
4.20 & 2.822E+00 & 1.918E+00 & 1.452E+00 & 1.289E+00 & 1.249E+00 & 1.249E+00 & 1.266E+00 & 1.302E+00 & 1.352E+00\
4.40 & 2.713E+00 & 1.852E+00 & 1.425E+00 & 1.281E+00 & 1.247E+00 & 1.246E+00 & 1.261E+00 & 1.293E+00 & 1.343E+00\
4.60 & 2.607E+00 & 1.791E+00 & 1.401E+00 & 1.273E+00 & 1.244E+00 & 1.244E+00 & 1.257E+00 & 1.285E+00 & 1.333E+00\
4.80 & 2.505E+00 & 1.734E+00 & 1.380E+00 & 1.267E+00 & 1.242E+00 & 1.241E+00 & 1.253E+00 & 1.278E+00 & 1.322E+00\
5.00 & 2.408E+00 & 1.682E+00 & 1.361E+00 & 1.262E+00 & 1.240E+00 & 1.240E+00 & 1.249E+00 & 1.271E+00 & 1.312E+00\
5.20 & 2.315E+00 & 1.634E+00 & 1.344E+00 & 1.257E+00 & 1.238E+00 & 1.238E+00 & 1.246E+00 & 1.266E+00 & 1.302E+00\
5.40 & 2.226E+00 & 1.590E+00 & 1.329E+00 & 1.253E+00 & 1.237E+00 & 1.237E+00 & 1.244E+00 & 1.261E+00 & 1.293E+00\
5.60 & 2.142E+00 & 1.550E+00 & 1.316E+00 & 1.250E+00 & 1.236E+00 & 1.236E+00 & 1.242E+00 & 1.256E+00 & 1.285E+00\
5.80 & 2.062E+00 & 1.513E+00 & 1.304E+00 & 1.247E+00 & 1.235E+00 & 1.235E+00 & 1.240E+00 & 1.252E+00 & 1.278E+00\
6.00 & 1.988E+00 & 1.480E+00 & 1.294E+00 & 1.244E+00 & 1.234E+00 & 1.234E+00 & 1.238E+00 & 1.249E+00 & 1.271E+00\
6.20 & 1.917E+00 & 1.451E+00 & 1.285E+00 & 1.242E+00 & 1.233E+00 & 1.233E+00 & 1.237E+00 & 1.246E+00 & 1.265E+00\
6.40 & 1.852E+00 & 1.424E+00 & 1.278E+00 & 1.240E+00 & 1.232E+00 & 1.232E+00 & 1.236E+00 & 1.244E+00 & 1.260E+00\
6.60 & 1.791E+00 & 1.400E+00 & 1.271E+00 & 1.238E+00 & 1.232E+00 & 1.232E+00 & 1.235E+00 & 1.242E+00 & 1.256E+00\
6.80 & 1.734E+00 & 1.379E+00 & 1.265E+00 & 1.237E+00 & 1.231E+00 & 1.231E+00 & 1.234E+00 & 1.240E+00 & 1.252E+00\
7.00 & 1.682E+00 & 1.360E+00 & 1.260E+00 & 1.236E+00 & 1.231E+00 & 1.231E+00 & 1.233E+00 & 1.238E+00 & 1.249E+00\
7.20 & 1.634E+00 & 1.343E+00 & 1.256E+00 & 1.235E+00 & 1.231E+00 & 1.231E+00 & 1.232E+00 & 1.237E+00 & 1.246E+00\
7.40 & 1.590E+00 & 1.328E+00 & 1.252E+00 & 1.234E+00 & 1.230E+00 & 1.230E+00 & 1.232E+00 & 1.236E+00 & 1.244E+00\
7.60 & 1.550E+00 & 1.315E+00 & 1.249E+00 & 1.233E+00 & 1.230E+00 & 1.230E+00 & 1.231E+00 & 1.235E+00 & 1.242E+00\
7.80 & 1.513E+00 & 1.304E+00 & 1.246E+00 & 1.232E+00 & 1.230E+00 & 1.230E+00 & 1.231E+00 & 1.234E+00 & 1.240E+00\
8.00 & 1.480E+00 & 1.294E+00 & 1.243E+00 & 1.232E+00 & 1.230E+00 & 1.230E+00 & 1.231E+00 & 1.233E+00 & 1.238E+00\
8.20 & 1.450E+00 & 1.285E+00 & 1.241E+00 & 1.231E+00 & 1.229E+00 & 1.229E+00 & 1.230E+00 & 1.232E+00 & 1.237E+00\
8.40 & 1.424E+00 & 1.277E+00 & 1.239E+00 & 1.231E+00 & 1.229E+00 & 1.229E+00 & 1.230E+00 & 1.232E+00 & 1.236E+00\
8.60 & 1.400E+00 & 1.271E+00 & 1.238E+00 & 1.231E+00 & 1.229E+00 & 1.229E+00 & 1.230E+00 & 1.231E+00 & 1.235E+00\
8.80 & 1.379E+00 & 1.265E+00 & 1.237E+00 & 1.230E+00 & 1.229E+00 & 1.229E+00 & 1.230E+00 & 1.231E+00 & 1.234E+00\
9.00 & 1.360E+00 & 1.260E+00 & 1.235E+00 & 1.230E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.231E+00 & 1.233E+00\
9.20 & 1.343E+00 & 1.256E+00 & 1.234E+00 & 1.230E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.230E+00 & 1.232E+00\
9.40 & 1.328E+00 & 1.252E+00 & 1.234E+00 & 1.230E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.230E+00 & 1.232E+00\
9.60 & 1.315E+00 & 1.248E+00 & 1.233E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.230E+00 & 1.231E+00\
9.80 & 1.304E+00 & 1.246E+00 & 1.232E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.230E+00 & 1.231E+00\
10.00 & 1.294E+00 & 1.243E+00 & 1.232E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.229E+00 & 1.231E+00\
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
log $\gamma^{2}$
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00
-8.00 5.944E+00 4.546E+00 3.162E+00 1.864E+00 8.500E-01 3.037E-01 9.803E-02 3.107E-02 9.828E-03
-7.80 5.944E+00 4.546E+00 3.162E+00 1.864E+00 8.501E-01 3.037E-01 9.804E-02 3.107E-02 9.829E-03
-7.60 5.944E+00 4.546E+00 3.162E+00 1.864E+00 8.501E-01 3.038E-01 9.805E-02 3.108E-02 9.830E-03
-7.40 5.944E+00 4.546E+00 3.162E+00 1.864E+00 8.502E-01 3.038E-01 9.806E-02 3.108E-02 9.832E-03
-7.20 5.944E+00 4.546E+00 3.162E+00 1.864E+00 8.503E-01 3.038E-01 9.808E-02 3.109E-02 9.834E-03
-7.00 5.944E+00 4.546E+00 3.162E+00 1.864E+00 8.504E-01 3.039E-01 9.811E-02 3.110E-02 9.836E-03
-6.80 5.944E+00 4.546E+00 3.162E+00 1.864E+00 8.505E-01 3.040E-01 9.813E-02 3.111E-02 9.839E-03
-6.60 5.944E+00 4.546E+00 3.162E+00 1.864E+00 8.507E-01 3.041E-01 9.817E-02 3.112E-02 9.843E-03
-6.40 5.944E+00 4.546E+00 3.163E+00 1.865E+00 8.509E-01 3.042E-01 9.822E-02 3.113E-02 9.848E-03
-6.20 5.944E+00 4.546E+00 3.163E+00 1.865E+00 8.511E-01 3.043E-01 9.827E-02 3.115E-02 9.854E-03
-6.00 5.944E+00 4.546E+00 3.163E+00 1.865E+00 8.515E-01 3.045E-01 9.835E-02 3.118E-02 9.862E-03
-5.80 5.944E+00 4.546E+00 3.163E+00 1.865E+00 8.519E-01 3.048E-01 9.844E-02 3.121E-02 9.872E-03
-5.60 5.944E+00 4.546E+00 3.163E+00 1.866E+00 8.524E-01 3.051E-01 9.855E-02 3.125E-02 9.885E-03
-5.40 5.944E+00 4.546E+00 3.163E+00 1.866E+00 8.530E-01 3.054E-01 9.870E-02 3.130E-02 9.900E-03
-5.20 5.944E+00 4.546E+00 3.164E+00 1.867E+00 8.539E-01 3.059E-01 9.888E-02 3.136E-02 9.920E-03
-5.00 5.944E+00 4.546E+00 3.164E+00 1.868E+00 8.549E-01 3.065E-01 9.911E-02 3.143E-02 9.945E-03
-4.80 5.944E+00 4.546E+00 3.165E+00 1.869E+00 8.562E-01 3.073E-01 9.940E-02 3.153E-02 9.976E-03
-4.60 5.944E+00 4.546E+00 3.165E+00 1.871E+00 8.578E-01 3.082E-01 9.977E-02 3.166E-02 1.002E-02
-4.40 5.944E+00 4.547E+00 3.166E+00 1.873E+00 8.599E-01 3.094E-01 1.002E-01 3.181E-02 1.007E-02
-4.20 5.944E+00 4.547E+00 3.167E+00 1.875E+00 8.625E-01 3.109E-01 1.008E-01 3.201E-02 1.013E-02
-4.00 5.944E+00 4.547E+00 3.169E+00 1.878E+00 8.658E-01 3.128E-01 1.016E-01 3.226E-02 1.021E-02
-3.80 5.944E+00 4.547E+00 3.170E+00 1.881E+00 8.700E-01 3.153E-01 1.025E-01 3.258E-02 1.031E-02
-3.60 5.943E+00 4.547E+00 3.172E+00 1.886E+00 8.753E-01 3.184E-01 1.037E-01 3.298E-02 1.044E-02
-3.40 5.943E+00 4.547E+00 3.174E+00 1.892E+00 8.819E-01 3.223E-01 1.053E-01 3.350E-02 1.061E-02
-3.20 5.942E+00 4.547E+00 3.176E+00 1.899E+00 8.904E-01 3.273E-01 1.072E-01 3.416E-02 1.082E-02
-3.00 5.940E+00 4.546E+00 3.179E+00 1.907E+00 9.011E-01 3.338E-01 1.097E-01 3.501E-02 1.110E-02
-2.80 5.937E+00 4.545E+00 3.182E+00 1.918E+00 9.146E-01 3.420E-01 1.130E-01 3.611E-02 1.145E-02
-2.60 5.934E+00 4.542E+00 3.184E+00 1.930E+00 9.316E-01 3.525E-01 1.172E-01 3.753E-02 1.191E-02
-2.40 5.928E+00 4.538E+00 3.186E+00 1.945E+00 9.529E-01 3.661E-01 1.226E-01 3.937E-02 1.251E-02
-2.20 5.919E+00 4.531E+00 3.187E+00 1.961E+00 9.793E-01 3.835E-01 1.297E-01 4.178E-02 1.329E-02
-2.00 5.907E+00 4.521E+00 3.185E+00 1.979E+00 1.012E+00 4.058E-01 1.389E-01 4.494E-02 1.431E-02
-1.80 5.891E+00 4.507E+00 3.179E+00 1.996E+00 1.051E+00 4.343E-01 1.509E-01 4.909E-02 1.566E-02
-1.60 5.868E+00 4.487E+00 3.169E+00 2.012E+00 1.096E+00 4.704E-01 1.665E-01 5.455E-02 1.744E-02
-1.40 5.837E+00 4.460E+00 3.152E+00 2.024E+00 1.147E+00 5.156E-01 1.869E-01 6.175E-02 1.979E-02
-1.20 5.798E+00 4.424E+00 3.128E+00 2.030E+00 1.202E+00 5.713E-01 2.133E-01 7.120E-02 2.291E-02
-1.00 5.747E+00 4.377E+00 3.095E+00 2.029E+00 1.258E+00 6.383E-01 2.470E-01 8.358E-02 2.700E-02
-0.80 5.684E+00 4.319E+00 3.050E+00 2.018E+00 1.309E+00 7.162E-01 2.897E-01 9.966E-02 3.237E-02
-0.60 5.609E+00 4.249E+00 2.995E+00 1.997E+00 1.353E+00 8.031E-01 3.427E-01 1.203E-01 3.936E-02
-0.40 5.520E+00 4.166E+00 2.929E+00 1.965E+00 1.383E+00 8.953E-01 4.071E-01 1.466E-01 4.837E-02
-0.20 5.420E+00 4.072E+00 2.853E+00 1.924E+00 1.399E+00 9.868E-01 4.832E-01 1.796E-01 5.988E-02
0.00 5.309E+00 3.969E+00 2.768E+00 1.876E+00 1.401E+00 1.071E+00 5.703E-01 2.202E-01 7.443E-02
0.20 5.190E+00 3.858E+00 2.678E+00 1.822E+00 1.392E+00 1.141E+00 6.665E-01 2.697E-01 9.268E-02
0.40 5.065E+00 3.742E+00 2.585E+00 1.766E+00 1.374E+00 1.192E+00 7.684E-01 3.288E-01 1.154E-01
0.60 4.937E+00 3.624E+00 2.490E+00 1.709E+00 1.352E+00 1.224E+00 8.711E-01 3.982E-01 1.433E-01
0.80 4.806E+00 3.504E+00 2.396E+00 1.653E+00 1.328E+00 1.239E+00 9.684E-01 4.780E-01 1.776E-01
1.00 4.674E+00 3.384E+00 2.304E+00 1.600E+00 1.304E+00 1.241E+00 1.054E+00 5.676E-01 2.192E-01
1.20 4.541E+00 3.266E+00 2.215E+00 1.549E+00 1.282E+00 1.236E+00 1.123E+00 6.652E-01 2.692E-01
1.40 4.409E+00 3.148E+00 2.128E+00 1.503E+00 1.260E+00 1.226E+00 1.173E+00 7.677E-01 3.286E-01
1.60 4.277E+00 3.033E+00 2.045E+00 1.459E+00 1.241E+00 1.215E+00 1.203E+00 8.704E-01 3.981E-01
1.80 4.146E+00 2.920E+00 1.967E+00 1.420E+00 1.224E+00 1.203E+00 1.217E+00 9.675E-01 4.780E-01
2.00 4.017E+00 2.809E+00 1.892E+00 1.384E+00 1.208E+00 1.192E+00 1.220E+00 1.053E+00 5.676E-01
2.20 3.888E+00 2.701E+00 1.821E+00 1.351E+00 1.194E+00 1.182E+00 1.216E+00 1.122E+00 6.652E-01
2.40 3.760E+00 2.597E+00 1.755E+00 1.321E+00 1.182E+00 1.172E+00 1.208E+00 1.171E+00 7.676E-01
2.60 3.634E+00 2.495E+00 1.692E+00 1.295E+00 1.171E+00 1.163E+00 1.199E+00 1.201E+00 8.703E-01
2.80 3.510E+00 2.397E+00 1.634E+00 1.271E+00 1.162E+00 1.155E+00 1.189E+00 1.215E+00 9.674E-01
3.00 3.388E+00 2.302E+00 1.581E+00 1.250E+00 1.154E+00 1.148E+00 1.179E+00 1.218E+00 1.053E+00
3.20 3.267E+00 2.211E+00 1.531E+00 1.231E+00 1.147E+00 1.142E+00 1.170E+00 1.214E+00 1.122E+00
3.40 3.149E+00 2.124E+00 1.485E+00 1.214E+00 1.141E+00 1.137E+00 1.162E+00 1.206E+00 1.171E+00
3.60 3.033E+00 2.041E+00 1.443E+00 1.200E+00 1.136E+00 1.132E+00 1.155E+00 1.197E+00 1.201E+00
3.80 2.919E+00 1.963E+00 1.405E+00 1.187E+00 1.131E+00 1.128E+00 1.148E+00 1.187E+00 1.215E+00
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[rccccccccc]{}\
\
\[-2ex\] log $\gamma^{2}$ &\
& -4.00 & -3.00 & -2.00 & -1.00 & 0.00 & 1.00 & 2.00 & 3.00 & 4.00\
4.00 & 2.809E+00 & 1.888E+00 & 1.370E+00 & 1.176E+00 & 1.127E+00 & 1.125E+00 & 1.142E+00 & 1.178E+00 & 1.218E+00\
4.20 & 2.701E+00 & 1.817E+00 & 1.339E+00 & 1.166E+00 & 1.124E+00 & 1.122E+00 & 1.137E+00 & 1.169E+00 & 1.213E+00\
4.40 & 2.596E+00 & 1.751E+00 & 1.311E+00 & 1.157E+00 & 1.121E+00 & 1.119E+00 & 1.132E+00 & 1.161E+00 & 1.206E+00\
4.60 & 2.494E+00 & 1.689E+00 & 1.285E+00 & 1.150E+00 & 1.118E+00 & 1.117E+00 & 1.128E+00 & 1.154E+00 & 1.197E+00\
4.80 & 2.396E+00 & 1.632E+00 & 1.263E+00 & 1.143E+00 & 1.116E+00 & 1.115E+00 & 1.125E+00 & 1.147E+00 & 1.187E+00\
5.00 & 2.302E+00 & 1.578E+00 & 1.243E+00 & 1.138E+00 & 1.114E+00 & 1.113E+00 & 1.122E+00 & 1.141E+00 & 1.178E+00\
5.20 & 2.211E+00 & 1.529E+00 & 1.225E+00 & 1.133E+00 & 1.113E+00 & 1.112E+00 & 1.119E+00 & 1.136E+00 & 1.169E+00\
5.40 & 2.124E+00 & 1.483E+00 & 1.209E+00 & 1.129E+00 & 1.111E+00 & 1.110E+00 & 1.117E+00 & 1.132E+00 & 1.161E+00\
5.60 & 2.041E+00 & 1.441E+00 & 1.195E+00 & 1.125E+00 & 1.110E+00 & 1.109E+00 & 1.115E+00 & 1.128E+00 & 1.154E+00\
5.80 & 1.962E+00 & 1.403E+00 & 1.183E+00 & 1.122E+00 & 1.109E+00 & 1.108E+00 & 1.113E+00 & 1.124E+00 & 1.147E+00\
6.00 & 1.887E+00 & 1.369E+00 & 1.172E+00 & 1.119E+00 & 1.108E+00 & 1.108E+00 & 1.112E+00 & 1.121E+00 & 1.141E+00\
6.20 & 1.817E+00 & 1.338E+00 & 1.163E+00 & 1.117E+00 & 1.107E+00 & 1.107E+00 & 1.111E+00 & 1.119E+00 & 1.136E+00\
6.40 & 1.751E+00 & 1.310E+00 & 1.155E+00 & 1.115E+00 & 1.107E+00 & 1.106E+00 & 1.109E+00 & 1.117E+00 & 1.132E+00\
6.60 & 1.689E+00 & 1.284E+00 & 1.148E+00 & 1.113E+00 & 1.106E+00 & 1.106E+00 & 1.109E+00 & 1.115E+00 & 1.128E+00\
6.80 & 1.631E+00 & 1.262E+00 & 1.142E+00 & 1.112E+00 & 1.106E+00 & 1.106E+00 & 1.108E+00 & 1.113E+00 & 1.124E+00\
7.00 & 1.578E+00 & 1.242E+00 & 1.136E+00 & 1.111E+00 & 1.105E+00 & 1.105E+00 & 1.107E+00 & 1.112E+00 & 1.121E+00\
7.20 & 1.528E+00 & 1.224E+00 & 1.132E+00 & 1.109E+00 & 1.105E+00 & 1.105E+00 & 1.106E+00 & 1.110E+00 & 1.119E+00\
7.40 & 1.483E+00 & 1.208E+00 & 1.128E+00 & 1.109E+00 & 1.105E+00 & 1.105E+00 & 1.106E+00 & 1.109E+00 & 1.117E+00\
7.60 & 1.441E+00 & 1.195E+00 & 1.124E+00 & 1.108E+00 & 1.104E+00 & 1.104E+00 & 1.106E+00 & 1.108E+00 & 1.115E+00\
7.80 & 1.403E+00 & 1.183E+00 & 1.121E+00 & 1.107E+00 & 1.104E+00 & 1.104E+00 & 1.105E+00 & 1.108E+00 & 1.113E+00\
8.00 & 1.369E+00 & 1.172E+00 & 1.119E+00 & 1.106E+00 & 1.104E+00 & 1.104E+00 & 1.105E+00 & 1.107E+00 & 1.112E+00\
8.20 & 1.338E+00 & 1.163E+00 & 1.116E+00 & 1.106E+00 & 1.104E+00 & 1.104E+00 & 1.105E+00 & 1.106E+00 & 1.110E+00\
8.40 & 1.310E+00 & 1.154E+00 & 1.114E+00 & 1.106E+00 & 1.104E+00 & 1.104E+00 & 1.104E+00 & 1.106E+00 & 1.109E+00\
8.60 & 1.284E+00 & 1.147E+00 & 1.113E+00 & 1.105E+00 & 1.104E+00 & 1.104E+00 & 1.104E+00 & 1.105E+00 & 1.108E+00\
8.80 & 1.262E+00 & 1.141E+00 & 1.111E+00 & 1.105E+00 & 1.104E+00 & 1.103E+00 & 1.104E+00 & 1.105E+00 & 1.108E+00\
9.00 & 1.242E+00 & 1.136E+00 & 1.110E+00 & 1.105E+00 & 1.103E+00 & 1.103E+00 & 1.104E+00 & 1.105E+00 & 1.107E+00\
9.20 & 1.224E+00 & 1.131E+00 & 1.109E+00 & 1.104E+00 & 1.103E+00 & 1.103E+00 & 1.104E+00 & 1.105E+00 & 1.106E+00\
9.40 & 1.208E+00 & 1.127E+00 & 1.108E+00 & 1.104E+00 & 1.103E+00 & 1.103E+00 & 1.104E+00 & 1.104E+00 & 1.106E+00\
9.60 & 1.195E+00 & 1.124E+00 & 1.108E+00 & 1.104E+00 & 1.103E+00 & 1.103E+00 & 1.103E+00 & 1.104E+00 & 1.105E+00\
9.80 & 1.182E+00 & 1.121E+00 & 1.107E+00 & 1.104E+00 & 1.103E+00 & 1.103E+00 & 1.103E+00 & 1.104E+00 & 1.105E+00\
10.00 & 1.172E+00 & 1.118E+00 & 1.106E+00 & 1.104E+00 & 1.103E+00 & 1.103E+00 & 1.103E+00 & 1.104E+00 & 1.105E+00\
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
log $\gamma^{2}$
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00
-8.00 5.704E+00 4.381E+00 3.070E+00 1.831E+00 8.456E-01 3.036E-01 9.802E-02 3.107E-02 9.827E-03
-7.80 5.704E+00 4.381E+00 3.070E+00 1.831E+00 8.457E-01 3.036E-01 9.803E-02 3.107E-02 9.828E-03
-7.60 5.704E+00 4.381E+00 3.070E+00 1.831E+00 8.457E-01 3.036E-01 9.804E-02 3.107E-02 9.829E-03
-7.40 5.704E+00 4.381E+00 3.070E+00 1.831E+00 8.458E-01 3.036E-01 9.805E-02 3.108E-02 9.831E-03
-7.20 5.704E+00 4.381E+00 3.070E+00 1.831E+00 8.458E-01 3.037E-01 9.807E-02 3.109E-02 9.832E-03
-7.00 5.704E+00 4.381E+00 3.070E+00 1.831E+00 8.459E-01 3.037E-01 9.809E-02 3.109E-02 9.835E-03
-6.80 5.704E+00 4.381E+00 3.070E+00 1.831E+00 8.460E-01 3.038E-01 9.812E-02 3.110E-02 9.838E-03
-6.60 5.704E+00 4.381E+00 3.070E+00 1.831E+00 8.462E-01 3.039E-01 9.815E-02 3.111E-02 9.841E-03
-6.40 5.704E+00 4.381E+00 3.070E+00 1.831E+00 8.464E-01 3.040E-01 9.819E-02 3.113E-02 9.846E-03
-6.20 5.704E+00 4.381E+00 3.070E+00 1.832E+00 8.466E-01 3.041E-01 9.825E-02 3.115E-02 9.852E-03
-6.00 5.704E+00 4.381E+00 3.070E+00 1.832E+00 8.469E-01 3.043E-01 9.832E-02 3.117E-02 9.859E-03
-5.80 5.704E+00 4.381E+00 3.070E+00 1.832E+00 8.472E-01 3.045E-01 9.840E-02 3.120E-02 9.869E-03
-5.60 5.704E+00 4.381E+00 3.071E+00 1.833E+00 8.477E-01 3.048E-01 9.851E-02 3.123E-02 9.880E-03
-5.40 5.704E+00 4.381E+00 3.071E+00 1.833E+00 8.483E-01 3.052E-01 9.865E-02 3.128E-02 9.895E-03
-5.20 5.704E+00 4.381E+00 3.071E+00 1.834E+00 8.490E-01 3.056E-01 9.882E-02 3.134E-02 9.914E-03
-5.00 5.704E+00 4.381E+00 3.072E+00 1.835E+00 8.499E-01 3.061E-01 9.904E-02 3.141E-02 9.937E-03
-4.80 5.704E+00 4.381E+00 3.072E+00 1.836E+00 8.511E-01 3.068E-01 9.931E-02 3.150E-02 9.967E-03
-4.60 5.704E+00 4.381E+00 3.072E+00 1.837E+00 8.525E-01 3.077E-01 9.966E-02 3.162E-02 1.000E-02
-4.40 5.704E+00 4.381E+00 3.073E+00 1.838E+00 8.544E-01 3.088E-01 1.001E-01 3.177E-02 1.005E-02
-4.20 5.704E+00 4.382E+00 3.074E+00 1.840E+00 8.567E-01 3.102E-01 1.006E-01 3.195E-02 1.011E-02
-4.00 5.704E+00 4.382E+00 3.075E+00 1.843E+00 8.596E-01 3.120E-01 1.013E-01 3.219E-02 1.019E-02
-3.80 5.703E+00 4.382E+00 3.076E+00 1.846E+00 8.633E-01 3.143E-01 1.022E-01 3.249E-02 1.029E-02
-3.60 5.703E+00 4.382E+00 3.078E+00 1.850E+00 8.680E-01 3.171E-01 1.034E-01 3.287E-02 1.041E-02
-3.40 5.703E+00 4.382E+00 3.079E+00 1.854E+00 8.739E-01 3.208E-01 1.048E-01 3.336E-02 1.057E-02
-3.20 5.702E+00 4.381E+00 3.081E+00 1.860E+00 8.814E-01 3.254E-01 1.066E-01 3.398E-02 1.077E-02
-3.00 5.700E+00 4.381E+00 3.084E+00 1.867E+00 8.909E-01 3.313E-01 1.090E-01 3.478E-02 1.102E-02
-2.80 5.698E+00 4.380E+00 3.086E+00 1.876E+00 9.028E-01 3.389E-01 1.120E-01 3.580E-02 1.135E-02
-2.60 5.695E+00 4.378E+00 3.088E+00 1.887E+00 9.178E-01 3.486E-01 1.159E-01 3.713E-02 1.178E-02
-2.40 5.690E+00 4.374E+00 3.089E+00 1.899E+00 9.367E-01 3.610E-01 1.210E-01 3.885E-02 1.234E-02
-2.20 5.683E+00 4.368E+00 3.089E+00 1.913E+00 9.600E-01 3.769E-01 1.275E-01 4.109E-02 1.307E-02
-2.00 5.673E+00 4.360E+00 3.087E+00 1.927E+00 9.886E-01 3.974E-01 1.361E-01 4.403E-02 1.402E-02
-1.80 5.659E+00 4.348E+00 3.083E+00 1.942E+00 1.023E+00 4.234E-01 1.472E-01 4.789E-02 1.528E-02
-1.60 5.640E+00 4.331E+00 3.074E+00 1.955E+00 1.063E+00 4.565E-01 1.617E-01 5.296E-02 1.693E-02
-1.40 5.614E+00 4.308E+00 3.059E+00 1.966E+00 1.109E+00 4.979E-01 1.806E-01 5.965E-02 1.912E-02
-1.20 5.580E+00 4.276E+00 3.038E+00 1.971E+00 1.158E+00 5.490E-01 2.050E-01 6.844E-02 2.202E-02
-1.00 5.535E+00 4.235E+00 3.008E+00 1.970E+00 1.208E+00 6.106E-01 2.364E-01 7.997E-02 2.584E-02
-0.80 5.480E+00 4.184E+00 2.968E+00 1.960E+00 1.255E+00 6.826E-01 2.761E-01 9.498E-02 3.086E-02
-0.60 5.412E+00 4.120E+00 2.917E+00 1.940E+00 1.295E+00 7.633E-01 3.257E-01 1.143E-01 3.740E-02
-0.40 5.332E+00 4.045E+00 2.856E+00 1.910E+00 1.323E+00 8.493E-01 3.860E-01 1.390E-01 4.587E-02
-0.20 5.240E+00 3.958E+00 2.785E+00 1.870E+00 1.338E+00 9.351E-01 4.575E-01 1.700E-01 5.670E-02
0.00 5.137E+00 3.861E+00 2.705E+00 1.823E+00 1.340E+00 1.014E+00 5.397E-01 2.084E-01 7.044E-02
0.20 5.026E+00 3.757E+00 2.618E+00 1.771E+00 1.331E+00 1.080E+00 6.306E-01 2.551E-01 8.768E-02
0.40 4.908E+00 3.647E+00 2.528E+00 1.715E+00 1.314E+00 1.129E+00 7.270E-01 3.110E-01 1.091E-01
0.60 4.787E+00 3.534E+00 2.436E+00 1.658E+00 1.292E+00 1.160E+00 8.241E-01 3.767E-01 1.356E-01
0.80 4.662E+00 3.419E+00 2.344E+00 1.602E+00 1.268E+00 1.174E+00 9.162E-01 4.522E-01 1.680E-01
1.00 4.537E+00 3.304E+00 2.253E+00 1.549E+00 1.244E+00 1.176E+00 9.975E-01 5.369E-01 2.073E-01
1.20 4.411E+00 3.190E+00 2.165E+00 1.498E+00 1.222E+00 1.171E+00 1.063E+00 6.293E-01 2.546E-01
1.40 4.285E+00 3.077E+00 2.080E+00 1.451E+00 1.200E+00 1.162E+00 1.110E+00 7.262E-01 3.108E-01
1.60 4.160E+00 2.965E+00 1.998E+00 1.407E+00 1.181E+00 1.151E+00 1.138E+00 8.234E-01 3.766E-01
1.80 4.035E+00 2.855E+00 1.919E+00 1.367E+00 1.164E+00 1.140E+00 1.152E+00 9.153E-01 4.522E-01
2.00 3.911E+00 2.748E+00 1.845E+00 1.330E+00 1.148E+00 1.129E+00 1.154E+00 9.962E-01 5.369E-01
2.20 3.788E+00 2.643E+00 1.774E+00 1.297E+00 1.134E+00 1.119E+00 1.150E+00 1.061E+00 6.293E-01
2.40 3.666E+00 2.541E+00 1.707E+00 1.267E+00 1.122E+00 1.109E+00 1.143E+00 1.108E+00 7.262E-01
2.60 3.545E+00 2.441E+00 1.645E+00 1.239E+00 1.112E+00 1.101E+00 1.134E+00 1.136E+00 8.234E-01
2.80 3.426E+00 2.345E+00 1.586E+00 1.215E+00 1.102E+00 1.094E+00 1.125E+00 1.149E+00 9.152E-01
3.00 3.308E+00 2.252E+00 1.532E+00 1.194E+00 1.094E+00 1.087E+00 1.116E+00 1.152E+00 9.961E-01
3.20 3.192E+00 2.163E+00 1.481E+00 1.174E+00 1.087E+00 1.081E+00 1.107E+00 1.148E+00 1.061E+00
3.40 3.077E+00 2.077E+00 1.435E+00 1.157E+00 1.081E+00 1.076E+00 1.099E+00 1.141E+00 1.108E+00
3.60 2.965E+00 1.994E+00 1.392E+00 1.142E+00 1.076E+00 1.072E+00 1.092E+00 1.132E+00 1.136E+00
3.80 2.855E+00 1.916E+00 1.353E+00 1.129E+00 1.071E+00 1.068E+00 1.086E+00 1.123E+00 1.149E+00
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[rccccccccc]{}\
\
\[-2ex\] log $\gamma^{2}$ &\
& -4.00 & -3.00 & -2.00 & -1.00 & 0.00 & 1.00 & 2.00 & 3.00 & 4.00\
4.00 & 2.748E+00 & 1.841E+00 & 1.318E+00 & 1.118E+00 & 1.067E+00 & 1.064E+00 & 1.080E+00 & 1.114E+00 & 1.152E+00\
4.20 & 2.643E+00 & 1.771E+00 & 1.286E+00 & 1.108E+00 & 1.064E+00 & 1.061E+00 & 1.075E+00 & 1.106E+00 & 1.148E+00\
4.40 & 2.540E+00 & 1.704E+00 & 1.257E+00 & 1.099E+00 & 1.061E+00 & 1.059E+00 & 1.071E+00 & 1.098E+00 & 1.141E+00\
4.60 & 2.441E+00 & 1.642E+00 & 1.231E+00 & 1.091E+00 & 1.059E+00 & 1.057E+00 & 1.067E+00 & 1.091E+00 & 1.132E+00\
4.80 & 2.345E+00 & 1.584E+00 & 1.208E+00 & 1.085E+00 & 1.057E+00 & 1.055E+00 & 1.064E+00 & 1.085E+00 & 1.123E+00\
5.00 & 2.252E+00 & 1.530E+00 & 1.187E+00 & 1.079E+00 & 1.055E+00 & 1.053E+00 & 1.061E+00 & 1.080E+00 & 1.114E+00\
5.20 & 2.162E+00 & 1.479E+00 & 1.169E+00 & 1.074E+00 & 1.053E+00 & 1.052E+00 & 1.059E+00 & 1.075E+00 & 1.106E+00\
5.40 & 2.076E+00 & 1.433E+00 & 1.152E+00 & 1.070E+00 & 1.052E+00 & 1.051E+00 & 1.057E+00 & 1.071E+00 & 1.098E+00\
5.60 & 1.994E+00 & 1.391E+00 & 1.138E+00 & 1.066E+00 & 1.051E+00 & 1.050E+00 & 1.055E+00 & 1.067E+00 & 1.091E+00\
5.80 & 1.915E+00 & 1.352E+00 & 1.125E+00 & 1.063E+00 & 1.050E+00 & 1.049E+00 & 1.053E+00 & 1.064E+00 & 1.085E+00\
6.00 & 1.841E+00 & 1.317E+00 & 1.114E+00 & 1.060E+00 & 1.049E+00 & 1.048E+00 & 1.052E+00 & 1.061E+00 & 1.080E+00\
6.20 & 1.770E+00 & 1.285E+00 & 1.105E+00 & 1.058E+00 & 1.048E+00 & 1.047E+00 & 1.051E+00 & 1.058E+00 & 1.075E+00\
6.40 & 1.704E+00 & 1.256E+00 & 1.096E+00 & 1.056E+00 & 1.047E+00 & 1.047E+00 & 1.050E+00 & 1.056E+00 & 1.071E+00\
6.60 & 1.642E+00 & 1.230E+00 & 1.089E+00 & 1.054E+00 & 1.047E+00 & 1.046E+00 & 1.049E+00 & 1.055E+00 & 1.067E+00\
6.80 & 1.583E+00 & 1.207E+00 & 1.083E+00 & 1.052E+00 & 1.046E+00 & 1.046E+00 & 1.048E+00 & 1.053E+00 & 1.064E+00\
7.00 & 1.529E+00 & 1.186E+00 & 1.077E+00 & 1.051E+00 & 1.046E+00 & 1.046E+00 & 1.047E+00 & 1.052E+00 & 1.061E+00\
7.20 & 1.479E+00 & 1.168E+00 & 1.073E+00 & 1.050E+00 & 1.045E+00 & 1.045E+00 & 1.047E+00 & 1.050E+00 & 1.058E+00\
7.40 & 1.433E+00 & 1.152E+00 & 1.069E+00 & 1.049E+00 & 1.045E+00 & 1.045E+00 & 1.046E+00 & 1.049E+00 & 1.056E+00\
7.60 & 1.391E+00 & 1.138E+00 & 1.065E+00 & 1.048E+00 & 1.045E+00 & 1.045E+00 & 1.046E+00 & 1.049E+00 & 1.055E+00\
7.80 & 1.352E+00 & 1.125E+00 & 1.062E+00 & 1.048E+00 & 1.045E+00 & 1.045E+00 & 1.046E+00 & 1.048E+00 & 1.053E+00\
8.00 & 1.316E+00 & 1.114E+00 & 1.059E+00 & 1.047E+00 & 1.045E+00 & 1.044E+00 & 1.045E+00 & 1.047E+00 & 1.052E+00\
8.20 & 1.285E+00 & 1.105E+00 & 1.057E+00 & 1.047E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00 & 1.047E+00 & 1.050E+00\
8.40 & 1.256E+00 & 1.096E+00 & 1.055E+00 & 1.046E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00 & 1.046E+00 & 1.049E+00\
8.60 & 1.230E+00 & 1.089E+00 & 1.054E+00 & 1.046E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00 & 1.046E+00 & 1.049E+00\
8.80 & 1.207E+00 & 1.083E+00 & 1.052E+00 & 1.045E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00 & 1.048E+00\
9.00 & 1.186E+00 & 1.077E+00 & 1.051E+00 & 1.045E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00 & 1.047E+00\
9.20 & 1.168E+00 & 1.073E+00 & 1.050E+00 & 1.045E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00 & 1.047E+00\
9.40 & 1.152E+00 & 1.068E+00 & 1.049E+00 & 1.045E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00 & 1.046E+00\
9.60 & 1.138E+00 & 1.065E+00 & 1.048E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00 & 1.046E+00\
9.80 & 1.125E+00 & 1.062E+00 & 1.047E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00\
10.00 & 1.114E+00 & 1.059E+00 & 1.047E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.044E+00 & 1.045E+00\
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
log $\gamma^{2}$
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00
-8.00 5.639E+00 4.336E+00 3.045E+00 1.822E+00 8.444E-01 3.035E-01 9.802E-02 3.107E-02 9.827E-03
-7.80 5.639E+00 4.336E+00 3.045E+00 1.822E+00 8.445E-01 3.035E-01 9.802E-02 3.107E-02 9.828E-03
-7.60 5.639E+00 4.336E+00 3.045E+00 1.822E+00 8.445E-01 3.036E-01 9.803E-02 3.107E-02 9.829E-03
-7.40 5.639E+00 4.336E+00 3.045E+00 1.822E+00 8.446E-01 3.036E-01 9.805E-02 3.108E-02 9.830E-03
-7.20 5.639E+00 4.336E+00 3.045E+00 1.822E+00 8.446E-01 3.036E-01 9.806E-02 3.108E-02 9.832E-03
-7.00 5.639E+00 4.336E+00 3.045E+00 1.822E+00 8.447E-01 3.037E-01 9.809E-02 3.109E-02 9.834E-03
-6.80 5.639E+00 4.336E+00 3.045E+00 1.822E+00 8.448E-01 3.038E-01 9.811E-02 3.110E-02 9.837E-03
-6.60 5.639E+00 4.336E+00 3.045E+00 1.822E+00 8.450E-01 3.038E-01 9.815E-02 3.111E-02 9.841E-03
-6.40 5.639E+00 4.336E+00 3.045E+00 1.823E+00 8.452E-01 3.040E-01 9.819E-02 3.113E-02 9.845E-03
-6.20 5.639E+00 4.337E+00 3.045E+00 1.823E+00 8.454E-01 3.041E-01 9.824E-02 3.114E-02 9.851E-03
-6.00 5.639E+00 4.337E+00 3.045E+00 1.823E+00 8.457E-01 3.043E-01 9.831E-02 3.117E-02 9.859E-03
-5.80 5.639E+00 4.337E+00 3.046E+00 1.823E+00 8.460E-01 3.045E-01 9.839E-02 3.119E-02 9.868E-03
-5.60 5.639E+00 4.337E+00 3.046E+00 1.824E+00 8.465E-01 3.047E-01 9.850E-02 3.123E-02 9.879E-03
-5.40 5.639E+00 4.337E+00 3.046E+00 1.824E+00 8.470E-01 3.051E-01 9.863E-02 3.128E-02 9.894E-03
-5.20 5.639E+00 4.337E+00 3.046E+00 1.825E+00 8.477E-01 3.055E-01 9.880E-02 3.133E-02 9.912E-03
-5.00 5.639E+00 4.337E+00 3.046E+00 1.825E+00 8.486E-01 3.061E-01 9.902E-02 3.140E-02 9.935E-03
-4.80 5.639E+00 4.337E+00 3.047E+00 1.826E+00 8.497E-01 3.067E-01 9.928E-02 3.150E-02 9.965E-03
-4.60 5.639E+00 4.337E+00 3.047E+00 1.828E+00 8.511E-01 3.076E-01 9.962E-02 3.161E-02 1.000E-02
-4.40 5.639E+00 4.337E+00 3.048E+00 1.829E+00 8.529E-01 3.087E-01 1.001E-01 3.175E-02 1.005E-02
-4.20 5.639E+00 4.337E+00 3.049E+00 1.831E+00 8.552E-01 3.101E-01 1.006E-01 3.194E-02 1.011E-02
-4.00 5.639E+00 4.337E+00 3.050E+00 1.833E+00 8.580E-01 3.118E-01 1.013E-01 3.217E-02 1.018E-02
-3.80 5.639E+00 4.337E+00 3.051E+00 1.836E+00 8.616E-01 3.140E-01 1.022E-01 3.247E-02 1.028E-02
-3.60 5.639E+00 4.337E+00 3.052E+00 1.840E+00 8.661E-01 3.168E-01 1.033E-01 3.284E-02 1.040E-02
-3.40 5.638E+00 4.337E+00 3.054E+00 1.844E+00 8.718E-01 3.204E-01 1.047E-01 3.332E-02 1.055E-02
-3.20 5.637E+00 4.337E+00 3.056E+00 1.850E+00 8.791E-01 3.249E-01 1.065E-01 3.393E-02 1.075E-02
-3.00 5.636E+00 4.336E+00 3.058E+00 1.857E+00 8.882E-01 3.307E-01 1.088E-01 3.471E-02 1.100E-02
-2.80 5.634E+00 4.335E+00 3.060E+00 1.865E+00 8.997E-01 3.381E-01 1.118E-01 3.572E-02 1.133E-02
-2.60 5.631E+00 4.333E+00 3.062E+00 1.875E+00 9.142E-01 3.475E-01 1.156E-01 3.702E-02 1.175E-02
-2.40 5.627E+00 4.330E+00 3.063E+00 1.887E+00 9.324E-01 3.597E-01 1.205E-01 3.871E-02 1.230E-02
-2.20 5.620E+00 4.325E+00 3.063E+00 1.900E+00 9.549E-01 3.752E-01 1.270E-01 4.091E-02 1.301E-02
-2.00 5.610E+00 4.317E+00 3.061E+00 1.914E+00 9.825E-01 3.951E-01 1.353E-01 4.379E-02 1.394E-02
-1.80 5.597E+00 4.305E+00 3.057E+00 1.928E+00 1.016E+00 4.205E-01 1.462E-01 4.757E-02 1.517E-02
-1.60 5.579E+00 4.289E+00 3.048E+00 1.940E+00 1.054E+00 4.527E-01 1.604E-01 5.254E-02 1.680E-02
-1.40 5.554E+00 4.266E+00 3.034E+00 1.950E+00 1.099E+00 4.932E-01 1.789E-01 5.909E-02 1.894E-02
-1.20 5.521E+00 4.236E+00 3.014E+00 1.955E+00 1.146E+00 5.430E-01 2.028E-01 6.771E-02 2.178E-02
-1.00 5.478E+00 4.197E+00 2.985E+00 1.954E+00 1.195E+00 6.032E-01 2.335E-01 7.901E-02 2.553E-02
-0.80 5.425E+00 4.147E+00 2.946E+00 1.944E+00 1.241E+00 6.736E-01 2.725E-01 9.373E-02 3.045E-02
-0.60 5.359E+00 4.085E+00 2.897E+00 1.924E+00 1.279E+00 7.527E-01 3.211E-01 1.127E-01 3.688E-02
-0.40 5.281E+00 4.012E+00 2.837E+00 1.895E+00 1.307E+00 8.370E-01 3.803E-01 1.370E-01 4.520E-02
-0.20 5.191E+00 3.927E+00 2.766E+00 1.856E+00 1.322E+00 9.213E-01 4.507E-01 1.675E-01 5.585E-02
0.00 5.090E+00 3.832E+00 2.687E+00 1.809E+00 1.324E+00 9.992E-01 5.315E-01 2.053E-01 6.937E-02
0.20 4.981E+00 3.730E+00 2.602E+00 1.757E+00 1.315E+00 1.064E+00 6.210E-01 2.512E-01 8.634E-02
0.40 4.866E+00 3.621E+00 2.512E+00 1.701E+00 1.297E+00 1.113E+00 7.159E-01 3.063E-01 1.075E-01
0.60 4.746E+00 3.510E+00 2.421E+00 1.645E+00 1.276E+00 1.143E+00 8.115E-01 3.709E-01 1.335E-01
0.80 4.624E+00 3.396E+00 2.330E+00 1.589E+00 1.252E+00 1.157E+00 9.023E-01 4.453E-01 1.654E-01
1.00 4.500E+00 3.283E+00 2.240E+00 1.535E+00 1.228E+00 1.159E+00 9.823E-01 5.288E-01 2.042E-01
1.20 4.376E+00 3.169E+00 2.152E+00 1.485E+00 1.205E+00 1.154E+00 1.047E+00 6.197E-01 2.507E-01
1.40 4.252E+00 3.057E+00 2.067E+00 1.437E+00 1.184E+00 1.145E+00 1.093E+00 7.152E-01 3.061E-01
1.60 4.128E+00 2.947E+00 1.985E+00 1.393E+00 1.165E+00 1.134E+00 1.121E+00 8.109E-01 3.709E-01
1.80 4.005E+00 2.838E+00 1.907E+00 1.353E+00 1.148E+00 1.123E+00 1.134E+00 9.014E-01 4.453E-01
2.00 3.882E+00 2.732E+00 1.832E+00 1.316E+00 1.132E+00 1.112E+00 1.137E+00 9.810E-01 5.288E-01
2.20 3.761E+00 2.627E+00 1.762E+00 1.282E+00 1.118E+00 1.102E+00 1.133E+00 1.045E+00 6.197E-01
2.40 3.640E+00 2.526E+00 1.695E+00 1.252E+00 1.106E+00 1.093E+00 1.126E+00 1.091E+00 7.152E-01
2.60 3.521E+00 2.427E+00 1.632E+00 1.225E+00 1.096E+00 1.084E+00 1.117E+00 1.119E+00 8.108E-01
2.80 3.403E+00 2.331E+00 1.574E+00 1.200E+00 1.086E+00 1.077E+00 1.108E+00 1.132E+00 9.013E-01
3.00 3.286E+00 2.239E+00 1.519E+00 1.179E+00 1.078E+00 1.070E+00 1.099E+00 1.135E+00 9.809E-01
3.20 3.171E+00 2.150E+00 1.468E+00 1.159E+00 1.071E+00 1.065E+00 1.090E+00 1.131E+00 1.045E+00
3.40 3.058E+00 2.064E+00 1.422E+00 1.142E+00 1.065E+00 1.060E+00 1.083E+00 1.124E+00 1.091E+00
3.60 2.947E+00 1.982E+00 1.379E+00 1.127E+00 1.060E+00 1.055E+00 1.076E+00 1.115E+00 1.119E+00
3.80 2.838E+00 1.903E+00 1.340E+00 1.114E+00 1.055E+00 1.051E+00 1.069E+00 1.106E+00 1.132E+00
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[rccccccccc]{}\
\
\[-2ex\] log $\gamma^{2}$ &\
& -4.00 & -3.00 & -2.00 & -1.00 & 0.00 & 1.00 & 2.00 & 3.00 & 4.00\
4.00 & 2.731E+00 & 1.829E+00 & 1.304E+00 & 1.102E+00 & 1.052E+00 & 1.048E+00 & 1.064E+00 & 1.097E+00 & 1.134E+00\
4.20 & 2.627E+00 & 1.758E+00 & 1.272E+00 & 1.092E+00 & 1.048E+00 & 1.045E+00 & 1.059E+00 & 1.089E+00 & 1.131E+00\
4.40 & 2.525E+00 & 1.692E+00 & 1.242E+00 & 1.083E+00 & 1.045E+00 & 1.043E+00 & 1.055E+00 & 1.082E+00 & 1.123E+00\
4.60 & 2.427E+00 & 1.629E+00 & 1.216E+00 & 1.076E+00 & 1.043E+00 & 1.041E+00 & 1.051E+00 & 1.075E+00 & 1.115E+00\
4.80 & 2.331E+00 & 1.571E+00 & 1.193E+00 & 1.069E+00 & 1.041E+00 & 1.039E+00 & 1.048E+00 & 1.069E+00 & 1.106E+00\
5.00 & 2.238E+00 & 1.517E+00 & 1.172E+00 & 1.063E+00 & 1.039E+00 & 1.037E+00 & 1.045E+00 & 1.063E+00 & 1.097E+00\
5.20 & 2.149E+00 & 1.466E+00 & 1.154E+00 & 1.058E+00 & 1.037E+00 & 1.036E+00 & 1.043E+00 & 1.059E+00 & 1.089E+00\
5.40 & 2.063E+00 & 1.420E+00 & 1.137E+00 & 1.054E+00 & 1.036E+00 & 1.035E+00 & 1.041E+00 & 1.054E+00 & 1.081E+00\
5.60 & 1.981E+00 & 1.377E+00 & 1.123E+00 & 1.050E+00 & 1.035E+00 & 1.034E+00 & 1.039E+00 & 1.051E+00 & 1.075E+00\
5.80 & 1.903E+00 & 1.338E+00 & 1.110E+00 & 1.047E+00 & 1.034E+00 & 1.033E+00 & 1.037E+00 & 1.048E+00 & 1.069E+00\
6.00 & 1.829E+00 & 1.303E+00 & 1.099E+00 & 1.044E+00 & 1.033E+00 & 1.032E+00 & 1.036E+00 & 1.045E+00 & 1.063E+00\
6.20 & 1.758E+00 & 1.270E+00 & 1.089E+00 & 1.042E+00 & 1.032E+00 & 1.031E+00 & 1.035E+00 & 1.042E+00 & 1.058E+00\
6.40 & 1.692E+00 & 1.241E+00 & 1.081E+00 & 1.040E+00 & 1.031E+00 & 1.031E+00 & 1.034E+00 & 1.040E+00 & 1.054E+00\
6.60 & 1.629E+00 & 1.215E+00 & 1.074E+00 & 1.038E+00 & 1.031E+00 & 1.030E+00 & 1.033E+00 & 1.039E+00 & 1.051E+00\
6.80 & 1.571E+00 & 1.192E+00 & 1.067E+00 & 1.037E+00 & 1.030E+00 & 1.030E+00 & 1.032E+00 & 1.037E+00 & 1.047E+00\
7.00 & 1.516E+00 & 1.171E+00 & 1.062E+00 & 1.035E+00 & 1.030E+00 & 1.030E+00 & 1.031E+00 & 1.036E+00 & 1.045E+00\
7.20 & 1.466E+00 & 1.153E+00 & 1.057E+00 & 1.034E+00 & 1.030E+00 & 1.029E+00 & 1.031E+00 & 1.034E+00 & 1.042E+00\
7.40 & 1.420E+00 & 1.137E+00 & 1.053E+00 & 1.033E+00 & 1.029E+00 & 1.029E+00 & 1.030E+00 & 1.033E+00 & 1.040E+00\
7.60 & 1.377E+00 & 1.122E+00 & 1.049E+00 & 1.032E+00 & 1.029E+00 & 1.029E+00 & 1.030E+00 & 1.033E+00 & 1.038E+00\
7.80 & 1.338E+00 & 1.110E+00 & 1.046E+00 & 1.032E+00 & 1.029E+00 & 1.029E+00 & 1.030E+00 & 1.032E+00 & 1.037E+00\
8.00 & 1.303E+00 & 1.099E+00 & 1.044E+00 & 1.031E+00 & 1.029E+00 & 1.028E+00 & 1.029E+00 & 1.031E+00 & 1.036E+00\
8.20 & 1.270E+00 & 1.089E+00 & 1.041E+00 & 1.031E+00 & 1.028E+00 & 1.028E+00 & 1.029E+00 & 1.031E+00 & 1.034E+00\
8.40 & 1.241E+00 & 1.081E+00 & 1.039E+00 & 1.030E+00 & 1.028E+00 & 1.028E+00 & 1.029E+00 & 1.030E+00 & 1.033E+00\
8.60 & 1.215E+00 & 1.073E+00 & 1.038E+00 & 1.030E+00 & 1.028E+00 & 1.028E+00 & 1.029E+00 & 1.030E+00 & 1.033E+00\
8.80 & 1.192E+00 & 1.067E+00 & 1.036E+00 & 1.029E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.030E+00 & 1.032E+00\
9.00 & 1.171E+00 & 1.062E+00 & 1.035E+00 & 1.029E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.029E+00 & 1.031E+00\
9.20 & 1.153E+00 & 1.057E+00 & 1.034E+00 & 1.029E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.029E+00 & 1.031E+00\
9.40 & 1.137E+00 & 1.053E+00 & 1.033E+00 & 1.029E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.029E+00 & 1.030E+00\
9.60 & 1.122E+00 & 1.049E+00 & 1.032E+00 & 1.029E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.029E+00 & 1.030E+00\
9.80 & 1.110E+00 & 1.046E+00 & 1.032E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.030E+00\
10.00 & 1.099E+00 & 1.044E+00 & 1.031E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.028E+00 & 1.029E+00\
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
log $\gamma^{2}$
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00
-8.00 5.592E+00 4.304E+00 3.027E+00 1.815E+00 8.436E-01 3.035E-01 9.801E-02 3.107E-02 9.827E-03
-7.80 5.592E+00 4.304E+00 3.027E+00 1.815E+00 8.436E-01 3.035E-01 9.802E-02 3.107E-02 9.827E-03
-7.60 5.592E+00 4.304E+00 3.027E+00 1.815E+00 8.437E-01 3.035E-01 9.803E-02 3.107E-02 9.829E-03
-7.40 5.592E+00 4.304E+00 3.027E+00 1.816E+00 8.437E-01 3.036E-01 9.805E-02 3.108E-02 9.830E-03
-7.20 5.592E+00 4.304E+00 3.027E+00 1.816E+00 8.438E-01 3.036E-01 9.806E-02 3.108E-02 9.832E-03
-7.00 5.592E+00 4.304E+00 3.027E+00 1.816E+00 8.439E-01 3.037E-01 9.808E-02 3.109E-02 9.834E-03
-6.80 5.592E+00 4.304E+00 3.027E+00 1.816E+00 8.440E-01 3.037E-01 9.811E-02 3.110E-02 9.837E-03
-6.60 5.592E+00 4.304E+00 3.027E+00 1.816E+00 8.441E-01 3.038E-01 9.814E-02 3.111E-02 9.841E-03
-6.40 5.592E+00 4.304E+00 3.027E+00 1.816E+00 8.443E-01 3.039E-01 9.818E-02 3.112E-02 9.845E-03
-6.20 5.592E+00 4.304E+00 3.027E+00 1.816E+00 8.445E-01 3.041E-01 9.824E-02 3.114E-02 9.851E-03
-6.00 5.592E+00 4.304E+00 3.027E+00 1.816E+00 8.448E-01 3.042E-01 9.830E-02 3.116E-02 9.858E-03
-5.80 5.592E+00 4.304E+00 3.027E+00 1.817E+00 8.451E-01 3.044E-01 9.839E-02 3.119E-02 9.867E-03
-5.60 5.592E+00 4.304E+00 3.027E+00 1.817E+00 8.456E-01 3.047E-01 9.849E-02 3.123E-02 9.878E-03
-5.40 5.592E+00 4.304E+00 3.028E+00 1.817E+00 8.461E-01 3.050E-01 9.862E-02 3.127E-02 9.893E-03
-5.20 5.592E+00 4.304E+00 3.028E+00 1.818E+00 8.468E-01 3.055E-01 9.879E-02 3.133E-02 9.911E-03
-5.00 5.592E+00 4.304E+00 3.028E+00 1.819E+00 8.477E-01 3.060E-01 9.900E-02 3.140E-02 9.934E-03
-4.80 5.592E+00 4.304E+00 3.029E+00 1.820E+00 8.488E-01 3.067E-01 9.927E-02 3.149E-02 9.963E-03
-4.60 5.592E+00 4.304E+00 3.029E+00 1.821E+00 8.501E-01 3.075E-01 9.960E-02 3.160E-02 9.999E-03
-4.40 5.592E+00 4.304E+00 3.030E+00 1.822E+00 8.519E-01 3.086E-01 1.000E-01 3.175E-02 1.005E-02
-4.20 5.592E+00 4.305E+00 3.030E+00 1.824E+00 8.541E-01 3.099E-01 1.006E-01 3.193E-02 1.010E-02
-4.00 5.592E+00 4.305E+00 3.031E+00 1.826E+00 8.568E-01 3.117E-01 1.012E-01 3.216E-02 1.018E-02
-3.80 5.592E+00 4.305E+00 3.032E+00 1.829E+00 8.603E-01 3.138E-01 1.021E-01 3.245E-02 1.027E-02
-3.60 5.591E+00 4.305E+00 3.034E+00 1.832E+00 8.647E-01 3.166E-01 1.032E-01 3.282E-02 1.039E-02
-3.40 5.591E+00 4.305E+00 3.035E+00 1.837E+00 8.703E-01 3.201E-01 1.046E-01 3.329E-02 1.054E-02
-3.20 5.590E+00 4.305E+00 3.037E+00 1.842E+00 8.773E-01 3.245E-01 1.064E-01 3.389E-02 1.074E-02
-3.00 5.589E+00 4.304E+00 3.039E+00 1.849E+00 8.862E-01 3.302E-01 1.086E-01 3.467E-02 1.099E-02
-2.80 5.587E+00 4.303E+00 3.041E+00 1.857E+00 8.974E-01 3.375E-01 1.116E-01 3.566E-02 1.131E-02
-2.60 5.584E+00 4.301E+00 3.043E+00 1.866E+00 9.116E-01 3.468E-01 1.153E-01 3.694E-02 1.172E-02
-2.40 5.580E+00 4.298E+00 3.044E+00 1.878E+00 9.292E-01 3.587E-01 1.202E-01 3.861E-02 1.226E-02
-2.20 5.574E+00 4.293E+00 3.044E+00 1.890E+00 9.511E-01 3.740E-01 1.265E-01 4.078E-02 1.297E-02
-2.00 5.564E+00 4.285E+00 3.042E+00 1.904E+00 9.780E-01 3.935E-01 1.348E-01 4.361E-02 1.389E-02
-1.80 5.552E+00 4.274E+00 3.038E+00 1.917E+00 1.010E+00 4.184E-01 1.455E-01 4.733E-02 1.510E-02
-1.60 5.534E+00 4.258E+00 3.029E+00 1.929E+00 1.048E+00 4.500E-01 1.594E-01 5.223E-02 1.670E-02
-1.40 5.510E+00 4.236E+00 3.016E+00 1.939E+00 1.091E+00 4.897E-01 1.776E-01 5.868E-02 1.881E-02
-1.20 5.478E+00 4.207E+00 2.996E+00 1.944E+00 1.138E+00 5.387E-01 2.012E-01 6.717E-02 2.161E-02
-1.00 5.437E+00 4.169E+00 2.967E+00 1.942E+00 1.185E+00 5.978E-01 2.314E-01 7.830E-02 2.530E-02
-0.80 5.384E+00 4.120E+00 2.930E+00 1.932E+00 1.230E+00 6.670E-01 2.698E-01 9.282E-02 3.015E-02
-0.60 5.320E+00 4.060E+00 2.881E+00 1.913E+00 1.268E+00 7.449E-01 3.178E-01 1.116E-01 3.649E-02
-0.40 5.244E+00 3.988E+00 2.822E+00 1.884E+00 1.295E+00 8.280E-01 3.762E-01 1.355E-01 4.471E-02
-0.20 5.155E+00 3.904E+00 2.753E+00 1.846E+00 1.310E+00 9.112E-01 4.457E-01 1.656E-01 5.523E-02
0.00 5.056E+00 3.811E+00 2.675E+00 1.799E+00 1.312E+00 9.882E-01 5.255E-01 2.029E-01 6.859E-02
0.20 4.949E+00 3.710E+00 2.590E+00 1.747E+00 1.303E+00 1.053E+00 6.140E-01 2.484E-01 8.537E-02
0.40 4.835E+00 3.602E+00 2.501E+00 1.691E+00 1.286E+00 1.100E+00 7.078E-01 3.028E-01 1.062E-01
0.60 4.716E+00 3.492E+00 2.410E+00 1.635E+00 1.264E+00 1.130E+00 8.024E-01 3.667E-01 1.320E-01
0.80 4.596E+00 3.380E+00 2.320E+00 1.579E+00 1.240E+00 1.144E+00 8.921E-01 4.402E-01 1.636E-01
1.00 4.473E+00 3.267E+00 2.230E+00 1.525E+00 1.216E+00 1.146E+00 9.713E-01 5.228E-01 2.019E-01
1.20 4.350E+00 3.154E+00 2.142E+00 1.475E+00 1.194E+00 1.141E+00 1.035E+00 6.127E-01 2.479E-01
1.40 4.227E+00 3.043E+00 2.057E+00 1.427E+00 1.173E+00 1.132E+00 1.081E+00 7.071E-01 3.026E-01
1.60 4.105E+00 2.933E+00 1.976E+00 1.383E+00 1.153E+00 1.121E+00 1.108E+00 8.017E-01 3.667E-01
1.80 3.983E+00 2.826E+00 1.898E+00 1.343E+00 1.136E+00 1.110E+00 1.122E+00 8.912E-01 4.402E-01
2.00 3.861E+00 2.720E+00 1.823E+00 1.305E+00 1.120E+00 1.100E+00 1.124E+00 9.699E-01 5.228E-01
2.20 3.741E+00 2.616E+00 1.752E+00 1.272E+00 1.107E+00 1.090E+00 1.120E+00 1.033E+00 6.127E-01
2.40 3.621E+00 2.515E+00 1.686E+00 1.241E+00 1.095E+00 1.081E+00 1.113E+00 1.079E+00 7.071E-01
2.60 3.503E+00 2.417E+00 1.623E+00 1.214E+00 1.084E+00 1.072E+00 1.104E+00 1.106E+00 8.016E-01
2.80 3.386E+00 2.321E+00 1.564E+00 1.189E+00 1.075E+00 1.065E+00 1.095E+00 1.119E+00 8.911E-01
3.00 3.270E+00 2.229E+00 1.510E+00 1.168E+00 1.067E+00 1.058E+00 1.086E+00 1.122E+00 9.698E-01
3.20 3.156E+00 2.140E+00 1.459E+00 1.148E+00 1.060E+00 1.053E+00 1.078E+00 1.118E+00 1.033E+00
3.40 3.044E+00 2.055E+00 1.412E+00 1.131E+00 1.054E+00 1.048E+00 1.070E+00 1.111E+00 1.078E+00
3.60 2.934E+00 1.973E+00 1.369E+00 1.116E+00 1.048E+00 1.043E+00 1.064E+00 1.102E+00 1.106E+00
3.80 2.825E+00 1.894E+00 1.330E+00 1.103E+00 1.044E+00 1.040E+00 1.057E+00 1.094E+00 1.119E+00
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[rccccccccc]{}\
\
\[-2ex\] log $\gamma^{2}$ &\
& -4.00 & -3.00 & -2.00 & -1.00 & 0.00 & 1.00 & 2.00 & 3.00 & 4.00\
4.00 & 2.719E+00 & 1.820E+00 & 1.294E+00 & 1.091E+00 & 1.040E+00 & 1.036E+00 & 1.052E+00 & 1.085E+00 & 1.121E+00\
4.20 & 2.616E+00 & 1.749E+00 & 1.261E+00 & 1.081E+00 & 1.037E+00 & 1.034E+00 & 1.047E+00 & 1.077E+00 & 1.118E+00\
4.40 & 2.515E+00 & 1.683E+00 & 1.232E+00 & 1.072E+00 & 1.034E+00 & 1.031E+00 & 1.043E+00 & 1.069E+00 & 1.111E+00\
4.60 & 2.416E+00 & 1.620E+00 & 1.206E+00 & 1.064E+00 & 1.031E+00 & 1.029E+00 & 1.039E+00 & 1.063E+00 & 1.102E+00\
4.80 & 2.321E+00 & 1.562E+00 & 1.182E+00 & 1.058E+00 & 1.029E+00 & 1.027E+00 & 1.036E+00 & 1.057E+00 & 1.093E+00\
5.00 & 2.229E+00 & 1.507E+00 & 1.161E+00 & 1.052E+00 & 1.027E+00 & 1.026E+00 & 1.033E+00 & 1.051E+00 & 1.085E+00\
5.20 & 2.140E+00 & 1.457E+00 & 1.143E+00 & 1.047E+00 & 1.026E+00 & 1.024E+00 & 1.031E+00 & 1.047E+00 & 1.077E+00\
5.40 & 2.054E+00 & 1.410E+00 & 1.126E+00 & 1.042E+00 & 1.024E+00 & 1.023E+00 & 1.029E+00 & 1.042E+00 & 1.069E+00\
5.60 & 1.972E+00 & 1.367E+00 & 1.112E+00 & 1.039E+00 & 1.023E+00 & 1.022E+00 & 1.027E+00 & 1.039E+00 & 1.063E+00\
5.80 & 1.894E+00 & 1.328E+00 & 1.099E+00 & 1.036E+00 & 1.022E+00 & 1.021E+00 & 1.025E+00 & 1.036E+00 & 1.057E+00\
6.00 & 1.820E+00 & 1.292E+00 & 1.088E+00 & 1.033E+00 & 1.021E+00 & 1.020E+00 & 1.024E+00 & 1.033E+00 & 1.051E+00\
6.20 & 1.749E+00 & 1.260E+00 & 1.078E+00 & 1.030E+00 & 1.020E+00 & 1.020E+00 & 1.023E+00 & 1.031E+00 & 1.046E+00\
6.40 & 1.683E+00 & 1.231E+00 & 1.070E+00 & 1.028E+00 & 1.020E+00 & 1.019E+00 & 1.022E+00 & 1.029E+00 & 1.042E+00\
6.60 & 1.620E+00 & 1.205E+00 & 1.062E+00 & 1.027E+00 & 1.019E+00 & 1.019E+00 & 1.021E+00 & 1.027E+00 & 1.039E+00\
6.80 & 1.562E+00 & 1.181E+00 & 1.056E+00 & 1.025E+00 & 1.019E+00 & 1.018E+00 & 1.020E+00 & 1.025E+00 & 1.036E+00\
7.00 & 1.507E+00 & 1.160E+00 & 1.050E+00 & 1.024E+00 & 1.018E+00 & 1.018E+00 & 1.020E+00 & 1.024E+00 & 1.033E+00\
7.20 & 1.457E+00 & 1.142E+00 & 1.045E+00 & 1.023E+00 & 1.018E+00 & 1.018E+00 & 1.019E+00 & 1.023E+00 & 1.031E+00\
7.40 & 1.410E+00 & 1.126E+00 & 1.041E+00 & 1.022E+00 & 1.018E+00 & 1.017E+00 & 1.019E+00 & 1.022E+00 & 1.028E+00\
7.60 & 1.367E+00 & 1.111E+00 & 1.038E+00 & 1.021E+00 & 1.017E+00 & 1.017E+00 & 1.018E+00 & 1.021E+00 & 1.027E+00\
7.80 & 1.328E+00 & 1.099E+00 & 1.035E+00 & 1.020E+00 & 1.017E+00 & 1.017E+00 & 1.018E+00 & 1.020E+00 & 1.025E+00\
8.00 & 1.292E+00 & 1.087E+00 & 1.032E+00 & 1.020E+00 & 1.017E+00 & 1.017E+00 & 1.018E+00 & 1.020E+00 & 1.024E+00\
8.20 & 1.260E+00 & 1.078E+00 & 1.030E+00 & 1.019E+00 & 1.017E+00 & 1.017E+00 & 1.017E+00 & 1.019E+00 & 1.023E+00\
8.40 & 1.231E+00 & 1.069E+00 & 1.028E+00 & 1.019E+00 & 1.017E+00 & 1.017E+00 & 1.017E+00 & 1.019E+00 & 1.022E+00\
8.60 & 1.205E+00 & 1.062E+00 & 1.026E+00 & 1.018E+00 & 1.017E+00 & 1.016E+00 & 1.017E+00 & 1.018E+00 & 1.021E+00\
8.80 & 1.181E+00 & 1.056E+00 & 1.025E+00 & 1.018E+00 & 1.016E+00 & 1.016E+00 & 1.017E+00 & 1.018E+00 & 1.020E+00\
9.00 & 1.160E+00 & 1.050E+00 & 1.023E+00 & 1.018E+00 & 1.016E+00 & 1.016E+00 & 1.017E+00 & 1.018E+00 & 1.020E+00\
9.20 & 1.142E+00 & 1.045E+00 & 1.022E+00 & 1.017E+00 & 1.016E+00 & 1.016E+00 & 1.017E+00 & 1.017E+00 & 1.019E+00\
9.40 & 1.126E+00 & 1.041E+00 & 1.021E+00 & 1.017E+00 & 1.016E+00 & 1.016E+00 & 1.016E+00 & 1.017E+00 & 1.019E+00\
9.60 & 1.111E+00 & 1.038E+00 & 1.021E+00 & 1.017E+00 & 1.016E+00 & 1.016E+00 & 1.016E+00 & 1.017E+00 & 1.018E+00\
9.80 & 1.098E+00 & 1.035E+00 & 1.020E+00 & 1.017E+00 & 1.016E+00 & 1.016E+00 & 1.016E+00 & 1.017E+00 & 1.018E+00\
10.00 & 1.087E+00 & 1.032E+00 & 1.019E+00 & 1.017E+00 & 1.016E+00 & 1.016E+00 & 1.016E+00 & 1.017E+00 & 1.018E+00\
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
log $\gamma^{2}$
-4.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 3.00 4.00
-8.00 5.559E+00 4.281E+00 3.014E+00 1.811E+00 8.430E-01 3.035E-01 9.801E-02 3.107E-02 9.826E-03
-7.80 5.559E+00 4.281E+00 3.014E+00 1.811E+00 8.430E-01 3.035E-01 9.802E-02 3.107E-02 9.827E-03
-7.60 5.559E+00 4.281E+00 3.014E+00 1.811E+00 8.431E-01 3.035E-01 9.803E-02 3.107E-02 9.828E-03
-7.40 5.559E+00 4.281E+00 3.014E+00 1.811E+00 8.431E-01 3.036E-01 9.804E-02 3.108E-02 9.830E-03
-7.20 5.559E+00 4.281E+00 3.014E+00 1.811E+00 8.432E-01 3.036E-01 9.806E-02 3.108E-02 9.832E-03
-7.00 5.559E+00 4.281E+00 3.014E+00 1.811E+00 8.433E-01 3.036E-01 9.808E-02 3.109E-02 9.834E-03
-6.80 5.559E+00 4.281E+00 3.014E+00 1.811E+00 8.434E-01 3.037E-01 9.811E-02 3.110E-02 9.837E-03
-6.60 5.559E+00 4.281E+00 3.014E+00 1.811E+00 8.435E-01 3.038E-01 9.814E-02 3.111E-02 9.840E-03
-6.40 5.559E+00 4.281E+00 3.014E+00 1.811E+00 8.437E-01 3.039E-01 9.818E-02 3.112E-02 9.845E-03
-6.20 5.559E+00 4.281E+00 3.014E+00 1.812E+00 8.439E-01 3.040E-01 9.823E-02 3.114E-02 9.850E-03
-6.00 5.559E+00 4.281E+00 3.014E+00 1.812E+00 8.442E-01 3.042E-01 9.830E-02 3.116E-02 9.858E-03
-5.80 5.559E+00 4.281E+00 3.014E+00 1.812E+00 8.445E-01 3.044E-01 9.838E-02 3.119E-02 9.867E-03
-5.60 5.559E+00 4.281E+00 3.014E+00 1.812E+00 8.449E-01 3.047E-01 9.849E-02 3.123E-02 9.878E-03
-5.40 5.559E+00 4.281E+00 3.015E+00 1.813E+00 8.455E-01 3.050E-01 9.862E-02 3.127E-02 9.892E-03
-5.20 5.559E+00 4.281E+00 3.015E+00 1.813E+00 8.461E-01 3.054E-01 9.878E-02 3.133E-02 9.910E-03
-5.00 5.559E+00 4.281E+00 3.015E+00 1.814E+00 8.470E-01 3.059E-01 9.899E-02 3.140E-02 9.933E-03
-4.80 5.559E+00 4.281E+00 3.016E+00 1.815E+00 8.481E-01 3.066E-01 9.925E-02 3.149E-02 9.962E-03
-4.60 5.559E+00 4.282E+00 3.016E+00 1.816E+00 8.494E-01 3.075E-01 9.959E-02 3.160E-02 9.998E-03
-4.40 5.559E+00 4.282E+00 3.017E+00 1.817E+00 8.511E-01 3.085E-01 1.000E-01 3.174E-02 1.004E-02
-4.20 5.559E+00 4.282E+00 3.017E+00 1.819E+00 8.533E-01 3.099E-01 1.005E-01 3.192E-02 1.010E-02
-4.00 5.559E+00 4.282E+00 3.018E+00 1.821E+00 8.560E-01 3.116E-01 1.012E-01 3.215E-02 1.018E-02
-3.80 5.559E+00 4.282E+00 3.019E+00 1.824E+00 8.594E-01 3.137E-01 1.021E-01 3.244E-02 1.027E-02
-3.60 5.558E+00 4.282E+00 3.021E+00 1.827E+00 8.637E-01 3.164E-01 1.031E-01 3.280E-02 1.039E-02
-3.40 5.558E+00 4.282E+00 3.022E+00 1.832E+00 8.692E-01 3.199E-01 1.045E-01 3.327E-02 1.054E-02
-3.20 5.557E+00 4.282E+00 3.024E+00 1.837E+00 8.761E-01 3.243E-01 1.063E-01 3.387E-02 1.073E-02
-3.00 5.556E+00 4.281E+00 3.026E+00 1.843E+00 8.848E-01 3.299E-01 1.085E-01 3.463E-02 1.098E-02
-2.80 5.554E+00 4.280E+00 3.027E+00 1.851E+00 8.959E-01 3.371E-01 1.114E-01 3.562E-02 1.130E-02
-2.60 5.551E+00 4.278E+00 3.029E+00 1.860E+00 9.097E-01 3.462E-01 1.152E-01 3.689E-02 1.171E-02
-2.40 5.547E+00 4.275E+00 3.030E+00 1.871E+00 9.270E-01 3.580E-01 1.200E-01 3.854E-02 1.224E-02
-2.20 5.541E+00 4.270E+00 3.030E+00 1.883E+00 9.485E-01 3.731E-01 1.262E-01 4.068E-02 1.294E-02
-2.00 5.532E+00 4.263E+00 3.029E+00 1.897E+00 9.749E-01 3.924E-01 1.344E-01 4.349E-02 1.385E-02
-1.80 5.520E+00 4.252E+00 3.024E+00 1.910E+00 1.007E+00 4.170E-01 1.450E-01 4.717E-02 1.505E-02
-1.60 5.502E+00 4.236E+00 3.016E+00 1.922E+00 1.044E+00 4.482E-01 1.588E-01 5.201E-02 1.663E-02
-1.40 5.479E+00 4.215E+00 3.003E+00 1.931E+00 1.086E+00 4.873E-01 1.768E-01 5.839E-02 1.872E-02
-1.20 5.448E+00 4.187E+00 2.983E+00 1.936E+00 1.132E+00 5.356E-01 2.000E-01 6.679E-02 2.149E-02
-1.00 5.407E+00 4.149E+00 2.955E+00 1.934E+00 1.178E+00 5.940E-01 2.300E-01 7.781E-02 2.514E-02
-0.80 5.356E+00 4.101E+00 2.918E+00 1.924E+00 1.223E+00 6.625E-01 2.680E-01 9.218E-02 2.995E-02
-0.60 5.293E+00 4.042E+00 2.871E+00 1.905E+00 1.260E+00 7.394E-01 3.154E-01 1.108E-01 3.623E-02
-0.40 5.218E+00 3.971E+00 2.812E+00 1.877E+00 1.287E+00 8.217E-01 3.733E-01 1.345E-01 4.436E-02
-0.20 5.130E+00 3.889E+00 2.743E+00 1.838E+00 1.302E+00 9.042E-01 4.421E-01 1.643E-01 5.480E-02
0.00 5.033E+00 3.796E+00 2.666E+00 1.792E+00 1.304E+00 9.804E-01 5.213E-01 2.013E-01 6.804E-02
0.20 4.926E+00 3.695E+00 2.582E+00 1.740E+00 1.294E+00 1.044E+00 6.090E-01 2.464E-01 8.468E-02
0.40 4.813E+00 3.589E+00 2.494E+00 1.685E+00 1.277E+00 1.092E+00 7.021E-01 3.004E-01 1.054E-01
0.60 4.696E+00 3.479E+00 2.403E+00 1.628E+00 1.256E+00 1.121E+00 7.959E-01 3.638E-01 1.309E-01
0.80 4.576E+00 3.368E+00 2.312E+00 1.572E+00 1.232E+00 1.135E+00 8.850E-01 4.367E-01 1.622E-01
1.00 4.454E+00 3.256E+00 2.223E+00 1.519E+00 1.208E+00 1.137E+00 9.635E-01 5.186E-01 2.002E-01
1.20 4.332E+00 3.144E+00 2.136E+00 1.468E+00 1.186E+00 1.132E+00 1.027E+00 6.078E-01 2.459E-01
1.40 4.210E+00 3.033E+00 2.051E+00 1.420E+00 1.164E+00 1.123E+00 1.072E+00 7.014E-01 3.002E-01
1.60 4.088E+00 2.924E+00 1.969E+00 1.376E+00 1.145E+00 1.113E+00 1.100E+00 7.953E-01 3.637E-01
1.80 3.967E+00 2.817E+00 1.891E+00 1.335E+00 1.128E+00 1.102E+00 1.113E+00 8.840E-01 4.367E-01
2.00 3.847E+00 2.711E+00 1.817E+00 1.298E+00 1.112E+00 1.091E+00 1.115E+00 9.622E-01 5.186E-01
2.20 3.727E+00 2.608E+00 1.746E+00 1.264E+00 1.099E+00 1.081E+00 1.111E+00 1.025E+00 6.078E-01
2.40 3.608E+00 2.507E+00 1.679E+00 1.234E+00 1.086E+00 1.072E+00 1.104E+00 1.070E+00 7.014E-01
2.60 3.491E+00 2.409E+00 1.617E+00 1.206E+00 1.076E+00 1.064E+00 1.095E+00 1.097E+00 7.952E-01
2.80 3.374E+00 2.314E+00 1.558E+00 1.182E+00 1.067E+00 1.057E+00 1.086E+00 1.110E+00 8.839E-01
3.00 3.259E+00 2.222E+00 1.503E+00 1.160E+00 1.059E+00 1.050E+00 1.078E+00 1.113E+00 9.620E-01
3.20 3.146E+00 2.134E+00 1.452E+00 1.140E+00 1.052E+00 1.044E+00 1.069E+00 1.109E+00 1.025E+00
3.40 3.034E+00 2.048E+00 1.405E+00 1.123E+00 1.045E+00 1.039E+00 1.062E+00 1.102E+00 1.070E+00
3.60 2.924E+00 1.966E+00 1.362E+00 1.108E+00 1.040E+00 1.035E+00 1.055E+00 1.094E+00 1.097E+00
3.80 2.817E+00 1.888E+00 1.323E+00 1.095E+00 1.036E+00 1.031E+00 1.049E+00 1.085E+00 1.110E+00
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[rccccccccc]{}\
\
\[-2ex\] log $\gamma^{2}$ &\
& -4.00 & -3.00 & -2.00 & -1.00 & 0.00 & 1.00 & 2.00 & 3.00 & 4.00\
4.00 & 2.711E+00 & 1.814E+00 & 1.287E+00 & 1.083E+00 & 1.032E+00 & 1.028E+00 & 1.043E+00 & 1.076E+00 & 1.112E+00\
4.20 & 2.608E+00 & 1.743E+00 & 1.254E+00 & 1.073E+00 & 1.028E+00 & 1.025E+00 & 1.039E+00 & 1.068E+00 & 1.109E+00\
4.40 & 2.507E+00 & 1.677E+00 & 1.225E+00 & 1.064E+00 & 1.025E+00 & 1.023E+00 & 1.035E+00 & 1.061E+00 & 1.102E+00\
4.60 & 2.409E+00 & 1.614E+00 & 1.198E+00 & 1.056E+00 & 1.023E+00 & 1.021E+00 & 1.031E+00 & 1.054E+00 & 1.093E+00\
4.80 & 2.314E+00 & 1.555E+00 & 1.175E+00 & 1.050E+00 & 1.021E+00 & 1.019E+00 & 1.028E+00 & 1.048E+00 & 1.085E+00\
5.00 & 2.222E+00 & 1.501E+00 & 1.153E+00 & 1.044E+00 & 1.019E+00 & 1.017E+00 & 1.025E+00 & 1.043E+00 & 1.076E+00\
5.20 & 2.133E+00 & 1.450E+00 & 1.135E+00 & 1.039E+00 & 1.017E+00 & 1.016E+00 & 1.023E+00 & 1.038E+00 & 1.068E+00\
5.40 & 2.048E+00 & 1.403E+00 & 1.118E+00 & 1.034E+00 & 1.016E+00 & 1.015E+00 & 1.020E+00 & 1.034E+00 & 1.061E+00\
5.60 & 1.966E+00 & 1.360E+00 & 1.104E+00 & 1.031E+00 & 1.015E+00 & 1.014E+00 & 1.019E+00 & 1.030E+00 & 1.054E+00\
5.80 & 1.888E+00 & 1.321E+00 & 1.091E+00 & 1.027E+00 & 1.014E+00 & 1.013E+00 & 1.017E+00 & 1.027E+00 & 1.048E+00\
6.00 & 1.813E+00 & 1.285E+00 & 1.080E+00 & 1.025E+00 & 1.013E+00 & 1.012E+00 & 1.016E+00 & 1.025E+00 & 1.043E+00\
6.20 & 1.743E+00 & 1.253E+00 & 1.070E+00 & 1.022E+00 & 1.012E+00 & 1.012E+00 & 1.015E+00 & 1.022E+00 & 1.038E+00\
6.40 & 1.676E+00 & 1.224E+00 & 1.062E+00 & 1.020E+00 & 1.012E+00 & 1.011E+00 & 1.014E+00 & 1.020E+00 & 1.034E+00\
6.60 & 1.614E+00 & 1.197E+00 & 1.054E+00 & 1.018E+00 & 1.011E+00 & 1.011E+00 & 1.013E+00 & 1.019E+00 & 1.030E+00\
6.80 & 1.555E+00 & 1.174E+00 & 1.048E+00 & 1.017E+00 & 1.011E+00 & 1.010E+00 & 1.012E+00 & 1.017E+00 & 1.027E+00\
7.00 & 1.501E+00 & 1.153E+00 & 1.042E+00 & 1.016E+00 & 1.010E+00 & 1.010E+00 & 1.012E+00 & 1.016E+00 & 1.025E+00\
7.20 & 1.450E+00 & 1.134E+00 & 1.037E+00 & 1.014E+00 & 1.010E+00 & 1.010E+00 & 1.011E+00 & 1.015E+00 & 1.022E+00\
7.40 & 1.403E+00 & 1.118E+00 & 1.033E+00 & 1.014E+00 & 1.010E+00 & 1.009E+00 & 1.011E+00 & 1.014E+00 & 1.020E+00\
7.60 & 1.360E+00 & 1.103E+00 & 1.030E+00 & 1.013E+00 & 1.009E+00 & 1.009E+00 & 1.010E+00 & 1.013E+00 & 1.018E+00\
7.80 & 1.321E+00 & 1.091E+00 & 1.027E+00 & 1.012E+00 & 1.009E+00 & 1.009E+00 & 1.010E+00 & 1.012E+00 & 1.017E+00\
8.00 & 1.285E+00 & 1.080E+00 & 1.024E+00 & 1.011E+00 & 1.009E+00 & 1.009E+00 & 1.009E+00 & 1.011E+00 & 1.016E+00\
8.20 & 1.253E+00 & 1.070E+00 & 1.022E+00 & 1.011E+00 & 1.009E+00 & 1.009E+00 & 1.009E+00 & 1.011E+00 & 1.015E+00\
8.40 & 1.224E+00 & 1.061E+00 & 1.020E+00 & 1.010E+00 & 1.009E+00 & 1.008E+00 & 1.009E+00 & 1.010E+00 & 1.014E+00\
8.60 & 1.197E+00 & 1.054E+00 & 1.018E+00 & 1.010E+00 & 1.008E+00 & 1.008E+00 & 1.009E+00 & 1.010E+00 & 1.013E+00\
8.80 & 1.174E+00 & 1.048E+00 & 1.017E+00 & 1.010E+00 & 1.008E+00 & 1.008E+00 & 1.009E+00 & 1.010E+00 & 1.012E+00\
9.00 & 1.153E+00 & 1.042E+00 & 1.015E+00 & 1.009E+00 & 1.008E+00 & 1.008E+00 & 1.009E+00 & 1.009E+00 & 1.011E+00\
9.20 & 1.134E+00 & 1.037E+00 & 1.014E+00 & 1.009E+00 & 1.008E+00 & 1.008E+00 & 1.008E+00 & 1.009E+00 & 1.011E+00\
9.40 & 1.118E+00 & 1.033E+00 & 1.013E+00 & 1.009E+00 & 1.008E+00 & 1.008E+00 & 1.008E+00 & 1.009E+00 & 1.010E+00\
9.60 & 1.103E+00 & 1.030E+00 & 1.012E+00 & 1.009E+00 & 1.008E+00 & 1.008E+00 & 1.008E+00 & 1.009E+00 & 1.010E+00\
9.80 & 1.091E+00 & 1.027E+00 & 1.012E+00 & 1.009E+00 & 1.008E+00 & 1.008E+00 & 1.008E+00 & 1.009E+00 & 1.010E+00\
10.00 & 1.079E+00 & 1.024E+00 & 1.011E+00 & 1.009E+00 & 1.008E+00 & 1.008E+00 & 1.008E+00 & 1.009E+00 & 1.009E+00\
Table of the total free-free Gaunt factor for different $\kappa$
================================================================
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
log $\gamma^{2}$
$\infty$ 2 3 5 10 15 25 50
-8.00 1.103E+00 1.221E+00 1.152E+00 1.125E+00 1.112E+00 1.109E+00 1.106E+00 1.104E+00
-7.80 1.103E+00 1.221E+00 1.152E+00 1.125E+00 1.112E+00 1.109E+00 1.106E+00 1.104E+00
-7.60 1.103E+00 1.222E+00 1.152E+00 1.125E+00 1.112E+00 1.109E+00 1.106E+00 1.104E+00
-7.40 1.103E+00 1.222E+00 1.152E+00 1.126E+00 1.112E+00 1.109E+00 1.106E+00 1.104E+00
-7.20 1.103E+00 1.222E+00 1.152E+00 1.126E+00 1.112E+00 1.109E+00 1.106E+00 1.105E+00
-7.00 1.103E+00 1.222E+00 1.152E+00 1.126E+00 1.113E+00 1.109E+00 1.106E+00 1.105E+00
-6.80 1.103E+00 1.222E+00 1.153E+00 1.126E+00 1.113E+00 1.109E+00 1.107E+00 1.105E+00
-6.60 1.103E+00 1.223E+00 1.153E+00 1.126E+00 1.113E+00 1.109E+00 1.107E+00 1.105E+00
-6.40 1.103E+00 1.223E+00 1.153E+00 1.126E+00 1.113E+00 1.109E+00 1.107E+00 1.105E+00
-6.20 1.103E+00 1.224E+00 1.153E+00 1.126E+00 1.113E+00 1.110E+00 1.107E+00 1.105E+00
-6.00 1.104E+00 1.224E+00 1.154E+00 1.127E+00 1.113E+00 1.110E+00 1.107E+00 1.105E+00
-5.80 1.104E+00 1.225E+00 1.154E+00 1.127E+00 1.114E+00 1.110E+00 1.108E+00 1.106E+00
-5.60 1.104E+00 1.226E+00 1.155E+00 1.128E+00 1.114E+00 1.111E+00 1.108E+00 1.106E+00
-5.40 1.105E+00 1.227E+00 1.156E+00 1.128E+00 1.115E+00 1.111E+00 1.109E+00 1.107E+00
-5.20 1.106E+00 1.229E+00 1.157E+00 1.129E+00 1.115E+00 1.112E+00 1.109E+00 1.107E+00
-5.00 1.106E+00 1.231E+00 1.158E+00 1.130E+00 1.116E+00 1.113E+00 1.110E+00 1.108E+00
-4.80 1.107E+00 1.234E+00 1.159E+00 1.131E+00 1.117E+00 1.114E+00 1.111E+00 1.109E+00
-4.60 1.108E+00 1.237E+00 1.161E+00 1.133E+00 1.119E+00 1.115E+00 1.112E+00 1.110E+00
-4.40 1.110E+00 1.241E+00 1.164E+00 1.135E+00 1.120E+00 1.117E+00 1.114E+00 1.112E+00
-4.20 1.112E+00 1.246E+00 1.167E+00 1.137E+00 1.123E+00 1.119E+00 1.116E+00 1.114E+00
-4.00 1.114E+00 1.253E+00 1.171E+00 1.140E+00 1.125E+00 1.121E+00 1.118E+00 1.116E+00
-3.80 1.118E+00 1.261E+00 1.176E+00 1.144E+00 1.129E+00 1.125E+00 1.122E+00 1.120E+00
-3.60 1.121E+00 1.272E+00 1.182E+00 1.149E+00 1.133E+00 1.129E+00 1.126E+00 1.123E+00
-3.40 1.126E+00 1.285E+00 1.190E+00 1.155E+00 1.139E+00 1.134E+00 1.131E+00 1.128E+00
-3.20 1.132E+00 1.302E+00 1.200E+00 1.163E+00 1.145E+00 1.141E+00 1.137E+00 1.135E+00
-3.00 1.140E+00 1.323E+00 1.213E+00 1.173E+00 1.154E+00 1.149E+00 1.145E+00 1.143E+00
-2.80 1.150E+00 1.349E+00 1.228E+00 1.185E+00 1.165E+00 1.159E+00 1.155E+00 1.153E+00
-2.60 1.162E+00 1.383E+00 1.248E+00 1.201E+00 1.178E+00 1.172E+00 1.168E+00 1.165E+00
-2.40 1.177E+00 1.423E+00 1.272E+00 1.220E+00 1.195E+00 1.188E+00 1.183E+00 1.180E+00
-2.20 1.195E+00 1.473E+00 1.301E+00 1.243E+00 1.215E+00 1.208E+00 1.202E+00 1.199E+00
-2.00 1.217E+00 1.531E+00 1.336E+00 1.270E+00 1.239E+00 1.231E+00 1.225E+00 1.221E+00
-1.80 1.242E+00 1.597E+00 1.377E+00 1.302E+00 1.268E+00 1.258E+00 1.252E+00 1.247E+00
-1.60 1.271E+00 1.671E+00 1.422E+00 1.339E+00 1.300E+00 1.289E+00 1.282E+00 1.276E+00
-1.40 1.304E+00 1.749E+00 1.472E+00 1.379E+00 1.335E+00 1.324E+00 1.315E+00 1.309E+00
-1.20 1.337E+00 1.827E+00 1.522E+00 1.420E+00 1.372E+00 1.359E+00 1.350E+00 1.344E+00
-1.00 1.370E+00 1.899E+00 1.571E+00 1.460E+00 1.408E+00 1.394E+00 1.384E+00 1.377E+00
-0.80 1.400E+00 1.960E+00 1.612E+00 1.495E+00 1.440E+00 1.426E+00 1.415E+00 1.407E+00
-0.60 1.424E+00 2.004E+00 1.644E+00 1.523E+00 1.465E+00 1.450E+00 1.439E+00 1.431E+00
-0.40 1.438E+00 2.031E+00 1.663E+00 1.539E+00 1.480E+00 1.465E+00 1.453E+00 1.445E+00
-0.20 1.441E+00 2.038E+00 1.668E+00 1.542E+00 1.483E+00 1.468E+00 1.456E+00 1.448E+00
0.00 1.432E+00 2.030E+00 1.659E+00 1.534E+00 1.475E+00 1.459E+00 1.448E+00 1.440E+00
0.20 1.414E+00 2.009E+00 1.640E+00 1.515E+00 1.457E+00 1.441E+00 1.430E+00 1.422E+00
0.40 1.389E+00 1.981E+00 1.613E+00 1.489E+00 1.431E+00 1.416E+00 1.404E+00 1.396E+00
0.60 1.359E+00 1.948E+00 1.582E+00 1.459E+00 1.401E+00 1.386E+00 1.374E+00 1.366E+00
0.80 1.327E+00 1.915E+00 1.550E+00 1.427E+00 1.369E+00 1.354E+00 1.343E+00 1.335E+00
1.00 1.296E+00 1.882E+00 1.518E+00 1.396E+00 1.338E+00 1.322E+00 1.311E+00 1.303E+00
1.20 1.265E+00 1.851E+00 1.488E+00 1.365E+00 1.307E+00 1.292E+00 1.281E+00 1.273E+00
1.40 1.237E+00 1.822E+00 1.459E+00 1.337E+00 1.279E+00 1.263E+00 1.252E+00 1.244E+00
1.60 1.210E+00 1.796E+00 1.433E+00 1.310E+00 1.252E+00 1.237E+00 1.226E+00 1.218E+00
1.80 1.186E+00 1.772E+00 1.409E+00 1.286E+00 1.228E+00 1.213E+00 1.202E+00 1.194E+00
2.00 1.164E+00 1.750E+00 1.388E+00 1.265E+00 1.207E+00 1.191E+00 1.180E+00 1.172E+00
2.20 1.145E+00 1.731E+00 1.368E+00 1.245E+00 1.187E+00 1.171E+00 1.160E+00 1.152E+00
2.40 1.127E+00 1.714E+00 1.351E+00 1.228E+00 1.169E+00 1.154E+00 1.142E+00 1.134E+00
2.60 1.111E+00 1.699E+00 1.336E+00 1.212E+00 1.154E+00 1.138E+00 1.127E+00 1.119E+00
2.80 1.097E+00 1.686E+00 1.322E+00 1.199E+00 1.140E+00 1.124E+00 1.113E+00 1.105E+00
3.00 1.085E+00 1.674E+00 1.310E+00 1.186E+00 1.128E+00 1.112E+00 1.100E+00 1.092E+00
3.20 1.074E+00 1.664E+00 1.300E+00 1.176E+00 1.117E+00 1.101E+00 1.090E+00 1.082E+00
3.40 1.064E+00 1.655E+00 1.290E+00 1.166E+00 1.107E+00 1.091E+00 1.080E+00 1.072E+00
3.60 1.056E+00 1.647E+00 1.282E+00 1.158E+00 1.099E+00 1.083E+00 1.072E+00 1.063E+00
3.80 1.048E+00 1.640E+00 1.275E+00 1.151E+00 1.091E+00 1.076E+00 1.064E+00 1.056E+00
------------------ ----------- ----------- ----------- ----------- ----------- ----------- ----------- -----------
[rcccccccc]{}\
\
\[-2ex\] log $\gamma^{2}$ &\
& $\infty$ & 2 & 3 & 5 & 10 & 15 & 25 & 50\
4.00 & 1.042E+00 & 1.634E+00 & 1.269E+00 & 1.144E+00 & 1.085E+00 & 1.069E+00 & 1.058E+00 & 1.050E+00\
4.20 & 1.036E+00 & 1.629E+00 & 1.263E+00 & 1.139E+00 & 1.079E+00 & 1.064E+00 & 1.052E+00 & 1.044E+00\
4.40 & 1.031E+00 & 1.624E+00 & 1.259E+00 & 1.134E+00 & 1.075E+00 & 1.059E+00 & 1.047E+00 & 1.039E+00\
4.60 & 1.027E+00 & 1.621E+00 & 1.255E+00 & 1.130E+00 & 1.070E+00 & 1.054E+00 & 1.043E+00 & 1.035E+00\
4.80 & 1.023E+00 & 1.617E+00 & 1.251E+00 & 1.126E+00 & 1.067E+00 & 1.051E+00 & 1.039E+00 & 1.031E+00\
5.00 & 1.020E+00 & 1.614E+00 & 1.248E+00 & 1.123E+00 & 1.063E+00 & 1.048E+00 & 1.036E+00 & 1.028E+00\
5.20 & 1.017E+00 & 1.612E+00 & 1.245E+00 & 1.120E+00 & 1.061E+00 & 1.045E+00 & 1.033E+00 & 1.025E+00\
5.40 & 1.015E+00 & 1.609E+00 & 1.243E+00 & 1.118E+00 & 1.058E+00 & 1.042E+00 & 1.031E+00 & 1.023E+00\
5.60 & 1.013E+00 & 1.608E+00 & 1.241E+00 & 1.116E+00 & 1.056E+00 & 1.040E+00 & 1.029E+00 & 1.021E+00\
5.80 & 1.011E+00 & 1.606E+00 & 1.239E+00 & 1.114E+00 & 1.054E+00 & 1.039E+00 & 1.027E+00 & 1.019E+00\
6.00 & 1.010E+00 & 1.604E+00 & 1.238E+00 & 1.112E+00 & 1.053E+00 & 1.037E+00 & 1.025E+00 & 1.017E+00\
6.20 & 1.008E+00 & 1.603E+00 & 1.236E+00 & 1.111E+00 & 1.052E+00 & 1.036E+00 & 1.024E+00 & 1.016E+00\
6.40 & 1.007E+00 & 1.602E+00 & 1.235E+00 & 1.110E+00 & 1.050E+00 & 1.034E+00 & 1.023E+00 & 1.015E+00\
6.60 & 1.006E+00 & 1.601E+00 & 1.234E+00 & 1.109E+00 & 1.049E+00 & 1.034E+00 & 1.022E+00 & 1.014E+00\
6.80 & 1.005E+00 & 1.601E+00 & 1.233E+00 & 1.108E+00 & 1.049E+00 & 1.033E+00 & 1.021E+00 & 1.013E+00\
7.00 & 1.004E+00 & 1.600E+00 & 1.233E+00 & 1.107E+00 & 1.048E+00 & 1.032E+00 & 1.020E+00 & 1.012E+00\
7.20 & 1.004E+00 & 1.599E+00 & 1.232E+00 & 1.107E+00 & 1.047E+00 & 1.031E+00 & 1.020E+00 & 1.012E+00\
7.40 & 1.003E+00 & 1.599E+00 & 1.232E+00 & 1.106E+00 & 1.047E+00 & 1.031E+00 & 1.019E+00 & 1.011E+00\
7.60 & 1.003E+00 & 1.598E+00 & 1.231E+00 & 1.106E+00 & 1.046E+00 & 1.030E+00 & 1.019E+00 & 1.011E+00\
7.80 & 1.002E+00 & 1.598E+00 & 1.231E+00 & 1.105E+00 & 1.046E+00 & 1.030E+00 & 1.018E+00 & 1.010E+00\
8.00 & 1.002E+00 & 1.598E+00 & 1.230E+00 & 1.105E+00 & 1.045E+00 & 1.030E+00 & 1.018E+00 & 1.010E+00\
8.20 & 1.002E+00 & 1.597E+00 & 1.230E+00 & 1.105E+00 & 1.045E+00 & 1.029E+00 & 1.018E+00 & 1.009E+00\
8.40 & 1.002E+00 & 1.597E+00 & 1.230E+00 & 1.104E+00 & 1.045E+00 & 1.029E+00 & 1.017E+00 & 1.009E+00\
8.60 & 1.001E+00 & 1.597E+00 & 1.230E+00 & 1.104E+00 & 1.045E+00 & 1.029E+00 & 1.017E+00 & 1.009E+00\
8.80 & 1.001E+00 & 1.597E+00 & 1.230E+00 & 1.104E+00 & 1.044E+00 & 1.029E+00 & 1.017E+00 & 1.009E+00\
9.00 & 1.001E+00 & 1.597E+00 & 1.229E+00 & 1.104E+00 & 1.044E+00 & 1.028E+00 & 1.017E+00 & 1.009E+00\
9.20 & 1.001E+00 & 1.597E+00 & 1.229E+00 & 1.104E+00 & 1.044E+00 & 1.028E+00 & 1.017E+00 & 1.009E+00\
9.40 & 1.001E+00 & 1.596E+00 & 1.229E+00 & 1.104E+00 & 1.044E+00 & 1.028E+00 & 1.017E+00 & 1.008E+00\
9.60 & 1.001E+00 & 1.596E+00 & 1.229E+00 & 1.103E+00 & 1.044E+00 & 1.028E+00 & 1.016E+00 & 1.008E+00\
9.80 & 1.001E+00 & 1.596E+00 & 1.229E+00 & 1.103E+00 & 1.044E+00 & 1.028E+00 & 1.016E+00 & 1.008E+00\
10.00 & 1.000E+00 & 1.596E+00 & 1.229E+00 & 1.103E+00 & 1.044E+00 & 1.028E+00 & 1.016E+00 & 1.008E+00\
|
---
abstract: 'Alexopoulos proved that on a finitely generated virtually nilpotent group, the restriction of a harmonic function of polynomial growth to a torsion-free nilpotent subgroup of finite index is always a polynomial in the Mal’cev coordinates of that subgroup. For general groups, vanishing of higher-order discrete derivatives gives a natural notion of polynomial maps, which has been considered by Leibman and others. We provide a simple proof of Alexopoulos’s result using this notion of polynomials, under the weaker hypothesis that the space of harmonic functions of polynomial growth of degree at most $k$ is finite dimensional. We also prove that for a finitely generated group the Laplacian maps the polynomials of degree $k$ surjectively onto the polynomials of degree $k-2$. We then present some corollaries. In particular, we calculate precisely the dimension of the space of harmonic functions of polynomial growth of degree at most $k$ on a virtually nilpotent group, extending an old result of Heilbronn for the abelian case, and refining a more recent result of Hua and Jost.'
address:
- 'Tom Meyerovitch, Idan Perl and Ariel Yadin: Department of Mathematics, Ben Gurion University of the Negev, Be’er Sheva, Israel.'
- ' Matthew Tointon: Laboratoire de Mathématiques, Université Paris-Sud 11, 91405 Orsay cedex, France'
author:
- 'Tom Meyerovitch, Idan Perl, Matthew Tointon and Ariel Yadin'
title: Polynomials and harmonic functions on discrete groups
---
[10]{} G. K. Alexopoulos. Random walks on discrete groups of polynomial volume growth, *Ann. Probab.* **30**(2) (2002), 723-801. H. Bass. The degree of polynomial growth of finitely generated nilpotent groups, *Proc. London Math. Soc. (3)* **25**(4) (1972), 603-614. I. Benjamini, H. Duminil-Copin, G. Kozma and A. Yadin. Disorder, entropy and harmonic functions, *Ann. Probab.* **43** (2015), 2332-2373. T. H. Colding and W. P. Minicozzi II. Harmonic functions on manifolds, *Ann. of Math. (2)* **146**(3) (1997), 725-747. B. J. Green and T. C. Tao. The quantitative behaviour of polynomial orbits on nilmanifolds, *Ann. of Math. (2)* **175**(2) (2012), 465-540. M. Gromov. Groups of polynomial growth and expanding maps, *Publ. Math. IHES* **53** (1981), 53-73. Y. Guivarc’h. Croissance polynomiale et périodes des fonctions harmoniques, *Bull. Soc. Math. France* **101** (1973), 333-379. M. Hall. *The theory of groups*, Amer. Math. Soc./Chelsea, Providence, RI (1999). H. A. Heilbronn. On discrete harmonic functions, *Math. Proc. Cambridge Philos. Soc.* **45** (1949), 194-206. B. Hua and J. Jost. Polynomial growth harmonic functions on groups of polynomial volume growth, *Math. Z.* **280** (2015), 551-567. B. Hua, J. Jost and X. Li-Jost. Polynomial growth harmonic functions on finitely generated abelian groups, *Ann. Global Anal. Geom* **44** (2013), 417-432. V. A. Kaimanovich and A. M. Vershik. Random walks on discrete groups: boundary and entropy, *Ann. Probab.* **11**(3) (1983), 457-490. B. Kleiner. A new proof of Gromov’s theorem on groups of polynomial growth, *J. Amer. Math. Soc.* **23** (2010), 815-829. M. Lazard. Sur les groupes nilpotents et les anneaux de Lie, *Ann. Sci. Ecole Norm. Sup. (3)* **71** (1954), 101-190. A. Leibman. Polynomial mappings of groups, *Israel J. Math.* **129** (2002), 29-60. A. Mal’cev. On a class of homogeneous spaces, *Izvestiya Akad. Nauk SSSR, Ser Mat.* **13** (1949), 9-32. T. Meyerovitch and A. Yadin. Harmonic functions of linear growth on solvable groups, *Israel J. Math.* **216** (2016), 149-180. M. S. Raghunathan. *Discrete subgroups of Lie groups*, Springer, New York (1972).
D. Robinson. [*A course in the theory of groups*]{}, Vol. 80. Springer Science & Business Media (1996).
M. C. H. Tointon. Freiman’s theorem in an arbitrary nilpotent group, *Proc. London Math. Soc.* **109** (2014), 318-352. M. C. H. Tointon. Characterisations of algebraic properties of groups in terms of harmonic functions, *Groups Geom. Dyn.* **10** (2016), 1007-1049.
|
---
abstract: 'The trigger for the short bursts observed in $\gamma$-rays from many magnetar sources remains unknown. One particular open question in this context is the localization of burst emission to a singular active region or a larger area across the neutron star. While several observational studies have attempted to investigate this question by looking at the phase dependence of burst properties, results have been mixed. At the same time, it is not obvious a priori that bursts from a localized active region would actually give rise to a detectable phase-dependence, taking into account issues such as geometry, relativistic effects, and intrinsic burst properties such brightness and duration. In this paper we build a simple theoretical model to investigate the circumstances under which the latter effects could affect detectability of a dependence of burst emission on rotational phase. We find that even for strongly phase-dependent emission, inferred burst properties may not show a rotational phase dependence depending on the geometry of the system and the observer. Furthermore, the observed properties of bursts with durations short as 10-20% of the spin period can vary strongly depending on the rotational phase at which the burst was emitted. We also show that detectability of a rotational phase dependence depends strongly on the minimum number of bursts observed, and find that existing burst samples may simply be too small to rule out a phase dependence.'
author:
- |
C. Elenbaas,$^{1}$[^1] A.L. Watts,$^{1}$ and D. Huppenkothen$^{2,3,4}$\
$^{1}$Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, the Netherlands\
$^{2}$DIRAC Institute, Department of Astronomy, University of Washington, Box 351580, Seattle, WA 98195-1580, USA\
$^{3}$Center for Data Science, New York University, 65 5h Avenue, 7th Floor, New York, NY 10003, USA\
$^{4}$Center for Cosmology and Particle Physics, Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA\
bibliography:
- 'biblio.bib'
title: The rotational phase dependence of magnetar bursts
---
\[firstpage\]
stars: magnetars – magnetic fields – X-rays: bursts – gamma-rays: stars
Introduction
============
Magnetars, the most highly magnetized neutron stars (with dipole fields $\gtrsim 10^{13}$ G), are isolated stars powered primarily by magnetic field decay [@Thompson1993; @Thompson1995; @Kouveliotou1998; @Kouveliotou1999]. One of their key characteristics is the sporadic emission of soft $\gamma$-ray bursts [for reviews of this specific aspect see @Woods2006; @Turolla2015; @Kaspi2017]. Durations and fluences vary, but most of these bursts are short, lasting $\sim 0.01 - 1$ s, less than a typical magnetar spin period $P\sim 6$ s. The slow decay of the strong magnetic field is assumed to build up stresses in the system: stress release must involve rapid reconfiguration of the external magnetic field, particle acceleration, and $\gamma$-ray emission. However what triggers the occurrence of individual bursts, the way in which a burst progresses, and the associated emission processes, remain very poorly understood [@Turolla2015]. The failure point could be internal, within the crust of the star, or in the external magnetosphere itself.
One question is whether the bursts are triggered within a specific active region, fixed in the rotating frame of the star. Some crust zones, for example, are expected to be particularly prone to magnetically-induced faulting and yielding [see for example @Lander2015; @Gourgouliatos2015; @Thompson2017]. Certain regions of the magnetosphere could also be more active than others, in which case there may also be a preferred height above the neutron star surface. Being able to identify whether this is the case would certainly help in efforts to determine the burst mechanism.
One way to determine this, suggested by @Lyutikov2002, is to look at whether there is any rotational phase dependence to the bursts. A number of observational studies have attempted to investigate this, using various different measures such as the phase-dependence of the time at which the burst peak is recorded, or the phase-distribution of all of the burst photons. The evidence for phase-dependence using these measures is mixed (see Section \[sec:Overview of published burst phase-dependence analysis\] for more details). What has never been done is to determine from a theoretical perspective the circumstances under which bursts from a localized active region would actually give rise to a detectable phase-dependence. This will depend on geometry, gravitational light-bending, any beaming factor associated with the burst emission, the size of the burst sample for a given source, and the intrinsic burst properties (e.g. brightness, duration). Under some circumstances, bursts may well be visible throughout most of the rotational phase cycle even if they do originate from a specific active region.
In this paper we address this fundamental question of the circumstances under which emission from a localized bursting region would be detectable as a rotational phase dependence according to the measures used in the literature. We then revisit the observational studies carried out to date to see what constraints they actually place on the degree to which bursting might be localized. We also consider the degree to which the rotational phase at which a burst is emitted might affect the properties measured by an observer.
First, we provide an overview of magnetar burst phase-dependence studies in the literature in Section \[sec:Overview of published burst phase-dependence analysis\]. Different methods have been applied to various sources in an effort to assess the (non-)phase dependence of magnetar bursts. Next in Section \[sec:Methodology\], we briefly outline the method through which we aim to answer the aforementioned questions. We choose to simulate sequences of elementary bursts of which we can control the input parameters and study any phase-dependent effects we may observe. The light curve model is treated in Section \[sec:Light curve model\] and the simulations are described in Section \[sec:Simulations\]. We discuss the results of the simulations and assess the claims made in the literature in Sections \[sec:Results\] and \[sec:Discussion and Conclusion\]. We find that under certain conditions the properties of the observed bursts may become significantly phase-dependent. However, we also find that for a large range of input burst parameters and configurations, a guaranteed detection of phase-dependence requires many more bursts than have commonly been observed.
Overview of published burst phase-dependence analysis {#sec:Overview of published burst phase-dependence analysis}
=====================================================
Here we provide a review of previous work where, given the acquired data, the phase dependence of magnetar bursts has been evaluated. We focus on how the data were obtained and processed, and what method was used to determine the absence or presence of a phase dependence in the burst occurrences or properties. In practice, three methods that have been applied: (i) searching for any significant deviations from uniformity of burst occurrence and photon arrival-time distributions against phase, (ii) searching for any correlation between the phase at which bursts occur and the pulse maxima of the (underlying) pulsed emission, and (iii) Fourier analysis on the burst occurrence times in an effort to search for significant periodicities. The latter has been applied only once; the first two are far more common. It is worth mentioning that the first two methods depend on the accuracy of the ascertained timing ephemeris. The longer the time baseline spanned by the bursts, the greater the risk of undetected time anomalies, such as glitches or spin-down deviations, that may undermine the inference of the phase. Table \[tab:literature review\] provides a summary of the references that have carried out phase dependence analysis of magnetar bursts.
----------- -------- -------------------------------------------------------------------------- -------------- ------------------ ------------------ -- --
Reference Source Dates of bursts *Satellite*/ $n_{\rm bursts}$ Method$^{\rm b}$
\[dd/mm/‘yy\] & Instrument$^{\rm a}$ &&&&\
@Gavriil2004 & 1E 2259+586 & 18/06/‘02 & *RXTE*/PCA & 80 & (i)\
@Savchenko2010 & SGR J1550–5428 & 22/01/‘09 & *INTEGRAL*/ACS & 84 & (i)\
@Scholz2011 & & (22/01–30/09)/‘09 & *Swift*/XRT & 303 & (i)\
@Lin2012 & & 22/01/‘09 & *Swift*/XRT & 31 & (i)\
&& 30/01/‘09 &&&&&\
@Collazzi2015 & & 03/10/‘08–17/04/‘09 & *Fermi*/GBM & 354 & (i)\
@Mus2015 & & 22/01/‘09 & *RXTE*/PCA & 4 & (ii)\
&& 06/02/‘09 &&&&&\
&& 30/03/‘09 &&&&&\
&& 11/01/‘10 &&&&&\
@Gavriil2002 & 1E 1048.1–5937 & 29/10/‘01 & *RXTE*/PCA & 2 & (ii)\
&& 14/11/‘01 &&&&&\
@Gavriil2006 & & 29/06/‘04 & *RXTE*/PCA & 1 & (ii)\
@Dib2009 & & 29/10/‘01 & *RXTE*/PCA & 4 & (ii)\
&& 14/11/‘01 &&&&&\
&& 29/06/‘04 &&&&&\
&& 28/04/‘08 &&&&&\
@An2014 & & (17–27)/07/‘13 & *NuSTAR* & 8 & (ii)\
@Woods2005 & XTE J1810–197 & 22/07/‘03 & *RXTE*/PCA & 6 & (ii)\
&& 16/02/‘04 &&&&&\
&& 19/04/‘04 &&&&&\
&& 19/05/‘04 &&&&&\
@Gavriil2011 & 4U 0142+61 & (06/04–26/06)/‘06 & *RXTE*/PCA & 6 & (ii)\
@Palmer1999 & SGR 1806–20 & (10–15)/11/‘83 & *ICE* & 33 & (iii)\
@Palmer2002 & SGR 1900+14 & - & - & - & -\
----------- -------- -------------------------------------------------------------------------- -------------- ------------------ ------------------ -- --
[$^{\rm a}$Spacecraft/instrument acronyms: *Rossi X-ray Timing Explorer* (*RXTE*), Proportional Counter Array (PCA), *Nuclear Spectroscopic Telescope Array* (*NuSTAR*), *International Cometary Explorer* (*ICE*), Anti-Coincidence Shield (ACS), X-ray Telescope (XRT), *Nuclear Spectroscopic Telescope Array* (*NuSTAR*), and Gamma-ray Burst Monitor (GBM).\
$^{\rm b}$The methods are specified in Section \[sec:Overview of published burst phase-dependence analysis\].]{}
\[tab:literature review\]
The active phase of 1E 2259+586 on 2002 June 18 consisted of 80 bursts and was studied using method (i) [@Gavriil2004]; it was claimed that the burst peak phase occurrences tended to correlate with the intensity of the pulsed emission, yet no phase dependencies were observed for the burst durations, fluences, peak fluxes, and rise/fall-times.
In excess of 300 bursts were observed from SGR J1550–5428 between 2008 March and 2010 January and many of those bursts were detected by multiple space-based telescopes simultaneously. @Savchenko2010 found that the burst start times (the moment the burst exceeds 5$\sigma$ above background) of 84 bursts, observed with the Anti-Coincidence Shield (ACS) aboard the INTEGRAL spacecraft, appear to be distributed randomly across phase, i.e. no significant departure from the mean bursts per phase bin was identified. @Scholz2011 and @Collazzi2015 studied the burst peak times of, respectively, 303 and 354 bursts, and both found no significant ($>3\sigma$) deviations from the mean number of burst peaks per phase bin. @Scholz2011 however do show that the phase-folded photon times of arrival of the bursts exhibit an apparent pulse which has an offset with respect to maximum of the associated quiescent pulse profile. @Lin2012 study a sample of 31 bursts and similarly find that the burst count distribution is not uniform across phase. Moreover, they find that the phase probability density anti-correlates with the phase profile of the persistent emission (with a correlation factor of -0.5 and chance probability of $3.4\times10^{-2}$), which may suggest that the burst emission region is distinct to that of the persistent emission. Contrary to these results however, @Collazzi2015 do not find a significant ($>3\sigma$) pulse shape in the epoch folded burst emission light curves. Note that the data set used by @Lin2012 constitutes a subset of the data used by @Collazzi2015.
A total of 12 bursts from 1E 1048.1–5937, were analyzed using method (ii); 4 of which were observed with *RXTE*/PCA between 2001 October 29 and 2008 April 28 [@Gavriil2002; @Gavriil2006; @Dib2009] and 8 of which were observed with *NuSTAR* in 2013 July 17–27 [@An2014]. It was determined that the majority of the bursts[^2] observed with *RXTE*/PCA had a probable chance alignment with pulse maxima of less than 0.01. For the latter 8 bursts from the same source however there is no evidence for a preferred phase occurrence. 6 bursts from XTE J1810–197 observed with *RXTE*/PCA were also studied with method (ii) [@Woods2005]. These bursts consisted of individual burst spikes, which in turn occurred near the corresponding pulse maxima of the source, either leading or trailing. A chance alignment of these spikes with the pulse maxima was estimated at roughly 0.004. @Gavriil2011 studied 6 bursts of 4U 0142+61 observed with *RXTE*/PCA in 2006 from April to June. They found that several bursts appear to occur near the maxima of contemporaneous folded pulse profiles (no significance criteria are specified in the reference). They argue that this may indicate that the bursts comprise extreme episodes of local transient emission sites.
@Palmer1999 [@Palmer2002] studied the burst properties of SGR 1806–20 and SGR 1900+14, where for the former source it was found that active bursting episodes emerge from local active regions characterized as ‘relaxation systems’. From a larger burst sample a group of 33 bursts were identified as belonging to a single relaxation system. Method (iii), i.e. Fourier analysis on the burst occurrences of this group revealed no apparent modulation at the rotation frequency of the NS, indicating the lack of a phase dependence[^3].
Here we will focus mainly on the (non-)uniform phase occurrence of burst peaks as a proxy for the phase dependency of magnetar bursts. We briefly discuss the use of alternative methods in Section \[sec:Discussion and Conclusion\].
Methodology {#sec:Methodology}
===========
To understand how the observed emission may depend on the rotational phase, we intentionally introduce a phase-dependency by fixing the burst location to a certain region or burst patch on the magnetar surface and then set out to describe and simulate the process from emission, where we control the input parameters, to detection and characterization. Subsequently, we can study the effects of a certain configuration on the burst parameters by investigating the phase distributions of the observed burst properties. Moreover, we can establish detectability criteria for the phase-dependency for certain input values/distributions and system configurations.
In order to do so, we require a light curve model that describes how the burst emission is modified depending on the location of the bursts and additional system parameters, e.g. the inclination angle to the observer and compactness of the source. The latter parameter will reshape the trajectory of emitted photons through gravitational light bending.
In Section \[sec:Light curve model\] we ascertain an expression that describes the fraction of rays, i.e. paths along which the emitted photons propagate, that extend out from the burst location and intersect with an observer at infinity. Subsequently, in Section \[sec:Simulations\], we simulate sequences of bursts and investigate how the burst properties are modified through the correction of the burst intensity by the aforementioned expression.
Light curve model {#sec:Light curve model}
=================
{width="\textwidth"}
In the following we adopt natural units, i.e. $G=c=1$, and the spatial spherical coordinates ($r,\varphi,\theta$), where $\varphi$ is the polar angle to the $y$-axis and $\theta$ is azimuthal angle to the $z$-axis. Since magnetars rotate slowly (typically $|\boldsymbol{\Omega}|\sim10^{-1}$ rad s$^{-1}$) they can be considered to be almost spherically symmetric. Accordingly, we may assume that the metric external to the star is approximately given by the Schwarzschild spacetime solution, $$ds^2=-\mathcal{A}(r)dt^2+\mathcal{A}^{-1}(r)dr^2+r^2 \left(d\varphi^2+\sin^2\varphi\,d\theta^2\right)$$ where $\mathcal{A}(r)=(1-R_S/r)$, and $R_S=2M$ the Schwarzschild radius, with $M$ corresponding to the gravitational mass of the compact object. To model the effect of gravitational light bending on the burst emission we consider the configuration illustrated in Fig. \[fig:light\_bending\_configuration\], which is based on work done by @Pechenick1983. The stellar surface is located at a distance $R$ from the origin; for neutron stars, $R$ lies roughly in the range $2.5-4~R_S$. For now we assume that the burst emission originates at [^4] $r = R$ over a circular patch of angular radius $\psi$ centered at point $\boldsymbol{p}(R,\varphi,\theta)$ with total intensity $I$. Depending on the burst emission mechanism, $I$ may be anisotropic and depend on $\delta\in[0,\pi)$, i.e. the angle between the normal vector to the stellar surface $\hat{\boldsymbol{n}}$ and the outgoing emission vector $\hat{\boldsymbol{k}}_R$. The latter lies at $r=R$ along the associated null geodesic $\mathcal{G}$ of the outgoing emission which in turn intersects with the observer at $r=r_0$ where $\hat{\boldsymbol{k}}_R\to\hat{\boldsymbol{k}}$. We presume that the region $r < R$ is opaque and $r > R$ is entirely transparent. Moreover, due to the comparatively long rotation period of magnetars, we may neglect certain corrections, such as oblateness of the stellar surface, light travel-time delays, and Doppler effects, which become significant for NSs with $|\boldsymbol{\Omega}|\gg1$ rad s$^{-1}$ [e.g. @Morsink2007].
The Schwarzschild solution admits four Killing vectors associated with conserved quantities that emerge from symmetries inherent in the solution for which, $$K_\mu\dot{x}^\mu=\text{constant},$$ where $\dot{x}^{\mu}=dx^{\mu}/d\lambda$, with $\lambda$ is some affine parameter. Considering the orbital motion of a photon with tangent 4-vector $V^\mu=\dot{x}^{\mu}$ in the equatorial plane, i.e. $\varphi = \pi/2$, the Schwarzschild solution admits two Killing vectors $$\begin{aligned}
&\epsilon_\mu=(\partial_t)_\mu=\big(-\mathcal{A}(r),0,0,0\big),\\
&J_\mu=(\partial_\varphi)_\mu=\big(0,0,0,r^2\big),\end{aligned}$$ associated respectively with conservation of energy and the magnitude of angular momentum. Accordingly, we may define $$\begin{aligned}
&\epsilon_\mu\dot{x}^\mu=-\mathcal{A}(r)V^t\equiv-1,\\
&J_\mu\dot{x}^\mu=r^2V^\theta\equiv b,\end{aligned}$$ where $b\geq0$ denotes the impact parameter of the photon trajectory, i.e. the null geodesic $\mathcal{G}$. Since $\dot{x}^\theta=0$ and $g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu=0$ for massless particles, we find that the tangent 4-vector of an outgoing photon is given by, $$V^{\mu}=\left(\mathcal{A}^{-1}(r),\sqrt{1-\frac{b^2}{r^2}\mathcal{A}(r)},0,\frac{b}{r^2}\right).$$ Setting $b=0$, we obtain the tangent 4-vector of a radially outgoing photon, $$W^{\mu}=\left(\mathcal{A}^{-1}(r),1,0,0\right).$$ A stationary observer with 4-velocity $$U^{\mu}=\left(\mathcal{A}^{-1/2}(r),0,0,0\right),$$ will observe an angle $$\label{eq:observed photon angle}
\cos\xi=\sqrt{1-\frac{b^2}{r^2}\mathcal{A}(r)},$$ between the photons prescribed by $V^\mu$ and $W^\mu$ [@Pechenick1983]. Note that $\hat{\boldsymbol{n}}\cdot\hat{\boldsymbol{k}}_R=\cos\delta = \cos\,(\xi|_{r=R})$, such that we may write $$\delta(b)=\arcsin\left(\frac{b}{b_{\rm max}}\right),~~~~\text{with}~~b_{\rm max}=\frac{R}{\mathcal{A}^{1/2}(r=R)}.$$ The total angular deflection of $\mathcal{G}$, which determines the ‘bending’ of the photon trajectory from the surface patch to the observer, is given by $$\label{eq:angular deflection}
\theta_*(b) =\int_R^{r_0}\frac{b}{r^2}\left[1-\frac{b^2}{r^2}\mathcal{A}(r)\right]^{-1/2}dr.$$ Incidentally, the total coordinate light travel time along $\mathcal{G}$ is given by $$T_*(b) =\int_R^{r_0}\mathcal{A}^{-1}(r)\left[1-\frac{b^2}{r^2}\mathcal{A}(r)\right]^{-1/2}dr.$$
The difference in travel time between radially emitted photons (with $b=0$) and those with an arbitrary impact parameter can be estimated accordingly, $$\Delta t_*(b) =\int_R^{r_0}\mathcal{A}^{-1}(r)\left\{\left[1-\frac{b^2}{r^2}\mathcal{A}(r)\right]^{-1/2}-1\right\}dr.$$ The maximum travel time delay then for a typical NS with $R = 2.5\,R_S$ ($R = 10^6$ cm, $M = 1.5\,M_\odot$), is $\Delta t_*(b_{\rm max})\simeq6.7\times10^{-2}$ ms $\ll P\sim6$ s.
To an observer in the $+z$-direction the system is axisymmetric around the $z$-axis, such that the location of the burst patch $\boldsymbol{p}$ can be uniquely described by $\theta_0$. The angle between the observer’s line of sight and the rotation axis of the neutron star $\boldsymbol{\Omega}$ is denoted by $\chi$. Furthermore, the angle between the location of the burst patch and $\boldsymbol{\Omega}$ is given by $\alpha$. Accordingly, depending on the rotational phase of the neutron star, $$\label{eq:phase}
\phi(t)=\frac{2\pi t}{P},$$ the angle between the observer and burst patch $\theta_0$ is given by the relation $$\label{eq:cosine theta0}
\cos[\theta_0(\phi)] = \cos\chi\cos\alpha + \sin\chi\sin\alpha\cos\phi.$$
The observed brightness is the integral of the intensity at the observer, $$dI_{\rm obs}=I(r_0,\Omega')d\Omega'=\mathcal{A}^2(r=R)I(R,\Omega')d\Omega',$$ over the solid angle subtended in the observer’s sky, $$d\Omega'=\sin\theta' d\theta' d\varphi'\simeq\theta' d\theta' d\varphi'.$$ Due to the axisymmetry $d\varphi'=d\varphi\equiv\Phi(\theta_*,\theta_0)$, where we define the polar differential distance as the function $\Phi(\theta_*,\theta_0)$, which under the conditions $\theta_0+\psi\leq\theta_*\leq\pi$ and $\theta_0-\psi\geq0$ is given by the following expression, $$\begin{aligned}
\Phi(\theta_*,\theta_0) = \left\{
\begin{array}{ll}
2\arccos\left(\frac{\cos\psi-\cos\theta_0\cos\theta_*}{\sin\theta_0\sin\theta_*}\right) & \text{if } \theta_0-\psi\leq\theta_*\leq\theta_0+\psi,\\
0 & \text{otherwise.}
\end{array} \right.\end{aligned}$$ Consequently, together with $\theta'=\xi$ and Eq. (\[eq:observed photon angle\]) evaluated at $r_0\to\infty$, we obtain $$d\Omega'\simeq \Phi[\theta_*(b),\theta_0]\,\xi d\xi\simeq\frac{\Phi(b,\theta_0)}{r_0^2}b\,db.$$ We write the brightness of the source as $$I(R,\Omega')=I_0f[\delta(b)],$$ with the beaming functions given by $f(b)$. Currently, we do not have a physical model for the shape of the beaming function, which will most likely depend on the radiative transfer properties of the local magnetic field. Accordingly, the influence of the magnetic field on the light trajectories is ignored for now. An example of a more realistic model was considered by @vanPutten2016 in the case of fireball beaming. Here, for descriptive purposes we consider a Gaussian shape for the beaming function, $$\label{eq:beaming function}
f[\delta(b)]= \left\{
\begin{array}{ll}
1 & \text{isotropic}, \\
\sqrt{\frac{\pi}{2\sigma_{\rm b}^2}}~{\rm erf}\left(\frac{\pi}{2\sqrt{2}\sigma_{\rm b}}\right)^{-1}\exp\left(-\frac{\delta(b)^2}{2\sigma_{\rm b}^2}\right)& \text{beamed},
\end{array} \right.$$ where $\sigma_{\rm b}$ parameterizes the beam width. We neglect rotational aberration of light effects, since the star rotates slowly. Finally, we find the expression for the observed intensity of the source $$I_{\rm obs}(\theta_0)=I_0\left(\frac{R}{r_0}\right)^2\kappa(\theta_0),$$ where we define $$\label{eq:kappa}
\kappa(\theta_0)\equiv\left(\frac{R^{1/2}}{b_{\rm max}}\right)^4\int^{b_{\rm max}}_0f[\delta(b)]\,\Phi(b,\theta_0)\,b\,db.$$
Fig. \[fig:kappa\] shows the observed burst emission $I_{\rm obs}$ as a function of burst patch location $\theta_0$ for a compact object with $R=2.5\,R_S$. In this case the size of the patch is $\psi=1^\circ$. We consider the observed intensity for the case of isotropic and beamed emission. Note that the location of the terminator lies ‘behind’ the star, i.e. beyond $\theta_0=\pi/2$, at $\theta_0\simeq0.72\pi$. At this angle, only photons with an impact parameter of $b_{\rm max}\simeq3.23 R_S$ reach the observer. Note that the beamed emission is more prominent at $\theta_0=0$ and drops off faster than the isotropic emission with increasing $\theta_0$. For comparison, we plotted the emission profiles of sources with $R=1.6\,R_S$ and $R\gg R_S$, where the former is close to the most extreme case, i.e. $R>1.5\,R_S$ and the latter approximates flat spacetime.
In the simulations we concentrate on the relative changes in intensity between the input and observed burst. Accordingly, from equations (\[eq:cosine theta0\]) and (\[eq:kappa\]) we define $$\label{eq:kappa star}
\kappa_*(\phi)\equiv\frac{\kappa[\theta(\phi)]}{\kappa_{\rm max}},$$ which depends on the angles $\chi$, $\alpha$, and the phase of the neutron star, and describes the fraction of rays that intersect with the observer at infinity, from the entire ray-bundle that extends outwards from the burst patch. In the following section, we use this expression as our measure for how the burst intensity is modulated. Varying $\chi$ separately from $\alpha$, or vice-versa, acts as a multiplicative factor to the absolute intensity. Since, we only consider the fractional intensity, we may explore the parameter space of these angles by setting $\chi=\alpha$. Fig. \[fig:kappa angles\] illustrates the shape of $\kappa_*(\phi)$ for $R = 2.5\,R_S$ in 4 different angle configurations, both in the case of isotropic and beamed emission.
Simulations {#sec:Simulations}
===========
Per simulation run we produce a sequence of $n$ bursts where we control the input parameters of the bursts and the system. Nonetheless, we treat the detection of individual photons, which are entirely described by their times-of-arrival (TOA), in a probabilistic fashion.
We assume for simplicity that a single magnetar burst can be modeled with a exponential rise/exponential decay profile that allows for asymmetry, i.e. the rise-time and fall-time may be different. When simulating photons emitted at the source, this corresponds to drawing $N$ random photon times-of-emission (TOE) from a skewed Laplace distribution, $$\label{eq:skewed Laplace probability density function}
p_{\rm TOE}(t)=\frac{1}{(1+s)\tau}\left\{
\begin{array}{ll}
\exp\left[(t-t_0)/\tau\right] & \text{if }~ t < t_0, \\
\exp\left[-(t-t_0)/(s\tau)\right] & \text{if }~ t \geq t_0,
\end{array} \right.$$ where $\tau$ denotes the exponential rise-time of the burst, ($s\tau$) the decay-time, with $s$ the skewness factor, and $t_0$ the peak time of the burst. The profile of a simulated burst is shown in Fig. \[fig:burst\_profile\]. We deliberately adopt an oversimplified burst profile to better understand the differences between the input and output data. @Huppenkothen2015 decompose complex magnetar bursts, observed from SGR J1550-5418, into several spike-like components, which in turn are modeled with a similar profile as in equation (\[eq:skewed Laplace probability density function\]). In this paper we assume that a burst can be represented as a single spike, as a simple model that lets us explore the relevant effects. Note that we define the duration of the burst, $T_{90}$, as the time it takes for the fluence to increase from 0.05 to 0.95 of the total burst fluence. We fix the compactness of the source to $R=2.5\, R_{\rm S}$, corresponding to a typical neutron star with $R=10^6$ cm and $M = 1.5 M_\odot$, and set the rotation period to $P=6$ s. We choose a light curve bin width, $\delta t$, and background count level, $b$, such that the background count rate approximates that from *Fermi*/GBM data [@Huppenkothen2015], i.e. $\zeta_{\rm GBM}\sim318$ counts s$^{-1}$. A list of the simulation parameters is given in Table \[tab:sim\_pars\].
General simulation procedure {#sec:General simulation procedure}
----------------------------
Here we proceed to describe in more detail the general form of a simulation run step-by-step:
(I) We decide on the number of bursts $n$ we wish to produce per simulation run and set the inclination angle of the source $\chi$. Next, we assign values to the burst parameters $\psi$, $\alpha$, $N$, $t_0$, $\tau$, $s$, and $T_{90}$, where the latter three parameters cannot be defined independently of each other. In the following simulation runs, described in Sections \[sec:Run 1\], \[sec:Run 2\], and \[sec:Run 3\], we only consider symmetric input bursts, $s=1$, with a size of $\psi=1^{\circ}$, and draw their peak time from a uniform distribution, i.e. $t_0\sim{\rm Uniform}(0,P)$, where $P$ is the rotation period of the magnetar which we set to $P=6$.
(II) We generate a single burst by drawing $N$ photon emission times (TOEs) from $p_{\rm TOE}(t)$: We draw random numbers from a uniform distribution ${\rm Uniform}(0,1)$ and transform these values to follow the required skewed Laplace distribution by using the inverse cumulative distribution function of the latter, i.e. the percent point function, $$\label{eq:skewed Laplace percent point function}
{\rm TOE}(x)=\left\{
\begin{array}{ll}
t_0 + \tau\ln\left[\left(1+s\right)x\right] & \text{if }~ x < (1+s)^{-1},\\
t_0 - s\tau\ln\left[\left(1+s^{-1}\right)\left(1-x\right)\right] & \text{if }~ x \geq (1+s)^{-1}.
\end{array} \right.$$
(III) Using equation (\[eq:phase\]) we determine the phase of each TOE$_{i}$, i.e. $\phi_i$. Whether an emitted photon reaches the observer depends on whether the ray, along which the photon propagates, intersects with the detector, given by $\kappa_*(\phi_i)$, which denotes the fraction of photons directed into our line of sight. In order to decide whether a given photon intersects with the detector, we use rejection sampling: for each TOE$_{i}$ we generate a latent variable $z$ drawn from $p(z) = {\rm Uniform}(0,1)$. We only keep the TOE$_{i}$ if $z < \kappa_*(\phi_i)$.[^5]
(IV) The TOEs that we save are detected by the observer. A detected photon is recorded as a count with a corresponding TOA$_{i}$; where the TOA$_{i}$ = TOE$_{i}$ of the respective photon, since we consider a perfect detector and may neglect the distance to the source and gravitational time-delay effects – see Section \[sec:Light curve model\]. Furthermore, we add background counts or TOAs uniformly to the detected TOA data from the burst (with length $N_{\rm det}\leq N$), such that the mean background count rate becomes approximately $\zeta$.
(V) We bin the total TOA data in $\mathcal{N}_{\rm bins}$ time bins of length $\delta t$, whereby the counts in each bin follow Poisson statistics. We proceed by applying a similar burst identification algorithm as used by [@Gavriil2004], assuming that we can infer the background count rate to be $\zeta$, yet have no prior knowledge of the burst and system input parameters. The probability of the number of counts $k_i$ in the $i$th bin occurring is given by the Poisson distribution, $$\label{eq:Poisson distribution}
P_i=\frac{\mu^{k_i}e^{-\mu}}{k_i!},$$ where $\mu$ represents the mean count level, which in our case will be $b=\zeta\,\delta t$. Bins for which $$\label{eq:burst detection threshold}
P_i \leq 3\times10^{-3}\mathcal{N}_{\rm bins}^{-1}$$ are recorded as significant departures from the mean, where we have corrected for the number of trails by dividing by $\mathcal{N}_{\rm bins}$, i.e. the total number of bins searched over. From these, the time bin containing the maximum departure $y^{\rm sig}$ is labeled as $t_0^{\rm sig}$. The burst edges, labeled as $t_{\rm in}$ and $t_{\rm out}$, are ascertained by making use of a running mean, i.e. when the mean count level of an interval $\mu^*$ of $\Delta T_{\rm interval}=0.25$ s, moving outwards in steps of $\delta t$ on both sides of $t_0^{\rm sig}$, falls below $b^*=1.1b$, the burst edges are then given by the center time of the respective intervals; $t_{\rm in}$ before and $t_{\rm out}$ after the burst. The duration of this interval is denoted $\Delta T=|t_{\rm in}-t_{\rm out}|$.
(VI) We fit the light curve of any identified bursts with the following burst model, $$\label{eq:burst model}
m(t) = N p_{\rm TOE}(t\,|\,t_0,\tau,s) + \zeta,$$ using a *L-BFGS-B* constrained optimizer [@Zhu1997] to determine the maximum (Poisson) likelihood, whereby we fix the background parameter $\zeta$ and provide initial guesses for the remaining parameters: $$\begin{aligned}
\label{eq:initial parameters}
t_0^{\rm init} &= t_0^{\rm sig},\\
\tau^{\rm init} &= \left(t_0^{\rm sig} - t_{\rm in}\right)\left[\ln\left(\frac{y^{\rm sig}}{b^*}\right)\right]^{-1} ,\\
s^{\rm init} &= 1,\end{aligned}$$ and $N^{\rm init}$ is defined as the number of counts in the interval $\Delta T$ minus the background counts[^6], i.e. $\zeta\Delta T$. After the fit, we delete the bins in $\Delta T$ from the light curve, and repeat steps (V) and (VI) until no significant departures from the mean, i.e. $\mu=b$, are recorded.
(VII) We return to step (II) until we have generated the pre-defined number of bursts $n$. Note that the number of observed bursts might be different, since bursts might go undetected, or be interpreted as multiple separate bursts. Moreover, some identified ‘bursts’ may simply be significant statistical deviations from the background level. However, according to the condition stated in equation (\[eq:burst detection threshold\]) we only expect this to be the case in $\sim 0.3$% of the input bursts.
Run 1: Initial simulation run {#sec:Run 1}
-----------------------------
We start with the simplest scenario, where we consider simulations of sequences of identical bursts, referred to as Run 1. Per simulation we fix the values for $\chi$, $\alpha$, $\psi$, $s$ and $T_{90}$. The latter two parameters determine the value of $\tau$. Subsequently, we define $N$ using the condition that the input burst amplitude $A$ is $10^4$ photons s$^{-1}$. We run simulations for three separate burst durations (see Fig \[fig:bursts\]) and vary the angles $\chi$ and $\alpha$, to study their effects on the observed quantities.
We concentrate on the difference in input and observed best-fit value for the time of the burst peak (respectively, $t_0$ and $t_0^{\rm bf}$), where the difference is parameterized as $\Delta t_0\equiv t_0^{\rm bf}-t_0$, the rise-time $\tau$, skewness factor $s$, and burst duration $T_{90}$. The input values of the latter three parameters are denoted as $\tau_0$, $s_0$, and $T_{90,0}$. All input parameters of Run 1 are listed in Table \[tab:RUN1\].
Parameter Value
----------------------------- ----------------
$n$ $10^4$
$\chi$, $\alpha$ ($^\circ$) 30, 45, 60, 90
$A$ (photons s$^{-1}$) $10^4$
$T_{90}$ (s) 0.15, 1.5, 3.0
$\delta t$ (s) $200^{-1}$
: Input parameters for Run 1, consisting of 12 separate simulations. We consider a constant input burst profile with peak times distributed uniformly across phase. Per simulation we vary the burst duration $T_{90}$, and angles $\chi$, $\alpha$; where we set $\chi=\alpha$ (see Section \[sec:Light curve model\]).
\[tab:RUN1\]
Run 2: $T_{90}$-distribution {#sec:Run 2}
----------------------------
Next we perform a simulation run, Run 2, where in step (I) of the general simulation procedure (Section \[sec:General simulation procedure\]), we draw the burst duration $T_{90}$ for each individual burst from a lognormal distribution centred at $\overline{T}_{90}=0.1$ s, with a width of $\sigma_{T_{90}}=1$ [e.g. @Gogus2001], and lower and upper burst duration cutoff at, respectively, $T_{90}^{\rm min}=300^{-1}$ s and $T_{90}^{\rm max}=3$ s $< P$. Fixing the input burst amplitude at $A=10^4$ photons s$^{-1}$, as done in Run 1, we find that the rise-time of the shortest admissible burst duration, i.e. $300^{-1}$ s, is $\tau^{\rm min}\sim7.2\times10^{-4}$ s. Accordingly, we set $\delta t = 1400^{-1}$ s for this simulation run. The input parameters are summarised in Table \[tab:RUN2\] and the results are presented in Section \[subsec:Run 2\].
Parameter Value
----------------------------- --------------------------------------------------------------------------
$n$ $10^4$
$\chi$, $\alpha$ ($^\circ$) 30, 45, 60, 90
$A$ (photons s$^{-1}$) $10^4$
$T_{90}$ (s) $\sim{\rm LogNormal(\overline{\emph{T}}_{90},\sigma^2_{\emph{T}_{90}})}$
$\overline{T}_{90}$ (s) $0.1$
$\sigma_{T_{90}}$ (s) $1$
$\delta t$ (s) $1400^{-1}$
: Input parameters for Run 2, consisting of 4 separate simulations. The input burst durations $T_{90}$ are drawn from a lognormal distribution with lower and upper cutoff, respectively, at $T^{\rm min}_{90}=300^{-1}$ s and $T^{\rm max}_{90}=3$ s. Per simulation run we vary the angles $\chi$, $\alpha$; where we set $\chi=\alpha$.
\[tab:RUN2\]
Run 3: Burst amplitude distribution {#sec:Run 3}
-----------------------------------
Parameter Value
----------------------------- ------------------------------
$n$ $10^4$
$\chi$, $\alpha$ ($^\circ$) 30, 45, 60, 90
$A$ (photons s$^{-1}$) $\sim{\rm Powerlaw(\Gamma)}$
$\Gamma$ $5/3$
$T_{90}$ (s) 1
$\delta t$ (s) $200^{-1}$
: Input parameters for Run 3, consisting of 4 separate simulations. The input burst amplitudes $A$ are drawn from a powerlaw distribution \[equation (\[eq:burst energy distribution\])\], with $A^{\rm min}=5\times10^2$ photons s$^{-1}$, and $A^{\rm max}=10^6$ photons s$^{-1}$. Per simulation run we vary the angles $\chi$, $\alpha$; where we set $\chi=\alpha$.
\[tab:RUN3\]
In this simulation run we fix the burst duration to $T_{90} = 1$ s and draw an amplitude $A$ for each individual burst from a powerlaw distribution, $$\label{eq:burst energy distribution}
\frac{dn}{dA}\propto A^{-\Gamma},~~~\text{with}~~~A^{\rm min} < A < A^{\rm max},$$ where $A^{\rm min}$ and $A^{\rm max}$ represent the limits of the distribution, and $\Gamma$ denotes the powerlaw index. Note that the number of emitted photons during the burst $N$ are linearly proportional to $A$, such that these are distributed in a similar fashion.
In accordance with the observation of the energy distributions of magnetar bursts, we choose $\Gamma = 5/3$ [e.g. @Cheng1996]. The input parameters are summarised in Table \[tab:RUN3\] and the results are presented in Section \[subsec:Run 3\].
Results {#sec:Results}
=======
Predictions for Run 1
---------------------
{width="80.00000%"}
To better understand the results of the simulations, we first examine how $\kappa_*(\phi)$ will affect the burst parameters. In Fig. \[fig:RUN1\_analytic\], we plot the predicted phase distributions of the burst parameters (rows) for the 3 separate burst durations (columns) of Run 1. These curves were obtained by fitting the burst model to the theoretical lightcurve that results when the input model (see Fig. \[fig:bursts\]) is modulated, for a given phase, by the appropriate $\kappa_*(\phi)$ but without taking into account any photon noise or detectability effects (which are treated properly in the full simulations). It gives an idea of the general trends expected, but no idea of the scatter. Furthermore, we only fit modulated burst profiles with a peak rate of $\gtrsim600$ counts s$^{-1}$, since ones with lower peak rates will likely go undetected in the simulations.
Based on the predicted curves we expect that the parameter distributions that we obtain from Run 1 will deviate from their input parameters more strongly for longer burst durations $T_{90}$ and larger angles $\chi$ and $\alpha$. Approaching $\phi = \pi$ from below (above), we find that the bursts will appear to occur earlier (later), rise slower (faster), to become more skewed, and last longer, than their input counterparts. Note furthermore that, in contrast to the predicted phase distributions of $N$, $\Delta t_0$, $s$, and $T_{90}$, the phase distribution of $\tau$ is neither symmetric nor perfectly anti-symmetric about $\phi = \pi$. The results of Run 1 are presented in Section \[subsec:Run 1\].
Burst properties from simulations
---------------------------------
### Run 1: Initial simulation run {#subsec:Run 1}
In Figures \[fig:RUN1\_0.15\], \[fig:RUN1\_1.50\], and \[fig:RUN1\_3.00\], we plot the phase distributions (left) and parameter densities (right) of the obtained bursts parameters for 3 separate input burst durations $T_{90,0}$, respectively, 0.15 s, 1.5 s, and 3.0 s. Table \[tab:sig\_bursts\_RUN1\] lists the amount of bursts that were identified per configuration.
$T_{\rm 90}$ (s) $\chi=\alpha$ ($^\circ$) $n_{\rm id}$
------------------ -------------------------- --------------
0.15 30 10034
45 10022
60 8875
90 6592
1.50 30 10012
45 10017
60 9940
30 7438
3.00 30 10017
45 10012
60 10007
90 9553
: Number of identified bursts $n_{\rm id}$ for Run 1, per configuration. The input number of bursts for each simulation run was $n=10^4$. We expect that $\sim30$ of the identified ‘bursts’ simply constitute statistical deviations that exceed the burst identification threshold (given by equation \[eq:burst detection threshold\]).
\[tab:sig\_bursts\_RUN1\]
As predicted, we find that especially for longer duration bursts and larger angles, the phase dependence of the burst parameters becomes more pronounced. Evidently this is much less the case for bursts with $T_{90}\ll P$ – the parameter densities remain strongly peaked around their input values (e.g. Fig. \[fig:RUN1\_0.15\]). Nevertheless, in those cases around $\phi\sim\pi$ the bursts still go undetected for large values of $\chi$ and $\alpha$, because either no rays extending from the burst patch intersect with the detector during the burst (i.e. the bursts are invisible) or they do not significantly stand out from the background level. The results confirm that when approaching $\phi = \pi$ from below (above), the bursts will appear to occur earlier (later), rise slower (faster), and last longer, than their input counterparts. Moreover, the predicted asymmetric profile of the rise-time phase distribution (most notably in the parameter densities of Figures \[fig:RUN1\_1.50\] and \[fig:RUN1\_3.00\]) is clearly observed. Looking at the parameter densities, it appears that the rise-times of bursts going out of view are more spread out, yet those of bursts coming into view are more clustered. This is in accordance with the predictions; the initial slope of the rise-time phase distribution (from $\sim0-4\pi/5$) is steeper compared to the final slope (from $\sim6\pi/5-2\pi$). The predicted values between $\sim4\pi/5-6\pi/5$ are produced less well in the simulations, since the amount of detected photons is minimal around $\phi = \pi$, complicating burst-identification and characterization. We find that both in the predictions and, even more so, in the simulations that the majority of observed bursts have $\tau/\tau_0<1$. Since, $T_{90}\propto\tau$ we also find for most observed bursts that $T_{90}/T_{90,0}<1$, i.e. the bursts seem to last shorter than their input counterparts.
In general, the observed scatter is likely due to photon noise effects, which become most significant near $\phi=\pi$. These effects influence the efficacy of the burst-identification algorithm and the observed burst morphology.
### Run 2: $T_{90}$ distribution {#subsec:Run 2}
$\chi=\alpha$ ($^\circ$) $n_{\rm id}$
-------------------------- --------------
30 10028
45 9871
60 7382
90 5971
: Number of identified bursts $n_{\rm id}$ for Run 2, per configuration. The input number of bursts for each configuration was $n=10^4$.
\[tab:sig\_bursts\_RUN2\]
The results of Run 2, where we draw the burst duration for each individual burst from a lognormal distribution are shown in Fig. \[fig:RUN2\]. Table \[tab:sig\_bursts\_RUN2\] lists the number of identified bursts per configuration. The results closely resemble those of Run 1, with $T_{90}=0.15$ s (see Fig. \[fig:RUN1\_0.15\]). We only find weak phase dependencies, that only become noticeable for large values of the angles, $\chi$ and $\alpha$. In Fig. \[fig:RUN2\_T90\_hist\], the input (dotted histogram) and best-fit (solid histogram) $T_{90}$ distributions are shown for the 4 separate configurations. Notice the small dearth of short duration bursts in each histogram; short duration bursts contain fewer counts and may therefore be missed by the burst-identification algorithm (step (V) in Section \[sec:General simulation procedure\]). We furthermore find that there is a slight excess at $T_{90}\sim0.6 s$ (although not apparent when $\chi=\alpha=90^\circ$), which is due to the fact that for most observed bursts $\tau/\tau_0<1$ and $T_{90}\propto\tau$.
### Run 3: Burst amplitude distribution {#subsec:Run 3}
The results of Run 3, where we draw the burst amplitude/number of emitted burst photons for each individual burst from a powerlaw distribution, are presented in Fig. \[fig:RUN3\]. Table \[tab:sig\_bursts\_RUN3\] lists the number of identified bursts per configuration. We find much less spread in the phase distributions of the parameters, compared to e.g. the results from the 1.5 s burst in Run 1 (Fig. \[fig:RUN1\_1.50\]), however we do observe a considerable amount of scatter. The latter is likely due to the fact that the majority of input bursts ($\sim0.87$) are low-amplitude bursts, i.e. $A \lesssim10^4$ photons s$^{-1}$, which are more difficult to characterize, i.e. their morphology is relatively heavily affected by Poisson noise. Fig. \[fig:RUN3\_y\_hist\] displays the input and observed burst amplitude distributions. Despite a slight offset at larger angles, we find that the slope of the distributions is reproduced by the observed bursts. Input bursts with an amplitude $\lesssim10^{3}$ photons s$^{-1}$ may go unidentified as they will likely fall below the significance threshold of the burst identification algorithm.
$\chi=\alpha$ ($^\circ$) $n_{\rm id}$
-------------------------- --------------
30 6645
45 5831
60 4734
90 3717
: Number of identified bursts $n_{\rm id}$ for Run 3, per configuration. The input number of bursts for each configuration was $n=10^4$.
\[tab:sig\_bursts\_RUN3\]
Detectability of burst phase dependence
---------------------------------------
Here we set out to test the main method used in studies of burst phase dependence to date, see Section \[sec:Overview of published burst phase-dependence analysis\]. During the simulations we determine and record the phase occurrence of the burst peak $\phi_0^{\rm bf}$, i.e. the phase occurrence of the best-fit burst peak time $t_0^{\rm bf}$. After each burst we compile a distribution of the values for $\phi_0^{\rm bf}$ of all previous bursts up to the most recent one, and compare this burst phase occurrence distribution to a uniform distribution using a Kolmogorov–Smirnov (K–S) test, from which we obtain a p-value. We set the significance threshold at a p-value of 0.003, corresponding to a 0.3% probability that the observed burst peak phase occurrences are distributed uniformly across phase. To be clear: we are simulating emission from a fixed point on the NS surface, from which bursts are being emitted at random rotational phase. The naive expectation that this would result in an observable phase-dependence most from the expected modulation of the intensity (see Fig. \[fig:kappa angles\]) that results, for example, in missing some bursts emitted on the dark side of the star.
In Fig. \[fig:RUN1\_pv\] we plot the evolution of the p-value against the number of observed bursts up to that point in the simulation for the 3 separate burst durations (from top to bottom) of Run 1. The different curves per subplot correspond to different values of $\chi$ and $\alpha$. The horizontal dashed lines denote the threshold level and the vertical dotted lines indicate the number of bursts at which the p-value of a given simulation drops below the threshold level. Note that of these 12 simulations, the p-value does not fall below the threshold before $10^3$ bursts for $\chi=\alpha \leq 45^{\circ}$. Nevertheless, we do find a decreasing trend for $T_{90}=3.0$ s after $\sim500$ bursts, reaching the threshold at $\sim10^4$ bursts (the length of the simulation), for those angles. Remarkably, the p-value associated with the simulation where $T_{90}=1.50$ s and $\chi=\alpha=60^\circ$, does not show a decreasing trend before $10^4$ bursts. The remaining configurations do drop below the threshold fairly soon, i.e. after $\sim 20-150$ bursts.
{width="\textwidth"}
To determine the minimum number of bursts required to guarantee that the p-value drops below the threshold, we ran the simulation per configuration $\mathcal{N}_{\rm s}$ times, each time until the threshold was reached, and recorded the number of bursts. Fig. \[fig:(max)\_pv\_evolution\] shows the evolution of the p-value for $\mathcal{N}_{\rm s}$ = 10, 100, and 400 simulations, with $T_{\rm 90}=0.15$ s and $\chi=\alpha=90^{\circ}$. We found that the maximum obtained p-value (out of all $\mathcal{N}_{\rm s}$ simulations for a given configuration) after a specific number of bursts decreases at a certain rate (denoted by the black markers). In Fig. \[fig:pv\_limits\], we plot the maximum p-value attained, over $\mathcal{N}_{\rm s}=400$ simulations per configuration, against the number of bursts. Subsequently, we fit a straight lines to the decreasing trends of the log of the p-value and record the number of bursts at which these lines intersect with the threshold level. Accordingly, we find an estimate for the minimum number of bursts $n_{\rm min}$ at which, assuming a certain configuration, the observed $\phi_0^{\rm bf}$ distribution should deviate significantly from a uniform distribution. If the $\phi_0^{\rm bf}$ distribution does not significantly deviate from a uniform distribution after $n_{\rm min}$ bursts, then the configuration will likely be such that the modulation in intensity is less strong than assumed. The latter is dependent on assumptions on the parameters that determine the shape of $\kappa(\theta_0)$ (equation \[eq:kappa\]).
For the burst durations and configurations that we study in Run 1, we find for $\chi=\alpha = 90^\circ$ that $n_{\rm min}\sim100$ bursts. For $\chi=\alpha = 60^\circ$, we find $n_{\rm min}=1446$ bursts and $n_{\rm min}=296$ bursts, for $T_{90}=0.15$ s and $T_{90}=3.0$ s, respectively. Yet, we do not find an $n_{\rm min}$ for $T_{90}=1.50$ s, since the attained maximum p-value does not exhibit a decreasing trend before $10^3$ bursts; consistent with the simulation run displayed in the middle panel of Fig. \[fig:RUN1\_pv\]. This is because the burst spot remains (partially) visible throughout the NS’s rotation, such that enough counts can be detected for the duration of the burst, and the fact that the $\Delta t_{0}$ remains comparatively small, i.e. the corresponding parameter density comprises a narrow peak (Fig. \[fig:RUN1\_1.50\]), in contrast to e.g. the parameter density of the 3.0 s burst, which is much more spread out (Fig. \[fig:RUN1\_3.00\]).
Discussion and Conclusion {#sec:Discussion and Conclusion}
=========================
We have studied, from a theoretical perspective, the conditions under which magnetar bursts from a predefined localized active region or burst patch on the NS surface would give rise to a detectable phase-dependence. By adopting a straightforward input burst model, we were able to examine the changes in the observed bursts after they were modulated by the phase-dependent function $\kappa_*(\phi)$, which takes into account the effects of gravitational light bending and depends on the configuration of the system.
We found that the degree to which the inferred burst properties become phase dependent is strongly contingent on the duration of the bursts and geometry of the system; we find a stronger phase dependency of the burst properties for longer duration bursts and larger values of the angles $\chi$ and $\alpha$. The former is because longer bursts sample a wider range of photon trajectories and the latter and the latter is due to the fact that for larger values of $\chi$ and $\alpha$ the GR effects become more significant. Furthermore, the majority of observed bursts turn out to have $\tau/\tau_0<1$ and $T_{90}/T_{90,0}<1$, i.e. they rise faster and appear shorter than their input counterparts. Attempts to infer the properties of individual bursts with durations greater than $\sim 10-20$ % of the spin period should certainly take into account potential distortion due to phase-dependent effects.
Adopting a lognormal burst duration distribution that peaks at $\overline{T}_{90}=0.1$ s (as observed for well-sampled sources), from which we draw the input duration for each individual burst, we found that phase distributions of the parameters closely resembled those of Run 1, for which $T_{90}=0.15$ s. When considering a powerlaw distribution for the input burst amplitudes and burst duration of $T_{90}=1$ s, we observed a weak phase-dependency of the burst parameters and a considerable amount of scatter, which in turn is caused by the large fraction of low-amplitude input bursts, which are more affected by Poisson noise. We conclude that the observed distributions of burst properties from well-sampled sources are likely not strongly distorted due to phase-dependent effects, by virtue of being dominated by short bursts.
We studied the detectability of phase-dependence, using the most commonly-used measure (see Section \[sec:Overview of published burst phase-dependence analysis\]) whereby one concentrates on the phase occurrence of the burst peaks. In our setup all bursts originate at a specific small active region – in some respects the most extreme phase-dependent scenario. However rotational phase dependence of the peak occurrences was not always apparent. We found that one would require a minimum number of bursts for certain input burst properties and a given system configuration, to *guarantee* observing a phase dependence. Only in the case of the most extreme geometries, i.e. $\chi=\alpha>60^\circ$, does this approach the burst sample sizes that were examined in the literature, which range from tens to several hundred bursts. Studies that have not found a phase-dependence in the distribution of the burst peak occurrences as yet [e.g. @Savchenko2010; @Scholz2011; @Collazzi2015], might simply require a larger burst sample in order rule it out for certain geometries. For other geometries, however, it will never be possible to rule out the presence of a burst phase-dependence.
In our study we have considered only a restricted range of scenarios, where the emission region is tied to the stellar surface. One factor that we have not simulated in detail is that of any potential beaming of the burst emission. To offer brief insights for such influences, we show in Fig. \[fig:RUN1\_analytic\_beam\] the theoretical phase distributions of the observed burst properties in the case of beamed emission (Equation \[eq:beaming function\] with $\sigma_{\rm b}=\pi/6$) from a burst with $T_{90}=1.5$ s; the corresponding shape of $\kappa_*(\phi)$ is shown by the dashed curves in Fig. \[fig:kappa angles\]. We compare it to its isotropic counterpart and find that the phase-dependency of the burst properties will be enhanced in the presence of beaming.
The detectability of a burst phase-dependence depends strongly on the shape of $\kappa_*(\phi)$: the stronger the variation with $\phi$ the greater the modification to the input burst profiles. Introducing additional bursts patches or allowing for active regions to occur at a certain height above the surface, will cause the phase-dependence of $\kappa_*$ to decrease. A burst phase-dependence in those cases may then only become detectable if the emission is also strongly beamed.
In this paper we have not studied the method whereby a phase-dependence is searched for in the epoch folded photon times of arrival. This method can, and should, be subjected to the same level of scrutiny. An additional challenge with this method, however, is to determine a proper false alarm rate. Straightforwardly looking for deviations from uniformity of the times of arrival does not work, since a single burst already consists a significant departure. One must instead quantify the conditions under which one would detect a burst photon phase-dependence, even if the bursts originated at random locations on or above the NS surface. We defer this topic to future studies.
Acknowledgements {#acknowledgements .unnumbered}
================
CE and ALW acknowledge support from NOVA, the Dutch Top Research School for Astronomy. DH was partially supported by the Moore-Sloan Data Science Environment at New York University, and the James Arthur Postdoctoral Fellowship at New York University. DH acknowledges support from the DIRAC Institute in the Department of Astronomy at the University of Washington. The DIRAC Institute is supported through generous gifts from the Charles and Lisa Simonyi Fund for Arts and Sciences, and the Washington Research Foundation. We thank Matthew Baring for useful comments.
Parameter table
===============
Here we list brief descriptions of the simulation parameters and their associated symbols (see Table \[tab:sim\_pars\]).
Symbol Description
----------------------------- -----------------------------------------------
$n$ number of input bursts
$n_{\rm id}$ number of identified bursts
$n_{\rm min}$ number of bursts at which the p-value is
guaranteed to drop below the threshold
$R$ neutron star radius
$P$ neutron star rotation period
$\psi$ size of the burst patch
$\chi$ angle between NS axis of rotation and
the line-of-sight
$\alpha$ colatitude of the burst patch
$\zeta$ background rate
$\delta t$ light curve bin width
$b$ background level
$N$ number of emitted photons
$N_{\rm det}$ number of detected photons
$t_0$ burst peak time
$\phi_0$ burst peak phase occurrence
$\tau$ rise-time
$s$ skewness factor
$A$ burst amplitude
$\mathcal{N}_{\rm bins}$ number of time bins
$y^{\rm sig}$ maximum departure amplitude
$t_0^{\rm sig}$ time bin with $y^{\rm sig}$
$\Delta T$ time interval of an observed burst
$t_{\rm in}$, $t_{\rm out}$ limits of $\Delta T$
$\mu$ mean count level $\sim b$
$\mu^*$ running mean
$\Delta T_{\rm interval}$ time interval over which $\mu^*$ is estimated
$\Delta t_0$ difference input and best-fit burst peak
time: $\Delta t_0\equiv t^{\rm bf}-t_0$
$T_{90}$ burst duration
$\overline T_{90}$ mode of the duration distribution
$\sigma_{T_{90}}$ width of the duration distribution
$T_{90}^{\rm min}$ minimum burst duration
$T_{90}^{\rm max}$ maximum burst duration
$\Gamma$ powerlaw index of the amplitude
distribution
$A^{\rm min}$ minimum burst amplitude
$A^{\rm max}$ maximum burst amplitude
$\mathcal{N}_{\rm s}$ number of simulations
subscript ‘0’ input parameter
superscript ‘init’ initial guess
superscript ‘bf’ best-fit parameter
: A table of the simulation parameters appearing in sections \[sec:Simulations\] and \[sec:Results\].
\[tab:sim\_pars\]
\[lastpage\]
[^1]: E-mail:[email protected]
[^2]: @Dib2009 discuss the 4 bursts from AXP 1E 1048.1–5937 and note that only 3 of them occur near pulse maximum, whereas the fourth burst does not.
[^3]: No further details of the analysis procedure on the SGR 1806–20, such as the applied nominal threshold to determine significance, are given in the article. The phase dependence analysis procedure of the SGR 1900+14 data is also not described.
[^4]: Note that here we only consider the case where burst emission escapes from the system at the stellar surface [as it would from a trapped fireball, for example, due to the reduced scattering opacity close to the surface @Thompson1995]. It is for surface emission that the effects of GR will be most significant. We argue that bursts that occur high-up in the magnetosphere will be much less affected by the effects of GR or occultation of the star itself, and thus may exhibit weak to no phase-dependent properties. In effect we are considering the most optimistic case for the detection of phase-dependent effects.
[^5]: If no burst photons are detected, i.e. $N_{\rm det}=0$, we move on to the next burst \[step (II)\].
[^6]: If $N^{\rm init}<y_0^{\rm sig}$, we set $N^{\rm init}=(1+s^{\rm init})\tau^{\rm init}y_0^{\rm sig}/\delta t$.
|
---
abstract: 'We take into account the sigma meson cloud effect in the meson cloud model to calculate the distributions of light flavor sea quarks in the proton. Our calculation gives a better description of the data for ${\overline{d}}(x)/{\overline{u}}(x)$. We also provide a picture that the probability of finding a physical proton in a Fock state $\left|N\omega\right> $ is reasonable small with a smaller cutoff $\Lambda_{\omega}$.'
address:
- 'Department of Physics, Peking University, Beijing 100871, China'
- |
CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China\
Department of Physics, Peking University, Beijing 100871, China
author:
- Feng Huang
- 'Rong-Guang Xu'
- 'Bo-Qiang Ma'
title: |
Sigma Meson Cloud\
And Proton’S Light Flavor Sea Quarks
---
,
,
Sigma meson ,Light flavor sea quarks ,Meson cloud model ,Meson exchange model\
11.30.Hv ,12.39.-x ,13.60.-r ,14.40.-n
By now, it has been experimentally established that the light-flavor sea quarks ${\overline{d}}(x)$ and ${\overline{u}}(x)$ in the proton differ substantially [@NMC; @KA; @AB; @E866]. The violation of the Gottfried sum rule, first found by NMC [@NMC], indicated that $D\equiv\int_0^1[{\overline{d}}(x)-{\overline{u}}(x)]\,{{\mathrm d}}x =0.148\pm0.039$, and similar result for ${\overline{d}}(x)-{\overline{u}}(x)$ was obtained by HERMES [@KA]. In Drell-Yan experiments, ${\overline{d}}/{\overline{u}}$ was determined to be more than 2 at $x=0.18$ by NA51 [@AB] and the Bjorken-$x$ dependence (moment fraction dependence) of the ratio has been measured by E866/NuSea [@E866].
While no known symmetry requires ${\overline{d}}(x)$ equal to ${\overline{u}}(x)$, the large asymmetry was out of naive expectations. The sea of light flavor quark-antiquark pairs produced perturbatively from gluon splitting is flavor symmetric. Thus the large asymmetry requires a nonperturbative origin. It was first predicted by Thomas [@first], and later suggested by Henley and Miller [@first2], and Kumano and Londergan [@SK], that including the effects of virtual mesons (recognized as meson cloud model) can naturally account for the asymmetry. In previous articles applying meson cloud model to explain the light flavor sea asymmetry in the nucleon, a good description of the data was obtained for the distribution functions [@JS]. The dominant role is played by the pion, which provides that ${\overline{d}}(x)/{\overline{u}}(x)$ either increases monotonically with $x$ or turns back towards unity too slowly [@JS; @MA]. Many explanations have been applied to attack this problem [@AS; @WK; @WM; @NN], such as effects of $\Delta$ [@AS; @WK; @WM], the influence of the Pauli exclusion principle [@WM], adjustment of parameters [@NN], but none of these explanations provides a satisfactory description of the ratio ${\overline{d}}(x)/{\overline{u}}(x)$. However, Alberg, Henley and Miller [@omega] found that the inclusion of the effect of the isoscalar vector meson $\omega$, with a coupling constant $g_{\omega}^2/4\pi\approx8.1$, allows a good description of the data. As mentioned in that article, adding the $\sigma$ effect will likely improve the description of the data. Here, we follow this suggestion and take into account the sigma meson effect in the meson cloud model.
The existence of the $\sigma$ meson has been obscure for many years, while many theorists and experimentalists were searching for $\sigma$ which may play an important role in nuclear physics, because the $\sigma$ meson can provide reasonable middle-range nuclear force [@meson]. Sigma meson was first introduced as the chiral partner of pion in the linear representation of chiral symmetry, which is an important ingredient in modern hadron physics. It also plays an important role in spontaneous chiral symmetry breaking to understand the present spectroscopy of hadrons. However, it was not well established experimentally mainly due to the negative results of extensive analysis of the $\pi$-$\pi$ scattering phase shift. While recent reanalysis of the $\pi$-$\pi$ scattering phase shift [@OP] strongly suggested a pole of the $\sigma$-meson. Though we do not have a confirmed conclusion that the sigma meson exists as a real physical particle, it is sufficient for considering its effect to calculate ${\overline{d}}(x)-{\overline{u}}(x)$ and ${\overline{d}}(x)/{\overline{u}}(x)$ in the proton in the meson cloud model.
In the meson cloud model [@SK; @JS; @MA], the nucleon can be viewed as a bare nucleon (core) plus a series of baryon-meson Fock states which result from the fluctuation of nucleon to baryon plus meson $N\rightarrow BM$ (a bare baryon surrounded by a meson cloud). The model assumes that the life-time of a virtual baryon-meson Fock state is much longer than the interaction time in deep inelastic scattering (DIS) or Drell-Yan processes, thus the quark and anti-quark in the virtual baryon-meson Fock states can contribute to the parton distribution of the nucleon. These non-perturbation contribution can be calculated in a convolution between the fluctuation function, which describes the microscopic process $N\rightarrow BM$, and the quark (anti-quark) distribution of hadrons in the Fock states $\left|BM\right> $. Here we provide the usual formula. First the physical proton wave function is composed of the following Fock states $$\begin{aligned}
\left|P\right>=\sqrt{Z}\left|P\right>_{\tiny{\textrm{bare}}}+\sum_{BM}\int
{{\mathrm d}}y {{\mathrm d}}^{2}\textbf{k}_{\perp}\Psi_{BM}(y,\textbf{k}_{\bot}^{2})
\left|B(y,\emph{\textbf{k}}_{\bot}),M(1-y,-\emph{\textbf{k}}_{\bot})\right>\nonumber.\end{aligned}$$ Here $Z$ is the wave function renormalization constant, $\Psi_{BM}(y,\textbf{k}_{\bot}^{2})$ is the probability amplitude for finding a physical nucleon in a state consisting of a baryon $B$ with longitudinal moment fraction $y$ and transverse momentum $\emph{\textbf{k}}_{\bot}$, and a meson $M$ with longitudinal moment fraction $1-y$ and transverse momentum $-\emph{\textbf{k}}_{\bot}$. Therefore, the quark distribution functions $q(x)$ in the proton are given by $$\begin{aligned}
q(x)=Z q_{\tiny{\textrm{bare}}}(x)+\delta q(x)\;,\end{aligned}$$ with $$\begin{aligned}
\delta q(x)=\sum_{MB}[\int_x^1\frac{{{\mathrm d}}y}{y}f_{MB}(y)q_{M}(\frac{x}{y})+\int_x^1\frac{{{\mathrm d}}y}{y}f_{BM}(y)q_{B}(\frac{x}{y})]\;,\end{aligned}$$ and $$\begin{aligned}
f_{MB}(y) = f_{BM}(1-y)\;.\label{sym}\end{aligned}$$ As shown in [@WK], the wave function renormalization constant reads $$\begin{aligned}
Z=[1+\sum_{BM}\int_0^1f_{BM}(y){{\mathrm d}}y]^{-1}\simeq1-\sum_{BM}\int_0^1f_{BM}(y){{\mathrm d}}y\;.\end{aligned}$$ The number of each type of meson, $n_{M}$, is obtained by integrating $f_{MB}(y)$ over $y$. Then we have $Z=1-\sum_{M}n_M$. The splitting function is $$\begin{aligned}
f_{BM}(y)=\int_0^\infty|\Psi_{BM}(y,\textbf{k}_{\bot}^{2})|^{2}{{\mathrm d}}^{2}\textbf{k}_{\bot}\;.\end{aligned}$$ We use time-ordered perturbation theory (TOPT) in the infinite momentum frame (IMF) to calculate this function, which was given by [@JS] as $$\begin{aligned}
f_{BM}(y)=\frac{1}{4\pi^{2}} \frac{m_{N}m_{B}}{y(1-y)}
\frac{|G_{M}(y,\textbf{k}_{\bot}^2)|^{2}|V_{\mathrm{IMF}}|^{2}}{[m_N^2-M_{BM}^2(y,\textbf{k}_{\bot}^2)]^2}\;,\end{aligned}$$ where $$\begin{aligned}
M_{BM}^2(y,\textbf{k}_{\bot}^2)=\frac{m_B^2+\textbf{k}_{\bot}^2}{y}+\frac{m_M^2+\textbf{k}_{\bot}^2}{1-y}\end{aligned}$$ is the invariant mass of the final state, and an exponential form for the cutoff is [@WK; @meson] $$\begin{aligned}
G_{M}(y,\textbf{k}_{\bot}^2)=\exp(\frac{m_N^2-M_{BM}^2(y,\textbf{k}_{\bot}^2)}{2\Lambda_M^2})\;,\end{aligned}$$ which is required to respect Eq.(\[sym\]). Here $\Lambda_M$ is a cutoff parameter for each meson.
The vertex function $V_{\mathrm{IMF}}(y,\textbf{k}_{\bot}^2)$ depends on the effective interaction Lagrangian that describes the fluctuation process $N\rightarrow BM$. Here we use the meson exchange model for hadron production [@meson] and the linear sigma model [@book] to perform the calculation.
In a full calculation, we should include all kinds of mesons and baryons. While the probability of baryon-meson fluctuation should decrease with the invariant mass of the baryon-meson Fock state increasing, we can neglect the effects of Fock states with higher invariant mass. We just include specifically $\pi$, $\sigma$ and $\omega$ mesons with the nucleon here. It has been discussed that the effect of the $\rho$ meson and the intermediate $\Delta$ can also contribute a lot each. But the $\rho$ meson increases ${\overline{d}}(x)-{\overline{u}}(x)$, whereas the intermediate $\Delta$ decreases it, so these effects tend to cancel each other. And we also omit the effect of $\eta$ for its small coupling constant. Thus it is safe to focus on the effects of $\pi$, $\sigma$ and $\omega$ meson clouds here.
First, we focus on the sigma meson. From the Bonn meson-exchange model for the nucleon-nucleon interaction [@meson], the effective interaction Lagrangian for the scalar meson $\sigma$ is $\emph{L}=g{\overline{\psi}}\sigma\psi$. Thus we can get the vertex function for $\sigma$ to obtain the fluctuation function as $$\begin{aligned}
f_{N\sigma}(y)=\frac{g_{\sigma}^2}{8\pi^2} \frac{1}{y^2(1-y)}
\int_0^\infty {{\mathrm d}}\textbf{k}_{\bot}^2 |G_{\sigma}(y,\textbf{k}_{\bot}^2)|^2
\frac{m_N^2(1-y)^2+\textbf{k}_{\bot}^2}{[m_N^2-M_{N\sigma}^2(y,\textbf{k}_{\bot}^2)]^2}\;.\end{aligned}$$Now, we should know the coupling constant $g_\sigma$, the mass $m_\sigma$ and the cutoff $\Lambda_\sigma$.
As mentioned above, sigma meson is introduced to the hadron physics as the chiral partner from the view of chiral symmetry. In the $SU(2)$ linear $\sigma$ model [@book], the effective Lagrangian is $$\begin{aligned}
\emph{L}=g{\overline{\psi}}(\sigma+\gamma_5\mathbf{\tau}\cdot\mathbf{\pi})\psi\;.\end{aligned}$$ Even though we do not have direct information about $g_\sigma$, $g_\sigma = g_\pi$ is imposed by the linear $\sigma$ model. According to all kinds of calculations to fit different experimental data [@meson; @RJ; @range], the coupling constant $g_\sigma^2/4\pi=g_\pi^2/4\pi=13.6$ is taken in our numerical computations. In 2002 Particle Data Group (PDG), $f_0(600)$ or $\sigma$ appears below $1~\textrm{GeV}$ mass region. Recently it was found that $m_\sigma=585\pm20~\textrm{MeV}$ by a reanalysis of $\pi\pi$ scattering phase shift [@OP]. Similarly, by analysing $\pi\pi$ production processes [@OP], we obtain $m_\sigma=580\pm30~\textrm{MeV}$ for *pp* central collision. Thus, we will set the mass as $600~\textrm{MeV}$ in the following calculations. It is generally believed that the cutoff $\Lambda_\sigma$ value is in a range around $1~\textrm{GeV}$ as a phenomenological parameter. Here we will examine the effect of varying $\Lambda_\sigma$ value in the range $1.0<\Lambda_\sigma<1.3~\textrm{GeV}$.
We have made some detailed discussions about the $\sigma$ meson cloud above. The $\pi$ meson cloud effect has been discussed enough in previous articles [@first2; @JS; @omega; @HN], here we just set the parameters as $m_\pi=139~\textrm{MeV}$ and $\Lambda_\pi=(0.88\pm0.05)~\textrm{GeV}$, which are chosen to reproduce the integrated asymmetry $D\equiv\int_0^1[{\overline{d}}(x)-{\overline{u}}(x)]\,dx$ [@NMC; @KA] as shown in Fig. \[fig:d-u\].
![\[fig:d-u\] Comparison of our meson cloud model with data [@KA; @E866] for ${\overline{d}}(x) - {\overline{u}}(x)$. The thick solid curve is for $\Lambda_\pi = 0.88~\textrm{GeV}$, which gives a best description here.](d-u.EPS)
Moreover, we should discuss the functions of $q_M(x)$ and $q_B(x)$. Those for the nucleon and pion are measured, but the quark distribution functions of the $\sigma$ meson and $\omega$ meson are unknown. It has been traditional [@same] to assume that the structure function of the $\rho$ and $\pi$ are the same. It is suggested by the bag model [@omega] that the structure functions of the $\omega$, $\rho$, and $\pi$ are the same. Due to that the $\sigma$ meson operator is adopted as $\frac{1}{\sqrt{2}}({\overline{u}}u+{\overline{d}}d)$, there is some reason to assume that the quark distribution functions of the $\sigma$ meson is equal to that of the $\pi$. Thus we use [@dis] $$\begin{aligned}
xq_{\tiny{\textrm{V}}}(x)&=&0.99x^{0.61}(1-x)^{1.02}\nonumber,\\
xq_{\tiny{\textrm{sea}}}(x)&=&0.2(1-x)^{5.0}\end{aligned}$$for the valence and sea quark distribution functions of all mesons considered here.
![\[fig:HN\] Comparison of a harder bare nucleon sea quarks (thick solid line) with a traditional bare nucleon sea quarks (dashed line). The thin solid line is only the $\pi$ contribution to the ratio ${\overline{d}}(x)/ {\overline{u}}(x)$.](HN.EPS)
The bare nucleon sea is traditionally parametrized as [@N] $$\begin{aligned}
x{\overline{q}}_{\tiny{\textrm{bare}}}(x)&=&0.11(1-x)^{15.8}\nonumber,\\
{\overline{q}}_{\tiny{\textrm{bare}}}=u_{\tiny{\textrm{sea}}}&=&{\overline{u}}_{\tiny{\textrm{sea}}}=d_{sea}={\overline{d}}_{\tiny{\textrm{sea}}}\;,\end{aligned}$$which is recognized as Holtmann’s parametrization of the bare nucleon symmetric sea. While Alberg and Henley [@HN] used a harder distribution for the bare sea quarks of the form found in the determination of the gluon distribution [@gluon] $$\begin{aligned}
x{\overline{q}}_{\tiny{\textrm{bare}}}(x)=0.0124x^{-0.36}(1-x)^{3.8}\;.\end{aligned}$$Their calculations indicate that the harder distribution for the bare sea quarks gives a better description of ${\overline{d}}(x)/{\overline{u}}(x)$ as shown in Fig. \[fig:HN\]. In the following calculations, we will use only the harder bare nucleon sea. Also we can see that the change is not enough to explain the data, which guarantees the necessary to consider other flavor symmetric contribution. The inclusion of isoscalar meson $\omega$ with a reasonable coupling constant produces a similar improvement in agreement between theory and experiment [@omega], so does the inclusion of isoscalar meson $\sigma$.
![\[fig:ratio2\] Comparison of our meson cloud model with data [@E866] for ${\overline{d}}(x)/{\overline{u}}(x)$. The thick solid curve shows our result only considering the $\pi$ meson contribution. The shadowed area shows the effect of adding $\sigma$ meson and varying $\Lambda_\sigma$ value in the range $1.0<\Lambda_\sigma<1.3 \textrm{GeV}$.](ratio2.EPS)
![\[fig:600\] Comparison of our meson cloud model with data [@E866] for ${\overline{d}}(x)/{\overline{u}}(x)$. The thick solid curve shows our result if contributions from the $\sigma$ and $\omega$ clouds are omitted. The dashed curve shows adding $\omega$ cloud contribution with $\Lambda_\omega=1.0~\textrm{GeV}$. The shadowed area shows the contributions of three mesons, and the effect of varying $\Lambda_\omega$ value in the range $1.0<\Lambda_\omega<1.5 \textrm{GeV}$. ](600,1000.EPS)
![\[fig:600\] Comparison of our meson cloud model with data [@E866] for ${\overline{d}}(x)/{\overline{u}}(x)$. The thick solid curve shows our result if contributions from the $\sigma$ and $\omega$ clouds are omitted. The dashed curve shows adding $\omega$ cloud contribution with $\Lambda_\omega=1.0~\textrm{GeV}$. The shadowed area shows the contributions of three mesons, and the effect of varying $\Lambda_\omega$ value in the range $1.0<\Lambda_\omega<1.5 \textrm{GeV}$. ](600,1300.EPS)
The results of our calculations for light flavor sea quarks are shown in Fig. \[fig:ratio2\] and Fig. \[fig:600\]. In Fig. \[fig:ratio2\], we examine the effect of varying $\Lambda_\sigma$ value in the range $1.0<\Lambda_\sigma<1.3~\textrm{GeV}$. The thin solid curve on the upper of the shadowed area is the result of setting $\Lambda_\sigma=1.0~\textrm{GeV}$, while the bottom curve corresponds to $\Lambda_\sigma=1.3~\textrm{GeV}$. It is clear that adding sigma meson effect here gives really better description of the data. Our calculations illustrate that the larger value of $\Lambda_{\sigma}$ tends to give small values of the ratio ${\overline{d}}/{\overline{u}}$ and that decreasing $\Lambda_{\sigma}$ causes the maximum value of ${\overline{d}}/{\overline{u}}$ to be larger and to appear at a higher value of $x$. In Fig. \[fig:600\], we illustrate the effects from three mesons with different cutoff values. As calculated in [@omega], we set $g_{\omega}^2/4\pi$ as $8.1$, which is from fitting to dispersion relation descriptions of forward nucleon-nucleon scattering [@meson; @range; @coupling]. The shadowed area shows the results of varying $\Lambda_\omega$ value in the range $1.0<\Lambda_\omega<1.5~\textrm{GeV}$ with $\Lambda_\sigma$ at $1.0~\textrm{GeV}$ on the left and $1.3~\textrm{GeV}$ on the right. The dashed curves omit the $\sigma$ contribution and only consider the value of $\Lambda_\omega$ as $1.0~\textrm{GeV}$. When the value of $\Lambda_\omega$ is $1.5~\textrm{GeV}$, the ratio ${\overline{d}}(x)/{\overline{u}}(x)$ hardly changes with adding $\sigma$ meson cloud, which is consistent with [@omega], in which $\Lambda_\omega$ is set as $1.5~\textrm{GeV}$ to obtain a good description of data by considering only $\pi$ and $\omega$ effects. In order to have a deep understanding of this, we present the numbers of the related mesons $n_M$ in the proton with the different cutoff values in the two works, which are shown in Table 1. The large value of cutoff leads to a large value of meson number in the proton. Apparently, the too large $n_\omega$ resulted from the large value of $\Lambda_\omega=1.5~\textrm{GeV}$ should not be the real picture of the proton. For this reason, $\Lambda_\omega$ favors the small value of the examined range. Adding $\sigma$ cloud effect modifies the parameters describing the omega-nucleon interaction. And such a revision provides a picture that the probability of finding a physical proton in a Fock state $\left|N\omega\right> $ is reasonably small. We also realize that the same rule holds true for the $\sigma$-meson from the comparison of the two different values of $\Lambda_\sigma$.
[|c|c|c|c|c|c|c|]{}\
& $\Lambda_{\pi} $& $\Lambda_{\sigma}$ & $\Lambda_{\omega}$ & $n_{\pi}$ & $n_{\sigma}$ & $n_{\omega}$\
& $0.88$ & $1.0\sim1.3$ & $1.0\sim1.5$ & $0.175$ & $0.023\sim 0.078$ & $0.063\sim 0.671$\
& $0.83$ & no sigma & $1.5$ & $0.150$ & no sigma &0.671\
In summary, the inclusion of the $\sigma$ meson cloud effect brings a better description for ${\overline{d}}(x)/{\overline{u}}(x)$ in the proton and also provides a picture of a reasonable small $n_\omega$ in the proton. Thus we conclude that the inclusion of both isoscalar meson sigma and omega cloud effects can give an improved description of the experimental data for ${\overline{d}}(x)/{\overline{u}}(x)$ in the meson cloud model.
This work is partially supported by National Natural Science Foundation of China under Grant Numbers 10025523 and 90103007.
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|
---
abstract: 'We prove the existence of a canonical ‘higher Kolyvagin derivative’ homomorphism between the modules of higher rank Euler systems and higher rank Kolyvagin systems, as has been conjectured to exist by Mazur and Rubin. This homomorphism exists in the setting of $p$-adic representations that are free with respect to the action of a Gorenstein order $\mathcal{R}$ and, in particular, implies that higher rank Euler systems control the $\mathcal{R}$-module structures of Selmer modules attached to the representation. We give a first application of this theory by considering the (conjectural) Euler system of Rubin-Stark elements.'
address:
- 'King’s College London, Department of Mathematics, London WC2R 2LS, U.K.'
- 'Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-Ku, Tokyo, 153-8914, Japan'
- |
Osaka City University, Department of Mathematics, 3-3-138 Sugimoto\
Sumiyoshi-ku\
Osaka\
558-8585, Japan
author:
- 'David Burns, Ryotaro Sakamoto and Takamichi Sano'
title: |
On the theory of higher rank\
Euler, Kolyvagin and Stark systems, II
---
Introduction
============
Background to the problem
-------------------------
Ever since its introduction by Kolyvagin [@kolyvagin], the theory of Euler systems has played a vital role in the proof of many celebrated results concerning the structure of Selmer groups of $p$-adic representations over number fields.
In an attempt to axiomatise, and extend, the use of Euler systems, Mazur and Rubin [@MRkoly] developed an associated theory of ‘Kolyvagin systems’ and showed both that Kolyvagin systems controlled the structure of Selmer groups and that Kolyvagin’s ‘derivative operator’ gave rise to a canonical homomorphism between the modules of Euler and Kolyvagin systems that are associated to a given representation.
In this way, it became clear that Kolyvagin systems play the key role in obtaining structural results about Selmer groups and that the link to Euler systems is pivotal for the supply of Kolyvagin systems that are related to the special values of $L$-series.
For many representations, however, families of cohomology classes (such as Euler or Kolyvagin systems) are not themselves sufficient to control Selmer groups and in such ‘higher rank’ cases authors have considered various collections of elements in higher exterior powers of cohomology groups.
The theory of ‘higher rank Euler systems’ has in principle been well-understood for some time by now, with the first general approach being described by Perrin-Riou in [@PR] after significant contributions were made by Rubin in an important special case (related to Stark’s Conjecture) in [@rubinstark].
In addition, a general method was introduced in [@rubinstark §6] whereby higher rank Euler systems could be used to construct, in a non-canonical way, classical (rank one) Euler or Kolyvagin systems to which standard techniques could then be applied.
However, whilst this ‘rank reduction’ method has since been used both often and to great effect, notably by Perrin-Riou in [@PR] and by Büyükboduk in [@Buyuk] and [@Buyuk2], it is intrinsically non-canonical and also requires several auxiliary hypotheses (such as, for example, the validity of Leopoldt’s Conjecture in the settings considered in [@rubinstark] and [@Buyuk]) that can themselves be very difficult to verify.
In an attempt to address these deficiencies, Mazur and Rubin [@MRselmer] have developed a theory of ‘higher rank Kolyvagin systems’, and an associated notion of ‘higher rank Stark systems’ (these are collections of cohomology classes generalizing the units predicted by Stark-type conjectures), and showed that, under suitable hypotheses, such systems can be used to control Selmer groups.
However, the technical difficulties encountered when computing with higher exterior powers meant that the theory developed in [@MRselmer] was insufficient in the following respects.
- Coefficient rings were restricted to be either principal artinian local rings or discrete valuation rings, whilst dealing with more general coefficient rings is essential if one is to deal effectively with questions arising, for example, in either deformation theory or Galois module theory.
- More importantly, whilst Mazur and Rubin conjectured the existence of a canonical link between the theories of higher rank Euler and Kolyvagin (or Stark) systems, they were unable to shed any light on the precise nature of this relationship (which they described as ‘mysterious’).
Overview of the solution
------------------------
The first of the above problems was resolved independently by the first and third authors in [@sbA] and by the second author in [@sakamoto], a key part of the solution being the introduction of ‘exterior power biduals’ as a functorially stronger version of exterior powers.
Building on these earlier articles, [*we shall now resolve the second problem, and hence prove the conjecture of Mazur and Rubin, by constructing a canonical ‘higher Kolyvagin derivative’ map between the modules of Euler and Kolyvagin systems of any given rank*]{}.
We shall construct this map in the setting of $p$-adic representations that are free with respect to the action of an arbitrary Gorenstein order ${\mathcal{R}}$, as one would expect to suffice for applications to deformation theory. In addition, by combining the construction with results from [@sbA], we are able to deduce that, under natural hypotheses, higher rank Euler systems determine all of the higher Fitting ideals over ${\mathcal{R}}$ of the relevant Selmer modules.
In this regard we also note that obtaining concrete structural information about natural arithmetic modules such as ideal class groups, Tate-Shafarevic groups and Selmer groups with respect to coefficient rings that are not regular is a notoriously difficult problem and, despite an extensive literature discussing special cases (for recent examples see for instance Kurihara [@kuri], Greither and Popescu [@GreitherPopescu] and Greither and Kučera [@GK] and the references contained therein), there has not hitherto been any general approach to this problem.
There are two further differences between our approach (using exterior power biduals) and that of earlier articles that seem worthy of comment.
Firstly, we are able to show that, under standard hypotheses, [*the module of higher rank Kolyvagin systems is canonically isomorphic to the corresponding module of higher rank Stark systems and is therefore free of rank one over the coefficient ring ${\mathcal{R}}$*]{}. This key fact allows us avoid the problem highlighted by Mazur and Rubin in [@MRselmer Rem. 11.9] that not all higher rank Kolyvagin systems defined in terms of higher exterior powers are, in their terminology, ‘stub’ systems.
Secondly, as a key step in our construction of the higher Kolyvagin derivative operator, we shall develop a variant of the (non-canonical) rank-reduction methods employed in [@Buyuk], [@PR] and [@rubinstark] that is both canonical in nature and also avoids difficult auxiliary hypotheses that are used in these earlier articles.
In a further article it will be shown that all of the key aspects of the theory developed here extend naturally to the analogous Iwasawa-theoretic setting.
In view of the range of existing applications of the classical theory of Euler and Kolyvagin systems, it thus seems plausible that the very general theory developed here will have significant applications in the future.
For the moment, however, to give an early indication of the usefulness of this approach we shall just restrict to the setting that was originally considered by Rubin in [@rubinstark].
In this setting, we find that our methods allow us, in a straightforward fashion, to extend, refine and remove any hypotheses concerning the validity of Leopoldt’s Conjecture from the results of Rubin in [@rubinstark] and of Büyükboduk in [@Buyuk].
A summary of results
--------------------
For the reader’s convenience, we now give a brief summary of the main results of this article. For simplicity, we shall only discuss a special case of the general setting considered in later sections. In addition, we shall omit stating explicitly hypotheses that are standard in the theory of Euler, Kolyvagin and Stark systems (since, in each case, they are made precise by the indicated results in later sections).
We thus fix an odd prime $p$, a number field $K$ and a Galois representation $T$ over a Gorenstein ${\mathbb{Z}}_p$-order ${\mathcal{R}}$ that is endowed with a continuous action of the absolute Galois group of $K$. We also fix a power $M$ of $p$ and set $A:=T/MT$ and $R:={\mathcal{R}}/(M)$. For each natural number $r$ we write ${\rm ES}_r(T)$, respectively ${\rm KS}_r(A)$ and ${\rm SS}_r(A)$, for the modules of Euler systems of rank $r$ for $T$, respectively of Kolyvagin and Stark systems of rank $r$ for $A$, that are defined in §\[euler sys sec 1\], respectively in §\[defkoly\] and §\[defstark\].
Our main result is then the following.
\[thm1\] Under standard hypotheses, there exists a canonical ‘higher Kolyvagin derivative’ homomorphism $${\mathcal{D}}_r: {\rm ES}_r(T) \to {\rm KS}_r(A).$$ Under certain mild additional hypotheses, this homomorphism is surjective.
By this result, one can associate a canonical Kolyvagin system $$\kappa(c):={\mathcal{D}}_r(c)$$ in ${\rm KS}_r(A)$ to every Euler system $c$ in ${\rm ES}_r(T)$.
A key aspect of the proof of Theorem \[thm1\] is the development of a suitable ‘rank-reduction’ method by which consideration is restricted to the case $r=1$ where the result can be established by the existing methods of Mazur and Rubin.
We shall also further develop the general theory of higher rank Kolyvagin systems in order to prove the next result.
In the sequel, for each commutative noetherian ring $\Lambda$, each non-negative integer $i$ and each finitely generated $\Lambda$-module $M$ we write ${\rm Fitt}_\Lambda^i(M)$ for the $i$-th Fitting ideal of $M$.
\[thm2\] Under standard hypotheses, the following claims are valid.
- The module ${\rm KS}_r(A)$ of Kolyvagin systems of rank $r$ is free of rank one over $R$.
- For each system $\kappa$ in ${\rm KS}_r(A)$ and each non-negative integer $i$ one has $$I_i(\kappa)\subseteq {\rm Fitt}_R^i({\rm Sel}(A)),$$ where $I_i(\kappa)$ is a canonical ideal associated with $\kappa$ (see Definition \[koly ideal\]) and ${\rm Sel}(A)$ is a natural (dual) Selmer module for $A$ (which is denoted by $H^1_{{\mathcal{F}}^\ast}(K,A^\ast(1))^\ast$ in Theorem \[main\]).
- If $R$ is a principal ideal ring and $\kappa$ is a basis of ${\rm KS}_r(A)$, then the inclusion in claim (ii) is an equality.
Upon combining Theorems \[thm1\] and \[thm2\], we immediately obtain the following result.
\[thm3\] The following claims are valid.
- Under standard hypotheses, for each Euler system $c$ in ${\rm ES}_r(T)$ and each non-negative integer $i$ one has $I_i(\kappa(c))\subseteq {\rm Fitt}_R^i({\rm Sel}(A)).$
- Under certain mild additional hypotheses, and if $R$ is a principal ideal ring, then for each non-negative integer $i$ one has $\langle I_i(\kappa(c)) \mid c \in {\rm ES}_r(T ) \rangle_R = {\rm Fitt}_R^i({\rm Sel}(A)).$
As a key step in the proof of Theorem \[thm2\], we must develop the theory of Stark systems in order to prove the next result.
\[thm5\] Under standard hypotheses, the following claims are valid.
- The module ${\rm SS}_r(A)$ of Stark systems of rank $r$ is free of rank one over $R$.
- For each system $\epsilon$ in ${\rm SS}_r(A)$ and each non-negative integer $i$ one has $$I_i(\epsilon)\subseteq {\rm Fitt}_R^i({\rm Sel}(A)),$$ where $I_i(\epsilon)$ is a canonical ideal associated with $\epsilon$ (see Definition \[stark ideal\]).
- If $\epsilon$ is a basis of ${\rm SS}_r(A)$, then the inclusion in claim (ii) is an equality.
To relate Stark and Kolyvagin systems we prove (in §\[sec regulator\]) that there exists a canonical ‘regulator’ homomorphism of $R$-modules $${\rm Reg}_r: {\rm SS}_r(A) \to {\rm KS}_r(A).$$ [*We are able to prove that this map is bijective, and as a consequence, that the module ${\rm KS}_r(A)$ is free of rank one*]{} (see Theorem \[main\](i)). By combining this fact with Theorem \[thm5\], we are then able to prove Theorem \[thm2\].
We remark here that, for each system $\epsilon$ in ${\rm SS}_r(A)$, it is natural to expect that for each non-negative integer $i$ one has $I_i(\epsilon)=I_i({\rm Reg}_r(\epsilon))$. However, this seems to be difficult to prove and at present we have only verified it in the case $R$ is a principal ideal ring (which leads to Theorem \[thm2\](iii)). For more details concerning this issue see Remark \[difficulty remark2\].
By simultaneously considering the representations $T/p^m T$ for all natural numbers $m$, one can define Kolyvagin and Stark systems ‘over ${\mathcal{R}}$’ by setting $${\rm KS}_r(T):=\varprojlim_m {\rm KS}_r(T/p^mT) \,\,\text{ and }\,\, {\rm SS}_r(T):=\varprojlim_m {\rm SS}_r(T/p^mT).$$ In this way we obtain analogues of Theorems \[thm2\], \[thm3\] and \[thm5\] for the Selmer modules of $T$ (see Theorem \[thm koly’\], Corollary \[main cor\] and Theorem \[thm stark’\] respectively).
To give a straightforward application of the general theory, we consider, for each one-dimensional $p$-adic character $\chi$ of the absolute Galois group of $K$, a certain twisted form $T_\chi$ of the representation ${\mathbb{Z}}_p(1)$. Since in this case the (dual) Selmer module coincides with the $\chi$-isotyic component of a suitable ideal class group, we obtain the following result.
\[thm 6\] Let $\chi$ be a one-dimensional $p$-adic character of the absolute Galois group of $K$ that is of finite prime-to-$p$ order. Let $L$ be the field fixed by $\ker(\chi)$. Assume that all archimedean places of $K$ split completely in $L$, that no $p$-adic place of $K$ splits completely in $L$, that $\chi$ is neither trivial nor equal to the Teichmüller character, and that either $p>3$ or $\chi^2$ is not equal to the Teichmüller character.
Let $r$ be the number of archimedean places of $K$ and set $\mathcal{O} := {\mathbb{Z}}_p[\operatorname{im}(\chi)]$. Then for each non-negative integer $i$ one has $$\langle I_i(\kappa(c)) \mid c \in {\rm ES}_r(T_\chi) \rangle_{\mathcal{O}} =
{\rm Fitt}_{\mathcal{O}}^i(({\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\rm Cl}(\mathcal{O}_L))^\chi),$$ where ${\rm Cl}(\mathcal{O}_L)$ is the ideal class group of $L$.
We recall that, in this setting, the Rubin-Stark conjecture predicts the existence of a canonical Euler system in ${\rm ES}_r(T_\chi)$ comprising ‘Rubin-Stark elements’ that are explicitly related to the values at zero of $r$-th derivatives of Dirichlet $L$-functions.
Finally, we note that, under mild additional hypotheses, a slightly more careful application of the methods used to prove Theorem \[thm 6\] shows that for abelian extensions $L$ of $K$ the higher Fitting ideals of ${\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\rm Cl}(\mathcal{O}_L)$ as a ${\mathbb{Z}}_p[\operatorname{Gal}(L/K)]$-module are determined by Euler systems of rank $r$ for induced forms of the representation ${\mathbb{Z}}_p(1)$. (For brevity, however, this result will be discussed in a separate article.)
Organization
------------
In a little more detail, the basic contents of this article is as follows. In §\[ext bidual sec\] we review the definitions and basic properties of exterior power biduals and then establish several new functorial properties that will play a crucial role in later sections. In §\[pre\] we establish various preliminary results concerning Selmer structures and Galois cohomology groups over zero-dimensional Gorenstein rings. In §\[stark sys sec\] we review the main results on higher rank Stark systems and Selmer groups that were established in our earlier articles [@sbA] and [@sakamoto] and also extend these results to the setting of representations over Gorenstein orders. In §\[koly sys sec\] we develop a theory of higher rank Kolyvagin systems in the same degree of generality and, in particular, show that under standard hypotheses, the modules of Stark systems and Kolyvagin systems (of any given rank) are both canonically isomorphic and free of rank one over the relevant ring of coefficients. In §\[euler sys sec\] we construct a canonical higher rank ‘Kolyvagin-derivative’ homomorphism between the module of Euler systems of any given rank and the module of Kolyvagin systems (defined with respect to the canonical Selmer structure) of the same rank. This is the key result of this article and, as an essential part of its proof, we establish precise links to the corresponding situation in rank one. Finally, in §\[app sec\] we show that our approach leads directly to new results concerning the Galois structure of ideal class groups.
Some general notation
---------------------
In this article, $K$ always denotes a (base) number field (that is, a finite degree extension of ${\mathbb{Q}}$). We fix an algebraic closure $\overline {\mathbb{Q}}$ of ${\mathbb{Q}}$, and every algebraic extension of ${\mathbb{Q}}$ is regarded as a subfield of $\overline {\mathbb{Q}}$. For a positive integer $m$, let $\mu_m$ denote the group of $m$-th roots of unity in $\overline {\mathbb{Q}}$. For any field $E$, we denote the absolute Galois group of $E$ by $G_E$. For each place $v$ of $K$, we fix a place $w$ of $\overline {\mathbb{Q}}$ lying above $v$, and identify the decomposition group of $w$ in $G_K$ with $G_{K_v}$. For a finite extension $F/K$, the ring of integers of $F$ is denoted by ${\mathcal{O}}_F$. For a finite set $\Sigma$ of places of $K$, we denote by $\Sigma_F$ the set of places of $F$ which lie above a place in $\Sigma$. The ring of $\Sigma_F$-integers of $F$ is denoted by ${\mathcal{O}}_{F,\Sigma}$. The set of archimedean places (resp. $p$-adic places) of $F$ is denoted by $S_\infty(F)$ (resp. $S_p(F)$). We denote the set of places of $K$ which ramify in $F$ by $S_{\rm ram}(F/K)$.
Non-archimedean places (or ‘primes’) of $K$ are usually denoted by ${\mathfrak{q}}$. The Frobenius element of ${\mathfrak{q}}$ is denoted by ${\rm Fr}_{\mathfrak{q}}$.
For a continuous $G_K$-module $A$, we denote the set of places of $K$ at which $A$ is ramified by $S_{\rm ram}(A)$. Suppose that there is a finite set $S$ of places of $K$ such that $$S_\infty(K)\cup S_p(K) \cup S_{\rm ram}(A) \subseteq S.$$ Let $K_S$ denote the maximal Galois extension of $K$ unramified outside $S$. Then we can consider Galois cohomology groups $$H^i({\mathcal{O}}_{K,S},A):=H^i(K_S/K,A).$$ If $A$ is a $p$-adic representation, then one can define, for each place $v $ of $K$, a canonical (so-called) ‘finite’ local condition $$H^1_f(K_v, A) \subseteq H^1(K_v,A)$$ (see [@R §1.3], for example). When $v \notin S$, this is the unramified cohomology $H^1_{\rm ur}(K_v,A)$, which is defined by the kernel of the restriction to the inertia group. We set $$H^1_{/f}(K_v,A):=H^1(K_v,A)/H^1_f(K_v,A).$$ More generally, for each index $\ast$ we set $H^1_{/\ast}(K_v,A):=H^1(K_v,A)/H^1_\ast(K_v,A).$
Exterior power biduals {#ext bidual sec}
======================
In this section we quickly review the basic theory of exterior power biduals and then prove several new results that will be very useful in the sequel.
At the outset we fix a commutative ring $R$. [*All rings are assumed to be noetherian*]{}. For each $R$-module $X$ we set $$X^\ast:=\operatorname{Hom}_R(X,R).$$ For each subset $\mathcal{X}$ of $X$ we write $\langle \mathcal{X}\rangle_R$ for the $R$-submodule of $X$ that is generated by $\mathcal{X}$.
If $X$ is finitely presented, then for each non-negative integer $i$ we write ${\rm Fitt}_R^i(X)$ for the $i$-th Fitting ideal of $X$ (as discussed by Northcott in [@north]).
Definition and basic properties
-------------------------------
We first recall the basic definitions.
For each $R$-module $X$, each positive integer $r$ and each map $\varphi$ in $X^\ast$, there exists a unique homomorphism of $R$-modules $${{\bigwedge}}_R^r X \to {{\bigwedge}}_R^{r-1} X$$ with the property that $$x_1\wedge\cdots\wedge x_r \mapsto \sum_{i=1}^{r} (-1)^{i+1} \varphi(x_i) x_1\wedge\cdots\wedge x_{i-1}\wedge x_{i+1} \wedge \cdots \wedge x_r$$
for each subset $\{x_i\}_{1\le i\le r}$ of $X$. By abuse of notation, we shall also denote this map by $\varphi$. For non-negative integers $r$ and $s$ with $r \leq s$, this construction induces a natural homomorphism $${{\bigwedge}}_R^r X^\ast \to \operatorname{Hom}_R\left({{\bigwedge}}_R^s X, {{\bigwedge}}_R^{s-r} X\right); \ \varphi_1\wedge \cdots \wedge \varphi_r \mapsto \varphi_r \circ \cdots \circ \varphi_1.$$ Here $\bigwedge_R^r X^\ast$ means $\bigwedge_R^r (X^\ast)$. (We often use such an abbreviation.) We use this map to regard any element of ${{\bigwedge}}_R^r X^\ast$ as an element of $\operatorname{Hom}_R({{\bigwedge}}_R^s X, {{\bigwedge}}_R^{s-r} X)$.
\[def exterior bidual\] For any non-negative integer $r$, we define the ‘$r$-th exterior bidual’ of $X$ to be the $R$-module obtained by setting $${{\bigcap}}_R^r X:=\left({{\bigwedge}}_R^r X^\ast \right)^\ast.$$
Note that ${\bigcap}_R^1 X= X^{\ast \ast}$. So, if $X$ is reflexive, i.e. the canonical map $$\begin{aligned}
\label{2 dual}
X \to X^{\ast \ast}; \ x\to (\varphi \mapsto \varphi(x))\end{aligned}$$ is an isomorphism, then one can regard ${\bigcap}_R^1 X=X$. In practice, we usually consider exterior biduals of reflexive modules. (The fact that any modules over self-injective rings (in other words, zero-dimensional Gorenstein rings) are reflexive is often used in this paper.)
Note also that there is a canonical homomorphism $$\xi_X^r: {{\bigwedge}}_R^r X \to {{\bigcap}}_R^r X; \ x \mapsto (\Phi \to \Phi(x)),$$ which is neither injective nor surjective in general. However, if $X$ is a finitely generated projective $R$-module, then one can show that $\xi_X^r$ is an isomorphism.
For non-negative integers $r$, $s$ with $r\leq s$ and $\Phi \in {{\bigwedge}}_R^r X^\ast$, define a homomorphism $$\begin{aligned}
\label{map bidual}
{{\bigcap}}_R^s X \to {{\bigcap}}_R^{s-r} X\end{aligned}$$ as the $R$-dual of $${{\bigwedge}}_R^{s-r} X^\ast \to {{\bigwedge}}_R^{s} X^\ast ; \ \Psi \mapsto \Phi\wedge \Psi.$$ We denote the map (\[map bidual\]) also by $\Phi$, by abuse of notation. One can check that the following diagram is commutative: $$\xymatrix{
{{\bigwedge}}_R^s X \ar[r]^-{\Phi} \ar[d]_{\xi_X^s} &
{{\bigwedge}}_R^{s-r}X \ar[d]^{\xi_X^{s-r}}
\\
{{\bigcap}}_R^s X \ar[r]^-{\Phi} & {{\bigcap}}_R^{s-r} X.
}$$ We now recall two results from [@sbA] and [@sakamoto] that we will frequently use in the sequel.
\[prop injective\]
- Let $\iota: X \to Y$ be an injective homomorphism of $R$-modules for which the group ${\rm Ext}_R^1({\rm coker}( \iota), R)$ vanishes. Then for each $r\ge 0$ the homomorphism $${{\bigcap}}_R^r X \hookrightarrow {{\bigcap}}_R^r Y$$ that is naturally induced by $\iota$ is injective.
- Suppose that we have an exact sequence of $R$-modules $$Y \xrightarrow{\bigoplus_{i=1}^s \varphi_i} R^{\oplus s} \to Z \to 0.$$ If $Y$ is free of rank $r+s$, then ${\rm Fitt}_R^0(Z)$ is generated over $R$ by the set $$\left\{ {\rm im}( F )\ \middle| \ F \in {\rm im}\left( {{\bigcap}}_R^{r+s} Y \xrightarrow{{{\bigwedge}}_{1\leq i \leq s} \varphi_i} {{\bigcap}}_R^{r} Y \right) \right\}.$$
\[prop self injective\] Suppose that $R$ is self-injective, i.e. $R$ is injective as an $R$-module, and that we have an exact sequence of $R$-modules $$0 \to X \to Y \stackrel{\bigoplus_{i=1}^s \varphi_i}{\to} R^{\oplus s},$$ where $s$ is a positive integer. Then, for every non-negative integer $r$, we have $${\rm im}\left({{\bigwedge}}_{1\leq i \leq s} \varphi_i: {{\bigcap}}_R^{r+s} Y \to {{\bigcap}}_R^{r} Y \right)\subseteq {{\bigcap}}_R^{r} X.$$ Here we regard ${{\bigcap}}_R^r X \subseteq {{\bigcap}}_R^r Y$ by Proposition \[prop injective\](i). In particular, ${{\bigwedge}}_{1\leq i\leq s} \varphi_i$ induces a homomorphism $${{\bigwedge}}_{1\leq i \leq s} \varphi_i: {{\bigcap}}_R^{r+s}Y \to {{\bigcap}}_R^r X.$$
Further functorialities
-----------------------
In this section we prove several new properties of exterior power biduals that will play a key role in subsequent sections.
We first establish an appropriate formalism in our setting of the ‘rank reduction methods’ that were initiated by Rubin in [@rubinstark] and subsequently used by Perrin-Riou and by Büyükboduk. This result will later play an essential role in the proof of Theorem \[thm1\].
\[reduction\] Suppose that $R$ is self-injective. Let $X$ be an $R$-module and $Y$ an $R$-submodule of $X$. Let $r$ be a non-negative integer and identify ${\bigcap}_R^r Y$ with an $R$-submodule of ${\bigcap}_R^{r} X$ by using Proposition \[prop injective\](i). Then one has $${\bigcap}_R^r Y = \left\{ x \in {\bigcap}^{r}_{R}X \ \middle| \ \Phi(x) \in Y\!\left(={\bigcap}_R^1 Y\right) \text{ for all } \Phi \in {\bigwedge}_R^{r-1} X^\ast \right\}.$$
Since $R$ is self-injective, upon taking the $R$-dual of the tautological exact sequence $0 \to Y \to X \to X/Y\to 0$ we obtain another exact sequence $$0 \to (X/Y)^\ast \to X^\ast \to Y^\ast \to 0.$$ Then, by applying the result of Lemma \[wedge-kernel\] below to this sequence, we deduce that $$\begin{aligned}
\ker \left( {\bigwedge}^r_R X^\ast \to {\bigwedge}^r_R Y^\ast \right)=\left\langle \varphi \wedge \Phi \ \middle| \ \varphi \in (X/Y)^\ast, \ \Phi \in {\bigwedge}^{r-1}_R X^\ast \right\rangle_R,\end{aligned}$$ and hence that the sequence $$\left\langle \varphi \wedge \Phi \ \middle| \ \varphi \in (X/Y)^\ast, \ \Phi \in {\bigwedge}^{r-1}_R X^\ast \right\rangle_R \to {\bigwedge}^r_R X^\ast \to {\bigwedge}^r_R Y^\ast \to 0$$ is exact. Taking $R$-duals of the latter exact sequence, we deduce that $$\begin{aligned}
\label{characterize}
{\bigcap}_R^r Y &= \left\{ x \in {\bigcap}_R^r X \ \middle| \ x (\varphi\wedge\Phi)=0 \text{ for every $\varphi \in (X/Y)^\ast$ and $\Phi \in {\bigwedge}^{r-1}_R X^\ast$} \right\}
$$
Now we suppose that $x \in {\bigcap}_R^r X$ satisfies $\Phi(x) \in Y $ for every $\Phi \in \bigwedge_R^{r-1} X^\ast $. Then we have $$x(\varphi \wedge \Phi)=\varphi(\Phi(x))=0$$ for every $\varphi \in (X/Y)^\ast\subseteq X^\ast$ and $\Phi \in \bigwedge_R^{r-1} X^\ast $. Hence, by (\[characterize\]), we have $x \in {\bigcap}_R^r Y$. This proves the proposition since the opposite inclusion is clear.
\[wedge-kernel\] We suppose given a short exact sequence of $R$-modules $$0\to X \xrightarrow{\iota} Y \xrightarrow{\pi} Z\to 0.$$ Then, for every non-negative integer $r$, the kernel of the natural homomorphism $${\bigwedge}^r_R Y \stackrel{\pi}{\to} {\bigwedge}^r_R Z$$ is generated over $R$ by the elements $\iota(\varphi) \wedge \Phi$ as $\varphi$ ranges over $X$ and $\Phi$ over ${\bigwedge}^{r-1}_R Y$.
We shall denote $\otimes_R$ simply by $\otimes$. For each integer $i$ with $1 \leq i \leq r$, we consider the $R$-module $$\Omega_{r,i} := Y \otimes \cdots \otimes Y \otimes X \otimes Y \otimes \cdots \otimes Y,$$ in which there are $r-1$ copies of $Y$ and the module $X$ occurs as the $i$-th module from the left hand end (so that $\Omega_{r,i}$ is isomorphic to $X \otimes Y^{\otimes(r-1)}$).
We set $$\Omega_r:=\bigoplus_{i=1}^{r} \Omega_{r,i}$$ and claim that the sequence of $R$-modules $$\begin{aligned}
\Omega_r \to Y^{\otimes r} \xrightarrow{\pi^{\otimes r}} Z^{\otimes r} \to 0,\end{aligned}$$ in which the first map is the direct sum of the maps $\Omega_{r,i}\to Y^{\otimes r}$ that are induced by the given map $\iota: X \to Y$, is exact.
To prove this claim we use induction on $r$. When $r=1$, the claim is clear. When $r>1$, by the inductive hypothesis, we have the following commutative diagram of $R$-modules, whose rows and columns are exact: $$\begin{aligned}
\xymatrix{
\Omega_{r-1} \otimes X \ar[r] \ar[d] & Y^{\otimes (r-1)} \otimes X \ar[r] \ar[d] &
Z^{\otimes(r-1)} \otimes X \ar[r] \ar[d] & 0
\\
\Omega_{r-1} \otimes Y \ar[r] \ar[d] & Y^{\otimes r} \ar[r] \ar[d] &
Z^{\otimes(r-1)} \otimes Y \ar[r] \ar[d] & 0
\\
\Omega_{r-1} \otimes Z \ar[r] \ar[d] & Y^{\otimes (r-1)} \otimes Z \ar[r] \ar[d] &
Z^{\otimes r} \ar[r] \ar[d] & 0
\\
0 & 0 & 0.
}\end{aligned}$$ Since $\Omega_r = (\Omega_{r-1} \otimes Y) \oplus (Y^{\otimes (r-1)} \otimes X)$, we see that $$\operatorname{im}\left( \Omega_r \to Y^{\otimes r} \right) =
\ker \left( Y^{\otimes r} \to Z^{\otimes r} \right)$$ by diagram chasing. Hence we have proved the claim.
Now we put $\Upsilon_r := \ker\left( {\bigwedge}^r_R Y \to {\bigwedge}^r_R Z \right)$ and consider the commutative diagram $$\begin{aligned}
\xymatrix{
& \Omega_r \ar[r] \ar[d] & Y^{\otimes r} \ar[r] \ar[d]^{f_1} & Z^{\otimes r} \ar[r] \ar[d]^{f_2} & 0
\\
0 \ar[r] & \Upsilon_r \ar[r] & {\bigwedge}_R^rY \ar[r] & {\bigwedge}_R^r Z \ar[r] & 0,
}\end{aligned}$$ whose rows are exact. Clearly, $f_1$ is surjective, and the map $\ker (f_1) \to \ker (f_2)$ is also surjective, so we see that the map $\Omega_r \to \Upsilon_r$ is surjective (by Snake lemma). This shows that the inclusion $$\Upsilon_r \subseteq \left\langle \varphi \wedge \Psi \mid \varphi \in X, \ \Psi \in {\bigwedge}^{r-1}_R Y \right\rangle_R$$ holds. Since the opposite inclusion clearly holds, we have proved the lemma.
\[bidual-ker\] Suppose that $R$ is self-injective. Let $X$ be an $R$-module and $f \colon X \to R$ an $R$-homomorphism. Then for any non-negative integer $r$, we have $$\begin{aligned}
{\bigcap}^{r}_{R}\ker\left( f \right) = \ker\left(f \colon {\bigcap}^{r}_{R}X \to {\bigcap}^{r-1}_{R}X \right).\end{aligned}$$
By definition, we see that the composition ${\bigcap}^{r}_{R}\ker\left( f \right) \hookrightarrow {\bigcap}^{r}_{R}X \xrightarrow{f} {\bigcap}^{r-1}_{R}X$ is the zero map. Hence we have ${\bigcap}^{r}_{R}\ker\left( f \right) \subseteq \ker\left(f \colon {\bigcap}^{r}_{R}X \to {\bigcap}^{r-1}_{R}X \right)$. To prove the opposite inclusion, take $x \in \ker\left(f \colon {\bigcap}^{r}_{R}X \to {\bigcap}^{r-1}_{R}X \right)$. Then, by definition, we have $$0 = f(x)(\Phi) = x(f \wedge \Phi)=f(\Phi(x))$$ for any $\Phi \in \bigwedge^{r-1}_{R}X^{*}$, and so $\Phi(x) \in \ker(f)$. By Proposition \[reduction\], we conclude that $x \in {\bigcap}^{r}_{R}\ker\left( f \right)$.
The following corollary will be used in §§\[stark sys sec\] and \[koly sys sec\] to define Kolyvagin and Stark systems over a Gorenstein order.
\[morph\] We suppose to be given a surjective homomorphism $R \to S$ of self-injective rings. Let $X$ be an $R$-module, $F$ a free $R$-module of finite rank, and $Y$ an $S$-module. We then suppose to be given a commutative diagram of $R$-modules $$\begin{aligned}
\xymatrix{
X \ar[r] \ar[d] & F \ar[d]^{\pi}
\\
Y \ar@{^{(}->}[r] & F \otimes_{R}S
}\end{aligned}$$ in which the lower horizontal arrow is injective and the map $\pi$ is induced by the given homomorphism $R \to S$.
Then for any positive integer $r$, there exists a natural homomorphism of $R$-modules $${{\bigcap}}^{r}_{R}X \to {{\bigcap}}^{r}_{S}Y$$ that is independent of the given maps $X \to F$ and $Y \hookrightarrow F \otimes_{R} S$ and is such that all squares of the following diagram $$\begin{aligned}
\xymatrix{
{{\bigwedge}}^{r}_{R}X \ar[r]^-{\xi^{r}_{X}} \ar[d] & {{\bigcap}}^{r}_{R}X \ar[d] \ar[r] & {{\bigcap}}^{r}_{R}F \ar[d]
\\
{{\bigwedge}}^{r}_{S}Y \ar[r]^-{\xi^{r}_{Y}} & {{\bigcap}}^{r}_{S}Y \ar@{^{(}->}[r] & {{\bigcap}}^{r}_{S}(F \otimes_{R}S)
}\end{aligned}$$ commute. Here the second upper and lower horizontal arrows denote the maps induced by the given maps $X \to F$ and $Y \hookrightarrow F \otimes_{R} S$.
Since the map $Y \hookrightarrow F \otimes_{R}S$ is injective, by replacing $X$ with $\operatorname{im}\left(X \to F\right)$, we may assume that the map $X \to F$ is injective. Since $F$ is free of finite rank, the maps $\xi_{F}^{r}$ and $\xi_{F \otimes_{R}S}^{r}$ are isomorphisms. Hence we get the following commutative diagram: $$\begin{aligned}
\xymatrix{
{{\bigwedge}}^{r}_{R}X \ar[r]^-{\xi^{r}_{X}} \ar[d] & {{\bigcap}}^{r}_{R}X \ar@{^{(}->}[r] & {{\bigcap}}^{r}_{R}F \ar[r]^-{(\xi_{F}^{r})^{-1}} & {{\bigwedge}}^{r}_{R}F \ar[d]^-{{\wedge}^{r}\pi}
\\
{{\bigwedge}}^{r}_{S}Y \ar[r]^-{\xi_{Y}^{r}} & {{\bigcap}}^{r}_{S}Y \ar@{^{(}->}[r] & {{\bigcap}}^{r}_{S}(F \otimes_{R}S) \ar[r]^-{(\xi_{F \otimes_{R}S}^{r})^{-1}} & {{\bigwedge}}^{r}_{S}(F \otimes_{R}S).
}\end{aligned}$$ Since the diagram commutes and the map ${{\bigcap}}^{r}_{S}Y \hookrightarrow {{\bigcap}}^{r}_{S}(F \otimes_{R}S)$ is injective by Proposition \[prop injective\](i), we only need to show that $$\operatorname{im}\left({{\bigcap}}^{r}_{R}X \to {{\bigwedge}}^{r}_{S}(F \otimes_{R}S) \right) \subseteq \operatorname{im}\left({{\bigcap}}^{r}_{S}Y \to {{\bigwedge}}^{r}_{S}(F \otimes_{R}S)\right).$$ To see this, take elements $x \in {{\bigwedge}}^{r}_{R}F$ with $\xi_{F}^{r}(x) \in {{\bigcap}}_{R}^{r}X$ and $\Phi \in {{\bigwedge}}^{r-1}_{R}F^{*}$. We also denote by $\Phi$ the image of $\Phi$ under the map $${{\bigwedge}}^{r}_{R}F^{*} \to {{\bigwedge}}^{r}_{S}(F^{*} \otimes_{R}S) \xrightarrow{\sim} {{\bigwedge}}^{r-1}_{S}(F \otimes_{R}S)^{*}.$$ Then by Proposition \[reduction\], we have $\Phi(x) \in X$. Hence we conclude that $\Phi((\wedge^{r}\pi)(x)) = \pi(\Phi(x)) \in Y$. Since the natural map $$\operatorname{Hom}_{R}(F, R) \otimes_{R}S \to \operatorname{Hom}_{S}(F \otimes_{R}S, S)$$ is bijective we can again apply Proposition \[reduction\] to deduce that $\xi_{F \otimes_{R}S}^{r}((\wedge^{r}\pi)(x))$ belongs to $ {{\bigcap}}^{r}_{S}Y$, as required.
Preliminaries concerning Galois cohomology {#pre}
==========================================
Notation and hypotheses
-----------------------
Let $K$ be a number field. Let $(R, {\mathfrak{p}})$ be a self-injective (commutative) local ring (in other words, a zero-dimensional Gorenstein local ring) with a finite residue field $\Bbbk=R/{\mathfrak{p}}$ of characteristic $p>0$. Note that $R$ is an artinian ring, which is finite of order a power of $p$. Let $A$ be a free $R$-module of finite rank with an $R$-linear continuous action of $G_K$. ($A$ is endowed with discrete topology.) Since $A$ is finite, the action of $G_K$ factors through a finite quotient, so $A$ is unramified outside a finite set of places of $K$.
### Selmer structures {#section sel}
We fix a Selmer structure $\mathcal{F}$ on $A$. Recall that a Selmer structure $\mathcal{F}$ on $A$ is a collection of the following data (see [@MRkoly Def. 2.1.1]):
- a finite set $S(\mathcal{F})$ of places of $K$ such that $S_\infty(K)\cup S_p(K) \cup S_{\rm ram}(A) \subseteq S(\mathcal{F})$;
- for every $v \in S(\mathcal{F})$, a choice of an $R$-submodule $H_\mathcal{F}^1(K_v, A) \subseteq H^1(K_v, A)$.
The Selmer module attached to $\mathcal{F}$ is defined by $$H_\mathcal{F}^1(K, A) := \ker \left( H^1({\mathcal{O}}_{K,S(\mathcal{F})}, A) \to \bigoplus_{v\in S(\mathcal{F})} H^1_{/{\mathcal{F}}}(K_v, A)\right).
$$ (Recall that for any index $\ast$ we set $H^1_{/\ast}(K_v,A):=H^1(K_v,A)/H^1_\ast(K_v,A).) $ Note that $$H^1({\mathcal{O}}_{K,S({\mathcal{F}})}, A)=\ker \left( H^1(K, A) \to \bigoplus_{v \notin S({\mathcal{F}})} H^1_{/f}(K_v, A)
\right),$$ so we have $$H^1_{{\mathcal{F}}}(K,A) = \ker \left( H^1(K, A) \to \bigoplus_{v} H^1_{/{\mathcal{F}}}(K_v, A)
\right),$$ where we set $H^1_{\mathcal{F}}(K_v,A):=H^1_f(K_v,A)$ ($=H^1_{\rm ur}(K_v,A)$) for $v \notin S({\mathcal{F}})$. In the following, we set $S:=S({\mathcal{F}})$ for simplicity. (Note that Selmer structures and modules can be defined for continuous representations of $G_K$ of various other types: in §\[one-dim case\], for example, we consider representations over one-dimensional Gorenstein rings and Selmer modules associated to them.) We next review the definition of the dual Selmer structure ${\mathcal{F}}^\ast$ on $A^\ast(1)$. By local Tate duality, we have a canonical isomorphism $$H^1(K_v, A) \simeq H^1(K_v, A^\ast(1))^\ast.$$ Using this, we define ${\mathcal{F}}^\ast$ to be the following data
- $S({\mathcal{F}}^\ast):=S({\mathcal{F}})(=S)$;
- for $v\in S$, $$H^1_{{\mathcal{F}}^\ast}(K_v, A^\ast(1)):=\ker(H^1(K_v, A^\ast(1)) \simeq H^1(K_v, A)^\ast \to H^1_{{\mathcal{F}}}(K_v,A)^\ast).$$
If $B$ is an $R[G_{K}]$-submodule (resp. quotient) of $A$, then the Selmer structure ${\mathcal{F}}$ on $A$ induces a Selmer structure on $B$ (which we also denote by ${\mathcal{F}}$) as follows: we define the local condition by $$H^1_{\mathcal{F}}(K_v, B):=\ker(H^1(K_v,B) \to H^1_{/{\mathcal{F}}}(K_v,A))$$ $$(\text{resp. }H^1_{{\mathcal{F}}}(K_v,B):=\operatorname{im}(H^1_{{\mathcal{F}}}(K_v,A) \to H^1(K_v,B))).$$
The following result is a consequence of the (Poitou-Tate) global duality and is proved by Mazur and Rubin in [@MRkoly Th. 2.3.4].
\[gd\] Let ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ be Selmer structures on $A$ such that at every place $v$ one has $H^1_{{\mathcal{F}}_1}(K_v,A)\subseteq H^1_{{\mathcal{F}}_2}(K_v,A)$. Then there exists a canonical exact sequence $$\begin{gathered}
0\to H^1_{{\mathcal{F}}_1}(K,A) \to H^1_{{\mathcal{F}}_2}(K,A) \to \bigoplus_{v } H^1_{{\mathcal{F}}_2}(K_v,A)/H^1_{{\mathcal{F}}_1}(K_v,A) \\
\to H^1_{{\mathcal{F}}_1^\ast}(K,A^\ast(1))^\ast \to H^1_{{\mathcal{F}}_2^\ast}(K,A^\ast(1))^\ast \to 0.\end{gathered}$$
### Hypotheses {#section hyp}
Let $K(A)$ be the minimal Galois extension of $K$ such that $G_{K(A)}$ acts trivially on $A$. Let $M := \min \{ p^n \mid p^n R = 0 \}$. We denote by $K(1)$ the maximal $p$-extension of $K$ inside the Hilbert class field of $K$. We set $$K_M := K (\mu_{M}, ({\mathcal{O}}^\times_K)^{1/M}) K(1)\,\,\text{ and }\,\,K(A)_M := K(A)K_M.$$ Here $({\mathcal{O}}_K^\times)^{1/M}$ denotes the set $\{u \in \overline {\mathbb{Q}}\mid u^M \in {\mathcal{O}}_K^\times\}$.
In the following, we assume the following hypotheses.
\[hyp1\]
- $A \otimes_R \Bbbk$ is an irreducible $\Bbbk[G_K]$-module;
- there exists $\tau \in G_{K_M}$ such that $A/(\tau - 1)A \simeq R$ as $R$-modules;
- $H^1(K(A)_M/K, A) = H^1(K(A)_M/K, A^*(1)) = 0$.
\[hyp2\] $(A \otimes_R \Bbbk)^{G_K} = ((A \otimes_R \Bbbk)^*(1))^{G_K} = 0$.
It is clear that if $A$ satisfies Hypothesis \[hyp1\], then so also does $A^*(1)$. Note also that the vanishing of $ ((A \otimes_R \Bbbk)^*(1))^{G_K}$ is equivalent to the vanishing of $ (A^\ast(1)\otimes_R \Bbbk)^{G_K}$ and so Hypothesis \[hyp2\] for $A$ is equivalent to the same hypothesis for $A^\ast(1)$.
Let $\mathcal{P}$ be the set of primes ${\mathfrak{q}}\not\in S $ of $K$ such that ${\rm Fr}_{\mathfrak{q}}$ is conjugate to $\tau$ in ${\rm Gal}(K(A)_M/K)$. In particular, by Hypothesis \[hyp1\](ii), we see that ${\mathfrak{q}}$ splits completely in $K_M$ and that $A/({\rm Fr}_{\mathfrak{q}}-1)A \simeq R$ for every ${\mathfrak{q}}\in {\mathcal{P}}$. For a set $\mathcal{Q}$ of primes of $K$, we denote by $\mathcal{N}(\mathcal{Q})$ the set of square-free products of primes in $\mathcal{Q}$. (We let $1 \in \mathcal{N}(\mathcal{Q})$ for convention.) For ${\mathfrak{n}}\in {\mathcal{N}}$, the number of primes which divide ${\mathfrak{n}}$ is denoted by $\nu({\mathfrak{n}})$. (When ${\mathfrak{n}}=1$, we set $\nu(1):=0$.) We often abbreviate $\mathcal{N}(\mathcal{P})$ to $\mathcal{N}$. For ${\mathfrak{q}}\in {\mathcal{P}}$, we denote by $K({\mathfrak{q}})$ the maximal $p$-extension of $K$ inside the ray class field modulo ${\mathfrak{q}}$. Put $G_{\mathfrak{q}}:=\operatorname{Gal}(K({\mathfrak{q}})/K(1))=\operatorname{Gal}(K({\mathfrak{q}})_{\mathfrak{q}}/K_{\mathfrak{q}})$, where $K({\mathfrak{q}})_{\mathfrak{q}}$ is the completion of $K({\mathfrak{q}})$ at the (fixed) place lying above ${\mathfrak{q}}$. For ${\mathfrak{n}}\in {\mathcal{N}}$, we denote by $K({\mathfrak{n}})$ the composite of $K({\mathfrak{q}})$’s with ${\mathfrak{q}}\mid {\mathfrak{n}}$. We set $$\begin{aligned}
G_{\mathfrak{n}}:= \bigotimes_{{\mathfrak{q}}\mid {\mathfrak{n}}} G_{\mathfrak{q}}.\end{aligned}$$ Note that, since ${\mathfrak{q}}\in {\mathcal{P}}$ splits completely in $K_M$, the order of $G_{\mathfrak{q}}$ is divisible by $M$.
### Modified Selmer structures {#section mod sel}
For ${\mathfrak{a}},{\mathfrak{b}},{\mathfrak{n}}\in {\mathcal{N}}$ which are pairwise relatively prime, we define a Selmer structure $\mathcal{F}_{\mathfrak{a}}^{\mathfrak{b}}({\mathfrak{n}})$ by
- $S(\mathcal{F}_{\mathfrak{a}}^{\mathfrak{b}}({\mathfrak{n}})):=S \cup \{ {\mathfrak{q}}\mid {\mathfrak{a}}{\mathfrak{b}}{\mathfrak{n}}\}$;
- for $v \in S(\mathcal{F}_{\mathfrak{a}}^{\mathfrak{b}}({\mathfrak{n}}))$, $$\begin{aligned}
H^1_\mathcal{F_{\mathfrak{a}}^{\mathfrak{b}}({\mathfrak{n}})}(K_v,A) :=
\begin{cases}
H^1_{\mathcal{F}}(K_v, A) &\text{ if $v \in S$}, \\
0 &\text{ if $v \mid {\mathfrak{a}}$,}\\
H^1(K_v , A) &\text{ if $v\mid {\mathfrak{b}}$},\\
H^1_{\rm tr}(K_v,A) &\text{ if $v \mid {\mathfrak{n}}$},
\end{cases}\end{aligned}$$
where $H^1_{\rm tr}(K_{\mathfrak{q}},A)$ is the ‘transverse’ submodule of $H^1(K_{\mathfrak{q}}, A)$ which fits in a canonical decomposition $$H^1(K_{\mathfrak{q}}, A)=H^1_f(K_{\mathfrak{q}}, A)\oplus H^1_{\rm tr}(K_{\mathfrak{q}}, A).$$ Explicitly, one defines $$H^1_{\rm tr}(K_{\mathfrak{q}}, A):=H^1(K({\mathfrak{q}})_{\mathfrak{q}}/K_{\mathfrak{q}}, A^{G_{K({\mathfrak{q}})_{\mathfrak{q}}}})=\operatorname{Hom}(G_{\mathfrak{q}}, A^{{\rm Fr}_{\mathfrak{q}}=1}),$$ which is regarded as a submodule of $H^1(K_{\mathfrak{q}}, A)$ by the inflation map. Since $H^1_{\rm tr}(K_{\mathfrak{q}}, A)$ is canonically isomorphic to $H^1_{/f}(K_{\mathfrak{q}},A)$, we sometimes identify them. Note that $({\mathcal{F}}_{\mathfrak{a}}^{\mathfrak{b}}({\mathfrak{n}}))^\ast={\mathcal{F}}_{\mathfrak{b}}^{\mathfrak{a}}({\mathfrak{n}})$. This follows from the fact that $$H^1_f(K_{\mathfrak{q}},A)\simeq H^1_{/f}(K_{\mathfrak{q}},A^\ast(1))^\ast$$ for every ${\mathfrak{q}}\in {\mathcal{P}}$. If ${\mathfrak{a}}=1$, we abbreviate ${\mathcal{F}}_{\mathfrak{a}}^{\mathfrak{b}}({\mathfrak{n}})$ to ${\mathcal{F}}^{\mathfrak{b}}({\mathfrak{n}})$. Similarly, if ${\mathfrak{b}}=1$ or ${\mathfrak{n}}=1$, they are omitted. The Selmer structures ${\mathcal{F}}^{\mathfrak{n}}$ and ${\mathcal{F}}({\mathfrak{n}})$ will often appear. We remark that the associated Selmer modules are $$H^1_{{\mathcal{F}}^{\mathfrak{n}}}(K,A) := \ker \left( H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}}, A) \to \bigoplus_{v\in S}H^1_{/{\mathcal{F}}}(K_v,A)
\right)$$ and $$H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A) := \ker \left( H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}}, A) \to \bigoplus_{v\in S}H^1_{/{\mathcal{F}}}(K_v,A)
\oplus \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}} H^1_{/ {\rm tr}}(K_v,A)\right)$$ respectively, where $S_{\mathfrak{n}}:=S\cup \{{\mathfrak{q}}\mid {\mathfrak{n}}\}$.
### Finite-singular comparison maps {#section fs}
For ${\mathfrak{q}}\in {\mathcal{P}}$, we recall the definition of the ‘finite-singular comparison map’ $$\begin{aligned}
\varphi^{\rm fs}_{\mathfrak{q}}\colon H^1_f(K_{\mathfrak{q}}, A) \simeq H^1_{\rm tr}(K_{\mathfrak{q}}, A) \otimes G_{\mathfrak{q}},\end{aligned}$$ which is important in the theory of Kolyvagin systems. We use the following canonical isomorphisms: $$\begin{aligned}
H^1_{f}(K_{\mathfrak{q}}, A) \simeq A/({\rm Fr}_{\mathfrak{q}}- 1)A; \ a \mapsto a({\rm Fr}_{\mathfrak{q}}) \ \text{(evaluation of ${\rm Fr}_{\mathfrak{q}}$ to the 1-cocycle $a$)},\end{aligned}$$ $$\begin{aligned}
H^1_{\rm tr}(K_{\mathfrak{q}}, A) \otimes G_{\mathfrak{q}}= \operatorname{Hom}(G_{\mathfrak{q}}, A^{{\rm Fr}_{\mathfrak{q}}= 1}) \otimes G_{\mathfrak{q}}\simeq A^{{\rm Fr}_{\mathfrak{q}}= 1}; \
f \otimes \sigma \mapsto f(\sigma).\end{aligned}$$ Since $A/({\rm Fr}_{\mathfrak{q}}-1)A\simeq R$, one has $\det(1 - {\rm Fr}_{\mathfrak{q}}\mid A) = 0$, so there exists a unique polynomial $Q_{\mathfrak{q}}(x) \in R[x]$ such that $(x - 1)Q_{\mathfrak{q}}(x) = \det(1 - {\rm Fr}_{\mathfrak{q}}x \mid A)$ in $R[x]$. By the Cayley-Hamilton theorem, we know that $({\rm Fr}_{\mathfrak{q}}^{-1} - 1)Q_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1})$ annihilates $A$, so we have a well-defined map $$Q_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1}): A/({\rm Fr}_{\mathfrak{q}}-1)A \to A^{{\rm Fr}_{\mathfrak{q}}= 1}; \ a \mapsto Q_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1})a.$$ (This is actually an isomorphism, see [@R Cor. A.2.7].)
Now we define $\varphi^{\rm fs}_{\mathfrak{q}}$ to be the composite homomorphism
$$\begin{aligned}
\varphi^{\rm fs}_{\mathfrak{q}}\colon H^1_{f}(K_{\mathfrak{q}}, A) \simeq A/({\rm Fr}_{\mathfrak{q}}- 1)A \xrightarrow{Q_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1})}
A^{{\rm Fr}_{\mathfrak{q}}= 1} \simeq H^1_{\rm tr}(K_{\mathfrak{q}}, A) \otimes G_{\mathfrak{q}}.\end{aligned}$$
Application of the Chebotarev density theorem
---------------------------------------------
In this subsection, we prove several basic results that will be used later.
For an $R$-module $X$ and an ideal $I$ of $R$ we consider the $R$-submodule $$X[I] := \{x \in X \mid ax=0\, \text{ for all } \, a \in I\}$$ of $X$ comprising elements that are annihilated by elements of $I$.
\[prop h1\] Assume $(A \otimes_R \Bbbk )^{G_K}$ vanishes. Then, for any ideal $I$ of $R$, the homomorphism $H^1(K, A[I]) \to H^1(K, A)[I]$ induced by the inclusion $A[I] \hookrightarrow A$ is bijective.
Although a proof of this result was given by Mazur and Rubin in [@MRkoly Lem. 3.5.3], it relies on the validity of a lemma, which contains an error (see [@MRkoly Lem. 2.1.4] and [@MRselmer ‘Erratum’ on p.182]), and so their induction argument apparently fails. For the reader’s convenience, we shall give a full proof of Proposition \[prop h1\].
To prove this proposition, we need the following algebraic lemmas.
\[lemma S\] Let $(S, {\mathfrak{p}}_{S})$ be a local ring, $G$ a group and $B$ an $S[G]$-module. Suppose that $B$ is a flat $S$-module and that $(B/{\mathfrak{p}}_{S}B)^{G} = 0$. Then we have $$B^G = {\bigcap}_{i = 0}^{\infty}({\mathfrak{p}}^{i}_{S}B)^{G}.$$ In particular, if $\mathfrak{p}_S^n B=0$ for a sufficiently large $n$, then we have $B^G=0$.
Since $B$ is a flat $S$-module, we have $S[G]$-isomorphisms $${\mathfrak{p}}^{i}_{S}B/{\mathfrak{p}}^{i+1}_{S}B \simeq {\mathfrak{p}}^i_{S}/{\mathfrak{p}}^{i+1}_{S} \otimes_{S} B \simeq (B/{\mathfrak{p}}_{S}B)^{\dim_{S/{\mathfrak{p}}_{S}}({\mathfrak{p}}^{i}_{S}/{\mathfrak{p}}^{i+1}_{S})}.$$ By the assumption $(B/{\mathfrak{p}}_{S}B)^{G} = 0$, we see that $({\mathfrak{p}}^i_{S} B)^{G} = ({\mathfrak{p}}^{i+1}_{S}B)^{G}$. This equality holds for arbitrary $i$, so we have $B^{G} = {\bigcap}_{i=0}^{\infty}({\mathfrak{p}}^{i}_{S}B)^{G}$.
\[fixed-cor\] If $(A \otimes_R \Bbbk )^{G_K}$ vanishes, then the following claims are valid.
- For any ideals $I_{1}, \ldots, I_{d}$ of $R$, we have $$\left({\rm coker}\left( A \to A/I_{1}A \times \cdots \times A/I_{d}A \right) \right)^{G_{K}} = 0,$$ where $A \to A/I_{1}A \times \cdots \times A/I_{d}A$ is the diagonal map.
- For any elements $r_{1}, \ldots, r_{d} \in R$, we have $$\left({\rm coker}\left( A \xrightarrow{ (r_{1}, \ldots, r_{d}) \times } A^{d} \right) \right)^{G_{K}} = 0.$$
We prove claim (i) by induction on $d$. When $d=1$, the claim is clear. When $d>1$, we set $$M_{d} := {\rm coker}\left( A \to A/I_{1}A \times \cdots \times A/I_{d}A \right)$$ and $$M_{d-1} := {\rm coker}\left( A \to A/I_{1}A \times \cdots \times A/I_{d-1}A \right).$$ Applying the ‘kernel-cokernel lemma’ (see [@NSW Exer. 2 in Chap. I, §3]) to the sequence $$A \stackrel{f}{\to } A/I_{1}A \times \cdots \times A/I_{d}A \stackrel{g}{\to} A/I_{1}A \times \cdots \times A/I_{d-1}A,$$ where $g$ is the natural projection, we obtain the exact sequence $$\ker(g\circ f)\to \ker (g) \to \operatorname{coker}(f)\to \operatorname{coker}(g\circ f) \to \operatorname{coker}(g).$$ Noting that $$\begin{aligned}
\ker (g\circ f) &= I_1\cap \cdots \cap I_{d-1}=:J,
\\
\ker(g) &= A/I_dA,
\\
\operatorname{coker}(f) &= M_d,
\\
\operatorname{coker}(g\circ f) &= M_{d-1},
\\
\operatorname{coker}(g) &= 0,\end{aligned}$$ we have an exact sequence $$0 \to A/(J+I_d)A \to M_d \to M_{d-1} \to 0.$$ By the inductive hypothesis, we have $M_{d-1}^{G_K}=0$. Also, by applying Lemma \[lemma S\] to $S:=R/(J+I_d)$, $B:=A/(J+I_d)A$ and $G:=G_K$, we see that $(A/(J+I_d)A)^{G_K}=0$. Hence we have $M_d^{G_K}=0$. This proves (i).
Next, we prove claim (ii). We set $I_{i} := R[r_{i}]$ for each $1 \leq i \leq d$ and consider the sequence $$A \stackrel{f}{\to} A/I_1 A \times \cdots \times A/I_d A \stackrel{g}{\to} A^d,$$ where $g$ is induced by the ‘multiplication by $(r_1,\ldots,r_d)$’: $$g: A/I_1 A \times \cdots \times A/I_d A \to A^d; \ (a_1,\ldots,a_d) \mapsto (r_1a_1,\ldots,r_da_d).$$ (Note that this is injective.) Using the kernel-cokernel lemma, we obtain the exact sequence $$0 \to \operatorname{coker}(f)\to \operatorname{coker}(g\circ f) \to A/r_1A \times \cdots \times A/r_dA \to 0.$$ Since we know that $\operatorname{coker}(f)^{G_K}=0$ by (i) and that $(A/r_iA)^{G_K}=0$ by Lemma \[lemma S\], we have $\operatorname{coker}(g\circ f)^{G_K}=0$, i.e. $$\left({\rm coker}\left( A \xrightarrow{ (r_{1}, \ldots, r_{d}) \times } A^{d} \right) \right)^{G_{K}} = 0.$$
Let $\{r_1, \ldots, r_d\}$ be a set of generators of an ideal $I$ of $R$. We have a short exact sequence $$\begin{aligned}
\label{ex1}
0 \to A[I] \to A \to A/A[I] \to 0\end{aligned}$$ and an injection $$\begin{aligned}
\label{inj1}
A/A[I] \hookrightarrow A^{d};\ a \mapsto (r_1 a ,\ldots, r_d a).\end{aligned}$$ Since $(A \otimes_R \Bbbk)^{G_K} = 0$, we see by Lemma \[lemma S\] that the sequence (\[ex1\]) induces an exact sequence $$\begin{aligned}
0 \to H^1(K, A[I]) \to H^1(K, A) \to H^1(K, A/A[I]).\end{aligned}$$ (We apply Lemma \[lemma S\] with $S:=R/R[I]$ and $B:=A/A[I]$.) One also sees by Lemma \[fixed-cor\](ii) that (\[inj1\]) induces an injection $$\begin{aligned}
H^1(K, A/A[I]) \hookrightarrow H^1(K, A^{d}) = H^1(K, A)^d.\end{aligned}$$ Since the composition map $$\begin{aligned}
H^1(K, A) \to H^1(K, A/A[I]) \to H^1(K, A)^d\end{aligned}$$ is multiplication by $(r_1, \ldots, r_d)$, we have $$\begin{aligned}
H^1(K, A[I]) &= \ker\left( H^1(K, A) \to H^1(K, A/A[I]) \right)
\\
&= \ker\left( H^1(K, A) \xrightarrow{ (r_1, \ldots, r_d)\times } H^1(K, A)^d \right)
\\
&= H^1(K, A)[I].\end{aligned}$$
\[dualselisom\] Suppose that $(A \otimes_R \Bbbk)^*(1)^{G_K}$ vanishes. Then, for any ideal $I$ of $R$, the inclusion map $A^\ast(1)[I] \hookrightarrow A^*(1)$ induces an isomorphism $$\begin{aligned}
H^1_{{\mathcal{F}}^*}(K, A^\ast(1)[I]) \simeq H^1_{{\mathcal{F}}^*}(K, A^*(1))[I].\end{aligned}$$ (Here ${\mathcal{F}}$ denotes the fixed Selmer structure on $A$, and ${\mathcal{F}}^\ast$ the dual Selmer structure on $A^\ast(1)$ of ${\mathcal{F}}$. Note that the Selmer structure on $A^\ast(1)[I]$ induced by ${\mathcal{F}}^\ast$ coincides with the dual of the Selmer structure on $A/IA$ induced by ${\mathcal{F}}$, so the notation $H^1_{{\mathcal{F}}^*}(K, A^\ast(1)[I])$ makes no confusion.)
We have the following commutative diagram: $$\begin{aligned}
\xymatrix{
0 \ar[r] & H^{1}_{{\mathcal{F}}^{*}}(K, A^{*}(1)[I]) \ar[r] \ar[d] & H^{1}({\mathcal{O}}_{K, S}, A^{*}(1)[I]) \ar[r] \ar[d] &
\ar[d] \bigoplus_{v \in S}H^1_{/{\mathcal{F}}^{*}}(K_v,A^\ast(1)[I])
\\
0 \ar[r] & H^{1}_{{\mathcal{F}}^{*}}(K, A^{*}(1))[I] \ar[r] & H^{1}({\mathcal{O}}_{K, S}, A^{*}(1))[I] \ar[r] &
\bigoplus_{v \in S}H^1_{/{\mathcal{F}}^{*}}(K_v,A^\ast(1)).
}\end{aligned}$$ Here each row is exact. The middle vertical arrow is an isomorphism by Proposition \[prop h1\]. The right vertical arrow is injective since the dual $H^{1}_{{\mathcal{F}}}(K_{v}, A) \to H^{1}_{{\mathcal{F}}}(K_{v}, A/IA)$ is surjective (by the definition of the induced Selmer structure). Hence we conclude that the left vertical arrow is an isomorphism.
Let $\tau \in G_{K_{M}}$ be the element in Hypothesis \[hyp1\](ii). Recall that ${\mathcal{P}}$ is the set of primes ${\mathfrak{q}}\not \in S $ of $K$ such that ${\rm Fr}_{{\mathfrak{q}}}$ is conjugate to $\tau$ in $\operatorname{Gal}(K(A)_{M}/K)$.
\[chebotarev\] Assume Hypothesis \[hyp1\]. Let $c_{1}, \ldots, c_{s} \in H^1(K, A)$ and $c_{1}^{*}, \ldots, c_{t}^{*} \in H^1(K, A^*(1))$ be non-zero elements. If $s+t < p$, then there is a subset ${\mathcal{Q}}\subseteq {\mathcal{P}}$ of positive density such that ${\rm loc}_{{\mathfrak{q}}}(c_{i}) $ and ${\rm loc}_{{\mathfrak{q}}}(c^{*}_{i}) $ are all non-zero for every ${\mathfrak{q}}\in {\mathcal{Q}}$, where ${\rm loc}_{{\mathfrak{q}}}$ denotes the localization map $H^{1}(K, -) \to H^{1}(K_{{\mathfrak{q}}}, -)$.
Let $B \in \{A, A^{*}(1) \}$. By Hypothesis \[hyp1\](iii), we know that the restriction map $${\rm Res}: H^{1}(K, B) \to H^{1}(K(A)_{M}, B)^{G_{K}} = \operatorname{Hom}(G_{K(A)_{M}}, B)^{G_{K}}$$ is injective. Since $B$ is an irreducible $R[G_{K}]$-module by Hypothesis \[hyp1\](i) and since $(\tau - 1)B$ is a free $R$-module of rank ${\rm rank}_{R}(A) - 1$ by Hypothesis \[hyp1\](ii), there is no non-trivial $G_{K}$-stable $R$-submodule of $(\tau - 1)B$. Hence the natural map $$f: H^{1}(K(A)_{M}, B)^{G_{K}} = \operatorname{Hom}(G_{K(A)_{M}}, B)^{G_{K}} \to \operatorname{Hom}(G_{K(A)_{M}}, B/(\tau-1)B)$$ is also injective.
Now suppose $B=A$. For $1 \leq i \leq s$, let $F_{i}$ be the field corresponding to the subgroup $\ker({\rm Res}(c_{i}))$ of $G_{K(A)_M}$. Note that $F_{i}$ is a finite Galois extension over $K$ (this follows from the fact that ${\rm Rec}(c_i)$ is a $G_K$-homomorphism). Let $\widetilde{c}_{i} \colon G_{K} \to A$ be a $1$-cocycle, which represents $c_{i}$, and set $a_{i} := - \widetilde{c}_{i}(\tau) \in A/(\tau-1)A$. By the definition of $1$-coboundary, note that $a_{i} \in A/(\tau-1)A$ is independent of the choice of the representative $\widetilde{c}_{i}$ of $c_{i}$. Define $H_{i} \subseteq G_{K(A)_{M}}$ by $$H_{i} = f({\rm Res}(c_{i}))^{-1}(a_{i}).$$ Since $c_i$ is non-zero and both ${\rm Res}$ and $f$ are injective, we have $$[G_{K(A)_M} \colon \ker\left( f({\rm Res}(c_{i}))\right)] \geq p.$$
Next, we suppose $B=A^\ast(1)$. For $1 \leq i \leq t$, define $F_{i}^{*}$, $a_{i}^{*}$, and $H_{i}^{*} \subseteq G_{K(A)_M}$ similarly by using $c_{i}^{*}$ instead of $c_{i}$.
Since $s+t <p$ and $[G_{K(A)_M} \colon \ker\left( f({\rm Res}(c))\right)] \geq p$ for each $c \in \{ c_{1}, \ldots, c_{s}, c_{1}^{*}, \ldots, c_{t}^{*} \}$, we have $$G_{K(A)_M} \neq H_{1} \cup \cdots \cup H_{s} \cup H_{1}^{*} \cup \cdots \cup H^{*}_{t}.$$ Now set $F := F_{1} \cdots F_{s}F_{1}^{*} \cdots F_{t}^{*}$ (this is a finite Galois extension of $K$) and define ${\mathcal{Q}}$ to be the set of primes ${\mathfrak{q}}\not \in S $ of $K$ such that ${\mathfrak{q}}$ is unramified in $F/K$ and ${\rm Fr}_{{\mathfrak{q}}}$ is conjugate to $\tau\gamma$ in $\operatorname{Gal}(F/K)$ for some $\gamma \in G_{K(A)_M} \setminus (H_{1} \cup \cdots \cup H_{s} \cup H_{1}^{*} \cup \cdots \cup H^{*}_{t})$. Then we see that ${\mathcal{Q}}\subseteq {\mathcal{P}}$ by construction, and that $\mathcal{Q}$ is of positive density by the Chebotarev density theorem. If ${\mathfrak{q}}\in {\mathcal{Q}}$, then for any $1 \leq i \leq s$ we have $${\rm loc}_{{\mathfrak{q}}}(c_{i}) = \widetilde{c}_{i}(\tau\gamma) = \tau \widetilde{c}_{i}(\gamma) + \widetilde{c}_{i}(\tau)
= f({\rm Res}(c_{i}))(\gamma) - a_{i} \neq 0 \text{ in } A/(\tau - 1)A$$ where we identify $H^{1}_{f}(K_{{\mathfrak{q}}}, A)=A/({\rm Fr}_{{\mathfrak{q}}} - 1)A = A/(\tau - 1)A$. Similarly, we have ${\rm loc}_{{\mathfrak{q}}}(c_{i}^{*}) \neq 0$ for any $1 \leq i \leq t$ and ${\mathfrak{q}}\in {\mathcal{Q}}$. This completes the proof.
\[injective\] Assume Hypothesis \[hyp1\]. Let $S$ be a self-injective ring and $R \to S$ a surjective ring homomorphism. Then for any free, finitely generated, $R$-submodule $X$ of $H^{1}(K, A)$ the natural homomorphism $X \otimes_{R} S \to H^{1}(K, A \otimes_{R}S)$ is injective.
We may assume $X \neq 0$. Let ${\mathcal{Q}}$ be the set of all primes ${\mathfrak{q}}$ in ${\mathcal{P}}$ such that ${\rm loc}_{{\mathfrak{q}}}(X) \not\subseteq H^{1}_{f}(K_{{\mathfrak{q}}}, A)$. Note that ${\mathcal{Q}}$ is a finite set. Let $e_{1} \in X$ with ${\mathrm{Ann}}_{R}(e_{1}) = 0$ and $x$ a generator of $R[{\mathfrak{p}}]$. (The assumption that $R$ is a zero-dimensional Gorenstein local ring ensures that $R[{\mathfrak{p}}]$ is a principal ideal.) Note that $a \in R$ is a unit if and only if $xa \neq 0$. Since $xe_{1} \neq 0$, there is a prime ${\mathfrak{q}}_{1} \in {\mathcal{P}}\setminus {\mathcal{Q}}$ with ${\rm loc}_{{\mathfrak{q}}_{1}}(xe_{1}) \neq 0$ by Lemma \[chebotarev\]. Since $H^{1}_{f}(K_{{\mathfrak{q}}_{1}}, A)$ is a free $R$-module of rank $1$ and ${\rm loc}_{{\mathfrak{q}}_{1}}(xe_{1}) \neq 0$, the composition $Re_{1} \to X \to H^{1}_{f}(K_{{\mathfrak{q}}_{1}}, A)$ is an isomorphism and we have $$X \simeq H^1_f(K_{{\mathfrak{q}}_1},A) \oplus \ker \left( {\rm loc}_{{\mathfrak{q}}_{1}}|_{X} \right) \simeq Re_{1} \oplus \ker\left( {\rm loc}_{{\mathfrak{q}}_{1}}|_{X} \right).$$ In particular, the $R$-module $\ker\left( {\rm loc}_{{\mathfrak{q}}_{1}}|_{X} \right)$ is free. Hence we can take an element $e_{2} \in \ker\left( {\rm loc}_{{\mathfrak{q}}_{1}}|_{X} \right)$ with ${\mathrm{Ann}}_{R}(e_{2}) = 0$. Similarly, by Lemma \[chebotarev\], we get a prime ${\mathfrak{q}}_{2} \in {\mathcal{P}}\setminus ({\mathcal{Q}}\cup \{{\mathfrak{q}}_{1}\} )$ such that the composition $Re_{2} \to X \to H^{1}_{f}(K_{{\mathfrak{q}}_{2}}, A)$ is an isomorphism. Since $e_{2} \in \ker\left( {\rm loc}_{{\mathfrak{q}}_{1}}|_{X} \right)$, the composition $$Re_{1} \oplus Re_{2} \to X \to H^{1}_{f}(K_{{\mathfrak{q}}_{1}}, A) \oplus H^{1}_{f}(K_{{\mathfrak{q}}_{2}}, A)$$ is an isomorphism and $$X = Re_{1} \oplus Re_{2} \oplus
\left(\ker\left( {\rm loc}_{{\mathfrak{q}}_{1}}|_{X} \right) \cap \ker\left( {\rm loc}_{{\mathfrak{q}}_{2}}|_{X} \right) \right).$$ By repeating this argument, we can find an ideal ${\mathfrak{m}}\in {\mathcal{N}}({\mathcal{P}}\setminus {\mathcal{Q}})$ such that the sum of localization maps $$X \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{m}}}H^{1}_{f}(K_{{\mathfrak{q}}}, A)$$ is an isomorphism. Since $H^{1}_{f}(K_{{\mathfrak{q}}}, A) \otimes_{R} S \simeq H^{1}_{f}(K_{{\mathfrak{q}}}, A \otimes_{R} S)$ for any prime ${\mathfrak{q}}\in {\mathcal{P}}$, the map $X \otimes_{R} S \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{m}}}H^{1}_{f}(K_{{\mathfrak{q}}}, A\otimes_R S ) $ is an isomorphism. This implies that $X\otimes_R S \to H^{1}(K, A \otimes_{R}S)$ is injective.
Stark systems {#stark sys sec}
=============
In this section, we review some basic results on Stark systems. We continue to use the notation introduced in the previous section.
Definition {#defstark}
----------
Let $r$ be a non-negative integer. We recall the definition of Stark systems of rank $r$. The module of Stark systems is defined by the inverse limit $${\rm SS}_r(A,{\mathcal{F}}):=\varprojlim_{{\mathfrak{n}}\in {\mathcal{N}}} {{\bigcap}}_{R}^{r + \nu({\mathfrak{n}})} H_{{\mathcal{F}}^{\mathfrak{n}}}^1(K, A),$$ where the transition maps $$\begin{aligned}
v_{{\mathfrak{m}},{\mathfrak{n}}}: {{\bigcap}}_{R}^{r + \nu({\mathfrak{m}})} H^1_{{\mathcal{F}}^{\mathfrak{m}}} (K, A) \to {{\bigcap}}_{R}^{r + \nu({\mathfrak{n}})} H_{{\mathcal{F}}^{\mathfrak{n}}}^1 (K, A)\end{aligned}$$ (${\mathfrak{m}}, {\mathfrak{n}}\in {\mathcal{N}}$, ${\mathfrak{n}}\mid {\mathfrak{m}}$) are defined as follows. For each ${\mathfrak{q}}\in {\mathcal{P}}$, we fix an isomorphism $H^1_{/f}(K_{\mathfrak{q}}, A)\simeq R$ and let $v_{\mathfrak{q}}$ be the composition map $$\begin{aligned}
v_{\mathfrak{q}}\colon H_{{\mathcal{F}}^{\mathfrak{m}}}^1(K,A) \stackrel{{\rm loc}_{\mathfrak{q}}}{\to} H^1(K_{\mathfrak{q}}, A) \to H_{/f}^1(K_{\mathfrak{q}}, A) \simeq R.\end{aligned}$$ Since we have the exact sequence $$\begin{aligned}
0 \to H_{{\mathcal{F}}^{\mathfrak{n}}}^1(K, A) \to H_{{\mathcal{F}}^{\mathfrak{m}}}^1(K,A)
\stackrel{\bigoplus_{{\mathfrak{q}}\mid {\mathfrak{m}}/ {\mathfrak{n}}} v_{\mathfrak{q}}}{\to} R^{\nu({\mathfrak{m}}/ {\mathfrak{n}})},\end{aligned}$$ we see that ${{\bigwedge}}_{{\mathfrak{q}}\mid {\mathfrak{m}}/{\mathfrak{n}}}v_{\mathfrak{q}}$ induces $$\begin{aligned}
{{\bigwedge}}_{{\mathfrak{q}}\mid {\mathfrak{m}}/{\mathfrak{n}}}v_{\mathfrak{q}}\colon {{\bigcap}}_R^{r+\nu({\mathfrak{m}})} H_{{\mathcal{F}}^{\mathfrak{m}}}^1(K, A) \to
{{\bigcap}}_R^{r+\nu({\mathfrak{n}})} H_{{\mathcal{F}}^{\mathfrak{n}}}^1(K, A),\end{aligned}$$ by Proposition \[prop self injective\]. We denote this map by $v_{{\mathfrak{m}},{\mathfrak{n}}}$. Note that $v_{{\mathfrak{m}},{\mathfrak{n}}}=\bigwedge_{{\mathfrak{q}}\mid {\mathfrak{m}}/{\mathfrak{n}}}v_{\mathfrak{q}}$ is well-defined up to sign, but one can explicitly choose a sign for each pair $({\mathfrak{m}},{\mathfrak{n}})$ so that one has $v_{{\mathfrak{m}}', {\mathfrak{n}}}=v_{{\mathfrak{m}}, {\mathfrak{n}}} \circ v_{{\mathfrak{m}}',{\mathfrak{m}}}$ when ${\mathfrak{n}}\mid {\mathfrak{m}}\mid {\mathfrak{m}}'$. (For an explicit choice of sign, see [@sbA §3.1].) Thus the collection $\{v_{{\mathfrak{m}},{\mathfrak{n}}}\}_{{\mathfrak{m}},{\mathfrak{n}}}$ forms an inverse system.
By definition, a Stark system of rank $r$ (for ($A,{\mathcal{F}}$)) is an element $$\begin{aligned}
(\epsilon_{\mathfrak{n}})_{\mathfrak{n}}\in \prod_{{\mathfrak{n}}\in{\mathcal{N}}} {{\bigcap}}_{R}^{r + \nu({\mathfrak{n}})} H_{{\mathcal{F}}^{\mathfrak{n}}}^1(K, A)\end{aligned}$$ that satisfies $v_{{\mathfrak{m}},{\mathfrak{n}}}(\epsilon_{\mathfrak{m}})=\epsilon_{{\mathfrak{n}}}$ for all ideals ${\mathfrak{m}}$ and ${\mathfrak{n}}$ in ${\mathcal{N}}$ with ${\mathfrak{n}}\mid {\mathfrak{m}}$.
Stark systems and Selmer modules
--------------------------------
We can now define the key invariants that are associated to a Stark system.
In this definition we use the fact that each element of ${\bigcap}_R^{r+\nu({\mathfrak{n}})} H^1_{{\mathcal{F}}^{\mathfrak{n}}}(K,A)$ is, by its very definition, a homomorphism of $R$-modules from ${\bigwedge}_R^{r+\nu({\mathfrak{n}})}H^1_{{\mathcal{F}}^{\mathfrak{n}}}(K, A)^*$ to $R$.
\[stark ideal\] Let $\epsilon = ( \epsilon_{\mathfrak{n}})_{\mathfrak{n}}\in {\rm SS}_r(A, {\mathcal{F}})$. Then for each non-negative integer $i$ we define an ideal $I_i(\epsilon)$ of $R$ by setting $$\begin{aligned}
I_i(\epsilon) := \sum_{{\mathfrak{n}}\in {\mathcal{N}},\ \nu({\mathfrak{n}}) = i} {\rm im}(\epsilon_{\mathfrak{n}}).\end{aligned}$$
We consider the following hypothesis.
\[hyp large\] There exists an ideal ${\mathfrak{n}}$ in ${\mathcal{N}}$ such that $H^1_{({\mathcal{F}}^\ast)_{\mathfrak{n}}}(K,A^\ast(1))$ vanishes and $H^1_{{\mathcal{F}}^{\mathfrak{n}}}(K, A)$ is a free $R$-module of rank $r +\nu({\mathfrak{n}})$.
\[free rem\] The following observation will frequently be used: if one assumes Hypothesis \[hyp large\], then, for any ideal ${\mathfrak{m}}$ in ${\mathcal{N}}$ for which the group $H^1_{({\mathcal{F}}^\ast)_{\mathfrak{m}}}(K,A^\ast(1))$ vanishes, the $R$-module $H^1_{{\mathcal{F}}^{\mathfrak{m}}}(K, A)$ is free of rank $r +\nu({\mathfrak{m}})$. In fact, let ${\mathfrak{n}}\in {\mathcal{N}}$ be an ideal as in Hypothesis \[hyp large\] and ${\mathfrak{m}}\in {\mathcal{N}}$ with $H^1_{({\mathcal{F}}^\ast)_{\mathfrak{m}}}(K,A^\ast(1))=0$. Take an ideal ${\mathfrak{d}}\in {\mathcal{N}}$ such that ${\mathfrak{m}}\mid {\mathfrak{d}}$ and ${\mathfrak{n}}\mid {\mathfrak{d}}$. Then, by the global duality, we have exact sequences $$0 \to H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{d}}}}(K, A) \to \bigoplus_{{\mathfrak{q}}\mid \frac{{\mathfrak{d}}}{{\mathfrak{n}}}}H^{1}_{/f}(K_{{\mathfrak{q}}}, A) \to 0$$ and $$0 \to H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, A) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{d}}}}(K, A) \to \bigoplus_{{\mathfrak{q}}\mid \frac{{\mathfrak{d}}}{{\mathfrak{m}}}}H^{1}_{/f}(K_{{\mathfrak{q}}}, A) \to 0.$$ By Hypothesis \[hyp large\] and the definition of ${\mathcal{P}}$, we have $H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \simeq R^{r+\nu({\mathfrak{n}})}$ and $H^{1}_{/f}(K_{{\mathfrak{q}}}, A) \simeq R$ for any ${\mathfrak{q}}\in {\mathcal{P}}$, and so $H^{1}_{{\mathcal{F}}^{{\mathfrak{d}}}}(K, A) \simeq R^{r + \nu({\mathfrak{d}})}$. Since $R$ is a local ring, a projective $R$-module is free. Hence we conclude that $H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, A) \simeq R^{r+\nu({\mathfrak{m}})}$ by using the second (split) exact sequence.
\[fitt-lemma\] Assume Hypothesis \[hyp large\]. Let $i$ be a positive integer and ${\mathfrak{n}}\in {\mathcal{N}}$ with $\nu({\mathfrak{n}}) \geq i$. Suppose that $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{n}}}}(K, A^{*}(1))$ vanishes. Then we have $$\begin{aligned}
\operatorname{Fitt}_{R}^{i}\left(H^{1}_{{\mathcal{F}}^{*}}(K, A^{*}(1))^{*}\right)
&= \sum_{ {\mathfrak{q}}\in {\mathcal{P}}, \ {\mathfrak{q}}\mid {\mathfrak{n}}} \operatorname{Fitt}_{R}^{i-1}\left(H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{q}}}}(K, A^{*}(1))^{*}\right)
\\
&= \sum_{{\mathfrak{q}}\in {\mathcal{P}}} \operatorname{Fitt}_{R}^{i-1}\left(H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{q}}}}(K, A^{*}(1))^{*}\right).\end{aligned}$$
By Hypothesis \[hyp large\], we see that, for any prime ${\mathfrak{q}}\in {\mathcal{P}}$, there is an ideal ${\mathfrak{n}}\in {\mathcal{N}}$ with ${\mathfrak{q}}\mid {\mathfrak{n}}$ such that $\nu({\mathfrak{n}})\geq i$ and $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{n}}}}(K, A^{*}(1)) = 0$. Hence the first equality implies the second equality. To show the first equality, take an ideal ${\mathfrak{n}}\in {\mathcal{N}}$ with $\nu({\mathfrak{n}}) \geq i$ such that $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{n}}}}(K, A^{*}(1)) = 0$. Note that $H^{1}_{{\mathcal{F}}^{{{\mathfrak{n}}}}}(K, A) \simeq R^{r+\nu({\mathfrak{n}})}$ by Remark \[free rem\]. Then, by the global duality, we have exact sequences $$H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}}H^{1}_{/f}(K_{{\mathfrak{q}}}, A) \to H^{1}_{{\mathcal{F}}^{*}}(K, A^{*}(1))^{*} \to 0$$ and $$H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \to \bigoplus_{{\mathfrak{q}}\mid \frac{{\mathfrak{n}}}{{\mathfrak{r}}}}H^{1}_{/f}(K_{{\mathfrak{q}}}, A) \to H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{r}}}}(K, A^{*}(1))^{*} \to 0$$ for any prime ${\mathfrak{r}}\mid {\mathfrak{n}}$. Since $H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \simeq R^{r + \nu({\mathfrak{n}})}$ and $H^{1}_{/f}(K_{{\mathfrak{q}}}, A) \simeq R$ for any prime ${\mathfrak{q}}\in {\mathcal{P}}$, these sequences give finite presentations of $H^{1}_{{\mathcal{F}}^{*}}(K, A^{*}(1))^{*}$ and $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{r}}}}(K, A^{*}(1))^{*}$ respectively. Hence we see that $$\operatorname{Fitt}_{R}^{i}\left(H^{1}_{{\mathcal{F}}^{*}}(K, A^{*}(1))^{*}\right)
= \sum_{ {\mathfrak{q}}\mid {\mathfrak{n}}} \operatorname{Fitt}_{R}^{i-1}\left(H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{q}}}}(K, A^{*}(1))^{*}\right).$$
Using Lemma \[fitt-lemma\] repeatedly, we deduce the following
Assume Hypothesis \[hyp large\]. Then, for any non-negative integer $i$, we have $$\begin{aligned}
\operatorname{Fitt}_{R}^{i}\left(H^{1}_{{\mathcal{F}}^{*}}(K, A^{*}(1))^{*}\right)
= \sum_{{\mathfrak{m}}\in {\mathcal{N}}, \ \nu({\mathfrak{m}}) = i} \operatorname{Fitt}_{R}^{0}\left(H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{m}}}}(K, A^{*}(1))^{*}\right).\end{aligned}$$
We can now state one of the main results in the theory of Stark systems.
\[thm stark\] Under Hypothesis \[hyp large\] all of the following claims are valid.
- Let ${\mathfrak{n}}\in {\mathcal{N}}$ with $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{n}}}}(K, A^{*}(1)) = 0$. Then the natural projection homomorphism $${\rm SS}_{r}(A, {\mathcal{F}}) \to {\bigcap}^{r+\nu({\mathfrak{n}})}_{R}H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A); \ \epsilon \mapsto \epsilon_{\mathfrak{n}}$$ is bijective. In particular, the $R$-module ${\rm SS}_r(A,{\mathcal{F}})$ is free of rank one.
- The following claims are valid for all $\epsilon$ in ${\rm SS}_{r}(A, {\mathcal{F}})$ and all $i \ge 0$.
- $I_i(\epsilon)\subseteq I_{i+1}(\epsilon)$, with equality for all sufficiently large $i$.
- $I_\infty(\epsilon):= \bigcup_{i \ge 0}I_i(\epsilon)$ is equal to $R$ if and only if $\epsilon$ is a basis of ${\rm SS}_{r}(A, {\mathcal{F}})$.
- $I_i(\epsilon) = I_\infty(\epsilon)\cdot {\rm Fitt}_{R}^i \left( H^1_{{\mathcal{F}}^\ast}(K, A^*(1))^* \right).$
Both claim (i) and, in the case that $\epsilon$ is a basis of ${\rm SS}_{r}(A, {\mathcal{F}})$, the equality $I_i(\epsilon) = {\rm Fitt}_{R}^i \left( H^1_{{\mathcal{F}}^\ast}(K, A^*(1))^* \right)$ in claim (ii)(c) were independently proved by the first and third authors in [@sbA Th. 3.17 and 3.19(ii)] and by the second author in [@sakamoto Th. 4.7 and 4.10].
To deduce the remainder of claim (ii) we fix a basis $\epsilon_0$ of ${\rm SS}_{r}(A, {\mathcal{F}})$ and then for each $\epsilon$ in ${\rm SS}_{r}(A, {\mathcal{F}})$ define $\lambda_\epsilon \in R$ by the equality $\epsilon = \lambda_\epsilon\cdot \epsilon_0$. Then for each $i \ge 0$ one has $$\label{gen case} I_i(\epsilon) = \lambda_\epsilon\cdot I_i(\epsilon_0) = \lambda_\epsilon\cdot {\rm Fitt}_{R}^i \left( H^1_{{\mathcal{F}}^\ast}(K, A^*(1))^* \right).$$ Claim (ii)(a) is thus true since the definition of higher Fitting ideal implies both that $${\rm Fitt}_{R}^i \left( H^1_{{\mathcal{F}}^\ast}(K, A^*(1))^* \right) \subseteq {\rm Fitt}_{R}^{i+1} \left( H^1_{{\mathcal{F}}^\ast}(K, A^*(1))^* \right)$$ and ${\rm Fitt}_{R}^i \left( H^1_{{\mathcal{F}}^\ast}(K, A^*(1))^* \right) = R$ for all sufficiently large $i$.
The latter fact also combines with (\[gen case\]) to imply $I_i(\epsilon) = (\lambda_\epsilon)$ for all sufficiently large $i$, and hence that $I_\infty(\epsilon) = (\lambda_\epsilon)$. This equality implies claim (ii)(b) directly and also shows that (\[gen case\]) implies claim (ii)(c).
Stark Systems over Gorenstein orders {#one-dim case}
------------------------------------
Let $Q$ be a finite extension of ${\mathbb{Q}}_{p}$ and ${\mathcal{O}}$ the ring of integers of $Q$. Let ${\mathcal{Q}}$ be a finite-dimensional semisimple commutative $Q$-algebra. Let $({\mathcal{R}}, {\mathfrak{p}})$ be a local Gorenstein ${\mathcal{O}}$-order in ${\mathcal{Q}}$ (for basic properties of Gorenstein orders, see [@sbA §A.3]). Note that ${\mathcal{R}}/(p^{m})$ is a zero-dimensional Gorenstein local ring since $p$ is a regular element of ${\mathcal{R}}$.
Let $T$ be a free ${\mathcal{R}}$-module of finite rank with an ${\mathcal{R}}$-linear continuous action of $G_{K}$ which is unramified outside a finite set of places of $K$. Let $K(T)$ denote the minimal Galois extension of $K$ such that $G_{K(T)}$ acts trivially on $T$. Recall that $K_{p^{m}} := K (\mu_{p^{m}}, ({\mathcal{O}}^\times_K)^{1/p^{m}}) K(1)$ and $K(T)_{p^{m}} := K(T)K_{p^{m}}$ for any positive integer $m$. Set $K(T)_{p^{\infty}} := \bigcup_{m >0}K(T)_{p^{m}}$ and $\overline{T} := T/{\mathfrak{p}}T$.
For an ${\mathcal{R}}$-module $X$ we endow the Pontryagin dual $X^\vee := \operatorname{Hom}_{\mathcal{O}}(X,Q/{\mathcal{O}})$ with the natural action of ${\mathcal{R}}$. For any positive integer $m$ and any ${\mathcal{R}}/(p^m)$-module $X$, the module $X^\vee $ is naturally isomorphic to $X^\ast:=\operatorname{Hom}_{{\mathcal{R}}/(p^m)}(X, {\mathcal{R}}/(p^m))$ and in such cases we often identify the functors $(-)^\vee$ and $(-)^\ast$.
In this subsection, we assume the following hypothesis:
\[hyp1’\]
- $\overline{T}$ is an irreducible $({\mathcal{R}}/{\mathfrak{p}})[G_{K}]$-module;
- there exists $\tau \in G_{K(T)_{p^{\infty}}}$ such that $T/(\tau-1)T \simeq {\mathcal{R}}$ as ${\mathcal{R}}$-modules;
- $H^{1}(K(T)_{p^{{\infty}}}/K, \overline{T}) = H^{1}(K(T)_{p^{{\infty}}}/K, \overline{T}^{\vee}(1)) = 0$;
\[hyp(iv)\] Hypothesis \[hyp1’\] implies that $\overline{T}^{G_{K}}$ and $\overline{T}^{\vee}(1)^{G_{K}}$ both vanish. To see this note that if $\overline{T}^{G_{K}}$ does not vanish, then Hypothesis \[hyp1’\](i) implies $\dim_{\Bbbk}(\overline{T}) = 1$ and hence that $\overline{T}$ is the trivial $G_{K}$-representation. But then, in this case, the module $$H^{1}(K(T)_{p^{{\infty}}}/K, \overline{T}) = \operatorname{Hom}\left(\operatorname{Gal}(K(T)_{p^{{\infty}}}/K), \overline{T}\right)$$ does not vanish since there is a non-trivial $p$-subextension of $K(T)_{p^{{\infty}}}/K$ and this contradicts Hypothesis \[hyp1’\](iii). The vanishing of $\overline{T}^{\vee}(1)^{G_{K}}$ is proved by a similar argument.
If we assume Hypothesis \[hyp1’\], then $T/p^{m}T$ satisfies Hypotheses \[hyp1\] and \[hyp2\] for any positive integer $m$. In fact, it clearly satisfies Hypotheses \[hyp1\](i) and (ii). By Remark \[hyp(iv)\], Hypothesis \[hyp2\] also holds true. We shall show Hypothesis \[hyp1\](iii), i.e. that, setting $A := T/p^{m}T$, one has $$H^{1}(K(A)_{p^{m}}/K, A) = H^{1}(K(A)_{p^{m}}/K, A^{*}(1)) = 0.$$
Since the inflation map $H^{1}(K(A)_{p^{m}}/K, \overline{T}) \to H^{1}(K(T)_{p^{\infty}}/K, \overline{T})$ is injective, we have $H^{1}(K(A)_{p^{m}}/K, \overline{T}) = 0$ by Hypothesis \[hyp1’\](iii). Let $i$ be a non-negative integer. By the exact sequence $0 \to {\mathfrak{p}}^{i+1}A \to {\mathfrak{p}}^{i}A \to {\mathfrak{p}}^{i}A/{\mathfrak{p}}^{i+1}A \to 0$ and Remark \[hyp(iv)\] we have an exact sequence $$0 \to H^{1}(K(A)_{p^{m}}/K, {\mathfrak{p}}^{i+1}A) \to H^{1}(K(A)_{p^{m}}/K, {\mathfrak{p}}^{i}A) \to H^{1}(K(A)_{p^{m}}/K, {\mathfrak{p}}^{i}A/{\mathfrak{p}}^{i+1}A).$$ Since the $({\mathcal{R}}/{\mathfrak{p}})[G_{K}]$-module ${\mathfrak{p}}^{i}A/{\mathfrak{p}}^{i+1}A$ is isomorphic to a direct sum of $\overline{T}$, the group $H^{1}(K(A)_{p^{m}}/K, {\mathfrak{p}}^{i}A/{\mathfrak{p}}^{i+1}A)$ vanishes and so the natural map $$H^{1}(K(A)_{p^{m}}/K, {\mathfrak{p}}^{i+1}A) \to H^{1}(K(A)_{p^{m}}/K, {\mathfrak{p}}^{i}A)$$ is bijective. Since $i$ is arbitrary and ${\mathfrak{p}}^{m}A = 0$, we conclude that $H^{1}(K(A)_{p^{m}}/K, A)$ vanishes. The vanishing of $H^{1}(K(A)_{p^{m}}/K, A^{*}(1))$ is proved by a similar argument.
We fix a Selmer structure ${\mathcal{F}}$ on $T$. For a positive integer $m$, let $\mathcal{P}_{m}$ denote the set of primes ${\mathfrak{q}}\not\in S({\mathcal{F}})$ of $K$ such that ${\rm Fr}_{\mathfrak{q}}$ is conjugate to $\tau$ in $\operatorname{Gal}(K(T/p^{m}T)_{p^{m}}/K)$. Put ${\mathcal{N}}_{m}={\mathcal{N}}({\mathcal{P}}_{m})$. Note that ${\mathcal{N}}_{m+1} \subseteq {\mathcal{N}}_{m}$ for any positive integer $m$.
We suppose that $(T/p^{m}T, {\mathcal{F}}, {\mathcal{P}}_{m})$ satisfies Hypothesis \[hyp large\] for any positive integer $m$. Let $m$ be a positive integer and let ${\mathfrak{m}}$ and ${\mathfrak{n}}$ be ideals of ${\mathcal{N}}_{m+1}$ such that ${\mathfrak{m}}\mid {\mathfrak{n}}$ and $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{n}}}}(K, (T/p^{m+1}T)^{\vee}(1))$ vanishes. Then $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{n}}}}(K, (T/p^{m}T)^{\vee}(1))$ vanishes by Corollary \[dualselisom\] and so $H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m+1}T) \otimes_{{\mathcal{R}}/(p^{m+1})} {\mathcal{R}}/(p^{m})$ and $H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m}T)$ are free ${\mathcal{R}}/(p^{m})$-modules of the same rank by Remark \[free rem\]. Hence by Lemma \[injective\], the natural homomorphism $$H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m+1}T) \otimes_{{\mathcal{R}}/(p^{m+1})} {\mathcal{R}}/(p^{m}) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m}T)$$ is an isomorphism. Hence there is a canonical injection $$H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, T/p^{m}T) \to
H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m+1}T) \otimes_{{\mathcal{R}}/(p^{m+1})} {\mathcal{R}}/(p^{m}).$$ Applying Corollary \[morph\] with $X = H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, T/p^{m+1}T)$, $Y = H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, T/p^{m}T)$, and $F = H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m+1}T)$, we get a natural homomorphism $${\bigcap}^{r+\nu({\mathfrak{m}})}_{{\mathcal{R}}/(p^{m+1})}H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, T/p^{m+1}T) \to {\bigcap}^{r+\nu({\mathfrak{m}})}_{{\mathcal{R}}/(p^{m})}H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, T/p^{m}T).$$ Since ${\mathfrak{m}}$ is any element of ${\mathcal{N}}_{m+1}$, we get a homomorphism $${\rm SS}_{r}(T/p^{m+1}T, {\mathcal{F}}) \to {\rm SS}_{r}(T/p^{m}T, {\mathcal{F}}).$$
\[compatible\] Let $m$ be a positive integer.
- The map ${\rm SS}_{r}(T/p^{m+1}T, {\mathcal{F}}) \to {\rm SS}_{r}(T/p^{m}T, {\mathcal{F}})$ is surjective.
- Fix $\epsilon^{(m+1)}$ in ${\rm SS}_{r}(T/p^{m+1}T, {\mathcal{F}})$ and write $\epsilon^{(m)}$ for its image in ${\rm SS}_{r}(T/p^{m}T, {\mathcal{F}})$. Then for each non-negative integer $i$ one has $I_{i}(\epsilon^{(m+1)}){\mathcal{R}}/(p^m) = I_{i}(\epsilon^{(m)})$.
Take an ideal ${\mathfrak{n}}\in {\mathcal{N}}_{m+1}$ such that $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{n}}}}(K, (T/p^{m+1}T)^{\vee}(1))$ vanishes. Then we have the following commutative diagram: $$\xymatrix{
{\rm SS}_{r}(T/p^{m+1}T, {\mathcal{F}}) \ar[d] \ar[r] & {{\bigcap}}^{r+\nu({\mathfrak{n}})}_{{\mathcal{R}}/(p^{m+1})}H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m+1}T) \ar[d]
\\
{\rm SS}_{r}(T/p^{m}T, {\mathcal{F}}) \ar[r] & {{\bigcap}}^{r+\nu({\mathfrak{n}})}_{{\mathcal{R}}/(p^{m})}H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m}T).
}$$ The horizontal maps are isomorphisms by Theorem \[thm stark\](i) and the right vertical map is surjective by the commutativity of the diagram in Corollary \[morph\] and that $$H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m+1}T) \otimes_{{\mathcal{R}}/(p^{m+1})} {\mathcal{R}}/(p^{m}) \simeq H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, T/p^{m}T) \simeq ({\mathcal{R}}/(p^{m}))^{r+\nu({\mathfrak{n}})}.$$ Hence the map ${\rm SS}_{r}(T/p^{m+1}T, {\mathcal{F}}) \to {\rm SS}_{r}(T/p^{m}T, {\mathcal{F}})$ is surjective.
We will show claim (ii). We may assume that $\epsilon^{(m+1)}$ is a basis of ${\rm SS}_{r}(T/p^{m+1}T, {\mathcal{F}})$. By claim (i), $\epsilon^{(m)}$ is also a basis of ${\rm SS}_{r}(T/p^{m}T, {\mathcal{F}})$. Then we have $$\begin{aligned}
I_{i}(\epsilon^{(m+1)}){\mathcal{R}}/(p^m) &= {\rm Fitt}_{{\mathcal{R}}/(p^{m+1})}^{i}\left(H^{1}_{{\mathcal{F}}^{*}}(K, (T/p^{m+1}T)^{\vee}(1))^{\vee}\right){\mathcal{R}}/(p^m)
\\
&= {\rm Fitt}_{{\mathcal{R}}/(p^{m})}^{i}\left( \left( H^{1}_{{\mathcal{F}}^{*}}(K, (T/p^{m+1}T)^{\vee}(1))[p^{m}]\right)^{\vee} \right)
\\
&= {\rm Fitt}_{{\mathcal{R}}/(p^{m})}^{i}\left( H^{1}_{{\mathcal{F}}^{*}}(K, (T/p^{m}T)^{\vee}(1))^{\vee}\right)
\\
&= I_{i}(\epsilon^{(m)})\end{aligned}$$ where the first and forth equality follows from Theorem \[thm stark\](ii) and the third equality follows from Corollary \[dualselisom\].
\[stark ideal gorenstein\] We define the module ${\rm SS}_{r}(T, {\mathcal{F}})$ of Stark systems of rank $r$ for $(T, {\mathcal{F}})$ to be the inverse limit $${\rm SS}_{r}(T, {\mathcal{F}}) := \varprojlim_{m \in {\mathbb{Z}}_{>0}}{\rm SS}_{r}(T/p^{m}T, {\mathcal{F}}).$$ Let $i$ be non-negative integer and $\epsilon = (\epsilon^{(m)})_{m} \in {\rm SS}_{r}(T, {\mathcal{F}})$. Then, by Lemma \[compatible\](ii), we see that the family $(I_{i}(\epsilon^{(m)}))_{m}$ is an inverse system. We define an ideal $I_{i}(\epsilon)$ of ${\mathcal{R}}$ to be the inverse limit $$I_{i}(\epsilon) := \varprojlim_{m} I_{i}(\epsilon^{(m)}).$$
\[thm stark’\]
- The ${\mathcal{R}}$-module ${\rm SS}_{r}(T, {\mathcal{F}})$ is free of rank one.
- The following claims are valid for all $\epsilon$ in ${\rm SS}_{r}(T,{\mathcal{F}})$ and all $i \ge 0$.
- $I_i(\epsilon)\subseteq I_{i+1}(\epsilon)$, with equality for all sufficiently large $i$.
- $I_\infty(\epsilon):= \bigcup_{i \ge 0}I_i(\epsilon)$ is equal to ${\mathcal{R}}$ if and only if $\epsilon$ is a basis of ${\rm SS}_{r}(A, {\mathcal{F}})$.
- $I_i(\epsilon) = I_\infty(\epsilon)\cdot {\rm Fitt}_{{\mathcal{R}}}^i \left( H^1_{{\mathcal{F}}^\ast}(K, T^{\vee}(1))^{\vee}\right).$
Claim (i) follows directly from Lemma \[compatible\](i) and Theorem \[thm stark\](i).
To prove claim (ii) it is enough, just as with the proof of Theorem \[thm stark\](ii), to show that if $\epsilon = (\epsilon^{(m)})_{m}$ is a basis of ${\rm SS}_{r}(T, {\mathcal{F}})$, then for each non-negative integer $i$ one has $
I_{i}(\epsilon) = \operatorname{Fitt}_{{\mathcal{R}}}^{i}(H^{1}_{{\mathcal{F}}^{*}}(K, T^{\vee}(1))^{\vee}).$
In this case, each element $\epsilon^{(m)}$ is a basis of ${\rm SS}_{r}(T/(p^m), {\mathcal{F}})$ over ${\mathcal{R}}/(p^m)$ and so Theorem \[thm stark\](ii) implies that $$\begin{aligned}
I_{i}(\epsilon) &= \varprojlim_{m} \operatorname{Fitt}_{{\mathcal{R}}/(p^{m})}^{i}\left(H^{1}_{{\mathcal{F}}^{*}}(K, (T/p^{m}T)^{\vee}(1))^{\vee}\right)
\\
&= \varprojlim_{m} \operatorname{Fitt}_{{\mathcal{R}}}^{i}\left(H^{1}_{{\mathcal{F}}^{*}}(K, T^{\vee}(1))^{\vee}\right){\mathcal{R}}/(p^{m})\\
&= \operatorname{Fitt}_{{\mathcal{R}}}^{i}\left(H^{1}_{{\mathcal{F}}^{*}}(K, T^{\vee}(1))^{\vee}\right),\end{aligned}$$ where the second equality follows from Corollary \[dualselisom\] and the last from the fact that ${\mathcal{R}}$ is a complete noetherian ring and so every ideal is closed.
Kolyvagin systems {#koly sys sec}
=================
In this section, we develop the theory of higher rank Kolyvagin systems. We continue to use the notation introduced in §\[pre\].
Definition {#defkoly}
----------
In the following, we suppose $r>0$. (We do not define Kolyvagin systems of rank zero).
We recall the definition of Kolyvagin systems. As in §\[defstark\], we fix an isomorphism $H_{/f}^1(K_{\mathfrak{q}},A)\simeq R$ for each ${\mathfrak{q}}\in {\mathcal{P}}$ and identify them. We will again use the map $v_{\mathfrak{q}}$, which is defined by $$v_{\mathfrak{q}}: H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A) \stackrel{{\rm loc}_{\mathfrak{q}}}{\to} H^1(K_{\mathfrak{q}}, A) \to H^1_{/f}(K_{\mathfrak{q}},A)=R.$$ If ${\mathfrak{q}}\mid {\mathfrak{n}}$, this map induces $$v_{\mathfrak{q}}: {\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A)\otimes G_{\mathfrak{n}}\to {\bigcap}_R^{r-1}H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}}/{\mathfrak{q}})}(K,A)\otimes G_{\mathfrak{n}}.$$ (For the definition of $G_{\mathfrak{n}}$, see §\[section hyp\].)
The finite-singular comparison map $$\varphi_{\mathfrak{q}}^{\rm fs}: H^1_{{\mathcal{F}}({\mathfrak{n}}/{\mathfrak{q}})}(K,A) \stackrel{{\rm loc}_{\mathfrak{q}}}{\to} H_f^1(K_{\mathfrak{q}}, A) \stackrel{\varphi_{\mathfrak{q}}^{\rm fs}}{\to} H^1_{\rm tr}(K_{\mathfrak{q}},A)\otimes G_{\mathfrak{q}}=H^1_{/f}(K_{\mathfrak{q}},A)\otimes G_{\mathfrak{q}}=R\otimes G_{\mathfrak{q}},$$ which is defined in §\[section fs\], induces $$\varphi_{\mathfrak{q}}^{\rm fs} :{\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}}/{\mathfrak{q}})}(K,A)\otimes G_{{\mathfrak{n}}/{\mathfrak{q}}} \to {\bigcap}_R^{r-1}H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}}/{\mathfrak{q}})}(K,A)\otimes G_{\mathfrak{n}}.$$ A Kolyvagin system of rank $r$ (for $(A,{\mathcal{F}})$) is an element $$(\kappa_{\mathfrak{n}})_{\mathfrak{n}}\in \prod_{{\mathfrak{n}}\in {\mathcal{N}}}{\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A)\otimes G_{\mathfrak{n}}$$ which satisfies the ‘finite-singular relation’ $$v_{\mathfrak{q}}(\kappa_{\mathfrak{n}})=\varphi_{\mathfrak{q}}^{\rm fs}(\kappa_{{\mathfrak{n}}/{\mathfrak{q}}}) \ \text{ in } \ {\bigcap}_R^{r-1}H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}}/{\mathfrak{q}})}(K,A)\otimes G_{\mathfrak{n}}.$$ The set of all Kolyvagin systems of rank $r$ is denoted by ${\rm KS}_r(A,{\mathcal{F}})$. This is an $R$-submodule of $\prod_{{\mathfrak{n}}\in {\mathcal{N}}}{\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A)\otimes G_{\mathfrak{n}}$.
Regulator maps {#sec regulator}
--------------
We quickly review the relation between Kolyvagin and Stark systems (see [@sbA §4.2]).
There exists a canonical homomorphism of $R$-modules $${\rm Reg}_r: {\rm SS}_r(A,{\mathcal{F}}) \to {\rm KS}_r(A,{\mathcal{F}}),$$ which is referred to in loc. cit. as a ‘regulator map’. The definition of this map is as follows. Let $\epsilon=(\epsilon_{\mathfrak{n}})_{\mathfrak{n}}\in {\rm SS}_r(A,{\mathcal{F}})$. For each ${\mathfrak{n}}\in {\mathcal{N}}$, we define $$\kappa(\epsilon_{\mathfrak{n}}):=\left({\bigwedge}_{{\mathfrak{q}}\mid {\mathfrak{n}}}\varphi_{\mathfrak{q}}^{\rm fs} \right)(\epsilon_{\mathfrak{n}})\in {\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A)\otimes G_{\mathfrak{n}},$$ where we regard ${\bigwedge}_{{\mathfrak{q}}\mid {\mathfrak{n}}}\varphi_{\mathfrak{q}}^{\rm fs} $ as a map $${\bigwedge}_{{\mathfrak{q}}\mid {\mathfrak{n}}}\varphi_{\mathfrak{q}}^{\rm fs} : {\bigcap}_R^{r+\nu({\mathfrak{n}})} H^1_{{\mathcal{F}}^{\mathfrak{n}}}(K,A) \to {\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A)\otimes G_{\mathfrak{n}}.$$ Then one sees that $(\kappa(\epsilon_{\mathfrak{n}}))_{\mathfrak{n}}$ is a Kolyvagin system. In fact, if ${\mathfrak{q}}\mid {\mathfrak{n}}$, then we have $$\begin{aligned}
v_{\mathfrak{q}}(\kappa(\epsilon_{\mathfrak{n}}))&=&v_{\mathfrak{q}}\left(\left({\bigwedge}_{{\mathfrak{q}}' \mid {\mathfrak{n}}}\varphi_{{\mathfrak{q}}'}^{\rm fs} \right)(\epsilon_{\mathfrak{n}})\right) \\
&=&\varphi_{\mathfrak{q}}^{\rm fs}\left( \left({\bigwedge}_{{\mathfrak{q}}' \mid {\mathfrak{n}}/{\mathfrak{q}}}\varphi_{{\mathfrak{q}}'}^{\rm fs} \right)(v_{\mathfrak{q}}(\epsilon_{\mathfrak{n}}))\right) \\
&=& \varphi_{\mathfrak{q}}^{\rm fs}\left( \left({\bigwedge}_{{\mathfrak{q}}' \mid {\mathfrak{n}}/{\mathfrak{q}}}\varphi_{{\mathfrak{q}}'}^{\rm fs} \right)(\epsilon_{{\mathfrak{n}}/{\mathfrak{q}}})\right) \\
&=& \varphi_{\mathfrak{q}}^{\rm fs}(\kappa(\epsilon_{{\mathfrak{n}}/{\mathfrak{q}}})).\end{aligned}$$ The regulator map is defined by setting ${\rm Reg}_r(\epsilon):=(\kappa(\epsilon_{\mathfrak{n}}))_{\mathfrak{n}}.$
Kolyvagin systems and Selmer modules
------------------------------------
For each $\kappa \in {\rm KS}_r(A,{\mathcal{F}})$, we can associate it with invariants $I_i(\kappa)$, similarly to the case of Stark systems (see Definition \[stark ideal\]).
\[koly ideal\] Let $\kappa \in {\rm KS}_{r}(A, {\mathcal{F}})$. We fix a generator of $G_{\mathfrak{q}}$ for each ${\mathfrak{q}}\in {\mathcal{P}}$ and regard $${\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A)\otimes G_{\mathfrak{n}}={\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A)$$ for every ${\mathfrak{n}}\in {\mathcal{N}}$. For a non-negative integer $i$, define an ideal $I_i(\kappa)$ of $R$ by $$\begin{aligned}
I_i(\kappa) := \sum_{{\mathfrak{n}}\in {\mathcal{N}}, \ \nu({\mathfrak{n}}) = i}{\rm im}(\kappa_{\mathfrak{n}}),\end{aligned}$$ where we regard $\kappa_{\mathfrak{n}}\in {\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A) =\operatorname{Hom}_R \left( {\bigwedge}^r_R H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A)^*, R \right)$. These ideals, of course, do not depend on the choice of a generator of $G_{\mathfrak{q}}$ for each ${\mathfrak{q}}$.
The aim of this subsection is to prove the following theorem, which is one of the main results in this paper.
\[main\] Assume Hypotheses \[hyp1\], \[hyp2\] and \[hyp large\], and also suppose $p > 3$.
- The regulator map $$\begin{aligned}
{\rm Reg}_r \colon {\rm SS}_r(A, {\mathcal{F}}) \to {\rm KS}_r(A, {\mathcal{F}})\end{aligned}$$ is an isomorphism. In particular, the $R$-module ${\rm KS}_r(A,{\mathcal{F}})$ is free of rank one.
- Fix $\kappa$ in ${\rm KS}_r(A,{\mathcal{F}})$. Then for each ideal ${\mathfrak{n}}$ in ${\mathcal{N}}$ one has $$\begin{aligned}
{\rm im}(\kappa_{\mathfrak{n}}) \subseteq {\rm Fitt}_R^0 ( H^1_{{\mathcal{F}}({\mathfrak{n}})^*}(K, A^*(1))^*),\end{aligned}$$ with equality if $\kappa$ is a basis of ${\rm KS}_r(A, {\mathcal{F}})$. In particular, if $\kappa$ is a basis, then $$\begin{aligned}
{\rm im}(\kappa_1) = {\rm Fitt}_R^0( H^1_{{\mathcal{F}}^*}(K, A^*(1))^* ).\end{aligned}$$
- Fix $\kappa$ in ${\rm KS}_r(A,{\mathcal{F}})$. Then for each non-negative integer $i$ one has $$I_i(\kappa)\subseteq {\rm Fitt}_R^i(H^1_{{\mathcal{F}}^\ast}(K,A^\ast(1))^\ast),$$ with equality if $R$ is a principal ideal ring and $\kappa$ is a basis of ${\rm KS}_r(A,{\mathcal{F}})$.
\[difficulty remark\] For each $\kappa$ in ${\rm KS}_r(A,{\mathcal{F}})$ and each non-negative integer $i$, Theorem \[main\](i) allows us to define a canonical ideal of $R$ by setting $$I'_i(\kappa) := I_i({\rm Reg}_r^{-1}(\kappa)),$$ where the right hand side is as defined in Definition \[stark ideal\]. If $R$ is a principal ideal ring and $\kappa$ is a basis of ${\rm KS}_r(A,{\mathcal{F}})$, then Theorems \[thm stark\](ii) and \[main\](iii) combine to imply that $I'_i(\kappa) = I_i(\kappa)$ for all $i$. It would be interesting to know if such an equality is true more generally but this question seems to be difficult.
The next subsection is devoted to the proof of Theorem \[main\].
The proof of Theorem \[main\]
-----------------------------
In this subsection, we always assume Hypotheses \[hyp1\], \[hyp2\], and \[hyp large\], and fix an injection $\Bbbk \hookrightarrow R$. Note that $$H^{1}_{?}(K_{{\mathfrak{q}}}, A) \otimes_{R} \Bbbk \xrightarrow{\sim} H^{1}_{?}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk)$$ and $$H^{1}_{?}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk) \xrightarrow{\sim} H^{1}_{?}(K_{{\mathfrak{q}}}, A)[{\mathfrak{p}}]$$ where ${\mathfrak{q}}\in {\mathcal{P}}$, $? \in \{\emptyset, f, /f, {\rm tr}, /{\rm tr} \}$, the first map is the natural map, and the second is induced by the fixed map $\Bbbk \hookrightarrow R$.
\[s-iso\] If ${\mathfrak{n}}\in {\mathcal{N}}$ is an ideal such that $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{n}}}}(K, A^{*}(1)) = 0$, then the natural map $$H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \otimes_{R} \Bbbk \to H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A \otimes_{R} \Bbbk)$$ and the map $$H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A \otimes_{R} \Bbbk) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A)[{\mathfrak{p}}]$$ induced by $\Bbbk \hookrightarrow R$ are isomorphisms.
Note that $H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \simeq R^{r+\nu({\mathfrak{n}})}$ by Hypothesis \[hyp large\]. Applying Lemma \[injective\] with $S = \Bbbk$ and $X = H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A)$, we see that the natural map $$H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \otimes_{R} \Bbbk \to H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A \otimes_{R} \Bbbk)$$ is injective. By Hypothesis \[hyp2\] and Lemma \[lemma S\], we have $\left(\operatorname{coker}\left(A \otimes_{R} \Bbbk \to A\right)\right)^{G_{K}} = 0$, and so the map $$H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A \otimes_{R} \Bbbk) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A)[{\mathfrak{p}}]$$ induced by $\Bbbk \hookrightarrow R$ is also injective. Hence we have $$\begin{aligned}
r + \nu({\mathfrak{n}}) &= \dim_{\Bbbk}H^1_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \otimes_{R} \Bbbk
\\
&\leq \dim_{\Bbbk}H^1_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A \otimes_R \Bbbk)
\\
&\leq \dim_{\Bbbk}H^1_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A)[{\mathfrak{p}}]
\\
&= r + \nu({\mathfrak{n}})
$$ where the last equality follows from the fact that $\dim_{\Bbbk}R[{\mathfrak{p}}] = 1$ and $H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \simeq R^{r+\nu({\mathfrak{n}})}$. Hence both the injections are isomorphisms.
\[selisom\] For any ideal ${\mathfrak{n}}\in {\mathcal{N}}$, the map $H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) \to H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A)[{\mathfrak{p}}]$ induced by the map $\Bbbk \hookrightarrow R$ is an isomorphism.
Let ${\mathfrak{n}}\in {\mathcal{N}}$. By using Lemma \[chebotarev\], we can take an ideal ${\mathfrak{m}}\in {\mathcal{N}}$ with ${\mathfrak{n}}\mid {\mathfrak{m}}$ and $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{m}}}}(K, A^{*}(1)) = 0$. Then by the definition of the Selmer structure ${\mathcal{F}}({\mathfrak{n}})$, we have the following diagram, whose rows are exact: $$\begin{aligned}
{\small
\xymatrix{
H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A_\Bbbk) \ar@{^{(}->}[r] \ar[d] & H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, A_\Bbbk) \ar[r] \ar[d] & \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}}H^{1}_{/ {\rm tr}}(K_{{\mathfrak{q}}}, A_\Bbbk) \oplus \bigoplus_{{\mathfrak{q}}\mid \frac{{\mathfrak{m}}}{{\mathfrak{n}}}}H^{1}_{/f}(K_{{\mathfrak{q}}}, A_\Bbbk) \ar[d]^{\simeq}
\\
H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A)[{\mathfrak{p}}] \ar@{^{(}->}[r] & H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, A) [{\mathfrak{p}}] \ar[r] & \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}}H^{1}_{/ {\rm tr}}(K_{{\mathfrak{q}}}, A)[{\mathfrak{p}}] \oplus \bigoplus_{{\mathfrak{q}}\mid \frac{{\mathfrak{m}}}{{\mathfrak{n}}}}H^{1}_{/f}(K_{{\mathfrak{q}}}, A )[{\mathfrak{p}}] .
}}\end{aligned}$$ Here we abbreviate $A \otimes_{R} \Bbbk$ to $A_{\Bbbk}$, the vertical maps are induced by $\Bbbk \hookrightarrow R$ and the rightmost vertical map is an isomorphism. By Lemma \[s-iso\], the middle vertical map is also an isomorphism, and so is the left.
For an ideal ${\mathfrak{n}}\in {\mathcal{N}}$, put $$\lambda({\mathfrak{n}}) := \dim_\Bbbk H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk)$$ and $$\lambda^*({\mathfrak{n}}) := \dim_\Bbbk H^1_{{\mathcal{F}}^\ast({\mathfrak{n}})}(K, (A \otimes_R \Bbbk)^*(1)).$$
\[indep-diff\] For any ideal ${\mathfrak{n}}$ in ${\mathcal{N}}$ one has $r = \lambda({\mathfrak{n}}) - \lambda^{*}({\mathfrak{n}})$ and hence also $\lambda^{*}({\mathfrak{n}}) < \lambda({\mathfrak{n}})$.
Let ${\mathfrak{n}}\in {\mathcal{N}}$. We take an ideal ${\mathfrak{m}}\in {\mathcal{N}}$ with ${\mathfrak{n}}\mid {\mathfrak{m}}$ and $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{m}}}}(K, A^{*}(1)) = 0$ by using Lemma \[chebotarev\]. Then $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{m}}}}(K, (A \otimes_{R} \Bbbk)^{*}(1)) = 0$ by Corollary \[dualselisom\]. Hence we have $$\begin{aligned}
\lambda({\mathfrak{n}}) - \lambda^{*}({\mathfrak{n}}) = \dim_\Bbbk H^1_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, A \otimes_R \Bbbk) - \nu({\mathfrak{m}}) = r\end{aligned}$$ where the first equality follows from the global duality and $$\dim_{\Bbbk}H^{1}_{/f}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk) = \dim_{\Bbbk}H^{1}_{/{\rm tr}}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk) = 1$$ for any prime ${\mathfrak{q}}\in {\mathcal{P}}$, and the second equality follows from Lemma \[s-iso\] and Hypothesis \[hyp large\].
\[rankind\] Let ${\mathfrak{n}}\in {\mathcal{N}}$ and ${\mathfrak{q}}\in {\mathcal{P}}$ with ${\mathfrak{q}}\nmid {\mathfrak{n}}$. Assume that the localization maps $$H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) \to H^1_{f}(K_{\mathfrak{q}}, A \otimes_R \Bbbk)$$ and $$H^1_{{\mathcal{F}}^\ast({\mathfrak{n}})}(K, (A \otimes_R \Bbbk)^*(1)) \to H^1_{f}(K_{\mathfrak{q}}, (A \otimes_R \Bbbk)^*(1))$$ are non-zero. Then we have $\lambda({\mathfrak{n}}{\mathfrak{q}}) = \lambda({\mathfrak{n}}) - 1$ and $\lambda^*({\mathfrak{n}}{\mathfrak{q}}) = \lambda^*({\mathfrak{n}}) -1$.
By the assumption and the fact that $$\dim_\Bbbk H^1_{f}(K_{\mathfrak{q}}, A \otimes_R \Bbbk) = \dim_\Bbbk H^1_{f}(K_{\mathfrak{q}}, (A \otimes_R \Bbbk)^*(1)) = 1,$$ the sequences $$\begin{aligned}
0 \to H^{1}_{{\mathcal{F}}_{{\mathfrak{q}}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) \to H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) \to H^1_{f}(K_{\mathfrak{q}}, A \otimes_R \Bbbk) \to 0\end{aligned}$$ and $$\begin{aligned}
0 \to H^{1}_{/f}(K_{{\mathfrak{q}}}, A) \to H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}})}(K, (A \otimes_{R} \Bbbk)^{*}(1))^{*} \to H^1_{({\mathcal{F}}^{*})_{{\mathfrak{q}}}({\mathfrak{n}})}(K, (A \otimes_R \Bbbk)^{*}(1))^{*} \to 0\end{aligned}$$ are exact. Hence by the global duality and the second exact sequence, we have $$H^{1}_{{\mathcal{F}}^{{\mathfrak{q}}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) = H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk),$$ and so $$H^{1}_{{\mathcal{F}}({\mathfrak{n}}{\mathfrak{q}})}(K, A \otimes_{R} \Bbbk) = H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) \cap H^{1}_{{\mathcal{F}}({\mathfrak{n}}{\mathfrak{q}})}(K, A \otimes_{R} \Bbbk) = H^{1}_{{\mathcal{F}}_{{\mathfrak{q}}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk).$$ Thus $\lambda({\mathfrak{n}}{\mathfrak{q}}) = \lambda({\mathfrak{n}}) - 1$ by the first exact sequence, and so $\lambda^*({\mathfrak{n}}{\mathfrak{q}}) = \lambda^*({\mathfrak{n}}) -1$ by Corollary \[indep-diff\].
According to Mazur and Rubin [@MRkoly], it is convenient to regard elements in ${\mathcal{N}}$ as ‘vertices’. In Definition \[graph\] below, we consider a graph, whose vertices consist of elements in ${\mathcal{N}}$. We first recall the notion of ‘core vertices’ introduced by Mazur and Rubin.
We say that ${\mathfrak{n}}\in {\mathcal{N}}$ is a core vertex if $H^1_{{\mathcal{F}}^\ast({\mathfrak{n}})}(K, A^*(1)) = 0$.
\[core rem\]
- By Corollary \[dualselisom\], ${\mathfrak{n}}\in {\mathcal{N}}$ is a core vertex if and only if $\lambda^{*}({\mathfrak{n}}) = 0$. In this case, we have $\lambda({\mathfrak{n}}) = r$ by Corollary \[indep-diff\].
- Let ${\mathfrak{n}}\in {\mathcal{N}}$ be a core vertex. By the global duality, we have $$0 \to H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}}H^{1}_{/ {\rm tr}}(K_{{\mathfrak{q}}}, A) \to 0.$$ Since $H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \simeq R^{r + \nu({\mathfrak{n}})}$ by Hypothesis \[hyp large\] and $H^{1}_{/{\rm tr}}(K_{{\mathfrak{q}}}, A) \simeq R$ for any prime ${\mathfrak{q}}\in {\mathcal{P}}$, the $R$-module $H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A)$ is free of rank $r$.
\[diff-lem\] Let ${\mathfrak{n}}\in {\mathcal{N}}$ and ${\mathfrak{q}}\in {\mathcal{P}}$ with ${\mathfrak{q}}\nmid {\mathfrak{n}}$. Then we have $|\lambda({\mathfrak{n}}{\mathfrak{q}}) - \lambda({\mathfrak{n}})| \leq 1$ and $|\lambda^{*}({\mathfrak{n}}{\mathfrak{q}}) - \lambda^{*}({\mathfrak{n}})| \leq 1$.
Since $\dim_{\Bbbk}H^{1}_{/f}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk) =
\dim_{\Bbbk}H^{1}_{/{\rm tr}}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk) = 1$, we have $$0 \leq \dim_{\Bbbk}H^{1}_{{\mathcal{F}}^{{\mathfrak{q}}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) - \lambda({\mathfrak{n}}) \leq 1$$ and $$0 \leq \dim_{\Bbbk}H^{1}_{{\mathcal{F}}^{{\mathfrak{q}}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) - \lambda({\mathfrak{n}}{\mathfrak{q}}) \leq 1.$$ Hence $|\lambda({\mathfrak{n}}{\mathfrak{q}}) - \lambda({\mathfrak{n}})| \leq 1$. The inequality $|\lambda^{*}({\mathfrak{n}}{\mathfrak{q}}) - \lambda^{*}({\mathfrak{n}})| \leq 1$ follows from Corollary \[indep-diff\] and $|\lambda({\mathfrak{n}}{\mathfrak{q}}) - \lambda({\mathfrak{n}})| \leq 1$.
\[diff-cor\] There is a core vertex ${\mathfrak{n}}\in {\mathcal{N}}$ with $\nu({\mathfrak{n}}) = \lambda^{*}(1)$. Furthermore, every core vertex ${\mathfrak{n}}\in {\mathcal{N}}$ satisfies an inequality $\nu({\mathfrak{n}}) \geq \lambda^*(1)$.
The existence of a core vertex ${\mathfrak{n}}\in {\mathcal{N}}$ with $\nu({\mathfrak{n}}) = \lambda^{*}(1)$ follows from Lemma \[chebotarev\], Corollary \[indep-diff\], and Proposition \[rankind\]. The second claim follows from Lemma \[diff-lem\].
\[graph\] We define a graph ${\mathcal{X}}^0$ as follows.
- The vertices of ${\mathcal{X}}^0$ are the core vertices.
- Let ${\mathfrak{n}}$ and ${\mathfrak{n}}{\mathfrak{q}}$ be core vertices. We join ${\mathfrak{n}}$ and ${\mathfrak{n}}{\mathfrak{q}}$ by an edge in ${\mathcal{X}}^0$ if and only if the localization map $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) \to H^1_f(K_{\mathfrak{q}}, A \otimes_{R} \Bbbk)$ is non-zero.
\[joincore\] Let ${\mathfrak{n}}\in {\mathcal{N}}$ and ${\mathfrak{q}}\in {\mathcal{P}}$ with ${\mathfrak{q}}\nmid {\mathfrak{n}}$.
- If ${\mathfrak{n}}$ is a core vertex and the map $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) \to H^1_f(K_{\mathfrak{q}}, A \otimes_R \Bbbk)$ is non-zero, then ${\mathfrak{n}}{\mathfrak{q}}$ is also a core vertex and ${\mathfrak{n}}$ and ${\mathfrak{n}}{\mathfrak{q}}$ are joined by an edge in ${\mathcal{X}}^0$.
- If ${\mathfrak{n}}{\mathfrak{q}}$ is a core vertex and the map $H^1_{{\mathcal{F}}({\mathfrak{n}}{\mathfrak{q}})}(K, A \otimes_R \Bbbk) \to H^1_{\rm tr}(K_{\mathfrak{q}}, A \otimes_R \Bbbk)$ is non-zero, then ${\mathfrak{n}}$ is also a core vertex and ${\mathfrak{n}}$ and ${\mathfrak{n}}{\mathfrak{q}}$ are joined by an edge in ${\mathcal{X}}^0$.
We first prove claim (i). If ${\mathfrak{n}}$ is a core vertex, we have an exact sequence by the global duality $$\begin{gathered}
0 \to H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) \to H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk )
\to H^1_f(K_{\mathfrak{q}}, A \otimes_R \Bbbk)\\ \to H^1_{({\mathcal{F}}^{*})^{{\mathfrak{q}}}({\mathfrak{n}})}(K, (A \otimes_R \Bbbk)^*(1))^* \to 0.\end{gathered}$$ Since $\dim_\Bbbk H^1_f(K_{\mathfrak{q}}, A \otimes_R \Bbbk) = 1$ and the map $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) \to H^1_f(K_{\mathfrak{q}}, A \otimes_R \Bbbk)$ is non-zero, we have $H^1_{({\mathcal{F}}^{*})^{{\mathfrak{q}}}({\mathfrak{n}})}(K, (A \otimes_R \Bbbk)^*(1)) = 0$, and so $H^1_{{\mathcal{F}}^*({\mathfrak{n}}{\mathfrak{q}})}(K, (A \otimes_R \Bbbk)^*(1))=0$.
To prove claim (ii), we can show in a similar way that ${\mathfrak{n}}$ is a core vertex, by showing that $H^1_{({\mathcal{F}}^{*})^{{\mathfrak{q}}}({\mathfrak{n}})}(K, (A \otimes_R \Bbbk)^*(1)) = 0$. By the global duality, the cokernel of the localization map $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) \to H^1_f(K_{\mathfrak{q}}, A \otimes_{R} \Bbbk)$ injects into $H^1_{({\mathcal{F}}^{*})^{{\mathfrak{q}}}({\mathfrak{n}})}(K, (A \otimes_R \Bbbk)^*(1))^{*} = 0$. Hence ${\mathfrak{n}}$ and ${\mathfrak{n}}{\mathfrak{q}}$ are joined by an edge in ${\mathcal{X}}^0$.
\[koly-join1\] If ${\mathfrak{n}}$ and ${\mathfrak{n}}{\mathfrak{q}}$ are core vertices, then there is a path in ${\mathcal{X}}^0$ from ${\mathfrak{n}}$ to ${\mathfrak{n}}{\mathfrak{q}}$.
We may assume that the map $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) \to H^1_{f}(K_{\mathfrak{q}}, A \otimes_R \Bbbk)$ is zero. Then we have $$\begin{aligned}
H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk)
= H^1_{{\mathcal{F}}({\mathfrak{n}}{\mathfrak{q}})}(K, A \otimes_R \Bbbk)\end{aligned}$$ where the second equality follows from $\lambda({\mathfrak{n}}) = r = \lambda({\mathfrak{n}}{\mathfrak{q}})$. Furthermore, by the global duality, we have $\dim_\Bbbk H^1_{({\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}}))^*}(K, (A \otimes_R \Bbbk)^*(1)) = 1$. By Lemma \[chebotarev\], there is a prime ${\mathfrak{r}}\in {\mathcal{P}}$ with ${\mathfrak{r}}\nmid {\mathfrak{n}}{\mathfrak{q}}$ such that the localization maps $$H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) \to H^1_f(K_{\mathfrak{r}}, A \otimes_R \Bbbk)$$ and $$H^1_{({\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}}))^*}(K, (A \otimes_R \Bbbk)^*(1)) \to H^1_{f}(K_{\mathfrak{r}}, (A \otimes_R \Bbbk)^*(1))$$ are non-zero. By Lemma \[joincore\](i) and $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}({\mathfrak{n}}{\mathfrak{q}})}(K, A \otimes_R \Bbbk)$, it follows from the fact that the map $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) \to H^1_f(K_{\mathfrak{r}}, A \otimes_R \Bbbk)$ is non-zero that both ${\mathfrak{n}}{\mathfrak{r}}$ and ${\mathfrak{n}}{\mathfrak{q}}{\mathfrak{r}}$ are core vertices and that there are paths in ${\mathcal{X}}^0$ from ${\mathfrak{n}}$ to ${\mathfrak{n}}{\mathfrak{r}}$ and from ${\mathfrak{n}}{\mathfrak{q}}$ to ${\mathfrak{n}}{\mathfrak{q}}{\mathfrak{r}}$. Hence again by Lemma \[joincore\](i), we only need to show that $$H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}}{\mathfrak{r}})}(K, A \otimes_R \Bbbk) \neq H^1_{{\mathcal{F}}({\mathfrak{n}}{\mathfrak{r}})}(K, A \otimes_R \Bbbk).$$ Since the map $H^1_{({\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}}))^*}(K, (A \otimes_R \Bbbk)^*(1)) \to H^1_{f}(K_{\mathfrak{r}}, (A \otimes_R \Bbbk)^*(1))$ is non-zero, we have $H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}_{\mathfrak{q}}^{\mathfrak{r}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk)$ by the global duality. Hence we get an equality $$\begin{aligned}
H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}}{\mathfrak{r}})}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}_{{\mathfrak{q}}{\mathfrak{r}}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk)
= H^1_{{\mathcal{F}}_{\mathfrak{r}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk)\end{aligned}$$ where the second equality follows from $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk)$. Since the map $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) \to H^1_f(K_{\mathfrak{r}}, A \otimes_R \Bbbk)$ is non-zero, we have $$\dim_\Bbbk H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}}{\mathfrak{r}})}(K, A \otimes_R \Bbbk) = \dim_{\Bbbk} H^1_{{\mathcal{F}}_{\mathfrak{r}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) =
\lambda({\mathfrak{n}}) - 1 = \lambda({\mathfrak{n}}{\mathfrak{r}}) -1.$$ This completes the proof.
\[change\] Let $s$ be a positive integer. For $1 \leq i \leq s$, let ${\mathfrak{n}}_{i} \in {\mathcal{N}}$ be a core vertex and ${\mathfrak{q}}_{i} \in {\mathcal{P}}$ with ${\mathfrak{q}}_{i} \mid {\mathfrak{n}}_{i}$. If $2s < p$ and ${\mathfrak{n}}_{i}/{\mathfrak{q}}_{i}$ is not a core vertex for any $1 \leq i \leq s$, then there is a prime ${\mathfrak{r}}\in {\mathcal{P}}$ with ${\mathfrak{r}}\nmid {\mathfrak{n}}_{1} \cdots {\mathfrak{n}}_{s}$ such that ${\mathfrak{n}}_{1}{\mathfrak{r}}/{\mathfrak{q}}_{1}, \ldots, {\mathfrak{n}}_{s}{\mathfrak{r}}/{\mathfrak{q}}_{s}$ are core vertices and that there is a path in ${\mathcal{X}}^{0}$ from ${\mathfrak{n}}_{i}$ to ${\mathfrak{n}}_{i}{\mathfrak{r}}/{\mathfrak{q}}_{i}$ for every $1 \leq i \leq s$.
Let $1 \leq i \leq s$ and put ${\mathfrak{m}}_{i} = {\mathfrak{n}}_{i}/{\mathfrak{q}}_{i}$. Since $\lambda^{*}({\mathfrak{n}}_{i}) = 0$ and ${\mathfrak{m}}_{i}$ is not a core vertex, we have $\lambda({\mathfrak{m}}_{i}) = r + 1$ and $\lambda^{*}({\mathfrak{m}}_{i}) = 1$ by Corollary \[indep-diff\] and Lemma \[diff-lem\]. Note that $$H^{1}_{{\mathcal{F}}({\mathfrak{n}}_{i})}(K, A \otimes_{R} \Bbbk) \subseteq H^{1}_{{\mathcal{F}}^{{\mathfrak{q}}_{i}}({\mathfrak{m}}_{i})}(K, A \otimes_{R} \Bbbk)
= H^{1}_{{\mathcal{F}}({\mathfrak{m}}_{i})}(K, A \otimes_{R} \Bbbk).$$ In fact, we have $\lambda({\mathfrak{m}}_{i}) \leq \dim_{\Bbbk}H^{1}_{{\mathcal{F}}^{{\mathfrak{q}}_{i}}({\mathfrak{m}}_{i})}(K, A \otimes_{R} \Bbbk) \leq \lambda({\mathfrak{n}}_{i}) + 1 = \lambda({\mathfrak{m}}_{i})$, where the second inequality follows from the exact sequence $$0 \to H^{1}_{{\mathcal{F}}({\mathfrak{n}}_{i})}(K, A \otimes_{R} \Bbbk) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{q}}_{i}}({\mathfrak{m}}_{i})}(K, A \otimes_{R} \Bbbk) \to H^{1}_{/{\rm tr}}(K, A \otimes_{R} \Bbbk)$$ and $\dim_{\Bbbk}H^{1}_{/{\rm tr}}(K, A \otimes_{R} \Bbbk) =1$. By Lemma \[chebotarev\] and $2s < p$, there is a prime ${\mathfrak{r}}\in {\mathcal{P}}$ with ${\mathfrak{r}}\nmid {\mathfrak{n}}_{1} \cdots {\mathfrak{n}}_{s}$ such that the maps $$H^{1}_{{\mathcal{F}}({\mathfrak{n}}_{i})}(K, A \otimes_{R} \Bbbk) \to H^{1}_{f}(K_{{\mathfrak{r}}}, A \otimes_{R} \Bbbk)$$ and $$H^{1}_{{\mathcal{F}}^{*}({\mathfrak{m}}_{i})}(K, (A \otimes_{R} \Bbbk)^{*}(1)) \to H^{1}_{f}(K_{{\mathfrak{r}}}, (A \otimes_{R} \Bbbk)^{*}(1))$$ are non-zero for any $1 \leq i \leq s$. Then $\lambda^{*}({\mathfrak{m}}_{i}{\mathfrak{r}}) = \lambda^{*}({\mathfrak{m}}_{i}) - 1 = 0$ by Proposition \[rankind\]. Hence ${\mathfrak{m}}_{i}{\mathfrak{r}}$ is a core vertex. Furthermore, by Lemma \[joincore\](i), ${\mathfrak{n}}_{i}{\mathfrak{r}}$ is also a core vertex. Therefore by using Lemma \[koly-join1\], there is a path in ${\mathcal{X}}^{0}$ from ${\mathfrak{n}}_{i}$ to ${\mathfrak{m}}_{i}{\mathfrak{r}}$ for any $1 \leq i \leq s$.
\[koly-join3\] Suppose that ${\mathfrak{n}}_{1}, {\mathfrak{n}}_{2} \in {\mathcal{N}}$ are core vertices and $\nu({\mathfrak{n}}_{1}) = \nu({\mathfrak{n}}_{2}) = \lambda^{*}(1)$. If $p>3$, then there is a path in ${\mathcal{X}}^0$ from ${\mathfrak{n}}_{1}$ to ${\mathfrak{n}}_{2}$.
We will prove this lemma by induction on $\lambda^{*}(1) - \nu(\gcd({\mathfrak{n}}_{1}, {\mathfrak{n}}_{2})) \geq 0$. When it is equal to zero, we have ${\mathfrak{n}}_{1} = {\mathfrak{n}}_{2}$ and there is nothing to prove. Suppose ${\mathfrak{n}}_{1} \neq {\mathfrak{n}}_{2}$, and fix distinct primes ${\mathfrak{q}}_{1} \mid {\mathfrak{n}}_{1}$ and ${\mathfrak{q}}_{2} \mid {\mathfrak{n}}_{2}$. Then ${\mathfrak{n}}_{1}/{\mathfrak{q}}_{1}$ and ${\mathfrak{n}}_{2}/{\mathfrak{q}}_{2}$ are not core vertices by Corollary \[diff-cor\]. By Lemma \[change\] and $p>3$, there is a prime ${\mathfrak{r}}\in {\mathcal{P}}$ with ${\mathfrak{r}}\nmid {\mathfrak{n}}_{1}{\mathfrak{n}}_{2}$ such that both ${\mathfrak{n}}_{1}{\mathfrak{r}}/{\mathfrak{q}}_{1}$ and ${\mathfrak{n}}_{2}{\mathfrak{r}}/{\mathfrak{q}}_{2}$ are core vertices and that there are paths in ${\mathcal{X}}^0$ connecting ${\mathfrak{n}}_{1}$ to ${\mathfrak{n}}_{1}{\mathfrak{r}}/{\mathfrak{q}}_{1}$ and ${\mathfrak{n}}_{2}$ to ${\mathfrak{n}}_{2}{\mathfrak{r}}/{\mathfrak{q}}_{2}$. Since $\nu(\gcd({\mathfrak{n}}_{1}{\mathfrak{r}}/{\mathfrak{q}}_{1}, {\mathfrak{n}}_{2}{\mathfrak{r}}/{\mathfrak{q}}_{2})) = \nu(\gcd({\mathfrak{n}}_{1}, {\mathfrak{n}}_{2})) + 1$, there is a path in ${\mathcal{X}}^0$ connecting ${\mathfrak{n}}_{1}{\mathfrak{r}}/{\mathfrak{q}}_{1}$ to ${\mathfrak{n}}_{2}{\mathfrak{r}}/{\mathfrak{q}}_{2}$ by the induction hypothesis. This completes the proof.
\[koly-join2\] Suppose that ${\mathfrak{n}}\in {\mathcal{N}}$ is a core vertex with $\nu({\mathfrak{n}}) > \lambda^*(1)$. Then there is a core vertex ${\mathfrak{m}}\in {\mathcal{N}}$ with $\nu({\mathfrak{m}}) = \nu({\mathfrak{n}}) - 1$ such that there is a path in ${\mathcal{X}}^0$ from ${\mathfrak{n}}$ to ${\mathfrak{m}}$.
By Lemma \[koly-join1\], we may assume that ${\mathfrak{n}}/{\mathfrak{q}}$ is not a core vertex for any ${\mathfrak{q}}\mid {\mathfrak{n}}$. Then by Lemma \[joincore\](ii), we have $$H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}_{\mathfrak{n}}}(K, A \otimes_R \Bbbk).$$ Since $\lambda(1) - \dim_{\Bbbk}H^{1}_{{\mathcal{F}}_{{\mathfrak{n}}}}(K, A \otimes_{R} \Bbbk) = \lambda(1) - \lambda({\mathfrak{n}}) = \lambda(1) - r = \lambda^{*}(1) < \nu({\mathfrak{n}})$ and $$H^{1}_{{\mathcal{F}}_{{\mathfrak{n}}}}(K, A \otimes_{R} \Bbbk) = \ker\left( H^{1}_{{\mathcal{F}}}(K, A \otimes_{R} \Bbbk) \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}}H^{1}_{f}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk) \right),$$ the map $H^{1}_{{\mathcal{F}}}(K, A \otimes_{R} \Bbbk) \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}}H^{1}_{f}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk)$ is not surjective. For any prime ${\mathfrak{q}}\mid {\mathfrak{n}}$, if $H^1_{{\mathcal{F}}_{\mathfrak{n}}}(K, A \otimes_R \Bbbk) \neq H^1_{{\mathcal{F}}_{{\mathfrak{n}}/{\mathfrak{q}}}}(K, A \otimes_R \Bbbk)$, there is an element $x \in H^{1}_{{\mathcal{F}}}(K, A \otimes_{R} \Bbbk)$ such that ${\rm loc}_{{\mathfrak{q}}}(x) \neq 0$ and ${\rm loc}_{{\mathfrak{q}}'}(x) = 0$ for each prime ${\mathfrak{q}}' \mid {\mathfrak{n}}/{\mathfrak{q}}$. Hence there is a prime ${\mathfrak{q}}\mid {\mathfrak{n}}$ such that $$H^1_{{\mathcal{F}}_{\mathfrak{n}}}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}_{{\mathfrak{n}}/{\mathfrak{q}}}}(K, A \otimes_R \Bbbk)$$ since the map $H^{1}_{{\mathcal{F}}}(K, A \otimes_{R} \Bbbk) \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}}H^{1}_{f}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk)$ is not surjective. Let ${\mathfrak{m}}= {\mathfrak{n}}/{\mathfrak{q}}$. Since ${\mathfrak{m}}$ is not a core vertex, we have $\lambda^*({\mathfrak{m}}) = 1$ by Lemma \[diff-lem\]. Hence by Lemma \[chebotarev\], there is a prime ${\mathfrak{r}}\in {\mathcal{P}}$ with ${\mathfrak{r}}\nmid {\mathfrak{n}}$ such that the maps $$\begin{aligned}
H^1_{{\mathcal{F}}_{\mathfrak{n}}}(K, A \otimes_R \Bbbk) \to H^1_{f}(K_{\mathfrak{r}}, A \otimes_R \Bbbk)\end{aligned}$$ and $$\begin{aligned}
H^1_{{\mathcal{F}}^*({\mathfrak{m}})}(K, (A \otimes_R \Bbbk)^*(1)) \to H^1_{f}(K_{{\mathfrak{r}}}, (A \otimes_R \Bbbk)^*(1))\end{aligned}$$ are non-zero. Then ${\mathfrak{m}}{\mathfrak{r}}$ and ${\mathfrak{n}}{\mathfrak{r}}$ are core vertices by Proposition \[rankind\] and Lemma \[joincore\](i). Hence by using Lemma \[koly-join1\], we see that there is a path in ${\mathcal{X}}^0$ from ${\mathfrak{n}}$ to ${\mathfrak{m}}{\mathfrak{r}}$. Since the map $H^1_{{\mathcal{F}}^*({\mathfrak{m}})}(K, (A \otimes_R \Bbbk)^*(1)) \to H^1_{f}(K_{\mathfrak{r}}, (A \otimes_R \Bbbk)^*(1))$ is surjective, we have $H^1_{{\mathcal{F}}({\mathfrak{m}})}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}^{{\mathfrak{r}}}({\mathfrak{m}})}(K, A \otimes_R \Bbbk)$ by the global duality. Hence we get $$\begin{aligned}
H^1_{{\mathcal{F}}({\mathfrak{m}}{\mathfrak{r}})}(K, A \otimes_R \Bbbk) &= H^1_{{\mathcal{F}}^{{\mathfrak{r}}}({\mathfrak{m}})}(K, A \otimes_R \Bbbk) \cap H^1_{{\mathcal{F}}({\mathfrak{m}}{\mathfrak{r}})}(K, A \otimes_R \Bbbk)
\\
&= H^1_{{\mathcal{F}}_{{\mathfrak{r}}}({\mathfrak{m}})}(K, A \otimes_R \Bbbk).\end{aligned}$$ Furthermore, we have $$\dim_{\Bbbk}H^{1}_{{\mathcal{F}}_{{\mathfrak{m}}{\mathfrak{r}}}}(K, A \otimes_{R} \Bbbk) = \dim_{\Bbbk}H^{1}_{{\mathcal{F}}_{{\mathfrak{n}}{\mathfrak{r}}}}(K, A \otimes_{R} \Bbbk) = r - 1$$ since the map $H^1_{{\mathcal{F}}_{\mathfrak{n}}}(K, A \otimes_R \Bbbk) \to H^1_{f}(K_{\mathfrak{r}}, A \otimes_R \Bbbk)$ is non-zero and $$H^{1}_{{\mathcal{F}}_{{\mathfrak{m}}}}(K, A \otimes_{R} \Bbbk) = H^1_{{\mathcal{F}}_{\mathfrak{n}}}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_R \Bbbk).$$ Since $\lambda({\mathfrak{m}}{\mathfrak{r}}) = r$, we conclude that the sum of localization maps $$H^1_{{\mathcal{F}}_{{\mathfrak{r}}}({\mathfrak{m}})}(K, A \otimes_R \Bbbk) = H^1_{{\mathcal{F}}({\mathfrak{m}}{\mathfrak{r}})}(K, A \otimes_R \Bbbk) \to
\bigoplus_{{\mathfrak{s}}\mid {\mathfrak{m}}}H^{1}_{\rm tr}(K_{{\mathfrak{s}}}, A \otimes_{R} \Bbbk)$$ is non-zero. Thus there is a prime ${\mathfrak{s}}\mid {\mathfrak{m}}$ such that the localization map $$H^1_{{\mathcal{F}}({\mathfrak{m}}{\mathfrak{r}})}(K, A \otimes_R \Bbbk) \to H^1_{\rm tr}(K_{\mathfrak{s}}, A \otimes_R \Bbbk)$$ is non-zero. By Lemma \[joincore\](ii), ${\mathfrak{m}}{\mathfrak{r}}/{\mathfrak{s}}$ is a core vertex. Hence by Lemma \[koly-join1\] there is a path in ${\mathcal{X}}^0$ from ${\mathfrak{m}}{\mathfrak{r}}/{\mathfrak{s}}$ to ${\mathfrak{m}}{\mathfrak{r}}$. This completes the proof.
\[connected\] Suppose that Hypotheses \[hyp1\], \[hyp2\], and \[hyp large\]. If $p >3$, then the graph ${\mathcal{X}}^0$ is connected.
Let ${\mathfrak{n}}_{1}, {\mathfrak{n}}_{2} \in {\mathcal{N}}$ be core vertices. By Corollary \[diff-cor\] and Lemma \[koly-join2\], there are core vertices ${\mathfrak{m}}_{1}, {\mathfrak{m}}_{2} \in {\mathcal{N}}$ with $\lambda({\mathfrak{m}}_{1}) = \lambda({\mathfrak{m}}_{2}) = \lambda^{*}(1)$ such that there are paths in ${\mathcal{X}}^0$ from ${\mathfrak{n}}_{1}$ to ${\mathfrak{m}}_{1}$ and ${\mathfrak{n}}_{2}$ to ${\mathfrak{m}}_{2}$. Since $p >3$, there is a path in ${\mathcal{X}}^{0}$ from ${\mathfrak{m}}_{1}$ to ${\mathfrak{m}}_{2}$ by Corollary \[koly-join3\]. Hence the graph ${\mathcal{X}}^{0}$ is connected.
\[inj-koly\] Let ${\mathfrak{n}}\in {\mathcal{N}}$ and ${\mathfrak{q}}\in {\mathcal{P}}$ with ${\mathfrak{q}}\nmid {\mathfrak{n}}$. If ${\mathfrak{n}}$ and ${\mathfrak{n}}{\mathfrak{q}}$ are core vertices and the localization map $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) \to H^1_f(K_{\mathfrak{q}}, A \otimes_{R} \Bbbk)$ is non-zero, then the maps $$\begin{aligned}
\varphi_{{\mathfrak{q}}}^{\rm fs} \colon {\bigcap}^r_R H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \otimes G_{{\mathfrak{n}}} \to {\bigcap}^{r-1}_R H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}})}(K, A) \otimes G_{{\mathfrak{n}}{\mathfrak{q}}}\end{aligned}$$ and $$\begin{aligned}
v_{{\mathfrak{q}}} \colon {\bigcap}^r_R H^1_{{\mathcal{F}}({\mathfrak{n}}{\mathfrak{q}})}(K, A) \otimes G_{{\mathfrak{n}}{\mathfrak{q}}} \to {\bigcap}^{r-1}_R H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}})}(K, A) \otimes G_{{\mathfrak{n}}{\mathfrak{q}}}\end{aligned}$$ are isomorphisms.
Note that $H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A)$ is free of rank $r$ by Remark \[core rem\]. The natural map $$H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \otimes_{R} \Bbbk \to H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk)$$ is an isomorphism by Lemma \[injective\] and the fact that $$\dim_{\Bbbk} H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \otimes_{R} \Bbbk = r = \lambda({\mathfrak{n}}) = \dim_{\Bbbk} H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk).$$ Thus the localization map $$H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \to H^1_f(K_{\mathfrak{q}}, A)$$ is surjective since $H^{1}_{f}(K_{{\mathfrak{q}}}, A) \otimes_{R} \Bbbk \simeq H^{1}_{f}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk)$ and the map $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A \otimes_{R} \Bbbk) \to H^1_f(K_{\mathfrak{q}}, A \otimes_{R} \Bbbk)$ is non-zero. Hence, by the global duality, we have a split exact sequence of free $R$-modules $$\begin{aligned}
0 \to H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}})}(K, A) \to H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \to H^1_f(K_{\mathfrak{q}}, A) \to 0\end{aligned}$$ and $H^{1}_{({\mathcal{F}}^{*})^{{\mathfrak{q}}}({\mathfrak{n}})}(K, A^{*}(1)) = H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}})}(K, A^{*}(1)) = 0$. Again, by the global duality and the fact that $H^{1}_{({\mathcal{F}}^{*})^{{\mathfrak{q}}}({\mathfrak{n}})}(K, A^{*}(1)) = 0$, we have a split exact sequence of free $R$-modules $$\begin{aligned}
0 \to H^1_{{\mathcal{F}}_{\mathfrak{q}}({\mathfrak{n}})}(K, A) \to H^1_{{\mathcal{F}}({\mathfrak{n}}{\mathfrak{q}})}(K, A) \to H^1_{\rm tr}(K_{\mathfrak{q}}, A) \to 0.\end{aligned}$$ Since the $R$-modules $H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A)$ and $H^1_{{\mathcal{F}}({\mathfrak{n}}{\mathfrak{q}})}(K, A)$ are free of rank $r$, the maps $\varphi_{{\mathfrak{q}}}^{\rm fs}$ and $v_{{\mathfrak{q}}}$ are isomorphisms.
\[thm koly\] Suppose Hypotheses \[hyp1\], \[hyp2\], and \[hyp large\]. Let ${\mathfrak{n}}\in {\mathcal{N}}$ be a core vertex. If $p>3$, then the projection map $$\begin{aligned}
{\rm KS}_r(A, {\mathcal{F}}) \to {\bigcap}^r_R H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \otimes G_{\mathfrak{n}}; \ \kappa \mapsto \kappa_{{\mathfrak{n}}}\end{aligned}$$ is an isomorphism. In particular, ${\rm KS}_r(A, {\mathcal{F}})$ is a free $R$-module of rank one.
Since ${\mathfrak{n}}\in {\mathcal{N}}$ is a core vertex, by the global duality, we have a split exact sequence of free $R$-modules: $$0 \to H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{n}}}}(K, A) \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}}H^{1}_{/{\rm tr}}(K_{{\mathfrak{q}}}, A) \to 0.$$ Hence, by Hypothesis \[hyp large\], the map $$\begin{aligned}
\label{phiisom}
{\bigwedge}_{{\mathfrak{q}}\mid {\mathfrak{n}}}\varphi_{\mathfrak{q}}^{\rm fs} \colon
{\bigcap}_R^{r+\nu({\mathfrak{n}})} H^1_{{\mathcal{F}}^{\mathfrak{n}}}(K,A) \to {\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,A) \otimes G_{\mathfrak{n}}\end{aligned}$$ is an isomorphism. By using the map ${\rm Reg}_r \colon {\rm SS}_r(A, {\mathcal{F}}) \to {\rm KS}_r(A, {\mathcal{F}})$ and by Theorem \[thm stark\](i), we conclude that the map ${\rm KS}_r(A, {\mathcal{F}}) \to {\bigcap}^{r}_{R} H^1_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \otimes G_{\mathfrak{n}}$ is surjective.
Let $\kappa \in {\rm KS}_r(A, {\mathcal{F}})$ with $\kappa_{\mathfrak{n}}= 0$. To prove injectivity, we will show that $\kappa_{\mathfrak{m}}= 0$ for any ideal ${\mathfrak{m}}\in {\mathcal{N}}$ by induction on $\lambda^*({\mathfrak{m}})$. If $\lambda^*({\mathfrak{m}}) = 0$, then ${\mathfrak{m}}$ is a core vertex. Hence by Theorem \[connected\], Lemma \[inj-koly\], and $\kappa_{\mathfrak{n}}= 0$, we have $\kappa_{\mathfrak{m}}= 0$. Suppose that $\lambda^*({\mathfrak{m}}) > 0$ and $\kappa_{\mathfrak{m}}\neq 0$. By Lemma \[chebotarev\], we can take an ideal ${\mathfrak{r}}\in {\mathcal{N}}$ coprime to ${\mathfrak{m}}$ such that the localization maps $$\begin{aligned}
H^1_{{\mathcal{F}}({\mathfrak{m}})}(K, A \otimes_R \Bbbk) \to H^1_f(K_{{\mathfrak{q}}}, A \otimes_R \Bbbk)\end{aligned}$$ and $$\begin{aligned}
H^1_{{\mathcal{F}}^*({\mathfrak{m}})}(K, (A \otimes_R \Bbbk)^*(1)) \to H^1_f(K_{{\mathfrak{q}}}, (A \otimes_R \Bbbk)^*(1))\end{aligned}$$ are non-zero for any prime ${\mathfrak{q}}\mid {\mathfrak{r}}$ and that the sum of localization maps $$H^1_{{\mathcal{F}}({\mathfrak{m}})}(K, A \otimes_R \Bbbk) \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{r}}}H^{1}_{f}(K_{{\mathfrak{q}}}, A \otimes_R \Bbbk)$$ is injective. Then the map $$H^1_{{\mathcal{F}}({\mathfrak{m}})}(K, A) \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{r}}}H^{1}_{f}(K_{{\mathfrak{q}}}, A)$$ is also injective by using Corollary \[selisom\] and the injectivity of the maps $H^{1}_{f}(K_{{\mathfrak{q}}}, A \otimes_{R} \Bbbk) \to H^{1}_{f}(K_{{\mathfrak{q}}}, A)$ induced by the map $\Bbbk \hookrightarrow R$ for any prime ${\mathfrak{q}}\in {\mathcal{P}}$. By taking the dual of this map, we see that $$0 = \bigcap_{{\mathfrak{q}}\mid {\mathfrak{r}}}\ker\left(\varphi^{\rm fs}_{{\mathfrak{q}}} \colon {\bigcap}^{r}_{R}H^1_{{\mathcal{F}}({\mathfrak{m}})}(K, A) \otimes G_{{\mathfrak{m}}} \to {\bigcap}^{r-1}_{R}H^1_{{\mathcal{F}}_{{\mathfrak{q}}}({\mathfrak{m}})}(K, A) \otimes {G_{{\mathfrak{m}}{\mathfrak{q}}}}\right)$$ by Corollary \[bidual-ker\]. Hence there is a prime ${\mathfrak{q}}\mid {\mathfrak{r}}$ such that $\varphi_{\mathfrak{q}}^{\rm fs}(\kappa_{\mathfrak{m}}) \neq 0$, since we suppose $\kappa_{{\mathfrak{m}}} \neq 0$. Furthermore, we have $\lambda^\ast({\mathfrak{m}}{\mathfrak{q}}) = \lambda^\ast({\mathfrak{m}}) - 1$ by Proposition \[rankind\], and so we conclude that $\kappa_{{\mathfrak{m}}{\mathfrak{q}}} = 0$ by the induction hypothesis. By the definition of Kolyvagin system, we have $0 = v_{{\mathfrak{q}}}(\kappa_{{\mathfrak{m}}{\mathfrak{q}}}) = \varphi_{\mathfrak{q}}^{\rm fs}(\kappa_{\mathfrak{m}}) \neq 0$. This is a contradiction. Thus $\kappa_{\mathfrak{m}}= 0$.
The proof of Theorem \[connected\] is parallel to that of [@MRkoly Th. 4.3.12]. However, the notion of exterior power bidual plays a critical role in the proof of Theorem \[thm koly\], allowing us to overcome the problem discussed by Mazur and Rubin in [@MRselmer Rem. 11.9]. More precisely, it is crucial for the proof of Theorem \[thm koly\] that if $\kappa_{\mathfrak{m}}$ does not vanish, then there exists a prime ideal ${\mathfrak{q}}$ such that $\varphi_{\mathfrak{q}}^{\rm fs}(\kappa_{\mathfrak{m}})$ does not vanish and the corresponding fact is not true if one defines Kolyvagin systems by using exterior powers rather than exterior power biduals. This is the reason why Mazur and Rubin could not prove any result that corresponds to Theorem \[thm koly\].
The following lemma will be used in the proof of Theorem \[main\](iii).
\[fitt-ind-1\] Let ${\mathfrak{n}}\in {\mathcal{N}}$. Then for each natural number $i$ one has $$\sum_{{\mathfrak{q}}\in {\mathcal{P}},\ {\mathfrak{q}}\nmid {\mathfrak{n}}}\operatorname{Fitt}_{R}^{i-1}(H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}}{\mathfrak{q}})}(K, A^{*}(1))^{*})\subseteq
\operatorname{Fitt}_{R}^{i}( H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}})}(K, A^{*}(1))^{*}),$$ with equality if ${\mathrm{Ann}}_{R}( H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A))$ vanishes.
Note that, if ${\mathcal{F}}$ satisfies Hypothesis \[hyp large\], then so does ${\mathcal{F}}({\mathfrak{n}})$. In fact, by Lemma \[chebotarev\], we can take an ideal ${\mathfrak{d}}\in {\mathcal{N}}$ coprime to ${\mathfrak{n}}$ such that $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{d}}}({\mathfrak{n}})}(K, A^{*}(1))$ vanishes. Then, by the global duality, we have an exact sequence $$0 \to H^{1}_{{\mathcal{F}}^{{\mathfrak{d}}}({\mathfrak{n}})}(K, A) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{d}}{\mathfrak{n}}}}(K, A) \to \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}} H^{1}_{/{\rm tr}}(K_{{\mathfrak{q}}}, A) \to 0.$$ Since $H^{1}_{{\mathcal{F}}^{{\mathfrak{d}}{\mathfrak{n}}}}(K, A) \simeq R^{r+\nu({\mathfrak{d}}{\mathfrak{n}})}$ by Remark \[free rem\] and $H^{1}_{/{\rm tr}}(K_{{\mathfrak{q}}}, A) \simeq R$ for any prime ${\mathfrak{q}}\mid {\mathfrak{n}}$, we conclude that $H^{1}_{{\mathcal{F}}^{{\mathfrak{d}}}({\mathfrak{n}})}(K, A) \simeq R^{r+\nu({\mathfrak{d}})}$. Hence we can apply Lemma \[fitt-lemma\] for the Selmer structure ${\mathcal{F}}({\mathfrak{n}})$ and we have $$\begin{aligned}
\operatorname{Fitt}_{R}^{i}( H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}})}(K, A^{*}(1))^{*}) =
\sum_{{\mathfrak{q}}\in {\mathcal{P}}, \ {\mathfrak{q}}\nmid {\mathfrak{n}}}\operatorname{Fitt}_{R}^{i-1}(H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{q}}}({\mathfrak{n}})}(K, A^{*}(1))^{*}). \end{aligned}$$ (Note that $S({\mathcal{F}}({\mathfrak{n}})) = S \cup \{{\mathfrak{q}}\mid {\mathfrak{n}}\}$ and so the primes running in the sum on the right hand side are restricted to ${\mathfrak{q}}\in {\mathcal{P}}$ with ${\mathfrak{q}}\nmid {\mathfrak{n}}$.) Since the natural maps $H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}}{\mathfrak{q}})}(K, A^{*}(1))^{*} \to
H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{q}}}({\mathfrak{n}})}(K, A^{*}(1))^{*}$ are surjective, we have $$\sum_{{\mathfrak{q}}\in {\mathcal{P}}, \ {\mathfrak{q}}\nmid {\mathfrak{n}}}\operatorname{Fitt}_{R}^{i-1}(H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}}{\mathfrak{q}})}(K, A^{*}(1))^{*})
\subseteq \sum_{{\mathfrak{q}}\in {\mathcal{P}}, \ {\mathfrak{q}}\nmid {\mathfrak{n}}}\operatorname{Fitt}_{R}^{i-1}(H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{q}}}({\mathfrak{n}})}(K, A^{*}(1))^{*}).$$
It is therefore enough to show that if ${\mathrm{Ann}}_{R}( H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A))$ vanishes, then the reverse inclusion is also valid.
Under this assumption, there exists an element $e$ of $H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A)$ with ${\mathrm{Ann}}_{R}(e) = 0$. Let $x$ be a generator of $R[{\mathfrak{p}}]$. Then by Lemma \[chebotarev\], there is an ideal ${\mathfrak{m}}\in {\mathcal{N}}$ coprime to ${\mathfrak{n}}$ such that ${\rm loc}_{{\mathfrak{q}}}(xe) \neq 0$ for any prime ${\mathfrak{q}}\mid {\mathfrak{m}}$ and $H^{1}_{({\mathcal{F}}^\ast)_{{\mathfrak{m}}}({\mathfrak{n}})}(K, A^{*}(1)) = 0$. Since $H^{1}_{f}(K_{{\mathfrak{q}}}, A) \simeq R$, it follows from the fact that ${\rm loc}_{{\mathfrak{q}}}(xe) \neq 0$ that the map $H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \to H^{1}_{f}(K_{{\mathfrak{q}}}, A)$ is surjective for any prime ${\mathfrak{q}}\mid {\mathfrak{m}}$. Hence, by the global duality, we have $H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}})}(K, A^{*}(1)) = H^{1}_{({\mathcal{F}}^{*})^{{\mathfrak{q}}}({\mathfrak{n}})}(K, A^{*}(1))$. Therefore we have $$H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}}{\mathfrak{q}})}(K, A^{*}(1)) = H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}})}(K, A^{*}(1)) \cap H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}}{\mathfrak{q}})}(K, A^{*}(1)) = H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{q}}}({\mathfrak{n}})}(K, A^{*}(1)).$$ Again by Lemma \[fitt-lemma\], we have $$\begin{aligned}
\operatorname{Fitt}_{R}^{i}( H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}})}(K, A^{*}(1))^{*} )
&= \sum_{{\mathfrak{q}}\in {\mathcal{P}}, \ {\mathfrak{q}}\mid {\mathfrak{m}}}\operatorname{Fitt}_{R}^{i-1}(H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{q}}}({\mathfrak{n}})}(K, A^{*}(1))^{*})
\\
&= \sum_{{\mathfrak{q}}\in {\mathcal{P}}, \ {\mathfrak{q}}\mid {\mathfrak{m}}}\operatorname{Fitt}_{R}^{i-1}(H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}}{\mathfrak{q}})}(K, A^{*}(1))^{*})
\\
&\subseteq \sum_{{\mathfrak{q}}\in {\mathcal{P}}, \ {\mathfrak{q}}\nmid {\mathfrak{n}}}\operatorname{Fitt}_{R}^{i-1}(H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}}{\mathfrak{q}})}(K, A^{*}(1))^{*}).\end{aligned}$$ This completes the proof.
\[fitt-ind\] For each natural number $i$ one has $$\sum_{{\mathfrak{m}}\in {\mathcal{N}},\ \nu({\mathfrak{m}}) = i}\operatorname{Fitt}_{R}^{0}(H^{1}_{{\mathcal{F}}^{*}({\mathfrak{m}})}(K, A^{*}(1))^{*})\subseteq \operatorname{Fitt}_{R}^{i}( H^{1}_{{\mathcal{F}}^{*}}(K, A^{*}(1))^{*}),$$ with equality provided that ${\mathrm{Ann}}_{R}(H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A))$ vanishes for all ideals ${\mathfrak{n}}$ in ${\mathcal{N}}$.
This result follows directly from Lemma \[fitt-ind-1\].
Claim (i) follows from Theorem \[thm stark\](i), Theorem \[thm koly\], and the isomorphism (\[phiisom\]).
To prove claim (ii) it is enough to consider the case that $\kappa$ is a basis of ${\rm KS}_{r}(A, {\mathcal{F}})$. To do this we fix an ideal ${\mathfrak{n}}$ in $ {\mathcal{N}}$ and a generator of each $G_{{\mathfrak{q}}}$ and we regard $\kappa_{{\mathfrak{n}}}$ as an element of ${\bigcap}^{r}_{R}H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A)$. By claim (i), there exists a basis $\epsilon$ of ${\rm SS}_{r}(A, {\mathcal{F}})$ such that ${\rm Reg}_{r}(\epsilon) = \kappa$.
By using Lemma \[chebotarev\], we can take an ideal ${\mathfrak{m}}\in {\mathcal{N}}$ with ${\mathfrak{n}}\mid {\mathfrak{m}}$ and $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{m}}}}(K, A^{*}(1)) = 0$. Then by global duality, we have an exact sequence $$\begin{aligned}
\label{fitt-exact}
0 \to H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A) \to H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, A) \to X \to H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}})}(K, A^{*}(1))^{*} \to 0\end{aligned}$$ where $X = \bigoplus_{{\mathfrak{q}}\mid {\mathfrak{n}}}H^{1}_{/ {\rm tr}}(K_{{\mathfrak{q}}}, A) \oplus \bigoplus_{{\mathfrak{q}}\mid \frac{{\mathfrak{m}}}{{\mathfrak{n}}}}H^{1}_{/f}(K_{{\mathfrak{q}}}, A )$. Note that, under Hypothesis \[hyp large\], the $R$-modules $X$ and $H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, A)$ are respectively free of ranks $\nu({\mathfrak{m}})$ and $r + \nu({\mathfrak{m}})$. Since $\epsilon_{{\mathfrak{m}}}$ is a generator of ${\bigcap}^{r+\nu({\mathfrak{m}})}_{R}H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, A)$ by Theorem \[thm stark\](i), $\kappa_{{\mathfrak{n}}}$ is a generator of the $R$-module $$\operatorname{im}\left({\bigwedge}_{{\mathfrak{q}}\mid {\mathfrak{n}}}\varphi_{{\mathfrak{q}}}^{\rm fs} \colon {\bigcap}^{r+\nu({\mathfrak{m}})}_{R}H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K,A) \to {\bigcap}^{r}_{R}H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A)\right)$$ by the definitions of Stark system and the map ${\rm Reg}_{r}$. Hence by Lemma \[prop injective\](ii) and the exact sequence (\[fitt-exact\]), we have $$\begin{aligned}
\operatorname{Fitt}_{R}^{0}( H^{1}_{{\mathcal{F}}^{*}({\mathfrak{n}})}(K, A^{*}(1))^{*})
= \operatorname{im}(\kappa_{{\mathfrak{n}}}).\end{aligned}$$
Next, we will prove claim(iii). In view of claim(ii) and Corollary \[fitt-ind\], we only need to show that ${\mathrm{Ann}}_{R}( H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A))$ vanishes for any ideal ${\mathfrak{n}}\in {\mathcal{N}}$ when $R$ is a principal ideal ring.
Let ${\mathfrak{n}}\in {\mathcal{N}}$. Then, by using Lemma \[chebotarev\], take an ideal ${\mathfrak{m}}\in {\mathcal{N}}$ with ${\mathfrak{n}}\mid {\mathfrak{m}}$ and $H^{1}_{({\mathcal{F}}^{*})_{{\mathfrak{m}}}}(K, A^{*}(1))$ vanishes. Then we have $${\rm rank}_{R}(X) = \nu ({\mathfrak{m}}) < r + \nu({\mathfrak{m}}) = {\rm rank}_{R}(H^{1}_{{\mathcal{F}}^{{\mathfrak{m}}}}(K, A)).$$ Since $R$ is principal, there is an injection $$R^{r} \hookrightarrow \ker(H^1_{{\mathcal{F}}^{\mathfrak{m}}}(K,A) \to X)=H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A),$$ by the elementary divisor theorem. Hence the ideal ${\mathrm{Ann}}_{R}( H^{1}_{{\mathcal{F}}({\mathfrak{n}})}(K, A))$ vanishes, as required.
Kolyvagin Systems over Gorenstein orders
----------------------------------------
In this subsection, we use the same notation as in §\[one-dim case\]. Furthermore, [*we assume that Hypothesis \[hyp1’\] and $(T/p^{m}T, {\mathcal{F}}, {\mathcal{P}}_{m})$ satisfies Hypothesis \[hyp large\] for any positive integer $m$*]{}.
Let $m$ be a positive integer and ${\mathfrak{n}}\in {\mathcal{N}}_{m+1}$ a core vertex for $(T/p^{m+1}T, {\mathcal{F}})$. Then ${\mathfrak{n}}$ is also a core vertex for $(T/p^{m}T, {\mathcal{F}})$ by Corollary \[dualselisom\]. Hence in the same way as in §\[one-dim case\], we can construct a map $${\rm KS}_{r}(T/p^{m+1}T, {\mathcal{F}}) \to {\rm KS}_{r}(T/p^{m}T, {\mathcal{F}}).$$ such that the diagram $$\begin{aligned}
\xymatrix{
{\rm SS}_{r}(T/p^{m+1}T, {\mathcal{F}}) \ar[d] \ar[r]^{{\rm Reg}_{r}} & {\rm KS}_{r}(T/p^{m+1}T, {\mathcal{F}}) \ar[d]
\\
{\rm SS}_{r}(T/p^{m}T, {\mathcal{F}}) \ar[r]^{{\rm Reg}_{r}} & {\rm KS}_{r}(T/p^{m}T, {\mathcal{F}})
}\end{aligned}$$ commutes.
We define the module ${\rm KS}_{r}(T, {\mathcal{F}})$ of Kolyvagin systems for $(T, {\mathcal{F}})$ to be the inverse limit $${\rm KS}_{r}(T, {\mathcal{F}}) := \varprojlim_{m}{\rm KS}_{r}(T/p^{m}T, {\mathcal{F}}).$$
The maps ${\rm Reg}_{r} \colon {\rm SS}_{r}(T/p^{m}T, {\mathcal{F}}) \to {\rm KS}_{r}(T/p^{m}T, {\mathcal{F}})$ induce a homomorphism (also denoted by ${\rm Reg}_{r}$) $${\rm Reg}_{r} \colon {\rm SS}_{r}(T, {\mathcal{F}}) \to {\rm KS}_{r}(T, {\mathcal{F}}).$$ By Corollary \[dualselisom\] and Theorem \[main\](ii), we have $I_{i}(\kappa^{(m+1)}){\mathcal{R}}/(p^{m}) \subseteq I_{i}(\kappa^{(m)})$ for any $\kappa = \{\kappa^{(n)}\} \in {\rm KS}_{r}(T, {\mathcal{F}})$ and non-negative integer $i$. Hence we can define an ideal $I_{i}(\kappa)$ of ${\mathcal{R}}$ to be the inverse limit $$I_{i}(\kappa) := \varprojlim_{m}I_{i}(\kappa^{(m)}).$$
\[thm koly’\] Suppose that $p>3$.
- The map ${\rm Reg}_{r} \colon {\rm SS}_{r}(T, {\mathcal{F}}) \to {\rm KS}_{r}(T, {\mathcal{F}})$ constructed above is an isomorphism. In particular, the ${\mathcal{R}}$-module ${\rm KS}_{r}(T, {\mathcal{F}})$ is free of rank one.
- For each $\kappa$ in ${\rm KS}_r(T,{\mathcal{F}})$ one has $$I_{0}(\kappa) \subseteq {\rm Fitt}_{\mathcal{R}}^0 \left(H^1_{{\mathcal{F}}^*}(K, T^\vee(1))^\vee \right),$$ with equality if $\kappa$ is a basis of ${\rm KS}_r(T,{\mathcal{F}})$.
- Fix $\kappa$ in ${\rm KS}_r(A,{\mathcal{F}})$. Then for each non-negative integer $i$ one has $$I_i(\kappa)\subseteq {\rm Fitt}_{{\mathcal{R}}}^i(H^1_{{\mathcal{F}}^\ast}(K,T^\vee(1))^\vee),$$ with equality if ${\mathcal{R}}$ is a principal ideal ring and $\kappa$ is a basis of ${\rm KS}_r(T,{\mathcal{F}})$.
Claims (i), (ii) and (iii) follow as direct consequences of the respective claims in Theorem \[main\].
\[difficulty remark2\] For each $\kappa$ in ${\rm KS}_{r}(T, {\mathcal{F}})$ and each non-negative integer $i$, Theorem \[thm koly’\](i) allows us to define an ideal of ${\mathcal{R}}$ by setting $$I'_i(\kappa) := I_i({\rm Reg}_r^{-1}(\kappa)),$$ where the right hand side is as defined in Definition \[stark ideal gorenstein\]. If ${\mathcal{R}}$ is a principal ring and $\kappa$ is a ${\mathcal{R}}$-basis of ${\rm KS}_r(T,{\mathcal{F}})$, then Theorems \[thm koly’\](iii) and \[thm stark’\](ii)(c) combine to imply that $I'_i(\kappa) = I_i(\kappa)$ for all $i$ but we do not know if this is true more generally.
Euler systems and Kolyvagin systems {#euler sys sec}
===================================
In this section, we shall give a natural construction of higher rank Kolyvagin systems from higher rank Euler systems (see Theorem \[derivable1\] and Corollary \[higher der\]).
By using results in previous sections, we shall then show that higher rank Euler systems control Selmer modules (see Corollaries \[derivable cor\], \[remark surjective\] and \[main cor\]).
Definition {#euler sys sec 1}
----------
Let $K$ be a number field. Let $p$ be a prime number. Let $Q$ be a finite extension of ${\mathbb{Q}}_p$, and ${\mathcal{O}}$ the ring of integers of $Q$. Let $\mathcal{Q}$ be a finite dimensional semisimple commutative $Q$-algebra. Let ${\mathcal{R}}$ be a semilocal Gorenstein ${\mathcal{O}}$-order in $\mathcal{Q}$. Let $T$ be a free ${\mathcal{R}}$-module of finite rank with a continuous ${\mathcal{R}}$-linear action of $G_K$. We assume that $S_{\rm ram}(T)$ is finite, and choose a finite set $S$ of places of $K$ such that $$S_{\infty}(K)\cup S_p(K) \cup S_{\rm ram}(T) \subseteq S.$$ For a prime ${\mathfrak{q}}\notin S$, we set $$P_{\mathfrak{q}}(x):=\det(1-{\rm Fr}_{\mathfrak{q}}^{-1}x \mid T^\ast(1)) \in {\mathcal{R}}[x],$$ where $T^\ast(1):=\operatorname{Hom}_{\mathcal{R}}(T, {\mathcal{R}}(1))$. Let ${\mathcal{K}}/K$ be an abelian pro-$p$ extension such that all infinite places $v \in S_\infty(K)$ split completely in ${\mathcal{K}}$. We define a set of subfields of ${\mathcal{K}}/K$ by $$\Omega({\mathcal{K}}/K):=\{F \mid K\subseteq F\subseteq {\mathcal{K}}, \text{ $F/K$ is finite}\}.$$ For a field $F$ in $\Omega({\mathcal{K}}/K)$, set $$\begin{aligned}
S(F) &:= S\cup S_{\rm ram}(F/K),
\\
{\mathcal{G}}_F &:= \operatorname{Gal}(F/K).\end{aligned}$$
In the following, we assume
\[hyp free\]
- $H^1({\mathcal{O}}_{F, S(F)}, T)$ is a reflexive ${\mathcal{R}}[{\mathcal{G}}_F]$-module for every $F \in \Omega({\mathcal{K}}/K)$.
- $H^0(F, T)=0$ for every $F \in \Omega({\mathcal{K}}/K)$.
\[hyp free rem\] Since ${\mathcal{R}}$ is a Gorenstein ${\mathcal{O}}$-order, Hypothesis \[hyp free\](i) is satisfied if and only if each group $H^1({\mathcal{O}}_{F, S(F)}, T)$ is free as an ${\mathcal{O}}$-module. (See [@bassgorenstein Th. 6.2].)
\[rem torsion free\] When ${\mathcal{R}}={\mathcal{O}}={\mathbb{Z}}_p$ and $T={\mathbb{Z}}_p(1)$, Hypothesis \[hyp free\](i) is equivalent to the condition that the $p$-completion of the unit group ${\mathcal{O}}_{F,S(F)}^\times$ is torsion-free for every $F \in \Omega({\mathcal{K}}/K)$. This condition often appears in the context of Stark conjectures, and we usually choose another set $\Sigma$ of places to avoid assuming the condition, by considering the ‘$\Sigma$-modified unit group’ ${\mathcal{O}}_{F,S(F), \Sigma}^\times$ (see [@rubinstark], where our $\Sigma$ is denoted by $T$). For a general $p$-adic representation $T$, we can consider the ‘$\Sigma$-modified cohomology’ in a similar way to avoid assuming Hypothesis \[hyp free\](i). For details, see [@sbA §2.3]. In this article, we do not consider such modified cohomology theory for simplicity.
The definition of higher rank Euler systems is as follows.
Let $r$ be a non-negative integer. An element $$(c_F)_F \in \prod_{F \in \Omega({\mathcal{K}}/K)} {{\bigcap}}_{{\mathcal{R}}[{\mathcal{G}}_F]}^rH^1({\mathcal{O}}_{F,S(F)},T)$$ is said to be an Euler system of rank $r$ for ($T,{\mathcal{K}}$) if $${\rm Cor}_{F'/F}(c_{F'})=\left(\prod_{{\mathfrak{q}}\in S(F')\setminus S(F)} P_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1})\right)c_F \ \text{ in } \ {{\bigcap}}_{{\mathcal{R}}[{\mathcal{G}}_F]}^r H^1({\mathcal{O}}_{F,S(F')},T)$$ for any $F,F' \in \Omega({\mathcal{K}}/K)$ with $F \subseteq F'$, where $${\rm Cor}_{F'/F}: {\bigcap}_{{\mathcal{R}}[{\mathcal{G}}_{F'}]}^r H^1({\mathcal{O}}_{F',S(F')}, T) \to {\bigcap}_{{\mathcal{R}}[{\mathcal{G}}_F]}^r H^1({\mathcal{O}}_{F,S(F')},T)$$ is the map induced by the corestriction map.
The set of Euler systems of rank $r$ (for $(T, {\mathcal{K}})$) is denoted by ${\rm ES}_r(T,{\mathcal{K}})$. This has a natural structure of ${\mathcal{R}}[[\operatorname{Gal}({\mathcal{K}}/K)]]$-module.
\[remark free\] If Hypothesis \[hyp free\](i) is satisfied, then we have $${\bigcap}_{{\mathcal{R}}[{\mathcal{G}}_F]}^1 H^1({\mathcal{O}}_{F, S(F)}, T)=H^1({\mathcal{O}}_{F, S(F)}, T)^{\ast \ast}=H^1({\mathcal{O}}_{F, S(F)}, T)$$ for every $F\in \Omega({\mathcal{K}}/K)$, so we can regard an Euler system of rank one as an element in $ \prod_{F \in \Omega({\mathcal{K}}/K)} H^1({\mathcal{O}}_{F,S(F)},T)$. Thus our definition generalizes the classical definition of Euler systems given in [@R Def. 2.1.1].
The canonical Selmer structure {#section canonical}
------------------------------
The canonical Selmer structure ${\mathcal{F}}_{\rm can}$ on $T$ (see [@MRkoly Def. 3.2.1]) is the following data:
- $S({\mathcal{F}}_{\rm can}):=S_\infty(K) \cup S_p(K) \cup S_{\rm ram}(T)$;
- for $v\in S({\mathcal{F}}_{\rm can})$, $$H_{{\mathcal{F}}_{\rm can}}^1(K_v,T):=\begin{cases}
\ker(H^1(K_v,T) \to H^1(K_v^{\rm ur}, T \otimes_{{\mathbb{Z}}_p}{\mathbb{Q}}_p)) &\text{if $v \notin S_\infty(K)\cup S_p(K)$,}\\
H^1(K_v,T) &\text{if $v \in S_\infty(K)\cup S_p(K)$.}
\end{cases}$$
Here $K_v^{\rm ur}$ denotes the maximal unramified extension of $K_v$. The significance of this Selmer structure is explained by the following well-known result (taken from [@R Cor. B.3.5]).
\[lemma unram\] Let $c $ be an Euler system of rank one for $(T,{\mathcal{K}})$. Assume that ${\mathcal{K}}$ contains a ${\mathbb{Z}}_p^d$-extension of $K$ for some $d\geq 1$, in which no finite place of $K$ splits completely. Then $c_F$ belongs to $H^1_{{\mathcal{F}}_{\rm can}}(F,T)$ for every $F$ in $\Omega({\mathcal{K}}/K)$.
In practice, one usually takes ${\mathcal{K}}$ to be a sufficiently large abelian pro-$p$ extension. For later purposes, we now state the usual assumptions on ${\mathcal{K}}$ as an explicit hypothesis.
\[hyp K\] The field ${\mathcal{K}}$ contains $K({\mathfrak{q}})$ for every ${\mathfrak{q}}\notin S$ and a ${\mathbb{Z}}_p^d$-extension of $K$ for some $d\geq 1$, in which no finite place of $K$ splits completely.
This hypothesis is included in the definition of Euler systems given in [@R Def. 2.1.1].
Kolyvagin derivatives {#koly sect}
---------------------
We review the construction of ‘Kolyvagin derivatives’ in the higher rank case (see [@sbA §4.3.1]). We fix $M$, a power of $p$. We also fix $E \in \Omega({\mathcal{K}}/K)$ such that $E/K$ is unramified outside $S$ and that $K(1) \subseteq E$. (Recall that $K(1)$ denotes the maximal $p$-extension inside the Hilbert class field of $K$.) We denote $\overline {\mathcal{R}}:={\mathcal{R}}/(M)$, $R:=\overline {\mathcal{R}}[\operatorname{Gal}(E/K)]$, $A:=T/MT$, ${\mathcal{T}}:={\rm Ind}_{G_K}^{G_E}(T)$, and ${\mathcal{A}}:={\rm Ind}_{G_K}^{G_E}(A)={\mathcal{T}}/M{\mathcal{T}}$.
We shall recall some notation from §\[section hyp\] and set some new notation. We consider the set ${\mathcal{P}}$ of primes ${\mathfrak{q}}\notin S$ such that
- ${\mathfrak{q}}$ splits completely in $K(\mu_M,({\mathcal{O}}_K^\times)^{1/M})K(1)$,
- ${\mathcal{A}}/({\rm Fr}_{\mathfrak{q}}-1){\mathcal{A}}\simeq R$ as $R$-modules.
Note that ${\mathcal{P}}$ contains that defined in §\[section hyp\] if we assume Hypothesis \[hyp1\](ii) for ${\mathcal{A}}$. Let ${\mathcal{N}}={\mathcal{N}}({\mathcal{P}})$ be the set of square-free products of primes in ${\mathcal{P}}$. We set $$G_{\mathfrak{q}}:=\operatorname{Gal}(K({\mathfrak{q}})/K(1)) \simeq \operatorname{Gal}(E({\mathfrak{q}})/E),$$ where $E({\mathfrak{q}}):=E\cdot K({\mathfrak{q}})$. For ${\mathfrak{n}}\in {\mathcal{N}}$, we set $K({\mathfrak{n}}):=\prod_{{\mathfrak{q}}\mid {\mathfrak{n}}} K({\mathfrak{q}})$ (compositum) and $E({\mathfrak{n}}):=E\cdot K({\mathfrak{n}})$. We also set ${\mathcal{G}}_{\mathfrak{n}}:=\operatorname{Gal}(E({\mathfrak{n}})/K)$ and ${\mathcal{H}}_{\mathfrak{n}}:=\operatorname{Gal}(E({\mathfrak{n}})/E)$. Note that we have natural identifications $${\mathcal{H}}_{\mathfrak{n}}=\operatorname{Gal}(K({\mathfrak{n}})/K(1)) =\prod_{{\mathfrak{q}}\mid {\mathfrak{n}}} G_{\mathfrak{q}}.$$
For an Euler system $c$ of rank $r$ and ${\mathfrak{n}}\in {\mathcal{N}}$, we set $$c_{\mathfrak{n}}:=c_{E({\mathfrak{n}})} \in {{\bigcap}}_{{\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^r H^1 ({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T),$$ where $S_{\mathfrak{n}}:=S\cup \{{\mathfrak{q}}\mid {\mathfrak{n}}\}(=S(E({\mathfrak{n}})))$. Fix a generator $\sigma_{\mathfrak{q}}$ of $G_{\mathfrak{q}}$ for each ${\mathfrak{q}}$, and consider the ‘derivative operator’ $$D_{\mathfrak{q}}:=\sum_{i=1}^{| G_{\mathfrak{q}}|-1}i\sigma_{\mathfrak{q}}^i \in {\mathbb{Z}}[G_{\mathfrak{q}}].$$ For ${\mathfrak{n}}\in {\mathcal{N}}$, we set $$D_{\mathfrak{n}}:=\prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}D_{\mathfrak{q}}\in {\mathbb{Z}}[{\mathcal{H}}_{\mathfrak{n}}].$$ (We set $D_1:=1$ for convention.)
Note that the natural ‘mod $M$’ map $T \to T/MT=A$ induces a map $$\begin{aligned}
\label{mod M map}
{{\bigcap}}_{{\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^r H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T) \to {{\bigcap}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^r H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A).\end{aligned}$$ We explain the construction of this map, since we need Hypothesis \[hyp free\](i) here. We set $H^1(T):= H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)$ and $H^1(A):=H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A)$ for simplicity. Also, for the moment we denote ${\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]$ and $\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]$ simply by ${\mathcal{R}}$ and $\overline {\mathcal{R}}$ respectively (by abuse of notation). First, note that Hypothesis \[hyp free\](i) implies that $$\operatorname{Ext}_{\mathcal{R}}^1(H^1(T),{\mathcal{R}})=0.$$ (See [@sbA §A.3].) From this, we see that $$H^1(T)^\ast/M=\operatorname{Hom}_{\mathcal{R}}(H^1(T), {\mathcal{R}})/M \simeq \operatorname{Hom}_{\overline {\mathcal{R}}}(H^1(T)/M, \overline {\mathcal{R}})= (H^1(T)/M)^\ast,$$ where we abbreviate $X/MX$ to $X/M$. Since there is a natural map $H^1(T)/M \to H^1(A)$, we obtain a map $$H^1(A)^\ast \to (H^1(T)/M)^\ast \simeq H^1(T)^\ast/M.$$ This map induces a map $$\begin{aligned}
\label{wedge induce}
{\bigwedge}_{\overline {\mathcal{R}}}^r H^1(A)^\ast \to {\bigwedge}_{\overline {\mathcal{R}}}^r (H^1(T)^\ast/M)=\left( {\bigwedge}_{{\mathcal{R}}}^r H^1(T)^\ast \right)/M.\end{aligned}$$ Then we obtain (\[mod M map\]) as the following map: $$\begin{aligned}
{\bigcap}_{\mathcal{R}}^r H^1(T) &\to& \operatorname{Hom}_{\mathcal{R}}\left( {\bigwedge}_{\mathcal{R}}^r H^1(T)^\ast, \overline {\mathcal{R}}\right) \\
&=& \operatorname{Hom}_{\overline {\mathcal{R}}} \left( \left({\bigwedge}_{ {\mathcal{R}}}^r H^1(T)^\ast\right)/M, \overline {\mathcal{R}}\right) \\
&\stackrel{(\ref{wedge induce})}{\to}& \operatorname{Hom}_{\overline {\mathcal{R}}} \left( {\bigwedge}_{\overline {\mathcal{R}}}^r H^1(A)^\ast, \overline {\mathcal{R}}\right) \\
&=& {\bigcap}_{\overline {\mathcal{R}}}^r H^1(A).\end{aligned}$$ We denote the image of $c_{\mathfrak{n}}$ under (\[mod M map\]) by $\bar c_{\mathfrak{n}}$. The following is well-known.
\[lem invariant\] The element $D_{\mathfrak{n}}\cdot \bar c_{\mathfrak{n}}$ lies in $\left({{\bigcap}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^r H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A)\right)^{{\mathcal{H}}_{\mathfrak{n}}}$.
We use the identity $$\begin{aligned}
\label{telescoping}
(\sigma_{\mathfrak{q}}-1)D_{\mathfrak{q}}= |G_{\mathfrak{q}}| - \operatorname{N}_{G_{\mathfrak{q}}},\end{aligned}$$ where $\operatorname{N}_{G_{\mathfrak{q}}}:=\sum_{\sigma \in G_{\mathfrak{q}}} \sigma$. (This is checked by direct computation.)
We shall show that $$(\sigma-1)D_{\mathfrak{n}}\cdot c_{\mathfrak{n}}\in M \cdot {\bigcap}_{{\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^r H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)$$ for every $\sigma \in {\mathcal{H}}_{\mathfrak{n}}$. We prove this by induction on $\nu({\mathfrak{n}})$. When $\nu({\mathfrak{n}})=0$, i.e. ${\mathfrak{n}}=1$, there is nothing to prove. When $\nu({\mathfrak{n}})>0$, it is sufficient to show that $$(\sigma_{\mathfrak{q}}-1)D_{\mathfrak{n}}\cdot c_{\mathfrak{n}}\in M \cdot {\bigcap}_{{\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^r H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)$$ for every ${\mathfrak{q}}\mid {\mathfrak{n}}$ (since the augmentation ideal of ${\mathbb{Z}}[{\mathcal{H}}_{\mathfrak{n}}]$ is generated by the elements $\sigma_{\mathfrak{q}}-1$ for any ${\mathfrak{q}}\mid {\mathfrak{n}}$). Using (\[telescoping\]), we compute $$\begin{aligned}
(\sigma_{\mathfrak{q}}-1)D_{\mathfrak{n}}c_{\mathfrak{n}}&=&(| G_{\mathfrak{q}}| -\operatorname{N}_{G_{\mathfrak{q}}})D_{{\mathfrak{n}}/{\mathfrak{q}}} c_{\mathfrak{n}}\\
&=&| G_{\mathfrak{q}}| D_{{\mathfrak{n}}/{\mathfrak{q}}}c_{\mathfrak{n}}- P_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1}) D_{{\mathfrak{n}}/{\mathfrak{q}}} c_{{\mathfrak{n}}/{\mathfrak{q}}},\end{aligned}$$ where the second equality follows from the definition of Euler systems. Here we regard $c_{{\mathfrak{n}}/{\mathfrak{q}}}$ as an element in ${\bigcap}_{{\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^r H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)$ via the restriction map. (To relate the corestriction map on $\bigcap^r$ with $\operatorname{N}_{G_{\mathfrak{q}}}$, a slightly delicate consideration is needed, but we omit the detail. See [@sano Rem. 2.12] concerning this issue.) Note that $$P_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1}) D_{{\mathfrak{n}}/{\mathfrak{q}}} c_{{\mathfrak{n}}/{\mathfrak{q}}} =(P_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1})-P_{\mathfrak{q}}(1)) D_{{\mathfrak{n}}/{\mathfrak{q}}} c_{{\mathfrak{n}}/{\mathfrak{q}}} + P_{\mathfrak{q}}(1) D_{{\mathfrak{n}}/{\mathfrak{q}}} c_{{\mathfrak{n}}/{\mathfrak{q}}},$$ and that we know by induction hypothesis that the first term in the right hand side vanishes modulo $M$. Hence, since both $| G_{\mathfrak{q}}|$ and $P_{\mathfrak{q}}(1)$ are divisible by $M$ (by the definition of ${\mathcal{P}}$), we conclude that $(\sigma_{\mathfrak{q}}-1)D_{\mathfrak{n}}c_{\mathfrak{n}}$ vanishes modulo $M$. This proves the lemma.
One can also show that $$\begin{aligned}
\left({{\bigcap}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^r H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A)\right)^{{\mathcal{H}}_{\mathfrak{n}}} =
{{\bigcap}}_{R}^r H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A)^{{\mathcal{H}}_{\mathfrak{n}}}.\end{aligned}$$ (See [@sbA Prop. A.4]). Furthermore, under Hypothesis \[hyp free\](ii), we have $$H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A)^{{\mathcal{H}}_{\mathfrak{n}}}=H^1({\mathcal{O}}_{E,S_{\mathfrak{n}}},A)=H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}},{\mathcal{A}})$$ (see [@sbA §4.3.1]). Combining these observations with Lemma \[lem invariant\], we have proved the following
Assume Hypothesis \[hyp free\]. Then for each ${\mathfrak{n}}\in {\mathcal{N}}$ one has $$\kappa'(c_{\mathfrak{n}}):=D_{\mathfrak{n}}\cdot \bar c_{\mathfrak{n}}\in {{\bigcap}}_R^rH^1 ({\mathcal{O}}_{K,S_{\mathfrak{n}}},{\mathcal{A}}).$$
The element $\kappa'(c_{\mathfrak{n}})$ is called a Kolyvagin derivative.
Construction of Kolyvagin systems {#construction koly}
---------------------------------
In this subsection, we modify the Kolyvagin derivatives $(\kappa'(c_{\mathfrak{n}}))_{\mathfrak{n}}$ to construct a Kolyvagin system. Theorem \[derivable1\] below is the main result of this section.
We write ${\mathcal{I}}_{\mathfrak{n}}$ for the augmentation ideal of ${\mathbb{Z}}[{\mathcal{H}}_{\mathfrak{n}}]$. We recall that the cyclic subgroup of ${\mathcal{I}}_{\mathfrak{n}}^{\nu({\mathfrak{n}})}/{\mathcal{I}}_{\mathfrak{n}}^{\nu({\mathfrak{n}})+1}$ generated by $\prod_{{\mathfrak{q}}\mid {\mathfrak{n}}} (\sigma_{\mathfrak{q}}-1)$ is a direct summand, and is isomorphic to $G_{\mathfrak{n}}:=\bigotimes_{{\mathfrak{q}}\mid {\mathfrak{n}}} G_{\mathfrak{q}}$: $$\begin{aligned}
G_{\mathfrak{n}}=\bigotimes_{{\mathfrak{q}}\mid {\mathfrak{n}}} G_{\mathfrak{q}}&\stackrel{\sim}{\to} \left\langle \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1)\right\rangle \subseteq {\mathcal{I}}_{\mathfrak{n}}^{\nu({\mathfrak{n}})}/{\mathcal{I}}_{\mathfrak{n}}^{\nu({\mathfrak{n}})+1}
\\
\bigotimes_{{\mathfrak{q}}\mid {\mathfrak{n}}} \sigma_{{\mathfrak{q}}} &\mapsto \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1).\end{aligned}$$ (See [@MR Prop. 4.2].) We often identify $G_{\mathfrak{n}}$ with $ \left\langle \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1)\right\rangle $. In particular, we regard a Kolyvagin system for $({\mathcal{A}}, {\mathcal{F}})$ (for the definition, see §\[defkoly\]) as an element in $\prod_{{\mathfrak{n}}\in {\mathcal{N}}} {\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}}) \otimes \left \langle \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1)\right \rangle$.
For ${\mathfrak{q}}\in {\mathcal{P}}$, we shall denote $P_{\mathfrak{q}}({\rm Fr}_{\mathfrak{q}}^{-1})$ simply by $P_{\mathfrak{q}}$. If ${\mathfrak{q}}$ does not divide ${\mathfrak{n}}\in {\mathcal{N}}$, then ${\mathfrak{q}}$ is unramified in $E({\mathfrak{n}})$, so we can regard $P_{\mathfrak{q}}\in {\mathcal{R}}[{\mathcal{H}}_{\mathfrak{n}}]$. Furthermore, since $P_{\mathfrak{q}}(1) \equiv 0 \text{ (mod $M$)}$, we can regard $P_{\mathfrak{q}}\in \overline {\mathcal{R}}\otimes {\mathcal{I}}_{\mathfrak{n}}/{\mathcal{I}}_{\mathfrak{n}}^2$, which we denote by $P_{\mathfrak{q}}^{\mathfrak{n}}$. (Recall that $\overline {\mathcal{R}}:={\mathcal{R}}/(M)$.) For ${\mathfrak{n}}\in {\mathcal{N}}$, we define an element ${\mathcal{D}}_{\mathfrak{n}}\in \overline {\mathcal{R}}\otimes \left\langle \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1)\right\rangle$ as follows. We write ${\mathfrak{n}}={\mathfrak{q}}_1\cdots {\mathfrak{q}}_\nu$ ($\nu=\nu({\mathfrak{n}})$). We define $${\mathcal{D}}_{\mathfrak{n}}:=\det\left(
\begin{array}{ccccc}
0 &P_{{\mathfrak{q}}_1}^{{\mathfrak{q}}_2} &\cdots & &P_{{\mathfrak{q}}_1}^{{\mathfrak{q}}_\nu} \\
P_{{\mathfrak{q}}_2}^{{\mathfrak{q}}_1} & 0 & P_{{\mathfrak{q}}_2}^{{\mathfrak{q}}_3} & \cdots & P_{{\mathfrak{q}}_2}^{{\mathfrak{q}}_\nu} \\
\vdots & P_{{\mathfrak{q}}_3}^{{\mathfrak{q}}_2} & \ddots & &\vdots\\
\vdots & \vdots & &\ddots &\vdots\\
P_{{\mathfrak{q}}_\nu}^{{\mathfrak{q}}_1} &P_{{\mathfrak{q}}_\nu}^{{\mathfrak{q}}_2} &\cdots & & 0
\end{array}
\right) \in \overline {\mathcal{R}}\otimes \left\langle \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1)\right\rangle.$$ (Compare [@MR Def. 6.1] and [@sanojnt Def. 4.4].) One checks that this does not depend on the choice of the labeling ${\mathfrak{q}}_1,\ldots,{\mathfrak{q}}_\nu$ of the prime divisors of ${\mathfrak{n}}$.
Now we consider the following modification of $(\kappa'(c_{\mathfrak{n}}))_{\mathfrak{n}}$: $$\kappa(c)_{\mathfrak{n}}:=\sum_{{\mathfrak{d}}\mid {\mathfrak{n}}} \left(\kappa'(c_{\mathfrak{d}}) \otimes \prod_{{\mathfrak{q}}\mid {\mathfrak{d}}}(\sigma_{\mathfrak{q}}-1)\right){\mathcal{D}}_{{\mathfrak{n}}/{\mathfrak{d}}} \in {\bigcap}_R^r H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}},{\mathcal{A}}) \otimes \left \langle \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1)\right \rangle.$$ (Each $\kappa'(c_{\mathfrak{d}}) \in {\bigcap}_R^r H^1({\mathcal{O}}_{K,S_{\mathfrak{d}}},{\mathcal{A}})$ is regarded as an element of ${\bigcap}_R^r H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}},{\mathcal{A}})$.) One easily checks that $$\sum_{{\mathfrak{d}}\mid {\mathfrak{n}}} \left(\kappa'(c_{\mathfrak{d}}) \otimes \prod_{{\mathfrak{q}}\mid {\mathfrak{d}}}(\sigma_{\mathfrak{q}}-1)\right){\mathcal{D}}_{{\mathfrak{n}}/{\mathfrak{d}}} =\sum_{\tau \in \mathfrak{S}({\mathfrak{n}})} {\rm sgn}(\tau)\kappa'(c_{{\mathfrak{d}}_\tau}) \otimes \prod_{{\mathfrak{q}}\mid {\mathfrak{d}}_\tau}(\sigma_{\mathfrak{q}}-1) \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}/{\mathfrak{d}}_\tau}P_{\tau({\mathfrak{q}})}^{\mathfrak{q}},$$ where $\mathfrak{S}({\mathfrak{n}})$ is the set of permutations of prime divisors of ${\mathfrak{n}}$, and ${\mathfrak{d}}_\tau:=\prod_{\tau({\mathfrak{q}})={\mathfrak{q}}}{\mathfrak{q}}$. So one can also write $$\kappa(c)_{\mathfrak{n}}= \sum_{\tau \in \mathfrak{S}({\mathfrak{n}})} {\rm sgn}(\tau)\kappa'(c_{{\mathfrak{d}}_\tau}) \otimes \prod_{{\mathfrak{q}}\mid {\mathfrak{d}}_\tau}(\sigma_{\mathfrak{q}}-1) \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}/{\mathfrak{d}}_\tau}P_{\tau({\mathfrak{q}})}^{\mathfrak{q}}.$$ Note that this construction is parallel to that given by Mazur and Rubin in [@MRkoly (33) in Appendix A].
We need the following hypothesis which corresponds to an assumption in [@MRkoly Th. 3.2.4].
\[hyp local\] ${\rm Fr}_{\mathfrak{q}}^{p^k}-1$ is injective on $T$ for every ${\mathfrak{q}}\in {\mathcal{P}}$ and $k \geq 0$.
Now we state the main theorem of this section.
\[derivable1\] Let $r$ be a positive integer and $c \in {\rm ES}_r(T,{\mathcal{K}})$. Let ${\mathcal{F}}:={\mathcal{F}}_{\rm can}$ be the canonical Selmer structure (see §\[section canonical\]). Assume Hypotheses \[hyp free\], \[hyp K\] and \[hyp local\]. Then, for every ${\mathfrak{n}}\in {\mathcal{N}}$, we have $$\kappa(c)_{\mathfrak{n}}\in {{\bigcap}}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}}) \otimes \left \langle \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1)\right \rangle$$ and $$v_{\mathfrak{q}}(\kappa(c)_{\mathfrak{n}})=\varphi_{\mathfrak{q}}^{\rm fs}(\kappa(c)_{{\mathfrak{n}}/{\mathfrak{q}}}). $$ for every ${\mathfrak{q}}\mid {\mathfrak{n}}$. In particular, $\kappa(c):=(\kappa(c)_{\mathfrak{n}})_{\mathfrak{n}}\in {\rm KS}_r({\mathcal{A}},{\mathcal{F}})$.
The proof of Theorem \[derivable1\] will be given in the next subsection. For the moment, however, we use the result to derive several important consequences.
\[higher der\] Let $r$ and ${\mathcal{F}}$ be as in Theorem \[derivable1\]. Assume Hypotheses \[hyp free\], \[hyp K\] and \[hyp local\]. Choose a subfield $F$ of $E/K$ and set $A_F:={\rm Ind}_{G_K}^{G_F}(T/MT)$. Then there is a canonical ‘higher Kolyvagin derivative’ homomorphism $${\mathcal{D}}_r={\mathcal{D}}_r^F: {\rm ES}_r(T,{\mathcal{K}}) \to {\rm KS}_r(A_F,{\mathcal{F}}).$$
The construction of Kolyvagin systems given in Theorem \[derivable1\] gives a homomorphism $${\rm ES}_r(T,{\mathcal{K}}) \to {\rm KS}_r({\mathcal{A}},{\mathcal{F}}); \ c \mapsto \kappa(c).$$ The map ${\mathcal{D}}_r^F$ is obtained by composing this map with the natural map $${\rm KS}_r({\mathcal{A}},{\mathcal{F}})\to {\rm KS}_r(A_F,{\mathcal{F}})$$ induced by ${\mathcal{A}}(={\rm Ind}_{G_K}^{G_E}(T/MT) ) \to A_F$.
\[remark E\] Although we take $F$ as a subfield of $E/K$ in Corollary \[higher der\], this condition is not essential, since for an arbitrary finite abelian ($p$-)extension $F/K$ one can take $E$ so that $F \cdot K(1) \subseteq E$ by enlarging ${\mathcal{K}}$ and $S$ if necessary. The role of the field $E$ is auxiliary (in fact, ${\mathcal{D}}_r^F$ is independent of the choice of $E$), and so one can think of $F$ in Corollary \[higher der\] as arbitrary.
\[derivable cor\] Suppose $p>3$. Let $r $ be a positive integer, $c \in {\rm ES}_r(T,{\mathcal{K}})$ and ${\mathcal{F}}:={\mathcal{F}}_{\rm can}$. Choose a subfield $F$ of $E/K$ and set $A_F:={\rm Ind}_{G_K}^{G_F}(T/MT)$. Assume Hypotheses \[hyp free\], \[hyp K\] and \[hyp local\], and Hypotheses \[hyp1\], \[hyp2\] and \[hyp large\] for $A_F$ and ${\mathcal{F}}$. Let $\kappa(c):={\mathcal{D}}_r^F(c) \in {\rm KS}_r(A_F,{\mathcal{F}})$ be the Kolyvagin system constructed in Corollary \[higher der\].
- For ${\mathfrak{n}}$ in ${\mathcal{N}}$ one has $\operatorname{im}(\kappa(c)_{\mathfrak{n}}) \subseteq {\rm Fitt}_{\overline {\mathcal{R}}[{\mathcal{G}}_F]}^0(H^1_{{\mathcal{F}}({\mathfrak{n}})^\ast}(K,A_F^\ast(1))^\ast)$. In particular, one has $$\operatorname{im}(c_F)\subseteq {\rm Fitt}_{\overline {\mathcal{R}}[{\mathcal{G}}_F]}^0(H^1_{{\mathcal{F}}^\ast}(K,A_F^\ast(1))^\ast).$$ Here we regard $c_F \in {\bigcap}_{{\mathcal{R}}[{\mathcal{G}}_F]}^r H^1({\mathcal{O}}_{F,S},T)$ as an element in ${\bigcap}_{\overline {\mathcal{R}}[{\mathcal{G}}_F]}^r H^1({\mathcal{O}}_{F,S},A)\simeq \bigcap_{\overline {\mathcal{R}}[{\mathcal{G}}_F]}^r H^1({\mathcal{O}}_{K,S},A_F)$ by using the natural map (\[mod M map\]).
- For every non-negative integer $i$, we have $$I_i(\kappa(c))\subseteq {\rm Fitt}_{\overline {\mathcal{R}}[{\mathcal{G}}_F]}^i(H^1_{{\mathcal{F}}^\ast}(K,A_F^\ast(1))^\ast).$$
This follows directly from Theorems \[main\](ii) and (iii), noting that $\kappa(c)_1=c_F$.
The set ${\mathcal{P}}$ defined in §\[koly sect\] for ${\mathcal{A}}$ is in general smaller than the corresponding set that is defined in §\[section hyp\] for $A_F$. However, this difference does not matter since, as long as we can choose a subset of ${\mathcal{P}}$ of positive density as in Lemma \[chebotarev\] (by the Chebotarev density theorem), the theory of Stark and Kolyvagin systems work. We shall implicitly consider such a smaller set ${\mathcal{P}}$ also in the statements of the results below.
In the following result we recall the ideals $I'_i(\kappa)$ from Remark \[difficulty remark\].
\[remark surjective\] Let $p$, $r$, ${\mathcal{F}}$, $F$ and $A_F$ be as in Corollary \[derivable cor\]. For $c \in {\rm ES}_r(T,{\mathcal{K}})$, we set $\kappa(c):={\mathcal{D}}_r^F(c) \in {\rm KS}_r(A_F,{\mathcal{F}})$. Assume Hypotheses \[hyp free\], \[hyp K\] and \[hyp local\], and Hypotheses \[hyp1\], \[hyp2\] and \[hyp large\] for ${\mathcal{A}}$ and ${\mathcal{F}}$. We also consider the following additional hypotheses.
- $Y_K(T):=\bigoplus_{v \in S_\infty(K)} H^0(K_v, T^\ast(1))$ is a free ${\mathcal{R}}$-module of rank $r$;
- $H^0(K_v,\mathcal{A}^*(1))$ vanishes for each prime $v \in S \setminus S_{\infty}(K)$.
Then the following claims are valid.
- One has $$\langle \operatorname{im}(c_F) \mid c \in {\rm ES}_r(T,{\mathcal{K}})\rangle_{\overline {\mathcal{R}}[{\mathcal{G}}_F]} \subseteq
{\rm Fitt}_{\overline {\mathcal{R}}[{\mathcal{G}}_F]}^0(H^1_{{\mathcal{F}}^\ast}(K,A_F^\ast(1))^\ast),$$ with equality if hypotheses (a) and (b) are satisfied.
- For each non-negative integer $i$ one has $$\langle I'_i(\kappa(c)) \mid c \in {\rm ES}_r(T,{\mathcal{K}})\rangle_{\overline {\mathcal{R}}[{\mathcal{G}}_F]} \subseteq
{\rm Fitt}_{\overline {\mathcal{R}}[{\mathcal{G}}_F]}^i(H^1_{{\mathcal{F}}^\ast}(K,A_F^\ast(1))^\ast),$$ with equality if hypotheses (a) and (b) are satisfied.
- If $\overline {\mathcal{R}}[{\mathcal{G}}_F]$ is a principal ideal ring, then for each non-negative integer $i$ one has $$\langle I_i(\kappa(c)) \mid c \in {\rm ES}_r(T,{\mathcal{K}})\rangle_{\overline {\mathcal{R}}[{\mathcal{G}}_F]} \subseteq {\rm Fitt}_{\overline {\mathcal{R}}[{\mathcal{G}}_F]}^i(H^1_{{\mathcal{F}}^\ast}(K,A_F^\ast(1))^\ast),$$ with equality if hypotheses (a) and (b) are satisfied.
By Theorems \[main\] and \[thm stark\] (and Remark \[difficulty remark\]), it is sufficient to show that the validity of the given hypotheses (a) and (b) imply the existence of an Euler system $c$ such that $\kappa(c)$ is a basis of ${\rm KS}_r(A_F,{\mathcal{F}})$, or equivalently, that the homomorphism $${\mathcal{D}}_r^F: {\rm ES}_r(T,{\mathcal{K}}) \to {\rm KS}_r(A_F,{\mathcal{F}})$$ is surjective.
Since the natural map $${\rm KS}_r({\mathcal{A}}, {\mathcal{F}}) \to {\rm KS}_r(A_F,{\mathcal{F}})$$ is surjective (by Theorem \[main\](i) and the argument in the proof of Lemma \[compatible\](i)), it is thus enough to show surjectivity of the map $${\mathcal{D}}_r={\mathcal{D}}_r^E: {\rm ES}_r(T,{\mathcal{K}}) \to {\rm KS}_r({\mathcal{A}},{\mathcal{F}}),$$ or equivalently, surjectivity of the composite $${\rm Reg}_r^{-1} \circ {\mathcal{D}}_r: {\rm ES}_r(T,{\mathcal{K}}) \to {\rm SS}_r({\mathcal{A}},{\mathcal{F}}).$$ We prove surjectivity of this map by using results of the first and the third author in [@sbA].
To do this we recall that the Selmer structure ${\mathcal{F}}_S$ on ${\mathcal{A}}$ that is considered in [@sbA] is defined by setting
- $S({\mathcal{F}}_S):=S$;
- for $v \in S$, $H^1_{{\mathcal{F}}_S}(K_v,{\mathcal{A}}):=H^1(K_v,{\mathcal{A}}).$
We also note that the stated hypotheses (a) and (b) above correspond to [@sbA Hyp. 2.11 and 3.9]. We shall recall some constructions given in [@sbA].
By [@sbA Th. 2.17], under hypotheses (a) and \[hyp free\], there is a homomorphism $$\theta_{T,{\mathcal{K}}}:{\rm VS}(T,{\mathcal{K}}) \to {\rm ES}_r(T,{\mathcal{K}}),$$ where ${\rm VS}(T,{\mathcal{K}})$ is the module of ‘vertical determinantal systems’ (see [@sbA Def. 2.8]). We define $\mathcal{E}^{\rm b}(T,{\mathcal{K}}):= \operatorname{im}(\theta_{T,{\mathcal{K}}})$. This is called the module of ‘basic Euler systems’ (see [@sbA Def. 2.18]).
By [@sbA Th. 3.11(ii)], under hypotheses (b) and \[hyp1\], there is an isomorphism $${\rm HS}({\mathcal{A}}) \stackrel{\sim}{\to} {\rm SS}_r({\mathcal{A}},{\mathcal{F}}_S),$$ where ${\rm HS}({\mathcal{A}})$ is the module of ‘horizontal determinantal systems’ (see [@sbA Def. 3.2]). There is a natural surjection ${\rm VS}(T,{\mathcal{K}}) \to {\rm HS}({\mathcal{A}})$ (as in [@sbA §4.3.2]).
The argument in the proof of [@sbA Th. 4.16] shows the following diagram is commutative: $$\xymatrix{
\mathcal{E}^{\rm b} (T,{\mathcal{K}}) \ar[rr]^{{\rm Reg}_r^{-1}\circ {\mathcal{D}}_r}& & {\rm SS}_r({\mathcal{A}},{\mathcal{F}}) \ar@{^{(}->}[r] &{\rm SS}_r({\mathcal{A}},{\mathcal{F}}_S)
\\
{\rm VS}(T,{\mathcal{K}}) \ar@{->>}[u]^{\theta_{T,{\mathcal{K}}}} \ar@{->>}[rr] & & {\rm HS}({\mathcal{A}}). \ar[ur]_{\simeq} &
}$$ This diagram shows that ${\rm Reg}_r^{-1}\circ {\mathcal{D}}_r$ is surjective, as required.
\[main cor\] Suppose $p>3$. Let $r $ be a positive integer, $c \in {\rm ES}_r(T,{\mathcal{K}})$ and ${\mathcal{F}}:={\mathcal{F}}_{\rm can}$. Choose a subfield $F$ of $E/K$ and set $T_F:={\rm Ind}_{G_K}^{G_F}(T)$. Assume Hypotheses \[hyp free\], \[hyp K\], \[hyp local\] and \[hyp1’\] for $T$ and Hypothesis \[hyp large\] for $(T_F/p^{m}T_F, {\mathcal{F}}, {\mathcal{P}}_{m})$ for all positive integers $m$.
- One has $\operatorname{im}(c_F) \subseteq {\rm Fitt}_{{\mathcal{R}}[{\mathcal{G}}_F]}^0(H^1_{{\mathcal{F}}^*}(K, T_F^\vee(1))^\vee )$.
- Let $\kappa(c)_m:={\mathcal{D}}_r^F(c) \in {\rm KS}_r(T_F/p^m T_F,{\mathcal{F}})$ be the Kolyvagin system constructed in Corollary \[higher der\] (with $M=p^m$). We set $$\kappa(c):=(\kappa(c)_m)_m \in \varprojlim_m {\rm KS}_r(T_F/p^mT_F,{\mathcal{F}})={\rm KS}_r(T_F,{\mathcal{F}}).$$ Then for each non-negative integer $i$ one has $I_i(\kappa(c))\subseteq \! {\rm Fitt}_{{\mathcal{R}}[{\mathcal{G}}_F]}^i(H^1_{{\mathcal{F}}^\ast}(K,T_F^\vee(1))^\vee). $
- Assume, in addition, that $\bigoplus_{v \in S_\infty(K)} H^0(K_v, T^\ast(1))$ is a free ${\mathcal{R}}$-module of rank $r$ and that $H^{0}(E_{w}, (T/pT)^\vee(1))$ vanishes for all non-archimedean places $w$ of $E$ above $S$. Then for each non-negative integer $i$ one has $$\langle I'_i(\kappa(c)) \mid c \in {\rm ES}_r(T,{\mathcal{K}})\rangle_{ {\mathcal{R}}[{\mathcal{G}}_F]} = {\rm Fitt}_{ {\mathcal{R}}[{\mathcal{G}}_F]}^i(H^1_{{\mathcal{F}}^\ast}(K,T_F^\vee(1))^\vee).$$ If ${\mathcal{R}}[{\mathcal{G}}_F]$ is a principal ideal ring, then one can replace the ideals $I'_i(\kappa(c))$ by $I_i(\kappa(c))$ in this equality.
Claims (i) and (ii) follow directly from Theorem \[thm koly’\](ii) and (iii) (after noting that $I_0(\kappa(c))=\operatorname{im}(c_F)$.)
In a similar way, since the additional hypotheses in claim (iii) are equivalent to the validity of the hypotheses (a) and (b) in Corollary \[remark surjective\] for the representations $T$ and ${\rm Ind}_{G_K}^{G_E}(T/pT)$ respectively, this claim is a straightforward consequence of Corollary \[remark surjective\](iii) and Remark \[difficulty remark2\].
\[remark p=3\] In results above, we excluded the case $p=3$. This is due to the technical condition on $p$ in Lemma \[chebotarev\]. However, it is possible to treat the case $p=3$, if we assume that $T$ is not ‘self-dual’. In fact, Mazur and Rubin proved the result corresponding to Lemma \[chebotarev\] in [@MRkoly Prop. 3.6.1] under their running hypotheses, one of which is ‘either $T$ is not self-dual or $p>4$’ (see [@MRkoly (H.4) in §3.5]).
The proof of Theorem \[derivable1\]
-----------------------------------
In this subsection, we prove Theorem \[derivable1\].
When $r=1$, this result can be proved by using the argument of Mazur and Rubin in [@MRkoly Th. 3.2.4]. More precisely, whilst our setting is more general than that of [@MRkoly], since we work over a general number field $K$ and the coefficient ring of $T$ is a general Gorenstein order, it can be checked that the method of the proof of [@MRkoly Th. 3.2.4] also applies in this more general setting. The essential idea is, therefore, to reduce the general case of Theorem \[derivable1\] to the case that $r=1$.
Throughout this subsection, we assume Hypotheses \[hyp free\], \[hyp K\] and \[hyp local\]. We remark that $$\Psi = (\Psi_F)_F \in \varprojlim_{F \in \Omega({\mathcal{K}}/K)} {{\bigwedge}}_{{\mathcal{R}}[{\mathcal{G}}_F]}^{r-1}H^1({\mathcal{O}}_{F,S(F)},T)^\ast$$ induces a homomorphism $$\Psi \colon {\rm ES}_r(T,{\mathcal{K}}) \to {\rm ES}_1(T,{\mathcal{K}}); \ (c_F)_F\to (\Psi_F(c_F))_F.$$ This construction was introduced by Rubin in [@rubinstark §6] and also used by Perrin-Riou in [@PR §1.2.3].
The following lemma is a key.
\[key lemma\] Let ${\mathfrak{n}}\in {\mathcal{N}}$. Then, for every $\Phi \in {\bigwedge}_R^{r-1}H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}},{\mathcal{A}})^\ast$, there exists $\Psi \in \varprojlim_F {\bigwedge}_{{\mathcal{R}}[{\mathcal{G}}_F]}^{r-1}H^1({\mathcal{O}}_{F,S(F)},T)^\ast$ such that for any ${\mathfrak{d}}\mid {\mathfrak{n}}$ we have $$\Phi(\kappa'(c_{\mathfrak{d}}))=\kappa'(\Psi(c)_{\mathfrak{d}}) \text{ in } H^1_{{\mathcal{F}}^{\mathfrak{n}}}(K,{\mathcal{A}}).$$ (Note that $\Psi(c) \in {\rm ES}_1(T,{\mathcal{K}})$, and $\kappa'(\Psi(c)_{\mathfrak{d}})$ denotes the Kolyvagin derivative of the rank one Euler system $\Psi(c)$.)
Before proving this lemma, we use it to give a proof of Theorem \[derivable1\].
We first show that, for each ${\mathfrak{n}}\in {\mathcal{N}}$, the element $$\kappa(c)_{\mathfrak{n}}=\sum_{{\mathfrak{d}}\mid {\mathfrak{n}}} \left(\kappa'(c_{\mathfrak{d}}) \otimes \prod_{{\mathfrak{q}}\mid {\mathfrak{d}}}(\sigma_{\mathfrak{q}}-1)\right){\mathcal{D}}_{{\mathfrak{n}}/{\mathfrak{d}}}$$ lies in ${{\bigcap}}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}}) \otimes \left \langle \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1)\right \rangle.$ By Proposition \[reduction\], it is sufficient to show that $$\label{trans}
\Phi(\kappa(c)_{\mathfrak{n}})=\sum_{{\mathfrak{d}}\mid {\mathfrak{n}}} \left(\Phi(\kappa'(c_{\mathfrak{d}})) \otimes \prod_{{\mathfrak{q}}\mid {\mathfrak{d}}}(\sigma_{\mathfrak{q}}-1)\right){\mathcal{D}}_{{\mathfrak{n}}/{\mathfrak{d}}} \in H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}}) \otimes \left \langle \prod_{{\mathfrak{q}}\mid {\mathfrak{n}}}(\sigma_{\mathfrak{q}}-1)\right \rangle$$ for every $\Phi \in {\bigwedge}_R^{r-1}H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}},{\mathcal{A}})^\ast$.
By Lemma \[key lemma\], there exists $\Psi \in \varprojlim_F {\bigwedge}_{{\mathcal{R}}[{\mathcal{G}}_F]}^{r-1}H^1({\mathcal{O}}_{F,S(F)},T)^\ast$ such that $$\Phi(\kappa'(c_{\mathfrak{d}}))=\kappa'(\Psi(c)_{\mathfrak{d}})$$ for every ${\mathfrak{d}}\mid {\mathfrak{n}}$, so (\[trans\]) follows from the fact that Theorem \[derivable1\] holds for the rank one Euler system $\Psi(c)$.
Next, we show that $$v_{\mathfrak{q}}(\kappa(c)_{\mathfrak{n}})=\varphi_{\mathfrak{q}}^{\rm fs}(\kappa(c)_{{\mathfrak{n}}/{\mathfrak{q}}}) $$ for every ${\mathfrak{n}}\in {\mathcal{N}}$ and ${\mathfrak{q}}\mid {\mathfrak{n}}$. As in Definition \[koly ideal\], we fix an identification $${\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})\otimes G_{\mathfrak{n}}={\bigcap}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})$$ for each ${\mathfrak{n}}\in {\mathcal{N}}$. We note that, by definition, $${{\bigcap}}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})=\left( {{\bigwedge}}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})^\ast\right)^\ast,$$ so $\kappa(c)_{\mathfrak{n}}\in {{\bigcap}}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})$ is a map $$\kappa(c)_{\mathfrak{n}}: {{\bigwedge}}_R^r H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})^\ast \to R.$$ We also note that $v_{\mathfrak{q}}(\kappa(c)_{\mathfrak{n}})$ is the map $${{\bigwedge}}_R^{r-1}H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})^\ast \to R; \ \Phi \mapsto \kappa(c)_{\mathfrak{n}}(v_{\mathfrak{q}}\wedge \Phi).$$ Since we identify ${\bigcap}_R^1 H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})=H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})$ and regard $\Phi(\kappa(c)_{\mathfrak{n}}) \in H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})$, we have $$\kappa(c)_{\mathfrak{n}}(v_{\mathfrak{q}}\wedge \Phi)=(-1)^{r-1} v_{\mathfrak{q}}(\Phi(\kappa(c)_{\mathfrak{n}})).$$ Similarly, $\varphi_{\mathfrak{q}}^{\rm fs}(\kappa(c)_{{\mathfrak{n}}/{\mathfrak{q}}}) \in {{\bigcap}}_R^{r-1}H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})$ is the map $${{\bigwedge}}_R^{r-1}H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})^\ast \to R; \ \Phi \mapsto \kappa(c)_{{\mathfrak{n}}/{\mathfrak{q}}}(\varphi_{\mathfrak{q}}^{\rm fs} \wedge \Phi)=(-1)^{r-1}\varphi_{\mathfrak{q}}^{\rm fs}(\Phi(\kappa(c)_{{\mathfrak{n}}/{\mathfrak{q}}})).$$ So, to prove the equality of maps $v_{\mathfrak{q}}(\kappa(c)_{\mathfrak{n}})=\varphi_{\mathfrak{q}}^{\rm fs}(\kappa(c)_{{\mathfrak{n}}/{\mathfrak{q}}})$, it is sufficient to prove that they send each $\Phi \in {{\bigwedge}}_R^{r-1}H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})^\ast$ to the same element, namely, $$\begin{aligned}
\label{fs}
v_{\mathfrak{q}}(\Phi(\kappa(c)_{\mathfrak{n}}))=\varphi_{\mathfrak{q}}^{\rm fs}(\Phi(\kappa(c)_{{\mathfrak{n}}/{\mathfrak{q}}}))\end{aligned}$$ for every $\Phi \in {{\bigwedge}}_R^{r-1}H^1_{{\mathcal{F}}({\mathfrak{n}})}(K,{\mathcal{A}})^\ast$.
By Lemma \[key lemma\], there exists $\Psi \in \varprojlim_F {\bigwedge}_{{\mathcal{R}}[{\mathcal{G}}_F]}^{r-1}H^1({\mathcal{O}}_{F,S(F)},T)^\ast$ such that $$\Phi(\kappa(c)_{\mathfrak{n}})=\kappa(\Psi(c))_{\mathfrak{n}}\text{ and }\Phi(\kappa(c)_{{\mathfrak{n}}/{\mathfrak{q}}})=\kappa(\Psi(c))_{{\mathfrak{n}}/{\mathfrak{q}}},$$ So (\[fs\]) follows again from the fact that Theorem \[derivable1\] holds for the rank one Euler system $\Psi(c)$.
In the rest of this section we prove Lemma \[key lemma\].
At the outset we fix $\Phi$ in ${\bigwedge}_{R}^{r-1}H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}},{\mathcal{A}})^\ast$. Then, since the restriction map $${{\bigwedge}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1} H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A)^\ast \to {{\bigwedge}}_R^{r-1}H^1({\mathcal{O}}_{E,S_{\mathfrak{n}}},A)^\ast= {{\bigwedge}}_{R}^{r-1}H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}},{\mathcal{A}})^\ast$$ is surjective (since rings we consider here are self-injective), we can choose a lift $\widetilde \Phi \in{{\bigwedge}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1} H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A)^\ast $ of $\Phi \in {\bigwedge}_{R}^{r-1}H^1({\mathcal{O}}_{K,S_{\mathfrak{n}}},{\mathcal{A}})^\ast$.
We regard $\widetilde \Phi$ as an element of ${{\bigwedge}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1}H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)^\ast/M$ via the map $${{\bigwedge}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1} H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A)^\ast \to{{\bigwedge}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1}H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)^\ast/M$$ induced by $T \to T/M=A$ (see (\[wedge induce\])). We also have a surjective homomorphism $${{\bigwedge}}_{{\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1} H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)^\ast \to {{\bigwedge}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1}H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)^\ast/M,$$ and we fix a lift $\Psi_{\mathfrak{n}}\in {{\bigwedge}}_{{\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1} H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)^\ast$ of $\widetilde \Phi \in {{\bigwedge}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1}H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},T)^\ast/M$. Then, since the transition maps of the inverse limit $$\varprojlim_{F\in \Omega({\mathcal{K}}/K)} {{\bigwedge}}_{{\mathcal{R}}[{\mathcal{G}}_F]}^{r-1} H^1({\mathcal{O}}_{F,S(F)},T)^\ast$$ are surjective (by Hypothesis \[hyp free\] and [@sano Lem. 2.10]), one can take $$\Psi=(\Psi_F)_F \in \varprojlim_{F\in \Omega({\mathcal{K}}/K)} {{\bigwedge}}_{{\mathcal{R}}[{\mathcal{G}}_F]}^{r-1} H^1({\mathcal{O}}_{F,S(F)},T)^\ast$$ such that $\Psi_{E({\mathfrak{n}})}=\Psi_{\mathfrak{n}}$.
We shall show that this $\Psi$ satisfies the condition in Lemma \[key lemma\], namely that $$\Phi(\kappa'(c_{\mathfrak{d}}))=\kappa'(\Psi(c)_{\mathfrak{d}})$$ for all ${\mathfrak{d}}\mid {\mathfrak{n}}$.
We first note that $$\kappa'(\Psi(c)_{\mathfrak{d}})=D_{\mathfrak{d}}\cdot \widetilde \Phi_{\mathfrak{d}}(\bar c_{\mathfrak{d}}) \text{ in }H^1({\mathcal{O}}_{E({\mathfrak{d}}),S_{\mathfrak{d}}},A),$$ where $\widetilde \Phi_{\mathfrak{d}}\in {\bigwedge}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{d}}]}^{r-1}H^1({\mathcal{O}}_{E({\mathfrak{d}}),S_{\mathfrak{d}}},A)^\ast$ is the restriction of $\widetilde \Phi \in{{\bigwedge}}_{\overline {\mathcal{R}}[{\mathcal{G}}_{\mathfrak{n}}]}^{r-1} H^1({\mathcal{O}}_{E({\mathfrak{n}}),S_{\mathfrak{n}}},A)^\ast$. Hence one has $$\kappa'(\Psi(c)_{\mathfrak{d}})=D_{\mathfrak{d}}\widetilde \Phi_{\mathfrak{d}}(\bar c_{\mathfrak{d}})=\widetilde \Phi_{\mathfrak{d}}(D_{\mathfrak{d}}\bar c_{\mathfrak{d}})=\Phi(\kappa'(c_{\mathfrak{d}})),$$ where the last equality follows from the fact that $\widetilde \Phi_{\mathfrak{d}}$ is a lift of $\Phi$ by construction. This completes the proof of Lemma \[key lemma\].
Rubin-Stark elements and ideal class groups {#app sec}
===========================================
In this final section, we give a straightforward application of our theory in the original setting considered by Rubin in [@rubinstark].
In this setting we shall (unconditionally) prove a strong refinement of previous results of Rubin and of Büyükboduk that were obtained under the assumed validity of Leopoldt’s Conjecture. In addition, by a slightly more careful application of the same methods one can also prove much stronger result in this direction (see Remark \[promise\] below).
At the outset we fix an odd prime number $p$ (see Remark \[remark p=3\]). We also fix a number field $K$ and a homomorphism $$\chi: G_K \to \overline {\mathbb{Q}}^\times,$$ that has finite prime-to-$p$ order. We fix an embedding $\overline {\mathbb{Q}}\hookrightarrow \overline {\mathbb{Q}}_p$ and set ${\mathcal{O}}:={\mathbb{Z}}_p[\operatorname{im}(\chi)]$. We write $L$ for the field extension of $K$ that corresponds to $\ker (\chi)$ and set $\Delta:=\operatorname{Gal}(L/K)$. [*We suppose that all archimedean places of $K$ split completely in $L$*]{}. (In particular, if $K$ is totally real, then we assume that $\chi$ is totally even.)
For a ${\mathbb{Z}}_p[\Delta]$-module $X$, we define its ‘$\chi$-part’ by $$X^\chi:=\{a \in {\mathcal{O}}\otimes_{{\mathbb{Z}}_p} X \mid \sigma (a) = \chi(\sigma) a \text{ for every $\sigma \in \Delta$}\}.$$ We note that, since $|\Delta|$ is prime to $p$, this module is naturally isomorphic to ${\mathcal{O}}\otimes_{{\mathbb{Z}}_p[\Delta]} X$, where ${\mathcal{O}}$ is regarded as a ${\mathbb{Z}}_p[\Delta]$-algebra via $\chi$.
We write $T_\chi$ for a free ${\mathcal{O}}$-module of rank one upon which $G_K$ acts by the rule $$\sigma \cdot a:=\chi_{\rm cyc}(\sigma)\chi^{-1}(\sigma) a\quad (\sigma \in G_K, \ a \in T),$$ where $\chi_{\rm cyc}:G_K \to {\mathbb{Z}}_p^\times$ denotes the cyclotomic character. This $T_\chi$ is usually denoted by ${\mathcal{O}}(1)\otimes \chi^{-1}$.
Let ${\mathcal{F}}_{\rm can}$ be the canonical Selmer structure on $T_\chi$ and recall that there is a natural identification $$\label{selmer ident} H^1_{({\mathcal{F}}_{\rm can})^{*}}(K,T_\chi^\vee(1))^\vee \simeq ({\mathbb{Z}}_p \otimes_{\mathbb{Z}}{\rm Cl}({\mathcal{O}}_L[1/p]))^\chi.$$ Here ${\rm Cl}({\mathcal{O}}_{L}[1/p])$ denotes the quotient of the ideal class group ${\rm Cl}(\mathcal{O}_L)$ of $L$ by the subgroup generated by the classes of all prime ideals that divide $p$. We set $r:=|S_\infty(K)|$ and fix a finite set $S$ of places of $K$ such that $$S_\infty(K)\cup S_{\rm ram}(L/K) \subseteq S.$$ We always assume that $|S| > r$. (In particular, if $L/K$ is ramified then we can take $S = S_\infty(K)\cup S_{\rm ram}(L/K)$.) We quickly recall some notations from §\[euler sys sec 1\]. We fix a pro-$p$ abelian extension ${\mathcal{K}}/K$ that is sufficiently large to ensure Hypothesis \[hyp K\] is satisfied (for $S\cup S_p(K)$), and we write $\Omega({\mathcal{K}}/K)$ for the set of subfields of ${\mathcal{K}}/K$ that are finite over $K$. For a finite abelian extension $F/K$ we set ${\mathcal{G}}_F:=\operatorname{Gal}(F/K)$ and $S(F):=S \cup S_{\rm ram}(F/K)$. For a set $\Sigma$ of places of $K$ with $S(F) \subseteq \Sigma$, we denote by $\Sigma_F$ the set of places of $F$ which lie above a place in $\Sigma$. The ring of $\Sigma_F$-integers of $F$ is denoted by ${\mathcal{O}}_{F,\Sigma}$ and we write $\theta_{F/K,\Sigma}(s)$ for the $\Sigma$-truncated equivariant $L$-function for $F/K$ (as defined, for example, in [@bks1 §3.1]).
Since $p$ is odd and ${\mathcal{K}}/K$ is a pro-$p$ extension, all places in $S_\infty(K)$ split completely in ${\mathcal{K}}$. We label, and hence order, the places in $S_\infty(K)$ as $\{v_1,\ldots,v_r\}$ and for each $v_i$ we fix a place $w_i$ of $\overline {\mathbb{Q}}$ that lies above $v_i$. Since $|S| > r$ we can also fix a non-archimedean place $v_0$ in $S$ and a place $w_0$ of $\overline {\mathbb{Q}}$ lying above $v_0$.
Then for each $F \in \Omega({\mathcal{K}}/K)$ and each finite set $\Sigma$ of places of $K$ with $S(F) \subseteq \Sigma$, the Dirichlet regulator map induces an isomorphism of ${\mathbb{R}}[{\mathcal{G}}_{LF}]$-modules $${\mathbb{R}}\otimes_{\mathbb{Z}}{\bigwedge}_{{\mathbb{Z}}[{\mathcal{G}}_{LF}]}^r {\mathcal{O}}_{LF,\Sigma}^\times \stackrel{\sim}{\to} {\mathbb{R}}\otimes_{\mathbb{Z}}{\bigwedge}_{{\mathbb{Z}}[{\mathcal{G}}_{LF}]}^r X_{LF,\Sigma}$$ where $X_{LF,\Sigma}$ denotes the kernel of the map $\bigoplus_{w \in \Sigma_{LF}}{\mathbb{Z}}\cdot w \to {\mathbb{Z}}$ that sends $\sum_w a_w w$ to $\sum_w a_w$.
Under the stated assumptions on $S$, the functions $s^{-r}\theta_{LF/K,\Sigma}(s)$ are holomorphic at $s=0$ (see [@tatebook Chap. I, Prop. 3.4]) and, following Rubin [@rubinstark], one defines the ‘Rubin-Stark element’ $\eta_{LF/K,\Sigma}$ to be the unique element of ${\mathbb{R}}\otimes_{\mathbb{Z}}{\bigwedge}_{{\mathbb{Z}}[{\mathcal{G}}_{LF}]}^r {\mathcal{O}}_{LF,\Sigma}^\times$ that the above isomorphism sends to $$\underset{s\to 0}{\lim} s^{-r}\theta_{LF/K,\Sigma}(s) \cdot (w_1-w_0)\wedge \cdots \wedge (w_r-w_0).$$ If $\Sigma$ contains $S_p(K)$, then Kummer theory induces canonical isomorphisms $$({\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\mathcal{O}}_{LF,\Sigma}^\times)^\chi\simeq H^1({\mathcal{O}}_{LF,\Sigma},{\mathbb{Z}}_p(1))^\chi \simeq H^1({\mathcal{O}}_{F,\Sigma}, T_\chi).$$ (Note that, since $[L:K]$ is prime to $p$, $L$ is disjoint from ${\mathcal{K}}$ and so one can define the $\chi$-component of a ${\mathbb{Z}}_p[{\mathcal{G}}_{LF}]$-module.)
In particular, after fixing an embedding ${\mathbb{R}}\hookrightarrow {\mathbb{C}}_p$ we define $\eta_{LF/K,\Sigma}^\chi$ to be the image of $\eta_{LF/K,\Sigma}$ under the composite $$\begin{aligned}
{\mathbb{R}}\otimes_{\mathbb{Z}}{\bigwedge}_{{\mathbb{Z}}[{\mathcal{G}}_{LF}]}^r {\mathcal{O}}_{LF,\Sigma}^\times &\subseteq& {\mathbb{C}}_p \otimes_{{\mathbb{Z}}_p} {\bigwedge}_{{\mathbb{Z}}_p[{\mathcal{G}}_{LF}]}^r H^1({\mathcal{O}}_{LF,\Sigma\cup S_p(K)},{\mathbb{Z}}_p(1)) \\
& \to& {\mathbb{C}}_p \otimes_{{\mathbb{Z}}_p} {\bigwedge}_{{\mathcal{O}}[{\mathcal{G}}_F]}^r H^1({\mathcal{O}}_{F,\Sigma\cup S_p(K)},T_\chi)\end{aligned}$$ with the second map induced by the projection $$H^1({\mathcal{O}}_{LF,\Sigma\cup S_p(K)},{\mathbb{Z}}_p(1))\to H^1({\mathcal{O}}_{LF,\Sigma\cup S_p(K)},{\mathbb{Z}}_p(1))^\chi=H^1({\mathcal{O}}_{F,\Sigma\cup S_p(K)},T_\chi).$$
For each $F$ in $\Omega({\mathcal{K}}/K)$ we set $S(F)_p:=S\cup S_{\rm ram}(F/K) \cup S_p(K)(=S(F)\cup S_p(K))$ and [*we assume that the group $({\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\mathcal{O}}_{LF,S(F)_p}^\times)^\chi$ is a free ${\mathcal{O}}$-module.*]{}
Under this hypothesis the Rubin-Stark conjecture [@rubinstark Conj. B$'$] predicts that the element $c_{F,\chi}^{\rm RS}:=\eta_{LF/K, S(F)}^\chi$ belongs to ${\bigcap}_{{\mathcal{O}}[{\mathcal{G}}_{F}]}^r H^1({\mathcal{O}}_{F,S(F)_p},T_\chi)$ (regarded as a submodule of ${\mathbb{C}}_p \otimes_{{\mathbb{Z}}_p} {\bigwedge}_{{\mathcal{O}}[{\mathcal{G}}_F]}^r H^1({\mathcal{O}}_{F,S(F)_p},T_\chi)$ via [@sbA Prop. A.7]).
In addition, if this conjecture is valid for every $F$ in $\Omega({\mathcal{K}}/K)$, then the collection $$c^{\rm RS}_\chi := (c_{F,\chi}^{\rm RS})_F$$ forms an Euler system of rank $r$ for the pair $(T_\chi,{\mathcal{K}})$ (for a proof of this see [@rubinstark Prop. 6.1] or [@sano Prop. 3.5]).
By applying Corollary \[main cor\] in this setting, we can now prove the following result (the context of which is explained in Remark \[promise\] below).
\[RS theorem\] Assume that
- $\chi$ is neither trivial nor equal to the Teichmüller character, that
- no place in $S_p(K)$ splits completely in $L/K$, and that
- either $p>3$ or $\chi^2$ is not equal to the Teichmüller character.
For each non-negative integer $i$ define an ideal of $\mathcal{O}$ by setting $$I_i(T_\chi) := \langle I_i(\kappa(c)) \mid c \in {\rm ES}_r(T_\chi,{\mathcal{K}})\rangle_\mathcal{O},$$ where $\kappa(c) \in {\rm KS}_r(T_\chi,{\mathcal{F}})$ is the Kolyvagin system constructed from $c$. Then the following claims are valid.
- For each $i$ one has $I_i(T_\chi)\subseteq {\rm Fitt}_{\mathcal{O}}^i(({\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\rm Cl}(\mathcal{O}_L))^\chi)$.
- If no place in $S\setminus S_\infty(K)$ splits completely in $L/K$, then the inclusions in claim (i) are equalities and there is an isomorphism of $\mathcal{O}$-modules $$({\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\rm Cl}(\mathcal{O}_L))^\chi \simeq \bigoplus_{i \ge 0} I_{i+1}(T_\chi)/I_i(T_\chi).$$
- If the Rubin-Stark Conjecture is valid for $LF/K$ for each $F$ in $\Omega({\mathcal{K}}/K)$, then one has $\operatorname{im}(\eta_{L/K,S}^\chi) \subseteq {\rm Fitt}_{{\mathcal{O}}}^0(({\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\rm Cl}(\mathcal{O}_L))^\chi).$
At the outset, note that assumption (c) is used simply so that we can include the case $p=3$ (see Remark \[remark p=3\]).
We next show that the conditions of Corollary \[main cor\] are satisfied under the hypotheses given above.
Firstly, under hypothesis (a) above, the $\mathcal{O}$-module $H^1({\mathcal{O}}_{F,S(F)_p},T_\chi) = ({\mathbb{Z}}_p \otimes_{\mathbb{Z}}{\mathcal{O}}_{LF,S(F)_p}^\times)^\chi$ is easily seen to be free for every $F$ in $\Omega({\mathcal{K}}/K)$ and so Hypotheses \[hyp free\](i) is satisfied. We also obviously have $H^0(F,T_\chi)=0$ and so Hypothesis \[hyp free\](ii) is satisfied.
The field ${\mathcal{K}}$ is chosen so that Hypothesis \[hyp K\] is satisfied and Hypothesis \[hyp local\] is also trivial in this case.
Hypotheses \[hyp1’\](i) and (ii) are trivial since $T_\chi$ is of rank one over ${\mathcal{O}}$ (and so one can take $\tau=1$). Hypothesis \[hyp1’\](iii) also follows from the stated assumption (a): in fact this is checked in [@R Lem. 3.1.1] (see also [@MRkoly Lem. 6.1.5]).
By [@MRkoly Cor. 4.1.9(iii)] and [@MRselmer Cor. 3.5(ii)], for all positive integers $m$, Hypothesis \[hyp large\] is satisfied for $(T_\chi/p^{m}T_\chi, {\mathcal{F}}_{\rm can}, {\mathcal{P}}_{m})$ since in this case the coefficient ring is ${\mathcal{O}}$. (Note that in this case, the hypothesis (b) given above implies that the ‘core rank’ is equal to $r=|S_\infty(K)|$.)
Finally we consider the conditions (a) and (b) in Corollary \[remark surjective\]. In this setting the validity of condition (a) follows directly from the assumption that all places in $S_\infty(K)$ split completely in $L/K$.
In addition, if no place in $S \setminus S_\infty(K)$ splits completely in $L/K$, then the ${\mathcal{O}}$-module $$H^{0}(E_{w}, (T_\chi/pT_\chi)^\vee(1)) = H^0(E_w, {\mathcal{O}}/(p) \otimes \chi)$$ is easily seen to vanish for each non-archimedean prime $w$ of $E$ above $S \cup S_{p}(K)$. (Recall, from §\[koly sect\], that $E$ is a fixed auxiliary field in $\Omega({\mathcal{K}}/K)$ that contains $K(1)$ and is such that $E/K$ is unramified outside $S\cup S_p(K)$. For example, one could take $E:=K(1)$.)
We have now verified that the hypotheses of Corollary \[main cor\](i) and (ii) are satisfied under the given conditions (a), (b) and (c), and that the larger set of hypotheses of Corollary \[main cor\](iii) is satisfied if, in addition, no place in $S\setminus S_\infty(K)$ splits completely in $L/K$.
Next we note that, since $\chi$ is both non-trivial and primitive, the natural projection map $({\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\rm Cl}(\mathcal{O}_L))^\chi \to ({\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\rm Cl}(\mathcal{O}_L[1/p]))^\chi$ is bijective under the given condition (b). Therefore the identification (\[selmer ident\]) implies that $
H^1_{({\mathcal{F}}_{\rm can})^\ast}(K,T_\chi^\vee(1))^\vee = \left( {\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\rm Cl}(\mathcal{O}_L) \right)^\chi.
$
Given these observations, all of the stated claims follow directly from the result of Corollary \[main cor\] and the fact that $c_{K,\chi}^{\rm RS} =\eta_{L/K, S}^\chi$.
If $K$ is totally real, then the assumption that places in $S_\infty(K)$ split completely in $L$ implies $\chi$ is not the Teichmüller character and so the condition in Theorem \[RS theorem\](a) reduces to requiring $\chi$ is not trivial. In regard to Theorem \[RS theorem\](ii), note that if $S = S_\infty(K)\cup S_{\rm ram}(L/K)$ (which is permissable if $L/K$ is ramified), then no place in $S\setminus S_\infty(K)$ splits completely in $L/K$. In all cases, it is straightforward to choose a set $S$ that contains $S_\infty(K)\cup S_{\rm ram}(L/K)$ and is such that no place in $S\setminus S_\infty(K)$ splits completely in $L/K$.
\[promise\] Theorem \[RS theorem\] both refines and extends the results of Rubin in [@rubincrelle] and [@rubinstark] and, more recently, of Büyükboduk in [@Buyuk]. (For example, the main result of the latter article deals only with the case $i=0$ and assumes, amongst other things, that $K$ is totally real, $L/K$ is unramified at $p$ and, crucially, that Leopoldt’s conjecture is valid.) In addition, under certain mild additional hypotheses, a more careful application of the methods used to prove Theorem \[RS theorem\] allows one to prove that for a wide range of abelian extensions $L$ of $K$ the higher Fitting ideals of ${\mathbb{Z}}_p\otimes_{\mathbb{Z}}{\rm Cl}(\mathcal{O}_L)$ as a ${\mathbb{Z}}_p[\operatorname{Gal}(L/K)]$-module are determined by Euler systems of rank $r$ for induced forms of the representation ${\mathbb{Z}}_p(1)$. For brevity, however, we defer further discussion of this result to a subsequent article.
The approach used to prove Theorem \[RS theorem\] also leads to analogous results for the twisted representations $T_\chi(a) := T_\chi\otimes_{{\mathbb{Z}}_p}{\mathbb{Z}}_p(a)$ for arbitrary integers $a$. Taken in conjunction with the known validity of the Quillen-Lichtenbaum conjecture, this in turn leads to concrete new information about the Galois structure of even dimensional higher algebraic $K$-groups.
To be a little more precise, in this setting the Rubin-Stark Euler system $c^{\rm RS}_\chi$ defined above can be replaced, modulo the generalized Rubin-Stark conjecture formulated by Kurihara and the first and third authors in [@bks2-2 Conj. 3.5(i)], by an Euler system that is constructed in just the same way after replacing Rubin-Stark elements by the ‘generalized Stark elements’ $\eta_{L/K,S}(-a)$ that are introduced in loc. cit.
The same argument as in the proof of Theorem \[RS theorem\] then shows that for any integer $a$, and all suitable characters $\chi$ of $\operatorname{Gal}(L/K)$, the validity of [@bks2-2 Conj. 3.5(i)] implies that ideals of the form $\operatorname{im}(\eta_{L/K,S}(-a)^\chi)$ are contained in ${\rm Fitt}^0_\mathcal{O}(H^2(\mathcal{O}_{L,S},{\mathbb{Z}}_p(a+1))^\chi)$, respectively in ${\rm Fitt}_{{\mathcal{O}}}^0(({\mathbb{Z}}_p\otimes_{\mathbb{Z}}K_{2a}(\mathcal{O}_{L}))^\chi)$ if $a > 0$.
However, since no essentially new ideas are involved in this argument, we prefer not to give any more details here.
[99999999]{}
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|
---
abstract: |
In this paper two new classes of stationary random simplicial tessellations, the so-called $\beta$- and $\beta'$-Delaunay tessellations, are introduced. Their construction is based on a space-time paraboloid hull process and generalizes that of the classical Poisson-Delaunay tessellation. The distribution of volume-power weighted typical cells is explicitly identified, establishing thereby a remarkable connection to the classes of $\beta$- and $\beta'$-polytopes. These representations are used to determine principal characteristics of such cells, including volume moments, expected angle sums and cell intensities.\
[**Keywords**]{}. [Angle sums, beta-Delaunay tessellation, beta’-Delaunay tessellation, beta-polytope, beta’-polytope, Laguerre tessellation, paraboloid convexity, paraboloid hull process, Poisson point process, Poisson-Delaunay tessellation, Poisson-Voronoi tessellation, random polytope, stochastic geometry, typical cell, weighted typical cell, zero cell]{}\
[**MSC**]{} 52A22, 52B11, 53C65, 60D05, 60G55.
author:
- 'Anna Gusakova, Zakhar Kabluchko, and Christoph Thäle'
title: |
**The $\beta$-Delaunay tessellation I:\
Description of the model and geometry of typical cells**
---
Introduction
============
Random tessellations in a Euclidean space are among the most central objects studied in stochastic geometry. Their analysis is motivated by their rich inner-mathematical structures, but in equal measure also by the wide range of applications in which they arise. For example, tessellations, and especially triangulations of a space, play a prominent role for finite element methods in numerical analysis, in computer vision, material science, ecology, chemistry, astrophysics, machine learning, network modelling or computational geometry; we refer to the monographs [@AurenhammerKlein; @BlaszEtAl; @ChegDeyEtAl; @Edelsbrunner; @Haenggi; @LoBook; @Mo94; @OkabeEtAl; @PreparataShamos; @SW; @SKM] as well as the references cited therein for an extensive overview. However, there are only very few mathematically tractable models for which rigorous results are available and which do not require an analysis purely by computer simulations. Among these models are the Poisson-Voronoi tessellations and their duals, the Poisson-Delaunay tessellations. Their construction can be described as follows. Given a stationary Poisson point process $X$ in $\RR^{d-1}$ we define for any point $v\in X$ the Voronoi cell $C(v,X)$ of $v$ as $$C(v,X) := \{w\in\RR^{d-1}:\|w-v\|\leq\|w-v'\|\text{ for all }v'\in X\},$$ that is, $C(v,X)$ contains all points that are closer to $v$ than to any other point from $X$. In applications, $C(v,X)$ might represent the domain of influence of a point $v$, for example the area in a communication network that a base station placed at $v$ may cover. All such Voronoi cells are random convex polytopes and the collection of all Voronoi cells is the Poisson-Voronoi tessellation of $\RR^{d-1}$. To define the dual tessellation we say that the convex hull ${\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)$ of $d$ distinct points from $X$ is a Delaunay simplex provided that $X$ has no points in the interior of the ball containing $v_1,\ldots,v_d$ on its boundary. The collection of all Delaunay simplices is what is known as the Poisson-Delaunay tessellation. It is possible to show that ${\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)$ is a Delaunay simplex if and only if the Voronoi cells of the points $v_1,\ldots,v_d$ meet at a common point which is then the center of the circumscribed sphere of the simplex.
It is the purpose of this series of papers to introduce and to initiate a systematic study of a generalization of Poisson-Delaunay tessellations, the so-called $\beta$-Delaunay tessellations denoted by $\cD_\beta$. This is a one-parametric family of tessellations of $\RR^{d-1}$, where the parameter $\beta$ satisfies $-1<\beta<\infty$. We will also introduce the dual tessellations of $\cD_\beta$, which are denoted by $\cV_\beta$ and called the $\beta$-Voronoi tessellations. The classical Poisson-Delaunay and Poisson-Voronoi tessellations arise as the limiting cases of these tessellations when $\beta\to -1$.
![Poisson point processes in $\RR^d$ with $d=3$ used to construct the tessellations on the plane. Left: $\eta_\beta$ with $\beta=2$. Right: $\eta_\beta'$ with $\beta=3$. The plane $h=0$ in which the tessellation is constructed is shown in yellow.[]{data-label="fig:beta_d=3"}](pic3dbeta.pdf "fig:"){width="48.00000%"} ![Poisson point processes in $\RR^d$ with $d=3$ used to construct the tessellations on the plane. Left: $\eta_\beta$ with $\beta=2$. Right: $\eta_\beta'$ with $\beta=3$. The plane $h=0$ in which the tessellation is constructed is shown in yellow.[]{data-label="fig:beta_d=3"}](pic3dbeta_prime.pdf "fig:"){width="48.00000%"}
As for the classical Poisson-Delaunay tessellation, the construction of the $\beta$-Delaunay tessellation is based on a Poisson point process as well. However, while the Poisson point process for the Poisson-Delaunay tessellation is located in $\RR^{d-1}$, for the $\beta$-Delaunay tessellation we start with a Poisson point process $\eta_{\beta}$ in the product space $\RR^{d-1}\times [0,\infty)$ whose intensity measure has the form $\text{const}\cdot h^\beta\,{\textup{d}}v{\textup{d}}h$, where $v\in\RR^{d-1}$ stands for the spatial coordinate and $h>0$ for the height coordinate of a point $x=(v,h)\in\RR^{d-1}\times [0,\infty)$. A realization of this Poisson point process for $d=3$ is shown in the left panel of Figure \[fig:beta\_d=3\]. In a next step, we construct the paraboloid hull process associated with $\eta_{\beta}$. This is a particular germ-grain process with paraboloid grains which in stochastic geometry was introduced by Schreiber and Yukich [@SY08], and further developed in Calka, Schreiber and Yukich [@CSY13] and Calka and Yukich [@CYGaussian] in order to study the asymptotic geometry of random convex hulls near their boundary. We shall use the same paraboloid hull process to construct a random tessellation $\cD_\beta$ of $\RR^{d-1}$ with only simplicial cells as follows. Given $d$ points $x_1=(v_1,h_1),\ldots,x_d=(v_d,h_d)$ of $\eta_\beta$ with affinely independent spatial coordinates $v_1,\ldots,v_d$, there is a unique shift of the standard downward paraboloid $$\Pi^\downarrow := \left\{(v,h)\in\RR^{d-1}\times\RR\colon h\leq -\|v\|^2\right\}$$ containing $x_1,\ldots,x_d$ on its boundary. We declare ${\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)$ to be a $\beta$-Delaunay simplex in $\RR^{d-1}$ if and only if the interior of the downward paraboloid determined by $x_1,\ldots,x_d$ is void of points of $\eta_{\beta}$. The collection of all $\beta$-Delaunay simplices is called the $\beta$-Delaunay tessellation of $\RR^{d-1}$. Two realizations of the paraboloid hull process in the case $d=2$ are shown in Figure \[fig:beta\_d=2\].
The $\beta$-Voronoi tessellation $\cV_\beta$ can be constructed as follows. Imagine that each atom $x=(v,h)\in \RR^{d-1}\times [0,\infty)$ of the Poisson point process $\eta_\beta$ gives rise to a crystallization process in $\RR^{d-1}$ which starts at spatial position $v\in \RR^{d-1}$ at time $h>0$. The speed of the crystallization process is not constant and assumed to be such that the process reaches a point $w\in \RR^d$ at time $h + \|w-v\|^2$. Then, the cell generated by $(v,h)$ is just the set of all points $w\in \RR^{d-1}$ that are reached by the crystallization process started at $(v,h)$ not later than by a crystallization process started at any other point $(v',h')$. It should be emphasized that the cell may be empty and that in the case when it is non-epmty, it need not contain the point $v$ (which is different from the case of the classical Poisson-Voronoi tessellation). The set of non-empty cells forms the $\beta$-Voronoi tessellation $\cV_\beta$. The tessellations $\cD_\beta$ and $\cV_\beta$ are dual to each other in the following sense. A simplex ${\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)$ is a cell of the $\beta$-Delaunay tessellation if and only if the cells generated by $x_1,\ldots,x_d$ in the $\beta$-Voronoi tessellation are non-empty and meet at a common point.
![Construction of the $\beta$-Delaunay tessellation $\cD_\beta$ for $d=2$. The figure shows the Poisson point process $\eta_{\beta}$ and the corresponding paraboloid hull. Top: $\beta=0$, bottom: $\beta=2$. Points on the horizontal axis are the vertices of the $\beta$-Delaunay tessellation of $\RR$.[]{data-label="fig:beta_d=2"}](pic_beta0.pdf "fig:"){width="98.00000%"}\
![Construction of the $\beta$-Delaunay tessellation $\cD_\beta$ for $d=2$. The figure shows the Poisson point process $\eta_{\beta}$ and the corresponding paraboloid hull. Top: $\beta=0$, bottom: $\beta=2$. Points on the horizontal axis are the vertices of the $\beta$-Delaunay tessellation of $\RR$.[]{data-label="fig:beta_d=2"}](pic_beta2.pdf "fig:"){width="98.00000%"}
The goal of the present paper (which is the first in a series of papers) is to show that these constructions do indeed lead to well-defined stationary random tessellation of $\RR^{d-1}$ and to study their geometric properties. Moreover and in parallel to the construction of the $\beta$-Delaunay tessellation we introduce the concept of a $\beta'$-Delaunay tessellation $\cD_{\beta}'$ (together with its dual $\beta'$-Voronoi tessellation $\cV_\beta'$), whose construction is based on a Poisson point process $\eta_\beta'$ on the product space $\RR^{d-1}\times (-\infty,0)$ with intensity measure $\text{const}\cdot (-h)^{-\beta}\,{\textup{d}}v{\textup{d}}h$ for $\beta>(d+1)/2$. A realization of the Poisson point process $\eta_\beta'$ for $d=3$ is shown on the right panel of Figure \[fig:beta\_d=3\], while the paraboloid hull process in the case $d=2$ is shown in Figure \[fig:betaprime\_d=2\].
Whenever possible, we will develop our results for both, $\beta$ and $\beta'$, random tessellation models in parallel. In particular, we are interested in what is known as the typical cell of $\cD_\beta$ and $\cD_\beta'$. Intuitively, one can think of such a cell as a randomly chosen cell, which is selected independently of its size and shape. More generally, we shall study volume-power weighted typical cells, where the weight is certain power $\nu$ of the volume. One of our main contributions is a precise distributional characterization of the weighted typical cell of $\cD_\beta$ and $\cD_\beta'$. Very remarkably, this provides a link between the $\beta$-and $\beta'$-Delaunay tessellation and the class of $\beta$-polytopes and $\beta'$-polytopes, which was under intensive investigation in recent times, see, for example, [@GKT17; @kabluchko_formula; @KTT; @beta_polytopes]. More precisely, we will prove that the weighted typical cell of $\cD_\beta$ and $\cD_\beta'$ is a randomly rescaled volume-power weighted $\beta$- or $\beta'$-simplex, respectively. This opens a way to study the geometry and the combinatorial properties of the typical cells of $\cD_\beta$ and $\cD_\beta'$. Among our results are explicit formulas for the moments of the volume as well as probabilistic representations in terms of independent gamma- and beta-distributed random variables. We also compute explicitly the $j$-face intensities and determine the expected angle sums of weighted typical cells.
Finally, we prove that, as $\beta\to\infty$, the expected angle sums of the volume-power weighted typical cells tend to those of a regular simplex in $\RR^{d-1}$. We will pick up this topic in detail in part II of this series of papers, where we describe the common limiting tessellation $\cD_{\infty}$ of $\cD_\beta$ and $\cD_\beta'$, as $\beta\to\infty$, after suitable rescaling. This will provide an explanation of the limit behaviour of the expected angle sums just described. In part III we will prove various high-dimensional limit theorems for the volume of weighted typical cells in $\cD_\beta$, $\cD_\beta'$ and $\cD_{\infty}$, that is, limit theorems where $d\to\infty$ (potentially in a coupled way with other parameters). We also describe there the shape of large weighted typical cells in the spirit of Kendall’s problem, generalizing thereby results of Hug and Schneider [@HugSchneiderDelaunay] on the classical Poisson-Delaunay tessellation.
![Construction of the $\beta'$-Delaunay tessellation $\cD_\beta'$ for $d=2$. The figure shows the Poisson point process $\eta_{\beta}$ and the corresponding paraboloid hull. Top: $\beta=2$, bottom: $\beta=3$.[]{data-label="fig:betaprime_d=2"}](pic_betaprime2.pdf "fig:"){width="98.00000%"}\
![Construction of the $\beta'$-Delaunay tessellation $\cD_\beta'$ for $d=2$. The figure shows the Poisson point process $\eta_{\beta}$ and the corresponding paraboloid hull. Top: $\beta=2$, bottom: $\beta=3$.[]{data-label="fig:betaprime_d=2"}](pic_betaprime3.pdf "fig:"){width="98.00000%"}
The remaining parts of this text are structured as follows. In Section \[sec:Preliminaries\] we recall the necessary notions and notation from random tessellation and point process theory, which are used throughout the paper. The detailed construction of $\beta$-Delaunay tessellations is presented in Section \[sec:Construction\]. The explicit distributions of their volume-power weighted typical cells is the content of Section \[sec:TypicalCells\]. These results are used in Section \[sec:Volume\] to derive explicit formulas and probabilistic representations for the moments of the volume of such cells. The final Section \[sec:AnglesFaceIntensities\] discusses expected angle sums of weighted typical cells as well as formulas for face intensities.
Preliminaries {#sec:Preliminaries}
=============
Frequently used notation
------------------------
Let $d\geq 1$ and $A\subset\RR^d$. We denote by ${\rm int}\,A$ the interior of $A$ and by $\partial A$ its boundary. A centred closed Euclidean ball in $\RR^d$ with radius $r>0$ is denoted by $\BB_r^d$ and we put $\BB^d:=\BB_1^d$. The volume of $\BB^d$ is given by $$\kappa_d:=\frac{\pi^{d/2}}{\Gamma(1+{d\over 2})}.$$ By $\sigma_{d-1}$ we denote the spherical Lebesgue measure on $(d-1)$-dimensional unit sphere $\SS^{d-1}=\partial \BB^d$, normalized in such a way that $$\omega_d:=\sigma_{d-1}(\SS^{d-1})={2\pi^{d\over 2}\over \Gamma\left({d\over 2}\right)}.$$
Given a set $C\in \RR^{d-1}$ denote by ${\mathop{\mathrm{conv}}\nolimits}(C)$ its convex hull. For points $v_0,\ldots,v_k\in\RR^{d-1}$ we write ${\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_k)$ for the affine hull of $v_0,\ldots,v_k$, which is at most $k$-dimensional affine subspace of $\RR^{d-1}$, $k\in\{0,1,\ldots,d-1\}$.
In what follows we shall represent points $x\in\RR^d$ in the form $x=(v,h)$ with $v\in\RR^{d-1}$ (called *spatial coordinate*) and $h\in\RR$ (called *height*, *weight* or *time coordinate*).
(Poisson) point processes
-------------------------
Let $(\XX,\cX)$ be a measurable space supplied with a $\sigma$-finite measure $\mu$. By $\sfN(\XX)$ we denote the space of $\sigma$-finite counting measures on $\XX$. The $\sigma$-field $\cN(\XX)$ is defined as the smallest $\sigma$-field on $\sfN(\XX)$ such that the evaluation mappings $\xi\mapsto\xi(B)$, $B\in\cX$, $\xi\in\sfN(\XX)$ are measurable. A **point process** on $\XX$ is a measurable mapping with values in $\sfN(\XX)$ defined over some fixed probability space $(\Omega,\cA,\PP)$. By a **Poisson point process** $\eta$ on $\XX$ with intensity measure $\mu$ we understand a point process with the following two properties:
- for any $B\in\cX$ the random variable $\eta(B)$ is Poisson distributed with mean $\mu(B)$;
- for any $n\in\NN$ and pairwise disjoint sets $B_1,\ldots,B_n\in\cX$ the random variables $\eta(B_1),\ldots,\eta(B_n)$ are independent.
We refer to [@LP; @SW] for more the existence and construction of Poisson point processes and for further details.
Tessellations
-------------
In this subsection we recall the concept of a random tessellation and include a brief overview of basic properties. For more detailed discussion we refer reader to [@SW Chapter 10]. Roughly speaking, a tessellation (or a mosaic) is a system of polytopes that cover the whole space and have disjoint interiors. We fix a space dimension $d\geq 2$. Since the tessellations we construct in Section \[sec:Construction\] are driven by a Poisson point process in $\RR^d$ and induce a tessellation in $\RR^{d-1}$, we consider this set-up in what follows.
\[def:Tessellation\] A **tessellation** $M$ in $\RR^{d-1}$ is a countable system of subsets of $\RR^{d-1}$ satisfying the following conditions:
1. $M$ is locally finite system of non-empty closed sets, where local finiteness means that every bounded subset of $\RR^{d-1}$ has non-empty intersection of only finitely many sets from $M$;
2. the sets $m\in M$ are compact, convex and have interior points;
3. the sets of $M$ cover the space, meaning that $$\bigcup\limits_{m\in M}m=\RR^{d-1};$$
4. if $m_1,m_2\in M$ and $m_1\neq m_2$, then ${\operatorname{int}}m_1\cap {\operatorname{int}}m_2=\varnothing$.
The elements of $M$ are called [**cells**]{} of $M$ and they are convex polytopes by [@SW Lemma 10.1.1]. Given a polytope $P$ we denote for $k\in\{0,1,\ldots,d-1\}$ by $\cF_k(P)$ the set of its $k$-dimensional faces and let $\cF(P):=\bigcup_{k=0}^{d-1}\cF_{k}(P)$. A tessellation $M$ is called [**face-to-face**]{} if for all $P_1,P_2\in M$ we have $$P_1\cap P_2\in(\cF(P_1)\cap\cF(P_2))\cup\{\varnothing\}.$$ A face-to-face tessellation in $\RR^{d-1}$ is called [**normal**]{} if each $k$-dimensional face of the tessellation is contained in precisely $d-k$ cells for all $k\in\{0,1,\ldots,d-2\}$. We denote by $\MM$ the set of all face-to-face tessellations in $\RR^{d-1}$. By a [**random tessellation**]{} in $\RR^{d-1}$ we understand a particle process $X$ in $\RR^{d-1}$ (in the usual sense of stochastic geometry, see [@SW]) satisfying $X\in\MM$ almost surely. Implicitly, we assume here and for the rest of this paper that all the random objects we consider are defined on a probability space $(\Omega,\cA,\PP)$.
Construction of $\beta^{(')}$-Voronoi and $\beta^{(')}$-Delaunay tessellations {#sec:Construction}
==============================================================================
Underlying point processes
--------------------------
Let us start by defining the $\beta$- or $\beta'$-Voronoi tessellation and its dual, the $\beta$- or $\beta'$-Delaunay tessellation. We will give two alternative definitions using the concept of Laguerre tessellations and notion of paraboloid hull process introduced by Schreiber and Yukich [@SY08], Calka, Schreiber and Yukich [@CSY13] and Calka and Yukich [@CYGaussian]. As an underlying process for the $\beta$-Voronoi and the $\beta$-Delaunay tessellations we consider a space-time Poisson point process $\eta=\eta_{\beta}$ in $\RR^{d-1}\times \RR_{+}$, where $\RR_{+}:=[0,+\infty)$ denotes the set of non-negative real numbers, with intensity measure having density $$\label{eq:BetaPoissonIntensity}
(x,h)\mapsto\gamma\,c_{d,\beta} \cdot h^{\beta},
\qquad
c_{d,\beta}:={\Gamma\left({d\over 2}+\beta+1\right)\over \pi^{d\over 2}\Gamma(\beta+1)},
\qquad
\gamma > 0,
\,
\beta>-1,$$ with respect to the Lebesgue measure on $\RR^{d-1}\times \RR_{+}$. Here, $\gamma>0$ is the intensity parameter which usually will be suppressed in our notation, $\beta>-1$ is the shape parameter, and the normalizing constant $c_{d,\beta}$ has been introduced to simplify some of the computations below. See Figure \[fig:beta\_d=3\] (left panel) and Figure \[fig:beta\_d=2\] for realizations of these point processes with $d=3$ and $d=2$, respectively.
For the $\beta'$-Voronoi and the $\beta'$-Delaunay tessellations we consider a space-time Poisson point process $\eta'=\eta^{\prime}_{\beta}$ in $\RR^{d-1}\times\RR_{-}^*$, where $\RR_{-}^*:=(-\infty,0)$ denotes the set of negative real numbers, with intensity measure having density $$\label{eq:BetaPrimePoissonIntensity}
(x,h)\mapsto\gamma\,c'_{d,\beta} \cdot (-h)^{-\beta},
\qquad
c'_{d,\beta}:={\Gamma\left(\beta\right)\over \pi^{d\over 2}\Gamma(\beta-{d\over 2})},
\qquad
\gamma > 0,
\,
\beta >{d+1\over 2},$$ with respect to the Lebesgue measure on $\RR^{d-1}\times\RR_{-}^*$. Again, $\gamma>0$ and $\beta>(d+1)/2$ are the parameters of the process. See Figure \[fig:beta\_d=3\] (right panel) and Figure \[fig:betaprime\_d=2\] for realizations of $\eta^{\prime}_{\beta}$ with $d=3$ and $d=2$, respectively.
As we will see later, the $\beta$- and $\beta'$-tessellations can often be treated in a unified way. To make this explicit and in order to shorten and simplify the presentation of the paper we introduce the following variable notation. We put $\kappa:=1$ if the $\beta$-model is considered an $\kappa:=-1$ if we work with the $\beta'$-model. Moreover, through the paper we will use almost the same notation for $\beta$- and $\beta'$-tessellations which will only differ by using the $\prime$-symbol in case of $\beta'$ model. When the formulas for the $\beta$- and $\beta'$-model are close enough to join them into one expression, we will indicate this by $\!\!\!\!\phantom{x}^{(\prime)}$, meaning that this sign should be omitted as far as a $\beta$-model is considered.
Using the convention just introduced we can consistently represent the density of the Poisson point process $\eta^{(')}$ on $\RR^{d-1}\times\RR$ as $$(x,h)\mapsto\gamma\,c_{d,\beta}^{(')} \cdot (\kappa h)^{\kappa\beta} \cdot {\bf 1}\{\kappa h>0\}$$ with $\beta>-1$ for the $\beta$-model and $\beta>(d+1)/2$ for the $\beta'$-model.
General Laguerre tessellations {#sec:Laguerre_tess}
------------------------------
Let us start by defining a Laguerre tessellation, which can be considered as a generalized (or weighted) version of a classical Voronoi tessellation. For two points $v,w \in \RR^{d-1}$ and $h\in\RR$ we define the power of $w$ with respect to the pair $(v,h)$ as $${\mathop{\mathrm{pow}}\nolimits}(w,(v,h)):=\|w-v\|^2+h.$$ In this situation $h$ is referred to as the weight of the point $v$. A closely related concept is known from elementary geometry, where for $h < 0$ the value ${\mathop{\mathrm{pow}}\nolimits}(w, (v, h))$ describes the square length of the tangent through a point $w$ at a circle with radius $\sqrt{-h}$ around $v$.
Let $X$ be a countable set of marked points of the form $(v,h)$ in $\RR^{d-1}\times \RR$. Then the **Laguerre cell** of $(v,h)\in X$ is defined as $$C((v,h),X):=\{w\in\RR^{d-1}\colon {\mathop{\mathrm{pow}}\nolimits}(w,(v,h))\leq {\mathop{\mathrm{pow}}\nolimits}(w,(v',h'))\text{ for all }(v',h')\in X\}.$$ Let us mention an intuitive interpretation of the notions introduced above. Imagine that $v\in \RR^{d-1}$ denotes a point at which certain crystallization process starts at time $h\in\RR$. Then, $X$ is a collection of random centers of crystallization together with the corresponding initial times. Suppose further that after a crystallization process has started at some point $v\in\RR^{d-1}$, it needs time $R^2$ to cover a ball of radius $R>0$ around $v$. In particular, the spreading speed of crystallization is non-constant and decreases with time. Then, ${\mathop{\mathrm{pow}}\nolimits}(w,(v,h))$ is just the time at which the point $w$ is covered by the crystallization process that started at $(v,h)$. Moreover, the Laguerre cell of $(v,h)$ is just the crystal with “center” $v$, that is the set of points which are covered by the crystallization process that started at $(v,h)$ before they are covered by any other crystallization process. It should be pointed out that in our model we assume that crystallization starts at point $(v,h)\in X$ even if this point is already covered by another crystallization process, which started earlier.
Note that Laguerre cells can have vanishing topological interior and, in our case, most of them will actually be empty. The collection of all Laguerre cells of $X$, which have non-vanishing topological interior, is called the **Laguerre diagram**: $$\cL(X):=\{C((v,h),X)\colon (v,h)\in X, C((v,h),X)\neq\varnothing\}.$$ Also, let us emphasize that the Laguerre cell generated by $(v,h)$, even if it is non-empty, need not contain the point $v$. Indeed, the cell of $(v,h)$ does not contain $v$ if at time $h$ the point $v$ has been already covered by a crystallization process that started at some other point $(v',h')\neq (v,h)$.
It should be mentioned that a Laguerre diagram is not necessarily a tessellation, at least as long as we do not impose additional assumptions on the geometric properties of the set $X$. We also note that the case when all weights are equal corresponds to the case of the classical Voronoi tessellation. The first formal description of geometric properties of Laguerre diagram is due to Schlottmann [@Sch93] and a thorough investigation of the case when all weights are negative has been made by Lautensack and Zuyev [@LZ08; @Ldoc]. In our situation we are interested in the case of positive, negative, as well as general weights $h\in\RR$. More precisely and in view of the applications in part II of this series of papers, we consider a point process $\xi$ in $\RR^{d-1}\times E$, where $E\subset \RR$ is a possibly unbounded interval, satisfying the following properties.
- For every $(w,t)\in\RR^{d-1}\times E$ there are almost surely only finitely many $(v,h)\in\xi$ satisfying $${\mathop{\mathrm{pow}}\nolimits}(w,(v,h)) = \|w-v\|^2+h \leq t.$$ In words, at the time when a crystallization process starts at some $(w,t)$, the point $w$ is already reached by at most finitely many crystallization processes.
- With probability $1$ we have $${\mathop{\mathrm{conv}}\nolimits}(v\colon (v,h)\in\xi)=\RR^{d-1}.$$
- With probability $1$ no $d+1$ points $(v_0,h_0),\ldots, (v_d,h_d)$ from $\xi$ lie on the same downward paraboloid of the form $$\{(v,h)\in \RR^{d-1}\times E: \|v - w\|^2 + h = t\}$$ with $(w,t)\in\RR^{d-1}\times E$. In words, with probability $1$ it is not possible that $d+1$ crystallization processes reach the same point in space simultaneously.
In the following two lemmas we prove that the Laguerre diagram constructed on the Poisson point process $\xi$ is a random face-to-face normal tessellation in $\RR^{d-1}$.
Let $\xi$ be a point process satisfying conditions (P1) and (P2). Then $\cL(\xi)$ is a random face-to-face tessellation in $\RR^{d-1}$.
The proof follows directly from [@Sch93 Proposition 1]. In this context, it should be noted that in [@Sch93] the function $${\mathop{\mathrm{pow}}\nolimits}^*(w,(v,h)) = \|w-v\|^2-h =\|w-v\|^2+(-h)={\mathop{\mathrm{pow}}\nolimits}(w,(v,-h))$$ was considered. However, this does not influence the proof and we prefer to use the function ${\mathop{\mathrm{pow}}\nolimits}(\,\cdot\,)$ in this paper, which is more convenient for our purposes.
\[lem:voronoi\_normal\] Let $\xi$ be a point process satisfying conditions (P1)–(P3). Then the random tessellation $\cL(\xi)$ is normal with probability $1$.
Let us start by showing that condition (P3) implies that with probability $1$ for any $k\in\{2,\ldots, d\}$ there are no $k+1$ distinct points $(v_0,h_0),\ldots, (v_k,h_k)$ from $\xi$ such that:
- $\dim{\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_k)\leq k-1$;
- ${\mathop{\mathrm{pow}}\nolimits}(z,(v_0,h_0))=\ldots ={\mathop{\mathrm{pow}}\nolimits}(z,(v_k,h_k))$ for some point $z\in {\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_k)$.
Consider the converse event, namely that for some $k\in\{2,\ldots, d\}$ there exist points $(v_0,h_0),\ldots, (v_k,h_k)$ from $\xi$ satisfying (a) and (b) with $\dim{\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_k)=m\leq k-1$. Without loss of generality assume that $v_0,\ldots, v_{m}$ are affinely independent. With probability $1$ we can find $d-1-m$ distinct points $(v^*_{m+1},h^*_{m+1}),\ldots,(v^*_{d-1},h^*_{d-1})$ from $\xi$ such that $v_0,\ldots, v_{m},v^*_{m+1},\ldots,v^*_{d-1}$ are affinely independent. Then there exists a point $w\in\RR^{d-1}$ such that $${\mathop{\mathrm{pow}}\nolimits}(w,(v_0,h_0))=\ldots ={\mathop{\mathrm{pow}}\nolimits}(w,(v_{m-1},h_{m-1}))={\mathop{\mathrm{pow}}\nolimits}(w,(v^*_m,h^*_m))=\ldots ={\mathop{\mathrm{pow}}\nolimits}(w,(v^*_{d-1},h^*_{d-1})),$$ which is a unique solution of the system of linear equations $$\begin{aligned}
2\langle v_i-v_0,w\rangle&=\|v_0\|^2-\|v_i\|^2+h_0-h_i,\qquad\; i=1,\ldots,m,\label{eq_26.03_1}\\
2\langle v^*_i-v_0,w\rangle&=\|v_0\|^2-\|v^*_i\|^2+h_0-h^*_i,\qquad i=m+1,\ldots,d-1.\notag\end{aligned}$$ On the other hand condition (b) is equivalent to $$2\langle v_i-v_0,z\rangle=\|v_0\|^2-\|v_i\|^2+h_0-h_i,\qquad i=1,\ldots,k,$$ which together with means that $w-z$ is orthogonal to ${\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_m)={\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_k)$. Thus $${\mathop{\mathrm{pow}}\nolimits}(w,(v_0,h_0))=\ldots ={\mathop{\mathrm{pow}}\nolimits}(w,(v_{k},h_{k}))={\mathop{\mathrm{pow}}\nolimits}(w,(v^*_m,h^*_m))=\ldots ={\mathop{\mathrm{pow}}\nolimits}(w,(v^*_{d-1},h^*_{d-1})),$$ and by (P3) this happens with probability $0$, since $k+d-m\ge d+1$.
Let $F\in\cF_{k}(\cL(\xi))$, $k\in\{0,1,\ldots, d-2\}$, be a $k$-dimensional face of $\cL(\xi)$ and let $F$ be in the boundary of precisely $\ell+1$ cells $C(x_0,\xi),\ldots, C(x_\ell,\xi)$. This means that ${\mathop{\mathrm{aff}}\nolimits}F$ is the set of points $w\in\RR^{d-1}$, such that $$\label{eq_25.03_1}
{\mathop{\mathrm{pow}}\nolimits}(w,(v_0,h_0))=\ldots={\mathop{\mathrm{pow}}\nolimits}(w,(v_\ell,h_\ell)).$$ Since $F\neq\varnothing$ there exists at least one point $w\in \RR^{d-1}$ satisfying , which is equivalent to $$\label{eq_25.03_2}
2\langle v_i-v_0,w\rangle=\|v_0\|^2-\|v_i\|^2+h_0-h_i,\qquad i=1,\ldots,\ell.$$ It is easy to see, that also holds for any point $w+z$, $z\in{\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_\ell)^{\perp}$ and, in particular, there exists a point $w^{*}\in {\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_\ell)$ satisfying . Hence, condition (b) holds and with probability $1$ we have $\dim{\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_\ell)=\ell$. Moreover, the point $w^{*}$ is a unique solution of in ${\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_\ell)$. Thus, ${\mathop{\mathrm{aff}}\nolimits}F$ is the set of solutions of the system , which coincides with the hyperplane, orthogonal to ${\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_\ell)$ and intersecting ${\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_\ell)$ in a unique point $w^*$. We conclude that $\dim F = d-1-\ell=k$.
It is easy to see that if a point process $\xi$ satisfies properties (P1) — (P3) then with probability $1$ there is no non-empty Laguerre cells $C(x_0,\xi)$ with vanishing interior. First of all it follows from the last paragraph of the proof of Proposition 1 in [@Sch93] that every Laguerre cell with vanishing interior is a face of some other Laguerre cell with non-vanishing interior. Assume that there is a non-empty Laguerre cell $C(x_0,\xi)$ of dimension $k\leq d-2$ and assume that it coincides with a $k$-dimensional face of cells $C(x_1,\xi),\ldots, C(x_{d-k},\xi)$. If $k=0$ we have $$h_0={\mathop{\mathrm{pow}}\nolimits}(v_0,(v_1,h_1))=\ldots ={\mathop{\mathrm{pow}}\nolimits}(v_0,(v_d,h_d))$$ and by property (P3) this happens with probability $0$. If $k\ge 1$ then for any $z\in C(x_0,\xi)$ we have $$2\langle v_i-v_0,z\rangle=\|v_0\|^2-\|v_i\|^2+h_0-h_i,\qquad i=1,\ldots,d-k.$$ Since the dimension of the cell $C(x_0,\xi)$ is equal to $k$, $\dim{\mathop{\mathrm{aff}}\nolimits}(v_0,\ldots,v_{d-k})\leq d-k-1$. As follows from the proof of Lemma \[lem:voronoi\_normal\] this happens with probability $0$ as well.
Definition of $\beta^{(')}$-Voronoi tessellations
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The $\beta^{(')}$-Voronoi tessellations we are interested in are defined as the Laguerre tessellations driven by the Poisson point processes $\eta_\beta$ and $\eta^{\prime}_\beta$ with intensities given by and .
\[lem:properties\_satisfied\] The point processes $\eta_\beta$ for $\beta>-1$ and $\eta^{\prime}_\beta$ for $\beta>(d+1)/2$ satisfy properties (P1)–(P3) with $E=[0,\infty)$ in the $\beta$-case and $E= (-\infty,0)$ in the $\beta'$-case.
Property (P1) holds because the projections of the Poisson point processes $\eta_\beta$ and $\eta_\beta^\prime$ to the space component $\RR^{d-1}$ are everywhere dense sets, with probability $1$. Indeed, the integrals of the intensities of these processes over any set of the form $B\times \RR$, where $B\subset \RR^{d-1}$ is a non-empty ball, are infinite, which means that infinitely many points project to $B$ almost surely.
Property (P3) holds for any Poisson point process $\eta$ in $\RR^{d-1}\times E$ whose intensity measure $\Theta_{\eta}$ is absolutely continuous with respect to the Lebesgue measure with density $\varrho$, say (see, for example, [@Mo94 Proposition 4.1.2] for a closely related result in the setting of a stationary Poisson point process). Let us verify (P3) applying the multivariate Mecke formula [@SW Corollary 3.2.3]. We have that $$\begin{aligned}
\EE&\sum_{(x_1,\ldots,x_{d+1})\in \eta^{d+1}_{\neq}}{\bf 1}(x_1,\ldots,x_{d+1}\text{ lie on the same paraboloid})\\
&={1\over (d+1)!}\int_{\RR^{d-1}}\ldots \int_{\RR^{d-1}}\int_{E}\ldots \int_{E}\;\;\prod_{i=1}^{d}\varrho(h_i,v_i)\,{{\rm d}}v_1\ldots {{\rm d}}v_{d}\,{{\rm d}}h_1\ldots {{\rm d}}h_{d}\\
&\qquad\times \int_{\RR^{d-1}}\int _{E} {\bf 1}((v_{d+1}, h_{d+1})\in\Pi((v_1,h_1),\ldots,(v_d,h_d)))\varrho(h_{d+1},v_{d+1})\,{{\rm d}}v_{d+1}\,{{\rm d}}h_{d+1} = 0,\end{aligned}$$ since the inner integral is equal to $0$.
Verification of property (P2) requires additional computations. To consider the case of $\eta_\beta$, fix $w\in\RR^{d-1}$ and $t>0$. The inequalities in (P2) describe the bounded domain $$D := \{(v,h)\in \RR^{d-1}\times \RR_+: \|v-w\|^2 + h \leq t\}$$ lying below the paraboloid $h = \|v-w\|^2-t$ and above the hyperplane $h=0$. Since the intensity measure of the Poisson point process $\eta_\beta$ is locally integrable due to the condition $\beta>-1$, there are only finitely many points $(v,h)$ of $\eta_\beta$ in $D$ and Property (P2) holds.
In order to check (P2) for $\eta^{\prime}_\beta$, we fix $w\in\RR^{d-1}$ and $t<0$. We need to show that the downward paraboloid $$D: = \{(v,h)\in \RR^{d-1}\times \RR_-^*: \|v-w\|^2 + h \leq t\}$$ contains only finitely many points of $\eta_\beta^\prime$ a.s. Using the stationarity of the process $\eta_\beta^\prime$ in the space coordinate, we can put $w=0$ without loss of generality. The expected number of points of $\eta_\beta^\prime$ in $D$ is then given by $$\begin{aligned}
\EE \sum_{(v,h)\in\eta^{\prime}_\beta} {\bf 1}(\|v\|^2+h\leq t)
&=
\gamma\,c_{d,\beta}^{\prime}
\int_{\RR^{d-1}}\int_{0}^\infty {\bf 1}(\|v\|^2-s\leq t) s^{-\beta}{{\rm d}}s\,{{\rm d}}v\\
&=
\frac{\gamma\,c_{d,\beta}^{\prime}}{1-\beta}
\int_{\RR^{d-1}} (\|v\|^2 + t)^{1-\beta} {{\rm d}}v\\
&=
{\gamma \, c_{d,\beta}^{\prime}\over (1-\beta) c_{d-1,\beta-1}^{\prime}}|t|^{d+1-2\beta\over 2}<\infty,\end{aligned}$$ where we used the condition that $\beta> (d+1)/2$ to ensure the finiteness of the integral over $\RR^{d-1}$. This completes the proof.
Summarizing, we conclude that the Laguerre tessellations $\cL(\eta_\beta)$ and $\cL(\eta^{\prime}_\beta)$ are with probability one stationary and normal random tessellations in $\RR^{d-1}$. We can thus state the following definition.
The random tessellation $\cV_\beta:=\cL(\eta_\beta)$ is called the **$\beta$-Voronoi tessellation** and the random tessellation $\cV^{\prime}_\beta:=\cL(\eta^{\prime}_\beta)$ is called the **$\beta'$-Voronoi tessellation** in $\RR^{d-1}$.
Let us emphasize that even though the Poisson point processes $\eta_\beta$, respectively $\eta^{\prime}_{\beta}$, are actually well-defined on $\RR^{d-1}\times (0,\infty)$, respectively $\RR^{d-1}\times (-\infty,0)$, for every $\beta\in \RR$, the corresponding tessellations are well-defined under conditions $\beta>-1$, respectively $\beta>(d+1)/2$, only (because otherwise condition (P2) is not satisfied).
Definition of $\beta^{(')}$-Delaunay tessellations
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Given a Laguerre diagram $\cL(\xi)$ we can associate to it a so-called dual Laguerre diagram $\cL^*(\xi)$, which can be defined in the same spirit as a classical Delaunay diagram for given Voronoi construction. This generalised construction was introduced in [@Sch93 Section 3].
Let $\xi$ be a Poisson point process in $\RR^{d-1}\times E$, $E\subset \RR$ satisfying properties (P1) — (P3). Then $\cL(\xi)$ is a random normal face-to-face tessellation and we denote by $\cF_0(\cL(\xi))$ the set of its vertices. Further, given a point $z\in \cF_0(\cL(\xi))$ we construct a Delaunay cell $D(z,\xi)$ as a convex hull of those $v$ for which $(v,h)\in\xi$ and $z\in C((v,h),\xi)$, namely $$D(z,\xi): = {\mathop{\mathrm{conv}}\nolimits}(v\colon (v,h)\in\xi, z\in C((v,h),\xi)).$$ Since the tessellation $\cL(\xi)$ is normal with probability $1$, for every vertex $z\in \cF_0(\cL(\xi))$ there exists exactly $d$ points $x_1,\ldots, x_d$ of $\xi$ such the corresponding cells $C(x_1,\xi),\dots, C(x_d,\xi)$ of the Laguerre tessellation $\cL(\xi)$ contain $z$. Thus, $D(z,\xi)$ is a simplex with probability $1$. We define the **dual Laguerre diagram** $\cL^*(\xi)$ as a collection of all Delaunay simplices $$\cL^*(\xi):=\{D(z,\xi)\colon z\in \cF_0(\cL(\xi))\}.$$ From the above construction it follows that for any $z\in \cF_0(\cL(\xi))$ there exists a number $K_{z}\in\RR$ such that with probability $1$ there exist exactly $d$ points $(v_1,h_1),\ldots, (v_d,h_d)$ of $\xi$ with $${\mathop{\mathrm{pow}}\nolimits}(z, (v_1,h_1)) = \ldots = {\mathop{\mathrm{pow}}\nolimits}(z, (v_d,h_d)) = K_{z}$$ and there is no $(v,h)\in\xi$ with ${\mathop{\mathrm{pow}}\nolimits}(z,(v,h))<K_{z}$. Consider the set $$\label{eq:ApexProcess}
\xi^*:=\left\{(z,-K_{z})\in\RR^{d-1}\times\RR\colon z\in \cF_0(\cL(\xi))\right\}.$$ It turned out to be that the dual Laguerre diagram $\cL^*(\xi)$ is a Laguerre diagram constructed for the set $\xi^*$ and that $\xi^*$ satisfies properties (P1) and (P2) if $\xi$ satisfies (for the proof of those facts see [@Sch93 Proposition 2]). Thus, by Lemma \[lem:properties\_satisfied\] we conclude that $\cL^*(\xi)=\cL(\xi^*)$ is random face-to-face simplicial tessellation.
![Realization of $\beta$-Delaunay tessellation in $\RR^2$. Left: $\beta=5$. Right: $\beta=15$. The pictures above have been created with the help of the software project “The Computational Geometry Algorithms Library” (CGAL) [@CGAL].[]{data-label="fig:beta-tessellations"}](Beta5v5.pdf "fig:"){width="48.00000%"} ![Realization of $\beta$-Delaunay tessellation in $\RR^2$. Left: $\beta=5$. Right: $\beta=15$. The pictures above have been created with the help of the software project “The Computational Geometry Algorithms Library” (CGAL) [@CGAL].[]{data-label="fig:beta-tessellations"}](Beta15v5.pdf "fig:"){width="48.00000%"}
We will be interested in the case when $\xi$ is one of the Poisson point process $\eta_\beta$ or $\eta_{\beta}^\prime$.
The random tessellation $\cD_\beta:=\cL^*(\eta_{\beta})$ is called the **$\beta$-Delaunay tessellation** in $\RR^{d-1}$, while the random tessellation $\cD^{\prime}_\beta:=\cL^*(\eta^{\prime}_{\beta})$ is called the **$\beta'$-Delaunay tessellation** in $\RR^{d-1}$.
Paraboloid hull process {#sec:ParabHullProc}
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The paraboloid hull process was first introduced in [@SY08] and [@CSY13] in order to study the asymptotic geometry of the convex hull of Poisson point processes in the unit ball. It is designed to exhibit properties analogous to those of convex polytopes with the paraboloids playing the role of hyperplanes, with the spatial coordinates $v$ playing the role of spherical coordinates and with the height coordinates $h$ playing the role of the radial coordinate. The numerous properties of the paraboloid hull process, which are analogous to standard statements of convex geometry, have been developed in [@CSY13 Section 3] and we refer to this paper for further information and background material. At this point let us mention without making the statement precise and proving it, that the $\beta$-Delaunay tessellation we are interested in describes the local asymptotic structure (near the boundary of the unit sphere) of the so-called beta random polytope [@beta_polytopes] in the $d$-dimensional unit ball generated by $n$ points, as $n\to\infty$. After rescaling, the unit sphere looks locally like $\RR^{d-1}$, while the boundary of the beta random polytope (projected to the sphere) looks locally like the $\beta$-Delaunay tessellation.
Let $\Pi$ be the standard downward paraboloid, defined as $$\Pi:=\left\{(v',h')\in\RR^{d-1}\times\RR\colon h'=-\|v'\|^2\right\}.$$ Further, let $\Pi_{x}$ be the translation of $\Pi$ by vector $x:=(v,h)\in\RR^d$, that is, $$\Pi_x:=\left\{(v',h')\in\RR^{d-1}\times\RR\colon h'=-\|v'-v\|^2+h\right\}.$$
![Realization of $\beta^{'}$-Delaunay tessellation in $\RR^2$. Left: $\beta=2.1$. Right: $\beta=2.5$. The pictures above have been created with the help of the software project “The Computational Geometry Algorithms Library” (CGAL) [@CGAL].[]{data-label="fig:betaprime-tessellations"}](BetaPrime2-1v5.pdf "fig:"){width="48.00000%"} ![Realization of $\beta^{'}$-Delaunay tessellation in $\RR^2$. Left: $\beta=2.1$. Right: $\beta=2.5$. The pictures above have been created with the help of the software project “The Computational Geometry Algorithms Library” (CGAL) [@CGAL].[]{data-label="fig:betaprime-tessellations"}](BetaPrime2-5v5.pdf "fig:"){width="48.00000%"}
Moreover, given a set $A\subset \RR^d$ we define the hypograph and the epigraph of $A$ as $$\begin{aligned}
A^{\downarrow}:&=\{(v,h')\in\RR^{d-1}\times\RR\colon (v,h) \in A \text{ for some } h\ge h'\},\\
A^{\uparrow}:&=\{(v,h')\in\RR^{d-1}\times\RR\colon (v,h) \in A \text{ for some } h\leq h'\}.\end{aligned}$$ The point $x$ is the apex of the paraboloid $\Pi_x$ and we write $${\mathop{\mathrm{apex}}\nolimits}\Pi^{\downarrow}_x={\mathop{\mathrm{apex}}\nolimits}\Pi_x:=x.$$
The idea is that the shifts of $\Pi^{\downarrow}$ are, in some sense, analogous to the half-spaces in $\RR^{d}$ not containing the origin $0$ in their boundary. For any collection $x_1:=(v_1,h_1),\ldots,x_k:=(v_k,h_k)$ of $k\leq d$ points in $\RR^{d-1}\times\RR$ with affinely independent coordinates $v_1,\ldots,v_k$, we define $\Pi(x_1,\ldots,x_k)$ as the intersection of ${\mathop{\mathrm{aff}}\nolimits}(v_1,\ldots,v_k)\times\RR$ with a translation of $\Pi$ containing all points $x_1,\ldots,x_k$. It should be noted that the set $\Pi(x_1,\ldots,x_k)$ is well-defined, although for $k<d$ the translation of $\Pi$ containing all $x_1,\ldots,x_k$ is not unique. Nevertheless, for $k=d$ and all tuples $x_1,\ldots,x_d$ with affinely independent spatial coordinates $v_1,\ldots,v_d$ such a translation is unique. Then we define $\Pi[x_1,\ldots,x_k]$ as $$\Pi[x_1,\ldots,x_k]:=\Pi(x_1,\ldots,x_k)\cap \left({\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_k)\times\RR\right).$$
We will say that a set $A\subset \RR^d$ has the paraboloid convexity property if for each $y_1,y_2\in A$ we have $\Pi[y_1,y_2]\subset A$. Clearly, $\Pi[x_1,\ldots,x_k]$ is the smallest set containing $x_1,\ldots,x_k$ and having the paraboloid convexity property. Next, we say the set $A\subset \RR^d$ is upwards paraboloid convex if and only if $A$ has the paraboloid convexity property and if for each $x=(v,h)\in A$ we have $\{x\}^{\uparrow}\subset A$.
Finally, given a locally finite point set $X\subset\RR^d$ we define its **paraboloid hull** $\Phi(X)$ to be the smallest upwards paraboloid convex set containing $X$. In particular, given a Poisson point process $\xi$ in $\RR^{d-1}\times E$, $E\subset\RR$, we define the **paraboloid hull process** $\Phi(\xi)$ in $\RR^{d-1}\times E$ as the paraboloid hull of $\xi$.
Using the arguments analogous to [@CSY13 Lemma 3.1] it is easy to derive an alternative and more convenient way to represent $\Phi(\xi)$, namely with probability $1$ we have that $$\Phi(\xi)=\bigcup\limits_{(x_1,\ldots,x_d)\in\xi_{\neq}^d}\left(\Pi[x_1,\ldots, x_d]\right)^{\uparrow},$$ where $\xi_{\neq}^d$ is the collection of all $d$-tuples of distinct points of $\xi$.
For $(x_1,\ldots,x_d)\in\xi_{\neq}^d$ the set $\Pi[x_1,\ldots,x_d]$ is called a paraboloid sub-facet of $\Phi(\xi)$ if $\Pi[x_1,\ldots,x_d] \in\partial \Phi(\xi)$. Two paraboloid sub-facets $\Pi[x_1,\ldots,x_d]$ and $\Pi[y_1,\ldots,y_d]$ are called co-paraboloid provided that $\Pi(x_1,\ldots,x_d)=\Pi(y_1,\ldots,y_d)$ and by **paraboloid facet** of $\Phi(\xi)$ we understand the collection of co-paraboloid sub-facets. Since $\xi$ is a Poisson point process process each paraboloid facet of $\Phi(\xi)$ with probability one consists of exactly one sub-facet. Thus, we can say, that $\Pi[x_1,\ldots,x_d]$ is a paraboloid facet of $\Phi(\xi)$ if and only if $\xi \cap \left(\Pi(x_1,\ldots,x_d)\right)^{\downarrow}=\{x_1,\ldots, x_d\}$.
Using the paraboloid hull processes $\Phi(\xi)$ we construct now a diagram $\cD(\xi)$ on $\RR^{d-1}$ in the following way: for any any collection $x_1:=(v_1,h_1),\ldots,x_k:=(v_d,h_d)$ of pairwise distinct points from $\xi$ we say that the simplex ${\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)$ belongs to $\cD(\xi)$ if and only if $\Pi[x_1,\ldots,x_k]$ is a paraboloid facet of $\Phi(\xi)$. Thus, if $\xi$ satisfies properties (P1)–(P3), then $\cD(\xi)=\cL^{*}(\xi)$ is a random simplicial tessellation.
It is clear now that the tessellations $\cD(\eta_{\beta})$ and $\cD(\eta^{\prime}_{\beta})$ coincide with $\beta$-Poisson-Delaunay and $\beta'$-Poisson tessellations (respectively), defined in the previous subsection.
Weighted typical cells in $\beta$- and $\beta^{\prime}$-Delaunay tessellations {#sec:TypicalCells}
==============================================================================
Definition of the $\nu$-weighted typical cell
---------------------------------------------
In this section we derive an explicit representation of the distribution of typical cells in a $\beta$-Delaunay tessellation $\cD_\beta$ with parameter $\beta > -1$ and a $\beta^{\prime}$-Delaunay tessellation $\cD^{\prime}_\beta$ with parameter $\beta > (d+1)/2$ as described in the previous sections, and, more generally, the distribution of typical cells weighted by the $\nu$-th power of their volume, with $\nu\ge-1$. On the intuitive level, the construction presented below can be understood as follows. Consider the collection of all cells of $\cD_\beta$ or $\cD^{\prime}_\beta$ and assign to each cell a weight equal to the $\nu$-th power of its volume. Then, pick one cell at random, where the probability of picking each cell is proportional its $\nu$-th volume power. The resulting random simplex is denoted by $Z_{\beta,\nu}$, respectively $Z_{\beta,\nu}^\prime$, and its probability distribution on the space of compact convex subsets of $\RR^{d-1}$ is denoted by $\PP_{\beta,\nu}$, respectively $\PP_{\beta,\nu}^\prime$. Since the number of cells in the tessellation is infinite, some work is necessary in order to define these objects in a mathematically rigorous way. The reader should keep in mind the following two important special cases:
- $\nu=0$: $Z_{\beta,0}$ and $Z_{\beta,0}^{\prime}$ coincide with the classical typical cell of $\cD_\beta$ and $\cD_\beta^{\prime}$, respectively;
- $\nu=1$: $Z_{\beta,1}$ and $Z_{\beta,1}^{\prime}$ coincide the volume-weighted typical cell of $\cD_\beta$ and $\cD_\beta^{\prime}$, respectively (which has the same distribution as the a.s. unique cell containing the origin, up to translation; see Theorem 10.4.1 in [@SW]).
To formally present the definition of volume-power weighted typical cells, we use the concept of generalized centre functions and Palm calculus for marked point processes as outlined in [@SW p. 116] and [@SWGerman Section 4.3]. Following the arguments from Subsection \[sec:Laguerre\_tess\] and Subsection \[sec:ParabHullProc\], a random tessellation $\cD(\xi)$, where $\xi$ is a point process in $\RR^{d-1}\times E$, $E\subset \RR$ satisfying (P1)–(P3), coincides with the Laguerre tessellation of the random set $\xi^*$ described by . In this section we additionally assume that $\xi$ is stationary with respect to the shifts of the $\RR^{d-1}$-component, which implies stationarity of the tessellation $\cD(\xi)$. Observe that $\xi^*$ can alternatively be described via the set of apexes of paraboloid facets of the paraboloid hull process $\Phi(\xi)$, that is, $$\xi^*=\{(v,h)\colon (v,-h)={\mathop{\mathrm{apex}}\nolimits}\Pi(x_1,\ldots,x_d),\, x_i\in\xi,1\leq i\leq d,\,{\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)\in\cD(\xi)\}.$$ Let $\cC'$ denote the space of non-empty compact subsets of $\RR^{d-1}$ endowed with the usual Hausdorff metric and the corresponding Borel $\sigma$-field $\cB(\cC')$. The random tessellation $\cD(\xi)$ (which is defined as a random subset of $\cC'$) can be identified with the particle process $\sum_{c\in\cD(\xi)}\delta_c$, see [@SW Chapter 4]. Formally, this is a simple point process on $\cC'$, or equivalently, a random element in the space $\sfN_s(\cC')$ of $\sigma$-finite simple counting measures on $\cC'$ (a counting measure $\zeta$ on $\cC'$ is simple if $\zeta(\{K\})\in\{0,1\}$ for all $K\in\cC'$). Next, we define the measurable set $\cC'\circ\sfN(\cC'):=\{(K,\zeta)\in\cC' \times \sfN_s(\cC'):K\in\zeta\}$ and recall that a generalized centre function is any Borel-measurable map $z:\cC'\circ\sfN(\cC')\to\RR^{d-1}$ such that $z(K+t, \zeta+t) = z(K,\zeta)+t$ for every $t\in\RR^{d-1}$ and $(K,\zeta)\in\cC'\circ\sfN(\cC')$. In our case we take $$z(K,\zeta):=\begin{cases}
v &: K=C((v,h), \xi^*),\,\zeta =\widehat{\cD}(\xi)\\
0 &: \text{otherwise},
\end{cases}$$ where we recall that $C((v,h), \xi^*)$ is the Laguerre cell of $(v,h)\in \xi^*$.
In a next step, we consider the random marked point process $\mu_{\xi}$ on $\RR^{d-1}$ with mark space $\cC'$, formed as follows: $$\mu_{\xi}:=\sum\limits_{(v,h)\in \xi^*}\delta_{(v,M)},\qquad M:=C((v,h), \xi^*)-v.$$ It is evident from the construction that the point process $\mu_{\xi}$ is stationary and that the intensity measure $\Theta$ of $\mu_{\xi}$ is locally finite. Thus, according to [@SW Theorem 3.5.1] it admits the decomposition $$\Theta=\lambda\,[{\rm Leb}(\RR^{d-1})\otimes\PP_{\xi,0}],$$ where $0<\lambda<\infty$, ${\rm Leb}(\RR^{d-1})$ is the Lebesgue measure on $\RR^{d-1}$ and $\PP_{\xi,0}$ is a probability measure on $\cC'$, the so-called mark distribution of $\mu_{\xi}$. By [@SW p. 84] it can be represented as $$\PP_{\xi,0}(A) := {1\over\lambda}\EE\sum_{(v,M)\in\mu_\xi}{\bf 1}_A(M){\bf 1}_{[0,1]^{d-1}}(v),$$ where $[0,1]^{d-1}$ denotes the $(d-1)$-dimensional unit cube. The probability measure $\PP_{\xi,0}$ describes the mark attached to the typical point of $\mu_{\xi}$, that is, the typical cell of the tessellation $\cD(\xi)$. This motivates the following definition.
For a given $\nu$ we define a probability measure $\PP_{\xi,\nu}$ on $\cC'$ by $$\label{eq_2}
\PP_{\xi,\nu}(A) := {1\over \lambda_{\xi,\nu}}\EE\sum_{(v,M)\in\mu_{\xi}}{\bf 1}_A(M){\bf 1}_{[0,1]^{d-1}}(v){\operatorname{Vol}}(M)^\nu
$$ for $A\in \cB(\cC')$, where $\lambda_{\xi,\nu}$ is the normalizing constant given by $$\label{eq:def_gamma_const}
\lambda_{\xi,\nu}:= \EE\sum_{(v,M)\in\mu_{\xi}}{\bf 1}_{[0,1]^{d-1}}(v){\operatorname{Vol}}(M)^\nu
$$ It should be mentioned, that for some values of $\nu$ the value $\lambda_{\xi,\nu}$ can be equal to infinity. This is the reason why for any point process $\xi$ one needs to specify possible values of $\nu$.
A random simplex $Z_{\beta,\nu}$, where $\nu \ge -1$ and $\beta>-1$, with distribution $\PP_{\beta,\nu}:=\PP_{\eta_\beta,\nu}$ is the [**${\operatorname{Vol}}^{\nu}$-weighted**]{} (or just [**$\nu$-weighted**]{}) [**typical cell**]{} of the $\beta$-Delaunay tessellation $\cD_\beta$.
A random simplex $Z_{\beta,\nu}^{\prime}$, where $\beta>(d+1)/2$ and $2\beta - d>\nu\ge -1$, with distribution $\PP_{\beta,\nu}^{\prime}:=\PP_{\eta_\beta^{\prime},\nu}$ is the [**${\operatorname{Vol}}^{\nu}$-weighted typical cell**]{} of the $\beta^{\prime}$-Delaunay tessellation $\cD^{\prime}_\beta$.
That the constants $\lambda_{\beta,\nu}:=\lambda_{\eta_{\beta},\nu}$ and $\lambda^{\prime}_{\beta,\nu}:=\lambda_{\eta_{\beta}^{\prime},\nu}$ are in fact finite for the ranges of $d$, $\beta$ and $\nu$ mentioned in the previous definition will be established in the proof of Theorem \[theo:typical\_cell\_stoch\_rep\].
We also conjecture that it is possible to enlarge the diapason of possible values for parameter $\nu$ to $\nu>-2$.
Stochastic representation of the $\nu$-weighted typical cell
------------------------------------------------------------
After having introduced the concept of weighted typical cells, we are now going to develop an explicit description of their distributions. In fact, the following theorem may be considered as our main contribution in this paper, since it is the principal device on which most of the results in this part, but also in part II and III of this series of papers are based on. To present it, let us recall our convention that $\kappa=1$ if we consider the $\beta$-model and that $\kappa=-1$ in case of the $\beta'$-model.
\[theo:typical\_cell\_stoch\_rep\] Fix $d\geq 2$, $\nu\ge-1$ and $\beta>-1$ for the $\beta$-model or $2\beta - d>\nu\ge -1$, $\beta>(d+1)/2$ for the $\beta^{\prime}$-model. Then for any Borel set $A\subset \cC'$ we have that $$\begin{aligned}
\PP_{\beta,\nu}^{(\prime)}(A)
&=
\alpha_{d,\beta,\nu}^{(\prime)}\int_{(\RR^{d-1})^d}{{\rm d}}y_1\ldots {{\rm d}}y_d \, \int_{0}^{\infty}{{\rm d}}r\,{\bf 1}_A({\mathop{\mathrm{conv}}\nolimits}(ry_1,\ldots,ry_d)) r^{2\kappa d\beta+d^2+\nu(d-1)}\notag\\
&\qquad\times e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa\beta}}
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0),\end{aligned}$$ where $\Delta_{d-1}(y_1,\ldots,y_d)$ is the volume of ${\mathop{\mathrm{conv}}\nolimits}(y_1,\ldots,y_d)$ and $\alpha_{d,\beta,\nu}$, $\alpha^{\prime}_{d,\beta,\nu}$, $m_{d,\beta}$ and $m^{\prime}_{d,\beta}$ are constants given by $$\begin{aligned}
m_{d,\beta}^{(\prime)}&=\gamma\,c_{d,\beta}^{(\prime)}(2\pi c_{d+1,\beta}^{(\prime)})^{-1},\label{eq:Constant}\\
\alpha_{d,\beta,\nu}&=\pi^{d(d-1)\over 2}\,{(d-1)!^{\nu+1}(d+1+2\beta)\Gamma({d(d+\nu+2\beta)-\nu+1\over 2})\over \Gamma({d(d+\nu+2\beta)\over 2}+1)\Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1})}\Big({\gamma\,\Gamma({d\over 2}+\beta+1)\over \sqrt{\pi}\Gamma({d+1\over 2}+\beta+1)}\Big)^{d+{(\nu-1)(d-1)\over d+2\beta+1}}\notag\\
&\qquad\qquad\times{\Gamma({d+\nu\over 2}+\beta+1)^d\over \Gamma(\beta+1)^d}\prod\limits_{i=1}^{d-1}{\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})},\label{eq:Alpha}\\
\alpha^{\prime}_{d,\beta,\nu}&=\pi^{d(d-1)\over 2}\,{(d-1)!^{\nu+1}(d+1-2\beta)\Gamma({d(2\beta-d-\nu)\over 2})\over \Gamma({d(2\beta-d-\nu)+\nu+1\over 2})\Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1})}\Big({\gamma\,\Gamma(\beta-{d+1\over 2})\over \sqrt{\pi}\Gamma(\beta-{d\over 2})}\Big)^{d+{(\nu-1)(d-1)\over d-2\beta+1}}\notag\\
&\qquad\qquad\times{\Gamma(\beta)^d\over \Gamma(\beta-{d+\nu\over 2})^d}\prod_{i=1}^{d-1}{\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})}.\label{eq:AlphaPrime}\end{aligned}$$
\[rem:rep\_typical\] In more probabilistic terms, the $\nu$-weighted typical cell of the $\beta$-Delaunay tessellation $\cD_\beta$ has the same distribution as the random simplex ${\mathop{\mathrm{conv}}\nolimits}(RY_1,\ldots,RY_d)$, where
1. $R$ is a random variable whose density is proportional to $r^{2d\beta+d^2+\nu(d-1)}e^{-m_{d,\beta} r^{d+1+2\beta}}$ on $(0,\infty)$;
2. $(Y_1,\ldots,Y_d)$ are $d$ random points in the unit ball $\BB^{d-1}$ whose joint density is proportional to $$\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1} \prod\limits_{i=1}^d(1-\|y_i\|^2)^{\beta},
\qquad y_1\in \BB^{d-1},\ldots, y_d\in \BB^{d-1};$$
3. $R$ is independent of $(Y_1,\ldots,Y_d)$.
In the same way, the $\nu$-weighted typical cell of the $\beta^{\prime}$-Delaunay tessellation $\cD^{\prime}_\beta$ has the same distribution as the random simplex ${\mathop{\mathrm{conv}}\nolimits}(RY_1,\ldots,RY_d)$, where
1. $R$ is a random variable whose density is proportional to $r^{-2d\beta+d^2+\nu(d-1)}e^{-m^{\prime}_{d,\beta} r^{d+1-2\beta}}$ on $(0,\infty)$;
2. $(Y_1,\ldots,Y_d)$ are $d$ random points in $\RR^{d-1}$ whose joint density is proportional to $$\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1} \prod\limits_{i=1}^d(1+\|y_i\|^2)^{-\beta},
\qquad y_1\in \RR^{d-1},\ldots, y_d\in \RR^{d-1};$$
3. $R$ is independent of $(Y_1,\ldots,Y_d)$.
Exact formulas for the constants needed to normalize the density of $(Y_1,\ldots,Y_d)$ appearing in (b) and (b$^\prime$) will be given in and .
\[rem:rep\_typical\_beta\_-1\] Let us point out that in the limiting case $\beta\to -1$ the beta distribution with density $c_{d-1,\beta}(1-\|x\|^2)^{\beta}{\bf 1}_{\BB^{d-1}}(x)$ weakly converges to the uniform distribution on the unit sphere $\SS^{d-2}$, denoted by $\sigma_{d-2}$. Thus, $\PP_{\beta,\nu}$ for fixed $\nu\ge-1$ and $\gamma>0$ weakly converges to a probability measure $\PP_{-1,\nu}$ with $$\begin{aligned}
\PP_{-1,\nu}(A)
&=
\alpha^{*}_{d,\nu}
\int_{(\SS^{d-2})^d} \sigma_{d-2}({{\rm d}}u_1)\ldots \sigma_{d-2}({{\rm d}}u_d) \int_{0}^{\infty}{{\rm d}}r\,{\bf 1}_A({\mathop{\mathrm{conv}}\nolimits}(ru_1,\ldots,ru_d))
\\
&\hspace{4cm}\times r^{d^2-2d+\nu(d-1)}e^{-{\gamma\kappa_{d-1}\over \omega_d} r^{d-1}} (\Delta_{d-1}(u_1,\ldots,u_d))^{\nu+1},\end{aligned}$$ where $$\alpha^{*}_{d,\nu}=(\gamma \omega_d^{-1})^{d+\nu-1}{(d-1)(d-1)!^{\nu+1}\pi^{(\nu-1)(d-1)\over 2}\over 2^d \Gamma(d+\nu-1)}{\Gamma({(d+\nu-1)(d-1)\over 2})\over \Gamma({d(d+\nu-2)\over 2}+1)}{\Gamma({d+\nu\over 2})^d\over \Gamma({d+1\over 2})^{d+\nu-1}}\prod\limits_{i=1}^{d-1}{\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})}.$$ This coincides with the formula for the distribution of the $\nu$-weighted typical cell of Poisson-Delaunay tessellation in $\RR^{d-1}$ corresponding to the intensity $\gamma\omega_d^{-1}$ of underlying Poisson point process, see [@GusakovaThaele Theorem 2.3] for general $\nu$ and [@SW Theorem 10.4.4] for the case $\nu=0$. In part II of this series of papers we will in detail consider the weak limit of the tessellation $\cD_\beta$, as $\beta\to -1$, and prove that it coincides with the Poisson-Delaunay tessellation on $\RR^{d-1}$.
Let us recall that for any collection of points $$x_1:=(v_1,h_1)\in\RR^{d-1}\times\RR\;\; \ldots \;\; x_d:=(v_d,h_d)\in\RR^{d-1}\times\RR$$ with affinely independent spatial coordinates $v_i$ we denote by $\Pi^{\downarrow}(x_1,\ldots,x_d)$ the unique translation of the standard downward paraboloid $\Pi^\downarrow$ containing $x_1,\ldots,x_d$ on its boundary. If $x_1,\ldots,x_d$ are distinct points of the Poisson point process $\eta_\beta^{(')}$, then the simplex $K := {\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)$ belongs to the tessellation $\cD_\beta^{(')}$ if and only if ${\operatorname{int}}(\Pi^{\downarrow}(x_1,\ldots,x_d))\cap\eta_\beta^{(')}=\varnothing$, that is if there are no points of $\eta_\beta^{(')}$ strictly inside $\Pi^{\downarrow}(x_1,\ldots,x_d)$. Let us denote the apex of the paraboloid $\Pi^{\downarrow}(x_1,\ldots,x_d)$ by $(w,t)\in \RR^{d-1} \times \RR$. We then have $$t - \|v_i-w\|^2 = h_i,\qquad i \in \{1,\ldots,d\}.$$ As the center of the simplex ${\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)$ we choose the point $w$ and therefore put $$z(x_1,\ldots,x_d):=z({\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d), \cD(\eta_\beta^{(')}))=w.$$
We are now ready to begin with the essential part of the proof of Theorem \[theo:typical\_cell\_stoch\_rep\]. Fix some Borel set $A\subset \cC'$. From and the definition of the the generalized centre function for the tessellation $\cD_\beta^{(')}$ we get $$\begin{aligned}
S_{\eta_\beta^{(')}}(A)
&
:=
\lefteqn{\EE\sum_{(v,M)\in \mu_{\eta_\beta^{(')}}}{\bf 1}_A(M)\,{\bf 1}_{[0,1]^{d-1}}(v)\,({\operatorname{Vol}}(M))^{\nu}}\\
&=
{1\over d!}\,
\EE\sum\limits_{(x_1,\ldots,x_d)\in (\eta_\beta^{(')})_{\neq}^d}{\bf 1}_A({\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)-z(x_1,\ldots,x_d))\\
&\qquad\times {\bf 1}_{[0,1]^{d-1}}(z(x_1,\ldots,x_d)){\bf 1}\left\{{\operatorname{int}}(\Pi^{\downarrow}(x_1,\ldots,x_d))\cap\eta_\beta^{(')}=\varnothing\right\}\\
&\qquad\times \Delta_{d-1}(v_1,\ldots,v_d)^{\nu}.\end{aligned}$$ Here, $(\eta_\beta^{(')})_{\neq}^d$ denotes the collection of all tuples of the form $(x_1,\ldots,x_d)$ consisting of pairwise distinct points $x_1,\ldots,x_d$ of the Poisson point process $\eta_\beta^{(')}$. We write $S_{\beta}(A):=S_{\eta_{\beta}}(A)$ and $S_{\beta}^{\prime}(A):=S_{\eta_{\beta}^{\prime}}(A)$. Note that $S_{\beta}^{(\prime)}(A)$ is in fact the same as $\lambda_{\beta,\nu}^{(\prime)}\,\PP_{\beta,\nu}^{(\prime)}(A)$, but since at the present moment we don’t know whether the normalizing constants $\lambda_{\beta,\nu}$ and $\lambda_{\beta,\nu}^{\prime}$, given by , are finite, we prefer to use the notation $S_{\beta}^{(\prime)}(A)$. Applying the multivariate Mecke formula [@SW Corollary 3.2.3] to the Poisson point process $\eta_{\beta}^{(\prime)}$ and taking into account and we obtain $$\begin{aligned}
S_{\beta}^{(\prime)}(A)
&=
{\gamma^d (c^{(\prime)}_{d,\beta})^d\over d!}
\int_{\RR^{d-1}} {{\rm d}}v_1 \, \ldots \, \int_{\RR^{d-1}} {{\rm d}}v_d\,
\int_{0}^{\infty}{{\rm d}}h_1\, \ldots \, \int_{0}^{\infty} {{\rm d}}h_d \,
\notag\\
&\qquad\phantom{\times} {\bf 1}_A({\mathop{\mathrm{conv}}\nolimits}(v_1,\ldots,v_d)-z(\tilde x_1,\ldots,\tilde x_d))\notag\\
&\qquad\times {\bf 1}_{[0,1]^{d-1}}(z(\tilde x_1,\ldots,\tilde x_d))\PP\left({\operatorname{int}}(\Pi^{\downarrow}(\tilde x_1,\ldots,\tilde x_d))\cap \eta_{\beta}^{(\prime)}=\varnothing\right)\notag\\
&\qquad\times h_1^{\kappa\beta}\ldots h_d^{\kappa\beta}\,\Delta_{d-1}(v_1,\ldots,v_d)^{\nu}, \label{eq_4}\end{aligned}$$ where $\tilde x_i = (v_i, \kappa h_i)$ for $i=1,\ldots,d$. In the above integral, we are going to pass from the integration over the variables $(v_1,\ldots,v_d,h_1,\ldots,h_d)\in (\RR^{d-1})^d \times(\RR_{+}^*)^d$, where $\RR_+^*=(0,\infty)$, to the integration over certain new variables $(w,r,y_1,\ldots,y_d)\in \RR^{d-1}\times\RR_{+}^*\times\left(\RR^{d-1}\right)^d$ introduced as follows. Take some tuple $(v_1,\ldots,v_d,h_1,\ldots,h_d)\in (\RR^{d-1})^d \times(\RR_{+}^*)^d$ and assume that $v_1,\ldots,v_d$ are affinely independent. Denote the apex of the unique downward paraboloid $\Pi^\downarrow(\tilde x_1,\ldots,\tilde x_d)$ whose boundary passes through the points $(v_1,\kappa h_1),\ldots,(v_d,\kappa h_d)$ by $(w,\kappa r^2)\in \RR^{d-1} \times \RR$ and note that $w= z(\tilde x_1,\ldots,\tilde x_d)$. Observe that in the $\beta'$-case the second coordinate of the apex can be positive, but since such downward paraboloid contains infinitely many points of the Poisson point process $\eta_\beta^{\prime}$ (because any point $(w,0)$ with vanishing second coordinate is an accumulation point for the atoms of $\eta_{\beta}^{\prime}$), we can ignore this possibility in the following. We can write $v_i = w + ry_i$ with some uniquely defined and pairwise distinct $y_1,\ldots,y_d\in \RR^{d-1}$. The condition that the boundary of the paraboloid passes through the point $(v_i,\kappa h_i)$ reads as $\kappa r^2 - \|v_i-w\|^2 = \kappa h_i$, or $h_i = r^2 (1-\kappa\|y_i\|^2)$. In the $\beta$-case it follows that $y_1,\ldots,y_d \in \BB^{d-1}$. Let us therefore introduce the transformation $T:\RR^{d-1}\times\RR_{+}^*\times\left(\BB^{d-1}\right)^d\rightarrow\left(\RR^{d-1}\times\RR_{+}^*\right)^d$ (in the $\beta$-case) or $T:\RR^{d-1}\times\RR_{+}^*\times\left(\RR^{d-1}\right)^d\rightarrow\left(\RR^{d-1}\times\RR_{+}^*\right)^d$ (in the $\beta'$-case) defined as $$(w,r,y_1,\ldots,y_d)\mapsto \left(ry_1+w,r^2(1-\kappa\|y_1\|^2),\ldots,ry_d+w, r^2(1-\kappa\|y_d\|^2)\right) = (v_1,h_1,\ldots,v_d,h_d).$$ This transformation is bijective (up to sets of Lebesgue measure zero and provided that in the $\beta$’-case we agree to exclude from the image set all combinations $(v_1,h_1,\ldots,v_d,h_d)$ which lead to a paraboloid whose apex has positive height). The absolute value of the Jacobian of $T$ is the absolute value of the determinant of the block matrix $$J(T):=\left|
\begin{matrix}
E_{d-1} & y_1 & rE_{d-1} & 0 & \dots & 0\\
0 & 2r(1-\kappa\|y_1\|^2) & -2r^2\kappa y_1^{\top} & 0 &\dots & 0\\
E_{d-1} & y_2 & 0 & rE_{d-1} & \dots & 0\\
0 & 2r(1-\kappa\|y_2\|^2) & 0 & -2r^2\kappa y_2^{\top} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots &\ddots & \vdots \\
E_{d-1} & y_d & 0 & 0 & \dots & rE_{d-1}\\
0 & 2r(1-\kappa\|y_d\|^2) & 0 & 0 & \dots & -2r^2\kappa y_d^{\top}\\
\end{matrix}
\right|,$$ where $E_{k}$ is the $k\times k$ unit matrix, vectors $y_1,\ldots,y_d$ are considered to be column vectors and $|M|$ stands for the absolute value of the determinant of the matrix $M$. We can compute $J(T)$ as follows: $$\begin{aligned}
{J(T)\over2^dr^{d^2}} &=\left|
\begin{matrix}
E_{d-1} & y_1 & E_{d-1} & 0 & \dots & 0\\
0 & 1-\kappa\|y_1\|^2 & -\kappa y_1^{\top} & 0 &\dots & 0\\
E_{d-1} & y_2 & 0 & E_{d-1} & \dots & 0\\
0 & 1-\kappa\|y_2\|^2 & 0 & -\kappa y_2^{\top} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots &\ddots & \vdots \\
E_{d-1} & y_d & 0 & 0 & \dots & E_{d-1}\\
0 & 1-\kappa\|y_d\|^2 & 0 & 0 & \dots & -\kappa y_d^{\top}\\
\end{matrix}
\right|=\left|
\begin{matrix}
E_{d-1} & 0 & E_{d-1} & 0 & \dots & 0\\
0 & 1 & -\kappa y_1^{\top} & 0 &\dots & 0\\
E_{d-1} & 0 & 0 & E_{d-1} & \dots & 0\\
0 & 1 & 0 & -\kappa y_2^{\top} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
E_{d-1} & 0 & 0 & 0 & \dots & E_{d-1}\\
0 & 1 & 0 & 0 & \dots & -\kappa y_d^{\top}\\
\end{matrix}\right|\\
&=\left|
\begin{matrix}
0 & 0 & E_{d-1} & 0 & \dots & 0\\
\kappa y_1^{\top} & 1 & -\kappa y_1^{\top} & 0 &\dots & 0\\
0 & 0 & 0 & E_{d-1} & \dots & 0\\
\kappa y_2^{\top} & 1 & 0 & -\kappa y_2^{\top} & \dots & 0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \dots & E_{d-1}\\
\kappa y_d^{\top} & 1 & 0 & 0 & \dots & -\kappa y_d^{\top}\\
\end{matrix}\right|
=|\kappa^d|\,\left|
\begin{matrix}
0 & E_{d(d-1)}\\
\begin{matrix}
y_1^{\top} & 1 \\
\vdots & \vdots\\
y_d^{\top} & 1\\
\end{matrix} & \mbox{\normalfont\Large\bfseries 0} \\
\end{matrix}\right|=\left|
\begin{matrix}
1 & \ldots & 1\\
y_1 &\ldots & y_d \\
\end{matrix}\right|\end{aligned}$$ Thus, $J(T)=2^dr^{d^2}(d-1)! \Delta_{d-1}(y_1,\ldots,y_d)$. Applying the transformation $T$ in we derive $$\begin{aligned}
S_{\beta}^{(\prime)}(A)
&=
{(2\gamma\,c^{(\prime)}_{d,\beta})^d\over d}
\int_{\RR^{d-1}}{{\rm d}}y_1\, \ldots\, \int_{\RR^{d-1}}{{\rm d}}y_d\, \int_{\RR^{d-1}} {{\rm d}}w\, \int_{0}^{\infty}{{\rm d}}r\,{\bf 1}_A({\mathop{\mathrm{conv}}\nolimits}(ry_1,\ldots,ry_d)) {\bf 1}_{[0,1]^{d-1}}(w)\notag\\
&\qquad\times\PP\left(\{(v,\kappa h)\in\RR^{d-1}\times \kappa \RR_{+}^*\colon \kappa h<-\|v-w\|^2+\kappa r^2\}\cap \eta_{\beta}^{(\prime)}
=\varnothing\right)\,r^{2\kappa d\beta+d^2+\nu(d-1)}\notag\\
&\qquad\times \Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0).\label{eq_5}\end{aligned}$$ Using now the stationarity of the Poisson point processes $\eta_\beta$ and $\eta_\beta^{\prime}$ with respect to the $v$-coordinate we conclude that, for any $w\in\RR^{d-1}$, $$\begin{aligned}
P^{(\prime)} &:=\PP\left(\{(v,\kappa h)\in\RR^{d-1}\times \kappa \RR_{+}^*\colon \kappa h<-\|v-w\|^2+\kappa r^2\}\cap \eta_{\beta}^{(\prime)}=\varnothing\right)\\
&=\PP\left(\{(v,\kappa h)\in\RR^{d-1}\times \kappa \RR_{+}^*\colon \kappa h<-\|v\|^2+\kappa r^2\}\cap \eta_{\beta}^{(\prime)}=\varnothing\right)\\
&=\exp\Big(-\gamma c^{(\prime)}_{d,\beta}\int_{0}^{\infty}\int_{\RR^{d-1}}{\bf 1}(\kappa h<-\|v\|^2+\kappa r^2)h^{\kappa \beta}{{\rm d}}v\,{{\rm d}}h\Big).\end{aligned}$$
For the further computations we need to distinguish the $\beta$- and the $\beta^{\prime}$-cases. For the $\beta$-model we have $$\begin{aligned}
P&:=\exp\Big(-\gamma c_{d,\beta}\int_{0}^{r^2}\int_{\{v\colon \|v\|\leq (r^2-h)^{1/2}\}}h^{\beta}{{\rm d}}v\,{{\rm d}}h\Big)\\
&=\exp\Big(-\gamma \kappa_{d-1}c_{d,\beta}\int_{0}^{r^2}(r^2-h)^{{d-1\over 2}}h^{\beta}{{\rm d}}h\Big)\\
&=\exp\Big(-\gamma \kappa_{d-1}c_{d,\beta}r^{d+1+2\beta}\int_{0}^{1}(1-h')^{{d-1\over 2}}h'^{\beta}{{\rm d}}h'\Big)\\
&
=\exp\left(-m_{d,\beta} r^{d+1+2\beta}\right),\end{aligned}$$ where $m_{d,\beta}={\gamma c_{d,\beta}\over \pi c_{d+1,\beta}}$. For the $\beta^{\prime}$-model we obtain $$\begin{aligned}
P^{\prime}&:=\exp\Big(-\gamma c^{\prime}_{d,\beta}\int_{r^2}^{\infty}\int_{\{v\colon \|v\|\leq (h-r^2)^{1/2}\}}h^{-\beta}{{\rm d}}v\,{{\rm d}}h\Big)\\
&=\exp\Big(-\gamma \kappa_{d-1}c^{\prime}_{d,\beta}\int_{r^2}^{\infty}(h-r^2)^{{d-1\over 2}}h^{-\beta}{{\rm d}}h\Big)\\
&=\exp\Big(-\gamma \kappa_{d-1}c^{\prime}_{d,\beta}r^{d+1-2\beta}\int_{0}^{1}(1-h')^{{d-1\over 2}}h'^{\beta-(d+1)/2-1}{{\rm d}}h'\Big)\\
&=\exp\left(-m^{\prime}_{d,\beta} r^{d+1-2\beta}\right),\end{aligned}$$ with $m^{\prime}_{d,\beta}={\gamma c^{\prime}_{d,\beta}\over \pi c^{\prime}_{d+1,\beta}}$. Substituting this into leads to $$\begin{gathered}
\label{eq_7}
S_{\beta}^{(\prime)}(A)
={(2\gamma\,c^{(\prime)}_{d,\beta})^d\over d}\int_{\RR^{d-1}}{{\rm d}}y_1\, \ldots\, \int_{\RR^{d-1}} {{\rm d}}y_d\, \int_{0}^{\infty} {{\rm d}}r\,{\bf 1}_A({\mathop{\mathrm{conv}}\nolimits}(ry_1,\ldots,ry_d))
\\
\qquad \times r^{2\kappa d\beta+d^2+\nu(d-1)}e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa \beta}}
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0).\end{gathered}$$ We are now in position to determine the normalizing constants $\lambda_{\beta,\nu}$ and $\lambda_{\beta,\nu}^{\prime}$ from . To this end, we plug $A=\cC'$ (the set of non-empty compact subsets of $\RR^{d-1}$) into the expression for $S_\beta^{(\prime)}$. Doing this and using the substitution $s=m^{(\prime)}_{d,\beta} r^{d+1+2\kappa \beta}$, we obtain $$\begin{aligned}
\lambda_{\beta,\nu}^{(\prime)}
&=
S_{\beta}^{(\prime)}(\cC')\\
&=
{(2\gamma\,c^{(\prime)}_{d,\beta})^d\over d}\int_{(\RR^{d-1})^d}\int_{0}^{\infty} r^{2\kappa d\beta+d^2+\nu(d-1)}e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa \beta}}\\
&\qquad\times \Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0){{\rm d}}r\,{{\rm d}}y_1\ldots {{\rm d}}y_d\\
&={(2\gamma\,c^{(\prime)}_{d,\beta})^d\over d(d+1+2\kappa \beta)}(m^{(\prime)}_{d,\beta})^{-d-{(\nu-1)(d-1)\over d+2\kappa \beta+1}}\int_{0}^{\infty} s^{d+{(\nu-1)(d-1)\over d+2\kappa\beta+1}-1}e^{-s}{{\rm d}}s\\
&\qquad\times\int_{(\RR^{d-1})^d}\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0){{\rm d}}y_1\ldots {{\rm d}}y_d\\
&={(2\gamma\,c^{(\prime)}_{d,\beta})^d\Gamma(d+{(\nu-1)(d-1)\over d+2\kappa\beta+1})\over d(d+1+2\kappa\beta)}(m^{(\prime)}_{d,\beta})^{-d-{(\nu-1)(d-1)\over d+2\kappa \beta+1}}\\
&\qquad\times\int_{(\RR^{d-1})^d}\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0){{\rm d}}y_1\ldots {{\rm d}}y_d.\end{aligned}$$
The last integral (which is finite for $\nu\ge-1$) is equal – up to a constant – to the $(\nu+1)$-th moment of the volume of random simplex with vertices having a $\beta$- or $\beta^{\prime}$-distribution. The exact values were calculated in [@GKT17 Theorem 2.3] or [@KTT Proposition 2.8] and are given (in the cases $\kappa=+1$ and $\kappa=-1$) as follows: $$\begin{aligned}
\int_{(\BB^{d-1})^d}&\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod\limits_{i=1}^d(1-\|y_i\|^2)^{\beta}{{\rm d}}y_1\ldots {{\rm d}}y_d \notag\\
&={1\over (d-1)!^{\nu+1}c_{d-1,\beta}^d}{\Gamma({d+1\over 2}+\beta)^d\over \Gamma({d+\nu\over 2}+\beta+1)^d}{\Gamma({d(d+\nu+2\beta)\over 2} +1)\over \Gamma({d(d+\nu+2\beta)-\nu +1 \over 2})}
\prod\limits_{i=1}^{d-1} {\Gamma({i+\nu+1\over 2})\over \Gamma({i\over 2})},
\label{eq:betamoments}\\
\int_{(\RR^{d-1})^d}&\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod\limits_{i=1}^d(1+\|y_i\|^2)^{-\beta}{{\rm d}}y_1\ldots {{\rm d}}y_d \notag\\
&={1\over (d-1)!^{\nu+1}(c^{\prime}_{d-1,\beta})^d}{\Gamma({d(2\beta-d-\nu)+\nu+1\over 2})\over\Gamma({d(2\beta-d-\nu)\over 2})}{\Gamma(\beta-{d+\nu\over 2})^d\over\Gamma(\beta-{d-1\over 2})^d}\prod_{i=1}^{d-1}{\Gamma({i+\nu+1\over 2})\over\Gamma({i\over 2})}.\label{eq:betaprimemoments}\end{aligned}$$ We have thus shown that $$\begin{aligned}
\lambda_{\beta,\nu}&={\Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1})m_{d,\beta}^{-d-{(\nu-1)(d-1)\over d+2 \beta+1}}\over d(d+1+2\beta)(d-1)!^{\nu+1}}\Big({2\gamma\,c_{d,\beta}\Gamma({d+1\over 2}+\beta)\over c_{d-1,\beta}\Gamma({d+\nu\over 2}+\beta+1)}\Big)^d{\Gamma({d(d+\nu+2\beta)\over 2} +1)\over \Gamma({d(d+\nu+2\beta)-\nu +1 \over 2})}\prod\limits_{i=1}^{d-1}{\Gamma({i+\nu+1\over 2})\over \Gamma({i\over 2})},\\
\lambda^{\prime}_{\beta,\nu}&={\Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1})(m^{\prime}_{d,\beta})^{-d-{(\nu-1)(d-1)\over d-2 \beta+1}}\over d(d+1-2\beta)(d-1)!^{\nu+1}}\Big({2\gamma\,c^{\prime}_{d,\beta}\Gamma(\beta-{d+\nu\over 2})\over c^{\prime}_{d-1,\beta}\Gamma(\beta-{d-1\over 2})}\Big)^d{\Gamma({d(2\beta-d-\nu)+\nu+1\over 2})\over\Gamma({d(2\beta-d\nu\over 2})}
\prod\limits_{i=1}^{d-1}{\Gamma({i+\nu+1\over 2})\over \Gamma({i\over 2})}.\end{aligned}$$ In particular, this implies that $\lambda_{\beta,\nu}, \lambda^{\prime}_{\beta,\nu}<\infty$ provided $\nu\ge-1$. From we conclude $$\begin{aligned}
\PP^{(\prime)}_{\beta,\nu}(A)
=
\frac{S_{\beta}^{(\prime)}}{\lambda_{\beta,\nu}^{(\prime)}}
&=
\alpha^{(\prime)}_{d,\beta,\nu}\int_{(\RR^{d-1})^d}{{\rm d}}y_1\ldots {{\rm d}}y_d \, \int_{0}^{\infty}{{\rm d}}r\,{\bf 1}_A({\mathop{\mathrm{conv}}\nolimits}(ry_1,\ldots,ry_d)) r^{2\kappa d\beta+d^2+\nu(d-1)}\notag\\
&\qquad\times e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa\beta}}
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0),\end{aligned}$$ with $\alpha_{d,\beta,\nu}$ and $\alpha^{\prime}_{d,\beta,\nu}$ given by and respectively. This completes the argument.
The volume of weighted typical cells {#sec:Volume}
====================================
Moment formulas
---------------
In this section we apply Theorem \[theo:typical\_cell\_stoch\_rep\] to compute all moments of the random variables ${\operatorname{Vol}}(Z_{\beta,\nu})$ and ${\operatorname{Vol}}(Z^{\prime}_{\beta,\nu})$. These explicit formulas will be the basis of some of the results in part III of this series of papers. In particular, they generalize the moment formulas in [@GusakovaThaele] for weighted typical cells in classical Poisson-Delaunay tessellations.
\[theo:volume\] Let $Z_{\beta,\nu}$ be the $\nu$-weighted typical cell of a $\beta$-Delaunay tessellation with $\beta\geq -1$ and $\nu\ge-1$, and let $Z^{\prime}_{\beta,\nu}$ be the $\nu$-weighted typical cell of the $\beta^{\prime}$-Delaunay tessellation with $\beta\geq (d+1)/2$ and $2\beta - d>\nu\ge-1$. Then, for any $s>-\nu-1$, we have $$\begin{aligned}
\EE {\operatorname{Vol}}(Z_{\beta,\nu})^s &= {1\over (d-1)!^s}\Big({ \sqrt{\pi}\Gamma({d+1\over 2}+\beta+1)\over \gamma\Gamma({d\over 2}+\beta+1)}\Big)^{{s(d-1)\over d+2\beta+1}}{\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2})\over\Gamma({d(d+2\beta)+(\nu+s)(d-1)+1\over 2})}{\Gamma({d(d+\nu+s +2\beta)\over 2}+1)\over\Gamma({d(d+\nu +2\beta)\over 2}+1)}\\
&\qquad\times{\Gamma(d+{(\nu+s-1)(d-1)\over d+2\beta+1})\over\Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1})}{\Gamma({d+\nu\over 2}+\beta +1)^d\over\Gamma({d+\nu+s\over 2}+\beta +1)^d}\prod\limits_{i=1}^{d-1}{\Gamma({i+\nu+s+1\over 2})\over \Gamma({i+\nu+1\over 2})},\end{aligned}$$ and for any $2\beta -d-\nu>s>-\nu-1$ we obtain $$\begin{aligned}
\EE {\operatorname{Vol}}(Z^{\prime}_{\beta,\nu})^s &= {1\over (d-1)!^s}\Big({ \sqrt{\pi}\Gamma(\beta -{d\over 2})\over \gamma\Gamma(\beta-{d+1\over 2})}\Big)^{{s(d-1)\over d-2\beta+1}}{\Gamma({d(2\beta-d)-(\nu+s)(d-1)+1\over 2})\over\Gamma({d(2\beta-d)-\nu(d-1)+1\over 2})}{\Gamma({d(2\beta-d-\nu)\over 2})\over\Gamma({d(2\beta-d-\nu-s)\over 2})}\\
&\qquad\times{\Gamma(d+{(\nu+s-1)(d-1)\over d-2\beta+1})\over\Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1})}{\Gamma(\beta -{d+\nu+s\over 2})^d\over\Gamma(\beta -{d+\nu\over 2})^d}\prod\limits_{i=1}^{d-1}{\Gamma({i+\nu+s+1\over 2})\over \Gamma({i+\nu+1\over 2})}.\end{aligned}$$
Applying Theorem \[theo:typical\_cell\_stoch\_rep\] we get $$\begin{aligned}
\EE {\operatorname{Vol}}(Z^{(\prime)}_{\beta,\nu})^s &= \alpha_{d,\beta,\nu}^{(\prime)}\int_{(\RR^{d-1})^d}{{\rm d}}y_1\ldots {{\rm d}}y_d \, \int_{0}^{\infty}{{\rm d}}r{\operatorname{Vol}}({\mathop{\mathrm{conv}}\nolimits}(ry_1,\ldots,ry_d))^s r^{2\kappa d\beta+d^2+\nu(d-1)}\notag\\
&\qquad\times e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa\beta}}
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0),\end{aligned}$$ Then from Fubini’s theorem we obtain $$\begin{aligned}
\EE {\operatorname{Vol}}(Z^{(\prime)}_{\beta,\nu})^s &= \alpha_{d,\beta,\nu}^{(\prime)}\int_{0}^{\infty}r^{2\kappa d\beta+d^2+\nu(d-1)+s(d-1)}e^{-m^{(\prime)}_{d,\beta} r^{d+1+2\kappa\beta}}\,{{\rm d}}r\\
&\qquad\times \int_{(\RR^{d-1})^d}\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1+s}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0)\,{{\rm d}}y_1\ldots {{\rm d}}y_d\\
&=\alpha_{d,\beta,\nu}^{(\prime)}{\Gamma(d+{(\nu-1)(d-1)\over d+2\kappa\beta+1}+{s(d-1)\over d+2\kappa\beta+1})\over (d+1+2\kappa\beta)}(m_{d,\beta}^{(\prime)})^{-d-{(\nu-1)(d-1)\over d+2\kappa\beta+1}-{s(d-1)\over d+2\kappa\beta+1}}\\
&\qquad\times \int_{(\RR^{d-1})^d}\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1+s}\prod_{i=1}^d(1-\kappa\|y_i\|^2)^{\kappa\beta}{\bf 1}(1-\kappa\|y_i\|^2\ge 0)\,{{\rm d}}y_1\ldots {{\rm d}}y_d.\end{aligned}$$ Finally, using , and definition of constants $\alpha_{\beta,d,\nu}$, $\alpha^{\prime}_{\beta,d,\nu}$, $m_{d,\beta}$ and $m^{\prime}_{d,\beta}$ we complete the proof.
Probabilistic representations
-----------------------------
Based on the formulas for the moments of the volume of the random simplices $Z_{\beta, \nu}$ and $Z_{\beta, \nu}'$ we can obtain probabilistic representations for the random variables ${\operatorname{Vol}}(Z_{\beta,\nu})^2$ and ${\operatorname{Vol}}(Z_{\beta,\nu}')^2$, which are similar in spirit to the ones for Gaussian or beta random simplices [@GKT17; @GusakovaThaele].
Let us first recall some standard distributions. A random variable has a Gamma distribution with shape $\alpha\in(0,\infty)$ and rate $\lambda \in(0,\infty)$ if its density function is given by $$g_{\alpha,\lambda}(t)={\lambda^{\alpha}\over\Gamma(\alpha)}t^{\alpha-1}e^{-\lambda t},\quad t\in(0,\infty).$$ A random variable has a Beta distribution with shape parameters $\alpha_1, \alpha_2\in(0,\infty)$ if its density function is given by $$g_{\alpha_1,\alpha_2}(t)={\Gamma(\alpha_1+\alpha_2)\over\Gamma(\alpha_1)\Gamma(\alpha_2)}t^{\alpha_1-1}(1-t)^{\alpha_2 - 1},\quad t\in(0,1).$$ A random variable has a Beta prime distribution with shape parameters $\alpha_1, \alpha_2\in(0,\infty)$ if its density function is given by $$g_{\alpha_1,\alpha_2}(t)={\Gamma(\alpha_1+\alpha_2)\over\Gamma(\alpha_1)\Gamma(\alpha_2)}t^{\alpha_1-1}(1+t)^{-\alpha_1-\alpha_2},\quad t>0.$$
We will use the notation $\xi\sim {\mathop{\mathrm{Gamma}}\nolimits}(\alpha,\lambda)$, $\xi\sim {\mathop{\mathrm{Beta}}\nolimits}(\alpha,\beta)$ and $\xi\sim {\mathop{\mathrm{Beta}}\nolimits}^{\prime}(\alpha,\beta)$ to indicate that random variable $\xi$ has a Gamma distribution with shape $\alpha$ and rate $\lambda$, a Beta distribution with shape parameters $\alpha_1, \alpha_2$, or a Beta prime distribution with shape parameters $\alpha_1, \alpha_2$, respectively. Moreover, $\xi\overset{D}{=}\xi'$ will indicate that two random variables $\xi$ and $\xi'$ have the same distribution.
\[thm:ProbabilisticRepresentation\] The following assertions hold.
- For $\beta\geq -1, \nu\ge-1, d\geq 2$ one has that $$\begin{aligned}
\xi(1-\xi)^{d-1}\left((d-1)!{\operatorname{Vol}}(Z_{\beta,\nu})\right)^2&\overset{D}{=} (m_{\beta,d}^{-1}\,\rho)^{{2(d-1)\over d+2\beta+1}}(1-\eta)^{d-1}\prod\limits_{i=1}^{d-1}\xi_i,\label{eq:probBeta}\end{aligned}$$
- and for $\beta\geq (d+1)/2,\, 2\beta - d>\nu\ge -1, d\geq 2$ one has that $$\begin{aligned}
(1+\eta^{\prime})^{d-1}\left((d-1)!{\operatorname{Vol}}(Z^{\prime}_{\beta,\nu})\right)^2&\overset{D}{=} ((m_{\beta,d}^{\prime})^{-1}\,\rho^{\prime})^{{2(d-1)\over d-2\beta+1}}(\xi^{\prime})^{-1}(1+\xi^{\prime})^{d}\prod\limits_{i=1}^{d-1}\xi^{\prime}_i,\label{eq:probBetaPrime}\end{aligned}$$
where $$\begin{aligned}
\xi&\sim{\mathop{\mathrm{Beta}}\nolimits}\Big({d+\nu\over 2}+\beta+1, {(d-1)(d+\nu+2\beta)\over 2}\Big),\qquad \xi^{\prime}\sim{\mathop{\mathrm{Beta}}\nolimits}^{\prime}\Big(\beta-{d+\nu\over 2}, {(d-1)(2\beta - d -\nu)\over 2}\Big)\\
\eta &\sim{\mathop{\mathrm{Beta}}\nolimits}\Big({d+2\beta+1\over 2}, {(d-1)(d+\nu+2\beta)\over 2}\Big),\qquad \eta^{\prime}\sim{\mathop{\mathrm{Beta}}\nolimits}^{\prime}\Big(\beta-{d-1\over 2}, {(d-1)(2\beta - d -\nu)\over 2}\Big)\\
\rho&\sim{\mathop{\mathrm{Gamma}}\nolimits}\Big(d+{(\nu-1)(d-1)\over d+2\beta+1},1\Big),\qquad \rho^{\prime}\sim{\mathop{\mathrm{Gamma}}\nolimits}\Big(d+{(\nu-1)(d-1)\over d-2\beta+1},1\Big)\\
\xi_i&\sim{\mathop{\mathrm{Beta}}\nolimits}\Big({\nu+i+1\over 2}, {d-1-i\over 2}+\beta+1\Big),\qquad \xi^{\prime}_i\sim{\mathop{\mathrm{Beta}}\nolimits}^{\prime}\Big({\nu+i+1\over 2}, \beta-{d+\nu\over 2}\Big),\quad i\in\{1,\ldots,d-1\},\end{aligned}$$ are independent random variables, independent of ${\operatorname{Vol}}(Z_{\beta,\nu})$ and ${\operatorname{Vol}}(Z^{\prime}_{\beta,\nu})$, and $m_{\beta,d}$, $m_{\beta,d}^{\prime}$ are defined in .
Equality generalizes [@GusakovaThaele Theorem 2.6] for $\beta=-1$ to general values of $\beta\geq -1$ and [@GKT17 Theorem 2.5 (b)] for $\nu = -1$ to general $\nu \ge -1$. Equality generalizes [@GKT17 Theorem 2.5 (c)] for $\nu = -1$ to general $2\beta - d>\nu\ge-1$.
First of all let us recall that for $\xi_i\sim{\mathop{\mathrm{Beta}}\nolimits}({\nu+i\over 2}, {d-i\over 2}+1+\beta)$ with $s>-{\nu+1\over 2}$ and $\xi^{\prime}_i\sim{\mathop{\mathrm{Beta}}\nolimits}^{\prime}({\nu+i\over 2}, \beta-{d+\nu\over 2})$ with $-{\nu+1\over 2}< s< \beta -{d+\nu\over 2}$ we have $$\EE[\xi_i^{s}]={\Gamma({d+\nu\over 2}+\beta+1)\Gamma({i+\nu+1\over 2}+s)\over \Gamma({i+\nu+1\over 2})\Gamma({d+\nu\over 2}+\beta+1+s)},\qquad \EE[(\xi_i^{\prime})^{s}]={\Gamma(\beta-{d+\nu\over 2}-s)\Gamma({i+\nu+1\over 2}+s)\over \Gamma(\beta-{d+\nu\over 2})\Gamma({i+\nu+1\over 2})},$$ respectively, and for $\rho\sim{\mathop{\mathrm{Gamma}}\nolimits}(d+{(\nu-1)(d-1)\over d+2\beta+1},1)$ and $\rho^{\prime}\sim{\mathop{\mathrm{Gamma}}\nolimits}(d+{(\nu-1)(d-1)\over d-2\beta+1},1)$ with $s>0$ we have $$\EE\Big[\rho^{{2s(d-1)\over d+2\beta+1}}\Big]={\Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1}+{2s(d-1)\over d+2\beta+1})\over \Gamma(d+{(\nu-1)(d-1)\over d+2\beta+1})},\qquad \EE\Big[(\rho^{\prime})^{{2s(d-1)\over d-2\beta+1}}\Big]={\Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1}+{2s(d-1)\over d-2\beta+1})\over \Gamma(d+{(\nu-1)(d-1)\over d-2\beta+1})}.$$ respectively. Moreover, for $s>-{\nu+1\over 2}$ we compute that $$\EE\Big[\xi^{s}(1-\xi)^{(d-1)s}\Big]={\Gamma({(d-1)(d+\nu+2\beta)\over 2}+s(d-1))\Gamma({d+\nu\over 2}+\beta+1+s)\Gamma({d(d+2\beta+\nu)\over 2}+1)\over\Gamma({(d-1)(d+\nu+2\beta)\over 2})\Gamma({d+\nu\over 2}+\beta+1)\Gamma({d(d+2\beta+\nu)\over 2}+1+sd)}$$ and $$\EE\Big[(1-\eta)^{(d-1)s}\Big]={\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2})\Gamma({(d-1)(d+\nu+2\beta)\over 2}+s(d-1))\over\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2}+s(d-1))\Gamma({(d-1)(d+\nu+2\beta)\over 2})}.$$ Combining this with Theorem \[theo:volume\] we conclude that, for all $s>-{\nu+1\over 2}$, $$(d-1)!^{2s}\,\EE\Big[\xi^{s}(1-\xi)^{(d-1)s}{\operatorname{Vol}}(Z_{\beta,\nu})^{2s}\Big]=m_{\beta,d}^{-{2s(d-1)\over d+2\beta+1}}\EE\Big[\rho^{{2s(d-1)\over d+2\beta+1}}(1-\eta)^{(d-1)s}\prod\limits_{i=1}^{d-1}\xi_i^s\Big],$$ which finishes the proof of . Analogously for $-{\nu+1\over 2}< s< \beta -{d+\nu\over 2}$ we have $$\EE\Big[(\xi^{\prime})^{-s}(1+\xi^{\prime})^{ds}\Big]={\Gamma({(d-1)(2\beta-d-\nu)\over 2}-s(d-1))\Gamma(\beta-{d+\nu\over 2}-s)\Gamma({d(2\beta-d-\nu)\over 2})\over\Gamma({(d-1)(2\beta-d-\nu)\over 2})\Gamma(\beta-{d+\nu\over 2})\Gamma({d(2\beta-d-\nu)\over 2}-sd)}$$ and $$\EE\Big[(1+\eta^{\prime})^{(d-1)s}\Big]={\Gamma({d(2\beta-d)-\nu(d-1)+1\over 2})\Gamma({(d-1)(2\beta-d-\nu)\over 2}-s(d-1))\over\Gamma({d(2\beta-d)-\nu(d-1)+1\over 2}+s(d-1))\Gamma({(d-1)(2\beta-d-\nu)\over 2})}.$$ By Theorem \[theo:volume\] for all $-{\nu+1\over 2}< s< \beta -{d+\nu\over 2}$ we obtain that $$(d-1)!^{2s}\,\EE\Big[(1+\eta^{\prime})^{(d-1)s}{\operatorname{Vol}}(Z_{\beta,\nu}^{\prime})^{2s}\Big]=(m_{\beta,d}^{\prime})^{-{2s(d-1)\over d-2\beta+1}}\EE\Big[\rho^{{2s(d-1)\over d-2\beta+1}}(\xi^{\prime})^{-s}(1+\xi^{\prime})^{ds}\prod\limits_{i=1}^{d-1}(\xi_i^{\prime})^s\Big],$$ and follows.
The next result specifies, for integer values of $\nu$, the connection between the distributions of ${\operatorname{Vol}}(Z_{\beta,\nu})$ and ${\operatorname{Vol}}(Z^{\prime}_{\beta,\nu})$ with those of the volume of a beta-simplex and the volume of a beta-prime-simplex as studied in [@GKT17], respectively.
- For $d\geq 2$, $\beta\geq -1$ and integers $\nu\geq -1$ we have $$(1-\xi)^{d-1}\,{\operatorname{Vol}}(Z_{\beta,\nu})^2\overset{D}{=} (m_{\beta,d}^{-1}\,\rho)^{{2(d-1)\over d+2\beta+1}}{\operatorname{Vol}}\left({\mathop{\mathrm{conv}}\nolimits}(X_0,\ldots, X_{d-1})\right)^2,$$ where $X_0,\ldots, X_{d-1}$ are independent and identically distributed random points in $\BB^{d+\nu}$ whose distribution has density $$c_{d+\nu,\beta}(1-\|x\|)^{\beta},\qquad x\in\BB^{d+\nu},$$ $\xi\sim{\mathop{\mathrm{Beta}}\nolimits}({\nu+1\over 2}, {d(d+2\beta)+\nu(d-1)+1\over 2})$ is independent of ${\operatorname{Vol}}(Z_{\beta,\nu})$ and $\rho\sim{\mathop{\mathrm{Gamma}}\nolimits}(d+{(\nu-1)(d-1)\over d+2\beta+1},1)$ is independent of $X_0,\ldots, X_{d-1}$.
- For $d\geq 2$, $\beta\geq (d+1)/2$ and integers $2\beta - d>\nu\ge -1$ we have $$(1+\xi^{\prime})^{d-1}\,{\operatorname{Vol}}(Z^{\prime}_{\beta,\nu})^2\overset{D}{=} ((m_{\beta,d}^{\prime})^{-1}\,\rho^{\prime})^{{2(d-1)\over d-2\beta+1}}{\operatorname{Vol}}\left({\mathop{\mathrm{conv}}\nolimits}(X^{\prime}_0,\ldots, X^{\prime}_{d-1})\right)^2,$$ where $X^{\prime}_0,\ldots, X^{\prime}_{d-1}$ are independent and identically distributed random points in $\RR^{d+\nu}$ whose distribution has density $$c^{\prime}_{d+\nu,\beta}(1+\|x\|)^{-\beta},\qquad x\in\RR^{d+\nu},$$ $\xi^{\prime}\sim{\mathop{\mathrm{Beta}}\nolimits}^{\prime}({\nu+1\over 2}, {d(2\beta - d -\nu)\over 2})$ is independent of ${\operatorname{Vol}}(Z^{\prime}_{\beta,\nu})$ and $\rho^{\prime}\sim{\mathop{\mathrm{Gamma}}\nolimits}(d+{(\nu-1)(d-1)\over d-2\beta+1},1)$ is independent of $X^{\prime}_0,\ldots, X^{\prime}_{d-1}$.
From [@GKT17 Theorem 2.3 (b)] we have $$\begin{aligned}
(d-1)!^{2s}\EE[{\operatorname{Vol}}\left({\mathop{\mathrm{conv}}\nolimits}(X_0,\ldots, X_{d-1})\right)^{2s}]&={\Gamma({d+2\beta+\nu+2\over 2})^d\over \Gamma({d+2\beta+\nu+2\over 2}+s)^{d}}{\Gamma({d(d+2\beta+\nu)+2\over 2}+ds)\over \Gamma({d(d+\beta+\nu)+2\over 2}+(d-1)s)}\\
&\qquad\qquad\qquad\qquad\times\prod\limits_{i=1}^{d-1}{\Gamma({\nu+1+i\over 2}+s)\over\Gamma({\nu+1+i\over 2})},\end{aligned}$$ for any $\beta-{d+\nu\over 2}>s>0$. Using the equality $$\begin{aligned}
\EE[(1-\xi)^{(d-1)s}]&={\Gamma({d(d+2\beta+\nu)\over 2}+1)\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2}+(d-1)s)\over \Gamma({d(d+2\beta+\nu)\over 2}+1+(d-1)s)\,\Gamma({d(d+2\beta)+\nu(d-1)+1\over 2})},\end{aligned}$$ and combining this with Theorem \[theo:volume\] we conclude, for all $\beta-{d+\nu\over 2}>s>0$, $$\EE\Big[(1-\xi)^{d-1}\,{\operatorname{Vol}}(Z_{\beta,\nu})^2\Big]=\EE\Big[(m_{\beta,d}^{-1}\,\rho)^{{2(d-1)\over d+2\beta+1}}{\operatorname{Vol}}\left({\mathop{\mathrm{conv}}\nolimits}(X_0,\ldots, X_{d-1})\right)^2\Big],$$ and (a) is proven. Analogously, from [@GKT17 Theorem 2.3 (b) and (c)] we have $$\begin{aligned}
&(d-1)!^{2s}\EE[{\operatorname{Vol}}\left({\mathop{\mathrm{conv}}\nolimits}(X^{\prime}_0,\ldots, X^{\prime}_{d-1})\right)^{2s}]\\
&\qquad\qquad={\Gamma(\beta-{d+\nu\over 2}-s)^d\over \Gamma(\beta-{d+\nu\over 2})^{d}}{\Gamma({d(2\beta-d-\nu)\over 2}-(d-1)s)\over \Gamma({d(2\beta-d-\nu)\over 2}-ds)}\prod\limits_{i=1}^{d-1}{\Gamma({\nu+1+i\over 2}+s)\over\Gamma({\nu+1+i\over 2})},\end{aligned}$$ and using that $$\EE[(1+\xi^{\prime})^{(d-1)s}]={\Gamma({d(2\beta-d-\nu)\over 2}-(d-1)s)\Gamma({d(2\beta - d)-\nu(d-1)+1\over 2})\over \Gamma({d(2\beta-d-\nu)\over 2})\,\Gamma({d(2\beta - d)-\nu(d-1)+1\over 2}-s(d-1))},$$ together with Theorem \[theo:volume\] we have for all $s>0$, $$\EE\Big[(1+\xi^{\prime})^{d-1}\,{\operatorname{Vol}}(Z^{\prime}_{\beta,\nu})^2\Big]=\EE\Big[((m_{\beta,d}^{\prime})^{-1}\,\rho^{\prime})^{{2(d-1)\over d-2\beta+1}}{\operatorname{Vol}}\left({\mathop{\mathrm{conv}}\nolimits}(X^{\prime}_0,\ldots, X^{\prime}_{d-1})\right)^2\Big],$$ which finishes the proof of (b) and theorem.
The formula for ${\operatorname{Vol}}(Z_{\beta,\nu})$ is the extension of [@GusakovaThaele Proposition 2.8] to general $\beta\geq -1$.
In [@GKT17 Theorem 2.7] it was shown that the random variable $\xi$ and the random variable $\xi^{\prime}$ in the previous proposition are equal by distribution to the squared distance from the origin to the $(d-1)$-dimensional affine subspace spanned by the random vectors $X_0,\ldots, X_{d-1}$ and by the random vectors $X^{\prime}_0,\ldots, X^{\prime}_{d-1}$, respectively.
Angle sums and face intensities {#sec:AnglesFaceIntensities}
===============================
The aim of this section is to compute the intensity of $j$-dimensional faces in the $\beta$-Delaunay tessellations $\cD_\beta$ and $\cD_{\beta}^\prime$, for all $j\in \{0,\ldots,d-1\}$. Intuitively, the face intensities can be understood as follows. In a stationary random tessellation $\mathcal T$, the expected number of $j$-dimensional faces in a large cube of volume $V$ is asymptotically equivalent to $\gamma_j(\mathcal T)V$, as $V\to\infty$, for a certain constant $\gamma_j(\mathcal T)$, called the [intensity]{} of $j$-dimensional faces of $\mathcal T$. A precise definition, using Palm calculus, will be given below. To evaluate these constants for the tessellations $\cD_\beta$ and $\cD_{\beta}^\prime$, we will first compute the expected angle sums of the volume-power weighted typical cells of $\cD_\beta$ and $\cD_{\beta}^\prime$.
Expected angle sums of weighted typical cells
---------------------------------------------
Let us recall that $Z_{\beta,\nu}$ and $Z_{\beta,\nu}^\prime$ denote the typical cells of $\cD_\beta$ and $\cD_{\beta}^\prime$ weighted by the $\nu$-th power of their volume. Our aim is to compute the expected angle sums of these random simplices. First we need to introduce the necessary notation.
Given a simplex $T := {\mathop{\mathrm{conv}}\nolimits}(Z_1,\ldots,Z_d) \subset \RR^{d-1}$, we denote by $\sigma_k(T)$ the **sum of internal angles** of $T$ at all its $k$-vertex faces of the form ${\mathop{\mathrm{conv}}\nolimits}(Z_{i_1},\ldots, Z_{i_k})$, that is $$\sigma_k(T) = \sum_{\substack{1\leq i_1<\ldots< i_k\leq d\\F := {\mathop{\mathrm{conv}}\nolimits}(Z_{i_1},\ldots,Z_{i_k})}} \beta(F,T), \qquad k\in \{1,\ldots,d\}.$$ Here, $\beta(F,T)$ is the internal angle of $T$ at its face $F$ normalized in such a way that the angle of the full space is $1$, see [@SW p. 458]. If $Z_1,\ldots,Z_d$ are $d$ i.i.d. random points in $\BB^{d-1}$ distributed according to the beta density $$f_{d-1,\beta}(z) = c_{d-1,\beta} (1-\|z\|^2)^{\beta}, \qquad z\in \BB^{d-1},$$ then ${\mathop{\mathrm{conv}}\nolimits}(Z_1,\ldots,Z_d)$ is called the beta simplex with parameter $\beta>-1$. The beta simplex with parameter $\beta=-1$ is defined as ${\mathop{\mathrm{conv}}\nolimits}(Z_1,\ldots,Z_d)$, where $Z_1,\ldots,Z_d$ are i.i.d. uniform on the unit sphere $\SS^{d-2}$. The expected angle sums of these simplices, denoted by $$\mathbb J_{d,k}(\beta) : = \EE \sigma_k({\mathop{\mathrm{conv}}\nolimits}(Z_1,\ldots,Z_d)), \qquad k\in \{1,\ldots,d\},$$ have been computed in [@kabluchko_formula], see Theorem 1.2 and the discussion thereafter. According to this formula, we have $$\label{eq:J_nk_integral}
\mathbb J_{d,k}\left(\frac{\alpha-d+1}{2}\right)
=
\binom dk \int_{-\infty}^{+\infty} c_{\frac{\alpha d}2} (\cosh u)^{-\alpha d - 2}
\left(\frac 12 + {{\rm{i}}}\int_0^u c_{\frac{\alpha-1}{2}} (\cosh v)^{\alpha}{{\rm d}}v \right)^{d-k} {{\rm d}}u,
$$ for all $d\geq 3$, $k\in \{1,\ldots,d\}$ and $\alpha \geq d-3$, where $$\label{eq:c_beta}
c_{\gamma} := c_{1,\gamma} = \frac{ \Gamma\left(\gamma + \frac{3}{2} \right) }{ \sqrt \pi\, \Gamma (\gamma+1)}, \qquad \gamma>-1.$$ Similarly, let $Z_1^\prime,\ldots,Z_d^\prime$ be $d$ i.i.d. random points in $\RR^{d-1}$ distributed according to the beta$^\prime$ density $$f^\prime_{d-1,\beta}(z) = c_{d-1,\beta}^\prime (1+\|z\|^2)^{-\beta}, \qquad z\in \RR^{d-1}.$$ Then, ${\mathop{\mathrm{conv}}\nolimits}(Z_1^\prime,\ldots,Z_d^\prime)$ is called the beta simplex with parameter $\beta> \frac{d-1}{2}$. The expected angle sums of the beta$^\prime$ simplices, denoted by $$\mathbb J_{d,k}^\prime(\beta) : = \EE \sigma_k({\mathop{\mathrm{conv}}\nolimits}(Z_1^\prime,\ldots,Z_d^\prime)), \qquad k\in \{1,\ldots,d\},$$ have been computed in [@kabluchko_formula], see Theorem 1.7 and the discussion thereafter: $$\label{eq:J_nk_integral_prime}
\mathbb J_{d,k}^\prime\left(\frac{\alpha+d-1}{2}\right)
=
\binom dk \int_{-\infty}^{+\infty} c_{\frac{\alpha d}2}^\prime (\cosh u)^{-(\alpha d - 1)}
\left(\frac 12 + {{\rm{i}}}\int_0^u c_{\frac{\alpha+1}{2}}^\prime (\cosh v)^{\alpha-1}{{\rm d}}v \right)^{d-k} {{\rm d}}u,
$$ for all $d\in\NN$, $k\in \{1,\ldots,d\}$ and for all $\alpha >0$ such that $\alpha d >1$. Here, $$\label{eq:c_beta}
c_{\gamma}^\prime := c_{1,\gamma}^\prime = \frac{ \Gamma (\gamma) }{ \sqrt \pi\, \Gamma \left(\gamma - \frac 12\right)}, \qquad \gamma > \frac 12.$$
We are now going to state a formula for the expected angle sums of the typical cells $Z_{\beta,\nu}$ and $Z_{\beta,\nu}^\prime$. Note that we include the case of $Z_{-1,\nu}$ which is interpreted as the $\nu$-weighted typical cell in the classical Poisson-Delaunay tessellation; see Remark \[rem:rep\_typical\_beta\_-1\].
\[theo:angle\_sum\_cell\] Let $Z_{\beta,\nu}$ be the $\nu$-weighted typical cell of the $\beta$-Delaunay tessellation with $\beta\geq -1$ and integer $\nu\ge -1$. Also, let $Z_{\beta,\nu}^\prime$ be the $\nu$-weighted typical cell of the $\beta^\prime$-Delaunay tessellation with $\beta > (d+1)/2$ and integer $\nu$ such that $2\beta-d > \nu \ge-1$. Then, for all $k\in \{1,\ldots,d\}$, $$\begin{aligned}
\EE \sigma_k(Z_{\beta,\nu})
&=
\mathbb J_{d,k}\left(\beta + \frac {\nu+1} 2\right)\\
&=
\binom dk \int_{-\infty}^{+\infty} c_{\frac{\alpha d}2} (\cosh u)^{-\alpha d - 2}
\left(\frac 12 + {{\rm{i}}}\int_0^u c_{\frac{\alpha-1}{2}} (\cosh v)^{\alpha}{{\rm d}}v \right)^{d-k} {{\rm d}}u,
\\
\EE \sigma_k(Z_{\beta,\nu}^\prime)
&=
\mathbb J_{d,k}^\prime\left(\beta - \frac {\nu+1} 2\right)\\
&=
\binom dk \int_{-\infty}^{+\infty} c_{\frac{\alpha^\prime d}2}^\prime (\cosh u)^{-(\alpha^\prime d - 1)}
\left(\frac 12 + {{\rm{i}}}\int_0^u c_{\frac{\alpha^\prime-1}{2}}^\prime (\cosh v)^{\alpha^\prime-1}{{\rm d}}v \right)^{d-k} {{\rm d}}u,\end{aligned}$$ where $\alpha = 2\beta + \nu + d$ and $\alpha' = 2\beta - \nu - d$.
We consider the $\beta$-Delaunay case. Let first $\beta>-1$. Since rescaling does not change angle sums, it follows from Remark \[rem:rep\_typical\] that $$\EE \sigma_k(Z_{\beta,\nu}) = \EE \sigma_k({\mathop{\mathrm{conv}}\nolimits}(Y_1,\ldots,Y_d)),$$ where $(Y_1,\ldots,Y_d)$ are $d$ random points in the unit ball $\BB^{d-1}$ whose joint density is proportional to $$\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1} \, \prod\limits_{i=1}^d(1-\|y_i\|^2)^{\beta},
\qquad
y_1\in \BB^{d-1},\ldots,y_d\in \BB^{d-1}.$$ On the other hand, let $Y_1^*,\ldots,Y_d^*$ be $d$ i.i.d. random points in $\BB^{d-1}$ with joint density proportional to $$\prod\limits_{i=1}^d(1-\|y_i^*\|^2)^{\beta+ \frac {\nu+1} 2}.$$ By Remark 4.2 of [@beta_polytopes], we have $$\EE \sigma_k({\mathop{\mathrm{conv}}\nolimits}(Y_1,\ldots,Y_d)) = \EE \sigma_k({\mathop{\mathrm{conv}}\nolimits}(Y_1^*,\ldots,Y_d^*)).$$ The expected angle sums of ${\mathop{\mathrm{conv}}\nolimits}(Y_1^*,\ldots,Y_d^*)$ are $$\label{eq:J_nk_integral_repeat}
\EE \sigma_k({\mathop{\mathrm{conv}}\nolimits}(Y_1^*,\ldots,Y_d^*))
=
\mathbb J_{d,k}\left(\beta + \frac {\nu+1} 2\right),
$$ which is given by with $\alpha = 2\beta + \nu + d$. Taking everything together, proves the theorem in the $\beta$-Delaunay case with $\beta>-1$.
For the $\beta$-Delaunay case with $\beta=-1$ (where $Z_{-1,\nu}$ is interpreted as the $\nu$-weighted cell in the classical Poisson-Delaunay tessellation), the starting point is Remark \[rem:rep\_typical\_beta\_-1\] which implies that $$\EE \sigma_k(Z_{-1,\nu}) = \EE \sigma_k({\mathop{\mathrm{conv}}\nolimits}(Y_1,\ldots,Y_d)),$$ where $(Y_1,\ldots,Y_d)$ are $d$ random points in the unit sphere $\SS^{d-2}$ whose joint probability law is proportional to $$\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1} \sigma_{d-2}({{\rm d}}y_1)\ldots \sigma_{d-2}({{\rm d}}y_d),
\qquad
y_1\in \SS^{d-2},\ldots, y_d\in \SS^{d-2}.$$ The rest of the proof is similar to the case $\beta>-1$. The $\beta^\prime$-case is similar as well.
We stated Theorem \[theo:angle\_sum\_cell\] for integer $\nu$ only because this assumption is required by the method of proof of Remark 4.2 in [@beta_polytopes]. Specifying Theorem \[theo:angle\_sum\_cell\] to $\nu=0$, respectively $\nu=1$, we obtain the expected angle sums of the typical cell, respectively, the cell containing the origin, of the $\beta$-Delaunay tessellation, for $\beta\geq -1$. For the typical cell ($\nu=0$) of the classical Poisson-Delaunay tessellation in $\RR^{d-1}$ (corresponding to $\beta=-1$), the angle sum is given by $$\EE \sigma_k (Z_{-1,0}) = \mathbb J_{d,k}\left(-\frac 12 \right), \qquad k\in \{1,\ldots,d\}.$$ Applying to the quantity on the right-hand side Theorem 1.3 of [@kabluchko_formula], we arrive at the following result.
\[theo:angle\_sum\_cell\_beta\_-1\] Let $Z=Z_{-1,0}$ be the typical cell of the classical Poisson-Delaunay tessellation $\cD_{-1}$ in $\RR^{d-1}$. Then, for all $k\in \{1,\ldots,d\}$ such that $(d-1)(k-1)$ is even, we have $$\EE \sigma_{k}(Z) =
\binom{d}{k} \left(\frac{\Gamma(\frac{d}{2})}{\sqrt{\pi}\, \Gamma(\frac{d-1}{2})}\right)^{d-k} \cdot \frac{\sqrt \pi\, \Gamma(\frac{(d-1)^2+2}{2}) }{\Gamma(\frac{(d-1)^2+1}{2})}
\cdot {\mathop{\mathrm{Res}}\nolimits}\limits_{x=0} \left[\frac{\left(\int_{0}^x (\sin y)^{d-2} {{\rm d}}y\right)^{d-k}}{(\sin x)^{(d-1)^2+1}}\right].$$
In the case when both $d$ and $k$ are even, Proposition 1.4 of [@kabluchko_formula] yields a formula for $\EE \sigma_k(Z)$ which is more complicated than the one given in Theorem \[theo:angle\_sum\_cell\_beta\_-1\].
Behaviour of $\beta$-Delaunay cells and their expected angle sums as $\beta\to\infty$
-------------------------------------------------------------------------------------
\[fig:ConvergenceToRegular\] {width="0.5\columnwidth"}
Figure \[fig:ConvergenceToRegular\] shows the numerical values for the expected angle sums $\EE\sigma_1(Z_{\beta,\nu})$ of the $\nu$-weighted typical $\beta$-Delaunay simplex in $\RR^{d-1}$ for $\beta\in\{-1,0,\ldots,20\}$, with $\nu=0$ and $d=4$. It suggests that, as $\beta$ grows, $\EE\sigma_1(Z_{\beta,\nu})$ approaches the value ${3\over \pi} \arccos \frac 13 -1\approx 0.1755$, which is the angle sum $\sigma_1(\Sigma_3)$ of a regular simplex $\Sigma_3$ in $\RR^3$. The next proposition confirms this conjecture in full generality, that is, for general dimensions $d$, weights $\nu$, and $k\in\{1,\ldots,d\}$. Moreover, it states that the weak limit of $Z_{\beta,\nu}$ as $\beta\to\infty$ is the volume -weighted Gaussian simplex whose angle sums, by coincidence, are the same as for the regular simplex. We will come back to this behaviour in part II of this series of papers, where we shall describe the limit of the whole $\beta$-Delaunay tessellations as $\beta\to\infty$.
Fix $d\geq 2$ and $\nu>-1$. Then, as $\beta\to\infty$, the distribution of $\sqrt{2\beta}\, Z_{\beta,\nu}$ as well as that of $\sqrt{2\beta}\, Z_{\beta,\nu}^\prime$, converges weakly on the space $\mathcal C'$ to the distribution of the volume-power weighted Gaussian random simplex ${\mathop{\mathrm{conv}}\nolimits}(G_1,\ldots, G_d)$, where $(G_1,\ldots,G_d)$ are $d$ random points in $\RR^{d-1}$ with the joint density given by $$\frac{(d-1)!^{\nu+1}}{ d^{\frac{\nu+1}2} 2^{(\nu+1)(d-1)/2}}
\left(\prod\limits_{i=1}^{d-1} {\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})}\right)
\Delta_{d-1}(g_1,\ldots,g_d)^{\nu+1}
\left(\frac {1}{\sqrt{2\pi}}\right)^{d(d-1)} \prod_{i=1}^d e^{-\|g_i\|^2/2},
$$ for $g_1\in\RR^{d-1},\ldots,g_d\in\RR^{d-1}$. Also, if $\nu\ge-1$ is an integer, then for all $k\in\{1,\ldots,d\}$ one has that $$\lim_{\beta\to\infty}\EE\sigma_k(Z_{\beta,\nu})
=
\lim_{\beta\to\infty}\EE\sigma_k(Z_{\beta,\nu}^\prime)
=
\sigma_k(\Sigma_{d-1}),$$ where $\Sigma_{d-1}$ stands for a regular simplex with $d$ vertices in $\RR^{d-1}$.
For concreteness, we consider the $\beta$-case. From Remark \[rem:rep\_typical\] and Equation we know that $Z_{\beta,\nu}$ has the same distribution as the random simplex ${\mathop{\mathrm{conv}}\nolimits}(RY_1,\ldots,RY_d)$, where
- $R$ is a random variable with density proportional to $r^{2d\beta+d^2+\nu(d-1)}e^{-m_{d,\beta}r^{d+1+2\beta}}$, $r\in(0,\infty)$,
- $Y_1,\ldots,Y_d$ are random vectors with joint density equal to $$\begin{gathered}
f(y_1,\ldots,y_d) := \left(
{1\over (d-1)!^{\nu+1}c_{d-1,\beta}^d}{\Gamma({d+1\over 2}+\beta)^d\over \Gamma({d+\nu\over 2}+\beta+1)^d}{\Gamma({d(d+\nu+2\beta)\over 2} +1)\over \Gamma({d(d+\nu+2\beta)-\nu +1 \over 2})}
\prod\limits_{i=1}^{d-1} {\Gamma({i+\nu+1\over 2})\over \Gamma({i\over 2})}\right)^{-1}
\\
\times\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}\prod_{i=1}^d(1-\|y_i\|^2)^\beta,\qquad y_1\in\BB^{d-1},\ldots,y_d\in\BB^{d-1},\end{gathered}$$
- $R$ is independent from $(Y_1,\ldots,Y_d)$.
Let us first show that $R\to 1$ in probability, as $\beta\to\infty$. Define $\alpha:=d+{(\nu-1)(d-1)\over d+1+2\beta}$, let $Z\sim\Gamma(\alpha,1)$ and observe that $R$ and $(m_{d,\beta}^{-1}Z)^{1/(d+1+2\beta)}$ are identically distributed. We thus compute that $$\begin{aligned}
\EE[R] &= {1\over m_{d,\beta}^{1/(d+1+2\beta)}}\EE[Z^{1/(d+1+2\beta)}] \\
&=m_{d,\beta}^{-1/(d+1+2\beta)}{1\over\Gamma(\alpha)}\int_0^\infty z^{1/(d+1+2\beta)}z^{\alpha-1}e^{-z}\,{\textup{d}}z\\
&= \Bigg({2\sqrt{\pi}\Gamma({d\over 2}+\beta+{3\over 2})\over\Gamma({d\over 2}+\beta+1)}\Bigg)^{1\over d+1+2\beta}{\Gamma(\alpha+{1\over d+1+2\beta})\over \Gamma(\alpha)}=1+O(\beta^{-1}\log\beta)\end{aligned}$$ and, similarly, $$\begin{aligned}
\VV[R] &= \Bigg({2\sqrt{\pi}\Gamma({d\over 2}+\beta+{3\over 2})\over\Gamma({d\over 2}+\beta+1)}\Bigg)^{2\over d+1+2\beta}\Bigg({\Gamma(\alpha+{2\over d+1+2\beta})\over \Gamma(\alpha)}-{\Gamma(\alpha+{1\over d+1+2\beta})^2\over \Gamma(\alpha)^2}\Bigg)={\psi^{(1)}(d+1)\over 4\beta^2}+O(\beta^{-3})\end{aligned}$$ as $\beta\to\infty$, by a multiple application of the asymptotic expansion of the gamma function (here, $\psi^{(1)}$ stands for the first polygamma function, that is, the first derivative of $\ln\Gamma(x)$). As a consequence, using Chebyshev’s inequality we have that, for any $\varepsilon>0$, $$\begin{aligned}
\PP(|R-\EE[R]|>\varepsilon) \leq {\VV[R]\over\varepsilon^2} \to 0,\end{aligned}$$ as $\beta\to\infty$. In other words, $R-\EE[R]$ converges to zero in probability, as $\beta\to\infty$. Since $\EE[R]\to 1$, we also have that $R$ converges in probability to the constant random variable $1$, as $\beta\to\infty$, by Slutsky’s theorem.
Next, we claim that $\sqrt{2\beta}(Y_1,\ldots,Y_d)$ converges in distribution, as $\beta\to\infty$, to a $d$-tuple $(G_1,\ldots,G_d)$ of random vectors in $\RR^{d-1}$ with certain joint density which we will compute. Indeed, the density of $\sqrt{2\beta}(Y_1,\ldots,Y_d)$ is given by $$\left(\frac 1 {\sqrt{2\beta}}\right)^{d(d-1)}f\left(\frac{y_1}{\sqrt{2\beta}},\ldots,\frac{y_d}{\sqrt{2\beta}}\right)$$ We now let $\beta\to\infty$. Using that $(1-\|y\|^2/(2\beta))^\beta\to e^{-\|y\|^2/2}$ and the standard asymptotics $\Gamma(\beta + c_1)/\Gamma(\beta+c_2) \sim \beta^{c_1-c_2}$, we obtain that the above density converges pointwise to $$\frac{(d-1)!^{\nu+1}}{ d^{\frac{\nu+1}2} 2^{(\nu+1)(d-1)/2}}
\left(\prod\limits_{i=1}^{d-1} {\Gamma({i\over 2})\over \Gamma({i+\nu+1\over 2})}\right)
\Delta_{d-1}(y_1,\ldots,y_d)^{\nu+1}
\left(\frac {1}{\sqrt{2\pi}}\right)^{d(d-1)} \prod_{i=1}^d e^{-\|y_i\|^2/2}.$$ By Scheffé’s lemma, the tuple $\sqrt{2\beta}(Y_1,\ldots,Y_d)$ converges weakly to the tuple $(G_1,\ldots,G_d)$ with the above joint density. A related result without the volume-power weighting can be found in Lemma 1.1 in [@beta_polytopes].
Using now Slutsky’s theorem again together with the continuous mapping theorem we conclude that, as $\beta\to\infty$, the random simplex $$\sqrt{2\beta}Z_{\beta,\nu}=\sqrt{2\beta}{\mathop{\mathrm{conv}}\nolimits}(RY_1,\ldots,RY_d)$$ converges in distribution (on the space of convex bodies in $\RR^{d-1}$ supplied with the Hausdorff distance) to a weighted Gaussian simplex ${\mathop{\mathrm{conv}}\nolimits}(G_1,\ldots,G_d)$; the continuity of the involved map is guaranteed by [@SW Theorem 12.3.5]. Using once again the continuous mapping theorem, this implies that, as $\beta\to\infty$, $$\sigma_k(\sqrt{2\beta}Z_{\beta,\nu}) \overset{d}{\longrightarrow}\sigma_k({\mathop{\mathrm{conv}}\nolimits}(G_1,\ldots,G_d))$$ for all $k\in\{1,\ldots,d\}$, since angle sums are invariant under rescaling. As they are also bounded, the sequence of random variables $\sigma_k(\sqrt{2\beta}Z_{\beta,\nu})$, $\beta>-1$, is uniformly integrable and we have that $$\lim_{\beta\to\infty}\EE[\sigma_k(\sqrt{2\beta}Z_{\beta,\nu})] = \EE[\sigma_k({\mathop{\mathrm{conv}}\nolimits}(G_1,\ldots,G_d))].$$ However, by taking the limit $\beta\to\infty$ in [@beta_polytopes Remark 4.2] we have that the expected angle sum of the weighted Gaussian simplex ${\mathop{\mathrm{conv}}\nolimits}(G_1,\ldots,G_d)$ coincides with the expected angle sum of a standard (unweighted) Gaussian simplex ${\mathop{\mathrm{conv}}\nolimits}(N_1,\ldots,N_d)$, where $N_1,\ldots,N_d$ are are i.i.d. standard Gaussian random vectors in $\RR^{d-1}$. Finally, let $\Sigma_{d-1}$ be a regular simplex in $\RR^{d-1}$ and recall from [@GoetzeKabluchkoZap; @GaussianSimplexAngles] that the expected angle sum $\EE[\sigma_k(N_1,\ldots,N_d)]$ coincides with $\sigma_k(\Sigma_{d-1})$. We have thus shown that $$\begin{aligned}
\lim_{\beta\to\infty}\EE\sigma_k(Z_{\beta,\nu}) &=\lim_{\beta\to\infty}\EE[\sigma_k(\sqrt{2\beta}Z_{\beta,\nu})] \\
&= \EE[\sigma_k({\mathop{\mathrm{conv}}\nolimits}(G_1,\ldots,G_d))]\\
&= \EE[\sigma_k({\mathop{\mathrm{conv}}\nolimits}(N_1,\ldots,N_d))]\\
&= \sigma_k(\Sigma_{d-1}),\end{aligned}$$ and the proof is complete.
Face intensities in $\beta$-tessellations {#subsec:intensities}
-----------------------------------------
Given a stationary tessellation $\cT$ on $\RR^{d-1}$, one can introduce the notion of face intensities for faces of all dimensions $j\in\{0,\ldots, d-1\}$ as follows [@SW p. 450 and § 4.1]. Fix some center function $z:\cC' \to \RR^{d-1}$. Let $\cF_{j}(\cT)$ be the set of all $j$-dimensional faces of the cells of the tessellation $\cT$. By convention, each face is counted once even if it is a face of two or more cells. Consider the point process $$\pi_{j}(\cT) := \sum_{F\in \cF_j(\cT)} \delta_{z(F)}$$ on $\RR^{d-1}$ and note that it is stationary because the center function is required to be translation invariant. The **intensity** of $j$-dimensional cells of $\cT$ is just the intensity of this point process, that is $$\gamma_j(\cT) := \EE \sum_{F\in \cF_j(\cT)} {\bf 1}_{[0,1]^{d-1}}(z(F)).$$ In the next theorem we compute the cell intensities in the $\beta$- and $\beta^\prime$-Delaunay tessellations $\cD_{\beta}$ and $\cD_\beta^\prime$ on $\RR^{d-1}$.
\[theo:cell\_intensities\] For all $j\in\{0,\ldots, d-1\}$ and $\beta\geq -1$ (in the $\beta$-case) or $\beta>(d+1)/2$ (in the $\beta^\prime$-case), we have $$\gamma_j(\cD_{\beta}) = \frac {\mathbb J_{d,j+1}\left(\beta + \frac {1} 2\right)} {\EE {\operatorname{Vol}}(Z_{\beta,0})},
\qquad
\gamma_j(\cD_{\beta}^\prime) = \frac {\mathbb J_{d,j+1}^\prime\left(\beta - \frac {1} 2\right)} {\EE {\operatorname{Vol}}(Z_{\beta,0}^\prime)},$$ where $\EE {\operatorname{Vol}}(Z_{\beta,0}^{(\prime)})$ is as in Theorem \[theo:volume\] and $$\begin{aligned}
\mathbb J_{d,j+1}\left(\beta + \frac {1} 2\right)
&=
\binom d{j+1} \int_{-\infty}^{+\infty} c_{\frac{(2\beta + d) d}2} (\cosh u)^{- (2\beta + d) d - 2}\\
&\hspace{3cm}\times\left(\frac 12 + {{\rm{i}}}\int_0^u c_{\frac{2\beta + d - 1}{2}} (\cosh v)^{2\beta + d}{{\rm d}}v \right)^{d-j-1} {{\rm d}}u,\\
\mathbb J_{d,j+1}^\prime\left(\beta - \frac {1} 2\right)
&=
\binom d{j+1} \int_{-\infty}^{+\infty} c_{\frac{(2\beta-d) d}2}^\prime (\cosh u)^{-(2\beta-d) d + 1}\\
&\hspace{3cm}\times
\left(\frac 12 + {{\rm{i}}}\int_0^u c_{\frac{2\beta - d - 1}{2}}^\prime (\cosh v)^{2\beta-d-1}{{\rm d}}v \right)^{d-j-1} {{\rm d}}u.\end{aligned}$$
For concreteness, we consider the beta case. According to Theorem 10.1.3 of [@SW], the cell intensities of $\cD_{\beta}$ satisfy $$\label{eq:angle_sums_intensities}
\gamma_j(\cD_{\beta}) = \gamma_{d-1} (\cD_{\beta}) \cdot \EE \sigma_{j+1}(Z_{\beta,0}), \qquad j\in \{0,\ldots,d-1\},$$ where $Z_{\beta,0}$ is the typical cell of the tessellation $\cD_{\beta}$ (that is, a random simplex distributed according to $\PP_{\beta,0}$). The intensity of the cells of maximal dimension $d-1$ in known to satisfy $$\gamma_{d-1}(\cD_{\beta}) = \frac 1 {\EE {\operatorname{Vol}}(Z_{\beta,0})},$$ see [@SW Equation (10.4)]. On the other hand, by Theorem \[theo:angle\_sum\_cell\], $$\begin{aligned}
\EE \sigma_{j+1}(Z_{\beta,0})
&=
\mathbb J_{d,j+1}\left(\beta + \frac {1} 2\right),
$$ and the same theorem yields also an explicit expression for $\mathbb J_{d,j+1}(\beta + \frac {1} 2)$. Taking these three equations together completes the proof in the $\beta$-case. The $\beta^\prime$-case is similar.
Using duality, we can also compute the face intensities of the $\beta$- and $\beta^\prime$-Voronoi tessellations.
\[prop:duality\_face\_intensities\] The face intensities of $\cD_{\beta}^{(\prime)}$ and $\cV_{\beta}^{(\prime)}$ are related via $$\gamma_{k-1}(\cD_{\beta}^{(\prime)}) = \gamma_{d-k}(\cV_{\beta}^{(\prime)}),
\qquad
k\in \{1,\ldots,d\}.$$
Since the $\beta^{(')}$-Voronoi tessellation $\cV_{\beta}^{(\prime)}$ is dual to the $\beta^{(\prime)}$-Delaunay tessellation $\cD_{\beta}^{(\prime)}$, each $(k-1)$-dimensional face of $\cD_\beta^{(\prime)}$ corresponds to a $(d-k)$-dimensional face of $\cV_\beta^{(')}$, and the claim follows.
Expected face numbers of the typical $\beta$-Voronoi cell
---------------------------------------------------------
In the next theorem we compute the expected **$f$-vector** of the typical cell of the $\beta$- and $\beta^\prime$-Voronoi tessellations $\cV_{\beta}$ and $\cV_\beta^\prime$.
\[theo:typical\_beta\_poi\_vor\_f\_vect\] Let $Y_{\beta}^{(\prime)}$ be the typical cell of the $\beta^{(\prime)}$-Voronoi tessellation $\cV_{\beta}^{(\prime)}$ in $\RR^{d-1}$, where, as usual, $\beta \geq -1$ in the $\beta$-case and $\beta>(d+1)/2$ in the $\beta^\prime$-case. Then, for all $k\in \{1,\ldots,d\}$, $$\EE f_{d-k}(Y_{\beta}) = k \gamma_{k-1} (\cD_\beta) = \frac {k \mathbb J_{d,k}\left(\beta + \frac {1} 2\right)} {\EE {\operatorname{Vol}}(Z_{\beta,0})},
\qquad
\EE f_{d-k}(Y_{\beta}^\prime) = k \gamma_{k-1} (\cD_\beta^\prime) = \frac {k \mathbb J_{d,k}^\prime\left(\beta - \frac {1} 2\right)} {\EE {\operatorname{Vol}}(Z_{\beta,0}^\prime)}.$$
Let us consider the $\beta$-case. Note that the $\beta$-Voronoi tessellation $\cV_\beta$ is normal by Theorem 10.2.3 of [@SW] (for $\beta=-1$) or by Lemmas \[lem:voronoi\_normal\] and \[lem:properties\_satisfied\] (for $\beta>-1$). Hence, Theorem 10.1.2 of [@SW] implies that $$\gamma_{d-k} (\cV_{\beta}) = \frac1 {k} \EE f_{d-k}(Y_{\beta}).$$ By Proposition \[prop:duality\_face\_intensities\], we also have $\gamma_{k-1}(\cD_{\beta}) = \gamma_{d-k}(\cV_{\beta}$. Taking these equalities together and recalling Theorem \[theo:cell\_intensities\] yields the claim in the $\beta$-case. The $\beta^\prime$-case is similar.
For $\beta=-1$, $Y_{-1}$ is the typical cell in the classical Poisson-Voronoi tessellation in $\RR^{d-1}$. The expected $f$-vector of $Y_{-1}$ has been determined in [@kabluchko_formula Theorem 2.8], where $Y_{-1}$ was denoted by $\cV_{d-1}$. In [@beta_polytopes], we showed that the expected $f$-vector of $Y_{-1}$ is related to the angle sums of $\beta^\prime$-simplices, whereas the above theorem expresses it in terms of the values $\mathbb J_{d,k}( - \frac {1} 2)$ originating from $\beta$-simplices. For the typical Voronoi cell on the sphere, there also exist similar representations in terms of, both, $\beta$- and $\beta^\prime$-simplices [@kabluchko_thaele_voronoi_sphere].
Acknowledgement {#acknowledgement .unnumbered}
---------------
We would like to thank Claudia Redenbach (Kaiserslautern) for pointing us to the *CGAL Project* in order to create the simulations shown in Figure \[fig:beta-tessellations\] and Figure \[fig:betaprime-tessellations\]. AG was partially supported by the the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 *High-dimensional Phenomena in Probability – Fluctuations and Discontinuity*.
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|
---
abstract: 'Soft x-ray emission and absorption spectroscopy of the O K-edge are employed to investigate the electronic structure of wurtzite ZnO(0001). A quasiparticle band structure calculated within the *GW* approximation agrees well with the data, most notably with the energetic location of the Zn 3$d$ – O 2$p$ hybridized state and the anisotropy of the absorption spectra. Dispersion in the band structure is mapped using the coherent **k**-selective part of the resonant x-ray emission spectra. We show that a more extensive mapping of the bands is possible in the case of crystalline anisotropy such as that found in ZnO.'
author:
- 'A. R. H. Preston'
- 'B. J. Ruck'
- 'L. F. J. Piper'
- 'A. DeMasi'
- 'K. E. Smith'
- 'A. Schleife'
- 'F. Fuchs'
- 'F. Bechstedt'
- 'J. Chai'
- 'S. M. Durbin'
title: 'Band structure of ZnO from resonant x-ray emission spectroscopy'
---
Introduction
============
The wide band gap semiconductor zinc oxide (ZnO) is a material with much potential in future electronic devices. [@Science315.1377] Despite significant interest in its fundamental electronic properties, [@JApplPhys.98.041301] few spectroscopic studies of the full band structure exist. A small number of angle-resolved photoemission spectroscopy (ARPES) studies have been reported, [@PhysRevB.26.3144; @JPhysCondMat.17.1271; @JApplPhys.98.041301] yielding some agreement with density functional theory-based (DFT) calculations and providing information on surface and defect states. However, ARPES is limited to probing occupied states, and charging effects, surface preparation and sample quality are serious issues that can limit the accuracy of results. For example, photoemission measurements have variously located the fully occupied Zn 3[*[d]{}*]{} semi-core levels at energies ranging from 7.5 to 8.8 eV below the Fermi level. [@PhysRevLett.27.97; @JPhysCondMat.17.1271; @PhysRevB.26.3144; @PhysRevB.5.2296; @PhysRevB.9.600; @JApplPhys.98.041301] The close proximity of the $3d$ level to the O 2[*[p]{}*]{}-derived valence band has an appreciable impact on band structure calculations. [@PhysRevB.52.13975; @PhysRevB.47.6971; @PhysRevB.37.8958; @PhysRevB.52.R14316; @PhysRevB.73.245212; @PhysRevB.74.045202] As a result, there is a need for bulk sensitive measurements of the ZnO shallow core-level and valence band dispersion.
Here, we present angular dependent resonant x-ray emission spectroscopy (RXES; also known as resonant inelastic x-ray scattering – RIXS) of the O K-edge of wurtzite ZnO crystals, for comparison with hybrid DFT calculations and quasiparticle energies obtained within the *GW* framework. Earlier studies used x-ray absorption and emission spectroscopy (XAS and XES) to directly measure the O 2[*[p]{}*]{} partial density of states (PDOS) of nano-structured ZnO [@PhysRevB.68.165104; @PhysRevB.70.195325] and its anisotropy in the conduction band. [@JPhysCondMat.17.235] We go further by exploiting the dipole- and **k**-selection rules of RXES to map the anisotropic valence band of single-crystal ZnO.
RXES band mapping has been reported in simple materials like graphite, BN, SiC, and more,[@PhysRevLett.74.1234; @PhysRevLett.76.4054; @JElecSpecRelPhe.110.335] and reports on more complicated compounds related to ZnO are appearing.[@PhysRevB.72.085221; @PhysRevB.73.115212; @PhysRevB.75.165207; @PhysRevB.77.125204] However, to our knowledge there is no literature examining the K-edge electronic structure of a post transition metal oxide in the detail we present, or on any material using our image-based technique. Further, we show crystal anisotropy to be a useful tool, allowing a larger part of the Brillouin zone to be uniquely accessed via RXES. Finally, compelling experimental evidence for use of the final-state rule in RXES is presented.
Experiment and Theory
=====================
The sample, a 500 nm ZnO epilayer, was grown on epi-ready (0001) sapphire substrates by plasma-assisted MBE, as described previously. [@JElecMat.35.1316] As the sample was part of a separate study on doping, approximately 0.01% Ag was incorporated during growth, resulting in a layer averaged electron concentration of $7.3\times 10^{16}$ cm$^{-3}$. The high crystalline quality of the film was confirmed by RBS under the channeling condition, [@JElecMat.36.472] a streaky RHEED pattern, and by low temperature photoluminescence, which shows donor bound exciton peaks are dominant with up to three LO phonon replicas.
The x-ray spectroscopy was performed on the undulator beamline X1B at the National Synchrotron Light Source at Brookhaven National Laboratory, which is equipped with a spherical grating monochromator and a Nordgren-type emission spectrometer. The energy resolution over the O K-edge was approximately 0.20 eV and 0.37 eV for the XAS and RXES respectively. XAS were recorded in total electron yield mode. The photon energy was calibrated using the O K-edge and Ti L-edges of rutile TiO$_2$ measured during the experiment. The XES was calibrated with the 2nd order L-edge of Zn from both a calibration metal sample and the ZnO sample.
Due to its wurtzite structure, the ZnO *p*-projected PDOS is strongly anisotropic. [@JPhysCondMat.17.235] To measure the anisotropy XAS and RXES were recorded for light incident at 20$^{\circ}$ and 70$^{\circ}$ relative to the sample normal, referred to as normal and grazing geometry, respectively. Dipole selection rules mean that XAS recorded in the normal (grazing) geometry is dominated by the contributions from O 2$p_{xy}$ (2$p_{z}$) orbitals (see inset, Fig. \[Fig1\]). The opposite is true for RXES as the emission spectrometer is oriented perpendicular to the incident light: x-rays incident normal (grazing) to the sample result in grazing (normal) RXES. Thus $p_{xy}$ RXES is analysed with $p_z$ XAS and vice versa. This correspondence has been observed for wurtzite GaN by Strocov *et al.*. [@PhysRevB.72.085221]
Electronic structure calculations were performed within the hybrid DFT (HDFT) framework using the HSE03 functional for exchange and correlation. [@JChemPhys.118.8207] Quasiparticle effects were taken into account by a subsequent *GW* correction of the HSE03 eigenvalues using many body perturbation theory. In the *GW* calculations, the Coulomb potential was fully screened using the random phase approximation dielectric function based on the HSE03 eigenvalues and functions. [@PhysRevB.74.035101] It has been shown that the *ab initio* combination of HDFT and *GW* calculations gives excellent values for the fundamental gaps and $d$-band positions of many semiconductors. [@PhysRevB.76.115109]. At present calculating a fully **k**-resolved *GW* band structure is too computationally intensive, so we compare XAS and XES to PDOS obtained from the *GW* calculation, and RXES to a modified HDFT band structure.
Results and Discussion
======================
Figure \[Fig1\] shows the O K-edge XAS (open symbols) and XES, taken well above threshold (closed symbols). Strong linear dichroism associated with the anisotropy in the unoccupied states is seen in the XAS, especially in the first 10 eV above the conduction band minimum (CBM) where the $p_{xy}$ (black) contribution is considerably stronger than the $p_{z}$ (red) contribution. The XES is comprised of O 2$p$ states near the valence band maximum (VBM) and a clear peak corresponding to O 2$p$ states hybridized with Zn 3$d$ orbitals at 519.9 eV.
These spectra are compared with PDOS (solid lines) obtained from the *GW* calculation. The PDOS have been convolved using gaussians with widths of 0.37 eV and 0.20 eV for the occupied and unoccupied states respectively. This is an estimate of the intrinsic instrumental resolution of the experiment; no lifetime effects have been considered. The PDOS have been aligned so that the main valence band peak matches with experiment. This yields excellent agreement with the entire XES, accurately locating the $3d$ hybrid. The weak anisotropy in the PDOS, combined with the lower resolution of the XES, explains the lack of contrast between $p_{xy}$ and $p_z$ XES. The final state in XAS includes a core hole that is not included in the calculation, and which prevents absolute comparison between the energy scales of the XAS and XES. [@PhysRevB.25.5150] Therefore, to compare to XAS, the theoretical unoccupied PDOS has been rigidly shifted by an additional $-1.0$ eV, providing an estimate of the core hole binding energy. With this adjustment the experimental agreement is excellent in terms of both the main peak locations and intensities, and the orbital anisotropy.
The success of the *GW* calculation is highlighted by a comparison with the HDFT result. We calculate a band gap of 3.2 eV, compared to 2.1 eV obtained from the HDFT calculation (and 3.4 eV experimentally. [@JApplPhys.98.041301]) The valence band PDOS is also greatly improved: Figure \[Fig2\] shows the total O $2p$-projected occupied PDOS calculated using HDFT (black line), and with the *GW* correction applied (red line). With *GW* the Zn 3$d$ – O 2$p$ hybrid peak is shifted to lower energy, peaking at $-7.1$ eV, in much closer agreement with the experimental peak at $-7.4$ eV (closed symbols), correcting a key flaw of standard DFT calculations. [@PhysRevB.76.115109] With the Zn 3$d$ states located lower in energy the rest of the valence band is widened towards lower energies due to reduced $p$-$d$ repulsion. We note that scaling the HDFT energy axis by a factor of 1.1 is enough to obtain relatively close agreement with the *GW* calculation (dashed line) for the occupied states. An additional upward shift of 0.5 eV to the conduction band is sufficient to achieve similar agreement for the unoccupied states. We have confirmed the accuracy of this approach approach by calculating the *GW* band structure at selected **k** points.
Figure \[Fig3\] shows the band structure obtained from the HDFT calculation, with the above corrections applied. The $p_{xy}$ (black) and $p_z$ (red) orbital contributions to the eigenvalues are plotted independently; error bars represent the relative weight of the contribution. To simplify RXES analysis, the $p_{xy}$ ($p_z$) unoccupied states are plotted with $p_z$ ($p_{xy}$) occupied states (see above). RXES can be viewed as a scattering process where the final state includes an electron in the conduction band and a hole in the valence band, but no core-hole. So, unlike Fig. \[Fig1\], no core hole correction is applied to the conduction band.
As expected, the band structure is that of a direct-gap semiconductor. The strongly dispersing band that forms the CBM at $\Gamma$ is also anisotropic: only $p_{xy}$ states are found along $\Gamma$-M and $\Gamma$-K; $p_z$ along $\Gamma$-A-H. This combination of strong dispersion and orbital selectivity make ZnO especially suitable for band mapping with RXES. For weakly correlated systems, and excitation energies close to the absorption edge, resonant effects contribute to XES, adding a **k**-conserving coherent contribution to the scattering. The crystal anisotropy, combined with orbital selection rules, adds further **k**-selectivity to the RXES. For ZnO, the use of these properties in concert allows valence band dispersion to be measured across L-M-$\Gamma$-A-H-K-$\Gamma$ of the Brillouin zone.
RXES were taken over a range of excitation energies, indicated by vertical bars on Fig. \[Fig1\], and horizontal bars on Fig. \[Fig3\]. The coherent fraction, extracted using standard techniques, [@JElecSpecRelPhe.110.335] ranged from 0.45 to 0.6.
![(color online) Normalized coherent RXES showing $p_{xy}$ (black) and p$_{z}$ (red) data. Labels indicate excitation energy ($h\nu$).[]{data-label="Fig6"}](Fig6.eps){width="8cm"}
![(color online) The coherent RXES overlaid with orbitally resolved band structure: (a) p$_{xy}$ and (b) p$_{z}$ orbitals. Intensity scales from zero (white) to one (black).[]{data-label="Fig5"}](Fig5.eps){width="8cm"}
Using the above we construct Figure \[Fig5\], which shows normalized intensity maps of the coherent fraction of each spectrum placed in **k**-space according to the position of the bars of Fig. \[Fig3\]. Figures \[Fig5\](a) and (b) show the $p_{xy}$ and $p_z$ data, respectively. The imaging approach highlights the information contained in the raw spectra. Dispersion toward higher energy at $\Gamma$, as expected for a direct-gap material, [@JElecSpecRelPhe.110.335] is immediately obvious for both states. For the $p_{xy}$ states dispersion along $\Gamma$-A-H-K is clearly resolved, with the upper band dispersing to lower energy. The bands around $-4$ eV decrease in energy and increase in intensity along A-H-K, in line with theory. The position of the Zn 3$d$ states is apparent, just below the theoretical bands. The spectrum obtained with an excitation energy of $537.6$ eV is located near both M and K, out of order with the other spectra, because of the strongly localised states in the conduction band at 10.5 eV. The $p_z$ states add M-$\Gamma$ and K-$\Gamma$ to the parts of the brillioun zone we can resolve. The lower energy band near $-4$ eV disperses to $-5.5$ eV approaching $\Gamma$, in direct contrast to the observed behaviour of the $p_{xy}$ band in this energy range, and in agreement with theory.
There has been much discussion about the correct use of intermediate and final states in RXES calculations. [@PhysRevB.59.7433; @PhysRevB.73.115212; @JElecSpecRelPhe.110.335] We note that if the core-hole correction were applied to the unoccupied states of Fig. \[Fig3\] (a downward shift of the theory by $1.0$ eV) the $p_{xy}$ spectrum taken with $537.6$ eV would be relocated to near $\Gamma$ and the strong intensity around $-4.5$ eV would be anomalous. This is clear evidence for the use of the final-state rule in RXES.
Conclusion
==========
In conclusion we have used the strong anisotropy and dispersion of wurtzite ZnO, the **k**-selectivity of the RXES technique, and an HDFT+*GW* calculation to construct a band mapping of ZnO across a wide range of high symmetry points. The energy of the Zn 3$d$ core level is located and evidence for the use of the final state rule in RXES is presented. It should be possible to use further high resolution RXES experiments to tune theoretical calculations across the technologically important post transition metal oxide and nitride series.
We acknowledge the assistance of R. Mendelsberg and V. J. Kennedy. Work at UOC was supported by the Marsden Fund (UOC0604) and the NZ TEC Doctoral Scholarship Program; at BU by the DOE (DE-FG02-98ER45680) and donors of the American Chemical Society Petroleum Research Fund; and at FSU by the European Community through NANOQUANTA (NMP4-CT-2004-500198), the Deutsche Forschungsgemeinschaft (BE1346/18-2 and /20-1) and the Carl-Zeiss-Stiftung. The NSLS is supported by the DOE (DE-AC02-98CH10886).
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---
abstract: 'Although SGD requires shuffling the training data between epochs, currently none of the word-level language modeling systems do this. Naively shuffling all sentences in the training data would not permit the model to learn inter-sentence dependencies. Here we present a method that partially shuffles the training data between epochs. This method makes each batch random, while keeping most sentence ordering intact. It achieves new state of the art results on word-level language modeling on both the Penn Treebank and WikiText-2 datasets.[^1]'
author:
- |
Ofir Press\
Paul G. Allen School of Computer Science & Engineering, University of Washington\
[[email protected] ]{}
bibliography:
- 'acl2019.bib'
title: Partially Shuffling the Training Data to Improve Language Models
---
Background
==========
A language model is trained to predict word $n+1$ given all previous $n$ words. A recurrent language model receives at timestep $n$ the $n$th word and the previous hidden state and outputs a prediction of the next word and the next hidden state.
The training data for word-level language modeling consists of a series of concatenated documents. The sentences from these documents are unshuffled. This lets the model learn long term, multi-sentence dependencies between words.
The concatenation operation results in a single long sequence of words. The naive way to train a language model would be to, at every epoch, use the entire training sequence as the input, and use the same sequence shifted one word to the left as target output. Since the training sequence is too long, this solution is infeasible.
To solve this, we set a back propagation through-time length ($b$), and split the training sequence into sub-sequences of length $b$. In this case, in each epoch the model is first trained on the first sub-sequence, and then on the second one, and so on. While gradients are not passed between different sub-sequences, the last hidden state from sub-sequence $m$ becomes the initial hidden state while training the model with sub-sequence $m+1$.
For example, if the training sequence of words is:
`[A B C D E F G H I J K L]`
for $b = 3$, the resulting four sub-sequences are:
`[A B C] [D E F] [G H I] [J K L]`
Note that we only present the input sub-sequences, as the target output sub-sequences are simply the input sub-sequences shifted one word to the left[^2]. This method works, but it does not utilize current GPUs to their full potential.
In order to speed up training, we batch our training data. We set a batch size $s$, and at every training step we train the model on $s$ sub-sequences in parallel.
To do this, we first split the training sequence into $s$ parts. Continuing the example from above, for $s=2$, this results in:
`[A B C D E F]`
`[G H I J K L]`
Then, as before, we split each part into sub-sequences of length $b$:
`[A B C] [D E F]`
`[G H I] [J K L]`
Then, during the first training step in each epoch we train on:
`[A B C] `
`[G H I] `
and during the second training step in each epoch we train on:
`[D E F] `
`[J K L]`
Note that at every step, all sub-sequences in the batch are processed in parallel.
Before we introduced batching, in each epoch the output for each word in the training sequence was dependant on all previous words. With batching, the output of the model for each word is only dependant on the previous words in that batch element (or equivalently, row in our example), and the other words are ignored.
In our example, the hidden state that is given when inputting `G` is the default initial hidden state, and not the one that resulted after the input of `F`. This is not optimal, but since batching reduces the training time by a significant amount, all current models use this method.
The Partial Shuffle Method
==========================
While SGD calls for random batches in each epoch, in existing language models, the data is not shuffled between epochs during training. This means that batch $i$ in every epoch is made up of the same sub-sequences.
The straightforward way to shuffle the data would be to shuffle all sentences in the training sequence between each epoch. This hurts the language model’s performance, since it does not learn inter-sentence dependencies.
Here we present the Partial Shuffle method, which improves the performance of the model.
Like before, we first separate the sequence of words into $s$ rows. Using the example sequence from above, this would result in (for $s=2$):
`[A B C D E F]`
`[G H I J K L]`
Then, for each row, we pick a random index between zero and the length of the row and we take the words that are located before this index and move them to the end of the row. So in our example, if the random index for row one was $2$ and for row two was $5$ this would result in (red marks the words which were moved):
`[C D E F A B]`
`[L G H I J K]`
Finally, as before, each row (or equivalently, batch element) is divided into back-propagation through time segments. For $b = 3$, this will result in:
`[C D E] [F A B]`
`[L G H] [I J K]`
This method randomizes the batches while still keeping most of the word ordering intact.
Model Validation Test
-------------------------------------- ------------ -------
MoS 58.27 56.18
MoS + Partial Shuffle 57.43 55.35
MoS + Finetune 56.76 54.64
MoS + Finetune + Partial Shuffle 55.89 53.92
DOC 55.39 53.44
DOC + Partial Shuffle 54.90 53.28
DOC + Finetune 54.62 52.87
DOC + Finetune + Partial Shuffle 54.30 52.58
DOC + Finetune$^*$ 54.18 52.38
DOC + Finetune$^*$ + Partial Shuffle 53.79 52.00
: \[PTB\] Model perplexity on the Penn Treebank, without and with the Partial Shuffle method. Finetune$^*$ denotes repeating the finetuning operation three times.
Model Validation Test
-------------------------------------- ------------ -------
MoS 65.94 63.35
MoS + Partial Shuffle 64.09 61.97
MoS + Finetune 63.98 61.49
MoS + Finetune + Partial Shuffle 62.38 59.98
DOC 61.68 59.64
DOC + Partial Shuffle 61.28 58.93
DOC + Finetune 60.97 58.55
DOC + Finetune + Partial Shuffle 60.58 58.20
DOC + Finetune$^*$ 60.29 58.03
DOC + Finetune$^*$ + Partial Shuffle 60.16 57.85
: \[PTB\] Model perplexity on WikiText-2, without and with the Partial Shuffle method. Finetune$^*$ denotes repeating the finetuning operation three times.
Results
=======
We evaluate our method on the current state of the art model, DOC [@doc], and the previous state of the art model, MoS [@mos], on the Penn Treebank [@ptb] and WikiText-2 [@wt2] language modeling datasets. For each model, the hyper-parameters (including $b$ and $s$) are not modified from their original values. In addition, we present results for finetuned [@awd] models, with and without the Partial Shuffle.
Our shuffling method improves the performance of all models, and achieves new state of the art results on both datasets. Our method does not require any additional parameters or hyper-parameters, and runs in less than $\frac{1}{100}$th of a second per epoch on the Penn Treebank dataset.
Acknowledgements
================
This note benefited from feedback from Judit Acs, Shimi Salant and Noah A. Smith, which is acknowledged with gratitude.
[^1]: Our code is available at [<https://github.com/ofirpress/PartialShuffle>]{}
[^2]: For example, the target output sub-sequences here are `[B C D] [E F G] [H I J] [K L *]`, where `*` is the end-of-sequence token.
|
---
abstract: 'Brane gas cosmology is a scenario inspired by string theory which proposes a simple resolution to the initial singularity problem and gives a dynamical explanation for the number of spatial dimensions of our universe. In this work we have studied analytically and numerically the late-time behaviour of these type of cosmologies taking a proper care of the annihilation of winding modes. This has help us to clarify and extend several aspects of their dynamics. We have found that the decay of winding states into non-winding states behaving like a gas of ordinary non-relativistic particles precludes the existence of a late expansion phase of the universe and obstructs the growth of three large spatial dimensions as we observe today. We propose a generic solution to this problem by considering the dynamics of a gas of non-static branes. We have also obtained a simple criterion on the initial conditions to ensure the small string coupling approximation along the whole dynamical evolution, and consequently, the consistency of an effective low-energy description. Finally, we have reexamined the general conditions for a loitering period in the evolution of the universe which could serve as a mechanism to resolve the [*brane problem*]{} - a problem equivalent to the [*domain wall problem*]{} in standard cosmology - and discussed the scaling properties of a self-interacting network of winding modes taking into account the effects of the dilaton dynamics.'
author:
- Antonio Campos
title: 'Late-time dynamics of brane gas cosmology'
---
Introduction\[sec:Intro\]
=========================
To understand the nature of the initial singularity and the origin of the number of dimensions of our universe are two fundamental problems in cosmology. Is it any physical law that allows the universe to avoid the initial singularity? Why we live in $3+1$ dimensions? Is it, some how, possible to explain dynamically the spatial dimensionality of spacetime? In the standard theory of general relativity the number of dimensions of the cosmological spacetime is not derived from any fundamental law but simply assumed to be four. Moreover, it cannot address the problem of the initial singularity either because it is quite reasonable to believe that Einstein equations will not be valid close to the Planck scale.
An exciting potential resolution of these two issues within the framework of superstring theory was proposed by Brandenberger and Vafa in the late 80’s [@Brandenberger:1989aj]. This proposal is based in a fundamental symmetry of string theory called T-duality which states that the physics of strings in a box of size $R$ do not change if we replace the length of the box by $l^2_{st}/R$ (where $l^2_{st}$ is the string length). This symmetry is not respected by the standard cosmological equations of general relativity but it is naturally implemented when the dynamics of a dilaton field is properly taking into account [@Tseytlin:1992xk; @Tseytlin:1992ss].
In this framework the background spacetime has the topology of a torus with nine dimensions, which are assumed to be of equal size, and the dynamics is driven by a gas of fundamental strings. The evolution of the universe is considered to be adiabatic, that is, there is no cosmological production of entropy, and, the string coupling constant is assumed to be small such that an effective tree-level approximation of the string theory is valid. The string gas supports different string states which can be decomposed as combinations of oscillatory modes (stationary vibrating strings), momentum modes (non-stationary strings), and winding modes (strings wrapping around the torus). The T-duality symmetry interchanges winding modes with momentum modes leaving the spectrum of the theory invariant under the inversion of the radius of the torus (the energy of oscillatory modes is independent of the size of the torus). Under this symmetry the temperature of the string gas is also invariant in the sense that it is the same when the size of the universe is $R$ or $l^2_{st}/R$. That means that no physical singularity will occur as the radius of the torus is made indefinitely small ($R\ll l_{st}$) avoiding the inherent temperature singularity of standard cosmology.
This scenario also offers a mechanism for explaining an upper bound for the spacetime dimensionality under the assumption of thermal equilibrium. The net cosmological effect of winding modes is to stop expansion even though their energy grows when the volume of space increases. The reason is because they contribute to the expansion with a total negative pressure. The evolution of winding modes can be though as equivalent to that of large classical cosmic strings. If thermal equilibrium is maintained energy from the winding states can be transferred to the rest of the states of the energy spectrum and we can keep the universe expanding. The key observation is to realize that the probability of interaction decreases with the dimensionality of spacetime. As a result, the annihilation of strings with winding number can only be efficient if the number of dimensions of the spacetime is not larger than four. Then, at some point in the evolution, six spatial dimensions of the torus must remain small while the other three become large relative to the string length. Which dimensions grow and which stay small will mostly depend on thermal and quantum fluctuations. These qualitative arguments have been confirmed numerically [@Sakellariadou:1996vk; @Cleaver:1995bw]. Some thermodynamical aspects of a string gas and their implications for cosmology have been reviewed in [@Bassett:2003ck].
Recently, these ideas have been revived by Brandenberger et al. [@Alexander:2000xv] in order to include extended degrees of freedom other than fundamental strings [@Polchinski:1995mt; @Polchinski:1996na; @Polchinski:1998]. In this scenario the early universe is assumed to have $D+1$ spacetime dimensions with an isotropic toroidal topology and filled with a gas consisting of all possible branes with spatial dimension $p < D$ that the spectrum of a particular string theory could admit. Because Dp-branes respect T-duality [@Sen:1996cf] the initial singularity problem can also be easily resolved within this extended scenario [@Alexander:2000xv] (for a more detailed discussion see [@Boehm:2002bm]). As with fundamental strings, the energy of winding modes of Dp-branes grows with expansion, giving the larger contribution to the total energy of the gas those with the largest $p$, and they also tend to prevent expansion. Assuming thermal equilibrium as well, the cosmology of brane gases may also provide an explanation for the dimensionality of spacetime by an analogous mechanism of self-annihilation of winding states. Modes with larger $p$ are more massive and they must decay first allowing only $2p+1$ spatial dimensions to grow. The late-time evolution will be dominated by the winding modes with the lower spatial dimension ($p=1$). Then, a hierarchy of small dimensions is generated and the observed dimensionality of our universe can be again explained. Generalisations of brane gas models to manifolds with non-toroidal topology and to anisotropic backgrounds have been studied in [@Easson:2001fy; @Easther:2002mi] and [@Watson:2002nx], respectively. Attends to extend and discuss these type of cosmological scenarios by including fundamental degrees of freedom of 11-dimensional M-theory have been given in [@Easther:2002qk; @Alexander:2002gj].
Even though the string considerations of [@Brandenberger:1989aj] generalise quite easily to branes gases, they face a problem similar to the standard [*domain wall problem*]{} of cosmological models admitting the spontaneous breaking of a discrete symmetry in the early universe [@Zeldovich:1974uw; @Vilenkin:1994]. Causality indicates that despite an efficient annihilation of the winding states at least one Dp-brane across a Hubble volume should have survived. As with domain walls defects, the presence of even only one brane in our present horizon with an energy larger than the electroweak scale would have introduced fluctuations in the temperature of the microwave background radiation incompatible with current experimental bounds. This observation pose a severe constraint on cosmological models filled with branes gases. A solution to this new [*brane problem*]{}, proposed in [@Alexander:2000xv] and further developed in [@Brandenberger:2001kj], is to invoke a sufficiently long period of cosmological loitering [@Lemaitre:1931; @Glanfield:1966; @Felten:1986; @Sahni:1991ks; @Feldman:1993ue] which might have allowed the whole spatial extend of the universe to be in causal contact so the actual absence of branes can be explained by microphysical processes.
The purpose of this work is to clarify and extend previous results on the late-time dynamics of brane gas cosmologies (BGC). In particular, we have been interested to analyse several aspects of the cosmological evolution of this type of scenarios by incorporating the annihilation of brane modes with winding number appropriately. First, we have discussed analytically the qualitative features which are not sensible to the details of the modelling of winding mode decay. We have found that spatial dimensions cannot grow large if the brane states without winding number are produced in the form of ordinary non-relativistic matter. We suggest that this obstruction to explain the dimensionality of the spacetime can be very easily resolved if a gas of non-static branes is considered. Additionally, we have obtained a simple criterion on the initial conditions that guarantee the smallness of the string coupling at all times and, consequently, the consistency of an effective low-energy description for BGC. We have seen, by studying numerically how the dynamics change with respect to the values of some representative parameters, that the particular characteristics of the decay of winding modes mainly affect intermediate stages of the evolution of the universe. Finally, we have been interested to check the robustness of the resolution of the above mentioned [*brane problem*]{} by a phase of loitering and to investigate the scaling properties of a network of self-interacting winding states driven by the dynamics of a dilaton field.
The rest of the paper is organised as follows. In Sec. \[sec:BGC\] we give a brief review of the main ingredients of BGC. Sec. \[sec:Late\] is divided into two main parts. In the first part we describe with some detail how the process of winding mode decay into small loops without winding number can be modelled. The second part is devoted to investigate the corresponding late-time cosmological dynamics. The conclusions of our analysis are summarised in Sec. \[sec:Conclusions\].
Cosmology of brane gases\[sec:BGC\]
===================================
The brane gas scenario assumes that the early universe is filled with a gas in thermal equilibrium containing all the Dp-branes supported by the compactification of 11-dimensional M-theory on $S^1$. All the nine spatial dimensions left after compactification are considered to be toroidal and to start expanding adiabatically with an initial size of the order of the string length. The cosmological dynamics of this set up is dictated by the low-energy effective action, $$S_B
= - \int d^{D+1}x\, \sqrt{-G}\, \mathrm{e}^{-2\phi}
\left[ R
+4 (\nabla_{\mu}\phi)^2
-\frac{1}{12}H^2_{\mu\nu\alpha}
\right],$$ where $D$ are the spatial dimensions, $R$ denotes the scalar curvature corresponding to the metric tensor $G_{\mu\nu}$, $\phi$ represents the dilaton field, which is related with the radius of compactification, and $H_{\mu\nu\alpha}$ is the field strength of a bulk two-form potential $B_{\mu\nu}$. We are going to employ units in which the string length is $l_{st}\sim 1$. The matter source in this scenario is given by a gas of Dp-branes. The dynamics of individual branes with $p$ spatial dimensions embbeded in a $(D+1)$-dimensional bulk spacetime is described by the Dirac-Born-Infeld action [@Polchinski:1998] (see also [@Leigh:1989jq]), $$S_p
= T_p \int d^{p+1}\xi\, \mathrm{e}^{-\phi}
\sqrt{-\mathrm{det}( G_{\alpha\beta}
+B_{\alpha\beta}
+2\pi\alpha' F_{\alpha\beta})},$$ where $T^{-1}_p=\sqrt{\alpha'}(2\pi\sqrt{\alpha'})^p$, $\sqrt{\alpha'}$ is the string length $l_{st}$, the set of coordinates $\xi^\alpha (\alpha=0,\cdots,p)$ parametrise the D-brane world-volume, $G_{\alpha\beta}$ and $B_{\alpha\beta}$ are the pull-backs of the $(D+1)$-spacetime metric $G_{\mu\nu}$ and the antisymmetric tensor field $B_{\mu\nu}$, respectively, while $F_{\alpha\beta}$ is the field strength of a $U(1)$ gauge field $A_{\mu}$ living on the D-brane. This action includes the dynamics of transverse mode fluctuations, governed by $(D-p)$ world-volume scalar fields, and longitudinal mode fluctuations, described by the gauge field, in addition to winding modes giving the background mass of the brane. In the standard picture of BGC it is assumed that brane mode fluctuations are small and only winding modes dominate the cosmological dynamics. The mass energy of these modes is given by (see for instance [@Maggiore:1998cz; @Boehm:2002bm]), $$E^{(p)}_w = \tau_p\, Vol_p.
\label{eq:brane_mass}$$ with $\tau_p=T_p/g$ a re-scaled brane tension, $Vol_p$ the physical spatial volume of the brane, and $g\equiv\exp(\phi)$ the string coupling which is considered to be small. The equation of state corresponding to winding modes with $p$ spatial dimensions is [@Alexander:2000xv; @Vilenkin:1994], $$P_w^{(p)}
=\gamma_p E_w^{(p)},
\textrm{\,\,\, with } \gamma_p=-\frac{p}{D}.
\label{eq:eq_state}$$
In this work we shall restrict our analysis to spatially flat homogeneous and isotropic spacetime backgrounds, $$ds^2
= - dt^2
+ \mathrm{e}^{2\lambda(t)}\sum^{D}_{i=1}dx^2_i.$$ In [@Watson:2002nx], it was shown that the isotropy of the three large spatial dimensions that remains after the decay of the winding degrees of freedom comes out directly as a consequence of the dynamics. Introducing the new dilaton variable $\varphi = 2\phi - D\lambda$, the background equations of motion derived from the total effective action can be written as [@Tseytlin:1992xk; @Tseytlin:1992ss; @Veneziano:1991ek; @Gasperini:2002bn], $$\begin{aligned}
\dot\varphi^2 - D\dot\lambda^2
&=& \mathrm{e}^\varphi E,
\nonumber \\
\ddot\varphi
-D\dot\lambda^2
&=& \frac{1}{2}\,\mathrm{e}^\varphi E,
\nonumber \\
\ddot\lambda
-\dot\varphi\dot\lambda
&=& \frac{1}{2}\,\mathrm{e}^\varphi P.
\label{eq:field_equations}\end{aligned}$$ where $E$ and $P$ are the total energy and pressure build on all (winding and non-winding) matter source contributions, $$\begin{aligned}
E &=& \sum_p E_w^{(p)}
+E_{nw},
\\
P &=& \sum_p \gamma_p E_w^{(p)}
+\gamma E_{nw}.\end{aligned}$$ For non-winding modes we will consider an ordinary equation of state $P_{nw}=\gamma E_{nw}$ with $0\leq \gamma \leq 1$. The above dynamical equations obey an energy conservation law, $\dot E + D\dot\lambda P =0$, which comes out as a result of the adiabatic approximation [@Tseytlin:1992xk].
Late-time dynamics\[sec:Late\]
==============================
The early-time dynamics of BGC is quite well understood and it has been extensively discussed [@Tseytlin:1992xk; @Alexander:2000xv; @Bassett:2003ck; @Tsujikawa:2003pn]. In this work we are mainly interested in investigating the late-time dynamics of this type of cosmologies. For that purpose it is fundamental to supplement the field equations given in (\[eq:field\_equations\]) with an appropriate description of the decay of winding modes into states without winding number. We will assume that all winding states with $p>1$ have already been completely annihilated and six spatial dimensions have been frozen. Then, we can consider that the cosmological evolution is only driven by the dynamics of winding modes with $p=1$ in three spatial dimensions ($D=3$).
Modelling the decay of winding modes\[subsec:decay\]
----------------------------------------------------
The cosmological evolution of the winding strings can be thought as analogous to that of a network configuration of long cosmic strings in an expanding universe with toroidal topology [@Bennett:1986qt; @Bennett:1986zn; @Vilenkin:1994; @Brandenberger:1994by]. At time $t$ the total energy of a gas of $N_w$ winding modes with individual mass-energy $\tau R(t)$, recall Eq. (\[eq:brane\_mass\]), can be expressed as, $$E_w(t) = N_w(t) \tau R(t)
= \tau l_c N_w(t)\mathrm{e}^{\lambda(t)}.
\label{eq:winding_energy}$$ The initial physical size of any of the spatial dimensions of the torus, $l_c\exp(\lambda(0))$, is usually assumed to be of the order of the string length. Using Eq. (\[eq:winding\_energy\]) one can find an evolution equation for the total energy of the winding string gas, $$\dot E_w
= \dot\lambda E_w
+\tau R(t) \dot N_w(t).
\label{eq:winding_energy_evolution}$$ The second term represents the loss of energy due to the production of small loops without winding number. If the winding mode decay is negligible the above equation corresponds to the classical conservation equation of an ideal gas of non-interacting strings with equation of state $P_w=-E_w/3$, recall Eq. (\[eq:eq\_state\]). Long strings decay into small loop strings through self-interactions. For such a network there exists a characteristic length scale $L_w(t)=R/\sqrt{N_w}$ which describes the typical separation between the string-like branes. The rate of energy loss by the decay of winding modes into the production of small loops can be roughly approximated by, $$\begin{aligned}
\left. \dot E_w \right|_\mathrm{loop}
&\sim& -c
\cdot
\frac{1}{L^4_w}
\cdot
\tau'L_l
\cdot
V
\nonumber\\
&\sim& -c\, \tau N^2_w(t) \left( \frac{L_l(t)}{R(t)}
\right).
\label{eq:decay_rate}\end{aligned}$$ The parameter $c$ measures the efficiency of loop production, the space volume is denoted by $V$, and $L_l(t)$ represents the typical size of the created loops. The contribution $L^{-4}_w$ estimates the number of collisions per unit volume in a network with length scale $L_w$ whereas the second contribution, $\tau'L_l$, expresses the fact that the energy loss has to be proportional to the energy scale at which the small loops are produced. In the last part of the above rate equation, we have assumed that the string tension of the created loops should be similar to that of the winding modes $\tau'\sim \tau$. From Eqs. (\[eq:winding\_energy\_evolution\]) and (\[eq:decay\_rate\]) it is easy to obtain an evolution equation for $N_w(t)$, $$\dot N_w(t)
= - \frac{c}{l^2_c} L_l(t) N^2_w(t) \mathrm{e}^{-2\lambda(t)}.
\label{eq:winding_energy_rate}$$
Now it is convenient to introduce a dimensionless function of cosmological time, $N_l(t)$, in order to write the total energy of small string loops as, $$E_l(t) = N_l(t) \tau R(t) \mathrm{e}^{-(1+3\gamma)\lambda(t)}
= \tau l_c N_l(t) \mathrm{e}^{-3\gamma\lambda(t)},
\label{eq:loop_energy}$$ For $\gamma=0$ the produced loops behave like ordinary static matter and for $\gamma=1/3$ as relativistic particles. The function $N_l(t)$ describes the production of loops, it is zero if there is no loops at all and constant if they are no longer created. Thus, the rate of change for the total energy of produced loops will be, $$\dot E_l
= -3\gamma\dot\lambda E_l
+\tau R(t) \dot N_l(t) \mathrm{e}^{-(1+3\gamma)\lambda(t)}.
\label{eq:loop_energy_evolution}$$ The first term corresponds to the variation of energy due to the expansion of the universe and the second to the energy gained through winding mode decay. Within our adiabatic approximation (recall that this assumption is equivalent to the energy conservation equation discussed at the end of Sec. \[sec:BGC\]), we can find an evolution equation for $N_l(t)$, $$\dot N_l(t)
= \frac{c}{l^2_c} L_l(t) N^2_w(t) \mathrm{e}^{(3\gamma-1)\lambda(t)}.
\label{eq:loop_energy_rate}$$ Obviously, $c=0$ corresponds to no loop creation. Note that in this case the energy density of winding modes simply scales as $\rho_w\sim\exp (-2\lambda)$ and, consequently, it should always dominate over ordinary matter or radiation components into the future of a expanding universe.
The field equations (\[eq:field\_equations\]) can now be cast into a close set of first-order differential equations which completely describes the evolution of the universe during the process of winding mode decay , $$\begin{aligned}
\dot\varphi
&=& f,
\label{eq:dot_varphi} \\
\dot\lambda
&=& l,
\label{eq:dot_lambda} \\
\dot f
&=& 3l^2
+\frac{1}{2}\,\mathrm{e}^\varphi
\left( E_w + E_l
\right),
\label{eq:dot_f} \\
\dot l
&=& fl
-\frac{1}{6}\,\mathrm{e}^\varphi
\left( E_w -3\gamma E_l
\right),
\label{eq:dot_l} \\
f^2
&=& 3l^2
+\mathrm{e}^\varphi
\left( E_w + E_l
\right),
\label{eq:constraint}\end{aligned}$$ together with Eqs. (\[eq:winding\_energy\]), (\[eq:winding\_energy\_rate\]), (\[eq:loop\_energy\]), and (\[eq:loop\_energy\_rate\]). Notice that $l$ is nothing but the Hubble parameter.
To complete the description of winding mode annihilation we will have to make an assumption about the formation of small loops. Following [@Bennett:1986zn] (see also [@Brandenberger:2001kj]) we will assume the loops are produced with a radius proportional to cosmic time, $L_l(t) \sim t$. This will imply that at late times loops are formed with a typical size of the order of the Hubble horizon. Another reasonable possibility would have been to assume that the radius of loops scales with the characteristic length of the winding string network, that is $L_l(t) \sim L_w$. We have, nevertheless, checked that both cosmological evolutions are not substantially different. All these points will be made more clear in a later section where we will discuss the scaling properties of the winding mode network. Certainly, more sophisticated models to describe the decay of winding modes could be conceived. For instance, one can take into account the possibility of the reconnection of small loops to winding strings and loop decay into gravitational radiation or ordinary matter (relativist or not). However, for our purpose we can keep the description of winding mode annihilation as simple as possible.
Cosmological dynamics\[subsec:dynamics\]
----------------------------------------
In the most simple cosmological picture of BGC the universe is assumed to be initially in a state with all its spatial dimensions expanding isotropically, $l>0$, and with a physical size of the order of the string length, $l_{st}\sim 1$. As we have explained, these small toroidal dimensions can only become large if the winding and anti-winding modes can self-annihilate and decay into fundamental string loops or relativistic matter. In what follows we will be interested to explore several dynamical aspects of these cosmological scenarios.
### Qualitative analysis
Let us start this section illustrating some general properties of the dynamical evolution of BGC including the effects of winding mode decay but independently of the particular modelling of small loop creation. Consider the two-dimensional phase space spanned by $(f,l)$. The constraint equation (\[eq:constraint\]) in conjunction with the condition of positivity of the total energy restrict the cosmological dynamics to values that satisfy the inequalities $$f^2-3l^2 \geqslant 0
\,\,\,\,\, \mathrm{or} \,\,\,\,\,
|l| \leqslant \frac{1}{\sqrt{3}}|f|.$$ The straight lines $l=\pm f/\sqrt{3}$ correspond to solutions of the equations of motion with zero total energy (that is $E_w=E_l=0$ except in the exceptional limiting case $\varphi\rightarrow -\infty$ where the total energy can take any finite positive value) passing through the origin. To probe the dynamical character of the fixed point $(f,l)=(0,0)$ it is sufficient to study the time evolution of these special solutions. From Eq. (\[eq:dot\_l\]) it is straightforward to check that $l$ increases or decreases depending exclusively on the sign of $fl$. In the first and third quadrant of the phase space $l$ must grow and in the second and fourth quadrant it has to diminish. As a consequence, the two straight trajectories lying in the first and fourth quadrant move away from the origin and those in the second and third quadrant move closer to the origin. Since self-consistency demands that there are no trajectories in phase space crossing these special lines, any other trajectory solution of the equations of motion originating in the half left side of phase space approaches the origin asymptotically whereas trajectories that start in the half right side diverge from it, see Fig. \[fig:phase\_space\]. Then, the fixed point $(f,l)=(0,0)$ is a saddle point and in fact it is also, as it can be easily shown, the only critical point of the equations of motion. This simple dynamical picture is substantially modified if higher order curvature terms are included in the low-energy effective action [@Campos:2003].
![Phase space for $(f,l)$. In the dark grey area ($f^2-3l^2<0$) the total energy of the matter sources is negative and therefore it is excluded from the dynamical analysis. The light grey dotted wedge, defined by the lines $l=-f/3$ and $l=-f/\sqrt{3}$, is a region where the smallness of the string coupling cannot be guaranteed. We have plotted the numerical solutions of the equations of motion for several values of the physical parameters $c$, the efficiency of the winding mode decay, and $\gamma$, the parameter characterising the equation of state of the loops created. The dashed lines correspond to solutions with $\gamma=0$ whereas the dark continuous lines to solutions with $\gamma=1/3$. In both cases $c$ takes values $(0.1,1.0,10)$ from bottom to top. For comparison we have also included the solution corresponding to no winding mode decay $c=0$ (light continuous line). \[fig:phase\_space\]](phase_space_wl.eps){width="\columnwidth"}
Now, let us see that for a given equation of state of the created loops all the solutions of the dynamical equations approach the origin asymptotically close to a particular straight line. In other words, there exists a straight line which is a local solution of the equations of motion near the origin that attracts all other dynamically allowed trajectories in phase space. When the annihilation of winding modes is not taking into account this straight line is $l=f/3$ [@Tseytlin:1992xk; @Alexander:2000xv], which is in fact a global solution of the equations of motion. Intuitively one should expect that very close to the critical point almost all winding modes have already disappeared and mainly the source energy is in the form of a gas of string loops. In this regime $f$ and $l$ obey two differential equations decoupled from the rest of the variables, $$\begin{aligned}
\dot f
&\simeq& \frac{3}{2}l^2
+\frac{1}{2}f^2,
\nonumber\\
\dot l
&\simeq& fl
+\frac{\gamma}{2}\left( f^2 -3l^2
\right).
\label{eq:close_origin}\end{aligned}$$ To check our statement we have to look for straight line solutions inside the energetically allowed region, that is solutions of the form $l=\alpha f$ with $\alpha$ being a constant which obeys $|\alpha| \leq 1/\sqrt{3}$. Substituting in the previous two equations we obtain a consistency algebraic equation for $\alpha$, $$(\alpha + \gamma) (3\alpha^2 - 1) = 0.
\label{eq:special_lines}$$ Apart from the straight global solutions already studied ($l=\pm f/\sqrt{3}$) we get a new solution which depends on the equation of state of the produced loops $l=-\gamma f$. For the static case we get the horizontal line $l=0$ and in the relativistic limit the line $l=-f/3$. Obviously, this local solution does not exist if $|\gamma| > 1/\sqrt{3}$. To see that these new special lines are attractors of the dynamics close to the origin we have to analyse the behaviour of small deviations $l=\alpha f + \epsilon$ with $|\epsilon |\ll 1$. Using Eqs. (\[eq:close\_origin\]) again, the evolution of $\epsilon$ can be determined by the differential equation, $$\dot\epsilon
= \left[ 1 - 3\alpha (\alpha + \gamma)
\right]f\epsilon
-\frac{3}{2}(\alpha + \gamma) \epsilon^2.$$ For $\alpha = -\gamma$ this equation reduces to $\dot\epsilon = f\epsilon$ and then, noting that $f$ is negative and cannot change sign, it is very easy to see that the absolute value of $\epsilon$ is always a decreasing function of time. To probe the dynamical behaviour of the two other straight lines solutions looks much more subtle because the evolution of $\epsilon$ depends on the value of $\gamma$. Moreover, second order effects will come to dominate at late times and cannot be neglected. In general, one can say that at early times the line $\alpha=+1/\sqrt{3}$ is an attractor of other trajectories and the line $\alpha=-1/\sqrt{3}$ is a repeller. At late times all other solutions are attracted by both lines.
The relative behaviour of the rest of trajectories with respect to the special line $l=-\gamma f$ away from the origin is also important to understand the global dynamics of the phase space. In particular we are going to show that the solutions of the equations of motion can only cross the line from values of $l$ with $l>-\gamma f$ to values with $l<-\gamma f$ and never in the opposite direction. As a consequence of this fact we will obtain two key dynamical properties of the equations of motion. Let us start by considering the function defined by $\Phi(\varphi,\lambda;\alpha)\equiv\lambda - \alpha\varphi$, which for $\alpha=-1/3$ is nothing but a rescaling of the original dilaton field $\Phi(\varphi,\lambda;-1/3)=(2/3)\phi$. The special straight line is completely defined by extremizing this function $\dot\Phi(f,l;\alpha)=l-\alpha f=0$ for $\alpha=-\gamma$. Now consider a solution of the equations of motion crossing this special line. If the trajectory cross from values of $(f,l)$ where $\dot\Phi > 0$ ($l-\alpha f>0$) to values where $\dot\Phi < 0$ ($l-\alpha f<0$), then $\Phi(\varphi,\lambda;\alpha)$ will have a maximum. To see that actually this is the only possibility and the crossing cannot happen the other way round we have to compute $\ddot\Phi(f,\alpha f;\alpha)$ and check that it is strictly negative. Taking the evolution equations for $f$ and $l$ (\[eq:dot\_f\])-(\[eq:dot\_l\]) and using the constraint (\[eq:constraint\]) we get, $$\ddot\Phi(f,\alpha f;\alpha)
= \frac{\mathrm{e}^\varphi}{6}
\left[ \left( 3\alpha-1
\right) E_w
+3\left( \alpha+\gamma
\right) E_l
\right],$$ which for $\alpha=-\gamma$ simplifies to, $$\ddot\Phi
= - \frac{\mathrm{e}^\varphi}{6}
\left( 1+ 3\gamma
\right) E_w.
\label{eq:crossing_condition}$$ Thus, if the loops are produced with an equation of state characterised by $\gamma > -1/3$ the negativity of $\ddot\Phi(f,\alpha f;\alpha)$ is always guaranteed. In the special case $\gamma = -1/3$, which is completely equivalent to consider a cosmological evolution without winding mode annihilation, the special line cannot be cross in neither direction. This is nothing else but a reflection of the fact that $l=f/3$ is a global solution of the equations of motion.
It is worth to emphasise that the previous analytical result is independent of the evolution of the individual energies and thus of the equations of motion for the variables $N_w$ and $N_l$, and the particular modelling of the decay of winding modes. Moreover, there are two significant physical consequences that follow immediately. First, consider the special line $l=-\gamma f$ with $\gamma = 1/3$. Below this curve the dilaton and then the string coupling are always decreasing functions of time. Then, if any trajectory in phase space, solution of the equations of motion, starts with initial conditions below this curve one can ensure that the small string coupling approximation is dynamically preserved at all times as long as the equation of motion for the produced loops obeys $\gamma \leqslant 1/3$. And second, if the loops are produced in the form of ordinary non-relativistic matter, $\gamma=0$, and the universe enters a phase of contraction it will never be able to re-expand. This simple analytical result is in contradiction with the solutions obtained in [@Brandenberger:2001kj]. Probably because they did not enforced the constraint equation (\[eq:constraint\]) properly in their numerical analysis. The late-time behaviour we have observed for $\gamma=0$ is very unsatisfactory from the phenomenological point of view because it precludes the growth of three large spatial dimensions as we see today. In a subsequent section we will discuss a possible way to resolve this problem by considering a gas of non-static branes.
### Numerical analysis
In the previous paragraph we have discussed some properties of the dynamics of BGC which are independent of how the decay of winding modes is described by the equations of motion and we have shown that the final fate of the such cosmologies only depends on the equation of state of the produced loops. Now we are interested in studying numerically how this modelling influence and determine the intermediate evolution. For that purpose we have use a fourth-order Runge-Kutta method [@Gerald:1984] to solve Eqs. (\[eq:winding\_energy\_rate\]) and (\[eq:loop\_energy\_rate\])-(\[eq:dot\_l\]) for several representative values of the parameter $c$, the efficiency of winding mode annihilation, and $\gamma$, the parameter characterising the equation of state of the small loops created. When solving numerically the equations of motion for this problem one has to be very careful and use an efficient numerical algorithm with a sufficiently small spacing for the mesh because as the light grey dotted wedge depicted in Fig. \[fig:phase\_space\] is approached the solutions are highly sensitive to numerical errors. This explains why the authors of [@Brandenberger:2001kj] could have incorrectly found a runaway solution crossing the vertical axis ($f=0$) which is completely inconsistent with the original equations of motion.
Our previous qualitative analysis permits to constraint possible dynamically interesting, from the cosmological point of view, initial conditions. Our small universe has to be initially expanding, thus we have to take a positive $l(0)$. Since we are not interested to be very close to the special lines with zero total energy to enforce the small string coupling approximation one cannot take a very negative initial $\varphi$. The initial values we have chosen for all our numerical simulations are $\varphi(0)=0$, $l(0)=0.25$, $N_w(0)=25$, $N_l(0)=0$. The constraint (\[eq:constraint\]) has been used to obtain the initial value for $f(0)$. The starting strength of the string coupling we have taking is $g(0)=0.1$ which together with $\varphi(0)$ permits to determine $\lambda(0)=(2\ln g(0)-\varphi(0))/3$. Finally, we have taken $\tau=1$ and fixed the parameter $l_c$ using the condition $l_c\exp{(\lambda(0))}\sim 1$. It is also important to recall at this point that all dimensional quantities are measured in units of $l_{st}$. It is easy to check that the small coupling approximation will always be guaranteed with this choice of initial conditions.
In Fig. \[fig:phase\_space\] we have plot the phase space for the variables $(f,l)$, that is $(\dot\varphi,\dot\lambda)$. The dark grey area is the region forbidden by the condition of positivity of the total energy of the system ($f^2-3l^2>0$) and therefore it is excluded from the dynamical analysis. The white area and the light grey dotted wedge are regions allowed by energetic considerations. The light grey dotted wedge is defined by the lines $l=-f/3$ and $l=-f/\sqrt{3}$ and is the region where the string coupling grows and its smallness cannot be guaranteed. As emphasised before, trajectories originating in the white area can never cross into this region if $\gamma \leqslant 1/3$. The dashed lines correspond to solutions with $\gamma=0$ (static loops) whereas the dark continuous lines to solutions with $\gamma=1/3$ (relativistic loops). In both cases $c$ takes values $(0.1,1.0,10)$ from bottom to top. For comparison we have also included the solution corresponding to the case in which the winding modes do not self-annihilate $c=0$ (light continuous line). The evolution of the Hubble parameter and the scale factor as a function of cosmic time has also been plotted in Fig. \[fig:hubble\] and Fig. \[fig:scale\], respectively.
![Hubble parameter $l=\dot\lambda$ as a function of cosmic time. The plotted curves represent solutions of the equations of motion with parameters $c$ and $\gamma$ chosen in the same manner as in Fig. \[fig:phase\_space\]. \[fig:hubble\]](hubble_wl.eps){width="\columnwidth"}
As one can immediately see from the numerical solutions studied the efficiency parameter $c$ only qualitatively influences the very early dynamics of the equations of motion. In general, as it grows the maximum contraction rate decrease for any value of $\gamma$ (see Fig. \[fig:hubble\]) and if it is sufficiently large the universe could never enter a phase of contraction.
![Scale factor as a function of cosmic time. The plotted curves represent solutions of the equations of motion with parameters $c$ and $\gamma$ chosen in the same manner as in Fig. \[fig:phase\_space\]. \[fig:scale\]](scale_factor_wl.eps){width="\columnwidth"}
As expected from our previous qualitative analysis the very late-time cosmological dynamics of this model is mostly independent of the parameters which characterises the decay of winding modes. In fact, the scale factor very soon behaves like, $$\mathrm{e}^\lambda
\sim t^{2\gamma/(1+3\gamma^2)},$$ which corresponds to a radiation-dominated universe for $\gamma=1/3$ and to a flat static universe for $\gamma=0$. It is interesting to note that the above expansion law only coincides with that of the standard cosmological scenario for $\gamma=1/3$. This is because as the winding modes self-annihilate all trajectories approach asymptotically the line $l=-f/3$ where $\dot\phi\sim 0$ and $\ddot\phi\sim 0$. Thus, the original dilaton freezes dynamically without the need of introducing a dilaton potential and a standard expansion law is recovered before reaching the point $(f,l)=(0,0)$. This is not the case if $\gamma\neq 1/3$.
With $\gamma=0$ and for any reasonable value of the efficiency parameter $c$ we find that the universe always enters a phase of contraction after a very short period of time and never re-expands despite the self-annihilation of winding modes (see Figs. \[fig:phase\_space\] and \[fig:hubble\]). Thus if the winding modes decay into static loops the spatial dimensions can never become large and the dimensionality of our spacetime cannot be explained. Apart from our previous qualitative arguments this result can also be physically understood in the following way. Eq. (\[eq:dot\_l\]), viewed as an equation for $\lambda$, is equivalent to an equation for a damped point particle moving in an an effective potential with the following form, $$U_{eff}(\lambda)
= \frac{\tau l_c}{6}
\mathrm{e}^\varphi
\left[ N_w(t)\, \mathrm{e}^\lambda
+N_l(t)\, \mathrm{e}^{-3\gamma\lambda}
\right].$$ In the particular case in which $\gamma=0$, the second term is no longer a function of $\lambda$ and it can be merely interpreted as a time modulation of the potential origin. For such a potential, a universe that starts expanding will reach a maximum size and, at some point, it might enter a contraction phase from which it will not be able to escape. As we have seen numerically the effect of the modulating factor $\exp(\varphi)$, which is always a decreasing function of time, is not relevant for this argument to hold. This picture changes drastically at late times if $\gamma >0$ (See Figs. \[fig:hubble\] and \[fig:scale\] for the case $\gamma=1/3$). Now the second term in the effective potential $U_{eff}$ comes to dominate the dynamics as the number of winding modes $N_w$ goes to zero and $N_l$ grows. Thus, if a universe that started in a expanding phase passes through a stage of contraction it will inevitably go back to a new period of expansion. As a conclusion, we have given analytical and numerical evidences to the fact that, in contrast to the numerical results of [@Brandenberger:2001kj], re-expansion in the cosmological evolution of the universe is only possible if the small loops produced do not behave like ordinary static matter.
In Fig. \[fig:energy\] we show the profiles of all the energy components for $\gamma = 0$ and $\gamma = 1/3$. In both cases, the energy of winding modes goes to zero very rapidly and, consequently, the energy of small loops tracks the total energy of the universe during most of the time evolution. One can see, in general, that the small loop energy very soon evolves as, $$E_l
= \tilde{E}_l\mathrm{e}^{-3\gamma\lambda},$$ where $\tilde{E}_l$ is a constant which depends on the parameters of the decay process. For $\gamma = 0$ that means that the total energy reaches a constant value. We have checked numerically that the asymptotic value of this constant grows with an increasing efficiency parameter $c$. In ordinary Einstein theory a constant energy would have meant a matter-like dominated expansion of the universe. However, in our brane gas model, the effect of the dilaton coupling drives the universe through an ever contracting phase. On the other hand, for $\gamma=1/3$ we have a radiation-dominated like cosmological expansion for the universe. In this particular case the total energy decreases with time more rapidly the bigger is the parameter $c$. This is mainly due to the fact that the expansion goes faster if $c$ is larger.
![Energy profiles for $\gamma = 0$ (top) and $\gamma = 1/3$ (bottom). In both plots the dark lines correspond to $c=1.0$, the light dashed lines to $c=0.1$, and the light continuous line to $c=0$, that is, the case without winding mode decay (Recall that in this particular case $E_l=0$ and then the total energy is equal to $E_w$). The curves that asymptotically approach the zero axis represent the energy of winding modes and the curves starting at zero the energy of the loops produced. The curves tracked at late times by the loop energy is the total energy of the system $E_l+E_w$. \[fig:energy\]](energy_wl.eps){width="\columnwidth"}
### Loitering and the [*brane problem*]{}
The hierarchical picture of Dp-brane decay explains the actual dimensionality of spacetime though it leads to a variant of the domain wall problem in cosmology. A solution to this problem is to invoke a stage of cosmic inflation before the branes can dominate the energy density of the universe. An alternative proposed in [@Alexander:2000xv], and further developed in [@Brandenberger:2001kj], was to advocate a late phase of loitering in the universe. This is a stage in which the universe halts and its spatial extend $R$ could become much smaller than the Hubble horizon $d_H\sim l^{-1}$, that is a period in which $Rl\ll 1$.
We have checked numerically that a sufficiently long period in which this condition holds is quite natural in BGC, see Figs. \[fig:hubble\] and \[fig:scale\]. At early times and as long as the efficiency parameter $c$ is not too large one can assure that the Hubble horizon always becomes extremely large allowing the whole universe to be in causal contact. However, it is not hard to show that at late times one has, $$Rl
\rightarrow
\left\{
\begin{array}{ll}
0 & \textrm{\,\,\, for } \gamma=0, \\
t^{-1/2} & \textrm{\,\,\, for } \gamma=\frac{1}{3},
\end{array}
\right.$$ and then, even if the early period of loitering is not long enough the Hubble horizon always becomes larger than the spatial extend of the universe and the [*brane problem*]{} can be solved.
### Scaling
An important characteristic of the evolution of a self-interacting network of strings in a expanding universe is that it reaches a stage in which its characteristic length remains constant with respect to the size of the Hubble horizon $d_H$ [@Bennett:1986zn; @Bennett:1988vf; @Bennett:1989ak; @Bennett:1990yp]. An interesting question that we can answer with our analysis is whether this scaling regime is also reached when the cosmological dynamics is driven by the effects of a dilaton field.
When the evolution of the string network is assumed not to affect the standard (say, radiation- or matter-dominated) expansion of the universe, it makes no difference to talk about scaling with respect to the Hubble horizon or to the cosmic time because they are always proportional $d_H\sim l^{-1}\sim t$. However, when the back reaction on the dynamics of the gravitational background is taking into account the distinction between these two quantities become important because, as it happens in our problem, $d_H$ could become extremely large at some finite time ($l\rightarrow 0$).
To study the scaling properties of string configurations it is usually convenient to introduce a dimensionless parameter $\zeta(t)=t/L_w(t)$ measuring how far is the system from a scaling behaviour relative to cosmic time. The scaling regime is reached if $\zeta(t)$ relaxes to a constant value $\zeta_\ast$. In this situation, the decay of the number of winding modes can be expressed as, $$N_w
= \zeta^2_\ast
\left( \frac{l_c}{t}
\right)^2 \mathrm{e}^{2\lambda}
\sim \left\{
\begin{array}{ll}
t^{-2} & \textrm{\,\,\, for } \gamma=0, \\
t^{-1} & \textrm{\,\,\, for } \gamma=\frac{1}{3}.
\end{array}
\right.
\label{eq:scaling_decay}$$ which yields an energy density of winding modes that decrease with time independently of the equation of state of the loops created, $$\rho_w
= \zeta^2_\ast
\left( \frac{\tau}{t^2}
\right).$$ By substituting Eq. (\[eq:scaling\_decay\]) into the dynamical equation for $N_w$ given in (\[eq:winding\_energy\_rate\]), it can be seen that the parameters of a scaling solution must obey the following condition, $$c
= \frac{2(1-2\gamma+3\gamma^2)}{1+3\gamma^2}\zeta^{-2}_\ast
= \left\{
\begin{array}{ll}
2\zeta^{-2}_\ast & \textrm{\,\,\, for } \gamma=0, \\
\zeta^{-2}_\ast & \textrm{\,\,\, for } \gamma=\frac{1}{3}.
\end{array}
\right.
\label{eq:slope_condition}$$
![Characteristic length of the winding mode network as a function of cosmic time. This graph shows the scaling behaviour of the decay of winding modes. The plotted curves represent solutions of the equations of motion with parameters $c$ and $\gamma$ chosen in the same manner as in Fig. \[fig:phase\_space\]. \[fig:scaling\]](scaling_wl.eps){width="\columnwidth"}
[\*[5]{}[|c]{}|]{} & &\
& & & &\
$c=0.1$ & 0.2234500 & 0.0499299 & 0.3136377 & 0.0983686\
$c=1.0$ & 0.7069920 & 0.4998377 & 0.9945297 & 0.9890893\
$c=10$ & 2.2359990 & 4.9996915 & 3.1567680 & 9.9651842\
From our numerical simulations (see Fig. \[fig:scaling\] and Table \[tab:slopes\]) we readily observed that $\zeta(t)$ relaxes very rapidly to a constant value and, then, $L_w$ scales with cosmic time $t$. Moreover, apart from numerical errors, our solutions also satisfy the above scaling relation with high accuracy. To check if the system also scales with the cosmological horizon one has to recall from the previous subsection that at very early times the size of the horizon can become extremely large but soon after the winding modes have disappeared it behaves as, $$d_H
\sim
l^{-1}
\sim
\frac{1+3\gamma^2}{2\gamma}\, t.$$ Consequently, this means that for values of $\gamma>0$ the characteristic length of the network configuration will inevitably evolve to a scale comparable to the cosmological horizon. On the contrary, if $\gamma=0$, the string network can never scale with $d_H$ despite it does with cosmic time. This is closely related with the fact that, in this case, the universe reaches an indefinitely large period of loitering.
We would also like to mention that we have check numerically that these scaling properties do not change significantly if we take the typical size of the loops produced to scale with the characteristic length of the winding mode network, $L_l \sim L_w$, instead of taking a size that scales with cosmic time.
### A gas of non-static branes
The equation of state for the winding modes we have considered in our previous analysis, Eq. (\[eq:eq\_state\]), corresponds to a gas of non-relativistic branes. Let us briefly comment how we expect the cosmological dynamics will be affected if the early evolution is dominated by a gas of non-static branes. The generalised equation of state is characterised by a new parameter $\gamma_p$ given by (see for instance [@Boehm:2002bm; @Vilenkin:1994]), $$\gamma_p
= \left( \frac{p+1}{D}\langle v^2 \rangle
-\frac{p}{D}
\right),$$ where $\langle v^2 \rangle$ is the average of the squared velocity at each point of a brane. For $\langle v^2 \rangle \rightarrow 1$ the gas of branes behaves as a gas of relativistic particles whereas the non-relativistic or static limit corresponds to $\langle v^2 \rangle \rightarrow 0$. In the case of winding modes with $p=1$ in $D=3$ spatial dimensions we have a modified evolution equation for the Hubble parameter $l$ depending on this new characteristic velocity of the brane gas, $$\dot l
= fl
-\frac{1}{6}\,\mathrm{e}^\varphi
\left[ (1-2\langle v^2 \rangle ) E_w -3\gamma E_l
\right].$$ The first point to stress is that the very late-time cosmological evolution will not be qualitatively altered and the special lines $l=-\gamma f$ will still continue to be solutions of the equations of motion and attractors of the rest of trajectories close to the critical point $(0,0)$ in the phase space spanned by $(f,l)$. On the other hand, the rules for crossing these special lines away from the origin are significantly modified and this has interesting dynamical consequences. Eq (\[eq:crossing\_condition\]) now reads, $$\ddot\Phi
= - \frac{\mathrm{e}^\varphi}{6}
\left( 1 + 3\gamma - 2\langle v^2 \rangle
\right) E_w,$$ and then, it is very easy to see, following the same argumentation of previous sections, that now crossings from values of $f$ and $l$ with $l+\gamma f<0$ to values with $l+\gamma f>0$ will be allowed if, $$\langle v^2 \rangle
> \frac{3\gamma+1}{2}.$$ For $\gamma=0$ this means that the average velocity $\langle v^2 \rangle$ has to be greater than $1/2$. However, if this condition is satisfied the Hubble parameter $l$ cannot have a turning point with $fl>0$, that is with a negative value of $l$ (contracting phase). Thus, contrary to what happens when the branes are static, a universe that was initially expanding will ever stay expanding. But more interesting, if the universe started in a contracting phase it can now, for moderate values of the efficiency parameter, enter a stage of expansion before asymptotically reaching $(f,l)=(0,0)$. This stage of late expansion when the branes of the gas are not static makes the case in which the loops produced behave like ordinary matter to regain a phenomenological interest because this solves the obstruction discussed in previous sections to explain the dimensionality of our spacetime.
One might also ask whether it is still possible to ensure the small string coupling approximation dynamically for solutions with initial conditions satisfying the inequality $l<-f/3$ as for the $\langle v^2 \rangle=0$ case since now crossings of the special lines are in principle allowed. Fortunately, this criterion is still valid because the line $l+f/3=0$ cannot be cross from the region with $l+f/3<0$ (decreasing dilaton) to the region $l+f/3>0$ (growing dilaton) unless the brane gas has an exotic characteristic velocity which exceeds the velocity of light, $\langle v^2 \rangle > 1$.
With respect to the scaling properties of the winding mode decay, we do not expect significative modifications relative to the picture already outline for a gas of static branes.
Conclusions\[sec:Conclusions\]
==============================
In this work we have elaborated a detailed discussion of the late-time behaviour of brane gas cosmologies which have served to clarify and better understand some relevant issues of their dynamics.
As a conclusion we have found that in order to obtain a phenomenologically interesting cosmological evolution the loops produced without winding number have to behave as a gas with an equation of state characterised by a parameter satisfying $0<\gamma\leq 1/3$. Otherwise, the string coupling becomes large and the low-energy effective description fails (for $\gamma > 1/3$), or the spatial dimensions do not grow and the dimensionality of our present universe cannot be explained (for $\gamma \leq 0$). One would expect that the first difficulty could be alleviated simply by introducing higher-order quantum corrections. As we have seen, one can avoid the second obstacle by considering the dynamics of a gas with non-static branes. An alternative to obtain a late phase of expansion for $\gamma \leq 0$ is to include the effects of an axion or a moduli field. The problem with this option is that the dilaton evolves in general towards regions in phase space with a strong string coupling [@Foffa:1999; @Tsujikawa:2003pn]. In our analysis we have assumed for simplicity that the characteristic velocity of the brane gas $\langle v^2 \rangle$ is constant. An interesting open question would be to find out whether the more realistic situation in which this velocity also varies with cosmic time could reveal new cosmological features.
On the other hand, we have also confirmed that an early phase of loitering appears generically in brane gas scenarios if the efficiency parameter of the decay of winding modes takes moderate values. The existence of this phase in which the Hubble horizon becomes larger than the spatial extend of the universe offers a simple resolution to the [*brane problem*]{} as suggested in [@Alexander:2000xv]. In fact, the final fate of the universe in these type of cosmologies is always that of a loitering universe because the Hubble parameter goes to zero very slowly at late times. Thus, even for those cases in which the early period of loitering does not occur, or is not long enough to provide a causal microphysical explanation for a complete disappearance of branes, the [*brane problem*]{} can also be solved.
Finally, we have investigated whether the evolution of a network of winding modes driven by the dynamics of a dilaton field reaches a stage in which its characteristic length remains constant with respect to the size of the Hubble horizon $d_H$ as for ordinary cosmic strings in a expanding universe. We have shown that for values of $\gamma>0$ the characteristic length of the network will always scale with the cosmological horizon. However, for the particular case in which $\gamma=0$, the network never scales with $d_H$ despite it does with cosmic time. This is closely related with the fact that, in this case, the universe reaches an indefinitely large period of loitering.
The author thanks the support of the Alexander von Humboldt Foundation and the Universität Heidelberg.
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|
---
abstract: |
[*Constant-time*]{} programming is a countermeasure to prevent cache based attacks where programs should not perform memory accesses that depend on secrets. In some cases this policy can be safely relaxed if one can prove that the program does not leak more information than the public outputs of the computation.
We propose a novel approach for verifying constant-time programming based on a new information flow property, called [*output-sensitive noninterference*]{}. Noninterference states that a public observer cannot learn anything about the private data. Since real systems need to intentionally declassify some information, this property is too strong in practice. In order to take into account public outputs we proceed as follows: instead of using complex explicit declassification policies, we partition variables in three sets: input, output and leakage variables. Then, we propose a typing system to statically check that leakage variables do not leak [*more information about the secret inputs than the public normal output*]{}. The novelty of our approach is that we track the dependence of leakage variables with respect not only to the initial values of input variables (as in classical approaches for noninterference), but taking also into account the final values of output variables. We adapted this approach to LLVM IR and we developed a prototype to verify LLVM implementations.
author:
- Cristian Ene
- Laurent Mounier
- 'Marie-Laure Potet'
bibliography:
- 'cstTime.bib'
title: 'Output-sensitive Information flow analysis[^1]'
---
\[subsection\]
Introduction
============
An important task of cryptographic research is to verify cryptographic implementations for security flaws, in particular to avoid so-called timing attacks. Such attacks consist in measuring the execution time of an implementation on its execution platform. For instance, Brumley and Boneh [@bru2005] showed that it was possible to mount remote timing attacks by against OpenSSL’s implementation of the RSA decryption operation and to recover the key. Albrecht and Paterson [@alb2016] showed that the two levels of protection offered against the Lucky 13 attack from [@alb2013] in the first release of the new implementation of TLS were imperfect. A related class of attacks are [*cache-based attacks*]{} in which a malicious party is able to obtain memory-access addresses of the target program which may depend on secret data through observing cache accesses. Such attacks allow to recover the complete AES keys [@gul2011].
A possible countermeasure is to follow a very strict programming discipline called [**constant-time programming**]{}. Its principle is to avoid branchings controlled by secret data and memory load/store operations indexed by secret data. Recent secure C libraries such as NaCl [@ber2012] or mbedTLS[^2] follow this programming discipline. Until recently, there was no rigorous proof that constant-time algorithms are protected to cache-based attacks. Moreover, many cryptographic implementations such as PolarSSL AES, DES, and RC4 make array accesses that depend on secret keys and are not constant time. Recent works [@bar2014; @almeida2016; @bla2017] fill this gap and develop the first formal analyzes that allow to verify if programs are correct with respect to the constant-time paradigm.
An interesting extension was brought by Almeida et al. [@almeida2016] who enriched the constant-time paradigm *“distinguishing not only between public and private input values, but also between private and publicly observable output values*”. This distinction raises interesting technical and theoretical challenges. Indeed, constant-time implementations in cryptographic libraries like OpenSSL include optimizations for which paths and addresses can depend not only on public input values, but also on publicly observable output values. Hence, considering only input values as non-secret information would thus incorrectly characterize those implementations as non-constant-time. [@almeida2016] also develops a verification technique based on [*symbolic execution*]{}. However, the soundness of their approach depends in practice on the soundness of the underlying symbolic execution engine, which is very difficult to guarantee for real-world programs with loops. Moreover, their product construction can be very expensive in the worst case.
In this paper we deal with [*statically checking programs*]{} for [ **output-sensitive constant-time**]{} correctness: programs can still do branchings or memory accesses controlled by secret data if the information that is leaked is subsumed by the normal output of the program. To give more intuition about the property that we want to deal with, let us consider the following example, where $ct\_eq$ is a constant time function that allows to compare the arguments:
good = 1;
for (i=0; i<B_Size; i++){good = good & ct_eq(secret[i],in_p[i]);}
if (!good) { for(i=0; i<B_Size; i++) secret[i] = 0; }
return good;
Let suppose that the array variable $secret$ is secret, and all the other variables are public. Intuitively this a sort of one-time check password verifying that $in\_p=secret$ and otherwise overwrites the array $secret$ with zero. Obviously, this function is not constant-time as the variable $good$ depends on $secret$, and hence branching on $good$ violates the principles of constant-time programming. It is easy to transform this program into an equivalent one which is constant time. For example one could replace
if (!good) { for(i=0; i<B_Size; i++) secret[i] = 0; }
by
for (i=0; i<B_Size; i++) {secret[i] = secret[i] & ct_eq(good,1);}
But branching on $good$ is a benign optimization, since anyway, the value of $good$ is the normal output of the program. Hence, even if the function is not constant-time, it should be considered [**output-sensitive constant time**]{} with respect to its specification. Such optimization opportunities arise whenever the interface of the target application specifies what are the publicly observable outputs, and this information is sufficient to classify the extra leakage as benign [@almeida2016].
The objective of this work is to propose a [*static method*]{} to check if a program is [*output-sensitive constant time secure*]{}. We emphasize that our goal is [**not**]{} to verify that the legal output leaks “too much”, but rather to ensure that the unintended (side-channel) output does not leak [**more than**]{} this legal output.
First, we propose a novel approach for verifying constant-time security based on a new information flow property, called [ *output-sensitive noninterference*]{}. Information-flow security prevents confidential information to be leaked to public channels. Noninterference states that a public observer cannot learn anything about the private data. Since real systems need to intentionally declassify some information, this property is too strong. An alternative is [*relaxed noninterference*]{} which allows to specify explicit [*downgrading policies*]{}. In order to take into account public outputs while staying independent of how programs intentionally declassify information, we develop an alternative solution: instead of using complex explicit policies for functions, we partition variables in three sets: input, output and *leakage variables*. Hence we distinguish between the legal public output and the information that can leak through side-channels, expressed by adding fresh additional leakage variables. Then we propose a typing system that can statically check that leakage variables do not leak more secret information than the public normal output. The novelty of our approach is that we track the dependence of leakage variables with respect to both the [*initial value of input variables*]{} (as classically the case for noninterference) and [*final values of output variables*]{}. Then, we show how to verify that a program written in a high-level language is output-sensitive constant time secure by using this typing system.
Since timed and cache-based attacks target the executions of programs, it is important to carry out this verification in a language close to the machine-executed assembly code. Hence, we adapt our approach to a generic unstructured assembly language inspired from LLVM and we show how we can verify programs coded in LLVM. Finally, we developed a prototype tool implementing our type system and we show how it can be used to verify LLVM implementations.
To summarize, this work makes the following contributions described above: - in section \[s:2\] we reformulate output-sensitive constant-time as a new interesting noninterference property and we provide a sound type system that guarantees that well-typed programs are output-sensitive noninterferent;\
- in section \[s:3\] we show that this general approach can be used to verify that programs written in a high-level language are output-sensitive constant time;\
- in section \[llvm:sec\] we adapt our approach to the LLVM-IR language and we develop a prototype tool that can be used to verify LLVM implementations.
Output-sensitive non-interference {#s:2}
=================================
The language and Output-sensitive noninterference
--------------------------------------------------
In order to reason about the security of the code, we first develop our framework in [*While*]{}, a simple high-level structured programming language. In section \[s:3\] we shall enrich this simple language with arrays and in section \[llvm:sec\] we adapt our approach to a generic unstructured assembly language. The syntax of While programs is listed below: $$\begin{array}{lll}
c & ::= & x:=e ~|~ {\textsf{skip}}~|~ c_1;c_2~|~ {\textsf{If } e \textsf{ then } c_1 \textsf{
else } c_2 \textsf{ fi }}~|~ {\textsf{While } e \textsf{ Do } c \textsf{
oD }}
\end{array}$$ Meta-variables $x, e$ and $c$ range over the sets of program variables $Var$, expressions and programs, respectively. We leave the syntax of expressions unspecified, but we assume they are deterministic and side-effect free. The semantics is shown in Figure \[fig:while:sem\]. The reflexive and transitive closure of $\longrightarrow$ is denoted by $\Longrightarrow$. A state $\sigma$ maps variables to values, and we write $\sigma(e)$ to denote the value of expression $e$ in state $\sigma$. A configuration $(c, \sigma)$ is a program $c$ to be executed along with the current state $\sigma$.
$$\begin{array}{ll}
\begin{minipage}[h]{0.5\linewidth}
\begin{prooftree}
\AxiomC{ }
\UnaryInfC{$(x:=e,\sigma)\longrightarrow \sigma[x\mapsto\sigma(e)]$}
\end{prooftree}
\end{minipage}
&
\begin{minipage}[h]{0.4\linewidth}
\begin{prooftree}
\AxiomC{ }
\UnaryInfC{$({\textsf{skip}},\sigma)\longrightarrow \sigma$}
\end{prooftree}
\end{minipage}
\\
\\
\begin{minipage}[h]{0.5\linewidth}
\begin{prooftree}
\AxiomC{$(c_1,\sigma)\longrightarrow \sigma'$}
\UnaryInfC{$( c_1;c_2,\sigma)\longrightarrow (c_2, \sigma')$}
\end{prooftree}
\end{minipage}
&
\begin{minipage}[h]{0.4\linewidth}
\begin{prooftree}
\AxiomC{$(c_1,\sigma)\longrightarrow (c'_1,\sigma')$}
\UnaryInfC{$( c_1;c_2,\sigma)\longrightarrow (c'_1;c_2, \sigma')$}
\end{prooftree}
\end{minipage}
\\
\\
\begin{minipage}[h]{0.5\linewidth}
\begin{prooftree}
\AxiomC{$\sigma(e) = 1 ~ ? ~ i=1 : \ i=2$}
\UnaryInfC{$({\textsf{If } e \textsf{ then } c_1 \textsf{
else } c_2 \textsf{ fi }},\sigma)\longrightarrow (c_i, \sigma)$}
\end{prooftree}
\end{minipage}
&
\begin{minipage}[h]{0.4\linewidth}
\begin{prooftree}
\AxiomC{$\sigma(e) \not= 1$}
\UnaryInfC{$({\textsf{While } e \textsf{ Do } c \textsf{
oD }},\sigma)\longrightarrow \sigma$}
\end{prooftree}
\end{minipage}
\\
\\
\begin{minipage}[h]{0.5\linewidth}
\begin{prooftree}
\AxiomC{$\sigma(e) = 1$}
\UnaryInfC{$({\textsf{While } e \textsf{ Do } c \textsf{
oD }},\sigma)\longrightarrow (c;{\textsf{While } e \textsf{ Do } c \textsf{
oD }},\sigma)$}
\end{prooftree}
\end{minipage}
\end{array}$$
Intuitively, if we want to model the security of some program $c$ with respect to side-channel attacks, we can assume that there are three special subsets of variables: $X_I$ the public input variables, $X_O$ the public output variables and $X_L$ the variables that leak information to some malicious adversary. Then, output sensitive nonintereference asks that every two complete executions starting with $X_I$-equivalent states and ending with $X_O$-equivalent final states must be indistinguishable with respect to the leakage variables $X_L$ .
\[iol-sec\](adapted from [@almeida2016]) Let $X_I, X_O, X_L\subseteq Var$ be three sets of variables, intended to represent the input, the output and the leakage of a program. A program $c$ is [**($X_I, X_O, X_L$)-secure**]{} when all its executions starting with $X_I $-equivalent stores and leading to $X_O$-equivalent final stores, give $X_L$-equivalent final stores. Formally, for all $\sigma, \sigma', \rho, \rho'$, if ${\langle c, \sigma \rangle \Longrightarrow \sigma'}$ and ${\langle c, \rho \rangle \Longrightarrow \rho'}$ and $\sigma =_{X_I} \rho $ and $\sigma'=_{X_O}\rho'$, then $\sigma'=_{X_L}\rho'$.
Typing rules {#s:w1}
------------
This section introduces a type-based information flow analysis that allows to check whether a While program is output-sensitive noninterferent, i.e. the program does not leak more information about the secret inputs than the public normal output.
As usual, we consider a flow lattice of security levels ${\mathcal{L}}$. An element $x$ of ${\mathcal{L}}$ is an atom if $x\not=\bot$ and there exists no element $y \in {\mathcal{L}}$ such that $\bot \sqsubset y \sqsubset x$. A lattice is called *atomistic* if every element is the join of atoms below it.
\[as1\] Let $({\mathcal{L}}, {\sqcap}, {\sqcup}, \bot, \top)$ be an atomistic continuous bounded lattice. As usual, we denote $t_1{\sqsubseteq}t_2$ iff $t_2 = t_1 {\sqcup}t_2$. We assume that there exists a distinguished subset ${\mathcal{T}}_O\subseteq {\mathcal{L}}$ of atoms. Hence, from the above assumption, for any $\tau_o, \tau'_o\in {\mathcal{T}}_O$ and for any $t_1,t_2\in {\mathcal{L}}$: it holds that
- \[a1\] $\tau_o{\sqsubseteq}\tau'_o$ implies $\tau_o = \tau'_o$,
- \[a2\] $\tau_o{\sqsubseteq}t_1 \sqcup t_2$ implies $\tau_o{\sqsubseteq}t_1$ or $\tau_o{\sqsubseteq}t_2$,
- \[a3\] $\tau_o{\sqsubseteq}t_1$ implies that there exists $t\in {\mathcal{L}}$ such that $t_1= t \sqcup \tau_o$ and $\tau_0 \not{\sqsubseteq}t$.
A type environment $\Gamma : Var \mapsto {\mathcal{L}}$ describes the [**security levels**]{} of variables and the dependency with respect to the [**current**]{} values of variables in $X_O$. In order to catch dependencies with respect to current values of output variables, we associate to each output variable $o\in X_O$ a fixed and unique symbolic type $\alpha(o)\in {\mathcal{T}}_O$. For example if some variable $x\in Var$ has the type $\Gamma(x)=Low {\sqcup}\alpha(o)$, it means that the value of $x$ depends only on public input and the current value of the output variable $o\in X_O$.
Hence, we assume that there is a fixed injective mapping $\alpha : X_0\mapsto {\mathcal{T}}_0$ such that $\displaystyle\bigwedge_{o_1,o_2\in X_O} \big(o_1\not=o_2 \Rightarrow \alpha(o_1)\not=\alpha(o_2)\big) \wedge \bigwedge_{o\in X_O} \big(\alpha(o)\in {\mathcal{T}}_O \big)$. We extend mappings $\Gamma$ and $\alpha$ to sets of variables in the usual way: given $A\subseteq Var$ and $B\subseteq X_O$ we note $\displaystyle\Gamma(A) {\stackrel{def}=}\bigsqcup_{x\in A} \Gamma(x)$ , $\displaystyle\alpha(B) {\stackrel{def}=}\bigsqcup_{x\in B} \alpha(x)$.
Our type system aims to satisfy the following output sensitive non-interference condition: if the *final* values of output variables in $X_O$ remain the same, only changes to *initial* inputs with types ${\sqsubseteq}t$ should be visible to *leakage* outputs with type ${\sqsubseteq}t \sqcup \alpha(X_O)$. More precisely, given a derivation [$\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{}, the final value of a variable $x$ with final type $\Gamma'(x) = t {\sqcup}\alpha(A)$ for some $t\in {\mathcal{L}}$ and $A\subseteq X_O$, should depend at most on the initial values of those variables $y$ with initial types $\Gamma(y){\sqsubseteq}t$ and on the final values of variables in $A$. We call “real dependencies” the dependencies with respect to initial values of variables and “symbolic dependencies” the dependencies with respect to the current values of output variables. Following [@Hunt91] we formalize the non-interference condition satisfied by the typing system using reflexive and symmetric relations.
We write $=_{A_0}$ for relation which relates mappings which are equal on all values in $A_0$ i.e. for two mappings $f_1,f_2 : A \mapsto B$ and $A_0\subseteq A$, ${\ensuremath{f_1=_{A_0}f_2}}$ iff $\forall a\in A_0, f_1(a)=f_2(a)$. For any mappings $f_1 : A_1 \mapsto B$ and $f_2 : A_2 \mapsto B$, we write $f_1[f_2]$ the operation which updates $f_1$ according to $f_2$, namely $$f_1[f_2](x) {\stackrel{def}=}\left\{
\begin{array}{l l}
f_2(x) & \quad \text{if $x\in A_1\cap A_2$}\\
f_1(x) & \quad \text{if $x\in A_1\setminus A_2$}\\ \end{array} \right.$$ Given $\Gamma : Var \mapsto {\mathcal{L}}$ , $X\subseteq Var$ and $t\in {\mathcal{L}}$, we write $=_{\Gamma,X,t}$ for the reflexive and symmetric relation which relates states that are equal on all variables having type $v\sqsubseteq t$ in environment $\Gamma$, provided that they are equal on all variables in $X$: $\sigma=_{\Gamma,X,t}\sigma'$ iff $\sigma=_X\sigma' \Rightarrow \big( \forall x, ( \Gamma(x) {\sqsubseteq}t \Rightarrow \sigma(x)= \sigma'(x))\big)$. When $X = \emptyset$, we omit it, hence we write $=_{\Gamma,t}$ instead of $=_{\Gamma,\emptyset, t}$.
[@hunt2006] Let ${\mathcal{R}}$ and ${\mathcal{S}}$ be reflexive and symmetric relations on states. We say that program $c$ maps ${\mathcal{R}}$ into ${\mathcal{S}}$, written $c: {\mathcal{R}}\Longrightarrow {\mathcal{S}}$, iff $\forall \sigma, \rho$, if ${\langle c, \sigma \rangle \Longrightarrow \sigma'}$ and ${\langle c, \rho \rangle \Longrightarrow \rho'}$ then $\sigma {\mathcal{R}}\rho \Rightarrow \sigma' {\mathcal{S}}\rho'$.
The type system we propose enjoys the following useful property:\
[$\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} c : =$_{\Gamma,\Gamma(X_I)} \;\Longrightarrow\; =_{\Gamma',X_O,\alpha(X_O) \sqcup \Gamma(X_I)}$\
This property is an immediate consequence of Theorem \[l:sec\].
Hence, in order to prove that the above program $c$ is output sensitive non-interferent according to Definition \[iol-sec\], it is enough to check that for all $x_l \in X_L$, $\Gamma'(x_l) {\sqsubseteq}\alpha(X_O) \sqcup \Gamma(X_I)$. Two executions of the program $c$ starting from initial states that coincide on input variables $X_I$, and ending in final states that coincide on output variables $X_O$, will coincide also on the leaking variables $X_L$.
We now formally introduce our typing system. Due to assignments, values and types of variables change dynamically. For example let us assume that at some point during the execution, the value of $x$ depends on the initial value of some variable $y$ and the current value of some output variable $o$ (which itself depends on the initial value of some variable $h$), formally captured by an environment $\Gamma$ where $\Gamma(o)=\Gamma_0(h)$ and $\Gamma(x) = \Gamma_0(y) {\sqcup}\alpha(o)$, where $\Gamma_0$ represents the initial environment. If the next to be executed instruction is some assignment to $o$, then the current value of $o$ will change, so we have to mirror this in the new type of $x$: even if the value of $x$ does not change, its new type will be $\Gamma'(x) = \Gamma_0(y) {\sqcup}\Gamma_0(h)$ (assuming that $\alpha(o)\not{\sqsubseteq}\Gamma_0(y)$). Hence $\Gamma'(x)$ is obtained by replacing in $\Gamma(x)$ the symbolic dependency $\alpha(o)$ with the real dependency $\Gamma(o)$.
\[def\_r\] \[triangle\] If $t^0\in {\mathcal{T}}_O$ is an atom and $t',t\in {\mathcal{L}}$ are arbitrary types, then we denote by $t[t'/t^0]$ the type obtained by replacing (if any) the occurrence of $t^0$ by $t'$ in the decomposition $At(t)$ in atoms of $t$ $$\displaystyle t[t'/t^0] {\stackrel{def}=}\left\{
\begin{array}{l l}
t &\quad \text{if $t^0\not\in \ At(t)$}\\
t' {\sqcup}{\sqcup}_{b\in At(t)\setminus \{t^0\}}b &\quad \text{if $t^0\in \ At(t)$}\\
\end{array} \right.$$ Now we extend this definition to environments: let $x\in X_O$ and $p\in {\mathcal{L}}$. Then $\Gamma_1{\stackrel{def}=}\Gamma\lhd_\alpha x$ represents the environment where the symbolic dependency on the last value of $x$ of all variables is replaced by the real type of $x$: $\Gamma_1(y) {\stackrel{def}=}(\Gamma(y))[\Gamma(x)/\alpha(x)]$. Similarly, $(p, \Gamma) \lhd_\alpha x{\stackrel{def}=}p[\Gamma(x)/\alpha(x)]$.
The following lemma is an immediate consequence of the Assumption \[as1\] and Definition \[def\_r\].
\[lrs\] Let $x\in X_O$, $p\in {\mathcal{L}}$ and let us denote $\Gamma_1{\stackrel{def}=}\Gamma\lhd_\alpha x$ and $p_1 {\stackrel{def}=}(p, \Gamma) \lhd_\alpha x$. If $\alpha(x)\not{\sqsubseteq}\Gamma(x)$, then,
1. \[lrs0\] For any $v\in X$, $\Gamma_1(v) = (\Gamma(v), \Gamma) \lhd_\alpha x$,
2. \[lrs1\] For all variables $y\in Var$, $\alpha(x)\not{\sqsubseteq}\Gamma_1(y)$.
3. \[lrs2\] $\alpha(x)\not{\sqsubseteq}p_1$.
We want now to extend the above definition from a single output variable $x$ to subsets $X\subseteq X_O$. Our typing system will ensure that each generated environment $\Gamma$ will not contain circular symbolic dependencies between output variables, i.e., there are no output variable $o_1,o_2 \in X_O$ such that $\alpha(o_1) {\sqsubseteq}\Gamma(o_2)$ and $\alpha(o_2) {\sqsubseteq}\Gamma(o_1)$. We can associate a graph ${\mathcal{G}}(\Gamma)=(X_O, E)$ to an environment $\Gamma$, such that $(o_1,o_2)\in E$ iff $\alpha(o_1){\sqsubseteq}\Gamma(o_2)$. We say that $\Gamma$ is [**well formed**]{}, denoted [$\mathcal{AC}(\Gamma)$]{}, if ${\mathcal{G}}(\Gamma)$ is an acyclic graph. For acyclic graphs ${\mathcal{G}}(\Gamma)$, we define a preorder over $X_O$, denoted ${\sqsubseteq}_\Gamma$, as the transitive closure of the relation $\{(o_1,o_2) \in X_O \times X_O \ \mid \ \alpha (o_1) {\sqsubseteq}\Gamma (o_2)\}$, i.e. $o_1{\sqsubseteq}_\Gamma o_2$ iff there is a path from $o_1$ to $o_2$ in ${\mathcal{G}}(\Gamma)$. We also define the [*reachable variables* ]{} of $x\in X_O$ w.r.t. $\Gamma$, denoted [**${\ensuremath{\mathcal{GT}_{\Gamma}(x)}}$**]{}, to be the set of all $o\in X_O$ such that $x{\sqsubseteq}_\Gamma o$. Now, for acyclic graphs ${\mathcal{G}}(\Gamma)$, we can extend Definition \[triangle\] to subsets $X\subseteq X_O$, by first fixing an ordering $X=\{x_1,x_2,\ldots x_n\}$ of variables in $X_O$ compatible with the graph (i.e. $j\leq k$ implies that $x_j \not{\sqsubseteq}_\Gamma x_k$), and then $(p, \Gamma) \lhd_\alpha X {\stackrel{def}=}(((p, \Gamma)\lhd_\alpha x_1 ) \lhd_\alpha x_2) \ldots \lhd_\alpha x_n$. We also denote $\Gamma \lhd_\alpha X {\stackrel{def}=}((\Gamma\lhd_\alpha x_1 ) \lhd_\alpha x_2) \ldots \lhd_\alpha x_n$ (in this case the ordering is not important, i.e. $(\Gamma\lhd_\alpha x_1 ) \lhd_\alpha x_2 = (\Gamma\lhd_\alpha x_2 ) \lhd_\alpha x_1$).
The following lemma can be proved by induction on the size of $X$ using Lemma \[lrs\].
\[lrsA\] Let $X\subseteq X_O$, $p\in {\mathcal{L}}$ and let us denote $\Gamma_2{\stackrel{def}=}\Gamma\lhd_\alpha X$ and $p_2 {\stackrel{def}=}(p, \Gamma) \lhd_\alpha X$. If ${\ensuremath{\mathcal{AC}(\Gamma)}}$, then,
1. \[lrsA0\] For any $v\in X$, $\Gamma_2(v) = (\Gamma(v), \Gamma) \lhd_\alpha X$,
2. \[lrsA1\] For all variables $x\in X$, and all variables $y\in Var$, $\alpha(x)\not{\sqsubseteq}\Gamma_2(y)$.
3. \[lrsA2\] For all variables $x\in X$, $\alpha(x)\not{\sqsubseteq}p_2$.
Next Lemma gives a precise characterization of the new preorder induced by the application of the operator $ \lhd_\alpha$.
\[lrsB\] Let $\Gamma$ be a well formed environment and let $x\in X_O$ and $X\subseteq X_O$. Let us denote $\Gamma_1{\stackrel{def}=}\Gamma\lhd_\alpha x$ and $\Gamma_2{\stackrel{def}=}\Gamma\lhd_\alpha X$. Then $\Gamma_1$ and $\Gamma_2$ are well formed. Moreover, ${\sqsubseteq}_{\Gamma_1} ={\sqsubseteq}_\Gamma \setminus \{(x, o) \mid \ o\in X_O \}$ and ${\sqsubseteq}_{\Gamma_2} ={\sqsubseteq}_\Gamma \setminus \{(x, o) \mid \ x\in X, o\in X_O\}$.
The key remark is that any edge of ${\mathcal{G}}(\Gamma_1)$ where $\Gamma_1{\stackrel{def}=}\Gamma\lhd_\alpha x$ corresponds to either an edge or to a path of length two in ${\mathcal{G}}(\Gamma)$. Indeed, let $x_1,x_2\in X_O$ such that there exists an edge from $x_1$ to $x_2$ in ${\mathcal{G}}(\Gamma_1)$, that is $\alpha(x_1){\sqsubseteq}\Gamma_1(x_2)=\Gamma(x_2)[\Gamma(x)/\alpha(x)]$. Then either $\alpha(x_1){\sqsubseteq}\Gamma(x_2)$ or $\alpha(x){\sqsubseteq}\Gamma(x_2)$ and $\alpha(x_1){\sqsubseteq}\Gamma(x)$. Hence either there is an edge from $x_1$ to $x_2$ in ${\mathcal{G}}(\Gamma)$ or there must exist edges from $x_1$ to $x$ and from $x$ to $x_2$ (and hence a path of length two from $x_1$ to $x_2$) in ${\mathcal{G}}(\Gamma)$. Now the assertion of the Lemma is an immediate consequence of the above remark and Lemma \[lrs\].
Let ${\mathbf{aff}}(c)$ be the set of assigned variables in a program $c$, formally defined by: $${\mathbf{aff}}(c) {\stackrel{def}=}\left\{
\begin{array}{l l}
\{x\} &\quad \text{if $c\equiv x:=e$}\\
\emptyset& \quad \text{if $c\equiv skip$}\\
{\mathbf{aff}}(c_1) \cup {\mathbf{aff}}(c_2) & \quad \text{if $c\equiv c_1;c_2$}\\
{\mathbf{aff}}(c_1) \cup {\mathbf{aff}}(c_2) & \quad \text{if $c\equiv {\textsf{If } e \textsf{ then } c_1 \textsf{
else } c_2 \textsf{ fi }}$}\\
{\mathbf{aff}}(c) &\quad \text{if $c\equiv {\textsf{While } e \textsf{ Do } c \textsf{
oD }}$}\\
\end{array} \right.$$ and let us denote ${\mathbf{aff^I}}(c) {\stackrel{def}=}{\mathbf{aff}}(c)\cap(Var\setminus X_O)$ and ${\mathbf{aff^O}}(c) {\stackrel{def}=}{\mathbf{aff}}(c)\cap X_O$.
[p[0.5]{} p[0.5]{}]{}
&
\
[p[0.3]{} p[0.7]{}]{}
&
\
[p[0.34]{} p[0.66]{}]{}
&
\
\
[c]{}
\
[c]{}\
We define the ordering over environments as usual: $$\displaystyle\Gamma_1 \sqsubseteq \Gamma_2 {\stackrel{def}=}\bigwedge_{x\in Var} \Gamma_1(x) {\sqsubseteq}\Gamma_2(x).$$
We also define a restricted ordering over environments: $$\displaystyle\Gamma_1 \sqsubseteq_r \Gamma_2 {\stackrel{def}=}\bigwedge_{x\in Var} \Gamma_1(x) {\sqsubseteq}\Gamma_2(x) \wedge \bigwedge_{o\in X_O, x\in Var} (\alpha(o) {\sqsubseteq}\Gamma_2(x) \Rightarrow \alpha(o) {\sqsubseteq}\Gamma_1(x)) .$$
Is is immediate that $\Gamma_1 \sqsubseteq_r \Gamma_2$ implies $\Gamma_1 \sqsubseteq \Gamma_2$. Intuitively, when enriching an environment using ${\sqsubseteq}_r$, we have the right to add only “real dependencies” (and not “symbolic” dependencies with respect to variables in $X_O$). We adapt this definition for elements $t_1,t_2\in {\mathcal{L}}$ as well: we denote $t_1{\sqsubseteq}_r t_2$ when $t_1{\sqsubseteq}t_2$, and for all $o\in X_O$, $\alpha(o){\sqsubseteq}t_2 \Rightarrow \alpha(o){\sqsubseteq}t_1$, i.e. $t_2$ does not contain new symbolic dependencies w.r.t. $t_1$.
Next lemma is immediate from the definitions.
\[pl2\] Let $\Gamma_1, \Gamma_2, \Gamma_3$ such that $\Gamma_1\sqsubseteq \Gamma_2$, $p_1,p_2\in {\mathcal{L}}$, $o\in X_O$ and $X\subseteq X_O$ with $p_1{\sqsubseteq}_r p_2$. Then $\Gamma_1 \lhd_\alpha o \sqsubseteq \Gamma_2 \lhd_\alpha o$, $\Gamma_1 \lhd_\alpha X \sqsubseteq \Gamma_2 \lhd_\alpha X$, $(p_1, \Gamma_1) \lhd_\alpha o \sqsubseteq (p_2, \Gamma_2) \lhd_\alpha o$ and $(p_1, \Gamma_1) \lhd_\alpha X \sqsubseteq (p_2, \Gamma_2) \lhd_\alpha X$. Moreover, $\Gamma_1 \lhd_\alpha X {\sqsubseteq}_r \Gamma_3$ implies that $\Gamma_3 \lhd_\alpha X = \Gamma_3$.
The last assertion of the lemma is a consequence of the remark that $\Gamma_1 \lhd_\alpha X {\sqsubseteq}_r \Gamma_3$ implies that $\Gamma_3$ does not contain more “symbolic dependencies” than $\Gamma_1 \lhd_\alpha X$, and $\Gamma_1 \lhd_\alpha X$ does not contain any “symbolic dependencies” with respect to variables in $X$. Obviously, using that ${\sqsubseteq}_r \subseteq {\sqsubseteq}$, all inequalities hold also when the premise $\Gamma_1\sqsubseteq \Gamma_2$ is replaced by $\Gamma_1\sqsubseteq_r \Gamma_2$.
For a command $c$, judgements have the form [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} where $p\in {\mathcal{L}}$ and $\Gamma$ and $\Gamma'$ are type environments well-formed. The inference rules are shown in Figure \[fig:T1\]. The idea is that if $\Gamma$ describes the security levels of variables which hold before execution of $c$, then $\Gamma'$ will describe the security levels of those variables after execution of $c$. The type $p$ represents the usual program counter level and serves to eliminate indirect information flows; the derivation rules ensure that all variables that can be changed by $c$ will end up (in $\Gamma'$) with types greater than or equal to $p$. As usual, whenever $p=\bot$ we drop it and write [$\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} instead of [$\bot\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{}. Throughout this paper the type of an expression $e$ is defined simply by taking the lub of the types of its free variables $\Gamma[\alpha](fv(e))$, for example the type of $x+y+o$ where $o$ is the only output variable is $\Gamma(x){\sqcup}\Gamma(y){\sqcup}\alpha(o)$. This is consistent with the typing rules used in many systems, though more sophisticated typing rules for expressions would be possible in principle. Notice that considering the type of an expression to be $\Gamma[\alpha](fv(e))$ instead of $\Gamma(fv(e))$ allows to capture the dependencies with respect to the current values of output variables. In order to give some intuition about the rules, we present a simple example in Figure \[fig:ex1\].
\[ex2:5\] Let $\{x,y,z,u\}\subseteq Var\setminus X_O$ and $\{o_1,o_2,o_3\}\subseteq X_O$ be some variables, and let us assume that $\forall i\in \{1,2,3\}$, $\alpha(o_i)={\overline{O_i}}$. We assume that the initial environment is $\Gamma_0= [x\rightarrow X, y\rightarrow Y, z\rightarrow Z, u\rightarrow U, o_1\rightarrow O_1, o_2\rightarrow O_2, o_3\rightarrow O_3]$. Since the types of variables $x,u$ and $o_3$ do not change, we omit them in the following. We highlighted the changes with respect to the previous environment.
$$\begin{array}{llr}
& & \! \! \! \! \! \! p=\bot, \ \ \ \Gamma_0= [y\rightarrow Y, z\rightarrow Z, o_1\rightarrow O_1, o_2\rightarrow O_2]\\
(1) & o_1:=x+1 & \\
& & \! \! \! \Gamma_1= [y\rightarrow Y, z\rightarrow Z, {\bf o_1\rightarrow X}, o_2\rightarrow O_2]\\
(2) & y:=o_1+z & \\
& & \! \! \! \Gamma_2= [{\bf y\rightarrow {\overline{O_1}}{\sqcup}Z}, z\rightarrow Z, o_1\rightarrow X, o_2\rightarrow O_2]\\
(3) & o_1:=u & \\
& & \! \! \! \Gamma_3= [{\bf y\rightarrow X{\sqcup}Z}, z\rightarrow Z, {\bf o_1\rightarrow U}, o_2\rightarrow O_2]\\
(4) & z:=o_1+o_3 & \\
& & \! \! \! \! \! \! \Gamma_4= [y\rightarrow X{\sqcup}Z,{\bf z\rightarrow {\overline{O_1}}{\sqcup}{\overline{O_3}}}, o_1\rightarrow U, o_2\rightarrow O_2]\\
(5) &\textsf{If } (o_2=o_3+x) & {\bf p={\overline{O_3}}{\sqcup}O_2{\sqcup}X}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
(6) & \ \textsf{then} \ \ o_1:=o_2 & \\
& & \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \Gamma_6= [y\rightarrow X{\sqcup}Z,{\bf z\rightarrow U{\sqcup}{\overline{O_3}}}, {\bf o_1\rightarrow {\overline{O_3}}{\sqcup}O_2{\sqcup}X{\sqcup}{\overline{O_2}}}, o_2\rightarrow O_2]\\
(7) & \ \textsf{else} \ \ o_2:=o_1 & \\
& & \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \Gamma_7= [y\rightarrow X{\sqcup}Z, z\rightarrow {\overline{O_1}}{\sqcup}{\overline{O_3}}, o_1\rightarrow U, o_2\rightarrow {\bf {\overline{O_3}}{\sqcup}O_2{\sqcup}X{\sqcup}{\overline{O_1}}}]\\
(8) & \ \textsf{fi} & \\
& & \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \Gamma_8= (\Gamma_6 \lhd_\alpha o_2) {\sqcup}(\Gamma_7 \lhd_\alpha o_1) = [y\rightarrow X{\sqcup}Z, {\bf z\rightarrow U{\sqcup}{\overline{O_3}}}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
& & o_1\rightarrow {\bf {\overline{O_3}}{\sqcup}O_2{\sqcup}X{\sqcup}U}, o_2\rightarrow {\bf {\overline{O_3}}{\sqcup}O_2{\sqcup}X{\sqcup}U}]\\
\end{array}$$
After the first assignment, the type of $o_1$ becomes $X$, meaning that the current value of $o_1$ depends on the initial value of $x$. After the assignment $y:=o_1+z$, the type of $y$ becomes ${\overline{O_1}}{\sqcup}Z$, meaning that the current value of $y$ depends on the initial value of $z$ and the current value of $o_1$. After the assignment $o_1=u$, the type of $y$ becomes $X{\sqcup}Z$ as $o_1$ changed and we have to mirror this in the dependencies of $y$, and the type of $o_1$ becomes $X$. When we enter in the $\textsf{If }$, the program counter level changes to $p={\overline{O_3}}{\sqcup}O_2{\sqcup}X$ as the expression $o_2=o_3+x$ depends on the values of variables $o_2,o_3,x$, but $o_2$ and $o_3$ are output variables and $o_2$ will be assigned by the $\textsf{If }$ command, hence we replace the “symbolic” dependency $\alpha(o_2)={\overline{O_2}}$ by its “real” dependency $\Gamma(o_2)=O_2$. At the end of the $\textsf{If }$ command, we do the join of the two environments obtained after the both branches, but in order to prevent cycles, we first replace the “symbolic” dependencies by the corresponding “real” dependencies for each output variable that is assigned by the other branch.
Well-formed environments
------------------------
In this section we prove that, if the initial environment is well-formed, then all the environments generated by the typing system are well-formed too. To do so, the following lemma states some useful properties, where for any $p\in {\mathcal{L}}$, we denote by ${\mathbf{atomO}}(p){\stackrel{def}=}\{o\in X_O \ \mid \ \alpha(o){\sqsubseteq}p\}$.
\[pl1\]For all $\Gamma$, if [$\mathcal{AC}(\Gamma)$]{} and [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} and ${\mathbf{atomO}}(p)\cap {\mathbf{aff^O}}(c) = \emptyset$ then
1. \[pl1:0\] [$\mathcal{AC}(\Gamma')$]{},
2. \[pl1:1\] for any $o\in X_O \setminus({\mathbf{aff^O}}(c)\cup {\mathbf{atomO}}(p))$, ${\ensuremath{\mathcal{GT}_{\Gamma'}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}$,
3. \[pl1:1B\] for any $o\in {\mathbf{atomO}}(p)$, ${\ensuremath{\mathcal{GT}_{\Gamma'}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}\cup {\mathbf{aff^O}}(c)$,
4. \[pl1:2\] for any $o\in {\mathbf{aff^O}}(c)$, ${\ensuremath{\mathcal{GT}_{\Gamma'}(o)}}\subseteq {\mathbf{aff^O}}(c)$,
5. \[pl1:3\] for any $x\not\in {\mathbf{aff}}(c)$, $(\Gamma\lhd_\alpha {\mathbf{aff^O}}(c))(x) {\sqsubseteq}_r \Gamma'(x)$.
Proof is by induction on the derivation of [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} for all assertions in the same time. We do a case analysis according to the last rule applied (case Skip is trivial).
- $c$ is an assignment $x:=e$ for some $x\not\in X_O$.\
Since $x\not\in X_O$, it follows that ${\mathcal{G}}(\Gamma')= {\mathcal{G}}(\Gamma)$ and obviously for any $x\not\in {\mathbf{aff}}(c)$, $\Gamma(x)=\Gamma'(x)$ and ${\sqsubseteq}_{\Gamma} = {\sqsubseteq}_{\Gamma'}$ and [$\mathcal{AC}(\Gamma)$]{} implies [$\mathcal{AC}(\Gamma')$]{}.
- $c$ is an assignment $o:=e$ for some $o\in X_O\setminus fv(e)$. Hence in this case ${\mathbf{aff}}(c)={\mathbf{aff^O}}(c)=\{o\}$, and by assumption, $\alpha(o)\not{\sqsubseteq}p$.
Then $\Gamma' =\Gamma_1[o \mapsto p{\sqcup}\Gamma_1[\alpha](fv(e))]$ with $\Gamma_1=\Gamma\lhd_\alpha o$. By Lemma \[lrsB\], ${\ensuremath{\mathcal{AC}(\Gamma_1)}}$, and using Lemma \[lrs\] we get that ${\ensuremath{\mathcal{AC}(\Gamma_1)}}$ does not contain any edge with origin $o$ and hence ${\ensuremath{\mathcal{AC}(\Gamma')}}$. The second part follows from the remark that using Lemma \[lrsB\] we get ${\sqsubseteq}_{\Gamma_1} ={\sqsubseteq}_\Gamma \setminus \{(o, o') \mid \ o'\in X_O \}$ and ${\sqsubseteq}_{\Gamma'} ={\sqsubseteq}_{\Gamma_1} \cup \{(o', o) \mid \ o'\in {\mathbf{atomO}}(p) \}$ , and for any $o'\not=o$, $\Gamma_1(o')=\Gamma(o')$. Hence, ${\ensuremath{\mathcal{GT}_{\Gamma'}(o)}}=\emptyset$. Finally, for any $x\not\in {\mathbf{aff}}(c)$, $(\Gamma\lhd_\alpha o)(x) = \Gamma_1 (x)=\Gamma'(x)$.
- $c$ is an assignment $o:=e$ for some $o\in X_O\cap fv(e)$. Similar to the previous case.
- $c$ is $c_1;c_2$. Then [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c_1;c_2\}\Gamma_2}}$]{} was inferred based on [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c_1\}\Gamma_1}}$]{} and [$p\vdash_{\alpha} {\ensuremath{\Gamma_1\{c_2\}\Gamma_2}}$]{}. Let us denote $U_i={\mathbf{aff^O}}(c_i)$, for $i=1,2$.
Using the induction hypothesis for [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c_1\}\Gamma_1}}$]{} we get that ${\ensuremath{\mathcal{AC}(\Gamma_1)}}$, and for any $o\in X_O\setminus( U_1\cup {\mathbf{atomO}}(p))$, ${\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}$, for any $o\in {\mathbf{atomO}}(p)$, ${\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}\cup U_1$ and for any $o\in U_1$, ${\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\subseteq U_1$. In addition, for any $x\not\in U_1$, $(\Gamma\lhd_\alpha U_1))(x) {\sqsubseteq}_r \Gamma_1(x)$.
Using the induction hypothesis for [$p\vdash_{\alpha} {\ensuremath{\Gamma_1\{c_2\}\Gamma_2}}$]{} we get that ${\ensuremath{\mathcal{AC}(\Gamma_2)}}$, and for any $o\in X_O\setminus (U_2\cup {\mathbf{atomO}}(p))$, ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}$, for any $o\in {\mathbf{atomO}}(p)$, ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\cup U_2$ and for any $o\in U_2$, ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq U_2$. In addition, for any $x\not\in U_2$, $(\Gamma_1 \lhd_\alpha U_2)(x) {\sqsubseteq}_r \Gamma_2(x)$.
Since $ {\mathbf{aff^O}}(c)=U_1\cup U_2$, for any $o\in X_O\setminus (U_1\cup U_2\cup {\mathbf{atomO}}(p))$, we have both $o\in X_O\setminus (U_1\cup {\mathbf{atomO}}(p))$ and $o\in X_O\setminus (U_2\cup {\mathbf{atomO}}(p))$ and hence ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}$.
For any $o\in {\mathbf{atomO}}(p)$, ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\cup U_2 \subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}\cup U_1 \cup U_2$.
Now, for any $o\in (U_1\cup U_2)$, either $o\in U_2$ or $o\in U_1\setminus U_2$. If $o\in U_2$, then ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq U_2 \subseteq U_1\cup U_2$. If $o\in U_1\setminus U_2$, then ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}} \subseteq U_1 \subseteq U_1 \cup U_2$.
Finally, for any $x\not\in {\mathbf{aff}}(c)$, we have $(\Gamma\lhd_\alpha (U_1 \cup U_2)(x) = ((\Gamma\lhd_\alpha U_1)\lhd_\alpha U_2)(x) = ((\Gamma\lhd_\alpha U_1)(x), \Gamma\lhd_\alpha U_1) \lhd_\alpha U_2 \stackrel{(1)}{\sqsubseteq}_r ((\Gamma\lhd_\alpha U_1)(x), \Gamma_1) \lhd_\alpha U_2 \stackrel{(2)}{\sqsubseteq}_r (\Gamma_1(x), \Gamma_1) \lhd_\alpha U_2 = (\Gamma_1 \lhd_\alpha U_2)(x) \stackrel{(3)}{\sqsubseteq}_r \Gamma_2(x)$. We used whenever necessarily Lemma \[pl1\]; in addition, in (1) we used that $(\Gamma\lhd_\alpha U_1)(x)$ does not depend on variables in $U_1$, and by induction hypothesis for all variables $v\not\in U_1$, $(\Gamma\lhd_\alpha U_1)(v) {\sqsubseteq}_r \Gamma_1(v)$, in (2) we used that by induction hypothesis $(\Gamma\lhd_\alpha U_1)(x) {\sqsubseteq}_r \Gamma_1(x)$, in (3) we used that $x\not\in U_2$, and hence by induction hypothesis, $(\Gamma_1 \lhd_\alpha U_2)(x) {\sqsubseteq}_r \Gamma_2(x)$.
- $c$ is ${\textsf{If } e \textsf{ then } c_1 \textsf{
else } c_2 \textsf{ fi }}$ . Then [$p\vdash_{\alpha} {\ensuremath{\Gamma\{{\textsf{If } e \textsf{ then } c_1 \textsf{
else } c_2 \textsf{ fi }}\}\Gamma'}}$]{} was inferred based on [$p{\sqcup}p'\vdash_{\alpha} {\ensuremath{\Gamma\{c_i\}\Gamma_i}}$]{}, where $p'=(\Gamma[\alpha](fv(e)), \Gamma) \lhd_\alpha({\mathbf{aff^O}}(c_1)\cup{\mathbf{aff^O}}(c_2))$ and $\Gamma'= \Gamma'_1 {\sqcup}\Gamma'_2$ where $\Gamma'_1 = \Gamma_1 \lhd_\alpha {\mathbf{aff^O}}(c_2)$ and $\Gamma'_2= \Gamma_2 \lhd_\alpha {\mathbf{aff^O}}(c_1)$. Let us denote $U_i={\mathbf{aff^O}}(c_i)$, for $i=1,2$. First notice that ${\mathbf{atomO}}(p)\cap (U_1\cup U_2) = \emptyset$ ensures that ${\mathbf{atomO}}(p{\sqcup}p')\cap (U_1\cup U_2) = \emptyset$ .
Using the induction hypothesis for [$p{\sqcup}p'\vdash_{\alpha} {\ensuremath{\Gamma\{c_1\}\Gamma_1}}$]{} we get that ${\ensuremath{\mathcal{AC}(\Gamma_1)}}$, and for any $o\in X_O\setminus( U_1\cup {\mathbf{atomO}}(p{\sqcup}p'))$, ${\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}$, for any $o\in {\mathbf{atomO}}(p{\sqcup}p')$, ${\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}\cup U_1$ and for any $o\in U_1$, ${\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\subseteq U_1$. In addition, for any $x\not\in U_1$, $(\Gamma\lhd_\alpha U_1))(x) {\sqsubseteq}_r \Gamma_1(x)$.
Using the induction hypothesis for [$p{\sqcup}p'\vdash_{\alpha} {\ensuremath{\Gamma\{c_2\}\Gamma_2}}$]{} we get that ${\ensuremath{\mathcal{AC}(\Gamma_2)}}$, and for any $o\in X_O\setminus( U_2\cup {\mathbf{atomO}}(p{\sqcup}p')$, ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}$, for any $o\in {\mathbf{atomO}}(p{\sqcup}p')$, ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}\cup U_2$ and for any $o\in U_2$, ${\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq U_2$. In addition, for any $x\not\in U_2$, $(\Gamma\lhd_\alpha U_2))(x) {\sqsubseteq}_r \Gamma_2(x)$.
For any $o\in X_O\setminus (U_1\cup U_2\cup {\mathbf{atomO}}(p{\sqcup}p'))$, ${\ensuremath{\mathcal{GT}_{\Gamma'_1\cup \Gamma'_2}(o)}} ={\ensuremath{\mathcal{GT}_{(\Gamma_1 \lhd_\alpha U_2)\cup (\Gamma_2\lhd_\alpha U_1)}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}$ since both ${\ensuremath{\mathcal{GT}_{\Gamma_1 \lhd_\alpha U_2}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}$ and ${\ensuremath{\mathcal{GT}_{\Gamma_2 \lhd_\alpha U_1}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}$.
For any $o\in U_1$, ${\ensuremath{\mathcal{GT}_{\Gamma_2\lhd_\alpha U_1}(o)}} = \emptyset$ and by induction hypothesis ${\ensuremath{\mathcal{GT}_{\Gamma_1\lhd_\alpha U_2}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\subseteq U_1$. For any $o\in U_2$, ${\ensuremath{\mathcal{GT}_{\Gamma_1\lhd_\alpha U_2}(o)}} = \emptyset$ and by induction hypothesis ${\ensuremath{\mathcal{GT}_{\Gamma_2\lhd_\alpha U_1}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq U_2$. This implies that for any $o\in (U_1\cup U_2)$, ${\ensuremath{\mathcal{GT}_{\Gamma'_1\cup \Gamma'_2}(o)}} ={\ensuremath{\mathcal{GT}_{(\Gamma_1 \lhd_\alpha U_1)\cup (\Gamma_2\lhd_\alpha U_2)}(o)}} \subseteq U_1 \cup U_2$.
For any $o\in {\mathbf{atomO}}(p{\sqcup}p')$, by induction hypothesis ${\ensuremath{\mathcal{GT}_{\Gamma_1\lhd_\alpha U_2}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma_1}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}\cup U_1$ and ${\ensuremath{\mathcal{GT}_{\Gamma_2\lhd_\alpha U_1}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma_2}(o)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}\cup U_2$. We already proved that for any $o\in (U_1\cup U_2)$, ${\ensuremath{\mathcal{GT}_{\Gamma'_1\cup \Gamma'_2}(o)}} \subseteq U_1 \cup U_2$. The previous inclusions imply that for any $o\in {\mathbf{atomO}}(p{\sqcup}p')$, ${\ensuremath{\mathcal{GT}_{\Gamma'_1\cup \Gamma'_2}(o)}} \subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(o)}}\cup U_1 \cup U_2$.
Now if we assume by contradiction that $\neg {\ensuremath{\mathcal{AC}(\Gamma')}}$, we get that there must exist $x_1,x_2\in X_O$ such that $x_1{\sqsubseteq}_{\Gamma'_1{\sqcup}\Gamma'_2} x_2$ and $x_2{\sqsubseteq}_{\Gamma'_1{\sqcup}\Gamma'_2} x_1$. We make an analysis by case:
- $x_1\in U_1 \cap U_2$. Impossible, since we have that ${\ensuremath{\mathcal{AC}(\Gamma_1 \lhd_\alpha U_2)}}$ and ${\ensuremath{\mathcal{AC}(\Gamma_2 \lhd_\alpha U_1)}}$, and for any $o\in X_O$, $x_1\not{\sqsubseteq}_{\Gamma_1 \lhd_\alpha U_2} o$ and $x_1\not{\sqsubseteq}_{\Gamma_2 \lhd_\alpha U_1} o$.
- $x_1, x_2\in U_1 \setminus U_2$. From induction hypothesis we get ${\ensuremath{\mathcal{GT}_{\Gamma'_1}(x_i)}}\subseteq U_1$ and ${\ensuremath{\mathcal{GT}_{\Gamma'_2}(x_i)}}=\emptyset$. This implies that for any $x\in {\ensuremath{\mathcal{GT}_{\Gamma'_1}(x_i)}}$, it holds $x\in U_1$ and hence ${\ensuremath{\mathcal{GT}_{\Gamma'_2}(x)}}=\emptyset$. It means that $x_1{\sqsubseteq}_{\Gamma'_1{\sqcup}\Gamma'_2} x_2$ and $x_2{\sqsubseteq}_{\Gamma'_1{\sqcup}\Gamma'_2} x_1$ implies that $x_1{\sqsubseteq}_{\Gamma'_1} x_2$ and $x_2{\sqsubseteq}_{\Gamma'_1} x_1$ which contradicts ${\ensuremath{\mathcal{AC}(\Gamma'_1)}}$.
- $x_1\in U_1 \setminus U_2$ and $x_2\not\in U_1$. From induction hypothesis we get ${\ensuremath{\mathcal{GT}_{\Gamma'_1}(x_1)}}\subseteq U_1$ and ${\ensuremath{\mathcal{GT}_{\Gamma'_2}(x_1)}}=\emptyset$. This implies that for any $x\in {\ensuremath{\mathcal{GT}_{\Gamma'_1}(x_1)}}$, it holds $x\in U_1$ and hence ${\ensuremath{\mathcal{GT}_{\Gamma'_2}(x)}}=\emptyset$. It means that $x_1{\sqsubseteq}_{\Gamma'_1{\sqcup}\Gamma'_2} x$ implies that $x\in {\ensuremath{\mathcal{GT}_{\Gamma'_1}(x_1)}}\subseteq U_1$, contradiction with $x_2\not\in U_1$ and $x_1{\sqsubseteq}_{\Gamma'_1{\sqcup}\Gamma'_2} x_2$.
- $x_1\not\in U_1 \cup U_2$ and $x_2\not\in U_1 \cup U_2$. In this case, by induction ${\ensuremath{\mathcal{GT}_{\Gamma'_1{\sqcup}\Gamma'_2}(x_1)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(x_1)}} \cup (U_1\cup U_2)$ and ${\ensuremath{\mathcal{GT}_{\Gamma'_1{\sqcup}\Gamma'_2}(x_2)}}\subseteq {\ensuremath{\mathcal{GT}_{\Gamma}(x_2)}} \cup (U_1\cup U_2) $. Hence $x_1{\sqsubseteq}_{\Gamma'_1{\sqcup}\Gamma'_2} x_2$ implies that $x_1{\sqsubseteq}_{\Gamma} x_2$, $x_2{\sqsubseteq}_{\Gamma'_1{\sqcup}\Gamma'_2} x_1$ implies that $x_2{\sqsubseteq}_{\Gamma} x_1$, contradiction with ${\ensuremath{\mathcal{AC}(\Gamma)}}$.
- The remaining cases are symmetrical ones with the previous cases.
- Similar to the rule (If).
- Trivial from the premises of the rule, using the induction hypothesis, the transitivity of ${\sqsubseteq}_r$ and that $\Gamma_1 {\sqsubseteq}_r \Gamma_2$ implies that ${\mathcal{G}}(\Gamma_1)= {\mathcal{G}}(\Gamma_2)$.
Soundness of the typing system
------------------------------
As already stated above, our type system aims to capture the following non-interference condition: given a derivation [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{}, the final value of a variable $x$ with final type $t{\sqcup}\alpha(X_O)$, should depend at most on the initial values of those variables $y$ with initial types $\Gamma(y){\sqsubseteq}t$ and on the final values of variables in $X_O$. Or otherwise said, executing a program $c$ on two initial states $\sigma$ and $\rho$ such that $\sigma(y)=\rho(y)$ for all $y$ with $\Gamma(y){\sqsubseteq}t$ which ends with two final states $\sigma'$ and $\rho'$ such that $\sigma'(o)=\rho'(o)$ for all $o\in X_O$ will satisfy $\sigma'(x)=\rho'(x)$ for all $x$ with $\Gamma'(x){\sqsubseteq}t{\sqcup}\alpha(X_O)$. In order to prove the soundness of the typing system, we need a stronger invariant denoted ${\mathcal{I}}(t,\Gamma)$: intuitively, $(\sigma, \rho)\in {\mathcal{I}}(t,\Gamma)$ means that for each variable $x$ and $A\subseteq X_O$, if $\sigma =_A \rho$ and $\Gamma(x){\sqsubseteq}t{\sqcup}\alpha(A)$, then $\sigma(x)=\rho(x)$. Formally, given $t\in {\mathcal{L}}$ and $\Gamma : Var \mapsto {\mathcal{L}}$, we define $\displaystyle{\mathcal{I}}(t,\Gamma) {\stackrel{def}=}\bigcap_{A\subseteq X_O} =_{\Gamma, A, \alpha(A) \sqcup t}.$
The following lemmas provide some useful properties satisfied by the invariant ${\mathcal{I}}(t,\Gamma)$.
\[p0\] If $\Gamma_1\sqsubseteq \Gamma_2$ then for all $t\in {\mathcal{L}}$, ${\mathcal{I}}(t,\Gamma_1) \subseteq {\mathcal{I}}(t,\Gamma_2)$.
Assume $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma_1)$. We prove that $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma_2)$. Let $A\subseteq X_O$ and let $y\in Var$ such that $\Gamma_2(y){\sqsubseteq}\alpha(A) \sqcup t$. Assume that $\sigma =_A \rho$. We have to prove that $\sigma(y)=\rho(y)$. We have $\Gamma_1(y) {\sqsubseteq}_r \Gamma_2(y){\sqsubseteq}\alpha(A) \sqcup t$, and since $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma_1)$ and $\sigma =_A \rho$, we get $\sigma(y)=\rho(y)$.
\[l:pwf\] Let $x\in X_O, X\subseteq X_O$ and let $\Gamma$ be well-formed. Let $\Gamma_1= \Gamma\lhd_\alpha x$ and $\Gamma_2= \Gamma\lhd_\alpha X$. Then for all $t\in {\mathcal{L}}$, it holds ${\mathcal{I}}(t,\Gamma) \subseteq {\mathcal{I}}(t,\Gamma_1)$ and ${\mathcal{I}}(t,\Gamma) \subseteq {\mathcal{I}}(t,\Gamma_2)$.
We prove only the first inclusion, the second one can be easily proved by induction using the first one.
Let $\sigma, \rho$ such that $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma)$ and let us prove that $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma_1)$.
Let $B\subseteq X_O$ and let $y\in Var$ such that $\Gamma_1(y){\sqsubseteq}\alpha(B) \sqcup t$. Assume that $\sigma =_B \rho$. We have to prove that $\sigma(y)=\rho(y)$.
Then $ \Gamma_1(y) {\sqsubseteq}\alpha(B) \sqcup t$ implies that $\big(\Gamma(y)\big)[\Gamma(x)/\alpha(x)]{\sqsubseteq}\alpha(B) \sqcup t$. That is, either $\Gamma(y) {\sqsubseteq}\alpha(B)\sqcup t$ or $\Gamma(y) {\sqsubseteq}\alpha(B) \sqcup \alpha(x) \sqcup t$ and $\Gamma(x){\sqsubseteq}\alpha(B) \sqcup t$.
- If $\Gamma(y) {\sqsubseteq}\alpha(B) \sqcup t$, since $\sigma =_{B}\rho$ and $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma)$ we get $\sigma(y)=\rho(y)$.
- If $\Gamma(y) {\sqsubseteq}\alpha(B) \sqcup \alpha(x) \sqcup t$ and $\Gamma(x){\sqsubseteq}\alpha(B) \sqcup t$, from $\sigma =_{B}\rho$ and $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma)$, we get $\sigma =_{\{x\}}\rho$, and hence $\sigma =_{B\cup\{x\}}\rho$. Since $\Gamma(y) {\sqsubseteq}\alpha(B) \sqcup \alpha(x) \sqcup t$ and $\sigma =_{\Gamma, B\cup\{x\}, \alpha(B\cup\{x\}) \sqcup t}\rho$, we get $\sigma(y)=\rho(y)$.
\[pl5\] Let $o\in X_O$ and let $\Gamma' = \Gamma \lhd_\alpha o$ with well formed [$\mathcal{AC}(\Gamma)$]{}. Let $t\in {\mathcal{L}}$ and let $(\sigma, \rho)\in {\mathcal{I}}(t,\Gamma). $ For any $A\subseteq X_O$ such that $\sigma =_{A\setminus\{o\}} \rho$ and for any $y\in Var$ such that $\Gamma'(y){\sqsubseteq}\alpha(A) \sqcup t$ it holds that $\sigma(y)=\rho(y)$.
[$\mathcal{AC}(\Gamma)$]{} and $ \Gamma'(y) {\sqsubseteq}\alpha(A) \sqcup t$ implies that $\big(\Gamma(y)\big)[\Gamma(o)/\alpha(o)]{\sqsubseteq}\alpha(A\setminus\{o\}) \sqcup t$. That is, either $\Gamma(y) {\sqsubseteq}\alpha(A\setminus\{o\})\sqcup t$ or $\Gamma(y) {\sqsubseteq}\alpha(A) \sqcup t$ and $\Gamma(o){\sqsubseteq}\alpha(A\setminus\{o\}) \sqcup t$.
- If $\Gamma(y) {\sqsubseteq}\alpha(A\setminus\{o\}) \sqcup t$, as $\sigma =_{A\setminus\{o\}}\rho$ and $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma)$ we get $\sigma(y)=\rho(y)$.
- If $\Gamma(y) {\sqsubseteq}\alpha(A) \sqcup t$ and $\Gamma(o){\sqsubseteq}\alpha(A\setminus\{o\}) \sqcup t$, from $\sigma =_{A\setminus\{o\}}\rho$ and $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma)$, we get $\sigma =_{\{o\}}\rho$, and hence $\sigma =_{A}\rho$. Since $\Gamma(y) {\sqsubseteq}\alpha(A) \sqcup t$ and $\sigma =_{\Gamma, A, \alpha(A) \sqcup t}\rho$, we get $\sigma(y)=\rho(y)$.
\[l:pl4\] Let $\Gamma' = \Gamma \lhd_\alpha U$ for a well formed ${\ensuremath{\mathcal{AC}(\Gamma)}}$ where $U=\{o_1,\ldots ,o_n\} \subseteq X_O$ . Let $t\in {\mathcal{L}}$ and let $(\sigma, \rho)\in {\mathcal{I}}(t,\Gamma)$. For any $A\subseteq X_O$ such that $\sigma =_{A\setminus U} \rho$ and for any $y\in Var$ such that $\Gamma'(y){\sqsubseteq}\alpha(A) \sqcup t$ it holds that $\sigma(y)=\rho(y)$.
By induction on $n$. The case $n=1$ follows from the lemma \[pl5\]. Let us denote $U_{n-1}= U\setminus\{o_n\}=\{o_1,\ldots ,o_{n-1}\}$. and let $\Gamma_{n-1} = \Gamma \lhd_\alpha U_{n-1}$. Hence $\Gamma' = \Gamma_{n-1} \lhd_\alpha o_n$.
By lemma \[lrsB\], $\Gamma_{n-1}$ is well formed too. [$\mathcal{AC}(\Gamma_{n-1})$]{} and $ \Gamma'(y) {\sqsubseteq}\alpha(A) \sqcup t$ implies that $\big(\Gamma_{n-1}(y)\big)[\Gamma_{n-1}(o_n)/\alpha(o_n)]{\sqsubseteq}\alpha(A\setminus\{o_n\}) \sqcup t$. That is, either $\Gamma_{n-1}(y) {\sqsubseteq}\alpha(A\setminus\{o_n\})\sqcup t$ or $\Gamma_{n-1}(y) {\sqsubseteq}\alpha(A) \sqcup t$ and $\Gamma_{n-1}(o_n){\sqsubseteq}\alpha(A\setminus\{o_n\}) \sqcup t$.
- If $\Gamma_{n-1}(y) {\sqsubseteq}\alpha(A\setminus\{o_n\}) \sqcup t$, since $\sigma =_{(A\setminus\{o_n\})\setminus U_{n-1}}\rho$, by induction (taking $U'=\{o_1,\ldots ,o_{n-1}\}$ and $A'=A\setminus\{o_n\}$) we get $\sigma(y)=\rho(y)$.
- If $\Gamma_{n-1}(y) {\sqsubseteq}\alpha(A) \sqcup t$ and $\Gamma_{n-1}(o_n){\sqsubseteq}\alpha(A\setminus\{o_n\}) \sqcup t$, since $\sigma =_{(A\setminus\{o_n\})\setminus U_{n-1}}\rho$, we get by induction (taking $U'=U_{n-1}$ and $A'=A\setminus\{o_n\}$) that $\sigma =_{\{o_n\}}\rho$, and hence $\sigma =_{A\setminus U_{n-1}}\rho$. From $\Gamma_{n-1}(y) {\sqsubseteq}\alpha(A) \sqcup t$ and $\sigma =_{A\setminus U_{n-1}}\rho$ using the induction again (taking $U'=U_{n-1}$ and $A'=A$) we get $\sigma(y)=\rho(y)$.
The following theorem states the soundness of our typing system.
\[l1\] Let us assume that [$\mathcal{AC}(\Gamma)$]{} and $\forall o\in X_O, \alpha(o)\not{\sqsubseteq}t$. If [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} then $c~:~{\mathcal{I}}(t,\Gamma) \Longrightarrow {\mathcal{I}}(t,\Gamma').$
Proof is by induction on the derivation of [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{}. Let $\sigma, \sigma', \rho, \rho'$ such that $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma)$ and ${\langle c, \sigma \rangle \Longrightarrow \sigma'}$ and ${\langle c, \rho \rangle \Longrightarrow \rho'}$. We prove that $(\sigma', \rho')\in{\mathcal{I}}(t,\Gamma')$.
Let $A\subseteq X_O$ and let $y\in Var$ such that $\Gamma'(y){\sqsubseteq}\alpha(A) \sqcup t$. Assume that $\sigma' =_A \rho'$. We have to prove that $\sigma'(y)=\rho'(y)$.
We do a case analysis according to the last rule applied (case Skip is trivial).
- $c$ is an assignment $x:=e$ for some $x\not\in X_O$.\
Then $\Gamma' =\Gamma[x\mapsto p{\sqcup}\Gamma[\alpha](fv(e))]$. Hence $\sigma' =_{A} \rho'$ implies $\sigma =_{A} \rho$.
If $y\not\equiv x$, then $\Gamma(y) = \Gamma'(y){\sqsubseteq}\alpha(A) \sqcup t$, and since $\sigma =_{\Gamma, A, \alpha(A) \sqcup t}\rho$, we get $\sigma'(y) = \sigma(y)=\rho(y)=\rho'(y)$.
Let us assume that $c$ is $y:=e$ for some $e$ and $y\not\in X_O$. Then $ \Gamma[\alpha](fv(e)){\sqsubseteq}\Gamma'(y) {\sqsubseteq}\alpha(A) \sqcup t$, and this implies that: 1) for all variables $v\in fv(e)\setminus X_O$, $\Gamma(v){\sqsubseteq}\alpha(A) \sqcup t$, hence $\sigma(v)=\rho(v)$ and 2) for all variables $v\in fv(e)\cap X_O$, $\alpha(v){\sqsubseteq}\alpha(A) \sqcup t$, hence $v\in A$ and $\sigma(v)=\rho(v)$. We get that $\sigma(e)=\rho(e)$, and hence $\sigma'(y)=\rho'(y)$.
- $c$ is an assignment $o:=e$ for some $o\in X_O\setminus fv(e)$.
Using Lemma \[l:pwf\], we get that $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma_1)$, where $\Gamma_1=\Gamma\lhd_\alpha o$ and $\Gamma'=\Gamma_1[o \mapsto p{\sqcup}\Gamma_1[\alpha](fv(e)\setminus o)]$.
If $y\not\equiv o$, then $\Gamma_1(y) = \Gamma'(y){\sqsubseteq}\alpha(A) \sqcup t$, and since $\sigma =_{\Gamma_1, A, \alpha(A) \sqcup t}\rho$, we get $\sigma'(y) = \sigma(y)=\rho(y)=\rho'(y)$.
If $y\equiv o$, then $p{\sqcup}\Gamma_1[\alpha](fv(e)\setminus o) = \Gamma'(o) {\sqsubseteq}\alpha(A) \sqcup t$, and this implies that: 1) for all variables $v\in fv(e)\setminus X_O$, $\Gamma_1(v){\sqsubseteq}\alpha(A) \sqcup t$, hence $\sigma(v)=\rho(v)$ and 2) for all variables $v\in fv(e)\cap X_O$, $\alpha(v){\sqsubseteq}\alpha(A) \sqcup t$, hence $v\in A$ and $\sigma(v)=\rho(v)$. We get that $\sigma(e)=\rho(e)$, and hence $\sigma'(y)=\rho'(y)$.
- $c$ is an assignment $o:=e$ for some $o\in X_O\cap fv(e)$. Similar to the previous case, using the remark that $\Gamma(o) {\sqsubseteq}\alpha(A) \sqcup t$ implies that $\sigma(o)=\rho(o)$.
- Trivial, using the transitivity of $\Longrightarrow$.
- Let us denote $U_i=\{o^i_1,o^i_2,\ldots ,o^i_n\}={\mathbf{aff^O}}(c_i)$ for $i=1,2$ for some good orderings $U_i$ of ${\mathbf{aff^O}}(c_i)$.
- If $(\Gamma[\alpha](fv(e)), \Gamma) \lhd_\alpha (U_1\cup U_2)\not{\sqsubseteq}\Gamma'(y)$, obviously we get $y\not\in {\mathbf{aff}}(c)$, i.e. $c$ contains no assignments to $y$. Hence $\sigma'(y) = \sigma(y)$ and $\rho(y)=\rho'(y)$. Then from Lemma \[l:pwf\], we get that $ (\Gamma \lhd_\alpha ( U_1\cup U_2))(y) {\sqsubseteq}_r \Gamma'(y) {\sqsubseteq}\alpha(A) \sqcup t$. Moreover, $\sigma' =_{A}\rho'$ implies that $\sigma =_{A\setminus (U_1\cup U_2)}\rho$ and using Lemma \[l:pl4\] we get $\sigma(y)=\rho(y)$, and hence $\sigma'(y) = \sigma(y) = \rho(y)=\rho'(y)$.
- Let us suppose that $(\Gamma[\alpha](fv(e)), \Gamma) \lhd_\alpha (U_1\cup U_2){\sqsubseteq}\Gamma'(y)$. We denote $\Gamma^i_{0} = \Gamma \lhd_\alpha U_i$. Then $(\Gamma[\alpha](fv(e)), \Gamma) \lhd_\alpha (U_1\cup U_2) {\sqsubseteq}\Gamma'(y) {\sqsubseteq}\alpha(A) \sqcup t$, and this implies that\
$(\Gamma[\alpha](fv(e)), \Gamma) \lhd_\alpha (U_1\cup U_2) {\sqsubseteq}\alpha(A\setminus (U_1\cup U_2)) \sqcup t$. Moreover, $\sigma' =_{A}\rho'$ implies that $\sigma =_{A\setminus (U_1\cup U_2)}\rho$. Using Lemma \[l:pl4\], we get for each variable $v\in fv(e)$ that $\sigma(v)=\rho(v)$. This proves that $\sigma(e)=\rho(e)$ and hence both executions ${\langle c, \sigma \rangle \Longrightarrow \sigma'}$ and ${\langle c, \rho \rangle \Longrightarrow \rho'}$ take the same branch $i$. Then we use the induction hypothesis applied to [$p{\sqcup}p'\vdash_{\alpha} {\ensuremath{\Gamma\{c_i\}\Gamma_i}}$]{} to get that $(\sigma', \rho')\in {\mathcal{I}}(t,\Gamma_i)$, and then, since $\Gamma'= \Gamma_1 \lhd_\alpha U_2 {\sqcup}\Gamma_2 \lhd_\alpha U_1$, we use Lemmas \[p0\] and \[l:pwf\] to conclude $(\sigma', \rho')\in {\mathcal{I}}(t,\Gamma')$.
- $c$ is ${\textsf{While } e \textsf{ Do } c_1 \textsf{
oD }}$. Then [$p\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} was inferred based on
[$p{\sqcup}p_e\vdash_{\alpha} {\ensuremath{\Gamma'\{c\}\Gamma_1}}$]{} and $(\Gamma \lhd_\alpha U){\sqcup}(\Gamma_1 \lhd_\alpha U) {\sqsubseteq}_r \Gamma'$ where $U= {\mathbf{aff^O}}(c)$ and $p_e=\Gamma'[\alpha](fv(e))$.
From $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma)$, using Lemma \[l:pwf\], we get $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma\lhd_\alpha U)$ and using now Lemma \[p0\] and the inequality $\Gamma\lhd_\alpha U {\sqsubseteq}(\Gamma \lhd_\alpha U){\sqcup}(\Gamma_1 \lhd_\alpha U) {\sqsubseteq}_r \Gamma'$, we obtain $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma')$. Using the induction hypothesis applied to [$p{\sqcup}p_e\vdash_{\alpha} {\ensuremath{\Gamma'\{c\}\Gamma_1}}$]{}, we get that $(\sigma', \rho')\in{\mathcal{I}}(t,\Gamma_1)$. We apply again Lemma \[l:pwf\] and we get $(\sigma', \rho')\in{\mathcal{I}}(t,\Gamma_1 \lhd_\alpha U)$, and using Lemma \[p0\] and that $(\Gamma \lhd_\alpha U){\sqcup}(\Gamma_1 \lhd_\alpha U) {\sqsubseteq}_r \Gamma'$, we obtain $(\sigma', \rho')\in{\mathcal{I}}(t,\Gamma')$.
- Trivial, from Lemma \[p0\] and the induction hypothesis.
Soundness w.r.t. to output-sensitive non-interference {#s:iol}
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In this section we show how we can use the typing system in order to prove that a program $c$ is output-sensitive noninterferent. Let ${\ensuremath{Var^{e}}}= Var \cup \{\overline{o} \ | \ o\in X_O \}$. Let us define ${\mathcal{L}}{\stackrel{def}=}\{\tau_A \ \mid \ A\subseteq {\ensuremath{Var^{e}}}\}$. We denote $\bot = \tau_\emptyset$ and $\top = \tau_{{\ensuremath{Var^{e}}}}$ and we consider the lattice $({\mathcal{L}}, \bot , \top, {\sqsubseteq})$ with $\tau_A {\sqcup}\tau_{A'} {\stackrel{def}=}\tau_{A\cup A'}$ and $\tau_A {\sqsubseteq}\tau_{A'}$ iff $A\subseteq A'$. The following Theorem is a consequence of the Definition \[iol-sec\] and Theorem \[l1\].
\[l:sec\] Let ${\mathcal{L}}$ be the lattice described above. Let $(\Gamma,\alpha)$ be defined by $\Gamma(x)=\{\tau_x\}$, for all $x\in Var$ and $\alpha(o)=\{\tau_{\overline{o}}\}$, for all $o\in X_O$. If [$\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} and for all $x_l\in X_L$, $\Gamma'(x_l) {\sqsubseteq}\Gamma(X_I) {\sqcup}\alpha(X_O)$, then $c$ is ($X_I, X_O, X_L$)-secure.
Let $t= \Gamma(X_I)$. First, we prove that if $\sigma =_{X_I} \rho $, then $(\sigma, \rho)\in{\mathcal{I}}(t,\Gamma)$. Let $A\subseteq X_O$ such that $\sigma =_{A} \rho $ and let $y\in Var$ such that $\Gamma(y){\sqsubseteq}\alpha(A) \sqcup t = \alpha(A) \sqcup \Gamma(X_I)$. This implies that $y\in X_I$, and since $\sigma =_{X_I} \rho$, we get $\sigma(y)=\rho(y)$.
Now let $\sigma, \sigma', \rho, \rho'$, such ${\langle c, \sigma \rangle \Longrightarrow \sigma'}$ and ${\langle c, \rho \rangle \Longrightarrow \rho'}$ and $\sigma =_{X_I} \rho $ and $\sigma'=_{X_O}\rho'$. Let $x_l\in X_L$. We have to prove that $\sigma'=_{X_l}\rho'$. Let us apply the Theorem \[l1\] with $t= \Gamma(X_I)$. Since [$\vdash_{\alpha} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} and $(\sigma, \rho)\in{\mathcal{I}}_1(t,\Gamma)$, we get that $(\sigma', \rho')\in{\mathcal{I}}(t,\Gamma')$. It means that $\sigma'=_{\Gamma', X_O, \alpha(X_O) \sqcup \Gamma(X_I)}\rho'$. Since by hypothesis we have that $\sigma'=_{X_O}\rho'$ and $\Gamma'(x_l) {\sqsubseteq}\alpha(X_O) \sqcup \Gamma(X_I)$, we get that $\sigma'=_{x_l}\rho'$.
Output-sensitive constant-time {#s:3}
==============================
Following [@aga2000; @almeida2016], we consider two types of cache-based information leaks: 1) disclosures that happen when secret data determine which parts of the program are executed; 2) disclosures that arise when access to memory is indexed by sensitive information. In order to model the latter category, we shall enrich the simple language from section \[s:w1\] with [*arrays*]{}: $$\begin{array}{lll}
c & ::= &x:=e ~|~ x[e_1]:=e ~|~ {\textsf{skip}}~|~ c_1;c_2~|~ {\textsf{If } e \textsf{ then } c_1 \textsf{
else } c_2 \textsf{ fi }}~|~ {\textsf{While } e \textsf{ Do } c \textsf{
oD }}
\end{array}$$ To simplify notations, we assume that array indexes $e_1$ are basic expressions (not referring to arrays) and that $X_O$ does not contain arrays. Moreover as in [@almeida2016], a state or store $\sigma$ maps array variables $v$ and indices $i\in{\mathbbm{N}}$ to values $\sigma(v,i)$. The labeled semantics of While programs is listed in Figure \[fig:T13\]. In all rules, we denote ${\overrightarrow}{f}=(f_i)_i$, where $x_i[f_i] \text{ are the indexed variables in } e$. The labels on the execution steps correspond to the information which is leaked to the environment (${\textbf{r}()}$ for a read access on memory, ${\textbf{w}()}$ for a write access and ${\textbf{b}()}$ for a branch operation). In the rules for (If) and (While) the valuations of branch conditions are leaked. Also, all indexes to program variables read and written at each statement are exposed.
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We give in Figure \[fig:T14\] the new typing rules. As above, we denote ${\overrightarrow}{f}=(f_i)_i$, where $x_i[f_i] \text{ are the indexed variables in } e$. We add a fresh variable $x_l$, that is not used in programs, in order to capture the unintended leakage. Its type is always growing up and it mirrors the information leaked by each command. In rule $(As1")$ we take a conservative approach and we consider that the type of an array variable is the lub of all its cells. The information leaked by the assignment $x[e_1]:=e$ is the index $e_1$ plus the set ${\overrightarrow}{f}=(f_i)_i$ of all indexes occurring in $e$. Moreover, the new type of the array variable $x$ mirrors the fact that now the value of $x$ depends also on the index $e_1$ and on the right-hand side $e$.
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\[d3.2\] An [**execution**]{} is a sequence of visible actions: ${\stackrel{a_1}{\longrightarrow}}{\stackrel{a_2}{\longrightarrow}}\ldots {\stackrel{a_n}{\longrightarrow}}$. A program $c$ is [**($X_I, X_O$)-constant time**]{} when all its executions starting with $X_I $-equivalent stores that lead to finally $X_O$-equivalent stores, are identical.
Following [@almeida2016], given a set $X$ of program variables, two stores $\sigma$ and $\rho$ are [*$X$-equivalent*]{} when $\sigma(x, i)=\rho(x, i)$ for all $x\in X$ and $i\in{\mathbbm{N}}$. Two executions ${\stackrel{a_1}{\longrightarrow}}\ldots {\stackrel{a_n}{\longrightarrow}}$ and ${\stackrel{b_1}{\longrightarrow}}\ldots {\stackrel{b_m}{\longrightarrow}}$ are [*identical*]{} iff $n=m$ and $a_j=b_j$ for all $1\leq j \leq n$.
We can reduce the ($X_I, X_O$)-constant time security of a command $c$ to the ($X_I, X_O, \{x_l\}$)-security (see section \[s:iol\]) of a corresponding command ${\omega}(c)$, obtained by adding a fresh variable $x_l$ to the program variables $fv(c)$, and then adding recursively before each assignment and each boolean condition predicate, a new assignment to the leakage variable $x_l$ that mirrors the leaked information. Let $:, {\textbf{b}(,)} {\textbf{r}(,)} \wt$ be some new abstract operators. The construction of the instrumentation ${\omega}(\bullet)$ is shown in Fig. \[fig:ga\]. As above, we denote ${\overrightarrow}{f}=(f_i)_i$, where $x_i[f_i] \text{ are the indexed variables in } e$.
First we can extend the language with array variables, then we need to extend the typing system from section \[s:w1\] with a rule corresponding to the new rule $Ast'$. Then, the following lemma gives the relationship between the type of a program $c$ using the new typing system and the type of the instrumented program ${\omega}(c)$ using the extended typing system from the previous section.
$$\begin{array}{l|l}
\bullet & {\omega}(\bullet) \\
\hline
x:=e & x_l:=x_l:{\textbf{r}({\overrightarrow}{f})}; \ \ x:=e \\
\hline
x[e_1]:=e & x_l:=x_l:{\textbf{w}(e_1)}:{\textbf{r}({\overrightarrow}{f})}; \ \ x[e_1]:=e\\
\hline
skip & skip \\
\hline
c_1;c_2 & {\omega}(c_1);{\omega}(c_2) \\
\hline
{\textsf{If } e \textsf{ then } c_1 \textsf{
else } c_2 \textsf{ fi }} & x_l:=x_l:{\textbf{b}(e)}:{\textbf{r}({\overrightarrow}{f})}; \ \ {\textsf{If } e \textsf{ then } {\omega}(c_1) \textsf{
else } {\omega}(c_2) \textsf{ fi }}\\
\hline
{\textsf{While } e \textsf{ Do } c \textsf{
oD }} &x_l:=x_l:{\textbf{b}(e)}:{\textbf{r}({\overrightarrow}{f})}; \textsf{While } e \textsf{ Do } {\omega}(c);x_l:=x_l:{\textbf{b}(e)}:{\textbf{r}({\overrightarrow}{f})} \textsf{ oD}\\
\end{array}$$
\[al:ct\] Let $c$ a command such that $x_l\not\in fv(c)$, $\sigma, \sigma'$ two stores , $tr$ some execution trace and $[]$ the empty trace.
[*1. *]{} [$p\vdash_{\alpha}^{ct} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} iff [$p\vdash_{\alpha} {\ensuremath{\Gamma\{{\omega}(c)\}\Gamma'}}$]{}.
[*2. *]{} $(c,\sigma)\stackrel{tr}{\longrightarrow}^* \sigma'$ iff $({\omega}(c),\sigma[x_l\mapsto []])\longrightarrow^* \sigma'[x_l\mapsto tr]$.
Now combining Theorem \[l:sec\] and Lemma \[al:ct\] we get the following Theorem which proves the soundness of the new typing system.
\[l:ctime\] Let ${\mathcal{L}}$ be the lattice defined in the section \[s:iol\]. Let $(\Gamma,\alpha)$ be defined by $\Gamma(x)=\{\tau_x\}$, for all $x\in Var$ and $\alpha(o)=\{\tau_{\overline{o}}\}$, for all $o\in X_O$ and $\Gamma(x_l)=\bot$. If [$p\vdash_{\alpha}^{ct} {\ensuremath{\Gamma\{c\}\Gamma'}}$]{} and $\Gamma'(x_l) {\sqsubseteq}\Gamma(X_I) {\sqcup}\alpha(X_O)$, then $c$ is ($X_I, X_O$)- constant time.
Application to low-level code {#llvm:sec}
=============================
We show in this section how the type system we proposed to express output-sensitive constant-time non-interference on the [*While*]{} language can be lifted to a low-level program representation like the LLVM byte code [@lattner2004].
LLVM-IR
-------
$r \leftarrow op(Op, \overrightarrow{v})$ assign to $r$ the result of $Op$ applied to operands $ \overrightarrow{v}$
------------------------------------------- -------------------------------------------------------------------------------
$r \leftarrow load(v)$ load in $r$ the value stored at address pointed by $v$
$store (v_1, v_2)$ store at address pointed by $v_2$ the value $v_1$
$cond (r, b_{then}, b_{else})$ branch to $b_{then}$ if the value of $r$ is true and to $b_{false}$ otherwise
$goto~b$ branch to $b$
We consider a simplified LLVM-IR representation with four instructions: assignments from an expression (register or immediate value) or from a memory block (load), writing to a memory block (store) and (un)conditional jump instructions. We assume that the program control flow is represented by a control-flow graph (CFG) $G = ({\mathcal{B}}, \rightarrow_E, b_{init}, b_{end})$ where ${\mathcal{B}}$ is the set of basic blocks, $\rightarrow_E$ the set of edges connecting the basic blocks, $b_{init} \in {\mathcal{B}}$ the entry point and $b_{end} \in {\mathcal{B}}$ the ending point. We denote by $Reach(b, b')$ the predicate indicating that node $b'$ is [*reachable*]{} from node $b$, i.e., there exists a path in $G$ from $b$ to $b'$. A program is then a (partial) map from control points $(b,n) \in {\mathcal{B}}\times \mathbb{N}$ to instructions where each basic block is terminated by a jump instruction. The memory model consists in a set of [*registers or temporary variables*]{} $R$ and a set of memory blocks $M$ (including the execution stack). $Val$ is the set of values and memory block addresses. The informal semantics of our simplified LLVM-IR is given in Figure \[fig:tab-llvm\], where $r \in R$ and $v \in R \cup Val$ is a register or an immediate value.
In the formal operational semantics, execution steps are labelled with leaking data, i.e., addresses of store and load operations and branching conditions. This formal semantics is defined in Figure \[fig:sem-llvm\].
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Type system {#sec:type-system-llvm}
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First, we introduce the following notations for an LLVM-IR program represented by a CFG $G = ({\mathcal{B}}, \rightarrow_E, b_{init}), b_{end}$:
1. Function $dep: {\mathcal{B}}\rightarrow 2^{\mathcal{B}}$ associates to each basic block its set of “depending blocks”, i.e., $b' \in dep(b)$ iff $b'$ dominates $b$ and there is no block $b"$ between $b'$ and $b$ such that $b"$ post-dominates $b'$. We recall that a node $b_1$ dominates ([*resp.*]{} post-dominates) a node $b_2$ iff every path from the entry node $b_{init}$ to $b_2$ goes through $b_1$ ([*resp.*]{} every path from $b_2$ to the ending node $b_{end}$ goes through $b_1$).
2. Partial function $br: {\mathcal{B}}\rightarrow R$ returns the “branching register”, i.e., the register $r$ used to compute the branching condition leading outside $b$ ($b$ is terminated by an instruction $cond (r, b_{then}, b_{else})$). Note that in LLVM branching registers are always [*fresh*]{} and assigned only once before to be used.
3. Function $PtsTo: ({\mathcal{B}}\times \mathbb{N}) \times (R\cup M) \rightarrow 2^M$ returns the set of (addresses of) memory blocks pointed to by a given register or memory block at a given control point. For example, $bl \in PtsTo(b,n)(r)$ means that at control point $(b,n)$, register $r$ may contain the address of block $bl\in M$.
We now define a type system (Figures \[fig:llvm-op\] to \[fig:llvm-jump\]) that allows to express the output-sensitive constant-time property for LLVM-IR -like programs. The main difference with respect to the rules given at the source level (Figures \[fig:T1\] and \[fig:T14\]) is that the control-flow is explicitly given by the CFG, and not by the language syntax. For a LLVM-like program, an environment $\Gamma : R \cup M \mapsto {\mathcal{L}}$, associates security types to registers and memory blocks.
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#### **Assignment from an operation $Op$ (Figure \[fig:llvm-op\]).**
In rules Op0 and Op2, the new type $\tau$ of the assigned register $r$ is the join of the type of operands $\overrightarrow{v}$ and the type of all the branching conditions dominating the current basic block ($\tau_0$). Note that since branching registers $r$ are assigned only once in LLVM there is no need to update their dependencies from output variables (using the $\lhd_\alpha$ operator), $\Gamma(r)$ being never changed once $r$ has been assigned.
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#### **Assignment from a load expression (Figure \[fig:llvm-load\]).**
Rules Ld1 and Ld2 update $\Gamma$ in a similar way as Op1 and Op2, the main difference being that since some of the memory locations accessed when dereferencing $v$ (i.e., $PtsTo(b,n)(v)$) are in $A_m$ (i.e., potentially in the cache) the dependencies of $v$ are added to the type of the leakage variable $x_l$.
#### **Store instruction (Figure \[fig:llvm-store\]).**
Rule St updates the dependencies of all memory locations pointed to by $v_2$ by adding the types of $v_1$ and $v_2$ itself. In addition, the type of the leakage variable $x_l$ is also updated with the dependencies of $v_2$ and with the dependencies of all branching registers that influenced the execution flow to reach the current block $b$.
-- --
-- --
#### **Conditional and unconditional jump (Figure \[fig:llvm-jump\]).**
Rule CJmp indicates that the leakage variable type is augmented with the type of the branching condition register. Unconditional jumps (Rule Jmp) leave the environment unchanged.
Well typed LLVM programs are output-sensitive constant-time
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\[wt0:llvm\] An LLVM-IR program $p$ is well typed with respect to an initial environment $\Gamma_0$ and final environment $\Gamma'$ (written $\vdash_\alpha p : \Gamma_0 \Rightarrow \Gamma'$) , if there is a family of [*well-defined*]{} environments $\{(\Gamma)_{(b,n)} \ \mid \ (b,n)\in ({\mathcal{B}}, {\mathbbm{N}})\}$, such that for all nodes $(b,n)$ and all its successors $(b',n')$, there exists a type environment ${\gamma}$ and $A\subseteq X_O$ such that $\vdash_\alpha (b,n) : \Gamma_{(b,n)} \Rightarrow {\gamma}\text{\ and \ } ({\gamma}\lhd_\alpha A) {\sqsubseteq}\Gamma_{(b',n')}.$
In the above definition the set $A$ is mandatory in order to prevent dependency cycles between variables in $X_O$.
The following Theorem is the counterpart of Theorem \[l:ctime\]. It shows the soundness of our type system for LLVM-IR programs with respect to output-sensitive constant-time.
\[llvm:ctime\] Let ${\mathcal{L}}$ be the lattice from the section \[s:iol\]. Let $(\Gamma,\alpha)$ be defined by $\Gamma(x)=\{\tau_x\}$, for all $x\in R\cup M$, $\alpha(o)=\{\tau_{\overline{o}}\}$, for all $o\in X_O$ and $\Gamma(x_l)=\bot$. If $\vdash_\alpha p : \Gamma \Rightarrow \Gamma' $ and $\Gamma'(x_l) {\sqsubseteq}\Gamma(X_I) {\sqcup}\alpha(X_O)$, then $p$ is ($X_I, X_O$)- constant time.
Example
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We illustrate below the effect of the LLVM-IR typing rules on a short example. The C code of this example is given on Figure \[fig:c-ex\], and the corresponding (simplified) LLVM-IR on Figure \[fig:llvm-ex\].
int p[10], q[10] ; // global variables
int main () {
int x, y ;
p[x] = q[y] ;
return 0 ; // output value is always 0
}
@q = common global
@p = common global
%x = alloca i32
%y = alloca i32
%1 = load i32* %y
%2 = getelementptr @q, 0, %1
%3 = load %2
%4 = load %x
%5 = getelementptr @p, 0, %4
store %3, %5
First, we assume that $x_l$ denotes the *leakage* variable and that the content of C variables [p]{}, [q]{}, [x]{} and [y]{} are stored in memory blocks $b_0$ to $b_3$, i.e. at the initial control point, $PtsTo(@p)=b_0, PtsTo(@q)=b_1, PtsTo(\%x)=b_2, PtsTo(\%y)=b_3$.
Now we consider the following initial environment:
$\Gamma_0(@p) = \Gamma_0(@q) = \Gamma_0(\%x) = \Gamma_0(\%y) = \bot $\
$\Gamma_0(b_0) = P, \Gamma_0(b_1) = Q, \Gamma_0(b_2) = X, \Gamma_0(b_3) = Y, \Gamma_0(x_l) = \bot$
This initial environment captures the idea that the values of variables $@p$, $@q$, $\%x$, $\%y$ are addresses (of memory blocks corresponding to the “high-level” C variables $p$, $q$, $x$ and $y$) and hence their security type is $\bot$, and the memory blocks $b_0$ to $b_3$ correspond to the C variables $p$, $q$, $x$ and $y$, and this is mirrored in the initial environment $\Gamma_0$. Moreover, initially, nothing is leaked yet.
We then update $\Gamma_0$ by applying our typing rules in sequence to each instruction of the LLVM-IR representation. Note that the [getelementptr]{} instruction, which is specific to LLVM, allows to compute an address corresponding to an indexed access in a buffer. Hence, it is treated by our typing system as an arithmetic ($Op$) instruction.
[%1 = load %y]{} $$\begin{aligned}
\Gamma_1 & = & \Gamma_0[x_l \rightarrow \Gamma_0(x_l) \sqcup \Gamma_0(\%y), \%1 \rightarrow \Gamma_0(\%y) \sqcup \Gamma_0(PtsTo(\%y))] \\
& = & \Gamma_0[x_l \rightarrow \bot, \%1 \rightarrow Y]\end{aligned}$$
[%2 = getelementptr @q, 0, %1]{} $$\begin{aligned}
PtsTo(\%2)& =& b_1 \\
\Gamma_2 & = & \Gamma_1[\%2 \rightarrow \Gamma_1(@q) \sqcup \Gamma_1(0) \sqcup \Gamma_1(\%1)] \\
& = & \Gamma_1[\%2 \rightarrow Y]\end{aligned}$$
[%3 = load %2]{} $$\begin{aligned}
\Gamma_3 & = & \Gamma_2[x_l \rightarrow \Gamma_2(x_l) \sqcup \Gamma_2(\%2), \%3 \rightarrow \Gamma_0(\%2) \sqcup \Gamma_0(PtsTo(\%2))] \\
& = & \Gamma_2[x_l \rightarrow Y, \%3 \rightarrow Y {\sqcup}Q] \end{aligned}$$
[%4 = load %x]{} $$\begin{aligned}
\Gamma_4 & = & \Gamma_3[x_l \rightarrow \Gamma_3(x_l) \sqcup \Gamma_3(\%x), \%4 \rightarrow \Gamma_3(\%x) \sqcup \Gamma_3(PtsTo(\%x))] \\
& = & \Gamma_3[\%4 \rightarrow X]\end{aligned}$$
[%5 = getelementptr @p, 0, %4]{} $$\begin{aligned}
PtsTo(\%5)& =& b_0 \\
\Gamma_5 & = & \Gamma_4[\%5 \rightarrow \Gamma_4(@p) \sqcup \Gamma_4(0) \sqcup \Gamma_4(\%4)] \\
& = & \Gamma_4[\%5 \rightarrow X]\end{aligned}$$
[store %3, %5]{} $$\begin{aligned}
\Gamma_6 & = & \Gamma_5[x_l \rightarrow \Gamma_5(x_l) \sqcup \Gamma_5(\%5), PtsTo(\%5) \rightarrow \Gamma_5(PtsTo(\%5)) \sqcup \Gamma_5(\%3) \sqcup \Gamma_5(\%5)] \\
& = & \Gamma_5[x_l \rightarrow Y \sqcup X, b_0 \rightarrow P {\sqcup}Y {\sqcup}Q \sqcup X]\end{aligned}$$
Making all the replacements, we get that the final environment is: $$\begin{aligned}
\Gamma_6 & = &[x_l \rightarrow Y \sqcup X, b_0 \rightarrow P {\sqcup}Y {\sqcup}Q \sqcup X, b_1 \rightarrow Q, b_2 \rightarrow X, b_3 \rightarrow Y, @p \rightarrow \bot, @q \rightarrow \bot,\\
& & \%x \rightarrow \bot, \%y \rightarrow \bot, \%1 \rightarrow Y, \%2 \rightarrow Y, \%3 \rightarrow Y{\sqcup}Q, \%4 \rightarrow X, \%5 \rightarrow X] \end{aligned}$$ In this final environment $\Gamma_6$, variable $x_l$ depends on the initial types $X$ and $Y$ assigned to memory blocks $b_2$ and $b_3$. This means that the addresses accessed when reading (resp. writing) buffer [p]{} (resp. [q]{}) may *leak* to an attacker. Hence, if one of the variables $x$ or $y$ is a secret, since neither $x$ nor $y$ is an output value, then this program is not output sensitive constant-time, which may lead to a security issue.
Implementation
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We are developing a prototype tool implementing the type system for LLVM programs. This type system consists in computing flow-sensitive dependency relations between program variables. Def. \[wt0:llvm\] provides the necessary conditions under which the obtained result is sound (Theorem \[llvm:ctime\]). We give some technical indications regarding our implementation.
Output variables $X_O$ are defined as function return values and global variables; we do not currently consider arrays nor pointers in $X_O$. Control dependencies cannot be deduced from the syntactic LLVM level, we need to explicitly compute the dominance relation between basic blocks of the CFG (the $dep$ function). Def. \[wt0:llvm\] requires the construction of a set $A \subseteq X_O$ to update the environment produced at each control locations in order to avoid circular dependencies (when output variable are assigned in [*alternative*]{} execution paths). To identify the set of basic blocks belonging to such alternative execution paths leading to a given block, we use the notion of [*Hammock regions*]{} [@ferrante1987]. More precisely, we compute function $Reg: ({\mathcal{B}}\times {\mathcal{B}}\times (\rightarrow_E)) \rightarrow 2^{\mathcal{B}}$, returning the set of [*Hammock regions*]{} between a basic block $b$ and its immediate dominator $b'$ with respect to an incoming edge $e_i$ of $b$. Thus, $Reg(b', b, (c,b))$ is the set of blocks belonging to CFG paths going from $b'$ to $b$ without reaching edge $e_i=(c,b)$: $Reg(b', b, (c,b)) = \{ b_i \mid b' \rightarrow_E b_1 \dots \rightarrow_E b_n \rightarrow_E b \wedge \forall i \in [1,n-1].\; \neg Reach (b_i, c) \}.$ Fix-point computations are implemented using Kildall’s algorithm. To better handle real-life examples we are currently implementing the $PtsTo$ function, an inter-procedural analysis, and a more precise type analysis combining both over- and under-approximations of variable dependencies (see section \[concl:sec\]).
Related Work
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#### [**Information flow.**]{}
There is a large number of papers on language-based security aiming to prevent undesired information flows using type systems (see [@sab2003]). An information-flow security type system statically ensures noninterference, i.e. that sensitive data may not flow directly or indirectly to public channels [@vol1996; @mye1999; @vau2007; @swa2010]. The typing system presented in section \[s:w1\] builds on ideas from Hunt and Sands’
As attractive as it is, noninterference is too strict to be useful in practice, as it prevents confidential data to have any influence on observable, public output: even a simple password checker function violates noninterference. Relaxed definitions of noninterference have been defined in order to support such intentional downward information flows [@sab2009]. Li and Zdancewic [@li2005] proposed an expressive mechanism called [*relaxed noninterference*]{} for declassification policies that supports the extensional specification of secrets and their intended declassification. A declassification policy is a function that captures the precise information on a confidential value that can be [*declassified*]{}. For the password checker example, the following declassification policy $\lambda p.\lambda x.h(p)==x$, allows an equality comparison with the hash of password to be declassified (and made public), but disallows arbitrary declassifications such as revealing the password.
The problem of information-flow security has been studied also for low level languages. Barthe and Rezk [@bar2003; @bar2007] provide a flow sensitive type system for a sequential bytecode language. As it is the case for most analyses, implicit flows are forbidden, and hence, modifications of parts of the environment with lower security type than the current context are not allowed. Genaim and Spoto present in [@gen2005] a compositional information flow analysis for full Java bytecode.
#### [**Information flow applied to detecting side-channel leakages.**]{}
Information-flow analyses track the flow of information through the program but often ignore information flows through side channels. Side-channel attacks extract sensitive information about a program’s state through its observable use of resources such as time or memory. Several approaches in language-based security use security type systems to detect timing side-channels [@aga2000; @hed2005]. Agat [@aga2000] presents a type system sensitive to timing for a small While-language which includes a transformation which takes a program and transforms it into an equivalent program without timing leaks. Molnar et al [@mol2005] introduce the program counter model, which is equivalent to path non-interference, and give a program transformation for making programs secure in this model.
FlowTracker [@rodrigues2016] allows to statically detect time-based side-channels in LLVM programs. Relying on the assumption that LLVM code is in SSA form, they compute control dependencies using a sparse analysis [@Choi1991] without building the whole Program Dependency Graph. Leakage at assembly-level is also considered in [@bar2014]. They propose a fine-grained information-flow analysis for checking that assembly programs generated by CompCert are constant-time. Moreover, they consider a stronger adversary which controls the scheduler and the cache.
All the above works do not consider publicly observable outputs. The work that is closest to ours is [@almeida2016], where the authors develop a formal model for constant-time programming policies. The novelty of their approach is that it is distinguishing not only between public and private input values, but also between private and publicly observable output values. As they state, this distinction poses interesting technical and theoretical challenges. Moreover, constant-time implementations in cryptographic libraries like OpenSSL include optimizations for which paths and addresses can depend not only on public input values, but also on publicly observable output values. Considering only input values as non-secret information would thus incorrectly characterize those implementations as non-constant-time. They also develop a verification technique based on the self-composition based approach [@bar2004]. They reduce the constant time security of a program P to safety of a product program Q that simulates two parallel executions of P. The tool operates at the LLVM bytecode level. The obtained bytecode program is transformed into a product program which is verified by the Boogie verifier [@bar2005] and its SMT tool suite. Their approach is complete only if the public output is ignored. Otherwise, their construction relies on identifying the branches whose conditions can only be declared benign when public outputs are considered. For all such branches, the verifier needs to consider separate paths for the two simulated executions, rather than a single synchronized path and in the worst case this can deteriorate to an expensive product construction.
Conclusion and Perspectives {#concl:sec}
===========================
\[sec:conclusion\]
In this paper we proposed a static approach to check if a program is output-sensitive constant-time, i.e., if the leakage induced through branchings and/or memory accesses do not overcome the information produced by (regular) observable outputs. Our verification technique is based on a so-called output-sensitive non-interference property, allowing to compute the dependencies of a leakage variable from both the initial values of the program inputs and the final values of its outputs. We developed a type system on a high-level [**While**]{} language, and we proved its soundness. Then we lifted this type system to a basic LLVM-IR and we developed a prototype tool operating on this intermediate representation, showing the applicability of our technique.
This work could be continued in several directions. One limitation of our method arising in practice is that even if the two snippets $x_l=h;o=h$ and $o=h;x_l=o$ are equivalent, only the latter can be typed by our typing system. We are currently extending our approach by considering also an under-approximation $\beta(\bullet)$ of the dependencies between variables and using “symbolic dependencies” also for non-output variables. Then the safety condition from Theorem \[l:sec\] can be improved to something like “$\exists V$ such that $(\Gamma'(x_l)\lhd_\alpha V) {\sqsubseteq}(\Gamma(X_I)\lhd_\alpha V) {\sqcup}(\beta'(X_O) \lhd_\alpha V) {\sqcup}\alpha(X_O)$”. In the above example, we would obtain $\Gamma'(x_l) = \alpha(h) = \beta'(o) {\sqsubseteq}\alpha(o) {\sqcup}\beta'(o)$, meaning that the unwanted maximal leakage $\Gamma'(x_l)$ is less than the minimal leakage $\beta'(o)$ due to the normal output. From the implementation point of view, further developments are needed in order to extend our prototype to a complete tool able to deal with real-life case studies. This may require to refine our notion of arrays and to take into account arrays and pointers as output variables. We could also consider applying a sparse analysis, as in FlowTracker [@rodrigues2016]. It may happen that such a pure static analysis would be too strict, rejecting too much “correct” implementations. To solve this issue, a solution would be to combine it with the dynamic verification technique proposed in [@almeida2016]. Thus, our analysis could be used to find automatically which branching conditions are benign in the output-sensitive sense, which could reduce the product construction of [@almeida2016]. Finally, another interesting direction would be to adapt our work in the context of quantitative analysis for program leakage, like in [@CacheAudit].
[^1]: This work is supported by the French National Research Agency in the framework of the “Investissements d’ avenir” program (ANR-15-IDEX-02)
[^2]: mbed TLS (formerly known as PolarSSL). https://tls.mbed.org/
|
---
abstract: 'Let $(X,\Delta)$ be a log canonical $4$-fold over an algebraically closed field of characteristic zero. Assume that the ${\mathbb{Q}}$-divisor $K_X+\Delta$ is pseudo-effective. We prove that any sequence of $(K_X+\Delta)$-flips terminates.'
address: ' Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, UT 84112'
author:
- Joaquín Moraga
title: 'Termination of pseudo-effective 4-fold flips'
---
[^1]
Introduction {#introduction .unnumbered}
============
Two of the main goals of the minimal model program are to prove the existence of flips and termination of any sequence of such birational transformations. The existence and termination of flips for terminal $3$-folds was achieved by Mori in [@Mor88]. This result was generalized to the log canonical case by Shokurov [@Shok96]. All these proofs rely on a careful analysis of the flipping contractions for $3$-folds. In dimension $4$, Kawamata settled the existence of smooth $4$-fold flips in [@Kaw89] and this result was generalized to the singular case by many authors (see, e.g., [@KMM87]). In [@Fuj04], Fujino proved the termination of canonical $4$-fold flips by studying an invariant that decreases with flips, the so-called [*difficulty function*]{}. Later, in [@AHK] Alexeev, Hacon, and Kawamata proved termination of many klt $4$-fold flips using the same invariant. In [@Shok09], Shokurov introduced a special sequence of flips for the minimal model program of a pair, called [*ordered flips*]{}, or [*flips with scaling*]{}, and proved that any sequence of flips with scaling for klt $4$-folds terminates. The existence of flips in arbitrary dimension was finally achieved by Birkar, Cascini, Hacon, and McKernan in [@BCHM], where the authors also prove termination of flips with scaling for klt pairs $(X,\Delta)$ with $\Delta$ a big ${\mathbb{Q}}$-divisor. In [@Bir07], Birkar proved termination of any sequence of flips for klt pairs with $K_X+\Delta \sim_{\mathbb{Q}}D\geq 0$ assuming termination of flips in dimension $\dim(X)-1$ and the ACC conjecture for log canonical thresholds. Such conjecture was proved by Hacon, McKernan, and Xu in [@HMX14].
The primary technique in the article [@Bir07] is to study the log canonical threshold of the divisor $D$ with respect to $K_X+\Delta$, prove that this invariant increases with flips, and eventually it strictly increases in any sequence of flips, up to passing to a quasi-projective variety. Then, the ascending chain condition for log canonical thresholds shows that the sequence of flips must terminate. In this article, we prove termination of flips for a pseudo-effective log canonical $4$-fold, over an algebraically closed field, using a similar invariant. We will consider a generalized log canonical threshold with respect to the pair $(X,\Delta)$, that we denote by ${\rm lct}(K_X+\Delta)$ (see Definition \[lctdefinition\]) and we will call it the [*log canonical threshold*]{} of the pair. We want to prove that it behaves well in a sequence of pseudo-effective Kawamata log terminal $4$-fold flips. More precisely, let $$\label{seqflips}\nonumber
\xymatrix{
(X,\Delta)\ar@{-->}^-{\pi_1}[r] & (X_1,\Delta_1)\ar@{-->}^-{\pi_2}[r] & (X_2,\Delta_2)\ar@{-->}^-{\pi_3}[r] & \dots \ar@{-->}^-{\pi_j}[r] & (X_j,\Delta_j)\ar@{-->}^-{\pi_{j+1}}[r] & \dots \\
}$$ be a sequence of flips of klt $4$-folds, such that $K_X+\Delta$ is a pseudo-effective ${\mathbb{Q}}$-divisor. It is straightforward to prove that the inequality $${\rm lct}(K_{X_j}+\Delta_j) \leq {\rm lct}(K_{X_{j+1}}+\Delta_{j+1})$$ holds for every $j\in \mathbb{Z}_{\geq 1}$ (see Lemma \[acc\]). We will prove that in any such sequence, after finitely small modifications, the log canonical threshold strictly increases (up to passing to a quasi-projective variety). Then, we use the ACC for generalized log canonical thresholds, concluding the following:
\[termination\] Let $(X,\Delta)$ be a log canonical $4$-fold over an algebraically closed field of characteristic zero. Assume that the ${\mathbb{Q}}$-divisor $K_X+\Delta$ is pseudo-effective. Then, any sequence of $(K_X+\Delta)$-flips terminates.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The author would like to thank Caucher Birkar, Christopher Hacon, Sébastien Boucksom, Stefano Filipazzi, Tommaso de Fernex, and Yoshinori Gongyo.
Description of the proof {#desc}
========================
Let $(X,\Delta)$ be a Kawamata log terminal $4$-fold with $K_X+\Delta$ a pseudo-effective ${\mathbb{Q}}$-divisor. In Definition \[lctdefinition\], we will attach an invariant $c={\rm lct}(K_X+\Delta)$ to the pair $(X,\Delta)$. This invariant, is a generalized log canonical threshold in the sense of [@BZ16]. We consider an ample divisor $A$ on $X$. For $\lambda \in {\mathbb{Q}}_{>0}$ small enough, we will denote by $G_\lambda$ a general element in the ${\mathbb{Q}}$-linear system $|K_X+\Delta+c_\lambda G_\lambda|_{\mathbb{Q}}$ and by $c_\lambda$ the log canonical threshold of $G_\lambda$ with respect to $K_X+\Delta$. We prove that the log canonical thresholds $c_\lambda$ converge to $c$ when $\lambda \rightarrow 0$. Moreover, we will prove that the non-klt locus of the log canonical pairs $(X,\Delta+c_\lambda G_\lambda)$ stabilize to a subvariety $W$ for $\lambda$ sufficiently small.
In Section \[section:adjunction\], we will prove an adjunction formula for the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $(c+1)(K_X+\Delta)$ to any common minimal log canonical center $W_i \subseteq W$ of the log canonical pairs $(X,\Delta+c_\lambda G_\lambda)$. More precisely, using Kawamata subadjunction Theorem (see, e.g., [@HK10 Theorem 13.13]) we can write $$(K_X+\Delta +c_\lambda G_\lambda)|_{W_i} \sim_{\mathbb{Q}}K_{W_i} + B_{W_i}^{G_{\lambda}} + M_{W_i}^{G_\lambda},$$ where $M_{W_i}^{G_\lambda}$ is a ${\mathbb{Q}}$-divisor which is the push-forward of a nef divisor on a higher birational model of $W_i$, and $B_{W_i}^{G_\lambda}$ is an effective ${\mathbb{Q}}$-divisor, such that the pair $$(W_i, B_{W_i}^{G_\lambda})$$ is log canonical whenever $M_{W_i}^{G_\lambda}$ is ${\mathbb{Q}}$-Cartier. We aim to define the limit when $\lambda$ converges to zero of the above subadjunction formula. However, both divisors depend of the choice of $G_{\lambda}$ in its ${\mathbb{Q}}$-linear system. So in order to take the limit we need to prove that the divisor $$B^{\lambda}_{W_i} = \bigwedge_{j\in J} B_{W_i}^{G^{j}_\lambda}$$ is a well-defined effective ${\mathbb{R}}$-divisor, where $J$ is a finite set, and the ${\mathbb{Q}}$-divisors $G^{j}_\lambda$ are general in their ${\mathbb{Q}}$-linear system. Thus, we have a subadjunction formula $$(K_X+\Delta +c_\lambda G_\lambda)|_{W_i} \sim_{\mathbb{Q}}K_{W_i} + B_{W_i}^{\lambda} + M_{W_i}^{\lambda},$$ where the support of $B_{W_i}^\lambda$ is independent of $\lambda$ and $M_{W_i}^{\lambda}$ is nef in codimension one. Since the support of $B_{W_i}^{\lambda}$ is independent of $\lambda$ it makes sense to construct a limit ${\mathbb{R}}$-divisor $B_{W_i}$ when $\lambda$ converges to zero. This limit divisor will be used to define the desired adjunction formula $$((c+1)(K_X+\Delta))|_{W_i} \sim_{\mathbb{Q}}K_{W_i} + B_{W_i} + M_{W_i}.$$ Therefore, a sequence of flips for the pair $(X,\Delta)$ which does not contain $W_i$ in a flipped loci will induce a sequence of quasi-flips for the triple $(W_i, B_{W_i}+M_{W_i})$. By construction, this triple is a generalized pair in the sense of [@BZ16]. Moreover, we will show that it is a generalized log canonical pair.
In Section \[section:termination\], we prove that any $(K_X+\Delta)$-flip that intersects $W_i$ non-trivially and does not contain $W_i$ in its flipping locus induces an ample strict quasi-flip for the generalized log canonical pair $(W_i,B_{W_i}+M_{W_i})$. Therefore, in order to prove termination of flips around $W_i$, it suffices to show termination of ample strict quasi-flips for generalized log canonical pairs of dimension at most three. Termination of these quasi-flips in codimension one is proved using the fact that the coefficients of $B_{W_i}$ belong to a DCC set. Then, we prove termination of the weak quasi-flips by using standard arguments of low-dimensional flips.
Once we prove termination around each minimal log canonical center $W_i$ of the log canonical pairs $(X,\Delta+c_\lambda G_\lambda)$ we deduce termination around $W$, meaning that in any sequence of flips eventually all flipping loci are disjoint from the strict transform of $W$ on $X_j$. Replacing the variety $X_j$ with the complement of the strict transform of $W$, and the divisor $\Delta_j$ with the restriction to this quasi-projective subvariety, we achieve that the generalized log canonical pair $(c+1)(K_{X_j}+\Delta_j)$ has strictly less generalized log canonical centers than the pair $(c+1)(K_X+\Delta)$. Proceeding inductively, we deduce that for $j$ large enough the generalized pair $(c+1)(K_{X_j}+\Delta_j)$ is Kawamata log terminal, so we have that ${\rm lct}(K_{X}+\Delta) < {\rm lct}(K_{X_j}+\Delta_j)$. Thus, we deduce that an infinite sequence of $(K_X+\Delta)$-flips induces a sequence of generalized log canonical thresholds violating the ACC, leading to a contradiction.
Preliminaries and notation {#section:preliminaries}
==========================
In this section, we recall classic results and notation. We will follow the notation of standard references algebraic geometry [@Laz04a; @Laz04b] and the minimal model program [@KM98; @HK10; @Kol13]. Throughout this paper, we will work over an algebraically closed field $\mathbb{K}$ of characteristic zero.
Given a projective birational morphism $p \colon Y\rightarrow X$ from a normal variety $Y$ and a prime divisor $D$ on $Y$, we say that $p(D)$ is the [*center of $D$ on $X$*]{}. In what follows, we may identify the class of prime divisors over $X$ with the class of [*divisorial valuations*]{} of the function field $\mathbb{K}(X)$. The [*center on $X$ of a divisorial valuation of $X$*]{} is just the center of the corresponding prime divisor. The center of the divisorial valuation $E$ on $X$ wil be denoted by $c_E(X)$.
Generalized pairs
-----------------
In this subsection, we recall the standard definitions of generalized pairs.
\[genpair\] A [*generalized sub-pair*]{} is a triple $(X,B+M)$, where $X$ is a quasi-projective normal algebraic variety, $K_X+B+M$ is ${\mathbb{R}}$-Cartier, and $M$ is the push-forward of a nef divisor on a higher birational model of $X$. More precisely, there exists a projective birational morphism $f\colon X'\rightarrow X$ from a normal quasi-projective variety $X'$ and a nef ${\mathbb{R}}$-Cartier ${\mathbb{R}}$-divisor $M'$ such that $M=f_*(M')$. We define $B'$ by the equation $$K_{X'}+B'+M'=f^*(K_X+B+M).$$ A generalized sub-pair is called a [*generalized pair*]{} if $B$ is an effective divisor. We say that $B$ is the [*boundary part*]{} and that $M$ is the [*nef part*]{} of the generalized pair $(X,B+M)$. We may call the sum $B+M$ a [*generalized boundary*]{}. Observe that $M'$ defines a nef b-Cartier ${\mathbb{R}}$-divisor in the sense of [@Cor07 Definition 1.7.3]. We will say that this is the nef b-divisor [*associated to the generalized pair*]{}.
\[logsmooth\] Observe that we can always take a projective birational morphism $g\colon X'' \rightarrow X'$, such that $(X'',B''+M'')$ is a log smooth sub-pair, where $K_{X''}+ B''=g^*(K_{X'}+B')$ and $M''=g^*(M')$. Since $M''$ is nef, we can always replace $X'$ by $X''$ in the above definition, and therefore we may assume that $(X',B'+M')$ is log smooth.
\[logresolutiongeneralized\] Given a generalized pair $(X,B+M)$ and a projective birational morphism $f \colon X'' \rightarrow X$, we will say that $f$ is a [*log resolution*]{} of the generalized pair, if $X'' \rightarrow X$ factors through the variety $X'$, the exceptional locus ${\rm Ex}(f)$ is a divisor, and ${\rm Ex}(f)+B''$ is a divisor with simple normal crossing support. We may say that $(X'',B''+M'')$ is the [*associated generalized pair*]{} on $X''$.
Given two generalized pairs $(X,B+M)$ and $(X',B'+M')$ such that $X$ and $X'$ are birational, we will say that the nef parts $M$ and $M'$ are [*trace of a common nef b-divisor*]{}, if there exists a log resolution $(X'',B''+M'')$ for both generalized pairs with projective birational maps $\pi \colon X''\rightarrow X$ and $\pi'\colon X''\rightarrow X'$, such that $M=\pi_*(M'')$ and $M'=\pi'_*(M'')$.
Let $(X,B+M)$ be a generalized pair, $g\colon Y\rightarrow X'$ be a projective birational morphism, and $E$ a prime divisor on $Y$. We denote by $h$ the composition $f\circ g$, and we define the ${\mathbb{R}}$-divisor $B_Y$ by the formula $$K_Y+B_Y+M_Y=h^*(K_X+B+M),$$ where $M_Y=g^*(M')$. The [*generalized discrepancy*]{} (resp. [*generalized log discrepancy*]{}) of the generalized pair $(X,B+M)$ along the divisor $E$ is $-{\rm coeff}_{E}(B_Y)$ (resp. $1-{\rm coeff}_{E}(B_Y)$). Given $\epsilon \in (0,1)$, we say that the generalized pair $(X,B+M)$ is [*generalized $\epsilon$-Kawamata log terminal*]{} (resp. [*generalized log canonical*]{}) if generalized log discrepancies with respect to prime divisors over $X'$ are greater than $\epsilon$ (resp. non-negative). As usual, we may write [klt]{} (resp. [lc]{}) to abbreviate Kawamata log terminal (resp. log canonical). Moreover, if $\epsilon$ is zero in the above definition, we will omit it from the notation.
Given a prime divisor $E$ over $X$, we denote the generalized discrepancy of $(X,B+M)$ at $E$ by $a_E(X,B+M)$. We also may say that the divisorial valuation $E$ has generalized discrepancy $a_E(X,B+M)$ with respect to the generalized pair $(X,B+M)$, or with respect to the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $K_X+B+M$. If the generalized log discrepancy along $E$ is non-positive (resp. zero) we say that the image of $E$ on $X$ is a [*generalized non-klt center*]{} (resp. [*generalized log canonical center*]{}) of the generalized pair. The union of all the generalized non-klt centers of a generalized pair $(X,B+M)$ is the [*non-klt locus of the pair*]{}.
By Remark \[logsmooth\], we may assume that $(X',B'+M')$ is itself a log resolution of the generalized pair $(X,B+M)$, and therefore $(X,B+M)$ is generalized klt if and only if its generalized log discrepancies with respect to any prime divisor on $X'$ are greater than zero, or equivalently, if the coefficients of $B'$ are less than or equal to one.
A divisorial valuation $E$ over the generalized pair $(X,B+M)$ is called [*terminal*]{} if $a_E(X,B+M)\in {\mathbb{R}}_{>0}$, and it is called [*non-terminal*]{} otherwise. A generalized pair $(X,B+M)$ is said to be [*terminal*]{} if all its exceptional divisorial valuations are terminal with respect to $K_X+B+M$.
If $M=0$ in the above definition, then $(X,B)$ is a [*pair*]{} in the usual sense of [@KM98]. Conversely, every pair can be considered as a generalized pair with trivial nef part. In what follows, we may denote the boundary part of a pair by $\Delta\geq 0$, following the standard notation of [@KM98]. If we work with pairs, we will drop the word [*generalized*]{} from the above definitions.
Quasi-flips
-----------
In this subsection, we recall the standard definitions of quasi-flips for generalized pairs.
\[qfgen\] Let $(X,B+M)$ be a generalized log canonical pair. A birational contraction $\phi \colon X \rightarrow Z$ is said to be a [*weak contraction*]{} for the generalized pair if $-(K_X+B+M)$ is nef over $Z$. A [*quasi-flip*]{} of $\phi$ is a birational map $\pi \colon X \dashrightarrow X^+$ with a birational contraction $\phi^+ \colon X^+ \rightarrow Z$ such that the following conditions hold:
- The triple $(X^+, B^{+}+M^{+})$ is generalized log canonical,
- the ${\mathbb{R}}$-Cartier ${\mathbb{R}}$-divisor $K_{X^+}+B^{+}+M^{+}$ is nef over $Z$,
- the inequality $\phi^{+}_*(B^{+}) \leq \phi_*(B)$ of Weil ${\mathbb{R}}$-divisors on $Z$ holds, and
- the nef parts $M$ and $M^+$ are the trace of a common nef b-divisor.
As usual, the morphism $\phi$ is called the [*flipping contraction*]{} and the morphism $\phi^+$ is called the [*flipped contraction*]{}. We say that the quasi-flip $\pi$ is [*weak*]{} if both $\phi$ and $\phi^+$ are small morphisms, and that the quasi-flip $\pi$ is [*ample*]{} if both $-(K_X+B+M)$ and $(K_{X^{+}}+B^{+}+M^{+})$ are ample over $Z$. A [*flip*]{} for a generalized log canonical pair is a weak ample quasi-flip such that both $\phi$ and $\phi^+$ have relative Picard rank one. In the case that we have a sequence of quasi-flips, we will further require that all nef parts are trace of the same Cartier b-divisor.
A sequence of quasi-flips for generalized log canonical pairs $(X_j,B_j+M_j)$ is said to be [*under a DCC set*]{} if the coefficients of the boundary parts $B_j$ belong to a DCC set.
\[qflopgen\] We say that a quasi-flip $\pi$ is a [*quasi-flop*]{} if $\pi$ and $\pi^{-1}$ are quasi-flips. In this case, the flipping contraction (resp. flipped contraction) is called the [*flopping contraction*]{} (resp. [*flopped contraction*]{}) and the flipping locus (resp. flipped locus) is called the [*flopping locus*]{} (resp. [*flopped locus*]{}).
\[definitionstrict\] Given a quasi-flip $\pi \colon X\dashrightarrow X^{+}$ with flipping and flipped contractions $\phi$ and $\phi^+$ respectively, we define the [*non-flopping locus*]{} to be the smallest Zariski closed subset $N$ of $Z$ such that $\pi$ is a quasi-flop over $Z\setminus N$. We say that a quasi-flip is [*strict*]{} if the subvariety $N$ is non-empty.
\[kltflip\] A quasi-flip for generalized log canonical pairs is said to be [*klt*]{} if its flipping locus, and therefore its flipped locus, does not intersect the non-klt locus of the generalized pair. A quasi-flip for generalized log canonical pairs is said to be [*terminal*]{} if its flipping and flipped locus do not contain the center of a non-terminal valuation.
The following proposition is well-known for pairs, see for example [@Shok04 Monotonicity]. The proof in the case of generalized pairs is analogous.
\[monotonicity\] Given a quasi-flip $\pi \colon X \dashrightarrow X^+$ for the generalized pairs $(X,B+M)$ and $(X^+,B^+ + M^+)$ over $Z$, and a prime divisor $E$ over $X$, we have that $$a_E(X,B+M) \leq a_E(X^+,B^+ + M^+)$$ and such inequality is strict if the center of $E$ on $Z$ is contained in the non-flopping locus. Moreover, the non-flopping locus of an ample quasi-flip is the image of the flipping or flipped locus on $Z$.
Minimal models
--------------
In this subsection, we recall the standard definitions of minimal models.
A pair $(X,\Delta)$ is called [*divisorially log terminal*]{}, or [*dlt*]{} for short, if the coefficients of $\Delta$ are in the invertal $[0,1]$, there exists a log resolution $p\colon Y\rightarrow X$ such that we can write $K_Y+\Delta_Y+E=p^*(K_X+\Delta)$, where $\Delta_Y$ is the strict transform of $\Delta$ on $Y$, and the coefficients of the prime divisors of $E$ are less than one.
An ample weak contraction $\phi \colon X\rightarrow Z$ of relative Picard rank one such that $K_Z+\phi_*(\Delta)$ is ${\mathbb{R}}$-Cartier is called a [*$(K_X+\Delta)$-divisorial contraction*]{}. Indeed, the ${\mathbb{R}}$-Cartier condition of the divisor $K_Z+\phi_*(\Delta)$ implies that the exceptional locus of the morphism $\phi \colon X\rightarrow Z$ is purely divisorial, and if the divisorial contraction has Picard rank one, then the exceptional divisor must be irreducible (see, e.g., [@HK10]).
\[definitionminimalmodel\] Given a dlt pair $(X,\Delta)$ and a rational map $\pi \colon X \dashrightarrow X_{\rm min}$ such that
- $X_{\rm min}$ is a quasi-projective normal variety,
- $\pi^{-1}$ contracts no divisors, or equivalently, $\pi$ extracts no divisors,
- $K_{X_{\rm min}}+\Delta_{\rm min}$ is nef ${\mathbb{R}}$-Cartier ${\mathbb{R}}$-divisor, where $\Delta_{\rm min}$ is the strict transform of $\Delta$ on $X_{\rm min}$, and
- $a_E(X,\Delta)<a_E(X_{\rm min},\Delta_{\rm min})$ for every $\pi$-exceptional irreducible divisor $E\subsetneq X$.
We say that $\pi$ is a [*log terminal model*]{} (or [*minimal model*]{}) of $(X,\Delta)$. When $\pi$ is clear from the context, we also may say that $(X_{\rm min},\Delta_{\rm min})$ is a [*log terminal model*]{} (or [*minimal model*]{}) of $(X,\Delta)$.
Given a dlt pair $(X,\Delta)$, we say that a sequence of $(K_X+\Delta)$-divisorial contractions and $(K_X+\Delta)$-flips is a [*minimal model program*]{} for $(X,\Delta)$ if the composition $\pi$ of such birational maps is a [*log terminal model*]{} (or [*minimal model*]{}) of $(X,\Delta)$.
Let $(X,\Delta)$ be a pair and $p\colon Y \rightarrow X$ a projective birational morphism from a normal variety $Y$, then we will denote $K^{\Delta}_{Y/X}= K_Y - p^*(K_X+\Delta)$ the [*relative canonical divisor*]{}.
\[dual\] Given a pseudo-effective klt pair $(X,\Delta)$ with a minimal model $(X_{\rm min},\Delta_{\rm min})$, we can realize $K_X+\Delta$ as a generalized boundary. Indeed, let $p\colon Y \rightarrow X$ and $q\colon Y \rightarrow X_{\rm min}$ be two projective birational morphisms that give a log resolution of the birational contraction $\pi \colon X \dashrightarrow X_{\rm min}$, then we can write $$p^*(K_X+\Delta) = q^*(K_{X_{\rm min}}+\Delta_{\rm min}) + E,$$ where $q^*(K_{X_{\rm min}}+\Delta_{\rm min})$ is a nef ${\mathbb{R}}$-Cartier ${\mathbb{R}}$-divisor and $E$ is an effective $q$-exceptional ${\mathbb{R}}$-divisor. Thus, we have that $$K_X+\Delta = p_*( q^*(K_{X_{\rm min}}+\Delta_{\rm min})) + p_*(E)$$ is a generalized boundary with nef part $ p_*( q^*(K_{X_{\rm min}}+\Delta_{\rm min}))$ and boundary part $p_*(E)$. Moreover, for every $\mu \in {\mathbb{R}}_{\geq 1}$ we can realize the ${\mathbb{R}}$-Cartier ${\mathbb{R}}$-divisor $\mu (K_X+\Delta)$ as a generalized pair by writting $$\label{klttogen}
\mu(K_X+\Delta) = K_X+ \Delta + (\mu-1)p_*(E)+ (\mu-1)p_*( q^*(K_{X_{\rm min}}+\Delta_{\rm min})).$$ Here the boundary part is $$\Delta + (\mu-1)p_*(E)$$ and the nef part is $$(\mu-1)p_*( q^*(K_{X_{\rm min}}+\Delta_{\rm min})).$$ From now on, given a pseudo-effective klt pair $(X,\Delta)$ with a minimal model and $\mu \in {\mathbb{R}}_{\geq 1}$ using equation we may consider $\mu(K_X+\Delta)$ as a generalized pair.
\[lctdefinition\] The maximum positive real number $c$, such that $(c+1)(K_X+\Delta)$ is a generalized log canonical pair is called the [*generalized log canonical threshold*]{} of $K_X+\Delta$. The generalized log canonical threshold of $K_X+\Delta$ will be denoted by ${\rm lct}(K_X+\Delta)$.
\[valueoflct\] Let $(X,\Delta)$ be a pseudo-effective pair with a minimal model $(X_{\rm min},\Delta_{\rm min})$. Let $p\colon Y \rightarrow X$ and $q\colon Y\rightarrow X_{\rm min}$ be two projective birational morphisms which give a log resolution of the minimal model program $\pi \colon X\dashrightarrow X_{\rm min}$, so we can write $$p^*(K_X+\Delta) = q^*(K_{X_{\rm min}}+\Delta_{\rm min}) + E,$$ for some exceptional divisor $E$ on $Y$. Then, the log canonical threshold ${\rm lct}(K_X+\Delta)$ of $K_X+\Delta$ equals the maximum positive real number $c$, such that $K^{\Delta}_{Y/X}- cE$ has coefficients greater or equal than negative one. Moreover, ${\rm lct}(K_X+\Delta)$ is independent of the minimal model and the resolution.
Observe that we have the following equality $$p^*(\mu(K_X+\Delta)) = p^*(K_X+\Delta) + p^*((\mu-1)(K_X+\Delta)) =$$ $$p^*(K_X+\Delta) + (\mu-1)E + (\mu-1)q^*(K_{X_{\rm min}}+\Delta_{\rm min}) =$$ $$K_Y + ( -K_{Y/X}^{\Delta} + (\mu-1)E) + (\mu-1)q^*(K_{X_{\rm min}}+\Delta_{\rm min}).$$ By definition, the generalized pair $(c+1)(K_X+\Delta)$ is a generalized log canonical pair if and only if the coefficients of $cE - K^{\Delta}_{Y/X}$ are greater or equal than negative one. The fact that ${\rm lct}(K_X+\Delta)$ is independent of the chosen resolution is a straightforward computation. Finally, the fact that ${\rm lct}(K_X+\Delta)$ is independent of the chosen minimal model is a consequence of the negativity lemma. Indeed, we can pick $Y$ dominating both minimal models, so that the pull-back of the canonical divisors of both minimal models coincide.
\[finiteness\] Let $(X,\Delta)$ be a pseudo-effective pair with a minimal model. Then, the log canonical threshold of $K_X+\Delta$ is finite if and only if $K_X+\Delta$ is not nef.
If $K_X+\Delta$ is nef, then $E$ is trivial in the above proof, and the coefficients of $-K^{\Delta}_{Y/X}$ are less than one since $(X,\Delta)$ is a klt pair. On the other hand, if $K_X+\Delta$ is not nef, then by strict monotonicity \[monotonicity\], at least one coefficient of $E$ is non-trivial, so ${\rm lct}(K_X+\Delta)$ is finite.
Notation
--------
In this subsection, we introduce further notation that will be used in the proof of the theorem.
\[generalinq\] Given $\lambda \in {\mathbb{Q}}_{>0}$, we will say that $G_\lambda$ is [*general in its ${\mathbb{Q}}$-linear system*]{} $|K_X+\Delta+\lambda A|_{\mathbb{Q}}$ if $G_\lambda$ is the average of $k$ general elements of the linear system $|m(K_X+\Delta+\lambda A)|$ for $m$ and $k$ big and divisible enough. Thus, we can write $$G_\lambda = \frac{1}{mk} \sum_{j\in J} G^j_\lambda,$$ where $J$ is a finite set of cardinality $k$ and the Cartier divisors $G^j_\lambda$ are general elements of the linear system $|m(K_X+\Delta+\lambda A)|$. We will assume that $G_{\lambda}$ is general in its ${\mathbb{Q}}$-linear system unless otherwise stated.
\[lesseffective\] For every $\lambda \in {\mathbb{Q}}_{>0}$, the [*fixed component*]{} or [*fixed divisor*]{} of the ${\mathbb{Q}}$-linear system $|K_X+\Delta+\lambda A|_{\mathbb{Q}}$ is the wedge of all the ${\mathbb{Q}}$-divisors in the ${\mathbb{Q}}$-linear system. Since the ${\mathbb{Q}}$-divisor $K_X+\Delta+\lambda A$ has finitely generated section ring we know that the fixed component is a well-defined effective ${\mathbb{Q}}$-divisor. The fixed component of the ${\mathbb{Q}}$-linear system of a pseudo-effective ${\mathbb{Q}}$-divisor $D$ will be denoted by ${\rm Fix}(D)$. As usual, the [*movable part*]{} of the ${\mathbb{Q}}$-linear system of a pseudo-effective ${\mathbb{Q}}$-divisor $D$ is denoted by ${\rm Mov}(D)$.
Consider $(X,\Delta)$ a klt pair such that $K_X+\Delta$ is pseudo-effective, and $A$ an ample divisor on $X$. We will denote by $G_\lambda \in |K_X+\Delta+\lambda A|_{\mathbb{Q}}$ a general element in the ${\mathbb{Q}}$-linear system. We will write $c_\lambda$ for the log canonical threshold of the effective divisor $G_\lambda$ with respect to the klt pair $(X,\Delta)$. Moreover, if the discussion is independent of the chosen divisor $G_\lambda$ when this is general in its ${\mathbb{Q}}$-linear system, we will just write $\Delta_\lambda = \Delta+c_\lambda G_\lambda$, and consider the corresponding log canonical pair $(X,\Delta_\lambda)$.
Adjunction to the minimal log canonical centers {#section:adjunction}
===============================================
In this section, we prove an adjunction formula for the ${\mathbb{Q}}$-divisor $(c+1)(K_X+\Delta)$ to a common log canonical center of the pairs $(X,\Delta_\lambda)$.
Stabilization of the log canonical places {#stabsec}
-----------------------------------------
In this subsection, we prove that there exists a model $Y$ over $X$ on which the log canonical places of $(X,\Delta_\lambda)$ stabilize for $\lambda$ small enough.
\[stab\] There exist $\lambda_1 \in {\mathbb{Q}}_{>0}$ and a resolution of singularities $p\colon Y \rightarrow X$ such that for $\lambda \in (0,\lambda_1)$ the following statements hold:
1. The set of log canonical places of $(X,\Delta_\lambda)$ on $Y$ is independent of $\lambda$,
2. the log canonical centers of $(X,\Delta_\lambda)$ on $X$ are independent of $\lambda$,
3. the log canonical threshold $c_\lambda$ is the inverse of a linear function on $\lambda$, and
4. we have an equality $\lim_{\lambda \rightarrow 0} c_\lambda={\rm lct}(K_X+\Delta)$.
Let $\pi \colon X \dashrightarrow X_{\rm min}$ be a minimal model for $K_X+\Delta$ which is obtained by running a minimal model program of $K_X+\Delta$ with scaling of a general ample divisor $A$. Assume that $\lambda_1\in {\mathbb{Q}}_{>0}$ is small enough such that $(X_{\rm min}, \Delta_{\rm min} +\lambda A_{\rm min})$ is a minimal model for every $K_X+\Delta+\lambda A$ with $\lambda \in [0,\lambda_1)\cap {\mathbb{Q}}$, where $A_{\rm min}$ is the strict transform of $A$ on $X_{\rm min}$. Let $p\colon Y \rightarrow X$ and $q\colon Y \rightarrow X_{\rm min}$ be a log resolution of $\pi$ so we can write $$p^*(K_X+\Delta+\lambda A) = q^*( K_{X_{\rm min}}+\Delta_{\rm min} + \lambda A_{\rm min}) + E_{\lambda},$$ where $E_\lambda$ is a $q$-exceptional effective divisor with simple normal crossing support, and its coefficients at the prime divisors of its support are linear with respect to $\lambda$. We claim that if $G_\lambda$ is general in its ${\mathbb{Q}}$-linear system $|K_X+\Delta +\lambda A|_{\mathbb{Q}}$, then we can write $p^*(G_\lambda) = G_{\lambda,Y}+E_\lambda$, where $G_{\lambda,Y}$ is a semiample ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor. Indeed, the ${\mathbb{Q}}$-divisor $K_{X_{\rm min}}+\Delta_{\rm min}+\lambda A_{\rm min}$ is nef and big, therefore it is semiample by the base point free Theorem.
Hence, we can take $G_{\lambda,Y}$ general enough, such that for any $c\in {\mathbb{R}}_{>0}$ we have that $$\mathcal{J}((X,\Delta), c G_\lambda ) =
\mathcal{J}((Y, - K^{\Delta}_{Y/X}), p^*(cG_\lambda)) =
\mathcal{J}((Y, -K^{\Delta}_{Y/X}), cG_{\lambda,Y} + cE_\lambda) =$$ $$\mathcal{J}((Y,0), cG_{\lambda,Y}+ cE_\lambda - K^{\Delta}_{Y/X}) =
p_* \mathcal{O}_Y ( \lceil K^{\Delta}_{Y/X} - cE_\lambda - cG_{\lambda,Y} \rceil) =
p_* \mathcal{O}_Y (\lceil K^{\Delta}_{Y/X} - cE_\lambda \rceil).$$ We deduce that the log canonical threshold $c_\lambda$ of $G_\lambda$ with respect to the klt pair $(X,\Delta)$ is the supremum of the positive real numbers $c$, such that the simple normal crossing divisor $K^{\Delta}_{Y/X}- c E_{\lambda}$ has coefficients strictly greater than negative one.
Now, we prove the third claim. Let $E_1,\dots, E_r$ be the irreducible components of ${\rm Ex}(p)$ which appear with non-trivial coefficient on the divisor $E_\lambda$ for some $\lambda \in (0, \lambda_1)$. Denote by $\alpha_i$ the coefficient of $K^{\Delta}_{Y/X}$ at $E_i$, and $\beta_i - \lambda \gamma_i$ the coefficent of $E_\lambda$ at $E_i$. By Remark \[thedifferenceispositive\], we know that $\beta-\lambda \gamma_i>0$, for every $E_i$ with $i\in \{1,\dots, r\}$ and $\lambda \in (0,\lambda_1)$. Then we can define the functions $$c_{\lambda, i } = \frac{ \alpha_i + 1}{\beta_i - \lambda \gamma_i}$$ and see that $$c_\lambda = \min\{ c_{\lambda, i } \mid i\in \{1,\dots, r\}\}.$$ Observe that $c_\lambda$ is the minimum between finitely many multiplicative inverses of linear functions on $\lambda$, so we deduce that we can take $\lambda_1 \in {\mathbb{Q}}_{>0}$ small enough, such that $c_\lambda$ is the inverse of a linear function for $[0,\lambda_1)\cap {\mathbb{Q}}$.
Now, we turn to prove the first claim. Consider the set $\mathcal{I}_{\lambda}=\{ i\mid c_\lambda = c_{\lambda,i}\}\subseteq \{1,\dots, r\}$. We know that $\mathcal{I}_{\lambda}$ is constant for $\lambda \in (0,\lambda_1)\cap {\mathbb{Q}}$ so we may denote it by $\mathcal{I}$. Since $$\mathcal{J}((X,\Delta), c G_\lambda) =p_*( \mathcal{O}_Y( K^{\Delta}_{Y/X} - \lfloor c E_\lambda \rfloor)),$$ we have that the log canonical places of $(X,\Delta_\lambda)$ on $Y$ are exactly the divisors $E_i$ with $i \in \mathcal{I}$.
The second claim is a consequence of the first claim, as we have the following equality $$\nonumber
V(\mathcal{J}((X,\Delta),c_\lambda G_\lambda))= p_*\left( \bigcup_{i \in \mathcal{I}} E_i \right),$$ where both sides are taken with reduced scheme structure. Finally, the fourth claim follows from the continuity of the function $c_\lambda$ at the origin, and Proposition \[valueoflct\].
\[clambdaincreasing\] The log canonical threshold $c_\lambda$ is rational whenever $\lambda \in {\mathbb{R}}_{\geq 0}$ is a rational number. Moreover, $c_\lambda$ is a monotone increasing function with respect to $\lambda \in [0,\lambda_1)$.
\[minimalnonklt\] Since the minimality of non-klt centers only depends on the inclusion of such subvarieties we deduce that the set of minimal log canonical centers of $(X,\Delta_\lambda)$ on $X$ is independent of $\lambda \in (0,\lambda_1)\cap {\mathbb{Q}}$.
\[compe\] Observe that for every $\lambda' < \lambda$ in the interval $(0,\lambda_1)$ we have that $E_{\lambda'} \geq E_{\lambda}$. Indeed, being $A$ an ample divisor we can write $p^*(\lambda A) = q^*(\lambda A_{\rm min}) - \lambda F$, where $F$ is an effective divisor. Then $E_\lambda = E - \lambda F \geq 0$, where $E =p^*(K_X+\Delta)-q^*(K_{X_{\rm min}}+\Delta_{\rm min})\geq 0$ is an effective ${\mathbb{Q}}$-divisor by the negativity lemma (see, e.g., [@KM98 Lemma 3.39]). In particular, we have that ${\rm Fix}(p^*(K_X+\Delta+\lambda A))= E_\lambda$ for every $\lambda \in (0,\lambda_1)$.
\[thedifferenceispositive\] Since the pair $(X,\Delta)$ is assumed to be klt we have that $\alpha_i + 1 >0$ for every $i \in \{1,\dots, r\}$. Moreover, the coefficient $\beta_i - \lambda \gamma_i$ is strictly positive for $\lambda \in [0,\lambda_1)$. Indeed, this follows from the definition of minimal model \[definitionminimalmodel\], and the strict monotonicity of discrepancies \[monotonicity\], as $c_{E_i}(X)$ is contained in the exceptional locus of $X\dashrightarrow X_{\rm min}$.
Common divisorially log terminal modification {#cdlt}
---------------------------------------------
In this subsection, we prove that all the log canonical pairs $(X,\Delta_\lambda)$ share a common [*divisorially log terminal modification*]{} for $\lambda \in {\mathbb{Q}}_{>0}$ small enough. Recall that dlt modifications exists by [@KK10 Theorem 3.1].
\[dltmod\] Let $(X,\Delta)$ be a log canonical pair, we say that a projective birational morphism $p_m \colon Y_m \rightarrow X$ is a [*${\mathbb{Q}}$-factorial divisorially log terminal modification*]{} of $(X,\Delta)$, or [*dlt modification*]{} for short, if the following conditions hold:
- $Y_m$ is ${\mathbb{Q}}$-factorial,
- $p_m$ only extracts divisors with log discrepancy $-1$ with respect to $(X,\Delta)$, and
- if $E$ is the sum of the irreducible $p_m$-exceptional divisors and $\Delta_{Y_m}$ is the strict transform of $\Delta$ on $Y_m$, then $(Y_m ,\Delta_{Y_m} +E)$ is divisorially log terminal and $$K_{Y_m} + \Delta_{Y_m} + E = p_m^*(K_X+\Delta).$$
The following lemma follows from the proof of the existence of dlt modifications for log canonical pairs. For the sake of completeness, we will give a proof of the statement which is not in [@KK10 Theorem 3.1].
\[dltmodificationlemma\] Let $(X,\Delta)$ be a log canonical pair and $p\colon Y \rightarrow X$ a log resolution which is composite of blow-ups of centers of codimension at least two. Then, there exists a dlt modification $p_m \colon Y_m \rightarrow X$ of the pair $(X,\Delta)$ such that the exceptional locus of the rational map $\pi_m \colon Y \dashrightarrow Y_m$ is contained in the union of the prime divisors $E\subsetneq Y$ which are exceptional over $X$ and for which $a_E(X,\Delta)>-1$.
We can write $$K_Y+\Delta_Y + E_Y^{+}+E_Y^{0}-E_Y^{-} = p^*(K_X+\Delta),$$ where $E_Y^{+}$ denotes the sum of all $p$-exceptional divisors with discrepancy equal to negative one, $E_Y^{0}$ denotes the sum of all $p$-exceptional divisors with discrepancy in the interval $(-1, 0]$ and $E_Y^{-}$ is the sum of all $p$-exceptional divisors with discrepancy greater than zero. We will denote by $E_Y^{\rm Cr}$ the sum of the reduced prime divisors on $Y$ with discrepancy zero over $(X,\Delta)$, or equivalently, the components of the support of $E_Y^0$ which appears with coefficient zero in the sum.
Since $p$ is composite of blow-ups of centers of codimension at least two we know that there exists a $p$-exceptional effective divisor $C$ such that $-C$ is $p$-ample. Let $H$ be a sufficiently ample divisor on $X$ such that $-C+p^*(H)$ is ample on $Y$ and the divisor $H' \sim_{\mathbb{Q}}-C+p^*(H)$ intersects $\Delta_Y + E_Y^{+}+E_Y^{0}+E_Y^{\rm Cr}+E_Y^{-}$ transversally. Therefore, taking parameters $\epsilon_1, \epsilon_2$ and $\epsilon_3$ in ${\mathbb{Q}}_{>0}$ which are small enough, we have that the following pair is dlt: $$\label{dltdivisor}
(Y, \Delta_Y + E_Y^{+}+ (1+\epsilon_1)E_Y^{0} + \epsilon_2 E_Y^{\rm Cr} + \epsilon_3 H').$$ If we assume that $0<\epsilon_3 \ll \epsilon_1 \ll 1$ and $0<\epsilon_3 \ll \epsilon_2 \ll 1$ we can run a minimal model for the pair with scaling of an ample divisor over $X$, so we obtain a minimal model $Y_m$ over $X$ which is a dlt modification of $(X,\Delta)$ (see, e.g., [@KK10 Theorem 3.1] for the details). Observe that we have $$K_Y + \Delta_Y + E_Y^{+}+ (1+\epsilon_1)E_Y^{0} + \epsilon_2 E_Y^{\rm Cr} + \epsilon_3 H'
= \epsilon_1 E_Y^{0} + \epsilon_2 E_Y^{\rm Cr} + E_Y^{-} + \epsilon_3 H' + p^*(K_X+\Delta),$$ Thus, we get that $$K_Y + \Delta_Y + E_Y^{+}+ (1+\epsilon_1)E_Y^{0} + \epsilon_2 E_Y^{\rm Cr}+ \epsilon_3 H'
\sim_{X,{\mathbb{Q}}} \epsilon_1 E_Y^{0} + \epsilon_2 E_Y^{\rm Cr}+ E_Y^{-} + \epsilon_3 H'.$$ So the diminished base locus over $X$ of the ${\mathbb{Q}}$-divisor $$K_Y+\Delta_Y + E^{+}_Y + (1+\epsilon_1)E_Y^0 + \epsilon_2 E_Y^{\rm Cr}+ \epsilon_3 H'$$ is contained in the union of the support of the divisors $E_Y^{0}, E_Y^{\rm Cr}$ and $E_Y^{-}$. Therefore, we conclude that the flipping locus or exceptional divisor of every step of the minimal model program $\pi_m \colon Y \dashrightarrow Y_m$ is contained in the strict transform of the union of support of $E_Y^{0}, E_Y^{\rm Cr}$, and $E_Y^{-}$. In particular, the exceptional locus of $\pi_m \colon Y \dashrightarrow Y_m$ is contained in this locus.
\[monotonedecreasing\] The coefficient of the divisor $p^*(K_X+\Delta_\lambda)$ at any prime divisor on $Y$, which is exceptional over $X$, is a monotone function with respect to $\lambda$ around the origin.
Since $G_\lambda$ is general in its ${\mathbb{Q}}$-linear system we have that $${\rm Fix}(p^*(K_X+\Delta_\lambda))=
{\rm Fix}(p^*(K_X+\Delta+c_\lambda G_\lambda)) =
{\rm Fix}(p^*(K_X+\Delta+c_\lambda(K_X+\Delta+\lambda A))) =$$ $$(c_\lambda+1){\rm Fix}\left( p^*\left(K_X+\Delta+ \left( \frac{c_\lambda}{c_\lambda+1}\right) \lambda A \right) \right)=
(c_\lambda+1) E_{s(\lambda)},$$ where $E_{s(\lambda)}$ is defined as in Remark \[compe\] and $$s(\lambda) = \left( \frac{c_\lambda}{c_\lambda+1} \right) \lambda$$ is a monotone increasing function for $\lambda \in [0,\lambda_1)$. Indeed, we can compute $$s(\lambda) = \left( \frac{\alpha_i +1}{ \alpha_i +1+\beta_i - \lambda \gamma_i} \right) \lambda,$$ where $\alpha_i+1>0$ and $\lambda \mapsto \beta_i -\lambda \gamma_i$ is a positive function which is monotone increasing with respect to $\lambda$. So we have that $$p^*(K_X+\Delta_\lambda) =
{\rm Mov}( p^*(K_X+\Delta_\lambda) ) + {\rm Fix}(p^*(K_X+\Delta_\lambda) ) =
{\rm Mov}(p^*(K_X+\Delta_\lambda) ) + (c_\lambda+1)E_{s(\lambda)}$$ By Remark \[compe\], we deduce that $$(c_\lambda+1) E_{s(\lambda)} = (c_\lambda+1)( E - s(\lambda) F) =
c_\lambda ( E - \lambda F) + E,$$ where $E$ and $F$ are the effective divisors defined in Remark \[compe\]. Pick $E_j$ to be an irreducible divisor on $Y$ which is exceptional over $X$. Denote by $\beta_j$ the coefficient of $E$ at $E_j$ and by $\gamma_j$ the coefficient of $F$ at $E_j$. Then, the coefficient of $p^*( K_X+\Delta_\lambda)$ at the prime divisor $E_j$ equals $$( \alpha_i +1) \left( \frac{ \beta_j - \lambda \gamma_j }{\beta_i - \lambda \gamma_i }\right) + \beta_j.$$ Recall from Remark \[thedifferenceispositive\] that $\beta_i -\lambda \gamma_i>0$ for $\lambda \in (0,\lambda_1)$, so the above function is monotone around the origin.
\[monotonecoeff\] From now on, we will assume that $\lambda_1\in {\mathbb{Q}}_{>0}$ is small enough such that the coefficient of the divisor $p^*(K_X+\Delta_\lambda)$ at any prime divisor on $Y$ which is exceptional over $X$ is a monotone function with respect to $\lambda \in [0,\lambda_1)$.
\[zerocase\] Observe that the prime divisor $E_j$ have coefficient zero in $p^*(X+\Delta_{\lambda_0})$ for some fixed $\lambda_0\in (0,\lambda_1)$ if and only if $\beta_j=0$ and $\beta_j-\lambda_0 \gamma_j =0$, which implies that $\gamma_j=0$. Assume that $\beta_j=\gamma_j=0$ for some $E_j$. Since $\gamma_j=0$ we know that the center of $E_j$ on $X$ is not contained in the exceptional locus of the birational map $X\dashrightarrow X_{\rm min}$. Moreover, $\beta_j=0$ implies that $E_j$ is a crepant divisorial valuation of the pair $(X,\Delta)$. Since $G_\lambda$ is general in its ${\mathbb{Q}}$-linear system we deduce that $E_j$ is a crepant divisorial valuation of the log canonical pair $(X,\Delta_\lambda)$ for every $\lambda \in (0,\lambda_1)$.
\[comdltmodification\] There exists $\lambda_2 \in (0,\lambda_1)$ and a projective birational morphism $p_m \colon Y_m \rightarrow X$ such that $p_m$ is a ${\mathbb{Q}}$-factorial dlt modification for every log canonical pair $(X,\Delta_\lambda)$ with $\lambda \in (0,\lambda_2)\cap {\mathbb{Q}}$.
Consider $p\colon Y \rightarrow X$ as in the proof of Proposition \[stab\]. Since all the klt pairs $(K_X+\Delta+\lambda A)$ with $\lambda \in (0,\lambda_1)$ have the same minimal model, we may assume that the support of the fixed component of $|p^*(K_X+\Delta+\lambda A)|_{\mathbb{Q}}$ is independent of $\lambda \in (0,\lambda_1)\cap {\mathbb{Q}}$. Write $$K_{Y}+ \Delta_{\lambda,Y}+E_{\lambda,Y} =
p^*(K_X+\Delta_{\lambda}),$$ where $\Delta_{\lambda,Y}$ is the strict transform of $\Delta_\lambda$ on $Y$ and $E_{\lambda,Y}$ is a $p$-exceptional divisor. We consider the following decomposition $$E_{\lambda,Y}= E^{+}_{\lambda,Y} + E^0_{\lambda,Y} - E^{-}_{\lambda,Y},$$ where $E^{+}_{\lambda,Y}$ is supported on the sum of all $p$-exceptional divisors with discrepancy $ -1$, the ${\mathbb{Q}}$-divisor $E^0_{\lambda,Y}$ is the sum of all $p$-exceptional divisors with discrepancy greater than $-1$ and less than or equal to $0$, and $E^{-}_{\lambda,Y}$ is the sum of all $p$-exceptional divisors with positive discrepancy. By Lemma \[monotonedecreasing\], we can pick $\lambda_2 \in (0, \lambda_1)\cap {\mathbb{Q}}$ small enough such that the support of the divisors $E_{\lambda,Y}^+, E_{\lambda,Y}^0$, and $E^{-}_{\lambda,Y}$ are independent of $\lambda \in (0,\lambda_2]$. By Remark \[zerocase\], we know that the set of crepant valuations over $(X,\Delta_\lambda)$ is independent of $\lambda \in (0,\lambda_1)$ so we will denote the sum of such divisors on $Y$ by $E_Y^{\rm Cr}$.
We claim that a dlt modification of $(X,\Delta_{\lambda_2})$ is a dlt modification of every log canonical pair $(X,\Delta_{\lambda})$ with $\lambda \in (0,\lambda_2]\cap {\mathbb{Q}}$. Indeed, let $p_m \colon Y_m \rightarrow X$ be a ${\mathbb{Q}}$-factorial dlt modification of $(X,\Delta_{\lambda_2})$. Since $Y_m$ is ${\mathbb{Q}}$-factorial then it suffices to check the second and third conditions of Definition \[dltmod\].
First, we check that $p_m$ only extracts divisors with log discrepancy equal to zero for the pairs $(X,\Delta_\lambda)$. By the construction of dlt modifications any divisor extracted by $p_m$ is also extracted by $p$. If a divisor $E$ is extracted by $p_m$ then its log discrepancy with respect to the pair $(X,\Delta_{\lambda_2})$ is zero, so its strict transform on $Y$ is a component of $E^{+}_{\lambda_2,Y}$, then it is a component of $E^{+}_{\lambda,Y}$ for any $\lambda \in (0,\lambda_2]\cap {\mathbb{Q}}$, which means that the log discrepancy of $E$ with respect to $(X,\Delta_\lambda)$ is zero for every $\lambda \in (0,\lambda_2]\cap {\mathbb{Q}}$.
Now, it suffices to check that $(Y_m, \Delta_{\lambda,Y_m}+E_{\lambda,Y_m})$ is a dlt pair for every $\lambda \in (0,\lambda_2]\cap {\mathbb{Q}}$. Observe that the coefficients of $\Delta_{\lambda,Y_m} + E_{\lambda,Y_m}$ are contained in the interval $[0,1]$. Indeed, all the irreducible components of the divisor $E_{\lambda,Y}^0, E_Y^{\rm Cr}$, and $E^{-}_{\lambda,Y}$ are contracted by the rational map $\pi_m \colon Y\dashrightarrow Y_m$ since the coefficients of $\Delta_{\lambda_2,Y_m}+E_{\lambda_2, Y_m}$ are contained in the interval $[0,1]$. So it suffices to prove that there exists a log resolution $q_Z \colon Z\rightarrow Y_m$ for the pairs $(Y_m, \Delta_{\lambda,Y_m}+E_{\lambda,Y_m})$ such that the discrepancies of any prime $q_Z$-exceptional divisor with respect to $(Y_m, \Delta_{\lambda,Y_m} + E_{\lambda,Y_m})$ is strictly greater than negative one for every $\lambda \in (0,\lambda_2]$. Indeed, let $p_Z \colon Z \rightarrow Y$ and $q_Z \colon Z \rightarrow Y_m$ be two projective morphisms that give a log resolution of the minimal model program $\pi_m \colon Y \dashrightarrow Y_m$. By Lemma \[dltmodificationlemma\], we know that the indeterminancy locus of the rational map $\pi_m \colon Y \dashrightarrow Y_m$ is contained in the support of the divisor $E^0_{\lambda_2, Y} + E_Y^{\rm Cr}+ E^{-}_{\lambda_2,Y}$, so it is contained in the support of the divisor $$E^0_{\lambda,Y} + E_Y^{\rm Cr}+ E^{-}_{\lambda,Y}$$ for arbitrary $\lambda \in (0,\lambda_2]$. Thus, we can obtain $Z$ by composite of blow-ups along centers contained in the support of $E^0_{\lambda,Y}+ E_Y^{\rm Cr}+E^{-}_{\lambda,Y}$. Therefore, using the formula to compute discrepancies over log smooth pairs [@KM98 Lemma 2.29] we conclude that any prime divisor on $Z$ which is $q_Z$-exceptional, has positive log discrepancy with respect to $(Y_m, \Delta_{\lambda,Y_m}+E_{\lambda,Y_m})$.
Divisorial adjunction on the dlt model {#divadjss}
--------------------------------------
In this subsection, we use divisorial adjunction of dlt pairs (see, e.g., [@Kol92] and [@HK10 Theorem 3.24]) to the log canonical places of $(X,\Delta_\lambda$) on $Y$ which are exceptional divisors over $X$. We decompose the [*different divisor*]{} which is the divisor induced by the adjunction formula, into a fixed part and a semiample part. From now on, we will always assume that $\lambda \in (0,\lambda_2)\cap {\mathbb{Q}}$, where $\lambda_2$ is constructed in the proof of Proposition \[comdltmodification\] unless otherwise stated.
\[semiamplecomp\] Observe that we have a commutative diagram $$\xymatrix@C=7em{
Y \ar@{-->}[r]^-{\pi_m} \ar[rd]^-{q} \ar[d]^-{p} & Y_m \ar[ld]^-{p_m} \ar@{-->}[d]^-{q_m}\\
X \ar@{-->}[r]^-{\pi} & X_{\rm min}
}$$ such that $p \colon Y \rightarrow X$ and $q\colon Y \rightarrow X_{\rm min}$ give a log resolution of the minimal model program $\pi \colon X \dashrightarrow X_{\rm min}$, and $p_m\colon Y_m \rightarrow X$ is a divisorially log terminal modification of the log canonical pairs $(X,\Delta_\lambda)$ for every $\lambda \in (0,\lambda_2)\cap {\mathbb{Q}}$. Moreover, the rational map $q_m \colon Y_m \dashrightarrow X_{\rm min}$ is defined outside a subvariety of codimension two.
We denote by $G_1,\dots, G_k$ the prime divisors of $X$ which are contracted in the minimal model program $\pi \colon X \dashrightarrow X_{\rm min}$. We claim that for $\lambda \in {\mathbb{Q}}_{>0}$ small enough a general divisor $G_\lambda$ in the ${\mathbb{Q}}$-linear system $|K_X+\Delta+\lambda A|_{\mathbb{Q}}$ can be written as $$G_\lambda = \sum_{i=0}^k G_{\lambda,i},$$ where $G_{\lambda,i}$ are divisors supported on the prime divisors $G_i$ for every $i\in \{1,\dots ,k\}$ and the strict transform of $G_{\lambda,0}$ on $Y$ is a semiample ${\mathbb{Q}}$-divisor. Indeed, for $\lambda \in (0,\lambda_2)\cap {\mathbb{Q}}$ we have that the ${\mathbb{Q}}$-divisor $G_{\lambda,0, X_{\rm min}}$ belongs to the ${\mathbb{Q}}$-linear system $|K_{X_{\rm min}}+\Delta_{\rm min} + \lambda A_{\rm min}|_{\mathbb{Q}}$ which is nef and big, therefore $G_{\lambda,0,X_{\rm min}}$ is a semiample ${\mathbb{Q}}$-divisor by the base point free Theorem (see, e.g., [@HK10 Theorem 5.1]). Thus, for $\lambda \in (0,\lambda_2)\cap {\mathbb{Q}}$ we have that the ${\mathbb{Q}}$-divisor $q^*(G_{\lambda,0,X_{\rm min}})$ equals $G_{\lambda,0,Y}$, where $G_{\lambda,0,Y}$ is the strict transform of $G_{\lambda,0}$ on $Y$.
\[divadj\] Let $p_m\colon Y_m \rightarrow X$ be the common dlt modification of the log canonical pairs $(X,\Delta_\lambda)$ constructed in Proposition \[comdltmodification\]. In the proof of Proposition \[stab\] we denoted by $\{ E_i \mid i\in \mathcal{I} \}$ the set log canonical places of the pair $(X,\Delta_\lambda)$ on $Y$. Observe that such log canonical places may be exceptional divisors over $X$ or non-exceptional divisors over $X$. We will denote by $\mathcal{I}' \subseteq \mathcal{I}$ the set of log canonical places on $Y_m$ which are exceptional divisors over $X$. For every $i\in \mathcal{I}'$ we will denote the prime divisor $E_{i,Y_m}$ by $\mathcal{E}_i$ in order to abbreviate the notation. Thus, we can write $$\label{dltdivisorialadjunction1}
K_{Y_m} + \Delta_{Y_m} + c_\lambda G_{\lambda,Y_m} + \sum_{i \in \mathcal{I}'} \mathcal{E}_i=
p_m^*(K_X+\Delta+c_\lambda G_\lambda),$$ where the subscript $Y_m$ on a divisor denotes its strict transform on the model $Y_m$. Using divisorial dlt adjunction for the ${\mathbb{Q}}$-divisor to the prime divisor $\mathcal{E}_i$ we can write $$p_m^*(K_X+\Delta+ c_\lambda G_\lambda)|_{\mathcal{E}_i}=
K_{\mathcal{E}_i} + \Phi_{\mathcal{E}_i}^{G_\lambda},$$ where the pair $( \mathcal{E}_i, \Phi_{\mathcal{E}_i}^{G_\lambda})$ is dlt. The divisor $\Phi_{\mathcal{E}_i}^{G_\lambda}$ is called the [*different*]{}. We use the superscript $G_\lambda$ in the notation of the different to make explicit that it depends on the choice of $G_\lambda$ in its ${\mathbb{Q}}$-linear system, even if the latter is chosen to be general in its ${\mathbb{Q}}$-linear system.
\[dltadjunction\] Given $i\in \mathcal{I}'$, there exists $\lambda_3 \in (0,\lambda_2)$ and a projective birational morphism $q_Z \colon \mathcal{E}_{i,Z} \rightarrow \mathcal{E}_i$ over $W_i$, such that for every $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$ we can write $$K_{\mathcal{E}_{i,Z}} + F_{\mathcal{E}_{i,Z}}^\lambda + N_{\mathcal{E}_{i,Z}}^{G_\lambda} \sim_{\mathbb{Q}}q_Z^* (K_{\mathcal{E}_i} + \Phi^{G_\lambda}_{\mathcal{E}_i}),$$ where the following statements hold:
- The ${\mathbb{Q}}$-divisor $F_{\mathcal{E}_{i,Z}}^\lambda$ is independent of the choice of $G_\lambda$ in its ${\mathbb{Q}}$-linear system,
- the support of $F_{\mathcal{E}_{i,Z}}^\lambda$ is independent of $\lambda$, and
- the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $N_{\mathcal{E}_{i,Z}}^{G_\lambda}$ is semiample.
Consider the following commutative diagram $$\xymatrix@C=7em{
Z \ar[d]^-{p_Z}\ar[rd]^-{q_Z} & \\
Y \ar@{-->}[r]^-{\pi_m} \ar[rd]^-{q} \ar[d]^-{p} & Y_m \ar[ld]^-{p_m} \ar@{-->}[d]^-{q_m}\\
X \ar@{-->}[r]^-{\pi} & X_{\rm min}
}$$ where the bottom square is the one introduced in \[semiamplecomp\]. The projective birational morphisms $p_Z\colon Z \rightarrow Y$ and $q_Z \colon Z \rightarrow Y_m$ give a log resolution of the minimal model program $\pi_m \colon Y \dashrightarrow Y_m$. Thus, we can write $$p_Z^*( G_{\lambda,0,Y}) = q_Z^*(G_{\lambda,0,Y_m}) + D_\lambda,$$ where $D_\lambda$ is a $q_Z$-exceptional anti-effective divisor with coefficients that vary continuously with respect to $\lambda$. Since $p_Z^*(G_{\lambda,0,Y})$ is semiample we can take $\lambda_3 \in (0,\lambda_2)$ such that the support of the ${\mathbb{Q}}$-divisor $D_{\lambda}$ is independent of $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$. Indeed, we can write $$-D_\lambda = {\rm Fix}( (q\circ q_z)^*( G_{\lambda,0} ) ) =
{\rm Fix}( (q\circ q_z)^*( {\rm Mov}(K_X+\Delta+\lambda A))),$$ so the coefficients of $-D_\lambda$ are monotone with respect to $\lambda$ around the origin. We denote by $\mathcal{E}_{i,Z}$ the strict transform of the prime divisor $\mathcal{E}_i$ on $Z$ and by abuse of notation we write $q_Z \colon \mathcal{E}_{i,Z} \rightarrow \mathcal{E}_i$ for the restriction of the morphism $q_Z$ to $\mathcal{E}_{i,Z}$. We define $N_{\mathcal{E}_{i,Z}}^{G_\lambda}$ to be the restriction of the semiample ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $G_{\lambda,0,Z}$ to $\mathcal{E}_{i,Z}$ then we can write $$K_{\mathcal{E}_{i,Z}}+ F_{\mathcal{E}_{i,Z}}^{G_\lambda} + N_{\mathcal{E}_{i,Z}}^{G_\lambda} =
q_Z^* (K_{\mathcal{E}_i} + \Phi^{G_\lambda}_{\mathcal{E}_i}),$$ where $F^{G_\lambda}_{\mathcal{E}_{i,Z}}$ is a ${\mathbb{Q}}$-divisor which is supported on $$\label{locus}
\left(
{\rm supp}(\Delta_Z) \cup
{\rm supp}(E_{\lambda,Z}^{0}) \cup
{\rm supp}(E_{\lambda,Z}^{-})\cup
{\rm supp}\left(\sum_{i=1}^k G_{\lambda,i,Z}\right) \cup
{\rm supp} \left(\sum_{j\neq i}\mathcal{E}_{j,Z}\right)
\right) \cap \mathcal{E}_{i,Z},$$ and the divisors $E_{\lambda,Z}^{0}$ and $E_{\lambda,Z}^{-}$ are defined as in the proof of Proposition \[comdltmodification\]. By Lemma \[monotonedecreasing\] and Remark \[monotonecoeff\], we can assume that these two divisors have support independent of $\lambda \in (0,\lambda_2)$. Clearly, the other divisors in the locus \[locus\] have support independent of $\lambda \in (0,\lambda_2)$. We deduce that the locus \[locus\] is independent of $\lambda$, therefore the support of $F^{G_\lambda}_{\mathcal{E}_{i,Z}}$ is independent of $\lambda \in (0,\lambda_2)$. Finally, we need to argue that $F^{G_\lambda}_{\mathcal{E}_{i,Z}}$ is independent of $G_\lambda$ when the latter is general in its ${\mathbb{Q}}$-linear system. Indeed, if we choose $G_{\lambda}$ general in its ${\mathbb{Q}}$-linear system such that $G_{\lambda,0,Z}$ intersects the locus transversally, then the divisor $F^{G_\lambda}_{\mathcal{E}_i,Z}$ only depends on the fixed component of the ${\mathbb{Q}}$-linear system $|G_{\lambda,Z}|_{\mathbb{Q}}$.
Ambro’s canonical bundle formula {#ambro}
--------------------------------
In this subsection, we will use Ambro’s canonical bundle formula (see, e.g., [@FG12]) to define a [*discriminant divisor*]{} and a [*moduli divisor*]{} on $W_i$, where $W_i$ is a common minimal non-klt center of the log canonical pairs $(X,\Delta_\lambda)$ of codimension at least two. In particular, $W_i$ is the image of $\mathcal{E}_i$ on $X$ for some $i\in \mathcal{I}'$. The aim of this subsection is to prove that for every pair $(X,\Delta_\lambda)$ we can construct a discriminant divisor on $W_i$ which is independent of the choice of $G_\lambda$ in its ${\mathbb{Q}}$-linear system and its support is independent of $\lambda$. By abuse of notation, we will denote by $p_m\colon \mathcal{E}_i \rightarrow W_i$ the restriction of the morphism $p_m\colon Y_m \rightarrow X$ to the prime divisor $\mathcal{E}_i$.
\[boundary\] We recall the construction of the [*boundary divisor*]{} (see, e.g., [@Kol13]). Consider a projective birational morphism $p_m\colon \mathcal{E}_i \rightarrow W_i$ of normal quasi-projective varieties and a ${\mathbb{Q}}$-divisor $F$ on $\mathcal{E}_i$ such that $(\mathcal{E}_i,F)$ is a sub-pair which is sub-log canonical near the generic fiber of $p_m$. We can define a [*boundary divisor*]{} as follows: Given a prime divisor $C\subsetneq W_i$, we define the real number $$\mu_C\left(F \right)= \sup \{ t\in {\mathbb{R}}\mid \left(\mathcal{E}_i ,F+tp_m^*(C)\right)
\text{ is sub-log canonical over a neighbourhood of $\eta_C$}\},$$ where $\eta_C$ is the generic point of $C$. Observe that the pull-back $p^*_m(C)$ is well-defined over a neighbourhood of $\eta_C$ since $W_i$ is normal. Then we can define the ${\mathbb{Q}}$-divisor $$B_{W_i}(F) = \sum_{C\subsetneq W_i } \left(1-\mu_C\left(F\right)\right)C,$$ where the sum runs over all the prime divisors $C$ of $W_i$. By [@Cor07 Section 8.2], the above sum is finite.
\[canonicalbundle\] In \[cdlt\],we constructed a dlt pair $(\mathcal{E}_i, \Phi_{\mathcal{E}_i}^{G_\lambda})$ for any $i\in \mathcal{I}'$ and $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$. Moreover, by construction the divisor $K_{\mathcal{E}_i} + \Phi_{\mathcal{E}_i}^{G_\lambda}$ is relatively trivial over the base $W_i$. Therefore, using Ambro’s canonical bundle formula we can write $$K_{\mathcal{E}_i} + \Phi_{\mathcal{E}_i}^{G_\lambda}
\sim_{\mathbb{Q}}p_m^*( K_{W_i} + B_{W_i}( \Phi_{\mathcal{E}_i}^{G_\lambda}) + M_{W_i}( \Phi_{\mathcal{E}_i}^{G_\lambda}) ),$$ where $B_{W_i}( \Phi_{\mathcal{E}_i}^{G_\lambda})$ is called the [*boundary divisor*]{} or [*discriminant divisor*]{} and $M_{W_i}( \Phi_{\mathcal{E}_i}^{G_\lambda}) $ is called the [*moduli divisor*]{} or [*j-divisor*]{}. Here, the boundary divisor $B_{W_i}( \Phi_{\mathcal{E}_i}^{G_\lambda}) $ is the one constructed in \[boundary\] for the pair $(\mathcal{E}_i, \Phi_{\mathcal{E}_i}^{G_\lambda})$ and the morphism $p_m \colon \mathcal{E}_i\rightarrow W_i$. By construction, the pair $(W_i ,B_{W_i}( \Phi_{\mathcal{E}_i}^{G_\lambda}) )$ is log canonical whenever the moduli divisor is ${\mathbb{Q}}$-Cartier and $W_i$ is normal (see, e.g., [@Cor07 Remark 8.6.2]). Observe that both divisors $M_{W_i}( \Phi_{\mathcal{E}_i}^{G_\lambda})$ and $B_{W_i}( \Phi_{\mathcal{E}_i}^{G_\lambda}) $ depend on the choice of $G_\lambda$ in its ${\mathbb{Q}}$-linear system. By \[boundary\], we know that the boundary divisor is uniquely determined by the pair $(\mathcal{E}_i, \Phi_{\mathcal{E}_i}^{G_\lambda})$. However, the moduli part is only defined up to ${\mathbb{Q}}$-linear equivalence and the ${\mathbb{Q}}$-divisor itself depends on the starting choice of a ${\mathbb{Q}}$-divisor $L$ on $W_i$ such that $$K_{\mathcal{E}_i} + \Phi_{\mathcal{E}_i}^{G_\lambda}
\sim_{\mathbb{Q}}p_m^*( L ).$$
\[canbundsplit\] Let $W_i$ be a common minimal non-klt center of $(X,\Delta_\lambda)$ of codimension at least two for $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$. Then we can write $$(K_X + \Delta_\lambda)|_{W_i} \sim_{\mathbb{Q}}K_{W_i} + B_{W_i}^\lambda + M_{W_i}^{\lambda},$$ where the following statements hold:
- The effective ${\mathbb{Q}}$-divisor $B^{\lambda}_{W_i}$ is independent of the choice of $G_\lambda$ in its ${\mathbb{Q}}$-linear system, and
- the support of $B^{\lambda}_{W_i}$ is independent of $\lambda$.
Moreover, if $K_{W_i}+B^{\lambda}_{W_i}$ is ${\mathbb{Q}}$-Cartier we have that:
- The pair $(W_i,B^{\lambda}_{W_i})$ is log canonical, and
- The ${\mathbb{Q}}$-divisor $M^{\lambda}_{W_i}$ is nef.
By Proposition \[dltadjunction\], for every such $W_i$ we have that $i\in \mathcal{I}'$ and there is a commutative diagram $$\nonumber
\xymatrix@C=2em{
\mathcal{E}_{i,Z}\ar[rr]^-{q_Z} \ar[rd]_{p_{m,Z}} & & \mathcal{E}_i\ar[dl]^-{p_m} \\
& W_i &
}$$ such that $$K_{\mathcal{E}_{i,Z}} + F_{\mathcal{E}_{i,Z}}^\lambda + N_{\mathcal{E}_{i,Z}}^{G_\lambda} \sim_{\mathbb{Q}}q_Z^* (K_{\mathcal{E}_i} + \Phi^{G_\lambda}_{\mathcal{E}_i}),$$ where $F^\lambda_{\mathcal{E}_{i,Z}}$ has support independent of $\lambda$ and $N^{G_\lambda}_{\mathcal{E}_{i,Z}}$ is a semiample ${\mathbb{Q}}$-divisor. Consider $\pi_i \colon V_i \rightarrow W_i$ to be a projective generically finite morphism from a smooth variety $V_i$ which factors through the semistable reduction in codimension one for both morphisms $p_{m,Z}$ and $p_m$ (see, e.g., [@KKMS]). Taking the base change of the above diagram we obtain the following commutative diagram $$\nonumber
\xymatrix@C=2em{
\mathcal{F}_{i,Z}\ar[rr]^-{q_Z} \ar[rd]_{p_{m,Z}} & & \mathcal{F}_i\ar[dl]^-{p_m} \\
& V_i &
}$$ where by abuse of notation we use the same symbols for $p_{m,Z}, q_Z$ and $p_m$ and the corresponding morphisms induced by the base change. Thus, the morphisms of $p_{m,Z}\colon \mathcal{F}_{i,Z}\rightarrow V_i$ and $p_m \colon \mathcal{F}_i \rightarrow V_i$ have slc fibers in codimension one. We denote by $N_{\mathcal{F}_{i,Z}}^{G_\lambda}$ the pull-back of the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $N_{\mathcal{E}_{i,Z}}^{G_\lambda}$ to $\mathcal{F}_{i,Z}$ and by $$K_{\mathcal{F}_{i,Z}}+ F_{\mathcal{F}_{i,Z}}^{\lambda}$$ the pull-back of the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $K_{\mathcal{E}_{i,Z}} + F_{\mathcal{E}_{i,Z}}^\lambda$ to $\mathcal{F}_{i,Z}$. Observe that the properties of the divisors $F_{\mathcal{E}_i,Z}^{\lambda}$ and $N_{\mathcal{E}_i,Z}^{G_\lambda}$ are preserved, meaning that the divisor $N_{\mathcal{F}_{i,Z}}^{G_\lambda}$ is a semiample ${\mathbb{Q}}$-divisor and $F_{\mathcal{F}_{i,Z}}^{\lambda}$ is a ${\mathbb{Q}}$-divisor which is independent of $G_\lambda$ in its ${\mathbb{Q}}$-linear system and its support is independent of $\lambda$. Let $L$ be a ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor on $W_i$ such that $$(K_X + \Delta +c_\lambda G_\lambda)|_{W_i} \sim_{\mathbb{Q}}L.$$ We will apply Ambro’s canonical bundle formula on $V_i$ with respect to the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $\pi_i^*(L)$. By [@Amb04 Lemma 2.4], we have the following equality $$\label{eff}
B_{V_i}(F_{\mathcal{F}_{i,Z}}^{\lambda} + N_{\mathcal{F}_{i,Z}}^{G_\lambda})=
B_{V_i}(\Phi^{G_\lambda}_{\mathcal{F}_i})$$ for every ${\mathbb{Q}}$-divisor $G_\lambda$. We claim that there exists a finite set $J$ and ${\mathbb{Q}}$-divisors $G_\lambda^j$ in the ${\mathbb{Q}}$-linear system $|K_X+\Delta +\lambda A|_{\mathbb{Q}}$ with $j\in J$, such that the following equality of ${\mathbb{R}}$-divisors holds $$\label{wedge}
B_{V_i}(F_{\mathcal{F}_{i,Z}}^{\lambda}) =
\bigwedge_{j \in J} B_{V_i}(F_{\mathcal{F}_{i,Z}}^{\lambda} + N_{\mathcal{F}_{i,Z}}^{G_\lambda^j}).$$ Indeed, for every prime divisor $C\subsetneq V_i$ we can choose $G_{\lambda}^j$ general in the ${\mathbb{Q}}$-linear system $|K_X+\Delta+\lambda A|_{\mathbb{Q}}$ so that $N^{G_\lambda^j}_{\mathcal{F}_{i,Z}}$ intersects the fibers of $p_{m,Z}$ transversally over a neighbourhood of the generic point of $C$. Therefore, we obtain the following equality of coefficients $$\mu_C (F_{\mathcal{F}_{i,Z}}^{\lambda}) =
\mu_C \left(F_{\mathcal{F}_{i,Z}}^{\lambda}+ N_{\mathcal{F}_{i,Z}}^{G_\lambda^j}\right).$$ Since the support of the ${\mathbb{R}}$-divisor $$B_{V_i}(F_{\mathcal{F}_{i,Z}}^{\lambda}+ N_{\mathcal{F}_{i,Z}}^{G_\lambda^j})$$ contains finitely many prime divisors for any $G_\lambda^j$ we conclude that we may take $J$ to be finite. Putting equation and equation together we have that $$B_{V_i}^{\lambda}=
B_{V_i}(F_{\mathcal{F}_{i,Z}}^{\lambda}) =
\bigwedge_{j\in J} B_{V_i}( N_{\mathcal{F}_{i,Z}}^{G^j_\lambda} + F_{\mathcal{F}_{i,Z}}^{\lambda}) =
\bigwedge_{j \in J} B_{V_i}(\Phi^{G^j_\lambda}_{\mathcal{F}_i})$$ is a ${\mathbb{R}}$-divisor which is independent of the choice of $G_\lambda$ in its linear system and its support is independent of $\lambda$. Since the pairs $(\mathcal{E}_i, \Phi_{\mathcal{E}_i}^{G_\lambda})$ are dlt for every ${\mathbb{Q}}$-divisor $G_\lambda$ which is general in its ${\mathbb{Q}}$-linear system we deduce that the divisors $$B_{W_i}(\Phi_{\mathcal{E}_i}^{G^j_\lambda}) =
\frac{1}{\deg(\pi_i)} {\pi_i}_* ( B_{V_i}(\Phi_{\mathcal{F}_i}^{G^j_\lambda}) ) \geq 0$$ are effective ${\mathbb{Q}}$-divisors for any $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$, and therefore $$B^{\lambda}_{W_i} = \bigwedge_{j \in J} B_{W_i}(\Phi^{G^j_\lambda}_{\mathcal{E}_i}) =
\frac{1}{\deg(\pi_i)} {\pi_i}_*(B^{\lambda}_{V_i}) \geq 0$$ is an effective ${\mathbb{Q}}$-divisor which is independent of $G_\lambda$ in its ${\mathbb{Q}}$-linear system and its support is independent of $\lambda$.
From now on, we assume that $K_{W_i}+B_{W_i}^{\lambda}$ is a ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor. By construction, the pair $$(W_i, B_{W_i}(\Phi_{\mathcal{E}_i}^{G_\lambda})),$$ is log canonical for every $G_\lambda$ general in its ${\mathbb{Q}}$-linear system, so we have that $(W_i, B_{W_i}^\lambda)$ is log canonical as well. Finally, we claim that the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $$M^{\lambda}_{V_i} = L- K_{V_i} - B^{\lambda}_{V_i}$$ is nef. Observe that for every $G_{\lambda}$ general in its ${\mathbb{Q}}$-linear system the ${\mathbb{Q}}$-divisor $M_{V_i}(\Phi_{\mathcal{F}_i}^{G_\lambda})$ is nef since we have that $\dim(X)=4$ and $\dim(W_i)\in \{1,2\}$, so either $\dim(V_i)=1$ or $\dim(\mathcal{F}_i)-\dim(V_i)=1$ (see, e.g., [@Amb04]). Let $C\subsetneq W_i$ be a prime divisor, then by construction we can choose $G^0_\lambda$ such that the effective ${\mathbb{Q}}$-divisor $$B_{V_i}(F_{\mathcal{F}_{i,Z}}^\lambda+N_{\mathcal{F}_{i,Z}}^{G_\lambda^0}) - B_{V_i}^\lambda$$ does not contain $C$ in its support. Morever, since the ${\mathbb{Q}}$-divisor $$M_{V_i}(\Phi_{\mathcal{F}_i}^{G_\lambda^0}) =
M_{V_i}( F_{\mathcal{F}_{i,Z}}^\lambda+ N_{\mathcal{F}_{i,Z}}^{G_\lambda^0})$$ is nef, the following inequalities hold $$M_{V_i}^{\lambda} \cdot C =
(L- K_{V_i}- B_{V_i}(F_{\mathcal{F}_{i,Z}}^\lambda+ N_{\mathcal{F}_{i,Z}}^{G_\lambda^0})) \cdot C
+ ( B_{V_i}(F_{\mathcal{F}_{i,Z}}^\lambda+ N_{\mathcal{F}_{i,Z}}^{G_\lambda^0})- B^{\lambda}_{V_i})\cdot C =$$ $$M_{V_i}(\Phi_{\mathcal{F}_i}^{G_\lambda^0}) \cdot C
+ (B_{V_i}(F_{\mathcal{F}_{i,Z}}^\lambda+ N_{\mathcal{F}_{i,Z}}^{G_\lambda^0}) - B_{V_i}^\lambda)\cdot C \geq 0,$$ proving the claim. Moreover, since $M_{W_i}^\lambda = {\pi_i}_*(M_{V_i}^\lambda)$ and ${\rm dim}(W_i)\leq 2$, we deduce that $M_{W_i}^\lambda$ is nef.
From the proof of Proposition \[canbundsplit\] we can see that the divisor $B_{W_i}^{\lambda}$ is uniquely determined by the dlt pair $(\mathcal{E}_i, \Phi_{\mathcal{E}_i}^{G_\lambda})$, but the moduli part $M_{W_i}^{\lambda}$ is only defined up to ${\mathbb{Q}}$-linear equivalence and different choices of the ${\mathbb{Q}}$-divisor $L$ induce different moduli divisors.
\[dec\] By Lemma \[monotonedecreasing\], we know that the coefficients of the irreducible components of the fixed divisor of $p^*(K_X+\Delta_\lambda)$ are monotone functions with respect to $\lambda \in [0,\lambda_3)$. In particular, the coefficients of the irreducible components of the fixed divisor of $(p\circ p_Z)^*(K_X+\Delta_\lambda)$ and $p_m^*(K_X+\Delta_\lambda)$ are monotone functions with respect to $\lambda \in [0,\lambda_3)$ as well. We denote by $N^{G_\lambda}_{\mathcal{E}_i}$ the push-forward of the semiample ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $N^{G_\lambda}_{\mathcal{E}_{i,Z}}$ by the birational morphism $q_Z$. Therefore, we have an adjunction formula $$K_{\mathcal{E}_i} + F^{\lambda}_{\mathcal{E}_i}+ N^{G_\lambda}_{\mathcal{E}_i} \sim_{\mathbb{Q}}p_m^*(K_X+\Delta_\lambda)|_{\mathcal{E}_i},$$ and the coefficients of $F^{\lambda}_{\mathcal{E}_i}$ are monotone with respect to $\lambda$ around the origin. Therefore, we have a well-defined ${\mathbb{R}}$-divisor $$F_{\mathcal{E}_i}^0 = \lim_{\lambda \rightarrow 0} F_{\mathcal{E}_i}^\lambda.$$ Analogously, the ${\mathbb{R}}$-divisors $F_{\mathcal{E}_{i,Z}}^0 = \lim_{\lambda \rightarrow 0} F_{\mathcal{E}_{i,Z}}^{\lambda}$ and $B_{W_i} = \lim_{\lambda \rightarrow 0} B_{W_i}^\lambda$ are well-defined.
Log canonical centers of codimension one {#cod1w}
----------------------------------------
In this subsection, we use divisorial adjunction of dlt pairs (see, e.g., [@Kol92] and [@HK10 Theorem 3.24]) to log canonical places of $(X,\Delta_\lambda$) which are divisors on $X$, and decompose the [*different divisor*]{} into a fixed component and the push-forward of a nef ${\mathbb{Q}}$-divisor on a higher birational model. From now on, we will always assume that $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$, where $\lambda_3$ is constructed in the proof of Proposition \[dltadjunction\] unless otherwise stated.
\[gidivisors\] Recall from \[semiamplecomp\] that we denote by $G_i$ with $i\in \{1,\dots, k\}$ the prime divisors that are contracted in the minimal model program $X\dashrightarrow X_{\rm min}$. Observe that every log canonical center of codimension one of $(X,\Delta_\lambda)$ is supported on one of the prime divisors $G_i$. Indeed, the diminished base locus of $(K_X+\Delta+\lambda A)$ equals the exceptional locus of the rational map $\pi \colon X \dashrightarrow X_{\rm min}$. Up to permuting the divisors $G_i$ we can assume that $G_1,\dots, G_{k_0}$ are the log canonical centers of codimension one of $(X,\Delta_\lambda)$ for every $\lambda \in {\mathbb{Q}}_{>0}$ sufficiently small.
\[adjlcc1\] Let $i\in \{1,\dots, k_0\}$ such that $G_i$ is normal and $\lambda \in (0,\lambda_3)$. Then, we can write $$(K_X+\Delta+c_\lambda G_\lambda)|_{G_i} \sim_{\mathbb{Q}}K_{G_i}+ B_{G_i}^\lambda + M_{G_i}^{\lambda},$$ where the following statements hold:
- The effective ${\mathbb{Q}}$-divisor $B_{G_i}^\lambda$ is independent of the choice of $G_\lambda$ in its ${\mathbb{Q}}$-linear system, and
- the support of $B_{G_i}^\lambda$ is independent of $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$.
Moreover, if $K_{G_i}+B_{G_i}^\lambda$ is ${\mathbb{Q}}$-Cartier we have that:
- The pair $(G_i, B_{G_i}^\lambda)$ is log canonical, and
- the ${\mathbb{Q}}$-divisor $M_{G_i}^{\lambda}$ is the push-forward of a nef ${\mathbb{Q}}$-divisor on a higher birational model.
In particular, $M_{G_i}^\lambda$ is nef in codimension one.
Let $p_m\colon Y_m \rightarrow X$ be the common dlt modification of the log canonical pairs $(X,\Delta_\lambda)$ constructed in Proposition \[comdltmodification\]. By \[semiamplecomp\], we can write $$K_{Y_m} + \Delta_{Y_m} + c_\lambda \left( \sum_{i=1}^k G_{\lambda,i,Y_m} \right) +c_\lambda G_{\lambda,0,Y_m} +
\sum_{i \in \mathcal{I}'} \mathcal{E}_i = p_m^*(K_X+\Delta+c_\lambda G_\lambda),$$ where the divisors $G_{\lambda,i,Y_m}$ are the strict transforms on $Y_m$ of the irreducible components of $G_\lambda$ which are contracted by the minimal model program $X\dashrightarrow X_{\rm min}$, and $G_{\lambda,0,Y_m}$ is the strict transform on $Y_m$ of the irreducible component of $G_\lambda$ which is not contracted on $X_{\rm min}$. Observe that the divisors $G_1,G_2,\dots, G_{k_0}$ are exactly those where $G_{\lambda,i}$ has coefficient one in $\Delta+c_\lambda G_\lambda$. We denote by $M^{\lambda}_{G_{i,Z}}$ the restriction of the nef ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $(p\circ p_Z)^*(K_X+\Delta_\lambda)$ to $G_{i,Z}$ and by $M^{\lambda}_{G_{i,Y_m}}$ the push-forward of $M^{\lambda}_{G_{i,Z}}$ via the morphism $q_Z \colon G_{i,Z}\rightarrow G_{i,Y_m}$. Therefore, we have a divisorial adjunction formula $$\label{divgeneralizedadjunction}
K_{G_{i,Y_m}} + B^{\lambda}_{G_{i,Y_m}} + M^{\lambda}_{G_{i,Y_m}} \sim_{\mathbb{Q}}p_m^*(K_X+\Delta_\lambda )|_{G_{i,Y_m}}.$$ By pushing-forward via the birational morphism $p_m \colon G_{i,Y_m}\rightarrow G_i$ we obtain the desired decomposition $$K_{G_{i}}+ B^\lambda_{G_{i}} + M^\lambda_{G_{i}} \sim_{\mathbb{Q}}(K_X+\Delta+c_\lambda G_\lambda)|_{G_i}.$$
\[dec2\] Arguing similarly as in Remark \[dec\], we have that for $\lambda \in [0,\lambda_3)$ the coefficient at the prime divisor $G_{\lambda, i, Y_m}$ with $i\in \{1, \dots, k_0\}$ of $p_m^*(K_X+\Delta_\lambda)$ is a monotone function with respect to $\lambda$ around the origin. Therefore, the coefficient at any prime divisor of $B_{G_i}^{\lambda}$ is a monotone function with respect to $\lambda$ around the origin. In particular, the limit $B_{G_i} = \lim_{\lambda \rightarrow 0} B_{G_i}^\lambda$ is a well-defined ${\mathbb{R}}$-divisor.
\[comparebirkar\] The above divisorial adjunction formula is a divisorial adjunction of generalized pairs in the sense of [@Bir17 Section 3] (see \[genpair\] for the definition of generalized pair). In [@Bir17 Lemma 3.3], Birkar proves that the coefficients of the divisor $B^{\lambda}_{G_i}$ belong to a DCC set which only depends on the coefficients of $\Delta_\lambda$ and the Cartier index of $M^{\lambda}_{G_{i,Z}}$.
Adjunction formula
------------------
In this subsection, we prove an adjunction formula for $(c+1)(K_X+\Delta)$ to each minimal log canonical center of $(X,\Delta_\lambda)$ for $\lambda\in (0,\lambda_3)$.
\[adjunction\] Let $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$ and $W_i$ a minimal log canonical center of $(X,\Delta_\lambda)$. Then we can write $$((c+1)(K_X+\Delta))|_{W_i} \sim_{\mathbb{Q}}K_{W_i} + B_{W_i} + M_{W_i},$$ where the following statements hold:
- The ${\mathbb{R}}$-divisor $M_{W_i}$ has numerical class in the cone of nef divisors in codimension one of $W_i$, and
- The pair $(W_i, B_{W_i})$ is log canonical whenever $M_{W_i}$ is ${\mathbb{R}}$-Cartier.
Being the case $\dim(W_i)=0$ trivial we need to prove the statement for $\dim(W_i) \in \{1,2,3\}$. We prove the case $\dim(W_i)\leq 2$ by using Proposition \[canbundsplit\]. The case $\dim(W_i)=3$ is analogous by using Proposition \[adjlcc1\].
We will denote by $L$ a ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor on $W_i$ such that $$((c+1)(K_X+\Delta))|_{W_i} \sim_{\mathbb{Q}}L,$$ and we aim to prove the existence of an effective ${\mathbb{R}}$-divisor $B_{W_i}$ such that the numerical class of the ${\mathbb{R}}$-divisor $L-K_{W_i}-B_{W_i}$ is contained in the cone of nef divisors in codimension one of $W_i$. By Proposition \[canbundsplit\], for every $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$ we may write $$(K_X+\Delta+c_\lambda G_\lambda)|_{W_i} \sim_{\mathbb{Q}}K_{W_i} + B_{W_i}^\lambda + M_{W_i}^\lambda,$$ where $M^{\lambda}_{W_i}$ is a ${\mathbb{Q}}$-divisor whose numerical class is contained in the cone of nef divisors in codimension one of $W_i$ and $B^{\lambda}_{W_i}$ is an effective divisor such that $(W_i,B^{\lambda}_{W_i})$ is log canonical whenever $K_{W_i}+B^{\lambda}_{W_i}$ is ${\mathbb{R}}$-Cartier. Moreover, the support of the divisor $B_{W_i}^\lambda$ is independent of $\lambda \in (0,\lambda_3)$. By Remark \[dec\] and Remark \[dec2\], we know that the ${\mathbb{R}}$-divisor $$B_{W_i} = \lim_{\lambda \rightarrow 0} B_{W_i}^{\lambda}$$ is well-defined. Moreover, the numerical class of the ${\mathbb{R}}$-divisor $$M_{W_i} = L - K_{W_i} - B_{W_i},$$ is the numerical limit of the divisors $$M_{W_i}^\lambda = L- K_{W_i}-B^{\lambda}_{W_i},$$ which are contained in the cone of nef divisors in codimension one of $W_i$. Being the cone of nef divisors in codimension one closed we infer that $M_{W_i}$ is contained in the cone of nef divisors in codimension one.
If the divisor $M_{W_i}$ is ${\mathbb{R}}$-Cartier, then it is nef in codimension one.
By construction, the divisor $K_{W_i}+B_{W_i}+M_{W_i}$ is a ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor even if $B_{W_i}$ is just an ${\mathbb{R}}$-divisor that may not be ${\mathbb{R}}$-Cartier. In the case $\dim(W_i)=3$, meaning that $W_i=G_i$ for some $i\in \{1,\dots, k_0\}$, we know that $M_{G_i}$ is the push-forward to $G_i$ of the restriction of the nef ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $(q\circ p_Z)^*((c+1)(K_{X_{\rm min}}+\Delta_{\rm min}))$ to $G_{i,Z}$.
The ${\mathbb{R}}$-divisor $M_{W_i}$ depends on the choice of a starting ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $L$ on $W_i$. However, changing the choice of $L$ in its ${\mathbb{Q}}$-linear system only changes $M_{W_i}$ in its ${\mathbb{R}}$-linear system.
Termination of quasi-flips on the minimal log canonical centers {#section:termination}
===============================================================
In this section, we prove that a sequence of flips of a pseudo-effective klt $4$-fold $(X,\Delta)$ terminates around any common minimal log canonical center $W_i$ of the log canonical pairs $(X,\Delta_\lambda)$ for $\lambda\in (0,\lambda_3)\cap {\mathbb{Q}}$.
Generalized pair on the minimal log canonical center
----------------------------------------------------
In this subsection, we prove that the triple $(W_i,B_{W_i}+M_{W_i})$ induced by the adjunction in \[adjunction\], is indeed a generalized pair.
\[betteradjunction\] Let $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$ and let $W_i$ be a minimal log canonical center of $(X,\Delta_\lambda)$. Then we can write $$((c+1)(K_X+\Delta))|_{W_i} \sim_{\mathbb{Q}}K_{W_i} + B_{W_i} + M_{W_i},$$ where $(W_i,B_{W_i}+M_{W_i})$ is a generalized log canonical pair.
First, lets assume that $\dim(W_i)\leq 2$. Consider $\pi_i \colon V_i \rightarrow W_i$ to be a resolution of singularities of the generalized pairs $(W_i, B_{W_i}^\lambda+ M_{W_i}^\lambda)$ in the sense of \[logresolutiongeneralized\]. We denote by $p_m \colon \mathcal{F}_i \rightarrow V_i$ be the morphism induced by $\mathcal{E}_i \rightarrow W_i$ with respect to the base change $V_i \rightarrow W_i$. For every $\lambda \in (0, \lambda_3)\cap {\mathbb{Q}}$ and $G_\lambda$ general in its ${\mathbb{Q}}$-linear system we know that the pair $(\mathcal{E}_i, \Phi_{\mathcal{E}_i}^{G_\lambda})$ is dlt, therefore we have that $${\rm coeff} (B_{V_i}(\Phi_{\mathcal{F}_i}^{G_\lambda})) \leq 1.$$ So we conclude that ${\rm coeff}(B_{V_i}^\lambda ) \leq 1$, being the latter the wedge of ${\mathbb{R}}$-divisors with coefficients bounded above by one. Thus, we have a resolution of singularities $p_i \colon V_i \rightarrow W_i$ of the pairs $(W_i,B_{W_i}^\lambda)$, such that $$K_{V_i}+ B_{V_i}^\lambda + M_{V_i}^\lambda = \pi_i^*(K_{W_i}+B_{W_i}^\lambda+M_{W_i}^\lambda),$$ and such resolution is independent of $\lambda$. Taking the limit $\lambda \rightarrow 0$ we obtain a divisor $B_{V_i}$ with ${\rm coeff}(B_{V_i})\leq 1$ and $$K_{V_i}+ B_{V_i}+ M_{V_i} = p_i^*(K_{W_i}+B_{W_i} +M_{W_i}),$$ so the generalized pair $(W_i,B_{W_i}+M_{W_i})$ is log canonical, where we are considering the higher model in \[genpair\] to be $(V_i,B_{V_i}+M_{V_i})$.
Now, it suffices to show the statement when the log canonical center has codimension one. By Remark \[gidivisors\], we know that any such log canonical center equals one of the prime divisors $G_i$ with $i\in \{1,\dots, k_0\}$. In this case, we are in the situation of the proof of Proposition \[adjlcc1\]. For every $\lambda \in (0,\lambda_3)$ we have a resolution of singularities $p_Z \colon G_{i,Z}\rightarrow G_i$ such that $$K_{G_i,Z} + B^\lambda_{G_i,Z} + M^\lambda_{G_i,Z} = p_Z^*( K_{G_i}+ B_{G_i}^\lambda + M_{G_i}^\lambda).$$ Moreover, since the pair $$(K_{G_{i,Y_m}}, B_{G_{i,Y_m}}^\lambda + M_{G_{i,Y_m}}^\lambda)$$ is dlt we know that ${\rm coeff}(B^\lambda_{G_i,Z})\leq 1$. Hence, the statement follows by taking the corresponding limit.
\[arb\] Observe that in the above proof, when $W_i$ is a surface we can take an arbitrary resolution of singularities $p_i \colon V_i \rightarrow W_i$ of the pairs $(W_i, B_{W_i}^\lambda)$ to define the generalized pair structure on the triple $(W_i, B_{W_i} +M_{W_i})$.
Quasi-flips for generalized pairs on the minimal log canonical centers
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In this subsection, we introduce quasi-flips for generalized log canonical pairs, and prove that a sequence of $(K_X+\Delta)$-flips that does not contain the minimal log canonical center $W_i$ in a flipping locus induces a sequence of quasi-flips for the induced generalized pair. Throughout the remainder of this section, we will consider a single log canonical center so we simply denote it by $W$.
\[qflipslcp\] Let $(X,\Delta)$ be a klt $4$-fold with $K_X+\Delta$ pseudo-effective, and $W$ a minimal log canonical center of $(X,\Delta_\lambda)$ with $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$. Consider a sequence of $(K_X+\Delta)$-flips $$\nonumber
\xymatrix{
(X,\Delta)\ar@{-->}^-{\pi_1}[r] & (X_1,\Delta_1)\ar@{-->}^-{\pi_2}[r] & (X_2,\Delta_2)\ar@{-->}^-{\pi_3}[r] & \dots \ar@{-->}^-{\pi_j}[r] & (X_j,\Delta_j)\ar@{-->}^-{\pi_{j+1}}[r] & \dots \\
}$$ which does not contain $W$ in a flipping locus. Then it induces a sequence of birational transformations $$\nonumber
\xymatrix{
(W,B+M)\ar@{-->}^-{\pi_1}[r] & (W_1,B_1+M_1)\ar@{-->}^-{\pi_2}[r] & (W_2,B_2+M_2)\ar@{-->}^-{\pi_3}[r] & \dots \ar@{-->}^-{\pi_j}[r] & (W_j,B_j+M_j)\ar@{-->}^-{\pi_{j+1}}[r] & \dots \\
}$$ where $(W,B+M)$ is the generalized log canonical pair obtained by adjunction in \[adjunction\]. The map $\pi_j$ is either a strict ample $(K_W+B_W+M_W)$-quasi-flip or the identity. In the latter case the flipping locus of $\pi_j \colon X_{j-1}\dashrightarrow X_j$ is disjoint from $W$.
It suffices to prove that $\pi_1$ induces a strict ample $(K_W+B+M)$-quasi-flip if its flipping locus intersect $W$ non-trivially. Let $\phi$ and $\phi^+$ be the flipping contraction and flipped contraction of $\pi_1$, respectively. By Proposition \[betteradjunction\], we know that both triples $(W,B+M)$ and $(W_1,B_1+M_1)$ are generalized lc, so the first condition of \[qfgen\] holds. Moreover, given a curve $C\subsetneq W_1$ which is being contracted by the flipped contraction, by the adjunction formula \[adjunction\] we have that $$(K_{W_1}+B_{1}+M_{1})\cdot C = (c+1)(K_{X_1}+\Delta_1)|_{W_1} \cdot C < 0,$$ concluding that the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $K_{W_1}+B_1+M_1$ is anti-ample over $Z$. Analogously, we can check that $K_{W}+B+M$ is ample over $Z$, so the second condition of \[qfgen\] holds. We claim that the projective birational map $W \dashrightarrow W_1$ is a $(K_{W}+B+M)$-negative map, meaning that we can find two projective birational morphisms $p \colon V \rightarrow W$ and $q\colon V \rightarrow W_1$ which give a log resolution of the above generalized pairs, and we can write $$p^*( K_{W}+B+M ) - q^*( K_{W_1}+B_{1}+M_{1}) \geq 0.$$ Indeed, we can take $V$ to be the strict transform of $W$ on a log resolution of the flip $X\dashrightarrow X_1$ given by the projective birational morphisms $p_X$ and $q_X$, so we can write $$p^*( K_{W}+B+M ) - q^*( K_{W_1}+B_{1}+M_{1}) =
(c+1)(p_X^*(K_X+\Delta)-q_X^*(K_{X_1}+\Delta_1))|_V = E \geq 0.$$ Therefore, we have the following inequality of Weil ${\mathbb{R}}$-divisors on $Z$ $$\phi^+_*(B_1) = \phi^+_*(B - E) \leq \phi_*(B),$$ where $\phi$ and $\phi^+$ are the flipping and flipped contraction of $\pi_1$ restricted to $W$ and $W_1$, respectively.
Now, we prove that the nef parts $M_{W_j}$ are the trace of a common Cartier b-divisor. In the case that $W$ has dimension three, this follows from the divisorial adjunction defining $M_{W_j}$. On the other hand, if $W$ has dimension at most two, we will prove that the set of divisorial valuations extracted by the sequence of quasi-flips form a finite set of non-terminal valuations over $W$. Indeed, by Proposition \[stab\] we know that the set of non-terminal valuations of the log canonical pairs $(X,\Delta_\lambda)$ stabilize for $\lambda$ sufficiently small, therefore by adjunction, the set of non-terminal valuations over $(K_X+\Delta_\lambda)|_{W}$ stabilize as well. Observe that for every $j\in \mathbb{Z}_{\geq 1}$, we can find $\lambda_j\in (0,\lambda_3)$ small enough, such that the sequence of quasi-flips $\pi_1,\dots,\pi_j$ is a sequence of $(K_X+\Delta_\lambda)|_{W}$-quasi-flips for every $\lambda \in (0,\lambda_j)$. Moreover, any divisor extracted by the quasi-flip $\pi_1,\dots,\pi_j$ is a non-terminal valuation over $W$ for the klt pairs $(K_X+\Delta_\lambda)|_{W}$ with $\lambda \in (0,\lambda_j)$. Thus, we conclude that there exists a smooth variety $V$ with surjective projective birational morphisms $p_j \colon V \rightarrow W_j$ for every $j\in \mathbb{Z}_{\geq 1}$. Hence, by Remark \[arb\] we can define the generalized pair structure of $(W_j,B_j+M_j)$ with a fixed nef divisor on $V$.
Finally, it suffices to check that the quasi-flip is strict, meaning that its non-flopping locus is non-empty in the sense of \[definitionstrict\]. Observe that the flip $\pi_1 \colon X \dashrightarrow X_1$ is ample in the sense of \[definitionstrict\], therefore by monotonicity of discrepancies \[monotonicity\] the coefficient of $$\label{ambientdifference}
p_X^*(K_X+\Delta)-q_X^*(K_{X_1}+\Delta_1)$$ at any $p_X$-exceptional or $q_X$-exceptional prime divisor with center on the flipping or flipped locus is strictly positive. It suffices to show that there exists a component $E_i$ with non-trivial coefficient in the divisor which intersect $V$ non-trivially. Indeed, by the negativity lemma [@KM98 Lemma 3.39 (2)] a fiber of $p$ is either disjoint from the support of or is contained in such support. Therefore, a fiber of $p$ over the intersection of $W$ and the flipping locus of $\pi_1 \colon X \dashrightarrow X_1$ must be contained in the union of prime divisors which have non-trivial coefficient in , concluding that there exists a prime divisor $E_i$ with non-trivial coefficient in which intersect $V$ non-trivially. So we deduce that the divisor $E$ is non-trivial, which implies that $\pi_1\colon W\dashrightarrow W_1$ is a quasi-flip that is not a quasi-flop.
In what follows, we will prove that the coefficients of the divisors $B_j$ in the sequence of quasi-flips of Proposition \[qflipslcp\] belong to a DCC set. To do so, we will construct a [*generalized boundary part*]{} for Ambro’s canonical bundle formula for generalized pairs and we will compare the divisors $B_j$ with such generalized boundary parts. Thus, we can apply the ACC for generalized log canonical thresholds by Birkar and Zhang [@BZ16 Theorem 1.5] to prove the statement.
\[genambro\] Consider a projective birational morphism $p_m \colon \mathcal{E}_j \rightarrow W_j$ of normal quasi-projective varieties and a generalized boundary $F+N$ on $\mathcal{E}_j$, with boundary part $F$ and nef part $N$, such that $(\mathcal{E}_j, F+N)$ is a generalized sub-pair which is generalized sub-log canonical near the generic fiber of $p_m$. We can define a [*boundary part*]{} as follows: Given a prime divisor $C\subsetneq W_i$, we define the real number $$\overline{\mu}_C = \sup \{ t\in {\mathbb{R}}\mid (\mathcal{E}_j, F+N+tp_m^*(C))
\text{ is generalized sub-log canonical over a neighbourhood of $\eta_C$}\},$$ where $\eta_C$ is the generic point of $C$. Observe that the pull-back $p_m^*(C)$ is well-defined over a neighbourhood of $\eta_C$ since $W_j$ is normal. Then we can define the ${\mathbb{Q}}$-divisor $$\overline{B}_{W_i}(F+N) = \sum_{C\subsetneq W_j} (1-\overline{\mu}_{C}(F)) C,$$ where the sum runs over all the prime divisors $C$ of $W_j$. Arguing as in [@Cor07 Section 8.2] we can see that the above sum is finite.
The following lemma is a version of [@Amb04 Lemma 2.4] for generalized pairs.
\[lemmafordcc\] Let $q_Z \colon \mathcal{E}_{j,Z}\rightarrow \mathcal{E}_j$ be a log resolution over $W_j$ of the generalized pair $(\mathcal{E}_j,F+N)$ and let $(\mathcal{E}_{j,Z}, F_Z+N_Z)$ the associated generalized pair on $\mathcal{E}_{j,Z}$. Then the following equalities of Weil ${\mathbb{R}}$-divisors holds $$\overline{B}_{W_j}(F+N) = \overline{B}_{W_j}(F_Z+N_Z)=B_{W_j}(F_Z).$$
Let $C\subsetneq W_j$ be an irreducible divisor and $t\in {\mathbb{R}}$. Consider $\mathcal{E}$ a log resolution of the generalized sub-pairs $(\mathcal{E}_j, F+t p_m^*(C)+N)$ and $(\mathcal{E}_{j,Z}, F_Z+ t p_{m,Z}^*(C) +N_Z)$ with projective birational morphisms $r\colon \mathcal{E}\rightarrow \mathcal{E}_{j}$ and $r_Z \colon \mathcal{E}\rightarrow \mathcal{E}_{j,Z}$. Then, we have that $$r^*( K_{\mathcal{E}_j}+ F+t p_m^*(C)+N ) = K_{\mathcal{E}} + B_{\mathcal{E}} + t (p_m \circ r)^*(C) + N_{\mathcal{E}} =
r_Z^*( K_{\mathcal{E}_{j,Z}} + F_Z+t p_{m,Z}^*(C)+N_Z ),$$ where $N_{\mathcal{E}}=r_Z^*(N_Z)$. Therefore, we have that $(\mathcal{E}_j, F+N+t p_m^*(C))$ is generalized sub-lc over a neighbourhood of $\eta_C$ if and only if $(\mathcal{E}_{j,Z}, F_Z+N_Z+t p_{m,Z}^*(C))$ is generalized sub-lc over a neighbourhood of $\eta_C$, concluding the first equality. The second equality follows from the fact that $$K_{\mathcal{E}} + B_{\mathcal{E}} + t (p_m \circ r)^*(C)
= r_Z^*( K_{\mathcal{E}_{j,Z}} + F_Z+t p_{m,Z}^*(C)).$$
\[wehaveDCC\] The sequence of quasi-flips of Proposition \[qflipslcp\] is a sequence of quasi-flips under a DCC set.
It suffices to prove that the coefficients of the ${\mathbb{Q}}$-divisors $B_j$ belong to a DCC set. In order to do so, we will apply Lemma \[lemmafordcc\] and the ACC for generalized log canonical thresholds [@BZ16 Theorem 1.5].
Assume that $W_j$ has codimension one on $X_j$ so we are in the situation of Proposition \[adjlcc1\]. By Remark \[comparebirkar\], the coefficients of $B_j$ are obtained by a divisorial adjunction of generalized pairs whose nef part is the trace of the nef b-Cartier divisor induced by and whose boundary part has the same coefficients as the ${\mathbb{Q}}$-divisor $$\label{fixedpartdlt}
K_{Y_m} + \Delta_{Y_m} + c\left( \sum_{i=1}^k G_{0,i,Y_m}\right) + \sum_{i \in \mathcal{I}'} \mathcal{E}_i.$$ Therefore, by [@Bir17 Lemma 3.3] the coefficients of the ${\mathbb{R}}$-divisors $B_j$, with $j\in \mathbb{Z}_{\geq 1}$, belong to a DCC set.
From now on, assume that $W_j$ has dimension at least two on $X_j$ so we are in the situation of Proposition \[canbundsplit\]. Denote by $N_{\mathcal{E}_{j,Z}}^0$ the restriction of the nef ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $$\label{nefbcartier}
(c+1)p_Z^*(q^*(K_{X_{\rm min}}+\Delta_{\rm min}))$$ to $\mathcal{E}_{j,Z}$ and by $N_{\mathcal{E}_j}^0$ the push-forward of $N^{0}_{\mathcal{E}_{j,Z}}$ to $\mathcal{E}_j$. Observe that $N^{0}_{\mathcal{E}_{j,Z}}$ is the nef part of a generalized boundary in the sense of \[genpair\] and the Cartier index of the ${\mathbb{Q}}$-divisor $N^{0}_{\mathcal{E}_{j,Z}}$ is independent of $j \in \mathbb{Z}_{\geq 1}$. By Remark \[dec\], we know that the ${\mathbb{R}}$-divisors $F_{\mathcal{E}_j}^0=\lim_{\lambda \rightarrow 0} F^\lambda_{\mathcal{E}_j}$ are well-defined. Moreover, the coefficients of the ${\mathbb{R}}$-divisors $F_{\mathcal{E}_j}^0$ belong to a DCC set, since these coefficients can be obtained by divisorial adjunction of generalized pairs of a divisor whose nef part is the trace of the nef b-Cartier divisor induced by and whose boundary part has set of coefficients equal to the set of coefficients of the ${\mathbb{Q}}$-divisor . Therefore, by the ACC for generalized log canonical thresholds [@BZ16 Theorem 1.5] it suffices to prove that $$B_{W_j} = \overline{B}_{W_j}(F^{0}_{\mathcal{E}_j}+N^{0}_{\mathcal{E}_j}).$$ Indeed, we have the following equalities $$B_{W_j} = \lim_{\lambda\rightarrow 0} B_{W_j}(F^\lambda_{\mathcal{E}_{j,Z}}) =
\lim_{\lambda \rightarrow 0} \overline{B}_{W_j}(F^\lambda_{\mathcal{E}_{j,Z}}+N^{G_\lambda}_{\mathcal{E}_{j,Z}}) =
\overline{B}_{W_j}(F^{0}_{\mathcal{E}_{j,Z}} + N^{0}_{\mathcal{E}_{j,Z}}) =
\overline{B}_{W_j}(F^{0}_{\mathcal{E}_j} + N^0_{\mathcal{E}_j}),$$ where the first equality follows from the definition, the second and fourth equalities follows from Lemma \[lemmafordcc\], and the third equality follows from Remark \[dec\].
The coefficients of the divisors $B_j$ belong to a DCC set which only depends on the coefficients of the ${\mathbb{Q}}$-divisor \[fixedpartdlt\] and the Cartier index of the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $K_{X_{\rm min}}+\Delta_{\rm min}$.
Generalized terminalization and small ${\mathbb{Q}}$-factorialization
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In this subsection, we prove that generalized klt pairs have a generalized ${\mathbb{Q}}$-factorial terminalization and a generalized small ${\mathbb{Q}}$-factorialization.
A [*generalized ${\mathbb{Q}}$-factorial terminalization*]{} of a generalized klt pair $(X,B+M)$ is a ${\mathbb{Q}}$-factorial generalized terminal pair $(Y,B_Y+M_Y)$ together with a projective morphism $p \colon Y \rightarrow X$ such that $K_Y+B_Y+M_Y=p^*(K_X+B+M)$. Moreover, we require that both nef parts $M$ and $M_Y$ are the trace of a common nef b-divisor on a higher birational model which dominates $Y$ and $X$.
A [*generalized small ${\mathbb{Q}}$-factorialization*]{} of a generalized klt pair $(X,B+M)$ is a ${\mathbb{Q}}$-factorial generalized klt pair $(Y,B_Y+M_Y)$ together with a small projective morphism $p\colon Y \rightarrow X$ such that $K_Y+B_Y+M_Y=p^*(K_X+B+M)$. Moreover, we require that both nef parts $M$ and $M_Y$ are the trace of a common nef b-divisor on a higher birational model which dominates $Y$ and $X$.
\[existence\] A generalized ${\mathbb{Q}}$-factorial terminalization and a generalized small ${\mathbb{Q}}$-factorialization exist for any generalized klt pair $(X,B+M)$.
Pick $A$ be an ample effective ${\mathbb{Q}}$-divisor which contains every center of a divisorial valuation of $(X,B+M)$ of generalized log discrepancy equal to one. If $\epsilon \in {\mathbb{R}}_{>0}$ is sufficiently small, then $(X,B+\epsilon A+M)$ is a generalized klt pair with boundary part $B+\epsilon A$ and nef part $M$. So replacing $B+M$ with $B+\epsilon A+M$ we may assume that there is no divisorial valuations of generalized log discrepancy equal to one.
Let $(X',B'+M')$ be a log resolution of $(X,B+M)$ so that $M$ is nef and write $$K_{X'}+B'+M' = f^*(K_X+B+M)$$ as in the definition of generalized pairs. Hence, we have that $$K_{X'}+B'_1+M' = f^*(K_X+B+M) + B'_2,$$ where $B'_1$ and $B'_2$ are effective divisors with no common components and $f_*(B'_1)=B$. Let $F_1$ be the sum of all the $f$-exceptional divisors which are not irreducible components of $B'_2$. Pick $\epsilon \in {\mathbb{R}}_{>0}$ sufficiently small such that the generalized pair $$K_{X'}+B'_0+M' = K_{X'}+B'_1+\epsilon F_1+ M'$$ is a generalized klt pair with boundary part $B'_1+\epsilon F_1$. By [@BZ16 Lemma 4.4 (2)] we can run a minimal model program over $X$, with scaling of an ample divisor, for the generalized klt pair $(X',B'_0+M')$ to obtain a minimal model $Y$ over $X$. We denote by $\pi \colon X' \dashrightarrow Y$ the corresponding minimal model program. The negativity of contractions implies that $$K_Y+B_Y+M_Y=f^*(K_X+B+M),$$ where $B_Y=\pi_*(B'_0)$. Therefore, all the irreducible components of $B_2'$ are contracted by the minimal model program $\pi$. Since flips preserve the ${\mathbb{Q}}$-factorial condition we conclude that $(Y,B_Y+M_Y)$ is a generalized ${\mathbb{Q}}$-factorial terminalization of $(X,B+M)$.
Analogously, by [@BZ16 Lemma 4.4 (2)] we can run a minimal model program with scaling of an ample divisor for the generalized klt pair $(X',B'_1+M')$ over $X$ to obtain a terminal model $(Y,B_Y+M_Y)$ which is ${\mathbb{Q}}$-factorial. We denote by $\pi \colon X'\dashrightarrow Y$ the corresponding minimal model program. Since $B_Y=\pi_*(B'_1)$ we may denote $E'=\pi_*(B'_2)$ to write $$K_Y+B_Y+M_Y=f^*(K_X+B+M)+E'.$$ By negativity of contractions we conclude that $E'=0$, so we contracted all $f$-exceptional divisors in the minimal model program $\pi$, which means that $(Y,B_Y+M_Y)$ with the projective birational morphism $f\colon Y \rightarrow X$ give a generalized small ${\mathbb{Q}}$-factorialization of $(X,B+M)$.
Generalized difficulty function
-------------------------------
In this subsection, we will use a generalized version of the difficulty function introduced in [@AHK] in order to prove the following proposition (compare with [@AHK Theorem 2.15])
\[cod2\] Consider a sequence $$\nonumber
\xymatrix{
(W,B+M)\ar@{-->}^-{\pi_1}[r] & (W_1,B_1+M_1)\ar@{-->}^-{\pi_2}[r] & (W_2,B_2+M_2)\ar@{-->}^-{\pi_3}[r] & \dots \ar@{-->}^-{\pi_j}[r] & (W_j,B_j+M_j)\ar@{-->}^-{\pi_{j+1}}[r] & \dots \\
}$$ of flips for generalized klt pairs. Then it cannot happen infinitely many times that the flipping or flipped locus has a component of codimension $2$ in $W_j$ which is contained in $B_j$.
Let $(X,B+M)$ be a ${\mathbb{Q}}$-factorial generalized terminal pair. We can pick $f\colon X' \rightarrow X$ a log resolution so that $M'$ is a nef ${\mathbb{R}}$-divisor. Then, by the negativity lemma we can write $f^*(M)=M'+E$, where $E$ is an effective divisor. Consider $C\subsetneq X$ a subvariety of codimension two which is contained in a unique irreducible component $B_i$ of $B$, is not contained in the image of $E$ on $X$, and is not contained in the singular locus of $B_i$. Since the triple $(X,B+M)$ is terminal it is smooth along the generic point of $C$. Let $E_1$ be the unique irreducible divisor of the blow up $\pi_1 \colon X_1 = {\rm Bl}_C X \rightarrow X$ which dominates $C$ and let $C_1 = E_1 \cap {\pi_1}^{-1}_*(B_i)$. By induction, for $k\in \mathbb{Z}_{\geq 2}$ we can define $E_k$ to be the unique irreducible divisor of the blow up $\pi_k \colon X_{k} = {\rm Bl}_{C_{k-1}} X_{k-1} \rightarrow X_{k-1}$ which dominates $C_{k-1}$ and $C_{k}= E_k \cap {\pi_k}^{-1}_*(B_{i})$. A simple computation shows that $a_{E_k}(X,B+M)=k(1-b_i)$, where $b_i$ is the coefficient of $B_i$ on $B$. The divisorial valuation corresponding to $E_k$ is called the [*k-th echo*]{} of $X$ along $C$. Any such divisorial valuation will be called an [*echo along a subvariety $C$ of codimension two*]{}.
\[finitediv\] Let $(X,B+M)$ be a ${\mathbb{Q}}$-factorial generalized terminal pair. Then the divisorial valuations $E$ over $X$ with generalized discrepancy $a_E(X,B+M)$ in the interval $(0,1)$ are the following:
- The echoes of $X$ along subvarieties $C$ of codimension two,
- and finitely many others.
Consider $(X',B'+M')$ a log resolution of $(X,B+M)$ so that $M'$ is nef. Then we can write $$K_{X'}+B'+M' = f^*(K_X+B+M),$$ where all the prime divisors of $B'$ with positive coefficients are disjoint. Therefore, we conclude using the formula to compute discrepancies over a log smooth pair (see, e.g., [@KM98 Lemma 2.29]).
\[boundN\] Let $(X,B+M)$ be a generalized klt pair. There exists $\epsilon >0$ and $N\in \mathbb{Z}_{\geq 1}$ such that there are exactly $N$ divisorial valuations $E$ over $X$ with generalized discrepancy $a_E(X,B+M)$ in the interval $(-1,\epsilon)$.
By Proposition \[existence\], we can construct a ${\mathbb{Q}}$-factorial terminalization $(Y,B_Y+M_Y)$ of the generalized pair $(X,B+M)$. Then, by Lemma \[finitediv\] we know that there are finitely many divisorial valuations over $Y$ with generalized discrepancy in the interval $(0,1)$ which are not echoes along subvarieties of codimension two. Observe that the echoes have generalized discrepancy at least $1-b$, where $b=\max \{ b_i \mid \text{$b_i$ is a coefficient of $B$ }\}$. Therefore, it suffices to take $0<\epsilon < 1-b$ and $N$ the number of divisorial valuations with generalized discrepancy in the interval $(-1,\epsilon)$.
\[weight\] Given $\alpha \in (0,1)$, we define the following [*weight functions*]{} $w : {\mathbb{R}}_{>0}\rightarrow {\mathbb{R}}_{\geq 0}$ to be
- $w_\alpha^{-}(x)=1-x$ for $x\leq \alpha$ and $w^{-}_\alpha(x)=0$ for $x>\alpha$, and
- $w_\alpha^{+}(x)=1-x$ for $x< \alpha$ and $w^{+}_\alpha(x)=0$ for $x\geq\alpha$.
The [*summed weight*]{} is the function $W\colon (-\infty, 1)\rightarrow {\mathbb{R}}_{\geq 0}$ defined by the formula $$W(b) = \sum_{k=1}^\infty w(k(1-b)),$$ where $w$ is one of the above weight funcions. Since the function $w$ have compact support the function $W$ is well-defined.
Let $\nu \colon \coprod \widetilde{B}_i \rightarrow \cup B_i$ be the normalization of the divisor ${\rm supp}(B)$. For any irreducible subvariety $C\subsetneq {\rm supp}(B)$ the preimage $\widetilde{C}=\nu^{-1}(C)$ splits into a finite union of irreducible components $\widetilde{C}_{i,j}\subsetneq \widetilde{B}_i$.
Let $(X,B+M)$ be a ${\mathbb{Q}}$-factorial generalized pair which is terminal. We define $$\begin{aligned}
\label{difficulty}
\overline{\delta}(X,B+M) = \sum W(b_i)\rho(\widetilde{B}_i) + \sum_{{\rm codim}(c_E(X))\neq 2} w(a_E(X,B+M)) \\
\nonumber + \sum_{\stackrel{C\subsetneq Y {\rm irreducible }}{{\rm codim}(C)=2}}
\left[ \sum_{c_E(X)=C} w(a_E(X,B+M)) - \sum_{\widetilde{C}_{i,j}}W(b_i) \right],\end{aligned}$$ where $B=\sum_i b_i B_i$ and the $B_i$’s are pairwise different prime divisors. The function $\overline{\delta}(X,B+M)$ will be called the [*generalized difficulty function*]{} while the function $\delta(X,B+M)=\overline{\delta}(X,B)$ will be called the [*difficulty function*]{}.
\[pos\] Since $(X,B+M)$ is ${\mathbb{Q}}$-factorial and terminal then $(X,B)$ is terminal as a pair. Therefore, by [@AHK Lemma 2.5] we know that $\delta(X,B+M)$ is a well-defined invariant in the sense that only finitely many summands are giving contribution in the formula . Moreover, by [@AHK Lemma 2.6] we know that $\delta(X,B+M) \geq 0$. Observe that for every divisorial valuation $E$ over $X$ we have an inequality $a_E(X,B+M)\leq a_E(X,B)$ and the difference $\overline{\delta}(X,B+M) - \delta(X,B+M)$ is supported in finitely many divisorial valuations over $X$ for which the inequality is strict. Moreover, since $w$ is a decreasing function we have that $\overline{\delta}(X,B+M)\geq \delta(X,B+M)\geq 0$ and that the invariant $\overline{\delta}(X,B+M)$ is finite.
\[welldefined\] Let $f\colon X'\rightarrow X$ be a projective birational morphism, $(X',B'+M')$ and $(X,B+M)$ be two generalized pairs which are ${\mathbb{Q}}$-factorial and terminal, such that $K_{X'}+B'+M'=f^*(K_X+B+M)$ and the divisors $M$ and $M'$ are the trace of a common nef b-divisor on a higher model. Then $\overline{\delta}(X',B'+M')=\overline{\delta}(X,B+M)$.
Let $\widetilde{B_i'} \rightarrow \widetilde{B_i}$ be the morphism induced by the birational map ${\rm supp}(B') \rightarrow {\rm supp}(B)$. The difference between $\overline{\delta}(X',B'+M')$ and $\overline{\delta}(X,B+M)$ is produced by the subvarieties $\widetilde{C'}_{i,j}\subsetneq \widetilde{B_i'}$ whose image on $\widetilde{B_i}$ has codimension at least three on $X$. The contribution of such subvarieties is measured by the first and last summands of the formula of the generalized difficulty function and they cancel out.
Let $(X,B+M)$ be a generalized klt pair. We define the [*generalized difficulty*]{} of $(X,B+M)$ denoted by $\overline{\delta}(X,B+M)$ to be $\overline{\delta}(Y,B_Y+M_Y)$, where $(Y,B_Y+M_Y)$ is any ${\mathbb{Q}}$-factorial generalized terminalization of $(X,B+M)$. By Proposition \[existence\] and Lemma \[welldefined\] the above is well-defined.
The following proposition is a consequence of the existence of generalized terminalizations \[existence\] and the monotonicity properties of the difficulty function proved in [@AHK Lemma 2.10] and [@AHK Lemma 2.11]. The monotonicity properties follow from formal conditions satisfied by the weight functions (see, e.g., [@AHK Conditions 2.1]).
\[evdec\] In any sequence of flips for generalized klt pairs the generalized difficulty function is eventually decreasing.
\[alpha\] Fix a number $\alpha \in (-1,1)$. Then in any sequence of flips for generalized klt pairs there cannot be infinitely many flips for which there exists a divisorial valuation with generalized discrepancy $\alpha$ whose center is in the flipping or flipped locus.
If $\alpha \in (-1,0)$, then we are done because there are finitely many of such divisorial valuations and after a flip the generalized discrepancy strictly increases. So we may assume that $\alpha \in (0,1)$. For the flipping locus we can use the weight function $w_{\alpha}^{-}$ as defined in \[weight\]. After a flip the corresponding generalized discrepancy changes from $a_E(X,B+M)=\alpha$ to $a_E(X^+,B^{+}+M^{+})>\alpha$ so the corresponding difficulty function decreases at least by $1-\alpha$. Then, by Remark \[pos\] and Proposition \[evdec\] we know that this cannot happen infinitely many times. A similar argument works for the flipped locus using the weight function $w_\alpha^+$ instead of $w_\alpha^{-}$.
In order to prove Proposition \[cod2\] we need to prove that eventually all components of codimension two of the flipping and flipped locus have their generic point contained in the smooth locus of the variety. This statement will be proved in Lemma \[nonsing\] using a surface computation. We will need the following version of the ACC for minimal generalized log discrepancies of generalized pairs of dimension two.
\[ACCsurfaces\] Let $\Lambda$ be a DCC set and $p \in {\mathbb{Z}}_{\geq 1}$. The set of minimal generalized log discrepancies of generalized klt pairs $(X,B+M)$ of dimension two, such that the coefficients of the boundary part $B$ belong to $\Lambda$ and the Cartier indices of $M$ and $M'$ are bounded above by $p$, satisfies the ACC.
First, observe that a generalized klt pair $(X,B+M)$ of dimension $2$ is ${\mathbb{Q}}$-factorial. Indeed, the small ${\mathbb{Q}}$-factorialization constructed in Proposition \[existence\] is an isomorphism, therefore the divisor $M$ is ${\mathbb{Q}}$-Cartier and then $(X,B)$ is a klt pair. Assume that there exists a sequence of generalized klt pairs $(X_j,B_j+M_j)$ of dimension two such that their minimal generalized log discrepancies do not belong to a set with the ACC, meaning that they form an infinite strictly increasing sequence in the interval $(0,1)$. Let $(X'_j, B'_j+M'_j)$ be a log resolution of $(X_j, B_j+M_j)$, so that $M'_j$ is a nef divisor and let $E_j$ be the prime divisor on $X'_j$ computing the minimal generalized log discrepancy, then we have that $$a_{E_j}(X_j,B_j+M_j)=a_{E_j}(X_j,B_j) + {\rm coeff}_{E_j}(f_j^*(M_j)) - {\rm coeff}_{E_j}(M'_j).$$ Observe that the set $a_{E_j}(X_j,B_j)$ holds the ACC (see, e.g., [@Ale93]). Moreover, ${\rm coeff}_{E_j}(f_j^*(M_j))$ and ${\rm coeff}_{E_j}(M'_j)$ belong to a discrete family by the bound $p$ on the Cartier indices of the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisor $M_j$ and $M'_j$. Thus, we infer that $a_{E_j}(X_j,B_j+M_j)$ holds the ACC as well.
\[nonsing\] In any sequence of flips for generalized klt pairs, there cannot happen infinitely many times that the flipping or flipped loci contain a component of codimension two which is contained in the singular locus of the generalized pair.
Let $(X_j,B_j+M_j)$ be the generalized klt pairs in the sequence of flips. Denote by $E_1,\dots, E_k$ the non-terminal divisorial valuations of $(X_j,B_j+M_j)$ for $j$ large enough. Assume that $X_j$ is generically singular along a codimension $2$ component of the flipping or flipped locus. By cutting with two generic hyperplanes we obtain a generalized klt surface $(Z,{B_j}|_Z + {M_j}|_Z)$ so that any curve in the terminalization of such surface corresponds to one of the divisors $E_1,\dots, E_k$. Observe that the Cartier index of $M'_j$ is bounded, so the Cartier index of the b-Cartier divisor associated to the nef part of $(Z,{B_j}|_Z+{M_j}|_Z)$ is bounded as well. We claim that the Cartier indices of the divisors $M_j|_Z$ are bounded. Indeed, the coefficients of $M_j|_Z$ belong to a discrete set. Moreover, we may assume that all the klt surfaces $(Z,{B_j}|_Z)$ are $\epsilon$-klt for some $\epsilon>0$ concluding that $M_j|_Z$ have bounded Cartier index (see, e.g., [@Ale93]). Since there are finitely many divisors $E_i$ there is one that appears as the minimal generalized log discrepancy of $(Z,{B_j}|_Z + {M_j}|_Z)$ infinitely many times and strictly increasing at every step, leading to a contradiction by Lemma \[ACCsurfaces\].
Let $C$ be a subvariety of codimension two which is contained in ${\rm supp}(B_j)$. By Lemma \[nonsing\], we may assume that the generic point of $C$ is contained in the smooth locus of $X_j$. The blow-up of $C$ produces a divisor with generalized discrepancy less than one which has the form $1-\sum m_ib_i$, where the $m_i$ are positive integers and the $b_i$ are the coefficients of the boundary part $B_j$ of the generalized pair. There are finitely many of such generalized discrepancies in the interval $(-1,1)$. Thus, we can apply Proposition \[alpha\] to finish the proof.
Termination of quasi-flips on the log canonical centers
-------------------------------------------------------
In this subsection, we prove that any sequence of flips of a pseudo-effective klt $4$-fold $(X,\Delta)$ terminates around $W$, where $W$ is a minimal log canonical center of $(X,\Delta_\lambda)$ for $\lambda \in (0,\lambda_3)\cap {\mathbb{Q}}$.
By Proposition \[qflipslcp\], in order to prove termination around $W$ it is enough to prove that any sequence of strict ample quasi-flips for generalized log canonical pairs terminates in dimension at most three. We start proving that any such sequence terminates in codimension one in the klt case.
We say that a sequence of birational transformations [*terminates in codimension one*]{} if after finitely many birational transformations all maps are isomorphisms in codimension one. In particular, a sequence of quasi-flips terminates in codimension one, if after finitely many quasi-flips all quasi-flips are weak.
\[tercod1\] Any sequence of strict ample klt quasi-flips for generalized log canonical pairs under a DCC set terminates in codimension one.
We claim that any divisor which is extracted in the sequence of quasi-flips has discrepancy at most zero. Indeed, the generalized discrepancy along the generic point of an irreducible component of the boundary part with coefficient $0 < b_i < 1$ is $-b_i$. Therefore such irreducible divisor is a non-terminal valuation over all previous models in the sequence of quasi-flips. We conclude that there are finitely many divisorial valuations that can be extracted in the sequence of strict ample quasi-flips.
Now, we prove that each of these finitely many divisorial valuations can be extracted at most finitely many times. Indeed, if a non-terminal valuation $E$ is extracted by the quasi-flip $\pi_{j_1}$ and $\pi_{j_2}$ for $j_1 < j_2$ we have that there exists a quasi-flip between $\pi_{j_1}$ and $\pi_{j_2}$ that is contracting such non-terminal valuation, therefore the generalized log discrepancy of the pairs $(X_j,B_j+M_j)$ strictly increases at $E$, which means that the coefficient at $E$ of $B_{j_1+1}$ is strictly greater than the coefficient at $E$ of $B_{j_2+1}$. Since the sequence of quasi-flips is under a DCC set the coefficients of $B_j$ belong to a DCC set, so we deduce that this can only happen finitely many times, meaning that every non-terminal valuation can be extracted only finitely many times.
We conclude that there are finitely many divisorial valuations that can be extracted in the sequence of quasi-flips, and each of these can be extracted at most finitely many times. Thus, after finitely many quasi-flips all quasi-flips do not extract divisors. By induction on the Picard rank of $X_j$, we conclude that after finitely many quasi-flips both the flipping and flipped contractions are isomorphisms in codimension one.
Recall from [@KMM87 Lemma 5.1.7] that given a flip $\pi_{j+1}\colon (X_j,\Delta_j) \dashrightarrow (X_{j+1},\Delta_{j+1})$ of klt pairs we have that ${\rm dim}({\rm Ex}(\phi_{j+1}))+{\rm dim}({\rm Ex}(\phi_{j+1}^+)) \geq {\rm dim}(X_j)-1$. In the case that $X_j$ is a $4$-fold we infer that the possible choices for the pair $({\rm dim}({\rm Ex}(\phi_{j+1})), {\rm dim}({\rm Ex}(\phi_{j+1}^+)))$ are $(2,1),(1,2)$, and $(2,2)$. Observe that we can use Proposition \[cod2\] to prove that the sequence of quas-flips induced on every log canonical center of codimension one of the pairs $(X,\Delta_\lambda)$ terminates in codimension one. Indeed, any such log canonical center is of the form $G_i$ for some $i\in\{1,\dots,k_0\}$. The prime divisor $G_i$ appears with coefficient one in the boundary part of the generalized log canonical pair $(c+1)(K_X+\Delta)$, therefore we can apply the proposition to conclude that eventually no component of codimension two of the flipping locus or the flipped locus in contained in the strict transform of $G_i$. In particular, after finitely many flips the generic point of any component of codimension two of the flipping or flipped locus of a flip of type $(2,2)$ lies in the complement of the strict transform of $G_i$.
\[kltlocus\] Given a generalized log canonical pair $(X,B+M)$, we denote by $X^0$ the complement of the non-klt locus of $X$. We will write $B^0$ and $M^0$ for the restriction of $B$ and $M$ to $X^0$, respectively. By definition the generalized pair $(X^0,B^0+M^0)$ is klt. The variety $X^0$ will often be called the [*klt locus*]{} of $(X,B+M)$.
\[tersurfaces\] Any sequence of strict ample quasi-flips for generalized log canonical pairs of dimension two under a DCC set terminates.
By Lemma \[tercod1\], it suffices to show that eventually such quasi-flips are klt in the sense of \[kltflip\]. In order to do so, we need to prove a special termination around the generalized log canonical centers (see [@Fuj07 Theorem 2.1] for the dlt case, and [@Shok04 Corollary 4] for the lc case). If the generalized log canonical center has dimension zero then a quasi-flip is either disjoint from the center or contains such center, so after finitely many quasi-flips we can assume that no quasi-flip contains a generalized log canonical center of dimension zero. If the generalized log canonical center has dimension one we can use divisorial adjunction for generalized pairs to induce a generalized pair on this curve, such that the coefficients of its boundary part belong to a DCC set (see, e.g., [@Bir17 Lemma 3.3]). By strict monotonicity \[monotonicity\], we conclude that every quasi-flip which intersect this curve in a zero-dimensional locus strictly decrease such coefficients. Thus, after finitely many flips a one-dimensional log canonical center is either disjoint from the flipping locus or is contained in it. By finiteness of log canonical centers, we deduce that eventually all flips are disjoint from the non-klt locus of the generalized pair.
\[ter3folds\] Any sequence of strict ample quasi-flips for generalized log canonical pairs of dimension three under a DCC set terminates.
First, we reduce to the case of klt quasi-flips. We proceed by proving special termination around the generalized log canonical centers. If the log canonical center has dimension zero or one, then the argument is analogous to the one in the proof of Corollary \[tersurfaces\]. If the log canonical center has dimension two then we can use divisorial adjunction for generalized pairs (see [@Bir17 Section 3]) to obtain a sequence of strict ample quasi-flips for generalized log canonical surfaces under a DCC set (see [@Bir17 Lemma 3.3]). Therefore, termination around such generalized log canonical center follows from Corollary \[tersurfaces\]. Thus, after finitely many quasi-flips all flipping loci are disjoint from the non-klt locus of the generalized pair. So the quasi-flips are klt in the sense of \[kltflip\].
Now, it suffices to prove that a sequence of strict ample klt quasi-flips for generalized lc pairs of dimension three under a DCC set terminates. By Lemma \[tercod1\], we conclude that such sequence terminates in codimension one so we reduce to the case of strict ample weak klt quasi-flips. Consider a strict ample weak klt quasi-flip for generalized log canonical pairs $\pi \colon (X,B+M)\dashrightarrow (X_1,B_1+M_1)$ with flipping contraction $\phi \colon X \rightarrow Z$. By Lemma \[existence\], we can take a small ${\mathbb{Q}}$-factorialization $(X',B'+M')$ of the klt locus of $(X,B+M)$ and by [@BZ16 Section 4] we can run a relative minimal model program for $(X',B'+M')$ over $Z$ to produce a minimal model $(X'_1,B'_1+M'_1)$ over $Z$, which is a small ${\mathbb{Q}}$-factorialization of the klt locus of $(X_1,B_1+M_1)$. Moreover, $(X_1,B_1+M_1)$ is the ample model of $(X',B'+M')$ over $Z$. Thus, by taking small ${\mathbb{Q}}$-factorializations we reduce to prove termination of klt flips for ${\mathbb{Q}}$-factorial generalized log canonical $3$-folds.
We claim that the ${\mathbb{Q}}$-Cartier ${\mathbb{Q}}$-divisors $K_{X^0_j}+B^0_j+M^0_j$ have bounded Cartier index independent of $j\in \mathbb{Z}_{\geq 1}$. Indeed, by Corollary \[boundN\] we know that there exists $\epsilon>0$ and $N\in \mathbb{Z}_{\geq 1}$ such that the ${\mathbb{Q}}$-factorial generalized klt $3$-folds $(X^0_j,B^0_j+M^0_j)$ have at most $N$ divisorial valuations with generalized discrepancy in the interval $(-1,\epsilon)$. In particular, the ${\mathbb{Q}}$-factorial klt pairs $(X^0_j,B^0_j)$ have at most $N$ divisorial valuations with generalized discrepancy in the interval $(-1,\epsilon)$. Then, we can apply [@Shok04 Lemma 4.4.1] to deduce that the ${\mathbb{Q}}$-divisors $$K_{X^0_j}+B^0_j+M^0_j$$ have bounded Cartier index. Therefore, since the flipping locus of $\pi_j$ is contained in $X_j^0$, there exists $\alpha>0$ independent of $j\in \mathbb{Z}_{\geq 1}$, such that the generalized discrepancy of every divisorial valuation over the generalized pair $(X_j,B_j+M_j)$ increases at least by $\alpha$ when its center is contained in the flipping or flipped locus.
Now, we reduce to the case of terminal flips. Recall that the generalized pairs $(X_j,B_j+M_j)$ have finitely many non-terminal divisorial valuations over $X^0_j$. By the existence of $\alpha>0$, we know that eventually no flip contains the center of a generalized non-terminal valuation on its flipping locus. Thus, the coefficients of the terminalizations of $(X^0_j,B^0_j+M^0_j)$ stabilize for $j$ large enough. By Proposition \[existence\], we can take a ${\mathbb{Q}}$-factorial terminalization of the klt locus of $(X_j,B_j+M_j)$ to reduce to the case of terminal flips for generalized log canonical $3$-folds.
We claim that we may assume that $B_j=0$. Indeed, if the flipping locus intersects $B_j$ positively then the flipped locus has a component of codimension two which is contained in $B_{j+1}$. On the other hand, if the flipping locus intersects $B_j$ non-positively then it is either contained in $B_j$ or disjoint from $B_j$. By Proposition \[cod2\], we deduce that eventually the flipping and flipped locus are disjoint from $B_j$.
Therefore, we obtain a sequence of terminal flips for ${\mathbb{Q}}$-factorial generalized lc $3$-folds $(X_j,M_j)$. By Lemma \[finitediv\] the generalized pair $(X^0_j,M^0_j)$ has finitely many divisorial valuations with generalized discrepancy in the interval $(0,1)$, and every such divisorial valuation increases its generalized discrepancy at least by $\alpha>0$ when its center is contained in the flipping or flipped locus. We conclude that eventually no flip contains the center of a divisorial valuation with generalized discrepancy in the interval $(0,1)$ in its flipping or flipped locus. Since $a_{E}(X_j) \geq a_{E}(X_j,M_j)$ for every divisorial valuation $E$ over $X_j$, we deduce that after finitely many flips, no flip contains in its flipping or flipped locus the center of a divisorial valuation with discrepancy in the interval $(0,1)$.
We reduce to the case of $K_{X_j}$-flops. Consider a flip $\pi_{j+1} \colon (X_j,M_j) \dashrightarrow (X_{j+1},M_{j+1})$ which is $K_{X_j}$-negative, meaning that $\pi_{j+1}\colon X_j \dashrightarrow X_{j+1}$ is a terminal $3$-fold flip. By the classification of terminal $3$-fold extremal contractions (see [@Mor88] or [@KM92]), we conclude that the flipping locus of $\pi_{j+1}$ must contain a terminal singular point of Cartier index $r>1$ and therefore by [@CH11 Theorem 2.9] it must contain the center of a divisorial valuation with discrepancy $0<\frac{1}{r}<1$ with respect to $K_{X_j}$, giving a contradiction. If the flip $\pi_{j+1}$ is $K_{X_j}$-positive, meaning that $\pi_{j+1}^{-1} \colon X_{j+1} \dashrightarrow X_j$ is a flip of terminal $3$-folds, the same argument applied to the flipped locus leads to a contradiction. Thus, we reduce to termination of terminal flips for ${\mathbb{Q}}$-factorial generalized lc pairs $(X_j,M_j)$ such that all such flips are $K_{X_j}$-flops.
Finally, we prove that the sequence of $(K_{X_j}+M_j)$-flips which are $K_{X_j}$-flops terminates. For every divisorial valuation $E$ over $X^0_j$ we define the number $$\eta_{E}(X_j,M_j) = a_{E}(X_j) - a_{E}(X_j,M_j) \geq 0.$$ Observe that such number is always positive by the negativity lemma and it decreases at least by $\alpha >0$ when the center of $E$ on $X_j$ is contained in the flipping or flipped locus of the flip for the generalized pair. Therefore, a curve $C\subseteq {\rm Bs}_{-}(M_j)$ which contains the center $c_E(X_j)$ of a divisorial valuation $E$ with $\eta_E(X_j,M_j)<\alpha$ cannot be contained in the flipping locus of a $(K_{X_j}+M_j)$-flip. Since $M_j$ is the push-forward of a nef divisor on a higher birational model then its diminished base locus consists of finitely many curves. Recall from [@Koll90 Proposition 2.1.12] that a terminal $3$-fold $K_{X_j}$-flop has the same number of irreducible curves in its flopping and flopped locus, so the number of irreducible components of the diminished base locus of $M_j$ can only decrease.
We proceed by induction on the number $k_j=k(X_j,M_j)$ of curves in the diminished base locus of $M_j$ which are contained in the flipping locus of $\pi_i$ for some $i\geq j+1$. If $k_j=1$ for some $j\in \mathbb{Z}_{\geq 1}$ we deduce that the $(K_{X_j}+M_j)$-flip contains a single curve in the flipping locus and there are no further flips. Observe that $k_{j+1}=k_j$ if and only if all the curves contained in the flipped locus of $\pi_{j+1}$ are contained in the flipping locus of $\pi_i$ for some $i\geq j+2$. Let $j_0\in \mathbb{Z}_{\geq 1}$ and assume that $k_{j_0}\geq 2$. Consider a curve $C\subsetneq {\rm Bs}_{-}(M_{j_0})$ contained in the flipping locus of $\pi_{j_0+1}$. Therefore, we have that $M_{j_0} \cdot C <0$, so there exists a divisorial valuation $E$ over $X_{j_0}^0$, such that $\eta_E(X_{j_0},M_{j_0})>0$ and $c_E(X_{j_0},M_{j_0})=C$. Since the center of $E$ on $X_{j_0}$ is contained in $C$ we conclude that the center of $E$ on $X_{{j_0}+1}$ is contained in the flipped locus of $\pi_{{j_0}+1}$. If $k_{{j_0}+1}=k_{j_0}$, we deduce that the center of $E$ is contained in the flipping locus of $\pi_{j_1}$ for some $j_1 \geq j_0+2$. Analogously, if $k_{{j_1}+1}=k_{j_1}$ we have that the center of $E$ is contained in the flipping locus of $\pi_{j_2}$ for some $j_2 \geq j_1+2$. Inductively, we produce a sequence of flips $\pi_{j_k}$ which contain the center of $E$ in their flipping loci. Since $\eta_E(X_{j_k},M_{j_k})$ is non-negative and decreases at least by $\alpha>0$ when its center is contained in a flipping locus, we conclude that such sequence is finite, meaning that there is $i\in \mathbb{Z}_{\geq 1}$ such that the center of $E$ is contained in the flipping locus of $\pi_{j_{i}}$ but is not contained in the flipping locus of $\pi_{l}$ for every $l\geq j_i+1$. Therefore, there exists a curve in the flipped locus of $\pi_{j_{i}}$ which contains the center of $E$ and this curve is not contained in the flipping locus of $\pi_{l}$ for every $l \geq j_i+1$. Thus, we have that $k_{j_i+1}<k_{j_i}$ and we obtain termination by the inductive hypothesis.
Proof of the theorem {#section:acc}
====================
In this section, we prove the main theorem. First, we prove two lemmas which will be used in the proof of the theorem.
\[infinitelct\] If $K_X+\Delta$ is not nef, then the log canonical pairs $(X,\Delta_\lambda)$ have at least one common log canonical center, which is a generalized log canonical center of the generalized log canonical pair $(c+1)(K_X+\Delta)$, where $c={\rm lct}(K_X+\Delta)$.
By Proposition \[stab\], we can take a log resolution $Y$ of the minimal model program with scaling of an ample divisor $\pi \colon X \dashrightarrow X_{\rm min}$, such that $c_\lambda$ is the maximum positive real number $\mu$ for which $K^{\Delta}_{Y/X}-\mu E_\lambda$ has coefficients greater or equal than negative one. Moreover, we know that the coefficients at the prime irreducible components of $K^{\Delta}_{Y/X}- cE_\lambda$ are linear functions on $c$ and $\lambda$. Since $K_X+\Delta$ is not nef, we can use Corollary \[finiteness\] to deduce that ${\rm lct}(K_X+\Delta)$ is finite, so that $c_\lambda$ is finite for $\lambda$ small enough, which means that the log canonical pairs $(X,\Delta_\lambda)$ have at least one common log canonical center. The log canonical place on $Y$ corresponding to such common log canonical center, is an irreducible component with coefficient negative one of $K^{\Delta}_{Y/X}- c_\lambda E_\lambda$, so we conclude that it is a component with coefficient negative one of $K^{\Delta}_{Y/X}- c E$. By Proposition \[valueoflct\], we deduce that such common log canonical center of $(X,\Delta_\lambda)$ is a generalized log canonical center of the generalized log canonical pair $(c+1)(K_X+\Delta)$.
\[acc\] Assume that $(X,\Delta)$ has a minimal model $(X_{\rm min},\Delta_{\rm min})$. Let $\pi_{j+1} \colon (X_j,\Delta_j)\rightarrow (X_{j+1},\Delta_{j+1})$ be a sequence of flips for $K_X+\Delta$. Then, the sequence ${\rm lct}(K_{X_j}+\Delta_j)$ is an increasing sequence satisfying the ACC.
First, we prove that the sequence ${\rm lct}(K_{X_j}+\Delta_j)$ is increasing. We can consider a common log resolution $Y$ of $(X_j,\Delta_j),(X_{j+1},\Delta_{j+1})$ and $(X_{\rm min},\Delta_{\rm min})$, with projective birational morphisms $p_j,p_{j+1}$ and $q$. By monotonicity \[monotonicity\], we know that we can write $$p_j^*(K_{X_j}+\Delta_j) = q^*(K_{X_{\rm min}}+\Delta_{\rm min}) + E_j$$ and $$p_{j+1}^*(K_{X_{j+1}}+\Delta_{j+1}) = q^*(K_{X_{\rm min}}+\Delta_{\rm min}) + E_{j+1},$$ where $E_j \geq E_{j+1}$. By Proposition \[valueoflct\], we conclude that ${\rm lct}(K_{X_j}+\Delta_j) \leq {\rm lct}(K_{X_{j+1}}+\Delta_{j+1})$. Now we turn to prove that the sequence ${\rm lct}(K_{X_j}+\Delta_j)$ satisfies the ACC. Recall that we can consider $K_{X_j}+\Delta_j$ as the boundary of a generalized pair by writting $$K_{X_j}+\Delta_j = {p_j}_*( q^*(K_{X_{\rm min}}+\Delta_{\rm min})) + {p_j}_*(E_j),$$ where ${p_j}_*(E_j)$ is the boundary part. Observe that the Cartier index of the divisor $q^*(K_{X_{\rm min}}+\Delta_{\rm min})$ is independent of $j$. We claim that the coefficients of the prime components of ${p_j}_*(E_j)$ belong to a finite set which is independent of $j$. Indeed, for every irreducible component $E$ of the exceptional locus of the rational map $X\dashrightarrow X_{\rm min}$, the coefficient of ${p_j}_*(E_j)$ at $E$ equals $$a_E(X_{\rm min}, \Delta_{\rm min}) - a_E(X_j,\Delta_j)$$ and such numbers are independent of $j$ in a sequence of flips. Therefore, we can apply [@BZ16 Theorem 1.5] to conclude that the real numbers $${\rm lct}(K_{X_j}+\Delta_j)={\rm lct}((K_{X_j}+\Delta_j),{p_j}_*( q^*(K_{X_{\rm min}}+\Delta_{\rm min})) + {p_j}_*(E_j))$$ satisfies the ACC.
Consider a sequence of $(K_X+\Delta)$-flips as follows $$\label{sequenceofflips}
\xymatrix{
(X,\Delta)\ar@{-->}^-{\pi_1}[r] & (X_1,\Delta_1)\ar@{-->}^-{\pi_2}[r] & (X_2,\Delta_2)\ar@{-->}^-{\pi_3}[r] & \dots \ar@{-->}^-{\pi_j}[r] & (X_j,\Delta_j)\ar@{-->}^-{\pi_{j+1}}[r] & \dots \\
}$$ By special termination around the log canonical center of the log canonical pair $(X,\Delta)$ (see, e.g., [@Shok04 Corollary 4]) we deduce that after finitely many flips all flips are disjoint from the strict transform of the log canonical centers of the pair $(X,\Delta)$. Then, we obtain a sequence of klt flips for log canonical $4$-folds $(X,\Delta)$. Hence, without loss of generality we may assume that the pair $(X,\Delta)$ is itself klt. By Lemma \[infinitelct\], we may assume that ${\rm lct}(K_X+\Delta)=c$ is finite. We claim that for $j\in \mathbb{Z}_{\geq 1}$ large enough, all flipping loci are disjoint from the generalized non-klt locus of $(c+1)(K_{X_j}+\Delta_j)$. Indeed, by Lemma \[infinitelct\] we know that the pairs $(X,\Delta_\lambda)$ have a common minimal log canonical center $W_i$ which is also a log canonical center of the generalized log canonical pair $(c+1)(K_X+\Delta)$. If $W_i$ has dimension zero, termination around $W_i$ follows from monotonicity of discrepancies \[monotonicity\]. If $W_i$ has dimension one, then by Proposition \[adjunction\] we can apply adjunction for $(c+1)(K_X+\Delta)$ to $W_i$ and by Proposition \[wehaveDCC\] we know that the induced boundary part belong to a DCC set. So termination around $W_i$ follows from the strict monotonicity \[monotonicity\]. If $W_i$ has dimension at least two, then by Proposition \[qflipslcp\] and Proposition \[wehaveDCC\] we obtain an induced sequence of strict ample quasi-flips of generalized log canonical pairs on $W_i$ under a DCC set. If $W_i$ has dimension two, by Corollary \[tersurfaces\] we conclude that such sequence of birational transformations terminates around $W_i$, on the other hand if $W_i$ has dimension three, the same statement holds by Proposition \[ter3folds\]. Thus, all flips are eventually disjoint from the strict transform of $W_i$ or there is a flip which contains the strict transform of $W_i$ in its flipping locus. In the latter case the strict transform of $W_i$ is not a non-klt center of $(c+1)(K_{X_j}+\Delta_j)$ for $j$ large enough. Thus, we deduce that eventually all flips are disjoint from the strict transform of $W_i$.
We conclude that after finitely many flips, all flipping loci are disjoint from the strict transform of $W_i$. Replacing $X$ with the complement of $W_i$ on $X$ we achieve that the number of generalized log canonical centers of $(c+1)(K_X+\Delta)$ strictly decrease after finitely many flips. Meaning that for $j$ large enough the number of log canonical centers of $(c+1)(K_{X_j}+\Delta_j)$ is strictly less than the number of log canonical centers of $(c+1)(K_X+\Delta)$. Arguing similarly for the generalized log canonical pair $(c+1)(K_{X_j}+\Delta_j)$, we deduce that after finitely many flips the number of log canonical centers of $(c+1)(K_{X_j}+\Delta_j)$ strictly decrease again. Since $(c+1)(K_X+\Delta)$ has finitely many log canonical centers, we conclude that there exists $j_0\in \mathbb{Z}_{\geq 1}$, such that $(c+1)(K_{X_j}+\Delta_j)$ is a generalized klt pair. Hence, we conclude that ${\rm lct}(K_X+\Delta)<{\rm lct}(K_{X_j}+\Delta_j)$.
Proceeding inductively, we produce a sequence of klt $4$-folds $(X_j,\Delta_j)$, such that $${\rm lct}(K_X+\Delta) < {\rm lct}(K_{X_1}+\Delta_1) < \dots < {\rm lct}(K_{X_j}+\Delta_j) < \dots$$ is an ascending sequence of real numbers. By Lemma \[acc\], we conclude that such sequence must be finite. Therefore, the sequence of flips is finite.
[^1]: The author was partially supported by NSF research grants no: DMS-1300750, DMS-1265285 and by a grant from the Simons Foundation; Award Number: 256202
|
---
abstract: 'We describe a simple $n$-dimensional quantum cellular automaton (QCA) capable of simulating all others, in that the initial configuration and the forward evolution of any $n$-dimensional QCA can be encoded within the initial configuration of the intrinsically universal QCA. Several steps of the intrinsically universal QCA then correspond to one step of the simulated QCA. The simulation preserves the topology in the sense that each cell of the simulated QCA is encoded as a group of adjacent cells in the universal QCA.'
author:
- Pablo Arrighi
- Jonathan Grattage
bibliography:
- 'biblio.bib'
title: 'A Simple $n$-Dimensional Intrinsically Universal Quantum Cellular Automaton'
---
Introduction {#sec:introduction}
============
\[subsec:CA\]
Cellular automata (CA), first introduced by Von Neumann [@Neumann], consist of an array of identical cells, each of which may take one of a finite number of possible states. The whole array evolves in discrete time steps by iterating a function $G$. This global evolution $G$ is shift-invariant (it acts everywhere the same) and local (information cannot be transmitted faster than some fixed number of cells per time step). Because this is a physics-like model of computation [@MargolusPhysics], Feynman [@FeynmanQCA], and later Margolus [@MargolusQCA], suggested that quantising this model was important, for two reasons: firstly, because in CA computation occurs without extraneous (unnecessary) control, hence eliminating a source of decoherence; and secondly because they are a good framework in which to study the quantum simulation of a quantum system. From a computation perspective there are other reasons to study QCA, such as studying space-sensitive problems in computer science, [*e.g. *]{}‘machine self-reproduction’ [@Neumann] or ‘Firing Squad Synchronisation’, which QCA allow in the quantum setting. There is also a theoretical physics perspective, where CA are used as toy models of quantum space-time [@LloydQG]. The first approach to defining QCA [@ArrighiMFCS; @DurrWell; @Watrous] was later superseded by a more axiomatic approach [@ArrighiUCAUSAL; @ArrighiLATA; @SchumacherWerner] together with the more operational approaches [@BrennenWilliams; @NagajWocjan; @PerezCheung; @Raussendorf; @VanDam; @Watrous].
The most well known CA is Conway’s ‘Game of Life’, a two-dimensional CA which has been shown to be universal for computation, in the sense that any Turing Machine (TM) can be encoded within its initial state and then executed by evolution of the CA. Because TM have long been regarded as the best definition of ‘what an algorithm is’ in classical computer science, this result could have been perceived as providing a conclusion to the topic of CA universality. This was not the case, because CA do more than just running any algorithm. They run distributed algorithms in a distributed manner, model phenomena together with their spatial structure, and allow the use of the spatial parallelism inherent to the model. These features, modelled by CA and not by TM, are all interesting, and so the concept of universality must be revisited in this context to account for space. This is achieved by returning to the original meaning of the word *universality* [@AlbertCulik; @Banks; @DurandRoka], namely the ability for one instance of a computational model to be able to simulate other instances of the same computational model. Intrinsic universality formalises the ability of a CA to simulate another in a space-preserving manner [@MazoyerRapaport; @OllingerJAC; @Theyssier], and was extended to the quantum setting in [@ArrighiUQCA; @ArrighiNUQCA; @ArrighiPQCA].
There are several related results in the CA literature. For example, refs. [@MargolusPhysics; @MoritaCompUniv1D; @MoritaCompUniv2D] provide computation universal Reversible Partitioned CA constructions, whereas ref. [@MoritaIntrinsicUniv1D] deals with their ability to simulate any CA in the one-dimensional case. The problem of minimal intrinsically universal CA was addressed, [*cf. *]{}[@OllingerRichard], and for Reversible CA (RCA) the issue was tackled by Durand-Lose [@Durand-LoseLATIN; @Durand-LoseIntrinsic1D]. The difficulty is in having an $n$-dimensional RCA simulate all other $n$-dimensional RCA and not, say, the $(n-1)$-dimensional RCA, otherwise a history-keeping dimension could be used, as by Toffoli [@ToffoliConstruction]. There are also several other QCA related results. Watrous [@WatrousFOCS] has proved that QCA are universal in the sense of QTM. Shepherd, Franz and Werner [@ShepherdFranz] defined a class of QCA where the scattering unitary $U_i$ changes at each step $i$ (CCQCA). Universality in the circuit-sense has already been achieved by Van Dam [@VanDam], Cirac and Vollbrecht [@VollbrechtCirac], Nagaj and Wocjan [@NagajWocjan] and Raussendorf [@Raussendorf]. In the bounded-size configurations case, circuit universality coincides with intrinsic universality, as noted by Van Dam [@VanDam]. QCA intrinsic universality in the one-dimensional case is resolved in ref. [@ArrighiFI]. Given the crucial role of this in classical CA theory, the issue of intrinsic universality in the $n$-dimensional case began to be addressed in refs. [@ArrighiNUQCA; @ArrighiPQCA], where it was shown that a simple subclass of QCA, namely Partitioned QCA (PQCA), are intrinsically universal. Having shown that PQCA are intrinsically universal, it remains to be shown that there exists a $n$-dimensional PQCA capable of simulating all other $n$-dimensional PQCA for $n>1$, which is presented here.
PQCA are QCA of a particular form, where incoming information is scattered by a fixed unitary $U$ before being redistributed. Hence the problem of finding an intrinsically universal PQCA reduces to finding some universal scattering unitary $U$ (this is made formal in section \[subsec:flat\], see Fig.\[fig:flattening34\]). Clearly the universality requirement on $U$ is much more difficult than just quantum circuit universality. This is because the simulation of a QCA $H$ has to be done in a parallel, space-preserving manner. Moreover we must simulate not only an iteration of $H$ but several ($H^2$, …), so after every simulation the universal PQCA must be ready for a further iteration.
From a computer architecture point of view, this problem can be recast in terms of finding some fundamental quantum processing unit which is capable of simulating any other network of quantum processing units, in a space-preserving manner. From a theoretical physics perspective, this amounts to specifying a scattering phenomenon that is capable of simulating any other, again in a space-preserving manner.
An Intrinsically Universal QCA {#sec:nuqca}
==============================
The aim is to find a particular $U$-defined PQCA which is capable of intrinsically simulating any $V$-defined PQCA, for any $V$. In order to describe such a $U$-defined PQCA in detail, two things must be given: the dimensionality of the cells (including the meaning attached to each of the states they may take), and the way the scattering unitary $U$ acts upon these cells. [First we discuss the general scheme used to solve this problem, and then describe the PQCA implementing it.]{} [The necessary definitions for $n$-dimensional QCA are given in refs. [@ArrighiNUQCA; @ArrighiPQCA].]{}
Circuit Universality versus Intrinsic Universality in Higher Dimensions
-----------------------------------------------------------------------
As already discussed, intrinsic universality refers to the ability for one CA to simulate any other CA [in a way which preserves the spatial structure of the simulated CA. Conversely, computation universality refers to the ability of a CA to simulate any TM, and hence run any algorithm.]{} Additionally, circuit universality is the ability of one CA to simulate any circuit. [These are <span style="font-variant:small-caps;">Nand</span> gate circuits for classical circuits and CA, and <span style="font-variant:small-caps;">Toffoli</span> gate circuits for reversible circuits and CA.]{} Informally, in a quantum setting, circuit universality is the ability of a PQCA to simulate [any unitary evolution expressed as a]{} combination of a universal set of quantum gates, such as the standard gate set: <span style="font-variant:small-caps;">Cnot, [R($\frac{\pi}{4})$]{}</span> (also known as the $\frac{\pi}{8}$ gate), and the <span style="font-variant:small-caps;">Hadamard</span> gate.
The relationships between these three concepts of CA universality have been noted previously [@DurandRoka]. A computation universal CA is also a circuit universal CA, because circuits are finitary computations. Moreover, an intrinsic universal CA is also a computation universal CA, because it can simulate any CA, including computation universal CA. Hence intrinsic universality implies computation universality, which implies circuit universality.
In one-dimension this is not an equivalence. Intuitively, computation universality requires more than circuit universality, namely the ability to loop the computation, which is not trivial for CA. Similarly, intrinsic universality requires more than computation universality, such as the ability to simulate multiple communicating TM. In the classical setting there are formal results that distinguish these ideas [@OllingerJAC].
In $n$-dimensions, it is often assumed in the classical CA literature that circuit universality implies intrinsic universality, and [hence these are all equivalent]{} [@OllingerJAC]. Strictly speaking this is not true. Consider a two-dimensional CA which runs one-dimensional CA in parallel. If the one-dimensional CA is circuit/computation universal, but not computation/intrinsically universal, then this is also true for the two-dimensional CA. Similarly, in the PQCA setting, the two-dimensional constructions in [@PerezCheung] and [@Raussendorf] are circuit universal but not intrinsically universal.
However, this remains a useful intuition: Indeed, CA admit a block representation, where these blocks are permutations for reversible CA, while for PQCA the blocks are unitary matrices. Thus the evolution of any (reversible/quantum) CA can be expressed as an infinite (reversible/quantum) circuit of (reversible/ quantum) gates repeating across space. If a CA is circuit universal, and if it is possible to wire together different circuit components in different regions of space, then the CA can simulate the block representation of any CA, and hence can simulate any CA in a way which preserves its spatial structure. It is intrinsically universal. [ This is the route followed next in constructing the intrinsically universal $n$-dimensional PQCA. First the construction of the ‘wires’, which can carry information across different regions of space, is considered. Here these are signals which can be redirected or delayed using barriers, with each signal holding a qubit of information. Secondly, the ‘circuit-pieces’ are constructed, by implementing quantum gates which can be combined. One and two qubit gates are implemented as obstacles to, and interactions of, these signals.]{}
Flattening a PQCA into Space {#subsec:flat}
----------------------------
[In the classical CA literature it is considered enough to show that the CA implements some wires carrying signals, and some universal gates acting upon them, to prove that an $n$-dimensional CA is in fact intrinsically universal. ]{} Any CA can be encoded into a ‘wire and gates’ arrangement following the above argument, but this has never been made explicit in the literature. This section makes more precise how to flatten any PQCA in space, so that it is simulated by a PQCA which implements quantum wires and universal quantum gates. Flattening a PQCA means that the infinitely repeating, two-layered circuit is arranged in space so that at the beginning all the signals carrying qubits find themselves in circuit-pieces which implement a scattering unitary of the first layer, and then all synchronously exit and travel to circuit-pieces implementing the scattering unitary of the second layer, etc. An algorithm for performing this flattening can be provided, however the process will not be described in detail, for clarity and following the classical literature, which largely ignores this process.
The flattening process can be expressed in three steps: Firstly, the $V$-defined PQCA is expanded in space by coding each cell into a hypercube of $2^n$ cells. This allows enough space for the scattering unitary $V$ to be applied on non-overlapping hypercubes of cells, illustrated in the two-dimensional case in Fig. \[fig:flattening12\].
[![Flattening a PQCA into a simulating PQCA. *Left*: Consider four cells (white, light grey, dark grey, black) of a PQCA having scattering unitary $V$. The first layer PQCA applies $V$ to these four cells, then the second layer applies $V$ at the four corners. *Right*: We need to flatten this so that the two-layers become non-overlapping. The first layer corresponds to the centre square, and the second layer to the four corner squares. At the beginning the signals (white, light grey, dark grey, black) coding for the simulated cells are in the centre square. [They undergo $V$, and are directed towards the bottom left, top left, bottom right, and top right squares respectively, where they undergo $V$ but paired up with some other signals, etc.]{} \[fig:flattening12\]](img/flattening1and2.pdf "fig:")]{}
Secondly, the hypercubes where $V$ is applied must be connected with wires, as shown in Fig. \[fig:flattening12\] $(right)$. Within these hypercubes wiring is required so that incoming signals are bunched together to undergo a circuit implementation of $V$, and are then dispatched appropriately, as shown in Fig. \[fig:flattening34\] $(left)$. This requires both time and space expansions, with factors that depend non-trivially (but uninterestingly) upon the size of the circuit implementation of $V$ and the way the wiring and gates work in the simulating PQCA.
[![Flattening a PQCA into a simulating PQCA (cont’d). *Left*: Within the central square the incoming signals are bunched together so as to undergo a circuit which implements $V$, and are then dispatched towards the four corners. This diagram does not make explicit a number of signal delays, which may be needed to ensure that they arrive synchronously at the beginning of the circuit implementing $V$. *Right*: Within the central rectangle, the circuit which implements $V$ is itself a combination of smaller circuits for implementing a universal set of quantum gates such as <span style="font-variant:small-caps;">Cnot</span>, <span style="font-variant:small-caps;">Hadamard</span> and the <span style="font-variant:small-caps;">[R($\frac{\pi}{4})$]{}</span>, together with delays. [These are implemented as explained in sections \[subsec:onequbit\] and \[subsec:gates\].]{}\[fig:flattening34\]](img/flattening3and4.pdf "fig:")]{}
Next, an encoding of the circuit description of the scattering unitary $V$ is implemented in the simulating PQCA upon these incoming bunched wires, as shown in Fig. \[fig:flattening34\] $(right)$. This completes the description of the overall scheme according to which a PQCA that is capable of implementing wires and gates is also capable of intrinsically simulating any PQCA, and hence any QCA. A particular PQCA that supports these wires and gates can now be constructed.
Barriers and Signals Carrying Qubits {#subsec:onequbit}
------------------------------------
Classical CA studies often refer to ‘signals’ without an explicit definition. In this context, a signal refers to the state of a cell which may move to a neighbouring cell consistently, from one step to another, by the evolution of the CA. Therefore a signal would appear as a line in the space-time diagram of the CA. These lines need to be implemented as signal redirections. A $2$D solution is presented here, but this scheme can easily be extended to higher dimensions. Each cell has four possible basis states: empty ($\epsilon$), holding a qubit signal ($0$ or $1$), or a barrier ($\blacksquare$). The scattering unitary $U$ of the universal PQCA acts on $2\times 2$ cell neighbourhoods.
Signals encode qubits which can travel diagonally across the 2D space (NE, SE, SW, or NW). Barriers do not move, while signals move in the obvious way if unobstructed, as there is only one choice for any signal in any square of four cells. Hence the basic movements of signals are given by the following four rules: $${
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
& \\ \hline
$s$ & \\ \hline
\end{tabular}\,
}
} \mapsto {
\centering
\Ket{
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\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
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& $s$ \\ \hline
& \\ \hline
\end{tabular}\,
}
}, \qquad {
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
$s$ & \\ \hline
& \\ \hline
\end{tabular}\,
}
} \mapsto {
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\Ket{
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\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
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& \\ \hline
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\end{tabular}\,
}
},$$ $${
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
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& $s$ \\ \hline
& \\ \hline
\end{tabular}\,
}
} \mapsto {
\centering
\Ket{
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\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
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& \\ \hline
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\end{tabular}\,
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}, \qquad {
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
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& \\ \hline
& $s$ \\ \hline
\end{tabular}\,
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} \mapsto {
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\Ket{
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\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
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$s$ & \\ \hline
& \\ \hline
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}
}.$$ where $s\in \{0,1\}$ denotes a signal, and blank cells are empty.
The way to interpret the four above rules in terms of the scattering unitary $U$ is just case-by-case definition, [*i.e. *]{}they show that $U{
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
& \\ \hline
$s$ & \\ \hline
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}={
\centering
\Ket{
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\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
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& $s$ \\ \hline
& \\ \hline
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}
}$. Moreover, each rule can be obtained as a rotation of another, hence by stating that the $U$-defined PQCA is isotropic the first rule above suffices. This convention will be used throughout.
The ability to redirect signals is achieved by ‘bouncing’ them off walls constructed from two barriers arranged either horizontally or vertically: $${
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
{\cellcolor{orange}}& $s$ \\ \hline
{\cellcolor{orange}}& \\ \hline
\end{tabular}\,
}
} \mapsto {
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
{\cellcolor{orange}}& \\ \hline
{\cellcolor{orange}}& $s$ \\ \hline
\end{tabular}\,
}
}.$$ where $s$ again denotes the signal and the shaded cells denote the barriers which causes the signal to change direction. If there is only one barrier present in the four cell square being operated on then the signal simply propagates as normal and is not deflected: $${
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
{\cellcolor{orange}}& \\ \hline
$s$ & \\ \hline
\end{tabular}\,
}
} \mapsto {
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
{\cellcolor{orange}}& $s$ \\ \hline
& \\ \hline
\end{tabular}\,
}
}.$$ Using only these basic rules of signal propagation and signal reflection from barrier walls, signal delay (Fig. \[fig:delays\]) and signal swapping (Fig. \[fig:swap\]) tiles can be constructed. All of the rules presented so far are permutations of some of the base elements of the vector space generated by $$\Set{{
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
$w$ & $x$ \\ \hline
$y$ & $z$ \\ \hline
\end{tabular}\,
}
}}_{w,x,y,z \in \{\epsilon,0,1,\blacksquare\}}$$ therefore $U$ is indeed unitary on the subspace upon which its action has so far been described.
[![The ‘identity circuit’ tile, an $8\times 14$ tile taking 24 time-steps, made by repeatedly bouncing the signal from walls to slow its movement through the tile. The dotted line gives the signal trajectory, with the arrow showing the exit point and direction of signal propagation. The bold lines show the tile boundary.[]{data-label="fig:delays"}](img/delayCirc.pdf "fig:")]{}
[![The ‘swap circuit’ tile, a $16\times 14$ tile, where both input signals are permuted and exit synchronously after 24 time-steps. As the first signal (*bottom left*) is initially delayed, there is no interaction.[]{data-label="fig:swap"}](img/swapCirc.pdf "fig:")]{}
Gates {#subsec:gates}
-----
To allow a universal set of gates to be implemented by the PQCA, certain combinations of signals and barriers can be assigned special importance. The Hadamard operation on a single qubit-carrying signal can be implemented by interpreting a signal passing through a diagonally oriented wall, analogous to a semitransparent barrier in physics. This has the action defined by the following rule: $${
\centering
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\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
{\cellcolor{orange}}& \\ \hline
0 & {\cellcolor{orange}}\\ \hline
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}
} \mapsto\frac{1}{\sqrt{2}}{
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
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{\cellcolor{orange}}& 0 \\ \hline
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}
} + \frac{1}{\sqrt{2}}{
\centering
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{\cellcolor{orange}}& 1 \\ \hline
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}$$ $${
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\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
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{\cellcolor{orange}}& \\ \hline
1 & {\cellcolor{orange}}\\ \hline
\end{tabular}\,
}
} \mapsto \frac{1}{\sqrt{2}}{
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
{\cellcolor{orange}}& 0 \\ \hline
& {\cellcolor{orange}}\\ \hline
\end{tabular}\,
}
} - \frac{1}{\sqrt{2}}{
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
{\cellcolor{orange}}& 1 \\ \hline
& {\cellcolor{orange}}\\ \hline
\end{tabular}\,
}
}$$ This implements the Hadamard operation, creating a superposition of configurations with appropriate phases. Using this construction a Hadamard tile can be constructed (Fig. \[fig:hadamard\]) by simply adding a semitransparent barrier to the end of the previously defined delay (identity) tile (Fig. \[fig:delays\]).
[![The ‘Hadamard gate’ tile applies the Hadamard operation to the input signal. It is a modification of the identity circuit tile, with a diagonal (semitransparent) barrier added at the end which performs the Hadamard operation.[]{data-label="fig:hadamard"}](img/hadCirc.pdf "fig:")]{}
A way of encoding two qubit gates in this system is to consider that two signals which cross paths interact with one another. The controlled-[R($\frac{\pi}{4})$]{}[ ]{} operation can be implemented by considering signals that cross each other as interacting only if they are both $1$, in which case a global phase of $e^{\frac{i\pi}{4}}$ is applied. Otherwise the signals continue as normal. This behaviour is defined by the following rule: $${
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
1 & \\ \hline
1 & \\ \hline
\end{tabular}\,
}
} \mapsto e^{\frac{i\pi}{4}}{
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
& 1 \\ \hline
& 1 \\ \hline
\end{tabular}\,
}
}, \qquad {
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
$x$ & \\ \hline
$y$ & \\ \hline
\end{tabular}\,
}
} \mapsto {
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
& $y$ \\ \hline
& $x$ \\ \hline
\end{tabular}\,
}
} otherwise$$ where $x,y \in \{0,1\}$. This signal interaction which induces a global phase change allows the definition of both a two signal controlled-[R($\frac{\pi}{4})$]{} tile (Fig. \[fig:cphase\]) and a single signal [R($\frac{\pi}{4})$]{} operation tile (Fig. \[fig:phase\]).
[![The ‘[controlled-[R($\frac{\pi}{4})$]{}]{} gate’ tile[ applies the controlled-[R($\frac{\pi}{4})$]{} operation to the two input qubits, by causing the signals to interact at the highlighted point (grey circle). The qubits are then synchronised so that they exit at the same time along their original paths. No swapping takes place.]{}[]{data-label="fig:cphase"}](img/cPhaseCirc.pdf "fig:")]{}
[![The ‘[R($\frac{\pi}{4})$]{} gate’ tile. This tile makes use of a signal, set to $\ket{1}$, which loops inside the grid every six time-steps, ensuring that it will interact with the signal that enters the tile, and causing it to act as the control qubit to a [controlled-[R($\frac{\pi}{4})$]{}]{} operation. It therefore acts as a phase rotation on the input qubit, which passes directly through. [After 24 time-steps the auxiliary control signal has returned to its origin, unchanged, hence the tile can be reused.]{}[]{data-label="fig:phase"}](img/phaseCirc.pdf "fig:")]{}
These rules are simply a permutation and phase change of base elements of the form: $$\Set{{
\centering
\Ket{
\,
\begin{tabular}{ | p{2.8mm} | p{2.8mm} | }
\hline
$x$ & \\ \hline
$y$ & \\ \hline
\end{tabular}\,
}
}}_{x,y \in \{0,1\}}$$ (and their rotations), therefore $U$ is a unitary operation on the subspace upon which its action has so far been described. Wherever $U$ has not yet been defined, it is the identity. Hence $U$ is unitary.
Circuits: Combining Gates {#subsec:circuits}
-------------------------
A signal is given an $8 \times 14$ tile ($16 \times 14$ for two signal operations) in which the action is encoded. The signals enter each tile at the fifth cell from the left, and propagate diagonally NE. Each time step finds the tile shifted one cell to the right to match this diagonal movement, giving a diagonal tile. The signal exits the tile $14$ cells North and East of where it entered. This allows these tiles to be composed in parallel and sequentially with the only other requirement being that the signal exits at the appropriate point, [*i.e. *]{}the fifth cell along the tile, after $24$ time-steps. This ensures that all signals are synchronised as in Fig. \[fig:flattening34\] (*right*), allowing larger circuits to be built from these elementary tiles by simply plugging them together. Non-contiguous gates can also be wired together using appropriate wall constructions to redirect and delay signals so that they are correctly synchronised.
The implemented set of quantum gates, the identity, Hadamard, swap, [R($\frac{\pi}{4})$]{} and [controlled-[R($\frac{\pi}{4})$]{}]{}, gives a universal set. Indeed the standard set of <span style="font-variant:small-caps;">cNot</span>, <span style="font-variant:small-caps;">H</span>, <span style="font-variant:small-caps;">[R($\frac{\pi}{4})$]{}</span> can be recovered as follows: $$\textsc{cNot}\ket{\psi}=(\mathbb{I}\otimes H)(\textsc{cR(${\pi}\slash{4}$)})^4(\mathbb{I}\otimes H)\ket{\psi}$$ where $\textsc{cR($\frac{\pi}{4}$})^4$ denotes four applications of the [controlled-[R($\frac{\pi}{4})$]{}]{} gate, giving the controlled-<span style="font-variant:small-caps;">Phase</span> operation.
Conclusion {#sec:discussion}
==========
This paper presents a simple PQCA which is capable of simulating all other PQCA, preserving the topology of the simulated PQCA. This means that the initial configuration and the forward evolution of any PQCA can be encoded within the initial configuration of this PQCA, with each simulated cell encoded as a group of adjacent cells in the PQCA, [*i.e. *]{}intrinsic simulation. The construction in section \[sec:nuqca\] is given in two-dimensions, which can be seen to generalise to $n>1$-dimensions. The main, formal result of this work can therefore be stated as:
There exists an $n$-dimensional $U$-defined PQCA, $G$, which is an intrinsically universal PQCA. Let $H$ be a $n$-dimensional $V$-defined PQCA such that $V$ can be expressed as a quantum circuit $C$ made of gates from the set $\textsc{Hadamard}$, $\textsc{Cnot}$, and $\textsc{{R($\frac{\pi}{4})${}}}$. Then $G$ is able to intrinsically simulate $H$.
Any finite-dimensional unitary $V$ can always be approximated by a circuit $C(V)$ with an arbitrary small error $\varepsilon=\max_{\ket{\psi}}||V\ket{\psi}-C\ket{\psi}||$. Assuming instead that $G$ simulates the $C(V)$-defined PQCA, for a region of $s$ cells over a period $t$, the error with respect to the $V$-defined PQCA will be bounded by $st\varepsilon$. This is due to the general statement that errors in quantum circuits increase, at most, proportionally with time and space [@NielsenChuang]. Combined with the fact that PQCA are universal [@ArrighiNUQCA; @ArrighiPQCA], this means that $G$ is intrinsically universal, up to this unavoidable approximation.
Discussion and Future Work
--------------------------
QC research has so far focused on applications for more secure and efficient computing, with theoretical physics supporting this work in theoretical computer science. The results of this interdisciplinary exchange led to the assumptions underlying computer science being revisited, with information theory and complexity theory, for example, being reconsidered and redeveloped. However, information theory also plays a crucial role in the foundations of theoretical physics. These developments are also of interest in theoretical physics studies where physical aspects such as particles and matter are considered; computer science studies tend to consider only abstract mathematical quantities. Universality, among the many computer science concepts, is a simplifying methodology in this respect. For example, if the problem being studied crucially involves some idea of interaction, universality makes it possible cast it in terms of information exchanges *together* with some universal information processing. This paper presents an attempt to export universality as a tool for application in theoretical physics, a small step towards the goal of finding and understanding a *universal physical phenomenon*, within some simplified mechanics. Similar to the importance of the idea of the spatial arrangement of interactions in physics, intrinsic universality has broader applicability than computation universality and must be preferred. In short, if only one physical phenomenon is considered, it should be an intrinsically universal physical phenomenon, as it could be used to simulate all others.
The PQCA cell dimension of the simple intrinsically universal construction given here is four (empty, a qubit ($\ket{0}$ or $\ket{1}$), or a barrier). In comparison, the simplest classical Partitioned CA has cell dimension two [@MargolusQCA]. Hence, although the intrinsically universal PQCA presented here is the simplest found, it is not minimal. In fact, one can also manage [@Arrigh2UQCA] an intrinsically universal PQCA with a cell dimension of three, in two different ways. One way is to encode the spin degree of freedom (0 and 1) into a spatial degree of freedom, so that now the semitransparent barrier either splits or combines signals. The second way is to code barriers as pairs of signals as in the Billiard Ball CA model [@MargolusQCA]. These constructions may be minimal, but are not as elegant as the one presented here. In future work we will show that there is a simple, greater than two-dimensional PQCA which is minimal, as it has a cell dimension of two.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank Jérôme Durand-Lose, Jarkko Kari, Jacques Mazoyer, Kenichi Morita, Nicolas Ollinger, Guillaume Theyssier and Philippe Jorrand.
|
---
abstract: 'The time lag between optical and near-infrared (IR) flux variability can be taken as a means to determine the sublimation radius of the dusty “torus” around supermassive black holes in active galactic nuclei (AGN). I will show that data from big *optical* survey telescopes, e.g. the *Large Synoptic Survey Telescope (LSST)*, can be used to measure dust sublimation radii as well. The method makes use of the fact that the Wien tail of the hot dust emission reaches into the optical and can be reliably recovered with high-quality photometry. Simulations show that dust sublimation radii for a large sample of AGN can be reliably established out to redshift $z\sim 0.1-0.2$ with the LSST. Owing to the ubiquitous presence of AGN up to high redshifts, they have been studies as cosmological probes. Here, I discuss how optically-determined dust time lags fit into the suggestion of using the dust sublimation radius as a “standard candle” and propose and extension of the dust time lags as “standard rulers” in combination with IR interferometry.'
author:
- 'Sebastian F. Hönig'
title: |
Dust reverberation mapping in the era of big optical surveys\
and its cosmological application
---
Introduction
============
The near-infrared (IR) light curves of active galactic nuclei (AGN) show variability that is consistent with the optical light curves, however lagging by tens to hundreds of days [e.g. @Cla89; @Gla92; @Okn99a; @Gla04; @Sug06; @Kos09]. The lag supposedly represents the distance from the central engine to the region where the temperature drops to about 1500K so that dust can marginally survive. This sublimation radius, ${R_\mathrm{sub}}$, has been found to scale with the square-root of the AGN luminosity using both time delay [e.g. @Okn01; @Min04; @Sug06; @Kis11a] and IR interferometric measurements[e.g. @Kis11a], as expected from dust in local thermal equilibrium [e.g. @Bar87].
Both IR interferometry and dust reverberation mapping are limited by their technical requirements of either sensitive long-baseline arrays or simultaneous availability of optical *and* IR instrumentation. Here, I will show that upcoming large *optical* surveys can recover time lags from hot dust emission for a huge number of AGN using optical bands, therefore overcoming some of these limitations.
Since AGN are ubiquitous in the universe, they may be attractive targets as cosmological probes. Both the broad-line region (BLR) radius [@Haa11; @Wat11; @Cze13] and hot-dust radius [e.g. @Kob98; @Okn99b; @Okn02; @Yos04] seem promising “standard candles”, based on the observationally well-established BLR lag-luminosity [e.g. @Kas00; @Pet04] and NIR lag-luminosity relations [e.g. @Okn01; @Min04; @Sug06]. For the BLR, @Elv02 proposed that a combination of interferometric observations of emission lines and lag times may be used as “standard rulers”, thus bypassing the cosmic distance ladder. Here, I suggest that the hot-dust lags and near-IR interferometric sizes can also serve as standard rulers, although requiring advancements in IR interferometry.
Principles of dust reverberation mapping at optical wavelengths {#sec:idea}
===============================================================
Dust around AGN absorbs the UV/optical radiation from the putative accretion disk and reemits in the IR. At about 1500K, the dust sublimates, corresponding to the hottest dust emission peaking at $\sim2\,\micron$. Despite the exponential decrease of the Wien tail, some contribution of the hot-dust emission will reach into optical wavebands. Indeed, such a dust contribution has been reported by @Sak10 in the $I$-band from analyzing the color variability of optical variability. Therefore, the optical emission consists of contributions from two different emission regions, leading to a relative lag.
Dust is considered to be in local thermal equilibrium (LTE), allowing us to calculate the dust temperature/emission directly from the absorbed incident radiation. In @Hon11b, we presented a theoretical framework showing that these LTE considerations can be used to calculate temperature changes $\mathrm{d}T$ for variable incident radiation $\mathrm{d}L$ as $\mathrm{d}T/T = 1/4\,\mathrm{d}L/L$[^1]. Based on this relation, changes in the temperature and emission of a pre-defined dust distribution can be calculated to derive near-IR light curves [@Hon11b].
The optical emission is dominated by the AGN central engine’s “big blue bump” (BBB). The BBB spectral energy can be approximated as $\lambda F_\lambda \propto \lambda^{-4/3}$ [e.g. @Elv94; @Ric06; @Ste12a]. For the purpose of this study, I will assume that the host galaxy is constant and readily removable by image decomposition and that broad- and narrow-lines do not contribute significantly ($\la1-10\%$) to the broad-band fluxes (see Sect. \[sec:real\]). Therefore, the AGN emission in the optical can be approximated by $$\begin{aligned}
F_\mathrm{AGN} &= F_V \cdot
\left(\frac{\lambda}{0.55\,\micron}\right)^{-7/3}
\\ \nonumber
&+ \frac{F_\mathrm{BBB}(1.2\,\micron)}{\pi
B_\mathrm{1.2\,\micron}(1400\,K)} \cdot \pi
B_\mathrm{\lambda}(1400\,K)\end{aligned}$$ $F_V$ denotes the $V$-band flux at $0.55\,\micron$, $F_\mathrm{BBB}(1.2\,\micron)$ is the BBB flux component at 1.2$\micron$, and $\pi B_\mathrm{1.2\,\micron}(1400\,K)$ represents the flux of a 1400K black body at 1.2$\micron$ [observed near-IR color temperature; @Kis07; @Kis11b]. The normalization accounts for the fact that AGN show a generic turnover from BBB-dominated emission to host-dust emission at about $1-1.2\,\micron$ [e.g. @Neu79; @Elv94].
[l c c c c c]{} redshift & $z=0$ & $z=0.05$ & $z=0.1$ & $z=0.2$ & $z=0.3$\
$i$ band & 0.019 & 0.012 & 0.007 & 0.003 & $\ldots$\
$z$ band & 0.073 & 0.052 & 0.031 & 0.014 & 0.004\
$y$ band ($y3$) & 0.206 & 0.158 & 0.109 & 0.053 & 0.020\
$y$ band ($y4$) & 0.168 & 0.126 & 0.085 & 0.041 & 0.015\
Fig. \[fig:agn\_sed\] shows the total AGN SED, the BBB, and hot-dust components for a simulated object at redshift $z=0.03$. Overplotted are the transmission curves for the LSST filters $u$, $g$, $r$, $i$, $z$, and $y$, where the latter may either be represented by the $y3$ (referred to as $y$ in the following) or $y4$ filter. The Wien tail of the hot dust reaches into the $z$ and $y$ bands. However, the fractional contribution of the dust is very sensitive to the object’s redshift. In Table \[tab:hotdustcont\], hot-dust contributions to the total flux in $i$, $z$, and $y$ are listed for $0 < z < 0.3$. Out to about $z\sim0.1$ the dust contribution to the $y$ band is $\ga$10% and drops to $\sim$5% at $z=0.2$. Therefore, the dust component may be detected above the BBB out to $z\sim0.1-0.2$.
A dust reverberation mapping experiment for optical surveys\[sec:sim\]
======================================================================
In this letter I propose to use optical wavebands for dust reverberation mapping of AGN. Suitable telescope projects are currently being explored or under construction. The most promising survey for this experiment will be the *LSST*. Its cornerstones are high photometric quality ($<$1%), high cadence, and multi-year operation. The feasibility for *LSST* will be illustrated in the following. It can be easily translated to other surveys.
Simulation of observed light curves and construction of a mock survey\[sec:simobs\]
-----------------------------------------------------------------------------------
First, it is necessary to simulate survey data and find a method that allows for recovering dust time lags. @Kel09 show that the optical variability is well reproduced by a stochastic model based on a continuous autoregressive process [Ornstein-Uhlenbeck process; see also @Kel13]. The model consists of a white noise process with characteristic amplitude $\sigma$ that drives exponentially-decaying variability with time scale $\tau$ around a mean magnitude $m_0$. The parameters $\sigma$ and $\tau$ have been found to scale with black hole mass $M_\mathrm{BH}$ and/or luminosity $L$ of the AGN [e.g. @Kel09; @Kel13]. For the simulations, $L$ and $M_\mathrm{BH}$ are chosen and $\sigma$ and $\tau$ are drawn from the error distribution of the respective relation given in @Kel09. Since the amplitude of variability depends on wavelength, it was adjusted by $\sigma(\lambda) = \sigma \times
(\lambda/5500\,\mathrm{\AA})^{-0.28}$ as empirically found by @Meu11.
The BBB light curves are then propagated outward into the dusty region. Its inner edge, ${R_\mathrm{sub}}$, and thus the dust time lag $\tau({R_\mathrm{sub}})$, scales with $L$ as ${R_\mathrm{sub}}\propto \tau({R_\mathrm{sub}}) \propto
L^{1/2}$. The reaction of the dust on BBB variability is modeled using the principles outlined in Sect. \[sec:idea\]. For that, the dust is distributed in a disk with a surface density distribution $\Sigma(r)
\propto r^a$, and the temperature of the dust at distance $r$ from the AGN is calculated using the black-body approximation for LTE, $T(r) = T_\mathrm{sub}
\times (r/{R_\mathrm{sub}})^{-0.5}$ (sublimation temperature $T_\mathrm{sub}=1500\,$K). The power law index $a$ represents the compactness of the dust distribution ($a$ very negative = compact; $a\sim0$ = extended) and results in smearing out the variability signal/transfer function. Since its actual value is rather unconstrained, a random value is picked in the range $-2.5 < a < -0.5$, motivated by observations [@Hon10a; @Kis11b; @Hon12; @Hon13]. Using the dust variability model on actual data showed that only a fraction of the incident variable BBB energy, $w_\mathrm{eff}$, is converted into hot-dust variability [for details see @Hon11b]. Thus, a random $w_\mathrm{eff}$ is picked in the interval $w_\mathrm{eff} \ \epsilon \ [0.2,0.8]$ for each simulated AGN. In summary, the hot dust emission and its variability is fully characterized by $a$ and $w_\mathrm{eff}$.
Magnitudes at all *LSST* wavebands are extracted for the combined BBB + hot-dust emission for AGN with luminosities $L$ at distances $D_L$. The “mock observations” take into account the expected statistical and systematic errors of the *LSST*[^2]. It is assumed that each AGN is observed once every 7 days in $u$, 3 days in $g$, 5 days in $r$, 10 days in $i$, 20 days in $z$, and 15 days in $y$.
For illustration of the proposed method, AGN properties observed in the local universe ($z\la 0.1$) were approximated as follows: First, a redshift is randomly picked from a $(1+z)^3$-distribution. Then, luminosities are drawn randomly from the interval $\log L(\mathrm{erg/s}) \epsilon
[42.7,44.3]$ and adjusted by $10\cdot z$ (quasars become more abundant with $z$). $M_\mathrm{BH}$ is determined based on $L$ and an Eddington ratio picked randomly around the $L$-dependent mean $\log
\left<\ell_\mathrm{Edd}\right> = -1.0 + 0.3\times \log
L/\left<L\right>$ (Gaussian with standard deviation $\sigma_{\log \ell} = 0.22\,$dex), producing an $L$-$\ell_\mathrm{Edd}$ correlation.
How to recover dust time lags {#sec:howto}
-----------------------------
A catalog with 301 AGN has been simulated[^3]. Example light curves in the $g$ and $y$ bands are presented in Fig. \[fig:mock\_obs\]. The circles with error bars are the observed epochs that will be used as input for the reverberation experiment. In the following, a very simple cross-correlation approach will be used to successfully recover dust lags. The intention is to provide a proof-of-concept, while optimization or tests of better approaches [e.g. @Che12; @Che13; @Zu13] are encouraged for future studies.
First, the observed photometric light curves in each band and with very different time resolution and inhomogeneous coverage were resampled to a common $\Delta t =1$day using the stochastic interpolation technique described in [@Pet98] and @Sug06. For each band, 10 random realizations of the resampled light curves were simulated. From the resampled $ugri$ light curves, a reference BBB light curve was extracted. For that, the mean fluxes and standard deviations over the 10 random realizations of each band and epoch were calculated and a simple power law $f_\mathrm{\nu} \propto \nu^\beta$ was fit to the resulting $ugri$ fluxes at each resampled epoch. Based on this fit, a $V$-band flux at 0.55$\micron$ was determined. This method uses the maximum information of all bands simultaneously and the resulting BBB light curve is very close to the input AGN variability pattern.
In the next step, the BBB light curve was subtracted from the $y$ band fluxes. This procedure can produce negative fluxes and leaves some BBB variability in the result because of the overestimation of the BBB underlying the $y$ band and the wavelength-dependence of the variability. However, as we are interested only in the time delay signal, this does not need to be of concern. The most important result from this procedure is that a large part of the BBB signal has been removed in this $y-$BBB light curve.
Finally, the *observed* epochs of the $y-$BBB light curve are (discretely) cross-correlated with the BBB light curve by interpolating the BBB flux at the observed $y$ band epochs and accounting for different lags. An example cross-correlation function (CCF) for a ten-year *LSST* survey, smoothed with a box-car kernel with a width of 6 days, is shown in Fig. \[fig:ccf\]. The CCF shows a distinct negative/anti-correlation peak at zero lag. This peak originates from the subtraction method discussed above. As such, it closely follows the auto-correlation functions of the BBB and $y-$BBB light curves. After linearily-combining both ACFs and scaling to the 0-lag negative peak, the BBB effect on the CCF can be effectively removed.
To recover the dust lag, the highest peak in the CCF after ACF subtraction was automatically identified. A positive detection is considered if the peak CCF $\ge$0.2. An error region is defined as the range over which the CCF is at least half the peak CCF. The maximum CCF defines the time lag of the peak $\tau_\mathrm{peak}$. A center-of-mass time lag $\tau_\mathrm{com}$ is also determined at half the integrated CCF within the error region.
As a final remark, a direct cross-correlation between the observed $y$ band and any optical band did not recover a time delay, although a corresponding peak is seen when cross-correlating the input model light curves. In addition, the “shifted reference” method by @Che12 developed to recover time lags of the BLR from photometric filters was not successful. This originates from the fact that the target signal does not vary with the same amplitude as the reference band and is significantly smeared, and may require the refinement presented in @Che13.
Discussion {#sec:discu}
==========
Time delays in a real survey {#sec:real}
----------------------------
The quality of lag recovery critically depends on the cadence as well as the continuity of observations. Typically, objects will not be observable year-round. In order to simulate this effect, annual gaps of 2, 4, and 6 months were introduced for 1/6, 1/3, and 1/2 of the catalog, respectively. Due to strong noise features for lags longer than $\sim$350days, the CCF was analyzed only for lags $<$300days. In general, lags are detected for about 70% or more of all AGN, independent of gap length. Out of these lags, about 80% are consistent within error bars with the sublimation radius of the input model. This is arguably a high rate given the simplicity of the lag recovery scheme and the completely blind analysis without any human interaction. The success rates might be boosted with more sophisticated methods and a way to deal with the longer lags. Note that these numbers are strictly valid only for the specific AGN sample parameters (see Sect. \[sec:simobs\]).
There are some issues that merit further attention. The current simulations do not take into account contributions of broad lines. This should be a minor issue for the recovery of the BBB light curve given the multi-filter fitting scheme. However, broad components of Paschen lines in the $y$-band may impose a secondary lag signal onto the dust light curve [see @Che12 for Balmer lines]. While their contribution is probably smaller than the dust contribution in this band [see spectra in @Lan11], they should be part of a more advanced recovery scheme (see Sect \[sec:howto\]). Furthermore, potential scattered light from the BBB [e.g. @Kis08; @Gas12] and the Paschen continuum are also neglected, but their contribution to the total $y$-band flux is estimated to be of the order of 1%.
The host flux has been neglected in the modeling (see Sect. \[sec:idea\]). Based on the optical AGN/host decomposition of @Ben09, the contribution of the host to the total flux can reach 50% at $L_\mathrm{AGN} \la 10^{43}$erg/s and will drop to $\sim$10% for $L_\mathrm{AGN} \la 10^{45}$erg/s in the $V$-band. Therefore, if the host contribution was considered, the presented simulations would apply to a sample that is brighter by $\sim0.1-0.7$ mag.
Cosmological applications of dust time lags and dust emission sizes {#sec:cosmo}
-------------------------------------------------------------------
Radius-luminosity relations open up the possibility to use AGN as “standard candles” in cosmology. Such applications have been discussed for both the BLR [@Haa11; @Wat11] and the dust [e.g. @Kob98; @Okn99b; @Okn02; @Yos04]. While the current AGN reverberation sample is larger for the BLR than for the hot dust, the proposed use of optical surveys may change this picture significantly. Fig. \[fig:cosmo\] shows the 231 objects from the 301-object mock AGN catalog for which time lags $\tau_\mathrm{peak}$ were recovered. For each of these AGN, the $V$-band flux $f_V$ was measured and a luminosity distance $D_L$ independent of redshift was calculated as $$\label{eq:cosmo}
D_L(\mathrm{Mpc}) = 0.075 \times f_V(\mathrm{Jy})^{-1/2} \times
\tau_\mathrm{peak}(\mathrm{days}) \ .$$ The scaling factor of 0.075 was obtained by fitting $f_V^{-1/2} \cdot \tau$ to the known distances of the objects in the mock survey (a real survey will require normalizing to the cosmic distance ladder). The errors of individual data points are dominated by the uncertainties in $\tau$ as long as $f_V$ can be determined with $\la30\%$ accuracy. Overplotted are $z-D_L$-relations for a standard cosmology according to the latest Planck results ($H_0 = 67.3$(km/s)/Mpc, $\Omega_m = 0.315$, $\Omega_\Lambda = 0.685$) as well as for a universe without cosmological constant ($H_0 = 75$(km/s)/Mpc, $\Omega_m = 1$, $\Omega_\Lambda = 0$). For comparison, $K$-band reverberation-mapped AGN from literature are also shown (NGC 3227, NGC 4051, NGC 5548, NGC 7469: @Sug06; NGC 4151: @Kos09; Fairall 9: @Cla89; GQ Com: @Sit93; NGC 3783: @Gla92; Mark 744: @Nel96). The limitation to optical bands does not allow for distinguishing between different cosmological parameters. However, such a nearby sample ($z \la 0.1$) can be used to determine the scaling factor. Under the same configuration as taken for the optical bands, a survey using near-IR filters will reach $z\sim0.3$ in the $J$-band, $z\sim0.7$ in the $H$-band, and $z\sim1.3$ in the $K$-band.
One of the disadvantages of standard candles are their reliance on the cosmic distance ladder. This dependence can be overcome with “standard rulers” for which an angular diameter distance $D_A$ is determined by comparing physical with angular sizes. @Elv02 proposed the use of AGN BLR time lags in combination with spatially- and spectrally-resolved interferometry of broad emission lines for this purpose. However, optical/near-IR interferometry of AGN needs 8-m class telescopes (e.g. at the VLTI or Keck) and is limited to about 130m baseline lengths. A broad emission line in the near-IR has only been successfully observed and resolved for one AGN to-date [3C273; @Pet12].
Here, it is proposed that the dust time lags may also be used as a “standard ruler” in combination with directly measured near-IR angular sizes, $\rho_K$, from interferometry to determine angular size distances $D_A$, via the relation $$D_A(\mathrm{Mpc}) = 0.126 \times
\frac{\tau(\mathrm{days})}{\xi_\mathrm{int} \cdot
\rho_K(\mathrm{mas})} \ .$$ $\xi_\mathrm{int}$ corrects the different sensitivities of the CCF and interferometry measurements to extended dust distributions [e.g. @Kis11a; @Kis14] and may be determined from spectral or interferometric modeling.
For both the standard candle and ruler, time lags for the hot dust have to be determined. As compared to the BLR, the dust lags are about a factor of 4 longer (comparing the $\mathrm{H\beta}$ relation in @Ben09 with the dust relation in @Kis07), resulting in longer monitoring campaigns. On the other hand, dust monitoring does not require spectroscopy or the use of specific narrow-band filters, but can be executed with any telescope that has broad-band filters and sufficient sensitivity. Moreover, the hot-dust emission and sublimation radius are extremely uniform across AGN, e.g. showing a narrow range of color temperatures [e.g. @Gla04], a universal turn-over at $\sim1-1.2\,\micron$ [e.g. @Neu79; @Elv94], and an emissivity close to order unity [e.g. @Kis11b]. This points toward simple physics (radiative reprocessing within $\sim\micron$-sized dust grains) involved in dust emission at ${R_\mathrm{sub}}$, which arguably removes some physical uncertainties that are inherent to other methods.
For hot dust radii as standard rulers, the emission region has to be spatially resolved without the need for spectral resolution. This has been achieved for 13 nearby AGN to-date [$z\le0.16$; e.g. @Kis11a; @Wei12; @Kis14] with the limiting factor being sensitivity of IR interferometers. However, using a potential km-sized heterodyne array (e.g. the Planet Formation Imager[^4] with sufficient wavelength coverage into the $L$ and $M$ bands, space-based or on the ground, may allow for direct distance estimates directly into the Hubble flow out to $z\ga1$, entirely bypassing the cosmological distance ladder (Hönig et al., in prep.).
**Acknowledgements —** I want to thank Darach Watson and Aaron Barth for helpful discussions, and the anonymous referee for many helpful comments and suggestions that improved the manuscript. The Dark Cosmology Centre is funded by The Danish National Research Foundation.
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[^1]: Without loss of generality, I will assume that the dust emission follows a black body [see also @Kis07; @Hon10b; @Kis11b; @Mor11].
[^2]: based on the descriptions at <http://ssg.astro.washington.edu/elsst/magsfilters.shtml> and <http://ssg.astro.washington.edu/elsst/opsim.shtml?skybrightness>
[^3]: The catalog and analysis are available at http://dorm.sungrazer.org
[^4]: http://planetformationimager.org
|
---
abstract: 'We study the stability of a recently proposed model of scalar-field matter called mimetic dark matter or imperfect dark matter. It has been known that mimetic matter with higher derivative terms suffers from gradient instabilities in scalar perturbations. To seek for an instability-free extension of imperfect dark matter, we develop an effective theory of cosmological perturbations subject to the constraint on the scalar field’s kinetic term. This is done by using the unifying framework of general scalar-tensor theories based on the ADM formalism. We demonstrate that it is indeed possible to construct a model of imperfect dark matter which is free from ghost and gradient instabilities. As a side remark, we also show that mimetic $F({\cal R})$ theory is plagued with the Ostrogradsky instability.'
author:
- 'Shin’ichi Hirano'
- Sakine Nishi
- Tsutomu Kobayashi
title: Healthy imperfect dark matter from effective theory of mimetic cosmological perturbations
---
Introduction
============
Standard cosmology based on Einstein’s theory of general relativity is very successful under the assumption that unknown components called dark matter and dark energy dominate the energy density of the Universe. The origins of those dark components are puzzles in modern cosmology and particle physics, and a number of scenarios have been developed and explored so far.
Recently, a novel interesting model of scalar-field matter dubbed [*mimetic dark matter*]{} was put forward [@Chamseddine:2013kea]. (For earlier work see Refs. [@Lim:2010yk; @Gao:2010gj; @Capozziello:2010uv], and for a review see Ref. [@Sebastiani:2016ras].) The mimetic scalar field is generated by a singular limit of the general disformal transformation where the transformation is not invertible [@Deruelle:2014zza; @Yuan:2015tta; @Domenech:2015tca], and its kinetic term is subject to the constraint $$\begin{aligned}
g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi =-1.\end{aligned}$$ With this mimetic constraint the theory reproduces the behavior of pressureless dust in Einstein gravity, and thereby yields a candidate of dark matter. In the original version of the mimetic scalar-field theory, there is no nontrivial dynamics for scalar-type fluctuations, but by introducing the higher-derivative term $(\Box\phi)^2$ the scalar degree of freedom can be promoted to a dynamical field propagating with a nonzero sound speed [@Chamseddine:2014vna; @Mirzagholi:2014ifa]. The higher-derivative term modifies the fluid properties of mimetic dark matter, and due to its imperfect nature it is called [*imperfect dark matter*]{}. (For the Hamiltonian analysis of mimetic matter, see Refs. [@Chaichian:2014qba; @Malaeb:2014vua; @Ali:2015ftw].) Since a nonzero sound speed affects the evolution of density perturbations on small scales, imperfect dark matter could be relevant to the missing-satellites problem and the core-cusp problem [@Capela:2014xta]. Essentially equivalent theories appear in different theoretical settings such as Hořava-Lifshitz gravity [@Horava:2009uw; @Ramazanov:2016xhp], the Einstein-aether theory [@Jacobson:2000xp; @Jacobson:2014mda], and non-commutative geometry [@Chamseddine:2014nxa]. Mimetic theories with more general higher derivative terms can resolve black hole and cosmological singularities [@Chamseddine:2016uef; @Chamseddine:2016ktu]. The mimetic constraint can also be implemented in the general second-order scalar-tensor theory (the Horndeski theory [@Horndeski:1974wa; @Deffayet:2011gz; @Kobayashi:2011nu]), allowing for a variety of cosmological expansion histories with fluctuations having a vanishing sound speed [@Haghani:2015iva; @Arroja:2015wpa; @Rabochaya:2015haa; @Arroja:2015yvd; @Cognola:2016gjy]. See also Refs. [@Barvinsky:2013mea; @Matsumoto:2015wja; @Ramazanov:2015pha; @Hammer:2015pcx; @Liu:2017puc] for further developments in mimetic dark matter.
Although mimetic dark matter (or, more generically, [*mimetic gravity*]{}) has thus received much attention, it is not free from problematic issues. As it is anticipated from the fact that mimetic gravity of [@Chamseddine:2014vna; @Mirzagholi:2014ifa] can be reproduced as a certain limit of Hořava-Lifshitz gravity [@Horava:2009uw], the two theories share the various aspects, some of which could signal the sickness. For instance, scalar perturbations exhibit gradient instabilities [@Sotiriou:2009bx; @Ijjas:2016pad]. To what extent such gradient instabilities are dangerous depends on the time scale on which perturbations grow, but it is better if one could remove this potential danger from the theory in the first place. Another problem is an appearance of a caustic singularity [@Babichev:2016jzg], though it is a generic nature of scalar-tensor theories [@Babichev:2016hys; @Tanahashi:2017kgn] rather than a problem specific to mimetic/Hořava-Lifshitz gravity, and there exist some mechanisms to avoid formation of the caustic [@Mukohyama:2009tp; @Babichev:2017lrx].
In the present paper, we explore the way of resolving one of the above fundamental problems, namely, the gradient instability in mimetic gravity. For this purpose, we consider a unifying framework of general scalar-tensor theories [@Gleyzes:2013ooa; @Tsujikawa:2014mba; @Gao:2014soa] with the mimetic constraint, based on which we develop an effective theory of cosmological perturbations in mimetic theories. This allows us to study the perturbation property of mimetic gravity systematically at the level of the action rather than at the level of the equations of motion. It turns out that avoiding instabilities is not so difficult. We present a concrete example of a simple extension of mimetic gravity that is free from gradient as well as ghost instabilities.
This paper is organized as follows. In the next section we give a short review on the mimetic dark matter model of [@Chamseddine:2013kea; @Chamseddine:2014vna; @Mirzagholi:2014ifa] and its instability. We also argue briefly how one can remedy this gradient instability. In Sec. III we derive an effective theory of mimetic cosmological perturbations from a general action of a mimetic scalar-tensor theory. After presenting some unstable examples, we demonstrate that a healthy extension of mimetic gravity is indeed possible. We summarize our results and discuss future prospects in Sec. IV. As a side remark, in the appendix we show that the $F({\cal R})$ extension of mimetic gravity exhibits Ostrogradsky instabilities and hence is not viable.
Instabilities in mimetic gravity
================================
Mimetic gravity
---------------
We start with a brief review on mimetic gravity and its cosmology [@Chamseddine:2013kea; @Chamseddine:2014vna; @Mirzagholi:2014ifa]. The action for mimetic gravity can be written in the form[^1] $$\begin{aligned}
S&=\int{{\rm d}}^4 x\,{\cal L},
\\
\frac{{\cal L}}{\sqrt{-g}}&=\frac{{\cal R}}{2}-\lambda\left(g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+1\right)
-V(\phi)+\frac{\alpha}{2}\left(\Box\phi\right)^2,\label{eq:original-Lagrangian}\end{aligned}$$ where ${\cal R}$ is the Ricci scalar, $\phi$ is the mimetic scalar field, and $\lambda$ is the Lagrange multiplier enforcing $$\begin{aligned}
g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+1=0.\label{eq:mimetic-constraint}\end{aligned}$$ The parameter $\alpha$ can be a function of $\phi$, but for simplicity we assume that it is a nonzero constant. Here and hereafter we use the units ${M_{\rm Pl}}= 1$.
If $V(\phi)=\Lambda=\,$const, this theory is equivalent to the IR limit of projectable Hořava-Lifshitz gravity [@Horava:2009uw; @Mukohyama:2009mz; @Ramazanov:2016xhp]. It is also equivalent to a particular class of the Einstein-aether theory if the gradient of the mimetic scalar field is identified as the aether vector field [@Jacobson:2000xp; @Jacobson:2014mda].
The mimetic constraint (\[eq:mimetic-constraint\]) shows that $\partial_\mu\phi$ must be a timelike vector. It is therefore convenient to take the unitary gauge, $$\begin{aligned}
\phi(t,\Vec{x}) = t,\label{eq:condition-phi=t}\end{aligned}$$ and express the action in the Arnowitt-Deser-Misner (ADM) form in terms of three-dimensional geometrical objects on constant time hypersurfaces, i.e., the extrinsic and intrinsic curvature tensors, $K_{ij}$ and $R_{ij}$. Such a method has been employed broadly in studies of scalar-tensor theories of modified gravity [@Gleyzes:2013ooa; @Tsujikawa:2014mba; @Gao:2014soa; @Gleyzes:2014dya]. The ADM decomposition of spacetime leads to the metric $$\begin{aligned}
{{\rm d}}s^2=-N^2{{\rm d}}t^2+\gamma_{ij}\left({{\rm d}}x^i+N^i{{\rm d}}t\right)
\left({{\rm d}}x^j+N^j {{\rm d}}t\right),\end{aligned}$$ where $N$ is the lapse function, $N^i$ is the shift vector, and $\gamma_{ij}$ is the three-dimensional metric. The extrinsic curvature is then given by $$\begin{aligned}
K_{ij}:=\frac{1}{N}E_{ij},\quad
E_{ij}:=\frac{1}{2}\left(\dot \gamma_{ij}-D_iN_j-D_jN_i\right),\end{aligned}$$ where a dot denotes the derivative with respect to $t$ and $D_i$ is the covariant derivative induced by $\gamma_{ij}$. The unit normal to constant time hypersurfaces is written as $n_\mu=-\nabla_\mu \phi/\sqrt{-\nabla_\nu\phi\nabla^\nu\phi} = -N\delta^0_\mu$. Noting that the trace of the extrinsic curvature is given by $K = \nabla_\mu n^\mu$, we have $$\begin{aligned}
\Box \phi = -\frac{K}{N}+\frac{\Xi}{N},\end{aligned}$$ where $$\begin{aligned}
\Xi:=\frac{\dot N}{N^2}-\frac{N^i\partial_iN}{N^2}.\end{aligned}$$ We thus obtain the Lagrangian for mimetic gravity in the ADM form as $$\begin{aligned}
\frac{{\cal L}}{\sqrt{\gamma }}&=
\frac{N}{2}\left(R+K_{ij}K^{ij}-K^2\right)+\lambda\left(\frac{1}{N}-N\right)
\notag \\ &\quad-NV(t) +\frac{\alpha}{2N}\left(K-\Xi\right)^2,
\label{eq:ADM-metirc-original}\end{aligned}$$
Variation with respect to $\lambda$ gives the mimetic constraint, $$\begin{aligned}
N(t,\Vec{x})=1.\label{eq:solution-N=1}\end{aligned}$$ Note that we have not used the temporal gauge degree of freedom to set $N=1$; rather it was already used to impose Eq. (\[eq:condition-phi=t\]). As a result of solving the Euler-Lagrange equation we have obtained Eq. (\[eq:solution-N=1\]).
Variation with respect to $N$ yields $$\begin{aligned}
0&= \frac{1}{2}\left(R-E_{ij}E^{ij}+E^2\right)-2\lambda-V
\notag \\ &\quad
-\frac{3\alpha}{2}E^2+\frac{\alpha}{2}
\left[\dot E-D_i\left(N^iE\right)\right],\label{eq:delNeq}\end{aligned}$$ where we substituted $N=1$ after taking the variation. In contrast to the case of Einstein gravity, this equation is [*not*]{} an initial value constraint. Equation (\[eq:delNeq\]) fixes the Lagrange multiplier $\lambda$ in terms of the other variables, and hence is not necessary for the purpose of determining the dynamics of the metric.
Varying Eq. (\[eq:ADM-metirc-original\]) with respect to $N_i$, we obtain $$\begin{aligned}
D_j\pi^{ij}=0, \quad \pi^{ij}:=E^{ij}-(1-\alpha)\gamma^{ij} E.
\label{eq:momentum-const-original}\end{aligned}$$ Finally, variation with respect to $\gamma^{ij}$ leads to the evolution equation for the three-dimensional metric, $$\begin{aligned}
\frac{1}{\sqrt{\gamma}}\frac{\partial}{\partial t}\left(\sqrt{\gamma}\pi^{kl}\right)
\gamma_{ik}\gamma_{jl}+\cdots = 0.\label{eq:evolution-original}\end{aligned}$$ In deriving these equations we substituted $N=1$ after taking the variation. One can use Eqs. (\[eq:momentum-const-original\]) and (\[eq:evolution-original\]) to determine the evolution of the dynamical variables in the unitary gauge. Equivalently, one may start from the reduced Lagrangian, $$\begin{aligned}
\frac{{\cal L}}{\sqrt{\gamma}} = \frac{1}{2}\left(R+E_{ij}E^{ij}-E^2\right)-V+\frac{\alpha}{2}E^2,
\label{eq:reduced-Lag}\end{aligned}$$ to derive Eqs. (\[eq:momentum-const-original\]) and (\[eq:evolution-original\]). The Lagrangian (\[eq:reduced-Lag\]) is much easier to handle.
Cosmology in mimetic gravity {#sec:cosmology-in-mim}
----------------------------
Let us first study the evolution of the cosmological background, for which $N_i=0$ and $\gamma_{ij}=a^2(t)\delta_{ij}$ on constant $\phi$ hypersurfaces. The mimetic constraint enforces $N(t)=1$. The evolution equation reads [@Chamseddine:2014vna] $$\begin{aligned}
3H^2+2\dot H = \frac{2}{2-3\alpha}V,\label{eq:a-EL}\end{aligned}$$ where $H:=\dot a/a$ is the Hubble parameter. This is the only equation we can use to determine the evolution of the scale factor. Given $V=V(\phi)=V(t)$, one can integrate Eq. (\[eq:a-EL\]) to obtain $H=H(t)$, which contains one integration constant. This integration constant is left undetermined because the Hamiltonian constraint is missing in mimetic gravity, as is clear from the fact that Eq. (\[eq:delNeq\]) can only be used to fix $\lambda$.
Of particular interest is the case with $V=\Lambda=\,$ const. In this case Eq. (\[eq:a-EL\]) can be integrated to give $$\begin{aligned}
\frac{3(2-3\alpha)}{2}H^2 = \Lambda+\frac{C}{a^3},\end{aligned}$$ where $C$ is an integration constant. This equation may be identified as the Friedmann equation in the presence of the energy component mimicking dark matter, $C/a^3$. This component is essentially equivalent to dust-like matter as an integration constant found earlier in the context of Hořava-Lifshitz gravity [@Mukohyama:2009mz]. When viewed as a fluid, this mimetic dark matter has the imperfect character [@Mirzagholi:2014ifa] with its clustering behavior modified from the usual dust, and hence it could have some impact on the missing-satellites problem and the core-cusp problem [@Capela:2014xta].
Cosmological perturbations in mimetic gravity
---------------------------------------------
Let us next consider cosmological perturbations in mimetic gravity. In the unitary gauge, the mimetic constraint imposes $N=1$ even away from the homogeneous background. Therefore, it is sufficient to consider $$\begin{aligned}
&N_i=\partial_i\chi,
\\
&\gamma_{ij}=a^2e^{2\zeta}
\left(e^{h}\right)_{ij}
=a^2e^{2\zeta}\left(\delta_{ij}+h_{ij}+\frac{1}{2}h_{ik}h^k_j+\cdots\right)
,
\label{eq:def:pert}\end{aligned}$$ where $\chi$ and $\zeta$ are scalar perturbations and $h_{ij}$ is a transverse and traceless tensor perturbation. Plugging Eq. (\[eq:def:pert\]) to the reduced Lagrangian (\[eq:reduced-Lag\]) and expanding it to second order in perturbations, we can derive the quadratic actions for the tensor and scalar perturbations.
For the tensor sector we simply have $$\begin{aligned}
S_h^{(2)}
=\frac{1}{8}\int{{\rm d}}^4x\, a^3 \left[\dot{h}_{ij}^{2}-\frac{(\partial_kh_{ij})^{2}}{a^{2}}\right].
\label{eq:tensor2}\end{aligned}$$ Clearly, there is no pathology in the tensor sector of mimetic gravity, as Eq. (\[eq:tensor2\]) is identical to the corresponding Lagrangian in Einstein gravity.
For the scalar sector we have $$\begin{aligned}
S^{(2)} =\int{{\rm d}}^4x &\,a^3\biggl[
\frac{1}{a^2}(\partial\zeta)^2 -\frac{3}{2}(2-3\alpha)\dot\zeta^2
\notag \\ &\quad
+(2-3\alpha)\dot\zeta\frac{\partial^2\chi}{a^2}+\frac{\alpha}{2}
\left(\frac{\partial^2\chi}{a^2}\right)^2\biggr],\label{eq:L_S^2}\end{aligned}$$ where we used the background equation (\[eq:a-EL\]). Varying $S^{(2)}$ with respect to $\chi$, we obtain $$\begin{aligned}
\alpha \frac{\partial^2\chi}{a^2}+(2-3\alpha)\dot \zeta=0.\label{eq:sol-chi}\end{aligned}$$ This equation explains the reason why we have assumed that $\alpha\neq 0$: if $\alpha = 0$ then the curvature perturbation $\zeta$ would be nondynamical. Substituting this back into Eq. (\[eq:L\_S\^2\]), we arrive at $$\begin{aligned}
S_\zeta^{(2)}=\int{{\rm d}}^4x\,a^3\left[\left(\frac{3\alpha -2}{\alpha}\right)\dot\zeta^2
-(-1) \frac{(\partial\zeta)^2}{a^2}\right].\end{aligned}$$ From this quadratic Lagrangian we see that ghost instabilities can be avoided for $(3\alpha -2)/\alpha > 0$. However, it is obvious from the wrong sign in front of the $(\partial\zeta)^2$ term that the scalar sector suffers from gradient instabilities. This issue has already been known in the context of Hořava-Lifshitz gravity [@Horava:2009uw; @Sotiriou:2009bx; @Blas:2009yd], and was reemphasized in Ref. [@Ijjas:2016pad] in the context of mimetic cosmology.
Curing mimetic gravity: A basic idea {#subsec:cure}
------------------------------------
The gradient instability explained above arises from the curvature term $+\sqrt{\gamma}R\sim +2(\partial\zeta)^2/a^2$. In Einstein gravity, the sign of the $(\partial\zeta)^2$ term is flipped when one removes the perturbation of the lapse function by using the constraint equations. However, this procedure is absent in mimetic gravity and thus the wrong sign remains.
A possible way of curing this instability is to extend the theory so that the quadratic Lagrangian for the scalar perturbations contains the term $\partial^2\chi\partial^2\zeta$. Then, the Euler-Lagrange equation for $\chi$ i.e., Eq. (\[eq:sol-chi\]), would contain $\partial^2\zeta$. Substituting $\partial^2\chi \sim \partial^2\zeta+\cdots$ back to the Lagrangian, we would have extra contributions of the form $(\partial\zeta)^2$ and $(\partial^2\zeta)^2$ that improve the stability. In the ADM language, such terms arise from $$\begin{aligned}
RK,\quad R_{ij}K^{ij},\end{aligned}$$ which, in the covariant language, correspond to the terms $$\begin{aligned}
{\cal R}\Box\phi,\quad {\cal R}_{\mu\nu}\nabla^\mu\nabla^\nu\phi.\end{aligned}$$ These terms appear in mimetic Horndeski gravity as $$\begin{aligned}
\left({\cal R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}{\cal R}\right)\nabla^\mu\nabla^\nu \phi
\sim R_{ij}K^{ij}-\frac{1}{2}RK.\end{aligned}$$ Unfortunately, in this case two $\partial^2\chi\partial^2\zeta$ terms cancel out. It is therefore necessary to detune the coefficients of the two terms.
This observation motivates us to study a general mimetic scalar-tensor theory in the unitary gauge whose action is given by $$\begin{aligned}
S&=\frac{1}{2}\int {{\rm d}}^4x\sqrt{\gamma}\biggl[
N\left(R+K_{ij}K^{ij}-K^2\right)
\notag \\ &\quad\quad
+2\lambda\left(\frac{1}{N}-N\right)
+c_0(t,N)
\notag \\&\quad\quad
+c_1(t,N)K^2+c_2(t,N)RK+\cdots\biggr],
\label{eq:mimeticXG3}\end{aligned}$$ namely, a spatially covariant theory [@Gao:2014soa] with the mimetic constraint. In fact, the Lagrangian (\[eq:mimeticXG3\]) is redundant. The mimetic constraint enforces $N(t,\Vec{x})=1$, and also in this general theory the Euler-Lagrange equation $\delta S/\delta N=0$ is used only to fix the Lagrange multiplier $\lambda$. Therefore, we may instead start from the reduced action $$\begin{aligned}
S&=\frac{1}{2}\int{{\rm d}}^4x\sqrt{\gamma}\{
R+E_{ij}E^{ij}-[1-c_1(t)]E^2
\notag \\ &\quad\quad + c_0(t)+c_2(t)RE+\cdots
\},\end{aligned}$$ as we did in usual mimetic gravity. To study cosmology and stability in such gravitational theories, in the next section, we construct an effective theory of cosmological perturbations for a general class of scalar-tensor theories with the mimetic constraint.
In the context of Hořava gravity, several ideas have been proposed to remedy the sickness of the theory [@Blas:2009yd; @Blas:2009qj; @Blas:2010hb; @Horava:2010zj]. For example, in [@Blas:2009qj] the lapse function is allowed to be position dependent: $N=N(t,\Vec{x})$. In the context of mimetic gravity, such generalization (in the unitary gauge) is possible only by giving up the mimetic constraint, which is no longer what we call mimetic gravity. With this generalization, the scalar degree of freedom ceases to behave as a dark matter component. In [@Horava:2010zj] the sick scalar mode is removed by imposing extra symmetry. In contrast to those previous attempts, we require that the lapse function remains subject to the mimetic constraint, $N=1$, and no extra symmetry is imposed to remove the dangerous scalar mode, but we introduce the coupling between the extrinsic and intrinsic curvature tensors.
Effective theory of mimetic cosmological perturbations
======================================================
General Lagrangian
------------------
Following Refs. [@Gleyzes:2013ooa; @Tsujikawa:2014mba], we consider a general action of the form $$\begin{aligned}
S=\int{{\rm d}}^4x \sqrt{\gamma}
L(E,{\cal S},R,{\cal Y},{\cal Z};t),\end{aligned}$$ where $$\begin{aligned}
{\cal S}:=E_{ij}E^{ij},\quad {\cal Y}:=R_{ij}E^{ij},\quad {\cal Z}:=R_{ij}R^{ij}.\end{aligned}$$ Note, however, that we set $N=1$ from the beginning, which is different from the situation considered in [@Gleyzes:2013ooa; @Tsujikawa:2014mba]. We do not include terms such as $E_i^jE_j^kE_k^i$ and $E_i^jE_j^kR_k^i$ for simplicity, but it turns out that the inclusion of them do not change the essential result on the quadratic action for cosmological perturbations. Splitting $E_{ij}$ into the background and perturbation parts as $E_i^j=H\delta_i^j+\delta E_i^j$, we expand $L$ as $$\begin{aligned}
L&=
L_0+{\cal F}\delta E+(L_R+HL_{\cal Y}) R+L_{\cal S}\delta E_{ij}\delta E^{ij}
\notag \\ &\quad
+\frac{{\cal A}}{2}\delta E^2
+\left({\cal C}-\frac{L_{\cal Y}}{2}\right)
R\delta E+L_{\cal Y}R_{ij}\delta E^{ij}
\notag \\ &\quad
+L_{\cal Z}R_{ij}R^{ij}
+\frac{{\cal G}}{2}R^2+\cdots,\label{eq:Lexpanded}\end{aligned}$$ where $$\begin{aligned}
L_0&:=L(3H,3H^2,0,0,0;t),
\\
{\cal F}&:=L_E+2HL_{\cal S},
\\
{\cal A}&:=L_{EE}+4HL_{E{\cal S}}+4H^2L_{{\cal SS}},
\\
{\cal C}&:=L_{ER}+2HL_{{\cal S}R}+\frac{L_{\cal Y}}{2}+HL_{E{\cal Y}}+2H^2L_{{\cal SY}},
\\
{\cal G}&:=L_{RR}+2HL_{R{\cal Y}}+H^2L_{{\cal YY}},\end{aligned}$$ with $L_E:=\partial L/\partial E$, $L_{ER}:=\partial^2L/\partial E\partial R$, etc. The ellipsis denotes higher order terms. Notice that $R_{ij}$ itself is a perturbative quantity.
From the background part of the action we obtain $$\begin{aligned}
{\cal P}(H,\dot H;t):=\frac{{{\rm d}}{\cal F}}{{{\rm d}}t}+3H{\cal F}-L_0=0.\end{aligned}$$ One can integrate this equation to find the background solution, $H=H(t)$. Since we do not have the constraint corresponding to the Friedmann equation, one integration constant remains undetermined in the solution.
Let us move to the perturbation part. By substituting Eq. (\[eq:def:pert\]) to Eq. (\[eq:Lexpanded\]) it is straightforward to compute the quadratic action for the tensor and scalar perturbations.
The action for the tensor perturbations is given by $$\begin{aligned}
S_h^{(2)}=\int{{\rm d}}^4x\frac{a^3}{4}\left[L_{\cal S}\dot h_{ij}^2
-\frac{{\cal E}}{a^2}\left(\partial_k h_{ij}\right)^2+\frac{L_{\cal Z}}{a^4}
\left(\partial^2h_{ij}\right)^2
\right].\end{aligned}$$ The stability conditions read $L_{\cal S}>0$, $L_{\cal Z}\le 0$, and $$\begin{aligned}
{\cal E}:=L_R+\frac{1}{2a^3}\frac{{{\rm d}}}{{{\rm d}}t}\left(a^3 L_{\cal Y}\right)
\ge 0.\end{aligned}$$ The action for the scalar perturbations is given by
$$\begin{aligned}
S^{(2)}=\int{{\rm d}}^4x \,& a^3 \biggl\{
2\left[{\cal E}-\frac{3}{a}\frac{{{\rm d}}}{{{\rm d}}t}(a{\cal C})\right]\frac{(\partial\zeta)^2}{a^2}
+\left(\frac{9}{2}{\cal A}+3L_{\cal S}\right)\dot \zeta^2
+2\left(3L_{\cal Z}+4{\cal G}\right)\left(\frac{\partial^2\zeta}{a^2}\right)^2
\notag \\ &
+\frac{1}{2}\left({\cal A}+2L_{\cal S}\right)\left(\frac{\partial^2\chi}{a^2}\right)^2
-\left(3{\cal A}+2L_{\cal S}\right)\dot \zeta\frac{\partial^2\chi}{a^2}
+4{\cal C}\frac{\partial^2\zeta\partial^2\chi}{a^4}
\biggr\}.\label{eq:qaction1}\end{aligned}$$
The Euler-Lagrange equation $\delta S^{(2)}/\delta \chi = 0$ implies $$\begin{aligned}
\frac{\partial^2\chi}{a^2}
=\left(\frac{3{\cal A}+2L_{\cal S}}{{\cal A}+2L_{\cal S}}\right)\dot \zeta
-\frac{4{\cal C}}{{\cal A}+2L_{\cal S}}
\frac{\partial^2\zeta}{a^2},\end{aligned}$$ where we assumed that $$\begin{aligned}
{\cal A}+2L_{\cal S}\neq 0.\end{aligned}$$ Using this one can remove $\chi$ from the action (\[eq:qaction1\]) to get $$\begin{aligned}
S_\zeta^{(2)}&=\int{{\rm d}}^4x\,a^3\left[
q_1\dot\zeta^2-q_2\frac{(\partial\zeta)^2}{a^2}-q_3\frac{(\partial^2\zeta)^4}{a^4}
\right],\end{aligned}$$ where $$\begin{aligned}
q_1&:=2\left(\frac{3{\cal A}+2L_{\cal S}}{{\cal A}+2L_{\cal S}}\right)L_{\cal S},
\\
q_2&:=-2{\cal E}+\frac{8}{a}\frac{{{\rm d}}}{{{\rm d}}t}\left(
\frac{a{\cal C}L_{\cal S}}{{\cal A}+2L_{\cal S}}
\right),
\\
q_3&:=-2\left(3L_{\cal Z}+4{\cal G}\right)+\frac{8{\cal C}^2}{{\cal A}+2L_{\cal S}}.\end{aligned}$$ To ensure the stability it is required that $$\begin{aligned}
q_1>0, \quad q_2\ge 0, \quad q_3\ge 0.\end{aligned}$$ As it follows from the stability of the tensor sector that ${\cal E}>0$, the presence of nonzero ${\cal C}$ is crucial for the stability of the scalar sector.
Let us comment on the special case where ${\cal A}+2L_{\cal S}\equiv 0$. This occurs for instance in mimetic Horndeski gravity [@Haghani:2015iva; @Arroja:2015wpa; @Rabochaya:2015haa; @Arroja:2015yvd], whose Lagrangian is of the form $L\sim A_3(t)E+ A_4(t)(E^2-{\cal S})+\cdots$. This case is analogous to the $\alpha = 0$ limit of the simplest mimetic theory presented in the previous section: we always have $\zeta=\,$const and hence the curvature perturbation is nondynamical. This result is consistent with the analysis of [@Arroja:2015yvd].
Unstable examples
-----------------
We have argued that without the ${\cal C}$ term mimetic scalar-tensor theories inevitably suffer from gradient instabilities. Let us present two concrete examples that are more general than (\[eq:original-Lagrangian\]) but still are [*not*]{} healthy.
The first example is mimetic gravity generalized to include higher derivatives in the form of a general function $f(\Box\phi)$. This class of theories allows us to resolve the cosmological and black hole singularities [@Chamseddine:2016uef; @Chamseddine:2016ktu] as well as to reproduce the dynamics of loop quantum cosmology [@Liu:2017puc]. (For the Hamiltonian analysis of $f(\Box\phi)$ mimetic gravity, see Ref. [@Kluson:2017iem].) The Lagrangian is given by [@Chamseddine:2016uef; @Chamseddine:2016ktu] $$\begin{aligned}
\frac{{\cal L}}{\sqrt{-g}}=\frac{{\cal R}}{2}-\lambda\left(g^{\mu\nu}\partial_\mu\phi
\partial_\nu\phi+1\right)
-V(\phi)+f(\Box\phi).\label{eq:fbpmodel}\end{aligned}$$ In the ADM language, this Lagrangian corresponds to $$\begin{aligned}
L=\frac{1}{2}\left(R+{\cal S}-E^2\right)-V(t)+f(E),\end{aligned}$$ for which we have $$\begin{aligned}
{\cal A}=-1+f_{EE},\quad
L_{\cal S}=L_R=\frac{1}{2},\quad
{\cal C}=0,\end{aligned}$$ with $f_{EE}:=\partial^2 f/\partial E^2$. This implies that $$\begin{aligned}
q_1=3-\frac{2}{f_{EE}},\quad q_2=-1<0,\end{aligned}$$ i.e., the scenarios based on the Lagrangian of the form (\[eq:fbpmodel\]) are plagued with gradient instabilities. This instability was pointed out for the first time in Ref. [@Firouzjahi:2017txv]. Our result consistently reproduces that of [@Firouzjahi:2017txv].
The second example is the Lagrangian of the form $$\begin{aligned}
L=B_4(t)R+B_5(t)\left({\cal Y}-\frac{1}{2}ER\right)+
L'(E,{\cal S};t).\label{eq:unstable2}\end{aligned}$$ Also in this case we have ${\cal C}=0$, and hence the two conditions ${\cal E}>0$ and $q_2=-2{\cal E}>0$ are not compatible. This confirms the argument in Sec. \[subsec:cure\].
The Lagrangian (\[eq:unstable2\]) typically appears in the case of mimetic Horndeski gravity [@Haghani:2015iva; @Arroja:2015wpa; @Rabochaya:2015haa; @Arroja:2015yvd]. However, as noted above, the condition ${\cal A}+2L_{\cal S}\neq 0$ is not satisfied for the Horndeski terms. If the $L'(E,{\cal S};t)$ part is detuned away from the Horndeski form, $\zeta$ is dynamical and unstable.
A healthy extension of imperfect dark matter
--------------------------------------------
To demonstrate that it is indeed possible to construct a stable cosmological model subject to the mimetic constraint, let us consider a following simple extension of imperfect dark matter [@Chamseddine:2014vna; @Mirzagholi:2014ifa], $$\begin{aligned}
L &=\frac{1}{2}\left[R+{\cal S}-(1-\alpha)E^2\right]
+\beta(t) RE +\frac{\beta^2}{2\alpha} R^2,\label{eq:Lag-minimal}\end{aligned}$$ where we assume for simplicity that $\alpha$ is a constant, but $\beta$ is a time-dependent function.
Since $R=0$ on the homogeneous background, the background equation remains the same as presented in Sec. \[sec:cosmology-in-mim\] (with $V(t)=0$): $$\begin{aligned}
3H^2+2\dot H = 0.\end{aligned}$$ Thus, we consider a universe dominated by mimetic dark matter, $$\begin{aligned}
3H^2=\frac{C}{a^3}
\quad
\Rightarrow
\quad
H=\frac{2}{3t}.\end{aligned}$$
For the Lagrangian (\[eq:Lag-minimal\]) we have $$\begin{aligned}
L_{\cal S}=\frac{1}{2}, \quad {\cal E}=L_R=\frac{1}{2}+3\beta H,\quad L_{\cal Z}=0,\end{aligned}$$ and $$\begin{aligned}
q_1 = \frac{3\alpha-2}{\alpha},
\quad
q_2=\frac{4\dot\beta}{\alpha}-\left(1+2q_1 \beta H\right),
\quad
q_3=0.\end{aligned}$$ Since $L_{\cal Z}=q_3\equiv 0$, we do not need to care about the higher spatial derivative terms. Note that the coefficient in front of the $R^2$ term ($\beta^2/2\alpha$) is tuned so that $q_3=0$. If one would instead want to have a higher spatial derivative term $\partial^4\zeta$ that could affect the perturbation evolution on small scales, one may introduce a slight deviation from this value.
To be more specific, suppose that $$\begin{aligned}
\alpha = -\varepsilon,\quad \beta= - \xi \varepsilon t=-\xi\varepsilon\phi,\end{aligned}$$ with $0<\varepsilon\ll 1$ and $\xi={\cal O}(1)$. (Recall that we use the units ${M_{\rm Pl}}=1$.) In this case, we have $L_R=1/2-2\xi\varepsilon>0$. The propagation speed of gravitational waves, $c_h$, is slightly subluminal if $\xi>0$, $$\begin{aligned}
c_h=\sqrt{\frac{L_R}{L_{\cal S}}}=1-{\cal O}(\varepsilon)<1.\end{aligned}$$ Depending on the assumption about the origin of the high energy cosmic rays, a lower bound on $c_h$ has been obtained from gravitational Cherenkov radiation [@Moore:2001bv; @Elliott:2005va; @Kimura:2011qn]. In the case of the galactic origin, the constraint reads $$\begin{aligned}
1-c_h<2\times 10^{-15},\end{aligned}$$ which in turn sets the upper bound of $\varepsilon$. For the scalar sector, we have $$\begin{aligned}
q_1=\frac{2+3\varepsilon}{\varepsilon}>0,
\quad
q_2=\frac{20}{3}\xi -1+4\xi\varepsilon.\end{aligned}$$ It can be seen that $q_2>0$ for $\xi\gtrsim 1$ and the sound speed, $c_s$, is very small, $$\begin{aligned}
c_s=\sqrt{\frac{q_2}{q_1}}={\cal O}(\varepsilon^{1/2})\ll 1.\end{aligned}$$
Thus, we see that a simple healthy extension of mimetic gravity is indeed possible. It is straightforward to promote the ADM Lagrangian (\[eq:Lag-minimal\]) to a manifestly covariant form, which will be reported, supplemented with cosmological applications, in a separate publication [@Hirano:nextwork].
Summary and future prospects
============================
In this paper, we have developed a general theory of cosmological perturbations in the unitary gauge in scalar-tensor theories with the mimetic constraint, focusing in particular on the gradient instability issue. In the unitary gauge ($\phi(t,\Vec{x})=t$), the mimetic constraint enforces $N(t,\Vec{x})=1$, which greatly simplifies the analysis based on the ADM formalism. We have presented a concrete stable example of a mimetic theory of gravity, in which the mimetic scalar field plays the role of dark matter. As the evolution of density perturbations on small scales is modified in our new mimetic dark matter model, it would be interesting to investigate the power spectrum, which we leave for a future study.
For the purpose of stabilizing the scalar perturbations, we have introduced the coupling between the curvature and the scalar degree of freedom. This could modify the nature of a weak gravitational field inside the solar system. For this reason the solar-system constraints on our mimetic dark matter models should be studied. It would also be interesting to address the problem of the formation of a caustic singularity in mimetic dark matter. We will come back to those issues in a future study.
We thank Yuji Akita, Tomohiro Harada, Ryotaro Kase, Shinji Mukohyama, Kazufumi Takahashi, Shuichiro Yokoyama, and Daisuke Yoshida for useful comments and fruitful discussion. This work was supported in part by the MEXT-Supported Program for the Strategic Research Foundation at Private Universities, 2014-2017, the JSPS Research Fellowships for Young Scientists No. 15J04044 (S.N.), and by the JSPS Grants-in-Aid for Scientific Research No. 16H01102 and No. 16K17707 (T.K.).
Ostrogradsky instability in mimetic $F({\cal R})$ gravity
=========================================================
In this appendix, we consider the $F({\cal R})$ extension of mimetic gravity [@Nojiri:2014zqa] and show that the theory is plagued with the Ostrogradsky instability.
The action for mimetic $F({\cal R})$ gravity is given by [@Nojiri:2014zqa] $$\begin{aligned}
S=\int{{\rm d}}^4x\sqrt{-g}\left[\frac{F({\cal R})}{2}
-\lambda\left(g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+1\right)
-V(\phi)\right].\label{mimeticFR}\end{aligned}$$ Following the usual procedure in $F({\cal R})$ gravity, we rewrite the first term by introducing an auxiliary scalar field $\varphi$ as $$\begin{aligned}
F({\cal R})\;\to\;F'(\varphi)({\cal R}-\varphi)+F(\varphi),\end{aligned}$$ where we assumed that $F''(\varphi)\neq 0$. Performing the conformal transformation $\widetilde g_{\mu\nu}=e^{2\sigma}g_{\mu\nu}$ with $e^{2\sigma}=F'(\varphi)$, we obtain the action in the Einstein frame as $$\begin{aligned}
S&=\int {{\rm d}}^4x \sqrt{-\widetilde{g}}
\biggl[
\frac{\widetilde{\cal R}}{2}-3\widetilde g^{\mu\nu}\partial_\mu\sigma\partial_\nu\sigma-U(\sigma)
\notag \\ &\quad
-e^{-2\sigma}\lambda\left(\tilde g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+e^{-2\sigma}\right)
-e^{-4\sigma}V(\phi)
\biggr],\end{aligned}$$ where $U=\left(\varphi F'-F\right)/2(F')^2$. We are thus lead to study a bi-scalar mimetic theory whose action is of the form $$\begin{aligned}
S&=\int{{\rm d}}^4x\sqrt{-g}\biggl\{\frac{{\cal R}}{2}
-3g^{\mu\nu}\partial_\mu\sigma\partial_\nu\sigma+P(\phi, \sigma)
\notag \\ &\quad
-\lambda\left[g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+2f(\sigma)\right]
\biggr\}.\label{ap:bi-scalar-S}\end{aligned}$$ Here the redefinition of the Lagrange multiplier $\lambda$ was made. The tildes were omitted for brevity from the Einstein frame metric. To keep generality, we allow for general functions $P(\phi,\sigma)$ and $f(\sigma)$. In this sense mimetic $F({\cal R})$ gravity is a specific case of the general theory described by (\[ap:bi-scalar-S\]).
It follows from $\delta S/\delta \lambda = 0$ that $$\begin{aligned}
g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi+2f(\sigma)=0,\end{aligned}$$ which can be solved for $\sigma$ as $$\begin{aligned}
\sigma = h(X), \quad X:=-\frac{1}{2}g^{\mu\nu}\partial_\mu\phi\partial_\nu\phi,\label{FR_solution}\end{aligned}$$ where $h$ is the inverse function of $f$. Substituting this to the action (\[ap:bi-scalar-S\]), we obtain $$\begin{aligned}
S=\int{{\rm d}}^4x\sqrt{-g}\left[
\frac{{\cal R}}{2}-3h_X^2g^{\mu\nu}\partial_\mu X \partial_\nu X +P(\phi, X)
\right].\label{ostaction1}\end{aligned}$$ Now it is obvious that the field equation for $\phi$ is of fourth order, implying the presence of the Ostrogradsky ghost. Hence, mimetic $F({\cal R})$ gravity [@Nojiri:2014zqa] is never viable.
Note that a singular disformal transformation (i.e., imposing the mimetic constraint) does not commute with a regular conformal transformation. In other words, the frame in which the mimetic constraint is imposed could be crucial. Indeed, transforming the $F({\cal R})$ action to the scalar-tensor theory in the [*Einstein frame*]{} first and then imposing the mimetic constraint do not lead to (\[ostaction1\]). The resultant theory is original mimetic gravity plus another canonical scalar field, which is free from the Ostrogradsky ghost.
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|
---
abstract: |
We further investigate a divisibility relation on the set $\beta N$ of ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a hierarchy in which a position of every ultrafilter depends on the set of prime ultrafilters it is divisible by. We also construct ultrafilters with many immediate successors in this hierarchy and find positions of products of ultrafilters.\
[*2010 Mathematics Subject Classification*]{}: 03E20, 54D35, 54D80. [*Key words and phrases*]{}: divisibility, Stone-Čech compactification, ultrafilter
---
[**$\widemid$-divisibility of ultrafilters**]{}
[**Boris Šobot**]{}\
[Faculty of Sciences, University of Novi Sad,\
Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia\
e-mail: [email protected]]{}
Introduction
============
Let $N$ denote the set of natural numbers (without zero). The Stone-Čech compactification of the discrete space $N$ is the space $\beta N$ of all ultrafilters over $N$. For each $n\in N$ the principal ultrafilter $p_n=\{A\ps N:n\in A\}$ is identified with the respective element $n$. The topology on $\beta N$ is generated by (clopen) base sets of the form ${\bar A}=\{x\in\beta N:A\in x\}$.
A family $\cF$ of subsets of $N$ has the finite intersection property (f.i.p.) if the intersection of every finitely many elements of $\cF$ is nonempty. $\cF$ has the uniform f.i.p. if the intersection of every finitely many elements of $\cF$ is infinite. Every family with f.i.p. is contained in an ultrafilter, and every family with u.f.i.p. is contained in a nonprincipal ultrafilter. A family $\cF$ with the f.i.p. generates a filter $F$ if for every $B\in F$ there are $A_1,A_2,\dots,A_n\in\cF$ such that $A_1\cap A_2\cap\dots\cap A_n\subseteq B$.
If $f:N\str N$ is a function, the direct and inverse image of a set $A\subseteq N$ are $f[A]=\{f(a):a\in A\}$ and $f^{-1}[A]=\{b:f(b)\in A\}$. For every $f:N\str N$ there is unique continuous function $\widetilde{f}:\beta N\str\beta N$ extending $f$. It is given by $\widetilde{f}(x)=\{A\ps N:f^{-1}[A]\in x\}$ for $x\in\beta N$. $\widetilde{f}(x)$ is also generated by sets $f[A]$ for $A\in x$. Since $\widetilde{g}\circ\widetilde{f}$ is a continuous extension of $g\circ f$, it follows that $\widetilde{g}\circ\widetilde{f}=\widetilde{g\circ f}$ for every two functions $f,g:N\str N$.
The multiplication $\cdot$ on $N$ can be extended to $\beta N$ (using the same notation $\cdot$ for the extension) in such way that $(\beta N,\cdot)$ is a compact Hausdorff right-topological semigroup: for $x,y\in\beta N$, $$\label{product}
A\in x\cdot y\dl\{n\in N:A/n\in y\}\in x,$$ where for $A\ps N$ and $n\in N$, $A/n=\{m\in N:mn\in A\}=\{\frac an:a\in A,n\mid a\}$. The known properties if this structure are described in detail in [@HS].
We fix some more notation. Throughout the paper $P$ is the set of prime numbers. The complement of $A\subseteq N$ with respect to $N$ is $A^c=N\setminus A$, and the complement of $A\subseteq P$ with respect to $P$ is $A'=P\setminus A$. $[X]^k$ is the set of subsets of $X$ of cardinality $k$. The set of functions $f:A\str B$ will be denoted by ${{{}^{A}\hspace{-1mm}B}}$.
For $x\in\beta N$, if $A\in x$ then $X\in x$ if and only if $X\cap A\in x$. Because of this we can identify each ultrafilter $x$ on $N$ containing $A$ with the ultrafilter $x\rest A=\{X\cap A:X\in x\}$ on $A$. Thus it is also common to think of $\overline{A}$ as a subspace of $\beta N$. Also, $A^*=\overline{A}\ra A$ is the set of nonprincipal ultrafilters containing $A$.
The Rudin-Keisler preorder on $\beta N$ is defined as follows: $x\leq_{RK} y$ if and only of there is $f:N\str N$ such that $\widetilde{f}(y)=x$.
An ultrafilter $x\in N^*$ is a P-point if, for every sequence $\langle A_n:n\in N\rangle$ of sets in $x$ such that $A_m\supseteq A_n$ for $m<n$ there is a set $B\in x$ (called a pseudointersection of sets $A_n$) such that $B\setminus A_n$ is finite for every $n\in N$.
A function $c:X\str\{0,1\}$ is called a $2$-coloring of $X$. An ultrafilter $x$ is called Ramsey (selective) if, for every $A\in x$, every $k\in N$ and every $2$-coloring $c$ of $[A]^k$ there is a set $M\in x$ that is monochromatic, i.e. such that $|c[[M]^k]|=1$. It is well-known that $x$ is a Ramsey ultrafilter if and only if $x$ is minimal in $\leq_{RK}$ (there are no nonprincipal ultrafilters $y\leq_{RK}x$).
If $A\subseteq N$, we will denote $A\gstr=\{n\in N:\po a\in A\;a\mid n\}$ and $A\dstr=\{n\in N:\po a\in A\;n\mid a\}$. $\cU=\{A\ps N:A=A\gstr\}$ and $\cV=\{A\ps N:A=A\dstr\}$ are the collections of subsets of $N$ upwards/downwards closed for divisibility.
In [@So1] four divisibility relations on $\beta N$ were introduced. In [@So1] and [@So2] some properties of these relations were investigated. In particular, the relation $\widemid$ was introduced as an extension of the usual divisibility relation $\mid$ on $N$ to $\beta N$ analogous to extensions of functions described above. It was proven that, for $x,y\in\beta N$, $$\begin{aligned}
x\widemid y &\dl& x\cap\cU\subseteq y\\
&\dl& y\cap\cV\subseteq x,\end{aligned}$$ In this paper we will use these characterizations and under divisibility of ultrafilters we will understand $\widemid$-divisibility. For example, “least upper bound for $x$ and $y$” will mean the $\widemid$-smallest ultrafilter $z$ (if such exists) such that $x\widemid z$ and $y\widemid z$. “$x$ is below $y$” will mean $x\widemid y$. $\widemid$ is a preorder and, if we define $x=_\sim y\Leftrightarrow x\widemid y\land y\widemid x$, then $=_\sim$ is an equivalence relation. We denote the equivalence class of $x$ by $[x]_\sim$ and think of $\widemid$ as an order on the set of these classes. Let us also call $x,y\in\beta N$ incompatible if there is no $z\in\beta N\setminus\{1\}$ such that $z\widemid x$ and $z\widemid y$.
The motivation for introducing divisibility of ultrafilters is to inspect the effects of existence of some infinite sets of natural numbers to ultrafilters and (hopefully) the opposite: to draw number-theoretic conclusions from the existence of certain ultrafilters. However, we also hope that some of the results in this paper will help to better understand the product (\[product\]) of ultrafilters by finding its place in the divisibility hierarchy.
Prime ultrafilters {#secwidemin}
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\[funkcija\] Let $x\in\beta N$, $A\in x$ and $f:N\str N$.
\(a) If $f(a)\mid a$ for all $a\in A$, then $\widetilde{f}(x)\widemid x$.
\(b) If $a\mid f(a)$ for all $a\in A$, then $x\widemid \widetilde{f}(x)$.
\(a) If $B\in{\widetilde f}(x)\cap\cU$ then $f^{-1}[B]\in x$. But $f^{-1}[B]\cap A\subseteq B$ (because $B\in\cU$) so $B\in x$. Hence we have ${\widetilde f}(x)\widemid x$.
\(b) Analogously to (a) we prove that every $B\in{\widetilde f}(x)\cap\cV$ is also in $x$.
If $A\in x$, in order to determine $\widetilde{f}(x)$ it is enough to know values of $f(a)$ for $a\in A$. Hence, when using the lemma above we will sometimes define functions only on a set in $x$.
\[minbelow\] For every $x\in\beta N\setminus\{1\}$ there is $p\in\overline{P}$ such that $p\widemid x$.
We define a function $f:N\setminus\{1\}\str N$: let $f(n)$ be the smallest prime factor of $n$. Now ${\widetilde f}(x)\in\overline{P}$ (for $x\in\beta N\setminus\{1\}$) and, by Lemma \[funkcija\](a), ${\widetilde f}(x)\widemid x$.
Clearly, 1 is the smallest element in $(\beta N,\widemid)$. We will call $p\in\beta N\setminus\{1\}$ [*prime*]{} (or $\widemid$-minimal) if it is divisible only by 1 and itself. We will reserve labels $p,q,r,\dots$ for prime ultrafilters and $x,y,z,\dots$ for ultrafilters in general.
\[widemin\] $p\in\beta N$ is prime if and only if $p\in\overline{P}$.
Assume first that $p$ is prime but $p\in\overline{P^c}$. By Lemma \[minbelow\] there is an element $q\in\overline{P}$ below $p$. But $q\neq_\sim p$ because the set of composite numbers $(P\cup\{1\})^c\in(p\cap\cU)\setminus q$; hence $q\neq p$, a contradiction.
Now assume $p\in\overline{P}$ and $x\widemid p$, but $x\neq p$. By Lemma \[minbelow\] there is $q\in\overline{P}$ such that $q\widemid x$. It follows that $q\widemid p$. If $A\in q\setminus p$, let $B=(A\cap P)\uparrow$. Then $B\in q\cap\cU$, so $B\in p$ and $A\supseteq B\cap P\in p$, a contradiction.
Comparing this with Lemmas 7.3 and 7.5 from [@So1] we conclude that $\widemid$-minimal ultrafilters are also $\mid_L$-minimal, but not vice versa. Also, a corollary of this theorem is that there is a family of $2^{\goth c}$ incompatible ultrafilters, which improves Theorem 3.8 from [@So2].
Let us also mention that, by Fact \[greatest\], there are ultrafilters with $2^{\goth c}$-many divisors so not every divisibility can be established by means of Lemma \[funkcija\](a). It follows that the Rudin-Keisler preorder is not stronger than $\widemid$. On the other hand, neither is $\widemid$ stronger than $\leq_{RK}$, since ultrafilters containing $P$ that are not Ramsey are $\widemid$-minimal but nor $\leq_{RK}$-minimal.
Now we define levels of the $\widemid$-hierarchy.
Let $L_0=\{1\}$ and $L_n=\{a_1a_2\dots a_n:a_1,a_2,\dots,a_n\in P\}$ for $n\geq 1$. We will say that $x$ is of level $n$ if $x\in\overline{L_n}$.
In particular, $\overline{L_1}=\overline{P}$ is the set of prime ultrafilters.
In this paper we will mostly deal with ultrafilters on finite levels (belonging to $\overline{L_n}$ for some $n\in N$). In Section \[above\] we will see that there are also ultrafilters above all these finite levels.
Ultrafilters with one prime divisor
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\[defstepen\] For $A\subseteq N$ and $n\in N\cup\{0\}$ we denote $A^n=\{a^n:a\in A\}$. If $pow_n:N\str N$ is defined by $pow_n(a)=a^n$ then, for $x\in N^*$, $\widetilde{pow_n}(x)$ is generated by sets $A^n$ for $A\in x$. We will denote $\widetilde{pow_n}(x)$ with $x^n$.
Of course, $A^0=\{1\}$, $x^0=1$ and $x^1=x$.
\[kvadrati\] If $p\in\overline{P}$, the only ultrafilters below $p^n$ are $p^k$ for $k\leq n$.
If $p\in P$, the lemma is obvious. So we prove it for $p\in P^*$.
Since $\bigcup_{k\leq n}P^k\in p^n\cap\cV$, every ultrafilter $x\neq 1$ below $p^n$ must contain $P^k$ for some $k\leq n$.
If $x\in(P^k)^*$ then for every $A\subseteq P$, if $A^k\in p^k$ then we have $\bigcup_{i\leq n}A^i\in p^n\cap\cV$ so $x\widemid p^n$ implies that $A^k=P^k\cap\bigcup_{i\leq n}A^i\in x$. Thus $x\rest P^k=p^k\rest P^k$, which means that $x=p^k$.
For $A,B\subseteq N$ let $AB=\{ab:a\in A,b\in B\land gcd(a,b)=1\}$. In particular, $A^{(n)}=\underbrace{A\cdot A\cdot\dots\cdot A}_n$. If $p\in\overline{P}$ let $F_n^p=\{A^{(n)}:A\in p\rest P\}$.
Note the difference between $A^{(n)}$ and $A^n$ from Definition \[defstepen\]: elements of $A^{(n)}$ must be products of mutually prime numbers. $A$ will almost always be a subset of $P$ in which case “mutually prime” will mean “distinct”.
If $p\in P$ (a principal ultrafilter), $\{p\}^{(n)}=\emptyset\in F_n^p$, so there is no ultrafilter $x\supseteq F_n^p$.
Let $p\in P^*$. For any ultrafilter $x\supseteq F_n^p$, $p$ is the only ultrafilter from $\overline{P}$ below $x$.
If $q\in\overline{P}$ is such that $q\neq p$, there is $A\subseteq P$ such that $A\in p\setminus q$. Then $\bigcup_{k\leq n}A^{(k)}\in x\cap\cV\setminus q$, so $q\widetilde{\nmid}x$.
\[broj\] Let $p\in P^*$.
\(a) $p\cdot p\supseteq F_2^p$.
\(b) There are either finitely many or $2^{\goth c}$ ultrafilters $x\supseteq F_n^p$.
\(a) Let $A\in p\rest P$. Since $A^{(2)}/a=A\setminus\{a\}$ for $a\in A$, $\{a\in N:A^{(2)}/a\in p\}\supseteq A\in p$ and so $A^{(2)}\in p\cdot p$.\
(b) The set of ultrafilters $x\supseteq F_n^p$ is actually $\bigcap_{X\in F_n^p}\overline{X}$ so it is closed. By [@W], Theorem 3.3, closed subsets of $\beta N$ are either finite or of cardinality $2^{\goth c}$. Hence, for every $p\in P^*$ there are either finitely many or $2^{\goth c}$ ultrafilters containing $F_n^p$.
\[ramsey\] Let $p\in P^*$. There is unique ultrafilter $x\supseteq F_n^p$ if and only if $p$ is Ramsey.
Assume $p$ is Ramsey. To prove that $x$ is unique it suffices to show that for every set $S\subseteq P^{(n)}$ one of the sets $S$ and $P^{(n)}\setminus S$ contains a set from $F_n^p$. Define a coloring of $[P]^n$ as follows: $$c(\{a_1,a_2,\dots,a_n\})=\left\{\begin{array}{ll}
0, \mbox{if }a_1a_2\dots a_n\in S\\
1, \mbox{otherwise.}
\end{array}\right.$$ Since $p$ is Ramsey there is a monochromatic set $M\in p$, say $c(\{a_1,a_2,\dots,a_n\})=0$ for all $a_1,a_2,\dots,a_n\in M$. This means that $M^{(n)}\subseteq S$.
Now assume $p$ is not Ramsey. Then there is a 2-coloring $c$ of $[P]^n$ such that $p$ does not contain a monochromatic subset. Let $S=\{a_1a_2\dots a_n:c(\{a_1,a_2,\dots,a_n\})=0\}$; then both $F_n^p\cup\{S\}$ and $F_n^p\cup\{P^{(n)}\setminus S\}$ have the f.i.p. so there are at least two ultrafilters containing $F_n^p$.
It is well-known that, under CH, there are Ramsey ultrafilters in $N^*$ ([@CN], Theorem 9.19). Also, not all ultrafilters in $N^*$ are Ramsey. If $f:N\str P$ is any bijection, $\widetilde{f}$ maps Ramsey to Ramsey, and non-Ramsey to non-Ramsey ultrafilters, so (under CH) in $P^*$ there are also ultrafilters of both types. On the other hand, each Ramsey ultrafilter is a P-point and Shelah proved that it is consistent with ZFC that there are no P-points in $N^*$ (the proof can be found in [@Sh]).
Blass proved in [@B2] that (under CH) there is a non-Ramsey ultrafilter $p$ such that for every 3-coloring $c:[N]^2\str\{0,1,2\}$ there is $A\in p$ such that $|c[[A]^2]|\leq 2$. By a simple modification of the proof of Theorem \[ramsey\] we get that, for such $p\in P^*$, there are exactly two ultrafilters $x\supseteq F_2^p$.
\[proizvod\] Let $x\in\overline{P^{(2)}}$.
\(a) If there are disjoint $A,B\ps P$ such that $AB\in x$ then $x$ is divisible by at least two ultrafilters from $\overline{P}$.
\(b) For any two distinct $p,q\in\overline{P}$, $x$ is divisible by $p$ and $q$ if and only if for every two disjoint $A\in p$, $B\in q$ holds $AB\in x$.
\(a) If $AB\in x$ for disjoint $A,B\ps P$ then we can define functions $f_A$ and $f_B$ on $AB$ such that $f_A(ab)=a$ and $f_B(ab)=b$ (for $a\in A$, $b\in B$) so, by Lemma \[funkcija\](a), $\widetilde{f_A}(x)\widemid x$ and $\widetilde{f_B}(x)\widemid x$. Finally, $\widetilde{f_A}(x)\neq\widetilde{f_B}(x)$ because $A=f_A[AB]\in\widetilde{f_A}(x)$ and $B=f_B[AB]\in\widetilde{f_B}(x)$.\
(b) First let $p\widemid x$ and $q\widemid x$. Let $A\in p\rest P$ and $B\in q\rest P$ be disjoint. Then $A\gstr\in p\cap\cU\subseteq x$ and $B\gstr\in q\cap\cU\subseteq x$, so $AB=A\gstr\cap B\gstr\cap P^{(2)}\in x$.
Now let $P=A\cup B$ be a partition of $P$ with $A\in p$, $B\in q$. As in (a) we define $f_A$ and $f_B$. Since $AB\in x$, $\widetilde{f_A}(x)$ and $\widetilde{f_B}(x)$ are ultrafilters below $x$. Assume $\widetilde{f_A}(x)\neq p$. Let $A_1\in\widetilde{f_A}(x)\setminus p$ be disjoint from $B$; by the first implication $A_1B\in x$. But $A_1B$ and $(P\setminus(A_1\cup B))B$ (belonging to $x$ by assumption) are disjoint so they can not both be in $x$, a contradiction. Thus $\widetilde{f_A}(x)=p$, and in the same way we prove $\widetilde{f_B}(x)=q$.
\[twodiv\] Let $p\in\overline{P}$. The ultrafilters such that their only proper divisors are 1 and $p$ are exactly $p^2$ and (for $p\in P^*$) ultrafilters containing $F_2^p$.
In Lemma \[kvadrati\] we proved that the only proper divisors of $p^2$ are 1 and $p$. The proof for $x\supseteq F_2^p$ is similar. Now assume that $x$ is any ultrafilter such that 1 and $p$ are its only proper divisors. $x$ belongs either to $\overline{P}$, $\overline{P^2}$, $\overline{P^{(2)}}$ or to $\overline{(P\cup P^2\cup P^{(2)}\cup\{1\})^c}$.
In the first case $x$ has only one proper divisor, 1.
In the second case, if we let $p=\{A\ps P:A^2\in x\}$, it is easy to prove $x=p^2$.
Let $x\in\overline{P^{(2)}}$ and $G=\{A\subseteq P:A^{(2)}\in x\}$. $G$ is closed for finite intersections and sets in $G$ are nonempty, so $G$ has the f.i.p. If $A\cup B$ is any partition of $P$, by Lemma \[proizvod\] $AB\notin x$, so it follows that $A^{(2)}\in x$ or $B^{(2)}\in x$ (because $P^{(2)}=A^{(2)}\cup B^{(2)}\cup AB$), hence $A\in G$ or $B\in G$. This means that $G$ is an ultrafilter on $P$ and $x\supseteq F_2^G$. Since $p$ is the only element of $\overline{P}$ below $x$, we have $x\supseteq F_2^p$.
Finally, if $x\notin\overline{P\cup P^2\cup P^{(2)}\cup\{1\}}$, we can define the function $g:N\setminus(P\cup\{1\})\str N$ by $g(a_1a_2\dots a_n)=a_1a_2$ (where $a_1\leq a_2\leq\dots\leq a_n$ are prime). Then $\widetilde{g}(x)\neq x$ is an element of $P^2\cup P^{(2)}$ below $x$, so $x$ has more than two proper divisors.
The following definition and lemmas will be used in the proof of Theorem \[2cFpn\].
Let $X\subseteq N$ and let $d=\{X_k:k\in N\}$ be a partition of $X^{(n)}$. A set $A\subseteq X$ is $d$-thick if for all $m\in N$ and all finite partitions $A=A_1\cup A_2\cup\dots\cup A_m$ there is $i\leq m$ such that for every $k\in N$ $A_i^{(n)}\cap X_k\neq\emptyset$.
The condition of $d$-thickness strengthens the condition that $A^{(n)}$ intersects every $X_k$. The idea of this strengthening is to satisfy (b) of the lemma below.
\[thick\] Let $A\subseteq X$, $B\subseteq X$ and let $d=\{X_k:k\in N\}$ be a partition of $X^{(n)}$.
\(a) If $A$ is $d$-thick and $A\subseteq B$, then $B$ is $d$-thick.
\(b) If neither $A$ nor $B$ are $d$-thick, then $A\cup B$ is not $d$-thick.
\(a) is obvious.\
(b) By (a) we may assume without loss of generality that $A\cap B=\emptyset$. Let partitions $A=A_1\cup A_2\cup\dots\cup A_{m_A}$ and $B=B_1\cup B_2\cup\dots\cup B_{m_B}$ and $k_i,l_i\in N$ be such that $A_i^{(n)}\cap X_{k_i}=\emptyset$ for all $i\leq m_A$ and $B_i^{(n)}\cap X_{l_i}=\emptyset$ for all $i\leq m_B$. Then the partition $A\cup B=A_1\cup A_2\cup\dots\cup A_{m_A}\cup B_1\cup B_2\cup\dots\cup B_{m_B}$ and the same $k_i,l_i$ witness that $A\cup B$ is not $d$-thick.
\[thick2\] If, for each $n\geq 2$, $d_n=\{X_{n,k}:k\in N\}$ is a partition of $X^{(n)}$ such that for all $k\in N$ $$\label{eqparticije2}
X_{n+1,k}\subseteq\{xa:x\in X_{n,k},a\in P\}$$ and $A\subseteq X$ is not $d_{n_0}$-thick then, for all $n\geq n_0$, $A$ is not $d_n$-thick.
It suffices to prove the theorem for $n=n_0+1$. Let the partition $A=A_1\cup A_2\cup\dots\cup A_m$ and $k_i\in N$ be such that $A_i^{(n_0)}\cap X_{n_0,k_i}=\emptyset$ for all $i\leq m$. Then the same partition witnesses that $A$ is not $d_{n_0+1}$-thick: if $xa\in A_i^{(n_0+1)}\cap X_{n_0+1,k_i}$ (for $x\in X_{n_0,k}$, $a\in P$), then $x\in A_i^{(n_0)}\cap X_{n_0,k_i}$, a contradiction.
\[Pthick\] (a) There is a coloring $c:[N]^2\str N$ such that for every $k\in N$:
\(b) There are partitions $d_n=\{X_{n,k}:k\in N\}$ of $P^{(n)}$ (for $n\geq 2$) such that $P$ is $d_n$-thick for every $n\geq 2$ and (\[eqparticije2\]) holds.
\(a) We define sets $A^0_i=\{i\}$ (for $i\in N$) and, by recursion on $n$, $A^n_i=A^{n-1}_{2i-1}\cup A^{n-1}_{2i}$. Note that $a\in A_{2i-1}^{n-1}$ are exactly numbers with residue $1\leq r\leq 2^{n-1}$ modulo $2^{n}$.
First we color the pairs of numbers in the same $A^1_i$: $c(\{2i-1,2i\})=1$. By recursion on $n$, pairs of numbers $a,b\in A^n_i$ such that $a\in A^{n-1}_{2i-1}$ and $b\in A^{n-1}_{2i}$ are colored according to the difference $b-a$: if $2^j\leq b-a<2^{j+1}$ then $c(\{a,b\})=n-j$.
Let us prove (\[progr\]). Let $s=\{a_0+md:0\leq m\leq 2^{k}\}$ and $j$ is such that $2^j\leq d<2^{j+1}$. At least one $a_0+md\in s$ (for $0\leq m<2^{k}$) has residue $2^{k+j-1}-d<r\leq 2^{k+j-1}$ modulo $2^{k+j}$, which means that $a_0+md\in A^{k+j-1}_{2i-1}$ and $a_0+(m+1)d\in A^{k+j-1}_{2i}$ for some $i\in N$. Then $c(\{a_0+md,a_0+(m+1)d\})=(k+j)-j=k$.\
(b) Let $c$ be the coloring of $[N]^2$ defined in (a). We define $c_n:[N]^n\str N$ (for $n\geq 2$) by $c_n(\{a_1,a_2,\dots,a_n\})=c(\{a_1,a_2\})$ for $a_1<a_2<\dots<a_n$. Then (\[progr\]) implies that for every $n\geq 2$ and every $k\in N$
Now enumerate $P=\{p_n:n\in N\}$ in the increasing order. Define the partitions $d_n=\{X_{n,k}:k\in N\}$ of $P^{(n)}$: $X_{n,k}=\{p_{a_1}p_{a_2}\dots p_{a_n}:c_n(\{a_1,a_2,\dots,a_n\})=k\}$. Obviously, (\[eqparticije2\]) holds.
To prove that $P$ is $d_n$-thick, let $P=A_1\cup A_2\cup\dots\cup A_m$ be any finite partition of $P$. If $B_i=\{a\in N:p_a\in A_i\}$, then $N=B_1\cup B_2\cup\dots\cup B_m$ is a partition of $N$. By the infinite Van der Waerden’s theorem there is $i\leq m$ such that $B_i$ contains arithmetic progressions of any length. So for every $k\in N$, there is a progression $s$ in $B_i$ of length $2^{k}+n-1$. By (\[progrn\]) there are $a_1,a_2,\dots,a_n\in s\subseteq B_i$ such that $c_n(\{a_1,a_2,\dots,a_n\})=k$ so $p_{a_1}p_{a_2}\dots p_{a_n}\in X_{n,k}\cap A_i^{(n)}$.
\[2cFpn\] (CH) There is $p\in P^*$ such that for every $n\geq 2$ there are $2^{\goth c}$ ultrafilters $x\supseteq F_n^p$.
Let, for $n\geq 2$, $d_n$ be the partition of $P^{(n)}$ given by Lemma \[Pthick\](b). By the Continuum Hypothesis we can enumerate all subsets of $P$ as $\langle S_\xi:\xi<\omega_1\rangle$; recall that $S_\xi'=P\setminus S_\xi$. By recursion on $\xi<\omega_1$ we define sets $A_\xi$ and families ${\cal F}_\xi$ such that:
(1$_\xi$) ${\cal F}_\xi$ is a countable family of infinite subsets of $P$, closed for finite intersections;
(2$_\xi$) ${\cal F}_\zeta\subseteq {\cal F}_\xi$ for $\zeta<\xi$;
(3$_\xi$) ${\cal F}_\xi=\bigcup_{\zeta<\xi}{\cal F}_\zeta$ for $\xi$ a limit ordinal;
(4$_\xi$) $A$ is $d_n$-thick for all $A\in {\cal F}_\xi$ and all $n\geq 2$;
(5$_\xi$) either $A_\xi=S_\xi$ or $A_\xi=S_\xi'$, and $A_\xi\in {\cal F}_{\xi+1}$.
First we let ${\cal F}_0=\{P\}$; by Lemma \[Pthick\] $P$ is $d_n$-thick for all $n\geq 2$. Now for every $\xi<\omega_1$, assuming we have already defined ${\cal F}_\xi$ satisfying (1$_\xi$)-(4$_\xi$), we define $A_\xi$ and ${\cal F}_{\xi+1}$.
We first prove that for at least one of the possibilities $A_\xi=S_\xi$ and $A_\xi=S_\xi'$ all sets $A_\xi\cap A$ for $A\in F_\xi$ are $d_n$-thick for all $n\geq 2$. Assume not: then there are $A,B\in {\cal F}_\xi$ and $m,n\in N$ such that $S_\xi\cap A$ is not $d_m$-thick and $S_\xi'\cap B$ is not $d_n$-thick. If, say, $m\leq n$, by Lemma \[thick2\] both those sets are not $d_n$-thick. By Lemma \[thick\](a), $S_\xi\cap(A\cap B)$ and $S_\xi'\cap(A\cap B)$ are not $d_n$-thick and, finally, by (b) of the same lemma, their union $P\cap(A\cap B)=A\cap B$ is not $d_n$-thick, which is impossible by (4$_\xi$) and (1$_\xi$).
Hence we define $A_\xi=S_\xi$ if all $S_\xi\cap A$ are $d_n$-dense for all $n\geq 2$ and all $A\in {\cal F}_\xi$, and $A_\xi=S_\xi'$ otherwise. Let ${\cal F}_{\xi+1}={\cal F}_\xi\cup\{A_\xi\cap A:A\in {\cal F}_\xi\}$. If $\xi$ is a limit ordinal, we define ${\cal F}_\xi$ as in (3$_\xi$). Clearly, all the properties (1$_\xi$)-(5$_\xi$) are now satisfied.
In the end, by (1$_\xi$) and (5$_\xi$) $p:=\bigcup_{\xi<\omega_1}{\cal F}_\xi$ is an ultrafilter on $P$. Let $n\geq 2$. For every $k\in N$ the family $F_n^p\cup\{X_{n,k}\}$ has the f.i.p. (since every $A\in p\rest P$ is $d_n$-thick, every $A^{(n)}\in F_n^p$ intersects $X_{n,k}$), so there are ultrafilters $x_k\supseteq F_n^p\cup\{X_{n,k}\}$. But $X_{n,k}$ (for $k\in N$) are disjoint, so all $x_k$ are distinct ultrafilters. By Lemma \[broj\](b) there are $2^{\goth c}$ ultrafilters $x\supseteq F_n^p$.
Ultrafilters with two prime divisors
====================================
\[atmosttwo\] No ultrafilter $x\in\overline{P^{(2)}}$ is divisible by more than two prime ultrafilters.
Assume the opposite, that $x$ is divisible by $p_1,p_2,p_3\in\overline{P}$. Let $P=A_1\cup A_2\cup A_3$ be a partition such that $A_1\in p_1$, $A_2\in p_2$ and $A_3\in p_3$. We consider three cases. $1^\circ$ If $A_1P\notin x$, then clearly $A_1\gstr\notin x$ (because $A_1\gstr\cap P^{(2)}=A_1P$), so $p_1\nwidemid x$. $2^\circ$ If $A_1(A_1\cup A_3)\in x$, then $A_2P\notin x$ so, as in $1^\circ$, $p_2\nwidemid x$. $3^\circ$ Finally, if $A_1A_2\in x$ then $A_3P\notin x$ so $p_3\nwidemid x$. In each case we reach a contradiction.
$F_{1,1}^{p,q}=\{AB:A\in p\rest P,B\in q\rest P,A\cap B=\emptyset\}$ for $p,q\in\overline{P}$.
By Lemma \[proizvod\] ultrafilters $x\supseteq F_{1,1}^{p,q}$ are exactly those ultrafilters in $P^{(2)}$ divisible by $p$ and $q$.
\[broj2\] For any distinct $p,q\in\overline{P}$:
\(a) $p\cdot q\supseteq F_{1,1}^{p,q}$ and $q\cdot p\supseteq F_{1,1}^{p,q}$.
\(b) there are either finitely many or $2^{\goth c}$ ultrafilters $r\supseteq F_{1,1}^{p,q}$.
\(a) Let $A\in p$ and $B\in q$ be disjoint. Since $AB/a=B$ for $a\in A$, $\{a\in N:AB/a\in q\}\supseteq A\in p$ and so $AB\in p\cdot q$. Analogously $AB\in q\cdot p$.\
(b) Analogous to the proof of Lemma \[broj\](b).
If $n\in N$ then $nq=qn$ for all $q\in\beta N$ and it is not hard to see that this is the only ultrafilter containing $x\supseteq F_{1,1}^{n,q}$: by Lemma 5.1 of [@So1] every ultrafilter divisible by $n$ must contain the set $nN=\{na:a\in N\}$ so $nP\in x$; but $F_{1,1}^{n,q}=\{n(B\setminus\{n\}):B\in q\}$ generates an ultrafilter on $nP$.
On the other hand, by [@HS], Corollary 6.51, for every $p\in P^*$ there is $q\in P^*$ such that $pq\neq qp$, and we have at least two ultrafilters containing $F_{1,1}^{p,q}$. We will improve this in Theorem \[dvaiznadpq\].
Let $p,q\in P^*$. If there is unique $x\supseteq F_{1,1}^{p,q}$ then both $p$ and $q$ are P-points.
First let $X_n\in q$ for $n\in N$ and, without loss of generality, assume $X_0=P$ and $X_m\subseteq X_n$ for $m>n$. Let $Y=\{mn\in P^{(2)}:m>n\mbox{ and }m\in X_n\}$. Then, for every $n\in P$, $Y/n\supseteq X_n\setminus\{1,2,\dots,n\}\in q$ so $\{n\in N:Y/n\in q\}=P\in p$ and $Y\in p\cdot q$.
Since $F_{1,1}^{p,q}$ generates the unique ultrafilter, there is $AB\in F_{1,1}^{p,q}$ such that $AB\subseteq Y$ ($A\in p$, $B\in q$). To prove that $q$ is a P-point it suffices to show that $B$ is a pseudointersection of the sets $X_n$. For every $n\in N$ there is $a\in A$ such that $n\leq a$. If $b\in B$ is such that $b>a$, then $ab\in Y$ implies $b\in X_a\subseteq X_n$. Hence $B\setminus X_n$ is finite.
Since $p\cdot q$ and $q\cdot p$ both contain $F_{1,1}^{p,q}$, we have $p\cdot q=q\cdot p$, so by interchanging the roles of $p$ and $q$ we prove in the same way that $p$ is a P-point.
\[dvaiznadpq\] For every $p\in P^*$ there is an ultrafilter $q\in P^*$ such that there are $2^{\goth c}$ ultrafilters $r\supseteq F_{1,1}^{p,q}$.
Let $p\in P^*$ be given. Let $f:P\str {{{}^{N}\hspace{-1mm}P}}$ be such that, if $f(i)=f_i$, then $\langle f_i:i\in P\rangle$ is the sequence of all eventually constant functions $f_i:N\str P$ (i.e. such that there are $n_0\in N$ and $a\in P$ so that $f_i(n)=a$ for $n\geq n_0$), ordered in such way that for all $n\in N$: $f_i(n)\leq i$ for $i\in\{2,3\}$ and $f_i(n)<i$ for $i>3$. The set $D=\{f_i:i\in P\}$ is dense in the space ${{{}^{N}\hspace{-1mm}P}}$ (with the usual Tychonoff topology). For $m,n\in N$ and $A\in p$ let $U_n(A)=\{x\in{{{}^{N}\hspace{-1mm}P}}:x(n)\in A\}$ and $V_{m,n}=\{x\in{{{}^{N}\hspace{-1mm}P}}:x(m)\neq x(n)\}$. Then the family $\{U_n(A):n\in N\land A\in p\}\cup\{V_{m,n}:m\neq n\}$ has the uniform f.i.p. so the family $F=\{f^{-1}[U_n(A)]:n\in N\land A\in p\}\cup\{f^{-1}[V_{m,n}]:m\neq n\}$ also has the uniform f.i.p. (because $f$ is one-to-one, all $U_n(A)$ and $V_{m,n}$ are open and $D$ is dense in ${{{}^{N}\hspace{-1mm}P}}$). Hence there is an ultrafilter $q\in P^*$ containing $F$.
Now, for $n\in N$, let $g_n:P\str N$ be defined by $g_n(i)=if_i(n)$, and let $r_n=\widetilde{g_n}(q)$. Since $f_i(n)\neq i$ for all $i>3$ and all $n$, we conclude that $r_n\in\overline{P^{(2)}\cup\{4,9\}}$. Moreover, by Lemma \[funkcija\](b) all $r_n$ are divisible by $q$, hence they are nonprincipal and $r_n\in(P^{(2)})^*$.
Let $h_1,h_2:P^{(2)}\cup\{4,9\}\str P$ be defined by $h_1(ab)=a$ and $h_2(ab)=b$ for $a,b\in P$ and $a\leq b$. We prove that $\widetilde{h_1}(r_n)=p$ and $\widetilde{h_2}(r_n)=q$ for every $n\in N$. First, $\widetilde{h_2}(r_n)=\widetilde{h_2}\circ\widetilde{g_n}(q)=\widetilde{h_2\circ g_n}(q)=q$ (because $h_2\circ g_n$ is the identity function).
On the other hand, $h_1\circ g_n(i)=f_i(n)$ for all $i$. For any $A\in p$ and all $i\in f^{-1}[U_n(A)]$ we have $f_i(n)\in A$. Hence $h_1\circ g_n[f^{-1}[U_n(A)]]\subseteq A$ and so $A\in\widetilde{h_1\circ g_n}(q)$. Thus $\widetilde{h_1}(r_n)=\widetilde{h_1\circ g_n}(q)=p$.
By Lemma \[funkcija\](a) $p\widemid r_n$ and $q\widemid r_n$ for each $n\in N$, so $r_n\in F_{1,1}^{p,q}$. It remains to prove that all $r_n$ are distinct so, by Lemma \[broj2\](b), the set of ultrafilters $r\supseteq F_{1,1}^{p,q}$ will be of cardinality $2^{\goth c}$. Let $m<n$. We prove that the sets $g_m[f^{-1}[V_{m,n}]]$ and $g_n[f^{-1}[V_{m,n}]]$ are disjoint: if we assume that $g_m(i)=g_n(j)$ for some $i,j\in f^{-1}[V_{m,n}]$ then $if_i(m)=jf_j(n)$, so since $f_i(m)\leq i$, $f_j(n)\leq j$ and all of the numbers $i,j,f_i(m),f_j(n)$ are prime, we have $i=j$ and $f_i(m)=f_i(n)$, a contradiction with the fact $f_i\in V_{m,n}$. But $g_m[f^{-1}[V_{m,n}]]\in r_m$ and $g_n[f^{-1}[V_{m,n}]]\in r_n$, so $r_m\neq r_n$.
The higher levels
=================
We call ultrafilters of the form $p^k$ for some $p\in\overline{P}$ and $k\in N$ [*basic*]{}. Let $\cB$ be the set of all basic ultrafilters, and let $\cA$ be the set of all functions $\alpha:\cB\str N\cup\{0\}$ with finite support (i.e. such that $\{b\in\cB:\alpha(b)\neq 0\}$ is finite).
We will abuse notation and write $\alpha=\{(b_1,n_1),(b_2,n_2),\dots,(b_m,n_m)\}$ if $\alpha(b)=0$ for $b\notin\{b_1,b_2,\dots,b_m\}$ (allowing also some of the $n_i$ to be zeros).
Let $\alpha=\{(p_1^{k_1},n_1),(p_2^{k_2},n_2),\dots,(p_m^{k_m},n_m)\}\in\cA$ ($p_i\in\overline{P}$). With $F_\alpha$ we denote the family of all sets $$\begin{aligned}
\label{eqhigher}
(A_1^{k_1})^{(n_1)}(A_2^{k_2})^{(n_2)}\dots(A_m^{k_m})^{(n_m)}=\{\prod_{i=1}^m\prod_{j=1}^{n_i}a_{i,j}^{k_i} &:& a_{i,j}\in A_i\mbox{ for all }i,j\\
&&\land\mbox{ all }a_{i,j}\mbox{ are distinct}\}\end{aligned}$$ such that: (i) $A_i\in p_i\rest P$, (ii) $A_i=A_j$ if $p_i=p_j$ and $A_i\cap A_j=\emptyset$ otherwise.
$F_{1,1}^{p,q}=F_\alpha$ for $\alpha=\{(p,1),(q,1)\}$ and $F_2^p=F_\beta$ for $\beta=\{(p,2)\}$. If $\gamma=\{(p^2,1)\}$ then the set $F_\gamma=\{A^2:A\in p\rest P\}$ generates $p^2$.
If $\alpha=\{(p_1^{k_1},n_1),(p_2^{k_2},n_2),\dots,(p_m^{k_m},n_m)\}\in\cA$ ($p_i\in\overline{P}$), we denote $\sigma(\alpha)=\sum_{i=1}^m(k_in_i)$.
\[nivoi\] The $n$-th level $\overline{L_n}$ (for $n\in N$) consists precisely of ultrafilters containing $F_\alpha$ for some $\alpha\in\cA$ such that $\sigma(\alpha)=n$.
To avoid cumbersome notation we prove the theorem only for $n=4$. This special case contains essentially all the ideas needed for the proof in general.
Ultrafilters in the 4th level contain $L_4=\{a_1a_2a_3a_4:a_1,a_2,a_3,a_4\in P\}$. We partition $L_4$ as $$L_4=P^4\cup P^3P\cup (P^2)^{(2)}\cup P^2P^{(2)}\cup P^{(4)}.$$ So for every ultrafilter $x\in\overline{L_4}$ we have 5 cases.
$1^\circ$ $x\ni P^4=\{a^4:a\in P\}$. Then $p:=\{A\ps P:A^4\in x\}$ is an ultrafilter in $\overline{P}$ and it is easy to see that $x=p^4$, so $x\supseteq F_\alpha$ for $\alpha=\{(p^4,1)\}$.
$2^\circ$ $x\ni P^3P=\{a^3b:a,b\in P,a\neq b\}$. We define functions $f_1(a^3b)=a^3$ and $f_2(a^3b)=b$ for $a^3b\in P^3P$. Let $p^3={\widetilde f_1}(x)$ (clearly, ${\widetilde f_1}(x)\in\overline{P^3}$) and $q={\widetilde f_2}(x)$. We prove that $x\supseteq F_\alpha$ for $\alpha=\{(p^3,1),(q,1)\}$. Let $A^3B\in F_\alpha$ (it may be that $p=q$ - then $A=B$, or $p\neq q$ and $A\cap B=\emptyset$). Then $A^3B=f_1^{-1}[A^3]\cap f_2^{-1}[B]\in x$.
$3^\circ$ $x\ni (P^2)^{(2)}=\{a^2b^2:a,b\in P,a\neq b\}$. We define functions $f_1(a^2b^2)=a^2$ and $f_2(a^2b^2)=b^2$ for $a^2b^2\in (P^2)^{(2)}$ and $a<b$. Let $p^2={\widetilde f_1}(x)$ and $q^2={\widetilde f_2}(x)$. If $p\neq q$ then $x\supseteq F_\alpha$ for $\alpha=\{(p^2,1),(q^2,1)\}$: for each disjoint $A\in p\rest P$ and $B\in q\rest P$, $A^2B^2\supseteq f_1^{-1}[A^2]\cap f_2^{-1}[B^2]\in x$. Otherwise, $x\supseteq F_\alpha$ for $\alpha=\{(p^2,2)\}$: for each $A\in p$, $(A^2)^{(2)}=f_1^{-1}[A^2]\cap f_2^{-1}[A^2]\in x$.
$4^\circ$ $x\ni P^2P^{(2)}=\{a^2bc:a,b,c\in P,a\neq b\neq c\neq a\}$. We define functions $f_1(a^2bc)=a^2$, $f_2(a^2bc)=b$ and $f_3(a^2bc)=c$ for $a^2bc\in P^2P^{(2)}$ and $b<c$. Let $p^2={\widetilde f_1}(x)$, $q={\widetilde f_2}(x)$ and $r={\widetilde f_3}(x)$. If $q=r$ then $x\supseteq F_\alpha$ for $\alpha=\{(p^2,1),(q,2)\}$, otherwise $x\supseteq F_\alpha$ for $\alpha=\{(p^2,1),(q,1),(r,1)\}$.
$5^\circ$ $x\ni P^{(4)}=\{abcd:a,b,c,d\in P,\;a,b,c,d\mbox{ all distinct}\}$. Analogously to previous cases, $x\supseteq F_\alpha$ for one of the following: $\alpha=\{(p,4)\}$, $\alpha=\{(p,3),(q,1)\}$, $\alpha=\{(p,2),(q,1),(r,1)\}$, $\alpha=\{(p,2),(q,2)\}$ or $\alpha=\{(p,1),(q,1),(r,1),(s,1)\}$ for some $p,q,r,s\in\overline{P}$.
For every $\alpha\in\cA$ and every $p\in\overline{P}$ let $\alpha\rest p=\langle\alpha(p^k):k\in N\rangle$. Clearly, all such sequences have finitely many non-zero elements. If $\vec{x}=\langle x_k:k\in N\rangle$ and $\vec{y}=\langle y_k:k\in N\rangle$ are two such sequences in $N\cup\{0\}$ we say that $\vec{y}$ [*dominates*]{} $\vec{x}$ if for every $m\in N$, $\sum_{k\geq m}x_k\leq\sum_{k\geq m}y_k$.
We define an order on $\cA$ as follows: $\alpha\leq\beta$ if for every $p\in\overline{P}$ $\beta\rest p$ dominates $\alpha\rest p$.
If $\alpha=\{(p,2)\}$ and $\beta=\{(p,1),(p^2,1)\}$ (for some $p\in\overline{P}$), then $\alpha\rest p=\langle 2,0,0,\dots\rangle$ and $\beta\rest p=\langle 1,1,0,0,\dots\rangle$ so $\beta\rest p$ dominates $\alpha\rest p$ and $\alpha\leq\beta$.
But if $\alpha=\{(p,2),(p^2,2),(q,2)\}$ and $\beta=\{(p,1),(p^3,3),(q^2,1)\}$, then $\beta\rest p=\langle 1,0,3,0,\dots\rangle$ dominates $\alpha\rest p=\langle 2,2,0,0,\dots\rangle$, but $\beta\rest q=\langle 0,1,0,0,\dots\rangle$ does not dominate $\alpha\rest q=\langle 2,0,0,\dots\rangle$ so $\alpha\leq\beta$ does not hold.
It is not hard to see that $\alpha\leq\beta$ implies $\sigma(\alpha)\leq\sigma(\beta)$.
\[poredak\] Let $\alpha,\beta\in\cA$.
\(a) If $x\supseteq F_\alpha$, $y\supseteq F_\beta$ and $x\widemid y$, then $\alpha\leq\beta$.
\(b) There is no $x\supseteq F_\alpha\cup F_\beta$ for $\alpha\neq\beta$.
\(a) Assume that $\alpha\leq\beta$ does not hold. This means that, for some prime $p$, $\beta\rest p=\langle y_k:k\in N\rangle$ does not dominate $\alpha\rest p=\langle x_k:k\in N\rangle$, i.e. there is $m\in N$ such that $\sum_{k\geq m}x_k>\sum_{k\geq m}y_k.$ Let $n=\max\{k:x_k>0\lor y_k>0\}$; then $$\label{eqsigma}
u:=\sum_{k=m}^nx_k>v:=\sum_{k=m}^ny_k.$$ For any $A\in p$, if we denote $B=((A^m)^{(x_m)}(A^{m+1})^{(x_{m+1})}\dots(A^n)^{(x_n)})\gstr$, we have $B\in x$, because $B\cap L_{\sigma(\alpha)}$ contains a set in $F_\alpha$. On the other hand, $B\notin y$, because $B$ is disjoint from any set in $F_\beta$: every element of $B$ has a divisor of the form $a_{m,1}^m\dots a_{m,x_m}^ma_{m+1,1}^{m+1}\dots a_{n,x_n}^n$ with $u$ prime factors from $A$ to powers $\geq m$ and, because of (\[eqsigma\]), no element from a set in $F_\beta$ does (they all have exactly $v$ prime factors from $A$ to powers $\geq m$). Thus, $B\in x\cap\cU\setminus y$. This is a contradiction with $x\widemid y$.\
(b) Since $x\widemid x$, by (a) $x\supseteq F_\alpha\cup F_\beta$ would imply $\alpha\leq\beta$ and $\beta\leq\alpha$. However, the relation $\leq$ on $\cA$ is clearly antisymmetric, so $\alpha=\beta$.
\[example\] The reverse of Theorem \[poredak\](a) does not hold: we will construct ultrafilters $x\supseteq F_\alpha$ and $y\supseteq F_\beta$ such that $\alpha\leq\beta$ but $x\nwidemid y$.
If $p\in P^*$ is not Ramsey, by Theorem \[ramsey\] there is $A\subseteq P^{(2)}$ such that neither $A$ nor $P^{(2)}\setminus A$ contain a set in $F_2^p$. Exactly one of these two sets is in $p\cdot p$, say $A\in p\cdot p$, i.e. $\{a\in P:A/a\in p\}\in p$. Then for every $X\in p$ we can choose $a\in X$ such that $A/a\in p$. If $bc\notin A$ for all distinct $b,c\in X\cap A/a$, then $(X\cap A/a)^{(2)}\subseteq P^{(2)}\setminus A$, a contradiction with our choice of $A$. So for every $X\in p$ there are $a,b,c\in X$ such that $ab,ac,bc\in A$. This means that, if we denote $S=\{abc:ab,ac,bc\in A\}$, the family $F_3^p\cup\{S\}$ has the f.i.p. Hence we can find ultrafilters $x\supseteq F_2^p\cup\{P^{(2)}\setminus A\}$ and $y\supseteq F_3^p\cup\{S\}$, so $S\dstr\in(y\cap\cV)\setminus x$ and thus $x\nwidemid y$.
\[ispodiznad\] Let $\alpha,\beta\in\cA$ be such that $\alpha\leq\beta$.
\(a) If $x\supseteq F_\alpha$ then there is at least one $y\supseteq F_\beta$ such that $x\widemid y$.
\(b) If $y\supseteq F_\beta$ then there is at least one $x\supseteq F_\alpha$ such that $x\widemid y$.
\(a) Let $\alpha=\{(b_1,n_1),(b_2,n_2),\dots,(b_m,n_m)\}$ and $\beta=\{(b_1,n_1'),(b_2,n_2')$, $\dots,(b_m,n_m')\}$. We prove that the family $F_\beta\cup(x\cap\cU)$ has the f.i.p. Since every intersection of finitely many elements of $F_\beta$ contains an element of $F_\beta$ and $x\cap\cU$ is closed for finite intersections, it suffices to prove that every set $B'=A_1^{(n_1')}A_2^{(n_2')}\dots A_m^{(n_m')}\in F_\beta$ intersects every $X\in x\cap\cU$. But $X$ intersects $B=A_1^{(n_1)}A_2^{(n_2)}\dots A_m^{(n_m)}\in F_\alpha$, say $l\in B\cap X$. It suffices to show that $B'$ contains numbers divisible by $l$.
Let $p$ be any prime ultrafilter which is a divisor of $x$. Let $l=a_1^{k_1}a_2^{k_2}\dots a_r^{k_r}b$ be the factorization of $l$ such that $a_1^{k_1},a_2^{k_2},\dots$ (where $a_i\neq a_j$ for $i\neq j$) are elements of those sets $A_1,A_2,\dots$ that belong to a power of $p$, ordered so that $k_1\geq k_2\geq\dots$, and $b$ is the product of the other factors of $l$. Let $l'=(a_1')^{k_1'}(a_2')^{k_2'}\dots (a_s')^{k_s'}b'$ be the factorization of any element $l'\in B'$ obtained in the same way. Then the fact that $\beta\rest p$ dominates $\alpha\rest p$ implies that $s\geq r$ and $k_i'\geq k_i$ for $i\leq r$. We let $l_p=a_1^{k_1'}a_2^{k_2'}\dots a_r^{k_r'}(a_{r+1}')^{k_{r+1}'}\dots(a_s')^{k_s'}$; this element is clearly divisible by $a_1^{k_1}a_2^{k_2}\dots a_r^{k_r}$. In the same way we construct, for every prime $q\widemid x$, a number $l_q$. The product of all such $l_q$ will be the desired element divisible by $l$.\
(b) Analogously to (a), we can prove that $F_\alpha\cup(y\cap\cV)$ has the f.i.p.
Theorem \[ramsey\] shows that the ultrafilter $y$ from Theorem \[ispodiznad\](a) need not be unique. The next example shows the same for $x$ from Theorem \[ispodiznad\](b).
As in Example \[example\], let $p\in P^*$ be non-Ramsey and let $A\subseteq P^{(2)}$ be such that both $F_2^p\cup\{A\}$ and $F_2^p\cup\{A"\}$ (where $A"=P^{(2)}\setminus A$) have the f.i.p. Then $F_4^p\cup\{AA"\}$ also has the f.i.p.: if $B^{(4)}\in F_4^p$ then $B^{(2)}$ intersects both $A$ and $A"$, say $a=b_1b_2\in B^{(2)}\cap A$ and $a"=b_1"b_2"\in B^{(2)}\cap A"$ (and without loss of generality $b_1",b_2"$ are different from $b_1,b_2$). But then $aa"\in B^{(4)}\cap AA"$. Hence there is $y\supseteq F_4^p\cup\{AA"\}$ and, if we define $f_1(aa")=a$ and $f_2(aa")=a"$ for $a\in A$, $a"\in A"$, then $\widetilde{f_1}(y)\supseteq F_2^p\cup\{A\}$ and $\widetilde{f_2}(y)\supseteq F_2^p\cup\{A"\}$ are both divisors of $y$.
If $x\supseteq F_\alpha$ and $y\supseteq F_\beta$, then $x\cdot y\supseteq F_{\alpha+\beta}$, where $\alpha+\beta=\{(b,n_b+n_b'):(b,n_b)\in x,(b,n_b')\in y\}$.
First, the theorem clearly holds if one of the ultrafilters $x$, $y$ is in $N$. If $x=kx'$ and $y=ly'$ for some $k,l\in N$, $x',y'\in N^*$, then $x\cdot y=kl(x'\cdot y')$, so we can assume without loss of generality that $x$ and $y$ are not divisible by any elements of $N$.
Now let $\alpha=\{(b_1,n_1),(b_2,n_2),\dots,(b_m,n_m)\}$ and $\beta=\{(b_1,n_1'),(b_2,n_2'),\dots$, $(b_m,n_m')\}$. Then the sets in $F_{\alpha+\beta}$ are of the form $A_{b_1}^{(n_1+n_1')}A_{b_2}^{(n_2+n_2')}\dots A_{b_m}^{(n_m+n_m')}$ (where $A_p\in p$ for prime divisors $p$ of $x$ or $y$, $A_p\cap A_q=\emptyset$ for $p\neq q$ and, for basic divisors of the form $p^k$, $A_{p^k}=A_p^k$). But if $a=a_{1,1}a_{1,2}\dots a_{1,n_1}a_{2,1}\dots a_{m,n_m}\in A_{b_1}^{(n_1)}A_{b_2}^{(n_2)}\dots A_{b_m}^{(n_m)}$, then $$(A_{b_1}^{(n_1+n_1')}A_{b_2}^{(n_2+n_2')}\dots A_{b_m}^{(n_m+n_m')})/a\supseteq B_{b_1}^{(n_1')}\dots B_{b_m}^{(n_m')},$$ where $B_{b_k}=A_{b_k}\setminus\{a_{1,1},a_{1,2},\dots a_{m,n_m}\}$. For every $a\in A_{b_1}^{(n_1)}A_{b_2}^{(n_2)}\dots A_{b_m}^{(n_m)}$ we have $B_{b_1}^{(n_1')}\dots B_{b_m}^{(n_m')}\in y$ so $$\{a\in N:(A_{b_1}^{(n_1+n_1')}A_{b_2}^{(n_2+n_2')}\dots A_{b_m}^{(n_m+n_m')})/a\in y\}\supseteq A_{b_1}^{(n_1)}A_{b_2}^{(n_2)}\dots A_{b_m}^{(n_m)}\in x,$$ i.e. $A_{b_1}^{(n_1+n_1')}A_{b_2}^{(n_2+n_2')}\dots A_{b_m}^{(n_m+n_m')}\in x\cdot y$.
In particular, if $x\in\overline{L_m}$ and $y\in\overline{L_n}$ then $xy\in\overline{L_{m+n}}$. A corollary of this theorem is that levels $\overline{L_n}$ contain no idempotents (ultrafilters such that $x\cdot x=x$), since if $x\in\overline{L_n}$, then $x\cdot x\in\overline{L_{2n}}$. Another corollary is that, if $p\in\overline{P}$ and $x$ and $y$ are on finite levels, then $p\widemid x\cdot y$ implies $p\widemid x$ or $p\widemid y$, which (partly) justifies our calling such ultrafilters prime.
$|[x]_\sim|=1$ for every $n\in N$ and all ultrafilters $x\in\overline{L_n}$.
Assume the opposite, that $y=_\sim x$. If $x\in F_\alpha$ and $y\in F_\beta$, then Theorem \[poredak\](a) implies that $\alpha=\beta$, so $y\in\overline{L_n}$ as well. Let $A\ps L_n$ be such that $A\in y$ and $L_n\setminus A\in x$. Then $A\gstr\in y\cap\cU$ and $A\gstr\notin x$, a contradiction.
All of the proofs in the next lemma are analogous to Lemma \[broj\](b).
\(a) For every $\alpha\in\cA$ there are either finitely many or $2^{\goth c}$ ultrafilters containing $F_\alpha$.
\(b) For every $x\in\beta N$ there are either finitely many or $2^{\goth c}$ ultrafilters above $x$, and either finitely many or $2^{\goth c}$ ultrafilters below $x$.
\(c) For every $x\supseteq F_\alpha$ and all $\beta\leq\alpha\leq\gamma$ there are either finitely many or $2^{\goth c}$ ultrafilters above $x$ in $F_\gamma$, and either finitely many or $2^{\goth c}$ ultrafilters below $x$ in $F_\beta$.
Above all finite levels {#above}
=======================
\[greatest\] ([@So2], Theorem 4.1) There is the $\widemid$-greatest class (consisting of ultrafilters divisible by all other ultrafilters).
Let $\langle x_n:n\in N\rangle$ be a sequence of ultrafilters such that $x_n\in\overline{L_n}$ and $x_m\widemid x_n$ for $m<n$. Then there is an ultrafilter divisible by all $x_n$ and not divisible by any ultrafilter which is not below any $x_n$.
Let $F_1=\{A\in\cU:A\in x_n\mbox{ for some }n\in N\}$ and $F_2=\{B\in\cV:B^c\notin x_n\mbox{ for all }n\in N\}$. We prove that $F_1\cup F_2$ has the f.i.p. First, both $F_1$ and $F_2$ are closed for finite intersections. So let $A\in F_1$ and $B\in F_2$. There is $n\in N$ such that $A\in x_n$. Since also $B^c\notin x_n$, we have $A\cap B\in x_n$, so $A\cap B\neq\emptyset$.
In particular, for every countable set $S\subseteq\overline{P}$ there is an ultrafilter divisible by all $p\in S$ but not divisible by any prime $p\in\overline{P}\setminus S$. Since there are $2^{2^{\goth c}}$ subsets of $\overline{P}$, this can not hold for all uncountable $S\subseteq\overline{P}$.
Let us also note that it was shown in [@So1] that $\widemid$ is not antisymmetric, so there are ultrafilters above all finite levels such that $|[x]_\sim|>1$.
Final remarks
=============
The ultrafilters containing $F_2^p$ and $F_{1,1}^{p,q}$ bear similarities with ultrafilters that are preimages under the natural map from $\beta(N\times N)$ to $\beta N\times\beta N$. Such ultrafilters were investigated, among other papers, in [@B1], [@H1] and [@BM]. Hence some of the proofs in this paper are modifications of ideas from these papers (most of which can also be found in Chapter 16 of [@CN]). Since some of these modifications were not quite trivial, and for the sake of completeness, we decided to include all the proofs here. Also, the proof of Lemma \[Pthick\](b) is more general, different and (hopefully) more intuitive then in [@H1].
[99]{} A. Blass, Orderings on ultrafilters. PhD thesis, 1970. A. Blass, Ultrafilter mappings and their Dedekind cuts, Trans. Amer. Math. Soc. 188 (1974), No.2, 327-340. A. Blass, G. Moche, Finite preimages under the natural map from $\beta(N\times N)$ to $\beta N\times\beta N$, Topology Proc. 26 (2002), 407-432. W. W. Comfort, S. Negrepontis, The theory of ultrafilters. Springer-Verlag, 1974. N. Hindman, Preimages of points undeer the natural map from $\beta(N\times N)$ to $\beta N\times\beta N$, Proc. Amer. Math. Soc. 37 (1973), 603-608. N. Hindman, D. Strauss: Algebra in the Stone-Čech compactification, theory and applications. 2nd revised and extended edition, De Gruyter, 2012. S. Shelah, Proper and improper forcing, Perspectives in Mathematical Logic, Springer, 1998. B. Šobot, Divisibility in the Stone-Čech compactification, Rep. Math. Logic 50 (2015), 53-66. B. Šobot, Divisibility orders in the Stone-Čech compactification, submitted. R. Walker, The Čech-Stone compactification. Springer, 1974.
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---
abstract: 'Recent approaches for instance-aware semantic labeling have augmented convolutional neural networks (CNNs) with complex multi-task architectures or computationally expensive graphical models. We present a method that leverages a fully convolutional network (FCN) to predict semantic labels, depth and an instance-based encoding using each pixel’s direction towards its corresponding instance center. Subsequently, we apply low-level computer vision techniques to generate state-of-the-art instance segmentation on the street scene datasets KITTI and Cityscapes. Our approach outperforms existing works by a large margin and can additionally predict absolute distances of individual instances from a monocular image as well as a pixel-level semantic labeling.'
author:
- 'Jonas Uhrig$^{1,2}$, Marius Cordts$^{1,3}$, Uwe Franke$^{1}$, Thomas Brox$^{2}$'
- 'Jonas Uhrig$^{1,2}$, Marius Cordts$^{1,3}$, Uwe Franke$^{1}$, Thomas Brox$^{2}$'
bibliography:
- 'bib.bib'
title:
- 'Pixel-level Encoding and Depth Layering for Instance-level Semantic Labeling'
- |
Supplementary Material for\
Pixel-level Encoding and Depth Layering for Instance-level Semantic Labeling
---
16SubNumber[13]{}
[0.332]{} ![Example scene representation as obtained by our method: instance segmentation, monocular depth estimation, and pixel-level semantic labeling.[]{data-label="fig:titlefigure"}](Resources/titleJpg/frankfurt_000001_031266_leftImg8bit_instoverlay "fig:"){width="\textwidth"}
[0.332]{} ![Example scene representation as obtained by our method: instance segmentation, monocular depth estimation, and pixel-level semantic labeling.[]{data-label="fig:titlefigure"}](Resources/titleJpg/frankfurt_000001_031266_leftImg8bit_distoverlay "fig:"){width="\textwidth"}
[0.332]{} ![Example scene representation as obtained by our method: instance segmentation, monocular depth estimation, and pixel-level semantic labeling.[]{data-label="fig:titlefigure"}](Resources/titleJpg/frankfurt_000001_031266_leftImg8bit_semanticoverlay "fig:"){width="\textwidth"}
Introduction {#sec:introduction}
============
The task of visual semantic scene understanding is mainly tackled from two opposing facets: pixel-level semantic labeling [@ChenPapandreou2014; @FCN2015; @Papandreou2015] and bounding-box object detection [@girshick15fastrcnn; @girshick2014rcnn; @YOLO2015; @ren2015faster]. The first assigns each pixel in an image with a semantic label segmenting the semantically connected regions in the scene. Such approaches work well with non-compact (*background*) classes such as buildings or ground, yet they do not distinguish individual object instances. Object detection aims to find all individual instances in the scene and describes them via bounding boxes. Therefore, the latter provides a rather coarse localization and is restricted to compact (*object*) classes such as cars or humans.
Recently, instance-level semantic labeling gained increasing interest [@Dai2015; @Liang2015; @Urtasun2015a; @Urtasun2015b]. This task is at the intersection of both challenges. The aim is to combine the detection task with instance segmentation. Such a representation allows for a precise localization, which in turn enables better scene understanding. Especially in the domain of robotics and autonomous vehicles, instance-level semantic segmentation enables an explicit occlusion reasoning, precise object tracking and motion estimation, as well as behavior modeling and prediction.
Most state-of-the-art methods build upon a fully convolutional network (FCN) [@FCN2015]. Recent approaches typically add post-processing, for example, based on conditional random fields (CRFs) [@Urtasun2015a; @Urtasun2015b]. Other methods score region proposals for instance segmentation [@Dai2015FeatureMasking; @Hariharan2014] or object detection [@girshick15fastrcnn; @girshick2014rcnn; @YOLO2015; @ren2015faster], or use a multi-stage neural network for these tasks [@Dai2015; @Liang2015].
In this work, we focus on street scene understanding and use a single monocular image to simultaneously obtain a holistic scene representation, consisting of a pixel-level semantic labeling, an instance-level segmentation of traffic participants, and a 3D depth estimation for each instance. We leverage an FCN that yields powerful pixel-level cues consisting of three output channels: a semantic class, the direction to the object center (where applicable) and the object distance (where applicable). Scene understanding is mainly due to the network and post-processing with standard computer vision methods is sufficient to obtain a detailed representation of an instance-aware semantic segmentation, \[fig:titlefigure,fig:methodsummary\]. Our method significantly outperforms state-of-the-art methods on the street scene datasets KITTI [@KITTI2012] and Cityscapes [@Cordts2015].
![From a single image, we predict FCN outputs: semantics, depth, and instance center direction. Those are used to compute template matching score maps for semantic categories. Using these, we locate and generate instance proposals and fuse them to obtain our instance segmentation.[]{data-label="fig:methodsummary"}](Resources/method/Summary){width="\textwidth"}
Related Work {#sec:relatedwork}
============
For the task of instance-level semantic labeling, there exist two major lines of research. The first leverages an over-complete set of object proposals that are either rejected, classified as an instance of a certain semantic class, and refined to obtain an instance segmentation. Common to all such methods is that the performance is depending on the quality of these proposals, since they cannot recover from missing instances in the proposal stage. Generally, such approaches tend to be slow since all proposals must be classified individually. These properties cause inaccurate proposals to limit the performance of such methods [@Cordts2015; @Hosang2015Pami]. Our method belongs to the category of proposal-free methods, where the segmentation and the semantic class of object instances are inferred jointly.
[**Proposal-based instance segmentation.**]{} Driven by the success of deep learning based object detectors such as R-CNN [@girshick2014rcnn] or its variants [@girshick15fastrcnn; @ren2015faster; @He2015], recent methods rely on these detections for instance segmentation. Either the underlying region proposals, such as MCG [@Arbelaez2014], are directly used as instance segments [@Cordts2015; @Dai2015FeatureMasking; @Hariharan2014], or the bounding boxes are refined to obtain instance masks [@Chen2015; @Hariharan2015]. Instead of bounding boxes, [@objcut2005] uses a layered pictorial structure (LPS) model, where shape exemplars for object parts are mapped to the image in a probabilistic way. This yields an initial proposal for the object’s pose and shape, which is refined using appearance cues. Using a bank of object detectors as proposals, [@yang2012layered] infers the instance masks via occlusion reasoning based on discrete depth layers. In [@Tighe2014], pixel-level semantic labels are used to score object candidates and vice versa in an alternating fashion, while also reasoning about occlusions and scene geometry. Based on proposals that form a segmentation tree, an energy function is constructed in [@Silberman2014] and its solution yields the instance segmentation.
Recently, [@Dai2015] extended the R-CNN for instance segmentation with a multi-task network cascade. A fully convolutional network combined with three classification stages produces bounding-box proposals, refines these to segments, and ranks them to obtain the final instance-level labeling. They achieve excellent performance on PASCAL VOC [@Pascal2010] and MS COCO [@MSCoco2014].
[**Proposal-free instance segmentation.**]{} Pixel-level semantic labeling based on neural networks has been very successful [@ChenPapandreou2014; @Rother2015; @FCN2015; @Yu2016; @ZhengJayasumana2015]. This triggered interest in casting also instance segmentation directly as a pixel labeling task. In [@Ronneberger2015], the network predicts for each pixel, whether it lies on an object boundary or not, however, requiring a rather delicate training. Using a long short-term memory (LSTM) network [@LSTM1997], instance segmentations can be sequentially sampled [@Romera2015].
In [@Urtasun2015a; @Urtasun2015b], instances are encoded via numbers that are further constrained to encode relative depth ordering in order to prevent arbitrary assignments. An FCN predicts these IDs at each pixel and a subsequent Markov Random Field (MRF) improves these predictions and enforces consistency. However, such a method is limited to scenes, where a clear depth ordering is present, a single row of parking cars, and the maximum number of instances is rather low.
The proposal-free network (PFN) [@Liang2015] is a CNN that yields a pixel-level semantic labeling, the number of instances in the scene, and for each pixel the parameters of a corresponding instance bounding box. Based on these predictions, instances are obtained by clustering. The network has a fairly complex architecture with many interleaved building blocks, making training quite tricky. Further, the overall performance highly depends on the correct prediction of the number of instances in the scene. In street scenes, there can be hundreds of instances per image [@Cordts2015]. Thus, the number of training samples per number of instances is low, mistakes in their estimation can be critical, and the available cues for clustering might not correlate with the estimated number of instances.
In this work, we focus on urban street scenes. Besides each pixel’s semantic class, our network estimates an absolute depth, which is particularly useful for instance separation in street scenes. We encode instances on a pixel-level by the direction towards their center point. This representation is independent of the number of instances per image and provides strong signals at the instance boundaries.
Method {#sec:method}
======
FCN Feature Representation {#subsec:fcnoutputrepresentation}
--------------------------
Our network extends the FCN-8s model [@FCN2015] with three output channels that together facilitate instance segmentation. All channels are jointly trained as pixel-wise discrete labeling tasks using standard cross-entropy losses. Our proposed representation consists of (1) a semantic channel that drives the instance classification, (2) a depth channel to incorporate scale and support instance separation, and (3) a 2D geometric channel to facilitate instance detection and segmentation.
We chose the upscaling part of our FCN such that we can easily change the number of classes for each of the three proposed channels without re-initializing all upsampling layers. To this end, after the largest downsampling factor is reached, we use Deconvolution layers together with skip layers [@FCN2015] to produce a representation of $\frac{1}{8}$ of the input resolution with a depth of throughout all intermediate layers. The number of channels of this abstract representation is then reduced through $1\!\times\!1$ convolutions to the proposed semantic, depth, and instance center channels. To reach full input resolution, bilinear upsampling is applied, followed by a separate cross-entropy loss for each of our three output channels.
[0.32]{} ![Ground truth examples of our three proposed FCN channels. Color overlay (a) as suggested by [@Cordts2015], (b) represents depth per object from red (close) to blue (distant), (c) represents directions towards corresponding instance centers.[]{data-label="fig:semanticdepthexamples"}](Resources/method/methodJpg/frankfurt_000001_000538_idGtColor_vis "fig:"){width="\textwidth"}
[0.32]{} ![Ground truth examples of our three proposed FCN channels. Color overlay (a) as suggested by [@Cordts2015], (b) represents depth per object from red (close) to blue (distant), (c) represents directions towards corresponding instance centers.[]{data-label="fig:semanticdepthexamples"}](Resources/method/methodJpg/frankfurt_000001_000538_distGtColor_vis "fig:"){width="\textwidth"}
[0.32]{} ![Ground truth examples of our three proposed FCN channels. Color overlay (a) as suggested by [@Cordts2015], (b) represents depth per object from red (close) to blue (distant), (c) represents directions towards corresponding instance centers.[]{data-label="fig:semanticdepthexamples"}](Resources/method/methodJpg/frankfurt_000001_000538_instanceGt_vis "fig:"){width="\textwidth"}
[**Semantics.**]{} To cope with different semantic classes, we predict a semantic label for each input pixel, \[fig:semGT\]. These predictions are particularly important as they are the only source of semantic information in our approach. Further, the predicted semantic labels allow us to separate objects from background as well as objects of different classes from each other.
[**Depth.**]{} Urban street scenes typically contain objects at various distances [@Cordts2015]. To guide the post-processing in terms of objects at different scales, we predict a depth label for each object pixel. We assign all pixels within an object instance to a constant depth value, e.g. the median over noisy measurements or the center of a 3D bounding box, \[fig:depthGT\]. These depth estimates also support instance separation, which becomes apparent when considering a row of parking cars, where the depth delta between neighboring cars is a full car length instead of a few centimeters in continuous space. The depth values are discretized into a set of classes so that close objects have a finer depth resolution than distant objects.
[**Direction.**]{} Object instances are defined by their boundary and class. Therefore, it seems natural to train an FCN model to directly predict boundary pixels. However, those boundaries represent a very delicate signal [@amfm_pami2011] as they have a width of only one pixel, and a single erroneously labeled pixel in the training data has a much higher impact compared to a region-based representation.
We introduce a class-based representation which implicitly combines information about an instance’s boundary with the location of its visible center. For each object pixel we compute the direction towards its corresponding center and discretize this angle to a set of classes, \[fig:dirGT\]. This information is easier to grasp within a local region and is tailored for an FCN’s capability to predict pixel-wise labels. Especially for pixels on the boundary between neighboring objects, our representation clearly separates the instances as predictions have nearly opposite directions. Since we predict the center of the visible area of an object and not its physical center, we can handle most types of occlusions very well. Furthermore, instance centers have a distinct pattern, \[fig:dirGT\], which we exploit by applying template matching, as described in \[subsec:templatematching\]. Even though our proposed representation does not directly yield instance IDs, it is well defined even for an arbitrary number of instances per image.
To obtain an accurate direction estimation for each pixel, we assign the average direction by weighting all direction vectors with their respective FCN score (after softmax normalization). This allows us to recover a continuous direction estimation from the few discretized classes.
Template Matching {#subsec:templatematching}
-----------------
To extract instance centers, we propose template matching on the direction predictions, where templates are rectangular and contain the distinct pattern visible in \[fig:dirGT\]. We adjust the template’s aspect ratio depending on its semantic class, so we can better distinguish between pedestrians and vehicles. In order to detect also distant objects with consistent matching scores, we scale the size of the templates depending on the predicted depth class.
To reduce induced errors from confusions between objects of similar semantic classes, we combine multiple semantic classes into the categories *human*, *car*, *large vehicle*, and *two wheeler*.
Normalized cross-correlation (NCC) is used to produce a score map for each category by correlating all pixels with their respective template. These maps indicate the likelihood of pixels being an instance center, \[fig:methodsummary\]. In the following, we predict instances for each category separately. After all instances are found, we assign them the majority semantic class label.
Instance Generation {#subsec:instancegeneration}
-------------------
[**Instance Centers.**]{} To determine instance locations, we iteratively find maxima in the generated template matching score maps via non-maximum suppression within an area that equals the template size. This helps avoid multiple detections of the same instance while incorporating typical object sizes. Those maxima represent our *temporary instance centers*, which are refined and merged in the following steps.
[**Instance Proposals.**]{} Each pixel with a predicted direction from the FCN is assigned to the closest temporary instance center where the relative location and predicted direction agree. Joining all assigned pixels per instance hypothesis yields a set of *instance proposals*.
[**Proposal Fusion.**]{} \[subsubsec:instancefusion\] Elongated objects and erroneous depth predictions cause an over-segmentation of the instances. Thus, we refine the generated instances by accumulating estimated directions within each proposal. When interpreting direction predictions as vectors, they typically compensate each other within instance proposals that represent a complete instance, there are as many predictions pointing both left and right. However, incomplete instance proposals are biased to a certain direction. If there is a neighboring instance candidate with matching semantic class and depth in the direction of this bias, the two proposals are fused.
To the remaining instances we assign the average depth and the most frequent semantic class label within the region. Further, we merge our instance prediction with the pixel-level semantic labeling channel of the FCN by assigning the argmax semantic label to all non-instance pixels. Overall, we obtain a consistent scene representation, consisting of object instances paired with depth estimates and pixel-level labels for background classes.
Experiments {#sec:experiments}
===========
Datasets and Metrics {#subsec:datasetsandmetrics}
--------------------
We evaluated our approach on the KITTI object detection dataset[@KITTI2012] extended with instance-level segmentations [@Chen2014; @Urtasun2015b] as well as Cityscapes [@Cordts2015]. Both datasets provide pixel-level annotations for semantic classes and instances, as well as depth information, which is essential for our approach. For the ground truth instance depths we used the centers of their 3D bounding box annotation in KITTI and the median disparity for each instance in Cityscapes based on the provided disparity maps. We used the official splits for training, validation and test sets.
We evaluated the segmentation based on the metrics proposed in [@Urtasun2015b] and [@Cordts2015]. To evaluate the depth prediction, we computed the mean absolute error (MAE), the root mean squared error (RMSE), the absolute relative difference (ARD), and the relative inlier ratios ($\delta_1$, $\delta_2$, $\delta_3$) for thresholds $\delta_i = 1.25^i$ [@Wang2015]. These metrics are computed on an instance level using the depths in meters. We only considered instances that overlap by more than with the ground truth.
Network Details
---------------
For Cityscapes, we used the semantic classes and combined the object classes into categories (*car*, *human*, *two-wheeler*, and *large vehicle*). For KITTI, only *car* instance segmentations are available. For both datasets, we used depth classes and an explicit class for background. We chose ranges for each depth class and template sizes differently for each dataset to account for different characteristics of present objects and used camera settings [@Cordts2015]. This is necessary as distances and semantic classes of objects differ remarkably. Details are provided in the supplementary material. The instance directions were split into equal parts, each covering an angle of for both datasets.
We use the 8-stride version of an FCN, which is initialized using the ImageNet dataset [@ImageNet]. After initializing the upsampling layers randomly, we fine-tune the network on KITTI and Cityscapes to obtain all three output channels.
[0.33]{} ![Example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI. We even detect objects at very large distances.[]{data-label="fig:kittiexamples"}](Resources/kittiJpg/001042_image "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI. We even detect objects at very large distances.[]{data-label="fig:kittiexamples"}](Resources/kittiJpg/001042_gt "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI. We even detect objects at very large distances.[]{data-label="fig:kittiexamples"}](Resources/kittiJpg/001042_instanceSegmentation_IDs "fig:"){width="\textwidth"}
\
[0.33]{} ![Example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI. We even detect objects at very large distances.[]{data-label="fig:kittiexamples"}](Resources/kittiJpg/001159_image "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI. We even detect objects at very large distances.[]{data-label="fig:kittiexamples"}](Resources/kittiJpg/001159_gt "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI. We even detect objects at very large distances.[]{data-label="fig:kittiexamples"}](Resources/kittiJpg/001159_instanceSegmentation_IDs "fig:"){width="\textwidth"}
\
[0.33]{} ![Example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI. We even detect objects at very large distances.[]{data-label="fig:kittiexamples"}](Resources/kittiJpg/001083_image "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI. We even detect objects at very large distances.[]{data-label="fig:kittiexamples"}](Resources/kittiJpg/001083_gt "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI. We even detect objects at very large distances.[]{data-label="fig:kittiexamples"}](Resources/kittiJpg/001083_instanceSegmentation_IDs "fig:"){width="\textwidth"}
\
Ablation Studies {#subsec:ablationstudies}
----------------
We evaluated the influence of each proposed component by leaving out one or more components from the complete processing pipeline (*Ours*). The performance was evaluated on the respective validation sets and is listed in \[tab:kittiVariantsZhang,tab:csVariants\] (top) for both datasets.
For *Ours-D*, we removed the depth channel and chose the template size scale-agnostic. It turned out that a rather small template size, which leads to a large number of instance proposals, produces the best results. This is possible when post-processing heavily relies on correct direction predictions, which induces successful instance fusion. However, the performance is significantly worse in most metrics on both datasets compared to our full system, which shows that the depth information is an essential component of our approach. When the fusion component was also removed (*Ours-D-F*), a larger template size was needed to prevent an over-segmentation. However, performance dropped by an even larger margin than for *Ours-D*. In our last variant we kept the depth information but directly used the instance proposals as final instance predictions (*Ours-F*). The performance was even slightly worse than *Ours-D*, which shows that all our components are important to obtain accurate object instances. These observations are consistent on both datasets.
Instance Evaluation {#subsec:instanceevaluation}
-------------------
[**KITTI.**]{} We clearly outperform all existing works on KITTI (*Best* [@Urtasun2015a]/[@Urtasun2015b]), \[tab:kittiVariantsZhang\] (bottom). Compared to the better performing work *Best* [@Urtasun2015b], we achieve a margin of relative improvement averaged over all metrics. Even when comparing our single variant with the best numbers over all existing variants for each metric individually (*Mix* [@Urtasun2015b]), we achieve a significantly better performance. We also evaluated our approach using the metrics introduced in [@Cordts2015] to enable comparisons in future publications, \[tab:csVariants\] (bottom). Qualitative results are shown in \[fig:kittiexamples\].
[0.33]{}
![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000001_050149_leftImg8bit){width="\textwidth"}
[0.33]{}
![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000001_050149_gtFine_instanceIds){width="\textwidth"}
[0.33]{}
![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000001_050149_leftImg8bit_instanceSegmentation_IDs){width="\textwidth"}
\
[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/lindau_000033_000019_leftImg8bit "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/lindau_000033_000019_gtFine_instanceIds "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/lindau_000033_000019_leftImg8bit_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000000_016286_leftImg8bit "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000000_016286_gtFine_instanceIds "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000000_016286_leftImg8bit_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000000_016286_semanticGtColor "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000000_016286_distanceGtColor "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000000_016286_directionsGtColor "fig:"){width="\textwidth"}
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[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000000_016286_semanticPredColor "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000000_016286_distancePredColor "fig:"){width="\textwidth"}
[0.33]{} ![Example results of our instance segmentation and corresponding ground truth (rows $1$–$3$) on Cityscapes. We also include the three FCN output channels (row $5$) and their ground truth (row $4$). It can be seen that even distant objects are segmented well and the approach can handle occlusions.[]{data-label="fig:cityscapesexamples"}](Resources/cityscapesJpg/frankfurt_000000_016286_directionsPredColor "fig:"){width="\textwidth"}
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[**Cityscapes.**]{} On the Cityscapes dataset, our approach outperforms the baseline *MCG+R-CNN* [@Cordts2015] in all proposed metrics as evaluated by the dataset’s submission server, \[tab:csVariants\] (center). We nearly double the performance in terms of the main score ${\text{AP}}$. Compared to the performance on KITTI, \[tab:csVariants\] (bottom), the numbers are significantly lower, indicating the higher complexity of scenes in Cityscapes. Qualitative results are shown in \[fig:cityscapesexamples\].
Depth Evaluation
----------------
As shown in \[tab:depthEval\], the average relative and mean absolute error of our predicted instances are as low as and , respectively, on the KITTI dataset. On the Cityscapes dataset, which contains much more complex scenes, with many and distant object instances, we achieve and , respectively. These results are particularly impressive, since we used only single monocular images as input for our network. We hope that future publications compare their depth estimation performance using the proposed metrics.
Evaluation of semantic class labels
-----------------------------------
Our method also yields a pixel-level semantic labeling including background classes that we evaluate on Cityscapes, \[tab:semanticEval\]. We compare to two baselines, *FCN 8s* [@FCN2015] that uses the same FCN architecture as our approach and *Dilation10* [@Yu2016], which is the currently best performing approach on Cityscapes [@Cordts2015]. It can be seen that our approach is on par with the state-of-the-art although this work focuses on the harder instance segmentation task.
Conclusion {#sec:conclusion}
==========
In this work, we present a fully convolutional network that predicts pixel-wise depth, semantics, and instance-level direction cues to reach an excellent level of holistic scene understanding. Instead of complex architectures or graphical models for post-processing, our approach performs well using only standard computer vision techniques applied to the network’s three output channels. Our approach does not depend on region proposals and scales well for arbitrary numbers of object instances in an image.
We outperform existing works on the challenging urban street scene datasets Cityscapes [@Cordts2015] and KITTI [@Urtasun2015a; @Urtasun2015b] by a large margin. On KITTI, our approach achieves relative improvement averaged over all metrics and we almost double the performance on Cityscapes. As our approach can reliably predict absolute depth values per instance, we provide an instance-based depth evaluation. Our depth predictions achieve a relative error of only a few meters, even though the datasets contain instances in more than one hundred meters distance. The main focus of this work is instance segmentation, but we also achieve state-of-the-art performance for pixel-level semantic labeling on Cityscapes, with a new best performance on an instance-based score over categories.
Qualitative Results
===================
show further qualitative examples of our instance segmentation on urban scenes from KITTI [@KITTI2012] and Cityscapes [@Cordts2015]. It can be seen that our approach can segment even high numbers of instances despite heavy occlusions and clutter.
Depth Ranges
============
As mentioned in \[subsec:fcnoutputrepresentation\], we discretized continuous instance depths into depth classes. Instead of equidistantly splitting them, we chose the ranges for each class such that the sizes of objects within each depth class are similar. We found this option to yield slightly better results, since the subsequent template matching is based on our FCN’s depth prediction and equal object sizes per depth class result in more reliable template matching scores.
We defined the values as in \[tab:depthRanges\] to provide a good trade-off between number of depth classes and depth resolution, as well as number of samples per depth class in the training data. As the Cityscapes dataset contains a lot of object instances labeled for very high distances of over meters [@Cordts2015], the depth ranges had to be chosen differently than for KITTI [@KITTI2012].
Class-level Evaluation {#sec:detailedevaluation}
======================
Instance-level Evaluation
-------------------------
We list class-level performances of our approach for instance-level semantic labeling (*Ours*) and the baseline *MCG+R-CNN* [@Cordts2015] in \[tab:instanceEval\]. Our approach has difficulties especially for semantic classes that are least reliably classified by our FCN, such as bus, truck, and train \[tab:pixelEval,tab:pixelInstanceEval,tab:pixelConfusion\]. Best results are achieved for cars and humans, while we outperform the proposal-based baseline for all other classes by large margins in all used metrics.
Pixel-level Evaluation
----------------------
A detailed evaluation of our performance for pixel-level semantic labeling can be found in \[tab:pixelEval,tab:pixelInstanceEval,tab:pixelConfusion\]. Even though our main focus lies on instance-level semantic labeling, we achieve competitive results for all classes compared to the baselines listed in [@Cordts2015]. Using the instance-aware metric ${\text{iIoU}}$, we even outperform most existing works by a few percent points for the object classes *person*, *car*, and *bicycle*.
The reason for a comparably low performance on the classes *bus*, *truck*, and *train* becomes evident by inspecting \[tab:pixelEval,tab:pixelConfusion\]. We achieve comparably low semantic labeling results on a pixel-level for these classes and therefore our template matching and instance generation steps perform significantly worse than on all other object classes.
[1]{} \[1\][`#1`]{}
Badrinarayanan, V., Kendall, A., Cipolla, R.: Segnet: [A]{} deep convolutional encoder-decoder architecture for image segmentation (2015)
Lin, G., Shen, C., Reid, I.D., van den Hengel, A.: Efficient piecewise training of deep structured models for semantic segmentation (2015)
Liu, Z., Li, X., Luo, P., Loy, C.C., Tang, X.: Semantic image segmentation via deep parsing network. In: ICCV. pp. 1377–1385 (2015)
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/000325_image "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/000325_gt "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/000325_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/000391_image "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/000391_gt "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/000391_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/001484_image "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/001484_gt "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/001484_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/001536_image "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/001536_gt "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/001536_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/002925_image "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/002925_gt "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/002925_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/003228_image "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/003228_gt "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/003228_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/006233_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/006503_image "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/006503_gt "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (middle) on KITTI.[]{data-label="fig:kittiadditionalexamples"}](Resources/kittiJpg/006503_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/frankfurt_000000_000576_leftImg8bit "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/frankfurt_000000_000576_gtFine_instanceIds "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/frankfurt_000000_000576_leftImg8bit_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/frankfurt_000001_008200_leftImg8bit "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/frankfurt_000001_008200_gtFine_instanceIds "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/frankfurt_000001_008200_leftImg8bit_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/frankfurt_000001_012699_leftImg8bit "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/frankfurt_000001_012699_gtFine_instanceIds "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/frankfurt_000001_012699_leftImg8bit_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/munster_000126_000019_leftImg8bit "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/munster_000126_000019_gtFine_instanceIds "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/munster_000126_000019_leftImg8bit_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/munster_000146_000019_leftImg8bit "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/munster_000146_000019_gtFine_instanceIds "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/munster_000146_000019_leftImg8bit_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/munster_000162_000019_leftImg8bit "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/munster_000162_000019_gtFine_instanceIds "fig:"){width="\textwidth"}
[0.33]{} ![Further example results of our instance segmentation (right) and corresponding ground truth (center) on Cityscapes *validation*.[]{data-label="fig:cityscapesadditionalexamples"}](Resources/cityscapesJpg/munster_000162_000019_leftImg8bit_instanceSegmentation_IDs "fig:"){width="\textwidth"}
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---
abstract: 'We review generalized Fluctuation-Dissipation Relations which are valid under general conditions even in “non-standard systems”, e.g. out of equilibrium and/or without a Hamiltonian structure. The response functions can be expressed in terms of suitable correlation functions computed in the unperperturbed dynamics. In these relations, typically one has nontrivial contributions due to the form of the stationary probability distribution; such terms take into account the interaction among the relevant degrees of freedom in the system. We illustrate the general formalism with some examples in non-standard cases, including driven granular media, systems with a multiscale structure, active matter and systems showing anomalous diffusion.'
author:
- 'A. Sarracino'
- 'A. Vulpiani'
bibliography:
- 'fluct.bib'
title: 'On the Fluctuation-Dissipation Relation in non-equilibrium and non-Hamiltonian systems'
---
> The Fluctuation-Dissipation Theorem is a central result in equilibrium statistical mechanics. It allows one to express the linear response of a system to an external perturbation in terms of the spontaneous correlations, and its derivation is based on the detailed balance condition. In the last decades a great effort has been devoted to generalize this fundamental result to systems where detailed balance does not hold, because of the presence of some forms of dissipation, or energy and particle currents, that induce non-equilibrium conditions. Among the huge variety of systems belonging to this class, let us mention active and biological matter, driven granular media, molecular motors, and slow relaxing glasses. The derivation of a generalized Fluctuation-Dissipation Relation, and the investigation of its peculiar features in non-standard systems play therefore a central role in the building of a general theory of statistical mechanics beyond equilibrium.
Introduction {#Introduction}
============
One of the general results of the near-equilibrium statistical mechanics is the existence of a precise relation between the spontaneous fluctuations of the system and the response to external perturbations of physical observables. This result allows for the possibility of studying the response to time-dependent external fields, by analyzing time-dependent correlations. The idea dates back to Einstein’s work on Brownian motion [@E05]. Later, Onsager [@O31; @O31b] stated his regression hypothesis according to which the relaxation of a macroscopic perturbation follows the same laws governing the dynamics of fluctuations in equilibrium systems.
The Fluctuation-Dissipation Relation (FDR) theory was initially obtained for Hamiltonian systems near thermodynamic equilibrium, but now it is known that a generalized FDR holds for a broader class of systems. Among these, the FDR has been widely investigated in the context of turbulence (and more generally statistical fluid mechanics): let us mention for instance the seminal work by Kraichnann [@Kr00; @Kr59], and his attempt to obtain a closure theory from an assumption on the FDR. In recent years, the FDR attracted the interest of the scientific community active in the modelling of geophysical systems, in particular for climate dynamics [@L75; @B80; @NBH93; @MW06; @AM08; @N1]. Moreover, another wide field of research where FDR plays a central role is the stochastic thermodynamics theory [@Seifert], including models for colloidal systems, granular and active matter.
Here we present a brief review of some recent results on the theory of FDR, with specific focus on non-standard situations: systems with negative temperature, systems with many degrees of freedom and a multiscale structure, or systems with anomalous transport properties. We discuss in particular some subtle, non-trivial aspects related to these peculiar cases. Our work represents an update of the previous review [@capklages]. The interested reader can find more exhaustive reviews of the general subject in [@CR03; @BPRV08; @cuglirev; @Seifert; @maes3; @PSV17]. For a more rigorous discussion see also [@baladi].
The paper is organized as follows. In Section \[GFDR\] we present two generalized FDRs valid also in non-equilibrium systems and discuss some subtle points. The first relation involves the probability distribution function (PDF) of the stationary state, while the second one includes a quantity which depends on the transition rates of the model. In Section \[real\] we illustrate some examples where FDR are applied in non-standard cases, such as anomalous diffusion, systems with different time scales, and some models of active matter. In Section \[conc\] we draw some conclusions and mention perspectives for future work.
Generalized Fluctuation-Dissipation Relation {#GFDR}
============================================
The FDR theory was initially developed within the context of equilibrium statistical mechanics of Hamiltonian systems. This led to misleading claims on the (supposed) limited validity of the FDR [@RS78]. Indeed, in the following we will see that it is possible to derive a generalized FDR, which holds under rather general hypotheses, also in non-Hamiltonian systems [@DH75; @FIV90; @BLMV03].
van Kampen’s objection to the FDR
---------------------------------
Let us briefly discuss an objection by van Kampen [@vK71] to the original (perturbative) derivation of the FDR. From a technical point of view such a criticism can be rejected: however it had the merit of stimulating a deeper understanding of the FDR and its validity range.
In the dynamical systems terminology, van Kampen’s argument is the following. Given an impulsive perturbation $\delta
{\bf x}(0)$ on the state [**x**]{} of the system at time $t=0$, the difference between the perturbed trajectory and the unperturbed one is $$\label{3.1}
\delta x_i(t)=
\sum_{j} \frac{\partial x_i(t)}{\partial x_j(0)} \delta x_j(0)
+O(|\delta {\bf x}(0)|^2).$$ Averaging over the initial conditions, one has the mean response function: $$\label{3.2}
R_{i,j}(t)= \Biggl \langle \frac{\delta x_i(t)} {\delta x_j(0)}
\Biggr \rangle =\int \frac{\partial x_i(t)} {\partial x_j(0)}
\rho({\bf x}(0)) d{\bf x}(0 ) \,\,.$$ In the case of the equilibrium statistical mechanics, since $\rho({\bf
x}) \propto \exp\Bigl( -\beta H(\bf{x}) \Bigr)$, after an integration by parts, one obtains $$\label{3.3}
R_{i,j}(t)= \beta \Biggl \langle x_i(t)
\frac {\partial H({\bf x}(0))}
{ \partial x_j(0)} \Biggr \rangle \,.$$ It is easy to realize that the above result is nothing but the differential form of the usual FDR.
In the presence of chaos, however, the terms ${\partial
x_i(t)}/{\partial x_j(0)}$ grow exponentially as $e^{\lambda t}$, where $\lambda$ is the Lyapunov exponent. Therefore the linear expansion (\[3.1\]) is not accurate for a time larger than $(1/
\lambda) \ln (L/|\delta{\bf x}(0)|)$, where $L$ is the typical fluctuation of the variable ${\bf x}$. Thus, the linear response theory is expected to be valid only for extremely small and nonphysical perturbations (or times). For instance, according to this argument, requiring that the FDR holds up to $1 s$ when applied to the electrons in a typical conductor, would imply a perturbing electric field smaller than $10^{-20} V/m$, in clear disagreement with the experience.
The success of the linear theory for the computation of transport coefficients (e.g. electric conductivity) in terms of correlation functions of the unperturbed system, is evident, and its validity has been, directly and indirectly, verified in a huge number of cases. Kubo suggested that the origin of the effectiveness of the FDR theory may reside in the “constructive role of chaos”: “[*instability \[of the trajectories\] instead favors the stability of distribution functions, working as the cause of the mixing*]{}” [@K86]. The following derivation [@FIV90] of a generalized FDR supports this intuition.
A Generalized FDR {#GFDRs}
-----------------
One of the most intense research fields in non-equilibrium statistical mechanics addressed the issue of the fluctuation-dissipation theorem when the system under study is out of equilibrium. This situation can be due both to the presence of external forcing and continuous dissipation, so that a stationary state is reached, and to a very slow relaxation, leading to a non-stationary transient dynamics. Standard examples of systems falling in the first class are vibrated granular materials [@Puglisi] or active particles [@RevModPhys.85.1143], while to the second class there belong, for instance, Ising spin models, or spin and structural glasses [@CR03].
Many results have been derived in the last decades, which extend the validity of the FDR to the non-equilibrium realm [@BPRV08; @Seifert]. Of course, these results do not share the same generality as the equilibrium FDR, and their explicit forms can depend on the considered model. Nevertheless, these relations can have important applications in different contexts. In general, their relevance relies on the possibility to obtain information on the non-equilibrium response of the system from the study of unperturbed fluctuations, or vice-versa, depending on the more suitable conditions in numerical or experimental settings.
Let us start by presenting a derivation of a generalized FDR. It is easy to understand, see below, that it is possible to derive such a FDR also for finite perturbations, in non equilibrium and non Hamiltonian systems, and therefore van Kampen’s critique has a marginal role.
Consider a dynamical system ${\bf x}(0) \to {\bf x}(t)=U^t {\bf x}(0)$ with states ${\bf x}$ belonging to a $N$-dimensional space. We can consider the case in which the time evolution is not deterministic (e.g., stochastic differential equations). Let us assume the existence of an invariant probability distribution $\rho({\bf x})$, for which some “absolutely continuity”-type conditions are required (see later), and the mixing character of the system (from which its ergodicity follows); no assumption is made on the dimensionality $N$ of the system. Our aim is to study the behaviour of one component of ${\bf x}$, say $x_i$, when the system, whose statistical features are described by $\rho({\bf x})$, is subjected to an initial (non-random) perturbation such that ${\bf x}(0) \to {\bf x}(0) + \Delta {\bf
x}_{0}$.
An instantaneous kick modifies the density of the system into $\rho'({\bf x})$, related to the invariant distribution by $\rho'
({\bf x}) = \rho ({\bf x} - \Delta {\bf x}_0)$. Let us introduce the probability of transition from ${\bf x}_0={\bf x}(0)$ at time $0$ to ${\bf x}$ at time $t$, $w ({\bf x}_0,0 \to {\bf x},t)$ (of course in a deterministic system, with evolution law $ {\bf x}(t)=U^{t}{\bf
x}(0)$, one has $w ({\bf x}_0,0 \to {\bf x},t)=\delta({\bf
x}-U^{t}{\bf x}_{0})$, where $\delta(\cdot)$ is the Dirac delta). We can write an expression for the mean value of the variable $x_i$, computed with the density of the perturbed system: $$\label{3.4}
\Bigl \langle x_i(t) \Bigr \rangle ' = \int\!\int x_i \rho' ({\bf
x}_0) w ({\bf x}_0,0 \to {\bf x},t) \, d{\bf x} \, d{\bf x}_0 \; .$$ For the mean value of $x_i$ during the unperturbed evolution one has: $$\label{3.5}
\Bigl \langle x_i(t) \Bigr \rangle =
\int\!\int x_i \rho ({\bf x}_0)
w ({\bf x}_0,0 \to {\bf x},t) \, d{\bf x} \, d{\bf x}_0 \; .$$ Therefore, defining $\overline{\delta x_i} = \langle x_i \rangle' -
\langle x_i \rangle$, we have: $$\begin{aligned}
\label{3.6}
\overline{\delta x_i} \, (t) &=&
\int \int x_i \;
\frac{\rho ({\bf x}_0 - \Delta {\bf x}_0) - \rho ({\bf x}_0)}
{\rho ({\bf x}_0) } \;
\rho ({\bf x}_0) w ({\bf x}_0,0 \to {\bf x},t)
\, d{\bf x} \, d{\bf x}_0 \nonumber \\
&=& \Bigl \langle x_i(t) \; F({\bf x}_0,\Delta {\bf x}_0) \Bigr \rangle,\end{aligned}$$ where $$\label{3.7}
F({\bf x}_0,\Delta {\bf x}_0) =
\left[ \frac{\rho ({\bf x}_0 - \Delta {\bf x}_0) - \rho ({\bf x}_0)}
{\rho ({\bf x}_0)} \right] \; .$$ Note that the mixing property of the system is required to guarantee the decay to zero of the time-correlation functions and, thus, the switching off of the deviations from equilibrium.
In the case of an infinitesimal perturbation $\delta {\bf x}(0) =
(\delta x_1(0) \cdots \delta x_N(0))$, if $\rho({\bf x})$ is non-vanishing and differentiable, the function in (\[3.7\]) can be expanded to first order and one obtains: $$\begin{aligned}
\label{3.8}
\overline{\delta x_i} \, (t) &=&
- \sum_j \Biggl
\langle x_i(t) \left. \frac{\partial \ln \rho({\bf x})}{\partial x_j}
\right|_{t=0} \Biggr \rangle \delta x_j(0) \nonumber \\
&\equiv&
\sum_j R_{i,j}(t) \delta x_j(0),\end{aligned}$$ which gives the linear response $$\label{3.9}
R_{i,j}(t) = - \Biggl \langle x_i(t) \left.
\frac{\partial \ln \rho({\bf x})} {\partial x_j} \right|_{t=0}
\Biggr \rangle$$ of the variable $x_i$ with respect to a perturbation of $x_j$. It is easy to repeat the computation for a generic observable $A({\bf x})$, yielding $$\overline{A(t)}=- \sum_j \langle A({\bf x}(t)) \left.\frac{\partial \ln \rho({\bf x})} {\partial x_j} \right|_{t=0} \rangle \delta x_j(0) \,.$$
Let us note that the study of an “impulsive” perturbation is not a limitation: e.g., in the linear regime from the (differential) linear response one can understand the effect of a generic perturbation. For instance, consider a system ruled by the evolution law $${d {\bf x} \over dt}= {\bf Q}({\bf x})$$ and apply an infinitesimal perturbation: ${\bf Q}({\bf x}) \to {\bf Q}({\bf x})+ \delta {\bf Q}(t)$ with $\delta {\bf Q}(t)=0$ for $t<0$. Then one has $$\overline{\delta x_i(t)}=
\sum_j \int_0^t R_{ij}(t-t') \delta Q_j(t') \, dt' \,.$$
Finite perturbations and relevance of chaos
-------------------------------------------
We note that in the above derivation of the FDR relation we never used any approximation on the evolution of $\delta {\bf x}(t)$. Starting with the exact expression (\[3.6\]) for the response, only a linearization on the initial time perturbed density is needed, and this implies nothing but the smallness of the initial perturbation.
It is easy to understand that it is possible to derive a FDR also for finite perturbations. One has $$\overline{\delta A(t)}=\langle A({\bf x}(t)) F({\bf x}(0), \Delta {\bf x}(0))
\rangle,$$ where the $F({\bf x}(0), \Delta {\bf x}(0))$, Eq. (\[3.7\]), depends on the initial perturbation $\Delta {\bf x}(0)$ and the invariant probability distribution [@BLMV03]. We stress again that, from the evolution of the trajectories difference, one can define the leading Lyapunov exponent $\lambda$ by considering the absolute values of $\delta {\bf x}(t)$: at small $|\delta {\bf x}(0)|$ and large enough $t$ one has $$\label{3.11}
\Bigl \langle \ln |\delta {\bf x}(t)|\Bigr \rangle \simeq
\ln |\delta {\bf x}(0)| + \lambda t \,\, .$$ On the other hand, in the FDR one deals with averages of quantities with sign, such as $\overline{\delta {\bf x}(t)}$. This apparently marginal difference is very important and allows for the possibility to derive the FDR relation avoiding van Kampen’s objection. However van Kampen’s remark can have a practical, but not conceptual, relevance. For instance in the presence of chaos the computational error on $\overline{\delta {\bf x}(t)}$ increases as $e^{\alpha
t}/\sqrt{M}$ where $M$ is the number of perturbations and $\alpha
\simeq 2 \lambda$; such a behaviour can be easily verified, see e.g. [@FIV90; @BLMV03].
About the invariant measure {#invariant}
---------------------------
Since the generalized FDR (\[3.9\]) explicitly involves the invariant measure $\rho({\bf x})$, it is useful to comment on some features that immediately derive from the properties of $\rho$.
First, we note that in Hamiltonian systems, taking the canonical ensemble as the equilibrium distribution, one has $ \ln \rho= -\beta
H({\bf Q},{\bf P})\, + const.$ Recalling Eq. , if we denote by $x_i$ the component $q_k$ of the position vector ${\bf Q}$ and by $x_j$ the corresponding component $p_k$ of the momentum ${\bf
P}$, from Hamilton’s equations ($dq_k/dt=\partial H/\partial p_k$) one immediately has the differential form of the usual FDR [@K66; @K86] $$\label{3.12}
\frac{ \overline{\delta q_k} \, (t)} {\delta p_k(0)}
=\beta \Biggl \langle q_k(t) \frac {dq_k(0)} {dt} \Biggr \rangle =
- \beta \frac{d}{dt} \Bigl \langle q_k(t) q_k(0) \Bigr \rangle \, ,$$ where time-translational invariance has been used.
### Systems with negative temperature
In the most common Hamiltonian systems one has $$H=\sum_n {p_n^2 \over 2 m} + V(q_1, ... , q_N) \,.$$ However, there are cases where the “kinetic term” is not parabolic and therefore the previous Hamiltonian is replaced by the more general form $$H=\sum_n K(p_n) + V(q_1, ... , q_N) \,.$$ If $\{ q_n \}$ and $\{ p_n \}$ are bounded variables, for a suitable form of $K(p)$ one can have negative absolute temperature, i,e. for some values of the energy $E$, $\partial S(E)/\partial E$ is negative, being $S(E)$ the microcanonical entropy [@PSV17]. Such systems are not mere curiosity, being rather relevant for instance in hydrodynamics and plasma physics [@Joyce]. In addition, recent experimental measurements showed the presence of a negative absolute temperature in cold atoms systems [@Braun].
The generalized FDR derived in Sec. \[GFDRs\] holds even for systems with negative temperature. Since the, somehow, peculiar features of such systems, the validity of the FDR can appear to be not obvious. As an example, in Fig. \[figFDR\] from [@N2], we show the comparison between the response function and the (proper) correlation functions numerically obtained in a model with long-range interactions $$\label{longrange}
H= \sum_{i=1}^N (1- \cos p_i) +
N\Big( {J \over 2} m^2+ {K \over 4} m^4 + const. \Big),$$ where $m$ is the modulus of the vector $${\bf m}= \Big( {1 \over N}\sum_{i=1}^N \cos q_i,
{1 \over N}\sum_{i=1}^N \sin q_i \Big),$$ for a value of $E$ corresponding to a negative temperature.
![Check of the FDR by a direct computation of the mean response and comparison with the theory in the model (\[longrange\]) for the variable $A=\sin p_j$, for $E/N=1.9$. Other parameters are $J=-0.5,\, K=-1.4, \, N=250, \,
M=10^5$ and $\delta p_i(0)=0.01$ [@N2]. Reproduced figure with permission from F. Miceli, M. Baldovin, and A. Vulpiani. Phys. Rev. E, 99:042152, 2019, Copyright (2019) by the American Physical Society.[]{data-label="figFDR"}](Long-Fig.eps)
### Gaussian distribution
When the stationary state is described by a multivariate Gaussian distribution, $$\ln \rho({\bf x})= -{1 \over 2} \sum_{i,j}\alpha_{ij}x_i x_j + const.$$ with $\{ \alpha_{ij} \}$ a positive matrix, the elements of the linear response matrix can be written in terms of the usual correlation functions: $$\label{3.14}
R_{i,j} (t) =\sum_k \alpha_{j,k}
{\Bigl \langle x_i(t) x_k(0) \Bigr \rangle } \; .$$ The above result is nothing but the Onsager regression originally obtained for linear Langevin equations.
It is interesting to note that there are important nonlinear systems with a Gaussian invariant distribution, e.g. the inviscid hydrodynamics [@Kr59; @Kr00], where the Liouville theorem holds, and a quadratic invariant exists. Therefore one has a quite simple relation between the responses and the correlations; in spite of this fact, the dynamics is not linear and the behavior of the correlation functions is not trivial at all.
### Non Hamiltonian systems
When the form of $\rho({\bf x})$ is not known, as usually in non-Hamiltonian systems, the relation (\[3.9\]) does not give detailed quantitative information. However, it represents a connection between the mean response function $R_{i,j}$ and a suitable correlation function, computed in the non perturbed systems: $$\label{3.13}
\Bigl \langle x_i(t)f_j({\bf x}(0)) \Bigr \rangle \; ,
\quad \textrm{with} \quad
f_j({\bf x})=- \frac{\partial \ln \rho}{\partial x_j} \,\, ,$$ where, in the general case, the function $f_j$ is unknown.
Let us stress that, in spite of the technical difficulty for the determination of the function $f_j$, which depends on the invariant measure, a FDR always holds in mixing systems whose invariant measure is “smooth enough”. In particular, we note that the nature of the statistical steady state (either equilibrium, or non-equilibrium) has no role at all for the validity of the FDR.
### Marginal distribution
Let us stress that the knowledge of a marginal distribution $$p_i(x_i)= \int \rho(x_1, x_2, ....)\prod_{j \neq i}dx_j
\label{projection}$$ does not allow for the computation of the auto-response: $$R_{i,i}(t)\neq
- \Biggl \langle x_i(t) \left.
\frac{\partial \ln p_i(x_i)} {\partial x_i} \right|_{t=0}
\Biggr \rangle \,\,, \label{falseresponse}$$ even in the case of Gaussian variables. As one can easily understand from Eq. (\[3.14\]), $R_{i,i}(t)$ in general, even in the Gaussian case, is not proportional to $\langle x_i(t)x_i(0) \rangle $. This observation is very relevant for what follows. Consider for instance that the description of our system has been restricted to a single *slow* degree of freedom, such as the coordinate of a colloidal particle following a Langevin equation. The above discussion has indeed made clear that other fluctuating variables can be coupled to the one we are interested in, and therefore it is not correct to project them out by the marginalization in Eq. (\[projection\]). Conversely, a stationary probability distribution with new variables coupled to the colloidal particle position must be taken into account. This point will be discussed in more detail in Sec. \[marginal\].
### Chaotic dissipative systems
At this point one could object that in a chaotic deterministic dissipative system the above machinery cannot be applied, because the invariant measure is not smooth at all. Typically the invariant measure of a chaotic attractor has a multifractal structure and its Renyi dimensions $d_q$ are not constant. In chaotic dissipative systems the invariant measure is singular. However the previous derivation of the FDR relation is still valid if one considers perturbations along the expanding directions. Due to the singular nature of the invariant probability distribution, a general response function has two contributions, parallel and perpendicular to the attractor, corresponding respectively to the expanding (unstable) and the contracting (stable) directions of the dynamics [@N3]. Each perturbation can be written as the sum of two contributions $$\delta F(t)= \delta F_{\parallel}(t)+ \delta F_{\perp}(t),$$ and the effect of such a perturbation on the response on an observable $A$ attains the form $$\begin{aligned}
\overline{ \delta A (t)}&=&
\int_0^t R_{\parallel}^{(A)}(t-t') \delta F_{\parallel}(t') dt' \nonumber \\
&+&
\int_0^t R_{\perp}^{(A)}(t-t') \delta F_{\perp}(t') dt' \, .\end{aligned}$$ It is easy to realise that only for the part $R_{\parallel}^{(A)}$ one can have a FDR, i.e. it can be expressed in terms of a correlation function computed in the unperturbed dynamics on the attractor. On the contrary for the second contribution (from the contracting directions), the response can be obtained only numerically [@CS07].
As a matter of fact, there are at least two good reasons which allow us to hope that the singular structure of the invariant measure, at practical level, can be not very relevant. First, we note that in systems with many degrees of freedom, for a non pathological observable the contribution of $R_{\perp}^{(A)}$ should be negligible (we will consider again in the following this point) [@N4]. In addition, a small amount of noise, that is always present in a physical system, smooths the $\rho({\bf x})$ and the FDR can be derived. We recall that this “beneficial” noise has the important role of selecting the natural measure, and, in the numerical experiments, it is provided by the round-off errors of the computer. We stress that the assumption on the smoothness of the invariant measure allows one to avoid subtle technical difficulties.
Other forms of non-equilibrium FDR
----------------------------------
In Sec. \[GFDRs\] we have derived a FDR which involves the invariant measure. Here we consider a somehow complementary approach, in the sense that one expresses the response functions in terms of correlations with a quantity that involves the transition rates of the model. The common features, namely the appearance of extra-terms related to the coupling with “hidden” variables, will be discussed in the next subsection.
We consider a system with dynamics described by a Markov process with transition rates $W({\bf x'}\to{\bf x})$ from state ${\bf x'}$ to state ${\bf x}$, with normalization $$\sum_{{\bf x}} W({\bf x'}\to {\bf x})=0.
\label{2.0}$$ Assuming a perturbation in the form of a time-dependent external field $h(s)$, which couples to the potential $V({\bf x})$ and changes the energy of the system from $H({\bf x})$ to $H({\bf
x})-h(s)V({\bf x})$, the linear response function of the observable $A$ is $$R(t,s)= \left .\frac{\delta
\langle A(t)\rangle_h}{\delta h(s)}\right|_{h=0},
\label{2.4}$$ where $\langle\ldots\rangle_h$ denotes an average on the perturbed dynamics. The perturbed transition rates $W^h({\bf x}|{\bf x'})$, to linear order in $h$, take the form $$\begin{aligned}
W^h({\bf x'}\to {\bf x})&=&W({\bf x'}\to {\bf x}) \times \nonumber \\
&&\left\{1-\frac{\beta h}{2}\left[V({\bf x})-V({\bf x'})\right]+M({\bf x},{\bf x'})\right\},
\label{2.3}\end{aligned}$$ where $\beta$ is the inverse temperature (with $k_B=1$) and $M({\bf
x},{\bf x'})$ is an arbitrary symmetric function. The diagonal elements are obtained from the normalization condition (\[2.0\]). This structure derives from the local detailed balance principle [@CM99]. Note, however, that there is not a univocal prescription for the choice of $W^h$ through the function $M$ [@LCZ05; @CLSZ10]. For a general discussion of different symmetric factors in the transition rates, see for instance [@Basu_2015], or [@PhysRevE.93.032128; @baiesinew] in the context of lattice gas models. For simplicity, here we take $M=0$. Then, the response function can be written as $$R(t,s)= \frac{\beta}{2}\left[\frac{\partial \langle A(t)
V(s)\rangle}{\partial s} -\langle A(t)B(s)\rangle \right],
\label{2.9}$$ where $$B(s) \equiv B[{\bf x}(s)]=\sum_{{\bf x''}}\{V({\bf x''})-V[{\bf x}(s)]\}W[{\bf x}(s)\to {\bf x''}]
\label{2.10}$$ is an observable quantity, namely depends only on the state of the system at a given time.
The two formulae, Eqs. (\[3.9\]) and (\[2.9\]), show that, in general, non-equilibrium is not a limit, in the sense that the response function can still be expressed in terms of unperturbed correlators. Similar forms of FDR have been derived in [@BMW09; @maes1; @maes2; @maes2; @ss06; @Seifert; @Verley_2011; @PhysRevLett.103.090601], and also experimentally verified [@GPCCG09; @GPCM11]. In particular, Eq. (\[2.9\]) extends to discrete systems the relation derived for overdamped Langevin equation with continuous variables [@CKP94]. Let us also mention the rigorous derivation of a similar FDR in the context of exclusion processes reported in [@olla] and the recent results for the response to temperature perturbations [@falasco; @falasco2].
A simple illustration of Eq. (\[2.9\]) is provided by the case of the Langevin dynamics of a particle diffusing in a potential $U(x)$, $$\dot{x}(t) = - \frac{\partial U(x)}{\partial x}+ \sqrt{2 T } \zeta(t),
\label{2.10bis}$$ with $\zeta(t)$ a white noise with zero mean and unit variance. The response formula, with respect to a perturbing force $F$, reads as: $$\frac{\delta \langle x(t) \rangle_F}{\delta F(s)} =
\frac{\beta}{2} \left[ \frac{\partial \langle x(t) x(s) \rangle}{\partial s} - \langle x(t) B[x(s)] \rangle \right],
\label{2.12}$$ with $B[x(s)] = -\partial U/\partial x|_{x(s)}$. At equilibrium it can be easily proved that $\langle x(t) B[x(s)] \rangle = - \partial
\langle x(t) x(s) \rangle / \partial s $, recovering the standard FDR formula. Differently, out of equilibrium the contribution coming from the local field $B[x(s)]$ must be explicitly taken into account.
Finally, we note that, in this theoretical approach, non-linear FDRs can be derived, relating non-linear response functions with high-order correlation functions [@bouchaud2005nonlinear; @LCSZ08a; @LCSZ08b]. These nonlinear responses find important applications in the context of disordered and glassy systems [@crauste2010evidence]. An important point to stress is that, for non-linear FDRs, even at equilibrium the model dependent quantity $B$ defined in Eq. (\[2.10\]) is involved. These results point out the central role played by kinetic factors in characterizing the non-equilibrium dynamics [@Basu].
Usually marginal PDF is not enough: the role of coupling {#marginal}
--------------------------------------------------------
The main message from the previous section is that out of equilibrium the response function can still be expressed in terms of unperturbed correlators but these correlators involve particular quantities that do not appear at equilibrium. These quantities indeed characterize the non-equilibrium dynamics.
There is still a deep debate on the general physical meaning of such terms: some ones have pointed out the role played by different entropy productions [@seifert05; @ss06], some others have pinpointed the necessity to introduce new perspectives in order to characterize non-equilibrium dynamics [@maesbook].
What is clear is that the extra-terms unveil the presence of relevant couplings that arise in non-equilibrium systems. In particular, even if one is interested in the response of a specific variable, the knowledge of its statistical properties, namely the marginalized distribution function, is in general not enough. Other degrees of freedom can be coupled to the observable under study, making the prediction of its response more involved.
![Velocity correlation $C(t)=\langle V(t)V(0)\rangle/\langle
V(0)V(0)\rangle$ (black circles), response function (red squares), and generalized FDR Eq. (\[gfdrtracer\]) (green diamonds) computed in molecular dynamics numerical simulations of a massive tracer in a granular gas with packing fraction $\phi=0.6$, coefficient of restitution $\alpha=0.6$ and collision time $\tau_c$, see [@SVGP10] for details.[]{data-label="fdrtracer"}](FDR_tracer.eps)
An illustrative example is provided by the motion of an intruder in a granular gas [@SVCP10; @SVGP10; @GPSV14]. A granular fluid is made of macroscopic particles subject to external forcing, and therefore is characterized by dissipative interactions (inelastic collisions) and non-equilibrium dynamics. In order to describe the velocity autocorrelation of the intruder and its linear response, one can introduce a two-variable model (in one dimension, for simplicity)
\[grintr\] $$\begin{aligned}
M \dot{V}(t)= - \Gamma [V(t)-U(t)] + \sqrt{2 \Gamma T_{tr}} \mathcal{E}_v(t)\\
M' \dot{U}(t) = -\Gamma' U(t) - \Gamma V(t) + \sqrt{2 \Gamma' T_b} \mathcal{E}_U(t),\end{aligned}$$
where $V$ is the velocity of the intruder with mass $M$, $U$ describes a local velocity field (a local average of the velocities of the particles surrounding the intruder) whose dynamics is coupled with that of the tracer, $\Gamma$ is a viscosity, $\Gamma'$ and $M'$ are effective parameters to be determined. $T_{tr}$ is the intruder kinetic temperature, while $T_b$ is the value of the kinetic temperature of the granular fluid, playing the role of a non-equilibrium bath. $\mathcal{E}_v$ and $\mathcal{E}_U$ are delta-correlated noises with zero mean and unitary variance. The dilute limit can be obtained with $\Gamma' \sim M' \to \infty$, which implies small $U$.
Eqs. represent a linear model for which analytical results can be obtained and can describe real systems for not too high density. In particular, in the elastic limit ($T_{tr} = T_b$), the coupling with $U$ can still be important, but the equilibrium FDR is recovered. Out of equilibrium, one can apply the formula (\[3.9\]) to express the response in terms of correlation functions. Since the system is linear, the stationary distribution is a bivariate Gaussian, and from Eq. (\[3.9\]) directly follows $$R_{VV}(t)= \frac{\overline{\delta V(t)}}{\delta V(0)}=\sigma_{VV}^{-1}\langle V(t)V(0)\rangle + \sigma_{UV}^{-1}\langle V(t)U(0)\rangle,
\label{gfdrtracer}$$ where $\sigma_{VV}^{-1}$ and $\sigma_{UV}^{-1}$ are the elements of the inverse covariance matrix and can be expressed in terms of the model parameters, see [@CPV12] for further details. In Fig. \[fdrtracer\] we check the validity of such an approach. The main message we want to stress here is that a central role is played by the correlations between the variable $V$ and the local velocity field $U$. At variance with equilibrium cases, in general the knowledge of the statistical properties of $V$ alone, e.g. the measure of its marginalized PDF, is not enough to reconstruct the response to an external perturbation.
Toward realistic systems {#real}
========================
In this Section we illustrate the use of FDRs introduced above, in different non-equilibrium and non-standard systems, ranging from athermal and active matter to systems with anomalous diffusion and multiple time-scale structure.
FDR and the effective temperature
---------------------------------
One of the applications of the FDR in non-equilibrium systems deals with the interesting concept of *effective* temperature [@CKP97]. Indeed, within the context of non-equilibrium phenomena, the first attempts to formulate a general theory start from extending concepts well defined in the consolidate equilibrium theory. Here, for instance, one assumes that, after a sufficient long time $t_{eq}$, an isolated and finite system reaches an equilibrium state that can be characterized by a small number of parameters, the state variables, such as temperature and pressure. Thus, an interesting question concerns the possibility that also for non-equilibrium phenomena, a characterization in terms of a few variables is still feasible, at least in some particular regimes.
The effective temperature can be introduced via the linear FDR, as the ratio between response function and correlation function [@CKP97]. The study of the behavior of such a quantity can be interesting in itself, but its real meaning as relevant parameter characterizing some features of the system strongly depends on the considered models. The shape of the fluctuation-dissipation ratio can be useful to grasp information on the presence of different relevant time scales in the system. However, the observed different time scales are not necessarily related to an underlying complex dynamics [@VBPV09]. The general issue of effective temperature has been the subject of recent reviews [@cuglirev; @PSV17] and is an open line of intense research, with applications for instance in granular [@DMGBLN03; @BBDLMP05; @keys2007measurement] and active matter [@PhysRevE.77.051111; @Wang15184; @PhysRevE.90.012111; @Levis_2015; @Han7513; @PhysRevE.97.032125; @dieterich2015single; @preisler2016configurational; @PhysRevLett.118.015702; @workamp2018symmetry; @seifert2019stochastic; @cugliandolo2019effective; @golestanian; @klongvessa] (see also Section \[actives\] below). Here, we focus on a few examples in the context of athermal systems.
First, we mention some cases where the idea of effective temperature has been successfully applied. It has been shown that, in a model of a sheared, zero-temperature foam, different definitions of temperature, that for equilibrium thermal systems would be equivalent, take on the same value and show the same behavior as a function of the shear-rate [@durian1]. This observation suggests that in this situation the concept of effective temperature is robust and its introduction can be useful to build a statistical mechanics description out of equilibrium. Similar conclusions followed for other athermal systems, such as a sphere placed in an upward flow of gas [@durian2] and a two-dimensional air-driven granular medium [@durian3].
![Response function $R(t)$, velocity autocorrelation $C(t)$ and predicted response with a factorization approximation $R_G(t)$, measured for a blade suspended in a strongly vibrated granular gas [@GPSV14] for three different packing fractions (0.05 (a), 0.1 (b), and 0.15 (c)), reproduced with permission from A. Gnoli, A. Puglisi, A. Sarracino, and A. Vulpiani. Plos One, 9:e93720, (2014). 2014 Creative Commons Attribution (CC BY) license[]{data-label="fdrexp"}](new_fig_dil.eps "fig:") ![Response function $R(t)$, velocity autocorrelation $C(t)$ and predicted response with a factorization approximation $R_G(t)$, measured for a blade suspended in a strongly vibrated granular gas [@GPSV14] for three different packing fractions (0.05 (a), 0.1 (b), and 0.15 (c)), reproduced with permission from A. Gnoli, A. Puglisi, A. Sarracino, and A. Vulpiani. Plos One, 9:e93720, (2014). 2014 Creative Commons Attribution (CC BY) license[]{data-label="fdrexp"}](new_fig_med.eps "fig:") ![Response function $R(t)$, velocity autocorrelation $C(t)$ and predicted response with a factorization approximation $R_G(t)$, measured for a blade suspended in a strongly vibrated granular gas [@GPSV14] for three different packing fractions (0.05 (a), 0.1 (b), and 0.15 (c)), reproduced with permission from A. Gnoli, A. Puglisi, A. Sarracino, and A. Vulpiani. Plos One, 9:e93720, (2014). 2014 Creative Commons Attribution (CC BY) license[]{data-label="fdrexp"}](new_fig_den.eps "fig:")
In a different experimental setup the response and autocorrelation functions of a blade suspended in a strongly vibrated granular system were measured [@GPSV14]. In this case, it has been shown that only in the dilute regime an effective temperature can be properly defined, being the response and autocorrelation proportional to each other. At higher densities, on the contrary, response and correlations show more complex behaviors, see Fig. \[fdrexp\], and the coupling with other degrees of freedom in the system starts to play a central role. In particular, inspired by the model described in Sec. \[marginal\], it was suggested that the relevant coupling quantities can be defined in terms of a local velocity field. Experimental measures confirmed such an explanation, even if a quantitative description of the system correlations with the simple two-variable model turned out to be not accurate.
FDR and anomalous diffusion {#comb}
---------------------------
Another non-standard class of systems where the FDR has been investigated is represented by systems showing anomalous transport properties. These systems are characterized by a non-linear behavior of the mean square displacement [@BG90; @GSGWS96; @CMMV99; @MK00; @BC05; @BO02], i.e. $$\label{4}
\langle x^2(t)\rangle \sim t^{2 \nu} \,\,\, \mbox{with} \,\,\, \nu \ne
1/2.$$ Formally this corresponds to have a diffusion coefficient $D=\infty$ if $\nu > 1/2$ (superdiffusion) and $D=0$ if $\nu < 1/2$ (subdiffusion). Note that in general the same model can show different behaviors depending on the considered time scale. It is interesting to wonder whether the FDR in the form of the Einstein relation is still valid, namely whether the quantity $\langle x^2(t) \rangle$ is still proportional to $\overline{\delta x}(t)$ at any time: $$\label{3}
{\langle x^2(t) \rangle \over \overline{\delta x}(t)}=\frac{2}{\beta F},$$ where $$\label{2}
\overline{\delta x}(t)=
\langle x(t) \rangle_F - \langle x(t) \rangle
\simeq \mu F t \,\, ,$$ with $\langle\ldots\rangle_F$ denoting the average on the system perturbed by a force $F$, and $\mu$ the mobility.
Quite remarkably, it has been shown that the Einstein relation is robust and holds even in models showing anomalous behaviors. This has been explicitly proved in systems described by a fractional-Fokker-Planck equation [@MBK99; @BF98; @CK09]. In addition there is clear analytical [@LATBL10] and numerical [@VPV08] evidences that (\[3\]) is valid for the elastic single file model, i.e. a one-dimensional gas of elastic particles on a ring, which exhibits subdiffusive behavior due to the confinement, $\langle x^2
\rangle \sim \sqrt{t}$ [@HKK96].
An important point to stress is that the validity of a FDR in the standard Einstein form depends on the equilibrium properties of the systems, rather than on its anomalous dynamics. Indeed, if non-equilibrium conditions are introduced in the system, a generalized FDR has to be considered, including the extra-terms previously discussed. This has been explicitly shown, for instance, for a particle diffusing on a comb lattice [@comb; @Forte_2013], driven by an external force. In particular, denoting by $x\in(-\infty,\infty)$ the position of the particle along the backbone of the comb and by $y\in[-L,L]$ the coordinate along a tooth, transition rates from $(x,y)$ to $(x',y')$ are $$\begin{aligned}
W^d[(x,0)\rightarrow (x\pm 1,0)]&=&1/4\pm d \nonumber \\
W^d[(x,0)\rightarrow (x,\pm 1)]&=&1/4 \nonumber \\
W^d[(x,y)\rightarrow (x,y\pm 1)]&=&1/2~~~ \textrm{for}~y\ne 0,\pm L,
\label{ww}\end{aligned}$$ where $d$ is the drift in the $x$ direction. Applying the generalized FDR Eq. (\[2.9\]) for the response to a perturbation $\varepsilon$ one has [@comb] $$\begin{aligned}
\hspace*{-2cm}\frac{\overline{\delta\mathcal{O}}_d}{h(\varepsilon)}&=&\frac{\langle \mathcal{O}(t)\rangle_{d+\varepsilon}- \langle
\mathcal{O}(t)\rangle_d}{h(\varepsilon)} \nonumber \\
&=&\frac{1}{2}\left[\langle\mathcal{O}(t)x(t)\rangle_d-\langle\mathcal{O}(t)x(0)\rangle_d
-\langle\mathcal{O}(t)A(t,0)\rangle_d\right],
\label{FDR}\end{aligned}$$ where $\mathcal{O}$ is a generic observable, and $A(t,0)=\sum_{t'=0}^t
B(t')$, with $B$ following from Eq. (\[2.10\]) $$B[(x,y)]=\sum_{(x',y')}(x'-x)W^d[(x,y)\rightarrow (x',y')]=2d\delta_{y,0}.$$ The sum on $B$ has an intuitive meaning: it counts the time spent by the particle on the backbone. In Fig. \[fdrcomb\] it is shown the validity of this approach.
Let us note that in these cases the FDR holds if the displacement is compared to the linear response but it does not if the self-correlation of the particles position is used, due to the lack of a confining potential.
![Response function (black line) and second cumulant (black dotted line) measured in the comb model [@comb]. The term including the correlation with the quantity $B$ (green dotted line) is necessary to recover the response function (see blue dotted line), in agreement with the FDR (\[2.9\]). []{data-label="fdrcomb"}](FDR3.eps)
Another example is the inelastic single-file model [@VPV08], where one introduces dissipative interactions among the particles diffusing on a ring. Again, due to the non-equilibrium conditions, strong correlations among particles are present and the factorization of the invariant measure fails. For small dissipation (namely, small inelasticity, small packing fraction and/or fast thermostats) the Einstein relation is recovered, because of the weak lack of factorization. In general, the FDR involves extra-terms that take into account correlations with other degrees of freedom in the system.
Generalized FDR have also been applied in the case of superdiffusion for instance in models of Lévy walks and Lévy flights [@Gradenigo_2012; @PhysRevE.87.030104; @godec2013linear; @kusmierz2018thermodynamics] or generalized Langevin equations [@costa2003fluctuation; @dieterich2015fluctuation], while the linear response of a particle showing anomalous diffusion in an aging medium has been studied in [@pottier; @pottier1]. A very interesting open problem is the validity of the Einstein relation in systems of interacting particles on comb structures, where the diffusion of a tracer can be studied with analytical approaches and shows different non-trivial anomalous behaviors [@PhysRevLett.115.220601]. Finally, let us mention that for a tracer advected by a steady laminar flow and subject to an external force, showing non-trivial anomalous behaviors, such as negative differential and absolute mobility, the validity of a generalized FDR [@maes2] has been recently discussed in [@PhysRevLett.117.174501].
FDR and multiscale systems
--------------------------
Let us now discuss some aspects of the FDR in systems with non-trivial temporal structures, for instance with many degrees of freedom whose characteristic times are very different. The relation (\[3.9\]) suggests that, in general, the choice of the observable $A({\bf x})$, and the size of the perturbation $\Delta {\bf x}(0)$, can correspond to different relaxation behaviours of $\overline{\delta A(t)}$.
In systems with a unique characteristic time, e.g. the celebrated $3-d$ Lorenz system or $1d$ Langevin equation, one expects that there are not significant differences at varying the observable and the size of the perturbation, and numerical computations confirm the intuition. Less trivial is the case of high dimensional systems with many different characteristic times, where one observes a more interesting scenario. At varying the observables one has different correlation functions, whose relaxation times can be very different; in an analogous way different response functions can be characterized by very different temporal behaviours.
The fact that finite perturbations can relax with characteristic times which can depend very much on the size of the initial perturbation, is rather relevant in geophysical context, e.g. in the study of climate dynamics, where many degrees of freedom are involved with characteristic times which vary from seconds (3D turbulence) to weeks (geostrophic turbulence) and thousand years (oceanic currents and ice shields dynamics). Numerical computations on simplified models, e.g. the so called shell models, show the following scenario: the relaxation time of the finite perturbations increases with the size at the initial time [@BLMV03].
As already discussed, the relations between response and correlation are not trivial at all: this because there appears the invariant probability density which is not known, a part very few special cases (e.g. Hamiltonian systems, inviscid hydrodynamics and some Langevin equations). In dissipative systems, as mentioned before, we have an additional technical difficulty due to the singular structure of $\rho$. Nevertheless, we have the positive fact that the generalized FDR (\[3.9\]) indicates the existence of a relation between the response and some correlation whose precise functional shape is not known. It seems natural to hope that the simple correlations, i.e. $\langle x_i(t)x_i(0) \rangle$ is enough to catch at least the qualitative behaviour of the response function $R_{i,i}(t)$.
In order to illustrate this issue, let us briefly discuss a system introduced by Lorenz as a simplified model for the atmospheric circulation [@N5]: $${d x_k \over dt}=-
x_{k-1}(x_{k-2}-x_{k+1}) -\nu x_k + F + c_1\sum_{j=1}^M y_{k,j}$$ $${d y_{k,j} \over dt}=-
c b y_{k, j+1}(y_{k,j+2}-y_{k,j-1}) -c \nu y_{k,j} + c_1 x_k$$ where the set $\{ x_k \}$ with $k=1, ... , N$ and $\{ y_{k,j} \}$ with $j=1, ... , M$ are the slow-large scale and the fast small-scale variables respectively; the above system is often called Lorenz-96 model. Roughly speaking, the $\{ x_n \}$’s represent the synoptic scales, while the $\{ y_n \}$’s represent the convective scales.
![Autocorrelation $C_{kk}(t)$ (full line) and self-response $R_{kk}(t)$ of the slow variable $x_{k}$ for $k=3$ for the Lorenz-96 model [@N5]; the parameters are $ F=10,\, \nu=1, \, c=10, \, c_1=1, \,
N=36$ and $M=10$. Reproduced with permission from G. Lacorata and A. Vulpiani. Fluctuation-response relation and modeling in systems with fast and slow dynamics. Nonlinear Proc. in Geophys., 14:681, (2007). Copyright 2007 European Geosciences Union[]{data-label="figFDR2"}](LV-Fig2.eps)
Of course it is impossible to write down the invariant measure $\rho$ and therefore to find the proper correlation function for a given response. Nevertheless, as shown in Figs. \[figFDR1\] and \[figFDR2\], it is well evident how, even in the absence of a precise quantitative agreement, one has a certain similarity between the autocorrelation and self-response function. The correlations of the slow (fast) variables have a clear qualitative resemblance with the response of the slow (fast) variables themselves. In particular, the relaxation times of the response of fast (slow) variables are of the same order of magnitude of the corresponding correlation functions [@N5].
In some applications it is rather common to wonder about the response of a global variable which depends on many variables: for instance in the study of climatic dynamics it is rather natural to study the global temperature. The intuition suggests that in the presence of many degrees of freedom, even for dissipative systems, if we are interested in the response of some global variable, one can hope to invoke the help of some statistical regularization. Numerical computations indicate that such an intuition is correct.
Indeed, in some cases the above intuition is confirmed by a rigorous analytical treatment, as described in a recent paper by Wormell and Gottwald [@Wormell]. In this work it is studied the macroscopic variable $Q$ ruled by the discrete time map $$Q_{t+1}=A Q_t(1- Q_t),$$ with $$A=A_0+{A_1 \over M^{\gamma}}\sum_{j=1}^Mx_t^{(j)},$$ where the independent variables $\{ x^{(j)} \}$ evolve according to a deterministic law $$x_{t+1}^{(j)}=g_{a_j}(x_t^{(j)}),$$ e.g. $g_a(x)=x^2- (a x(1-x))^2$, the $\{ a_j \}$ are sampled by a smooth PDF and $\gamma \ge 1/2$. The stationary joint PDF of $(Q, \{
x^{(j)} \})$ does not depend in a continuous way on the $\{ a_j \}$ and therefore a FDR cannot hold for the variables $\{ x^{(j)} \}$. In spite of this, one has that the marginal PDF of the macroscopic variable $Q$ varies in a continuous (and differentiable) way with the $\{ a_j \}$: this allows to show a FDR for the variable $Q$.
FDR and active matter {#actives}
---------------------
In recent years, the field of active matter has drawn the attention of physicist on the study of a new form of intrinsically non-equilibrium systems, made by elementary constituents that continuously convert energy into motion [@DEMAGISTRIS201565]. This class of systems shows phenomena similar to those characterizing granular matter, such as clustering, segregation, non-equilibrium phase transitions, flocking, collective motion and so on, and many models have been proposed to describe such a huge variey of behaviors. Recently, the generalized FDRs, in different forms, have been applied to these models, see for instance [@PhysRevLett.117.038103; @PhysRevLett.119.258001; @PhysRevE.98.020604; @e19070356; @PhysRevE.90.052130; @maggi; @Szamel_2017; @chong; @burk; @pagona]. Here we mention two specific examples, in order to illustrate some peculiarities of this context.
As a first example we consider the active Ornstein-Uhlenbeck model, which describes the persistent motion characterizing a single active particle, with the introduction of a coloured noise [@maggi1]. The particle dynamics is modeled as $$\begin{aligned}
\label{eq:dim_AOUP}
\dot{x}& =& \frac{f(x)}{\gamma}+a, \qquad f(x)=-\frac{d}{dx}\phi(x), \nonumber \\
\dot{a}&=&-\frac{a}{\tau} +\frac{\sqrt{2D}}{\tau}\eta,\end{aligned}$$ where $x(t)$ is the position of the particle, $\tau$ is the persistence time, $\gamma$ the drag coefficient, $\phi(x)$ the potential acting on the system, $a(t)$ is the active bath, and $\eta(t)$ a delta-correlated white noise, with zero mean and unitary variance. The parameter $D$ fixes the amplitude of the active bath fluctuations $$\left<a(t)a(t') \right> = \frac{D}{\tau} \exp{\left[-\frac{t-t'}{\tau}\right]}.$$ The response to an external perturbation of a one-dimensional system of non-interacting particles described in this framework has been recently studied in [@Caprini_2018]. A direct application of Eq. (\[3.9\]) is possible in this case because the stationary distribution can be computed with some approximations. In particular, it has been shown that the non-equilibrium coupling between particle velocity and position has to be taken into account, playing a central role when the particle persistence time is large. Moreover, the analysis showed that although the approximation for the stationary distribution gives good results for the static properties, the dynamical behavior can be well described only in the limit of small persistence time. An FDR for Langevin equations with memory has been derived in [@safaverdi] from a different approach.
A different approach is the active Brownian particle model with energy depots [@PhysRevLett.80.5044], derived from the Rayleigh-Helmholtz treatment of sustained sound waves [@strutt1945theory]. We consider the equation for the motion of a particle of mass $m=1$, with position $x$ and velocity $v$, in an external potential $U(x)$ $$\begin{aligned}
\label{kramers}
\dot{x}(t) &=& v(t) \nonumber \\
\dot{v}(t)&=&-F[v(t)]-U'[x(t)] +\eta(t), \end{aligned}$$ where $\eta(t)$ is a white noise, with zero mean and $\langle
\eta(t)\eta(t')\rangle=2\gamma T\delta(t-t')$, $\gamma$ and $T$ being two parameters and $U(x)$ is a potential. The function $F[v]$ is given by $$\label{active}
F[v(t)]=-\gamma_1 v(t)+\gamma_2 v^3(t),$$ with $\gamma_1$ and $\gamma_2$ positive constants. This means that motion of the particle is accelerated at small $v$ and is damped at high $v$, taking into account the internal energy conversion of the active particles coupled to other energy sources. The stochastic equations (\[kramers\]) are out of equilibrium and a generalized FDR similar to Eq. (\[2.9\]) can be applied [@PhysRevE.88.052124]. In particular, the response function $R(t,t')$ of the velocity $v(t)$ to a perturbation $h(t')$ applied at a previous time $t'$ reads $$R(t,t')=\left .\frac{\delta \langle v(t)\rangle}{\delta
h(t')}\right|_{h=0},
\label{fdr.00}$$ and exploiting the relation valid for Gaussian noise [@CKP94] $$R(t,t')=\frac{1}{2\gamma T}\langle v(t)\eta(t')\rangle,
\label{fdr.0}$$ we get the FDR $$\begin{aligned}
R(t)&=&\frac{1}{2\gamma T}\Big\{\langle v(t)F[v(0)]\rangle
+\langle F[v(t)]v(0)\rangle \nonumber \\
&+&\langle v(t)U'[x(0)]\rangle+
\langle U'[x(t)]v(0)\rangle\Big\}.
\label{resp.0}\end{aligned}$$ In Fig. \[fdractive\] we show the validity of the above formula, where, again, the non-equilibrium coupling between velocity and position has to be taken into account.
![FDR Eq. (\[resp.0\]) for the model of active particles (\[active\]) with parameters $\gamma_1=5, \gamma_2=1,
\gamma T=0.5$ for a potential $U(x)=(1/2)x^2+(1/3)x^3+(1/4)x^4$.[]{data-label="fdractive"}](fdr.eps)
Conclusions and perspectives {#conc}
============================
We have reviewed a series of results on FDRs for out-of-equilibrium systems. The [*leitmotiv*]{} of our discussion is the importance of correlations among different variables for non-equilibrium response, much more relevant than in equilibrium systems: indeed the generalized FDRs discussed in Section \[GFDR\] deviate from the equilibrium counterpart for the appearance of additional contributions coming from correlated degrees of freedom. This is the case, for instance, of the linear response in the general two-variable Langevin model for a granular tracer: additional contributions to the equilibrium linear response appear when the main field $V$ is coupled to the auxiliary field $U$, which only appear out of equilibrium, see Section \[marginal\], or in the diffusion model on a comb-lattice, where – in the presence of a net drift – the linear response takes a non-negligible additive contribution, see Section \[comb\]. Other examples have been discussed in Section \[actives\].
Remarkably, in some cases, one may explicitly verify that the coupled field which contributes to the linear response in non-equilibrium setups is also involved in the violation of detailed balance: such a violation is measured by the fluctuating entropy production, whose connection with non-equilibrium couplings represents another interesting line of research [@Seifert].
In conclusion, many results point in the same direction, suggesting a general framework for linear-response in systems with non-zero entropy production. Even in out-of-equilibrium dynamics, a clear connection between response and correlation in the unperturbed system exists. A further step is looking for more accessible observables for the prediction of linear response: indeed, both the discussed formula, Eq. and Eq. , require the measurement of variables which, in general, depend on full phase-space (microscopic) observables and can be strictly model-dependent. Such a difficulty also explains why, in a particular class of slowly relaxing systems with several well separated time-scales, such as spin or structural glasses in the aging dynamics, i.e. after a sudden quench below some dynamical transition temperature, approaches involving “effective temperatures” have been used in a more satisfactory way [@CR03]. Other interpretations of the additional non-equilibrium contributions to the FDR have been proposed recently [@maesbook], but the predictive power of this approach is not yet fully investigated and represents an interesting line of ongoing research.
We thank A. Baldassarri, G. Boffetta, F. Corberi, M. Falcioni, G. Gradenigo, G. Lacorata, E. Lippiello, U. Marini Bettolo Marconi, A. Puglisi, L. Rondoni, D. Villamaina and M. Zannetti for a long collaboration on the issues here discussed. AS acknowledges support from “Programma VALERE” from University of Campania “L. Vanvitelli”.
|
---
abstract: 'A large fraction of the electronic health records consists of clinical measurements collected over time, such as blood tests, which provide important information about the health status of a patient. These sequences of clinical measurements are naturally represented as time series, characterized by multiple variables and the presence of missing data, which complicate analysis. In this work, we propose a surgical site infection detection framework for patients undergoing colorectal cancer surgery that is completely unsupervised, hence alleviating the problem of getting access to labelled training data. The framework is based on powerful kernels for multivariate time series that account for missing data when computing similarities. Our approach show superior performance compared to baselines that have to resort to imputation techniques and performs comparable to a supervised classification baseline.'
address:
- 'Dept. of Mathematics and Statistics, UiT The Arctic University of Norway, Tromsø, Norway'
- UiT Machine Learning Group
- 'Dept. of Signal Theory and Comm., Telematics and Computing, Universidad Rey Juan Carlos, Fuenlabrada, Spain'
- 'Dept. of Physics and Technology, UiT, Tromsø, Norway'
- 'Dept. of Gastrointestinal Surgery, University Hospital of North Norway (UNN), Tromsø, Norway'
- 'Clinic for Surgery, Cancer and Women’s Health, UNN, Tromsø, Norway'
- 'Institute of Clinical Medicine, UiT, Tromsø, Norway'
author:
- Karl Øyvind Mikalsen
- 'Cristina Soguero-Ruiz'
- Filippo Maria Bianchi
- Arthur Revhaug
- Robert Jenssen
bibliography:
- 'bibliography.bib'
title: An Unsupervised Multivariate Time Series Kernel Approach for Identifying Patients with Surgical Site Infection from Blood Samples
---
Surgical site infection ,Electronic health records ,Multivariate time series ,Kernel methods ,Missing data ,Unsupervised learning
Introduction
============
Surgical Site Infection (SSI) is one of the most common types of nosocomial infections [@lewis2013], representing up to 30% of hospital-acquired infections [@magill2012prevalence; @de2009surgical]. SSI can be divided into different types depending on the anatomical location of the infection [@ko2015american]. *Superficial* infections can be treated with local surgical debridement and antibiotics. On the other hand, *deep* infections are more complex and require lapratomies and/or percutaneous drainage and intravenous antibiotics. Recently, SSI risk factors such as advanced age, overweight, smoking, open surgery or disseminated cancer have been reported [@lawson2013risk]. Depending on the location of the infection, some factors contribute more to increase the risk of SSI. For example, longer lasting surgeries are associated with deep SSI, whereas a high body mass index is related with superficial SSI [@lawson2013risk; @blumetti2007surgical].
Along with increased mortality rate (3%), SSI also prolongs hospitalization up to two weeks [@whitehouse2002] and increases the risk of readmission [@shah2017evaluation]. This, in average, doubles the expenses per patient and increases the chances of readmission, with further additional costs up to 27,000 USD [@whitehouse2002; @owens2014surgical]. Hence, a reduction in the number of postoperative complications like SSI will be of great benefit both for the patients and the society.
Many recent studies have focused on the analysis of SSI from blood tests, both before and after surgery [@silvestre2014; @soguero2015data; @medina2016; @angiolini2016]. The advantage of using blood tests for this purpose is that they are recorded frequently with low burden for the patients and contain much information about their actual health status. Among blood tests results, the high predictive power of C-reactive protein (CRP) test has been evaluated and emphasized in several works. For example, authors in [@angiolini2016] and [@gans2015diagnostic], demonstrated the relation between different CRP cutoff values and the risk of SSI on postoperative days 3 and 4. Others have combined the results from blood tests with other structured and unstructured clinical data to predict SSI complications [@soguero2016support; @hu2017strategies; @Sanger2016259].
Since blood samples are collected over time they can naturally be represented as *time series*. Such clinical time series data have some special characteristics that distinguish them from time series from other domains. One key property is *missing data*, which can occur because of e.g. lack of documentation or lack of collection [@wells2013strategies]. Another characteristic is that the time series usually are *multivariate*. For example could the patients be described by the measurements of many different blood tests, such as e.g. hemoglobin, CRP, etc., where many of them exhibit relationships or dependencies, which can be non-trivial (non-linear).
In order to apply a machine learning algorithm on such time series, one can work directly in the input space using a *similarity measure* that accounts for dependencies in time and among the variables [@mikalsen2017time]. An advantage is that a time consuming feature learning process, or a manual feature design process that requires user intervention and domain expertise, can be then avoided. On the other hand, a key problem with classical similarity measures is that most of them are not able to cope with missing data in clinical time series data without applying a preprocessing step such as imputation in order to obtain complete data [@mitsa2010temporal]. However, important information about the clinical condition of the patient and the decisions of the caregiver may disappear in such a preprocessing step since the information that some data are missing is lost, and replaced by biased estimates.
In addition to the problem of analyzing multivariate time series (MTS) containing missing data, another key challenge associated with data originating from electronic health records (EHRs) is that getting access to labelled data for training the machine learning models is difficult. It is well known that manual annotation of labels, especially in the healthcare domain, is a cumbersome process that could be both time consuming and expensive [@MIKALSEN2017105], since clinical expertise is needed to create the training sets. Indeed, the workload for the clinicians is tremendous already, and with an aging and more diseased population we cannot expect these tasks to be prioritized in the future. To overcome this annotation problem, recently there have been proposed several different methods within the framework of semi-supervised learning [@chapelle2009semi], of which the so-called anchor learning is an example [@MIKALSEN2017105; @Halpernocw011].
In this work, we propose a disease classification framework for time series originating from EHR data such as e.g. blood tests. We address the key challenges described above by taking an unsupervised approach where we utilize powerful kernels for MTS containing missing data. These kernels have emerged due to recent advances in time series analysis as new family of methods that can cope with incomplete and multivariate data. Prominent examples include the *learned pattern similarity* (LPS) [@baydogan2016time] and the *time series cluster kernel* (TCK) [@mikalsen2017time]. These methods are able to handle missing data without having to resort to imputation methods. Another important property of the LPS and TCK kernels is that they can be trained in an unsupervised way and do not require tuning of critical hyperparameters. We take advantage of their robustness by using the kernels as input to an unsupervised *spectral clustering* framework for identifying patients with SSI. By doing so, the entire framework is grounded within the theoretically well understood kernel methods. Moreover, spectral clustering is considered a state-of-the-art clustering algorithm and has been successfully utilized in many applications [@ng2002spectral; @krzakala2013spectral; @lokse2017spectral].
The proposed methodology consists of two steps, namely to first compute the kernel and then apply spectral clustering (see Fig. \[fig:Scheme\]). We use the proposed framework to identify patients undergoing colorectal cancer surgery with SSI based on only blood samples, and illustrate its power by comparing to a similar framework where we use kernels that have to resort to imputation techniques. In addition, we compare to a supervised baseline for detecting SSI.
The rest of the paper is as follows. Section \[Methods\] describes materials and methods. Results are presented in Section \[Sec:Results\]. In Section \[sec: discussion\] we provide a discussion, whereas a conclusion is drawn in the last section.
Materials and methods {#Methods}
=====================
Data description {#Database}
----------------
Ethics approval for the present study was obtained from the Data Inspectorate and the Ethics Committee at the University Hospital of North Norway (UNN) [@jensen2017analysis; @soguero2016predicting]. The dataset we consider consists of 7741 patients that underwent a gastrointestinal surgical procedure at UNN in the years 2004–2012. The SSI persistent in-hospital morbidity is particularly associated with colorectal cancer surgery [@lawson2013risk; @blumetti2007surgical; @lawson2013reliability] and therefore patients that did not undergo this type of surgery were excluded, reducing the size of the cohort to 1137 patients.
The International Classification of Diseases (ICD10) or NOMESCO Classification of Surgical Procedures (NCSP) codes related to severe postoperative complications, both superficial and deep infections, were considered to extract a cohort. For the purpose of this work and similarly to earlier studies [@LIMON2014127; @Gibbons20111; @horan_gaynes_martone_jarvis_emori_1992; @BERGER2013974; @soguero2015data; @Sanger2016259], we do not distinguish between deep and superficial SSI. In collaboration with the clinician (author A. R.), a set of 11 different types of blood tests were defined as clinically relevant and extracted for all patients from their EHRs, namely, hemoglobine, leukocytes, CRP, potassium, sodium, creatininium, thrombocytes, albumin, carbamide, glucose and amylase. The blood tests can be considered as continuous variables over time and hence represented as MTS. For the purpose of the current analysis, we discretize time and let each time interval be one day. However, all 11 blood tests are not available every day for each patient, meaning that the dataset contains missing data. We focus on classification of SSI within 20 days after surgery and therefore define the *postoperative window* as the period from postoperative day 1 until day 20. Patients with less than two measurements during the postoperative window are removed from the cohort, which leads to a dataset with 232 infected patients (cases) and 651 non-infected ones (control).
Strategies for dealing with missing data
----------------------------------------
Since EHR data might be missing for several different reasons, in many cases the missingness mechanism cannot be described exclusively as either *missing completely at random* (MCAR), *missing at random* (MAR) or *missing not at random* (MNAR), but rather as a combination of all these three traditional schemes [@wells2013strategies; @hu2017strategies]. Indeed, one reason for missing data is *lack of documentation*, which occurs for example when a clinician orders a blood test but the test for some reason is not performed, or it is performed but not documented because of an error by the physician that performs the test or the data is lost during extraction. Such type of missing data is usually MCAR or MAR. However, missingness can also be caused by *lack of collection*. This happens, for example, when the clinician that is treating the patient thinks the health condition of the patients is so good that there is no reason to order a blood test on that particular day. In this case, data are MNAR.
Most machine learning methods, and in particular discriminative learning algorithms, work only with complete datasets [@little2014statistical]. However, in a clinical setting, creating a complete dataset by simply discarding patients with missing data may lead to incorrect assessments or prognostics, since the fraction of missing data is typically large [@soguero2016predicting]. Hence, one could end up discarding a lot of information and get a weak model. Moreover, complete case analysis as a result of discarding data only give unbiased predictions if the missingness mechanism is MCAR [@rubin1976inference; @schafer1997analysis].
As an alternative, a preprocessing step involving *imputation* of missing values with some estimated value is common. For example one could fill the missing values for each variable of interest with the mean or median value of the observed samples. A simple, but sometimes efficient approach, is to impute all missing values with zeros. Other so-called *single imputation* methods include machine learning based methods such as multilayer perceptrons, self-organizing maps, k-nearest neighbors, recurrent neural networks and regression-based imputation [@garcia2010pattern; @RAHMAN2015198].
The strategies for handling missing data discussed so far are general and can be applied to both vectorial and time series data. On the other hand, methods that only apply to time series data include to impute missing values using smoothing and interpolation techniques such as the well-known last observation carried forward (LOCF) scheme that imputes the last non-missing value for the following missing values. Further approaches for MTS are linear interpolation, moving average and Kalman smoothing, to name a few [@ENGELS2003968]. The list of possible imputation methods for MTS data is almost endless and a comprehensive overview of all these is beyond the scope of this paper. For a more detailed overview, we refer the interested reader to [@garcia2010pattern; @donders2006review; @durbin2012time].
A drawback common to all imputation methods discussed above is that the information about which values are actually missing is lost. Moreover, imputation typically also introduces additional bias into the data due to strong assumptions made by the imputation method. For example mean imputation often leads to biased (shorter) estimates of the distances between data points than what they actually are [@hu2017strategies]. To resolve the bias problem it has been suggested to correct for the bias by introducing binary indicator variables to account for the missingness pattern [@hu2017strategies]. An additional problem with single imputation is that the uncertainty associated with the missing values is ignored since they are replaced with “certain" estimates, which in turn may lead to smaller estimated standard errors than the true standard errors. *Multiple imputation* [@white2011multiple] resolves this problem by estimating the missing values multiple times and thereby creating multiple complete datasets. Thereafter for example a classifier is applied to all datasets and the results are combined to obtain the final predictions. However, applying multiple imputation to MTS in a clustering setting is a non-trivial task that involves several challenges [@basagana2013framework].
Despite that multiple imputation and other imputation methods can give satisfying results in some scenarios, these are ad-hoc solutions that lead to a multi-step procedure where the missing data are handled separately and independently from the rest of the analysis [@wells2013strategies]. This is not an optimal solution, and therefore several research efforts have been devoted over the last years to process incomplete temporal data without relying on imputation [@DBLP:journals/corr/ChePCSL16; @bianchi2018time; @DBLP:journals/corr/abs-1711-06516; @mikalsen2016learning; @pmlr-v56-Lipton16; @Marlin:2012:UPD:2110363.2110408]. In this regard, powerful kernel methods have been recently proposed, of which the TCK and LPS are prominent examples. Even though there are many similarities between these two kernels, the way missing data are dealt with is very different. In LPS the missing data handling abilities of decision trees are exploited. Along with ensemble methods, fuzzy approaches and support vector solutions, decision trees can be categorized as *machine learning approaches for handling missing data* [@garcia2010pattern]. Common to these approaches is that the missing data are handled naturally by the machine learning algorithm. One can also argue that the way missing data are dealt with in the TCK belongs to this category, since an ensemble approach is exploited. However, it can also be categorized as a *likelihood-based approach* since the underlying models in the ensemble are Gaussian mixture models. In the likelihood-based approaches the full, incomplete dataset is analyzed using maximum likelihood (or maximum a posteriori, equivalently), typically in combination with the expectation-maximization (EM) algorithm [@schafer2002missing; @little2014statistical].
The main advantage of these methods, compared to imputation methods, is that the missing data are handled automatically and no additional tasks are left to the user. For example in multiple imputation, a careful selection of the imputation model and other variables is needed to do the imputation [@schafer2002missing], which in particular in an unsupervised setting can turn out to be problematic. Moreover, similarly to multiple imputation, unbiased predictions are guaranteed if data are MAR.
A more detailed description of the TCK and LPS kernels is provided in the next subsection, along with a description of the other kernels for MTS used in this work.
Multivariate time series kernels
--------------------------------
Kernel methods have been of great importance in machine learning for several decades and have applications in many different fields [@Jenssen2010; @Jenssen2013; @camps2009kernel; @soguero2016support]. Within the context of time series, a *kernel* is a similarity measure that also is positive semi-definite [@shawe2004kernel]. Once defined, such similarities between pairs of time series may be utilized in a wide range of applications, such as classification or clustering, benefiting from the vast body of work in the field of kernel methods.
#### Linear kernel
The simplest of all kernel functions is the linear kernel, which for two data points represented as vectors, $x$ and $y$, is given by the inner product $\langle x, y \rangle$, possibly plus a constant $c$. One can also apply a linear kernel to pairs of MTS once they are unfolded into vectors. However, by doing so the information that they are MTS and there might be inherent dependencies in time and between attributes, is then lost. Nevertheless, in some cases such a kernel can be efficient, especially if the MTS are short [@chen2013model]. If the MTS contain missing data, the linear kernel requires a preprocessing step involving e.g. imputation.
#### Global alignment kernel
The most widely used time series similarity measure is *dynamic time warping* (DTW) [@Berndt:1994:UDT:3000850.3000887], where the similarity is quantified as the alignment cost between the MTS. More specifically, in DTW the time dimension of one or both of the time series is warped to achieve a better alignment. Despite the success of DTW in many applications, similar to many other similarity measures, it is non-metric and therefore cannot non-trivially be used to design a positive semi-definite kernel [@marteau2015recursive]. Hence, it is not suited for kernel methods in its original formulation. Because of its popularity there have been attempts to design kernels exploiting the DTW. For example Cuturi et al. designed a DTW-based kernel using global alignments [@cuturi2007kernel]. An efficient version of the global alignment kernel (GAK) is provided in [@cuturi2011fast]. The latter has two hyperparameters, namely the kernel bandwidth and the triangular parameter. These are usually set using some heuristics. GAK does not naturally deal with missing data and incomplete datasets, and therefore also requires a preprocessing step involving imputation.
#### Time series cluster kernel {#sec:TCK}
The TCK is based on an ensemble learning approach [@Strehl:2003] wherein the robustness to hyperparameters is ensured by joining the clustering results of many Gaussian mixture models (GMM) to form the final kernel. Hence, no critical hyperparameters have to be tuned by the user, and the TCK can be learned in an unsupervised manner. To ensure robustness to sparsely sampled data, the GMMs that are the base models in the ensemble, are extended using informative prior distributions such that the missing data is explicitly dealt with.
More specifically, the TCK matrix is built by fitting GMMs to the set of MTS for a range of number of mixture components. The idea is that by generating partitions at different resolutions, one can capture both the local and global structure of the data. Moreover, to capture diversity in the data, randomness is injected by for each resolution (number of components) estimating the mixture parameters for a range of random initializations and randomly chosen hyperparameters. In addition, each GMM sees a random subset of attributes and segments in the MTS. The posterior distributions for each mixture component are then used to build the TCK matrix by taking the inner product between all pairs of posterior distributions. Finally, given an ensemble of GMMs, the TCK is created in an additive way by using the fact that the sum of kernels is also a kernel. In this work, we have modified the kernel slightly from the way it was originally proposed in [@mikalsen2017time] by normalizing the vectors of posteriors to have unit length in the $l_2$-norm. This provides an additional regularization that may increase the generalization capability of the learned model. A more detailed description of the method is provided in \[appendix: TCK\].
#### Learned pattern similarity
LPS is a similarity measure that satisfies the requirements of a kernel, as shown in [@mikalsen2017time], which can naturally deal with MTS. Similar to the TCK, the LPS is also based on extracting random segments. Additionally, the LPS is similar to the TCK in the sense that one in an unsupervised way can learn a similarity between time series that is robust to hyperparameter choices and can deal with missing data using the missing data handling properties of tree-based learning. It generalizes the well-known autoregressive models [@shumway] to local autopatterns using multiple lag values for autocorrelation. These autopatterns are supposed to capture the local dependency structure in the time series and are learned using a tree-based learning strategy.
More specifically, a time series is represented as a matrix of segments. Randomness is injected to the learning process by randomly choosing time segment (column in the matrix) and lag $p$ for each tree in the random forest. A bag-of-words type compressed representation is created from the output of the leaf-nodes for each tree. The final time series representation is created by concatenating the representation obtained from the individual trees, which in turn are used to compute the similarity using a histogram intersection kernel [@barla2003histogram].
Given two MTS $X^{(n)}$ and $X^{(m)}$, a formal expression for the LPS-kernel is $$\label{eq: LPS}
K(X^{(n)},X^{(m)}) = \frac{1}{R J} \sum\limits_{k=1}^{R J} \min (h^n_k, h^m_k),$$ where $h^n_k$ is the $k$th entry of the concatenated bag-of-words representation $H(X^{(n)})$. More precisely, $H(X^{(n)})$ is a concatenation of $R$-dimensional frequency vectors of instances in the terminal nodes from all $J$ trees.
Model development
-----------------
The kernels that we described in the previous section are used to compute a kernel matrix on a training set, which is created by randomly selecting 80 percent of the dataset. The remaining 20 percent is used as test set. The LPS and TCK kernels are computed on the incomplete dataset containing missing data, whereas the GAK and linear kernel cannot work on incomplete datasets, and we therefore compute these on 6 different complete datasets obtained using mean imputation, LOCF imputation, 0-imputation, and replicates of these corrected for bias. In the bias corrected (BC) datasets we double the number of attributes in each MTS by stacking a binary MTS, representing imputed elements, to the imputed MTS. When using imputation of the mean, we calculate the mean for each attribute in the MTS, across all time intervals in the postoperative window and all patients in the training set. If an element is missing in the first time interval, we replace it with the mean when we do LOCF imputation.
After having computed the different kernels, we take an unsupervised approach to classifying the patients with SSI using the *spectrum* of the kernel matrices. We employ a variant of spectral clustering consisting of two steps, namely kPCA followed by k-means. In the first step, kPCA with the learned MTS kernel is used to compute a low dimensional representation of the MTS. Thereafter we cluster the learned representations using *k-means*. We assume that the number of clusters is known and set it to 2. Out-of-sample data are assigned to clusters according to the cluster labels of the *k-nearest neighbors* (kNN) in the training set. The processing pipeline we have described here is also illustrated in Fig. \[fig:Scheme\].
Model evaluation
----------------
The different models are evaluated both on the training and test set. Because of the imbalanced classes we decide to use *F1-score* [@hripcsak2005agreement] instead of accuracy as performance measure. F1-score is a function of two metrics, namely *precision* and *recall*. These two metrics are also commonly referred to as positive predictive value and sensitivity, respectively. Precision is the fraction of true positives (have infection) among all those that are classified (clustered) as positive cases, whereas recall is the fraction of positive cases in the gold standard classified as positive. F1-score can be expressed in terms of these two metrics as follows: $$F_1 = \frac{\text{recall} \times \text{precision}}{\text{recall} + \text{precision}}$$ In order to adapt this to an unsupervised regime, we define *clustering F1-score* similarly to how *clustering accuracy* is defined. i.e. we use the permutation of the labels provided by the clustering algorithm that gives the highest score. In the following, we refer to both clustering F1-score and classification F1-score simply as F1-score.
The procedure described in the previous section is repeated 10 times such that we can compute both mean and standard errors for the F1-score, i.e. we randomly select 10 different training and test sets (80 and 20 percent, respectively), and repeat the same process on all of them. In addition, in order to study how stable and robust the different methods are to varying length of the MTS, we vary the size of the postoperative window from 7 to 20 days.
Results {#Sec:Results}
=======
The TCK and LPS are run using default hyperparameters [@MikalsenTCK; @baydogan2016time], with the exception for the LPS that we increase the minimal segment length from $5 \%$ to $15 \%$ percent of the length of the MTS to account for the short time series. In accordance with [@Cuturi], for GAK we set the bandwidth $\sigma$ to two times the median distance of all MTS in the training set scaled by the square root of the median length of all MTS, and the triangular parameter [@Cuturi] to 0.2 times the median length of all MTS. Distances are measured using the canonical metric induced by the Frobenius norm. In the linear kernel we set the constant $c$ to 0.
[ ![Mean F1-score over 10 runs on training (left) and test set (right) and standard errors obtained using four different MTS kernels, followed by kPCA and k-means. Out-of-sample data is clustered using a kNN classifier. The green line represents a linear kernel, yellow the global alignment kernel, red the learned pattern similarity kernel and blue the time series cluster kernel. []{data-label="fig: clustering results"}](cluster_tr.pdf "fig:"){width="0.48\linewidth"} \[fig: train clust\] ]{} [ ![Mean F1-score over 10 runs on training (left) and test set (right) and standard errors obtained using four different MTS kernels, followed by kPCA and k-means. Out-of-sample data is clustered using a kNN classifier. The green line represents a linear kernel, yellow the global alignment kernel, red the learned pattern similarity kernel and blue the time series cluster kernel. []{data-label="fig: clustering results"}](cluster_te.pdf "fig:"){width="0.48\linewidth"} \[fig: test clust\] ]{}
Fig. \[fig: clustering results\] shows mean F1-score over 10 runs on the training (left) and test set (right), and standard errors, obtained using LPS and TCK kernels, followed by kPCA to 10 dimensions and k-means, where test data are clustered using a kNN classifier with $k=5$ and the cluster assignments as labels. Initial experiments showed that the clustering results are stable to varying values of these hyperparameters. For easier comparison, in the same figure we have also added results obtained with the GAK and linear kernel on the imputed dataset that gives the highest F1-score, namely 0-imputation, whereas the results for all 6 complete datasets are shown in Fig. \[fig: clustering results lin vs gak\].
[ ![Mean F1-score over 10 runs on training (left) and test set (right) obtained using the linear kernel and GAK with six different imputation methods, followed by kPCA and k-means. Out-of-sample data is clustered using a kNN classifier. Standard errors are not shown for increased readability.[]{data-label="fig: clustering results lin vs gak"}](linear_tr.pdf "fig:"){width="0.48\linewidth"} \[fig: train clust lin\] ]{} [ ![Mean F1-score over 10 runs on training (left) and test set (right) obtained using the linear kernel and GAK with six different imputation methods, followed by kPCA and k-means. Out-of-sample data is clustered using a kNN classifier. Standard errors are not shown for increased readability.[]{data-label="fig: clustering results lin vs gak"}](linear_te.pdf "fig:"){width="0.48\linewidth"} \[fig: test clust lin\] ]{}\
[ ![Mean F1-score over 10 runs on training (left) and test set (right) obtained using the linear kernel and GAK with six different imputation methods, followed by kPCA and k-means. Out-of-sample data is clustered using a kNN classifier. Standard errors are not shown for increased readability.[]{data-label="fig: clustering results lin vs gak"}](gak_tr.pdf "fig:"){width="0.48\linewidth"} \[fig: train clust gak\] ]{} [ ![Mean F1-score over 10 runs on training (left) and test set (right) obtained using the linear kernel and GAK with six different imputation methods, followed by kPCA and k-means. Out-of-sample data is clustered using a kNN classifier. Standard errors are not shown for increased readability.[]{data-label="fig: clustering results lin vs gak"}](gak_te.pdf "fig:"){width="0.48\linewidth"} \[fig: test clust gak\] ]{}
We observe that the two kernels, TCK and LPS, that can explicitly deal with the missing data, give very similar results both on the training and test set. When the postoperative window is 7 days the two methods yield an F1-score of approximately 0.63, then the performance increases almost linearly from day 7 to 15 where it stabilizes around a F1-score of 0.80. Further, it can be seen that TCK, LPS and GAK perform worse than the linear kernel when the postoperative window is short ($< 10$ days). However, as the size of the postoperative window increases, the relative performance of the TCK and LPS with respect to both GAK and the linear kernel improve. Even if the differences between LPS and TCK are small, it can be seen that the latter yields slightly better performance (in average) for all sizes of the postoperative window both in training and test. Moreover, the standard errors with the TCK are smaller, in particular in training when the postoperative window is short. This is particularly important in unsupervised frameworks like the one we are proposing.
For the two kernels that work on the imputed data, we observe that GAK (0 and 0+BC) and Linear (0, 0+BC, LOCF+BC) perform quite similarly, and these are the five imputed data methods that give the best performance. A common pattern for these five methods, and especially for GAK (0 and 0+BC), is that the F1-score increases when the postoperative window is increased from 7 to around 11 days (at least on the training set), but then the F1-score slowly starts decrease after that. The performance of the seven other imputation methods is considerably worse.
To further investigate the differences between LPS and TCK, beside F1-score, in Fig. \[fig: KPCA TCK 20\] and Fig. \[fig: KPCA LPS 20\] we show the kPCA representation corresponding to the two largest eigenvalues obtained using these two kernels on the training set with postoperative window size equal to 20. Interestingly the representations created by the LPS and TCK are very different. The LPS has a clearer manifold structure, whereas the TCK is more spread out in the plane. Even though it is difficult to argue that one of these two representations is superior to the other, the TCK at least more clearly reflects that the cohort of patients is very diverse and complex because of large individual differences. To better understand the performance of the GAK and linear kernel we also plot the 2D kPCA representation obtained on 0-imputed data in Fig. \[fig: KPCA GAK 20\] and Fig. \[fig: KPCA linear 20\]. We note that these are heavily influenced by outliers. Apart from the outliers, the other datapoints become very compact and close to each other, and it is therefore not strange that the clustering algorithm does not identify the groups correctly.
[ ![Mean F1-score over 10 runs on training (left) and test set (right) obtained using the static, manually extracted features in combination with a linear kernel and six different imputation methods, followed by kPCA and k-means. Out-of-sample data is clustered using a kNN classifier. Standard errors are not shown for increased readability. []{data-label="fig: clustering results manual"}](manual_tr.pdf "fig:"){width="0.48\linewidth"} \[fig: manual tr\] ]{} [ ![Mean F1-score over 10 runs on training (left) and test set (right) obtained using the static, manually extracted features in combination with a linear kernel and six different imputation methods, followed by kPCA and k-means. Out-of-sample data is clustered using a kNN classifier. Standard errors are not shown for increased readability. []{data-label="fig: clustering results manual"}](manual_te.pdf "fig:"){width="0.48\linewidth"} \[fig: manual te\] ]{}
As a baseline to compare the performance of our method, we follow the idea proposed by [@hu2017strategies] and compute higher level non-temporal features from the longitudinal data. Specifically, in addition to the average value of each of the blood tests, we compute two extreme values (the maximum and minimum) over the postoperative window. In this setting we consider a missing value as a specific blood test that is missing entirely during the postoperative window. This baseline cannot naturally deal with missing data and therefore we use the same six missing data imputation strategies as described for the temporal features. We apply a linear kernel to the manual features and then we follow the same scheme as in the temporal case. Fig. \[fig: clustering results manual\] shows mean F1-score over 10 runs on training (left) and test set (right) and standard errors obtained using this baseline on the six imputed datasets. Similarly to the results obtained with the linear kernel and GAK on the temporal data, the results obtained with this baseline heavily depend on the type of imputation method, and are also very fluctuating as function of length of the postoperative window.
[ ![Mean F1-score over 10 runs on training (left) and test set (right) and standard errors obtained using the TCK and LPS kernels, followed by kPCA to 10 dimensions and a kNN-classifier with with $k=5$. The dashed lines represent the mean clustering F1-score (from Figure \[fig: clustering results\]).[]{data-label="fig: classification results"}](class_tr.pdf "fig:"){width="0.48\linewidth"} \[fig: train class\] ]{} [ ![Mean F1-score over 10 runs on training (left) and test set (right) and standard errors obtained using the TCK and LPS kernels, followed by kPCA to 10 dimensions and a kNN-classifier with with $k=5$. The dashed lines represent the mean clustering F1-score (from Figure \[fig: clustering results\]).[]{data-label="fig: classification results"}](class_te.pdf "fig:"){width="0.48\linewidth"} \[fig: test class\] ]{}
In addition to comparing different kernels and methods for dealing with missing data, to see how well the unsupervised classification part works, we also benchmark the proposed unsupervised framework against a supervised baseline where the two first steps are the same, namely to compute the kernel and thereafter do kPCA. However, on the 10 dimensional kPCA representation we employ a kNN classifier with $k=5$ using the true labels. Figure \[fig: classification results\] shows mean classification F1-score and standard errors over the same 10 randomly drawn training (left) and test sets (right). Also with this baseline the performance of the TCK and LPS kernel is very similar. Moreover, in general, the supervised baseline performs better on the training set than the unsupervised method. This is expected. However, on the test set, the F1-scores obtained using the supervised baseline is almost identical to the proposed method for all sizes of the postoperative window. This implies that labelling is not a required task for identifying patients with surgical site infection from blood samples.
Discussion {#sec: discussion}
==========
In this work, we have proposed a framework for SSI identification based on secondary use of EHR data. Our first objective was to alleviate the problem of getting access to labelled EHR data. The results presented in the previous section clearly show that the proposed framework can detect accurately SSI, without relying on supervised training. In fact, the supervised baseline did not improve the performance compared to the proposed method. The second objective has been to deal with missing data effectively. We tackled this problem by introducing robust time series kernels as a main component in our framework and in the following we discuss our findings.
For the two kernels that work on the imputed data, GAK and linear, the choice of imputation method heavily affects the performance more than the choice of kernel itself. Both the linear kernel and GAK give very bad results especially with mean imputation, but also with LOCF imputation. In those cases, the mean F1-score is far from smooth over time. On the other hand, 0-imputation gives good results compared to the other imputation methods for both kernels. This can be surprising, since 0-imputation introduces a strong bias because blood test values are positive and therefore the value 0 is very rarely a good estimate. However, a possible explanation might be that 0-imputation provides some auto-correction for the bias, since the 0s now describe the missingness pattern. This result is in accordance with what Hu et al. found in [@hu2017strategies] when taking a manual feature design approach to the problem.
Regarding bias correction, it is maybe not so surprising that it does not lead to improved performance on the 0-imputed dataset for any of the two kernels, since 0-imputation in itself seems to have the same effect (ref. previous paragraph). On the other hand, since the performance obtained using the linear kernel combined with mean or LOCF imputation is substantially improved, whereas nothing happens for the GAK kernel, it is difficult to draw conclusions about the effect of bias correction.
We also note that the best imputed data methods all share the same pattern; the performance peaks around postoperative window size 11. The natural behaviour should be that the F1-score should increase as more information is added when the length of the postoperative window is increased. However, when the opposite happens, this indicates that all these imputation methods introduce a bias into the data that affects the performance. Hence, due to the high variance in the performance across different imputation methods and length of postoperative window, and since is difficult to do model selection in the unsupervised setting of our framework, we conclude that the linear kernel and GAK are not suitable for the task under analysis.
Not surprisingly, the time series kernels, TCK, LPS and GAK perform worse than the linear kernel when the postoperative window is short ($< 10$ days). When the MTS are shorter than 10 days the time dependency and time structure is probably not clear enough to be fully utilized by the specialized time series kernels. However, as the size of the postoperative window increases, the F1-score obtained using LPS and TCK increase smoothly, before it stabilizes around a F1-score of 0.80 when the length is around 14-15 days, considerably higher than the F1-score obtained using the GAK and linear kernel. This behaviour verifies the robustness of these two kernels with respect to varying length of postoperative window and missingness patterns.
We note that for the TCK and LPS kernels an underlying assumption is that values are MAR, whereas the missingness for EHR-data could be partly due to MNAR. However, it has been demonstrated by several authors that a slightly wrong assumption of MAR in many realistic scenarios do not have a big impact in terms of biased predictions [@collins2001comparison; @schafer2002missing; @hu2017strategies]. In our case we do not know by how much the assumption of MAR is broken. However, experiments in [@mikalsen2017time] demonstrated these kernels’ (and in particular TCK’s) robustness to large fractions of missing data – also in the case of MNAR data, whereas the imputation methods suffered in cases with much missing data. Even though, the data considered in this paper is completely different, it is interesting to observe a similar behaviour here.
Limitations and further work
----------------------------
Although the results obtained in this paper are promising just based on blood results, previous studies [@soguero2016predicting] have shown that the combination of heterogeneous data sources (e.g. free text, drugs, ICD-9 or vital signs) from the EHR might provide better performance. Including more data, however, comes at the cost of a more complicated and computationally demanding analysis. In addition, we did not differentiate between deep and superficial SSI in this work. These issues are subject to further work. Moreover, in this work, we focused on identifying patients with SSI based on postoperative data. In further work we would like to do prediction of SSI based on preoperative data and investigate if we can predict that the patient gets SSI before he or she is diagnosed with the complication, which is a framework that would be very valuable to operationalize in the clinic. The framework used in this work is general, and not restricted to SSI, it will therefore be interesting to also see how well it works on detecting other complications or diseases.
We could have benchmarked the proposed method against other kernels as well, but chose to not do that for the clarity of the presentation, and because the results obtained using GAK and the linear kernel indicate that results are more dependent on the imputation method than the type of kernel. We also tried with different clustering algorithms in the last step such as hierarchical clustering and kNN mode seeking ensemble clustering [@NordhaugMyhre2018491], but initial experiments did not improve performance, and we have therefore, for the conciseness of the presentation, not included them.
Conclusions {#sec: conclusion}
===========
Hospital acquired infections in general, and surgical site infection in particular, are major problems at modern hospitals nowadays. To be able to reduce this problem, accurate prediction of SSI is of utmost importance. In this study, we showed that analyzing EHR data as MTS within a kernel framework can be very powerful in that respect. In particular, the LPS and TCK kernels that explicitly can deal with the missing data, turn out to be more robust and work better than those kernels that require the incomplete data to be pre-processed using some imputation method.
Moreover, because of the two kernels’ robustness to hyperparameters, we showed that we can completely unsupervised identify patients with SSI and perform similarly to a supervised baseline, hence alleviating the problem of a time consuming and expensive manual label annotation process, often unfeasible for large datasets. Worth mentioning in that respect, is that in this paper we have also illustrated the power of using only blood tests as the data source, hence also reducing the burden for the patients and data engineers.
Conflict of interest {#conflict-of-interest .unnumbered}
====================
The authors have no conflict of interest related to this work.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was partially funded by the Norwegian Research Council FRIPRO grant no. 239844 on developing the *Next Generation Learning Machines*. Cristina Soguero-Ruiz is partially supported by project TEC2016-75361-R from Spanish Government and by project DTS17/00158 from Institute of Health Carlos III (Spain).
The authors would like to thank Kristian Hindberg from UiT The Arctic University of Norway for his assistance on preprocessing and extracting the data from the EHR system. We would also like to thank Rolv-Ole Lindsetmo and Knut Magne Augestad from the University Hospital of North Norway, Fred Godtliebsen from UiT, together with Stein Olav Skrøvseth from the Norwegian Centre for E-health Research for helpful discussions throughout the study and manuscript preparation.
TCK {#appendix: TCK}
===
Notation {#notation .unnumbered}
--------
The following notation is used. A multivariate time series (MTS) $X$ is defined as a (finite) sequence of univariate time series (UTS), $
X = \{ x_v \in \mathbb{R}^T \: | \: v = 1,2,\dots,V\},
$ where each attribute, $x_v$, is a UTS of length $T$. The number of UTS, $V$, is the *dimension* of $X$. The length $T$ of the UTS $x_v$ is also the length of the MTS $X$. Hence, a $V$–dimensional MTS, $X$, of length $T$ can be represented as a matrix in $\mathbb{R}^{V \times T}$. Given a dataset of $N$ MTS, we denote $X^{(n)}$ the $n$-th MTS. An incompletely observed MTS is described by the pair $(X^{(n)}, R^{(n)})$, where $R^{(n)}$ is a binary MTS with entry $r_v^{(n)}(t) = 0$ if the realization $x_v^{(n)}(t)$ is missing and $r_v^{(n)}(t) = 1$ if it is observed.
DiagGMM {#diaggmm .unnumbered}
-------
To build the TCK kernel matrix, we first fit different diagonal covariance GMM (DiagGMM) to the MTS dataset. In the DiagGMM one assumes time-dependent means, expressed by $\mu_g = \{ \mu_{gv} \in \mathbb{R}^T \: | \: v = 1,...,V\}$, where $\mu_{gv}$ is a UTS, and a time-constant covariance matrix is $\Sigma_g = diag\{\sigma_{g1}^2,...,\sigma_{gV}^2\}$, being $\sigma_{gv}^2$ the variance of attribute $v$. Moreover, the data is assumed to be *missing at random* (MAR), i.e. the missing elements are only dependent on the observed values. Under these assumptions, missing data can be analytically integrated away, such that imputation is not needed [@rubin1976inference], and the pdf for the incompletely observed MTS $(X, R)$ is given by $$\label{eq: p(x) gmm diag}
p(X \: | \: R, \: \Theta ) = \sum_{g=1}^G \theta_g \prod_{v=1}^V \prod_{t=1}^T \mathcal{N} (x_v(t) \: | \: \mu_{gv}(t), \sigma_{gv})^{r_v(t) }$$ The conditional probability of $Z$ given $X$, can be found using Bayes’ theorem, $$\label{eq: p(z|x) posterior}
\pi_{g} \equiv P(Z_g = 1 \: | \: X, \: R, \: \Theta )
= \frac{ \theta_g \prod_{v=1}^V \prod_{t=1}^T \mathcal{N} \left(x_v(t) \: | \: \mu_{gv}(t), \sigma_{gv}\right)^{r_v(t) }}{\sum_{g=1}^G \theta_g \prod_{v=1}^V \prod_{t=1}^T \mathcal{N} \left(x_v(t) \: | \: \mu_{gv}(t), \sigma_{gv}\right)^{r_v(t) }}.$$ The parameters of the DiagGMM are learned using a maximum a posteriori expectation maximization algorithm, as described in [@mikalsen2017time].
Ensemble strategy {#ensemble-strategy .unnumbered}
-----------------
To ensure diversity, each GMM model uses a number of components from the interval $[2,C]$, where $C$ is the maximal number of mixture components. For each number of components, we apply $Q$ different random initial conditions and hyperparameters. We let $\mathcal{Q} = \{ q = (q_1,q_2) \: | \: q_1=1,\dots Q, \: q_2 = 2,\dots, C \} $ be the index set keeping track of initial conditions and hyperparameters ($q_1$), and the number of components ($q_2$). Moreover, each model is trained on a random subset of MTS, accounting only a random subset of variables $\mathcal{V}$, with cardinality $ |\mathcal{V}| \leq V$, over a randomly chosen time segment $\mathcal{T}, |\mathcal{T}| \leq T$. The inner products of the posterior distributions from each mixture component are then added up to build the TCK kernel matrix, according to the ensemble strategy [@ensemble]. Algorithm \[alg:algorithm\] describes the details of the method.
Training set of MTS $ \{ X^{(n)} \}_{n=1}^N$ , $Q$ initializations, $C$ maximal number of mixture components. Initialize kernel matrix $K = 0_{N \times N} $. Compute posteriors $ \Pi^{(n)}(q) \equiv ( \pi_1^{(n)},\dots,\pi_{q_2}^{(n)} )^T $, by applying maximum a posteriori expectation maximization [@mikalsen2017time] to the DiagGMM with $q_2$ clusters and by randomly selecting,
- hyperparameters $\Omega(q) $,
- a time segment $ \mathcal{T}(q) $ of length [$T_{min} \leq |\mathcal{T}(q)| \: \leq \: T_{max}$ ]{},
- attributes $\mathcal{V}(q)$, with cardinality [$V_{min} \leq |\mathcal{V}(q)| \leq V_{max}$]{},
- a subset of MTS, $\eta(q) $, with [$N_{min} \leq |\eta(q)| \leq N$]{},
- initialization of the mixture parameters $ \Theta(q) $.
Update kernel matrix, $K_{nm} = K_{nm} + \frac{\Pi^{(n)}(q)^T \Pi^{(m)}(q)}{ \| \Pi^{(n)}(q) \| \| \Pi^{(m)}(q) \| } $. $K$ TCK matrix, time segments $\mathcal{T}(q) $, subsets of attributes $\mathcal{V}(q)$, subsets of MTS $\eta(q)$, parameters $ \Theta(q)$ and posteriors $\Pi^{(n)}(q) $.
Method details {#method-details .unnumbered}
--------------
Algorithm \[alg:algorithm\] describes the details of the method. $\mathcal{Q} = \{ q = (q_1,q_2) \: | \: q_1=1,\dots Q, \: q_2 = 2,\dots, C \} $ is the index set keeping track of initial conditions and hyperparameters ($q_1$), and the number of components ($q_2$).
In order to be able to compute similarities with MTS not available at the training phase, one needs to store the time segments $\mathcal{T}(q)$, subsets of attributes $\mathcal{V}(q)$, DiagGMM parameters $ \Theta(q)$ and posteriors $\mathbf{\Pi}^{(n)}(q)$. Then, the TCK for such out-of-sample MTS is evaluated according to Algorithm \[alg:algorithm out of sample\].
Test set $\big \{ X^{*(m)} \big \}_{m=1}^M$, time segments $\mathcal{T}(q)$, attributes $\mathcal{V}(q)$, subsets of MTS $\eta(q)$, parameters $ \Theta(q) $ and posteriors $\Pi^{(n)}(q) $. Initialize kernel matrix $K^* = 0_{N \times M} $. Compute posteriors $\Pi^{*(m)}(q) $, $m=1,\dots,M$ using the mixture parameters $ \Theta(q)$. Update kernel matrix, $K^*_{nm} = K^*_{nm} + \frac{\Pi^{(n)}(q)^T \Pi^{*(m)}(q)}{ \| \Pi^{(n)}(q) \| \| \Pi^{*(m)}(q) \| } $. $K^*$ TCK test kernel matrix.
In the LPS a time series is represented as a matrix of segments. Randomness is injected to the learning process by randomly choosing time segment (column in the matrix) and lag $p$ for each tree in the Random Forest. For each tree a Bag-of-Words type compressed representation is created from the output of the leaf-nodes. The final time series representation is created by concatenating the representation obtained from the individual trees and is in turn used to design the similarity using a histogram intersection kernel.
Given two multivariate time series $X^{(n)}$ and $X^{(m)}$, a formal expression for the LPS-kernel is $$\label{eq: LPS}
K(X^{(n)},X^{(m)}) = \frac{1}{R J} \sum\limits_{k=1}^{R J} \min (h^n_k, h^m_k),$$ where $h^n_k$ is the $k$th entry of the concatenated Bag-of-Words representation $H(X^{(n)})$. More precisely, $H(X^{(n)})$ is a concatenation of $R$-dimensional frequency vectors of instances in the terminal nodes from all trees.
Each tree generates a representation and the final time series representation is obtained via concatenation. For simplicity, assume that all trees contain the same number of terminal nodes R. The general case is easily handled. Let H j (x n ) denote the R-dimensional frequency vector of instances in the terminal nodes from tree g j for time series x n . We concatenate the frequency vectors over the trees to obtain the final representation of each time series, denoted as H (x n ), of length R J (and modified obviously for non-constant R). Our representation summarizes the patterns in the time series based on the terminal node distribution of the instances over the trees.
For each location, the subseries in the original data are concatenated to form a new attribute. The internal model selects a random attribute as the response variable then constructs a regression tree. A collection of these regression trees are processed to form a new set of instances based on the counts of the number
of subseries at each leaf node of each tree. Algorithm 8 describes the process. LPS can be summarised as follows: Stage 1: Construct an ensemble of r regression trees. 1. Randomly select a segment length l (line 3) 2. Select w segments of length l from each series storing the locations in matrix A (line 4). 3. Select w segments of length l from each di↵erence series storing the locations in matrix B (line 5). 4. Generate the n · l cases each with 2w attributes (line 6). 5. Choose a random column from W as the response variable then build a random regression tree (i.e. a tree that only considers one randomly selected attribute at each level) with maximum depth of d (line 7). Stage 2: Form a count distribution over each tree’s leaf node. 1. For each case x in the original data, get the number of rows of W that reside in each leaf node for all cases originating from x. 2. Concatenate these counts to form a new instance. Thus if every tree had t terminal nodes, the new case would have r · t features. In reality, each tree will have a di↵erent number of terminal nodes. Classification of new cases is based on a 1-nearest neighbour classification on these concatenated leaf node counts.
References {#references .unnumbered}
==========
|
[Exploring the SO(32) Heterotic String]{}
0.5cm
**Hans Peter Nilles$^a$, Saúl Ramos-Sánchez$^a$, Patrick Vaudrevange$^a$, Akin Wingerter$^b$**
$^a$*Physikalisches Institut, Universität Bonn*
*Nussallee 12, D-53115 Bonn, Germany*
0.5cm
$^b$ *Department of Physics, The Ohio State University*
*191 W. Woodruff Ave., Columbus, OH 43210, USA*
**Abstract**
.3cm
We give a complete classification of $\mathbb{Z}_N$ orbifold compactification of the heterotic SO(32) string theory and show its potential for realistic model building. The appearance of spinor representations of SO(2$n$) groups is analyzed in detail. We conclude that the heterotic SO(32) string constitutes an interesting part of the string landscape both in view of model constructions and the question of heterotic-type I duality.
.3cm
Introduction
============
String theory might provide a framework to describe all particle physics phenomena. Still we do not know how to derive the standard model of strong and electro-weak interactions from first principles. Apparently many roads seem to be possible: the so-called landscape of string vacua. Progress might be made by exploring this landscape in detail to understand possible phenomenological patterns that might be mapped to experimental observations. Such patterns might include concepts like supersymmetry, grand unification and extra dimensions.
In the present paper we would like to explore the heterotic SO(32) string theory and its suitability for model building. There has been less effort spent on the SO(32) theory than its $\text{E}_8\times\text{E}_8'$ brother, which was considered as the prime candidate initially. A detailed analysis of the SO(32) theory shows, however, that model building within this framework could be as exciting as in the $\text{E}_8\times\text{E}_8'$ case.
An additional motivation to consider the SO(32) heterotic theory is the exploration of the conjectured duality to the SO(32) type I string theory [@Polchinski:1995df]. This might prove useful to understand connections between heterotic model constructions and those based on type II orientifolds.
Our analysis considers the orbifold compactification[@Dixon:1985jw] of the SO(32) heterotic theory[^1], as it combines the complexity of Calabi-Yau compactification with the calculability of torus compactification. Many phenomenological properties find a geometric explanation in this framework [@Forste:2004ie; @Forste:2005rs; @Forste:2005gc]. We derive a complete classification of the four-dimensional heterotic SO(32) orbifold constructions. This is necessary as previous attempts to do so have been found to be incomplete. We explain the subtleties of the construction and give a detailed presentation of the $\mathbb{Z}_4$-orbifold. The remaining cases are given in detail on a web page [@SO32:webseite] that will be made available to the public.
Having achieved this goal of classification we explore properties that might be important for explicit model building. One aspect e.g. is the question of the appearance of spinor representations of SO(2$n$) gauge groups (with $n=5,6,7$). Spinors of SO(10) [@Georgi:1974my; @Fritzsch:1974nn] e.g. would be very suitable for a description of families of quarks and leptons, as argued in ref. [@Nilles:2004ej]. In addition, the appearance of these spinors might be relevant to understand the nature of the heterotic-type I duality in four space time dimensions [@Angelantonj:1996uy; @Kakushadze:1997wx; @Lalak:1999bk; @Blumenhagen:2005pm].
Using this information we provide a few explicit examples of 3-family models in this framework to illustrate the ease with which such models can be constructed. One example is obtained even in the absence of Wilson lines. Our results can be used as a starting point for a full classification of models including Wilson lines, the inclusion of which is, however, beyond the scope of this paper. Nonetheless, some useful patterns of possible spectra can be deduced from our results with the concept of fixed-point equivalent models [@Gmeiner:2002es]. It thus appears that the heterotic SO(32) theory is a fertile part of the string theory landscape.
The paper is organized as follows. In section \[subsec:general\_classification\] we present the strategy to classify all orbifolds of the SO(32) heterotic string. In section \[subsec:Z4\_classification\] and \[subsec:ineq\_models\] we illustrate the method for the $\mathbb{Z}_4$ orbifold explicitly and give the list of models for the $\mathbb{Z}_N$ orbifolds. Section 3 is devoted to the discussion of the spinorial representations of SO(2$n$) gauge groups for various $n$. Two explicit examples of 3-family models will be presented in section 4, followed by concluding remarks in section 5. Some technical details and tables are given in the appendices.
Classification of Orbifolds {#sec:classification_of_orbifolds}
===========================
Classification of $\boldsymbol{\text{SO}(32)}$ Orbifold Models {#subsec:general_classification}
--------------------------------------------------------------
To introduce the relevant notation [@Ibanez:1987pj; @Forste:2004ie] and to set the stage for the following calculations, we briefly summarize some of the concepts in orbifold constructions, before proceeding to describe the classification of inequivalent models.\
An orbifold is defined to be the quotient of a torus[^2] by a discrete set of its isometries, called the [*point group*]{} $P$. Modular invariance requires the action of the point group to be accompanied by a corresponding action $G$ ([*gauge twisting group*]{}) on the 16 gauge degrees of freedom: $$\mathcal{O} = T^6 \big{/}P \otimes T^{16} \big{/} G
\label{eq:define_orbifold_by_point_group}$$ Modular invariance and the homomorphism property of the gauge embedding $P \hookrightarrow G$ put further restrictions on $G$, which will be discussed later. Consistency with ten-dimensional anomaly cancellation requires $T^{16}$ to be an even, integral and self-dual lattice. In 16 dimensions, there are only 2 admissible choices, namely the root lattice of $\text{E}_8\times\text{E}_8'$ and the weight lattice of $\text{Spin}(32)/\mathbb{Z}_2$. Here, we focus our attention on the latter case.\
The representations of $\text{Spin}(32)$ fall into 4 conjugacy classes, corresponding to the adjoint, vector, spinor and conjugate spinor representation, respectively [@Green:1987sp; @Slansky:1981yr]. Two representations are said to be conjugate, if their weight vectors differ by an element of the root lattice $\Lambda_R$. Consequently, the weight lattice $\Lambda_W$ can be written as the sum of 4 disjoint sublattices, given by the highest weight of the respective representation modulo $\Lambda_R$. By $\text{Spin}(32)/\mathbb{Z}_2$ we shall understand the symmetry corresponding to the adjoint and spinor conjugacy classes, and denote the respective lattice by $\Lambda_{\text{Spin}(32)/\mathbb{Z}_2}$.\
The action of $G$ on $T^{16}$ can be described as a shift $X_L\mapsto X_L+V$ [@Dixon:1986jc], which induces the transformations $$\sigma_V(H_i) = H_i, \qquad \sigma_V(E_{\alpha}) = \exp\left(2\pi i \,\alpha\cdot V \right) E_{\alpha}
\label{eq:action_shift_on_operators}$$ on the Cartan generators and step operators of $\text{SO}(32)$, and these transformations clearly describe an automorphism of the algebra[^3]. The automorphisms of semi-simple Lie algebras have been classified [@Kac:1969xxx1], and it is straightforward to obtain the corresponding shifts, as we will now describe.\
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### Automorphisms of $\boldsymbol{\text{SO}(32)}$ {#automorphisms-of-boldsymboltextso32 .unnumbered}
To this end, consider the extended Dynkin diagram of SO(32) given in fig. \[fig:extended\_dynkin\_diagram\_of\_SO32\]. The numbers which have been adjoined to the nodes are the Kač labels $k_i$, which are by definition the expansion coefficients of the highest root $\alpha_H$ in terms of the simple roots, i.e. $$\alpha_H = k_1 \alpha_1 + \ldots + k_\ell \alpha_\ell,
\label{eq:def_kac_labels}$$ where $\ell$ is the rank[^4] of the algebra. For convenience, the Kač label of the [*most negative root*]{} $\alpha_0 \equiv -\alpha_H$ is set to $k_0 = 1$. Then, by a theorem due to Kač [@Kac:1969xxx1], all order-N automorphisms of an algebra up to conjugation are given by $$\sigma_{s,m}(E_{\alpha_j}) = \mu\,\exp\left(2\pi i s_j/N\right) E_{\alpha_j}, \quad j = 0, \ldots, \ell,
\label{eq:action_kac_on_operators}$$ where the sequence $s = (s_0, \ldots, s_\ell)$ may be chosen arbitrarily subject to the conditions that the $s_i$ are non-negative, relatively prime integers and $$N = m\sum_{i=0}^{\ell} k_i s_i.
\label{eq:order_N_intermsof_s_i}$$ Hereby, $\mu$ is an automorphism of the Dynkin diagram, and $m$ is the smallest integer such that $(\sigma_{s,m})^m$ is inner. Since in this context we are only interested in inner automorphisms, we set $\mu = \mathds{1}$ and $m=1$. Furthermore, it should be noted that two automorphisms $\sigma_{s}$ and $\sigma_{s'}$ are conjugate if and only if the sequence $s$ can be transformed into the sequence $s'$ by a symmetry of the extended diagram. In section \[subsec:Z4\_classification\], we will encounter an interesting example which shows that two such automorphisms of $\text{SO}(32)$ must not be identified.
### The Shift Vector {#the-shift-vector .unnumbered}
To derive the shift vector corresponding to a given automorphism is now particularly easy. Comparing eq. (\[eq:action\_shift\_on\_operators\]) to eq. (\[eq:action\_kac\_on\_operators\]), it immediately follows that $$\alpha_i \cdot V = \frac{s_i}{N}, \quad i=1, \ldots, \ell,
\label{eq:lineq_for_V}$$ for the $\ell$ linearly independent roots $\alpha_i$. Expanding $V$ in terms of the dual simple roots and substituting this expression in the previous equation gives $$V = \frac{1}{N} \left( s_1 \alpha_1^* + \ldots + s_\ell \alpha_\ell^* \right),
\label{eq:V_in_Dynkin_basis}$$ i.e. the integers $s_i$ divided by the order $N$ are the Dynkin labels of $V$. It is checked by a direct calculation that this $V$ also gives the correct transformation for the step operator corresponding to the most negative root.\
Determining the unbroken gauge group is now particularly simple. Looking at eq. (\[eq:V\_in\_Dynkin\_basis\]) we see that in the extended Dynkin diagram, the root $\alpha_i$ ($i=0,\ldots,\ell$) is projected out, if and only if the coefficient $s_i$ in eq. (\[eq:order\_N\_intermsof\_s\_i\]) does not vanish. To calculate the spectrum of the orbifold, we need an explicit expression for the shift vector $V$, which is easily obtained once the simple roots and their duals are given. For a standard choice of roots, see e.g. ref. [@Green:1987sp].
### Restrictions on the Shift Vector {#restrictions-on-the-shift-vector .unnumbered}
Not every shift vector $V$ which describes an automorphism of the algebra is an admissible choice for model construction. For a twist $\theta \in P$ of order N, $\theta^N = \mathds{1}$ implies that $N\,V$ should act as the identity on $T^{16}$, and hence, from the self-duality of the lattice, it immediately follows that $$N\,V \in \Lambda_{\text{Spin}(32)/\mathbb{Z}_2}.
\label{eq:gauge_embedding_hom_property}$$ By eq. (\[eq:V\_in\_Dynkin\_basis\]), $N\,V$ is only guaranteed to lie in the weight lattice $\Lambda_W$, so that some of the shift vectors will be ruled out.\
For the partition function of a $\mathbb{Z}_N$ orbifold to be modular invariant, the relation $$N\left( V^2 - v^2 \right) = 0 \text{ mod } 2
\label{eq:mod_inv_general}$$ has to be satisfied [@Dixon:1986jc], where $v$ is a 3-dimensional vector describing the action of the twist on the complexified, compact coordinates. This condition severely restricts the number of shift vectors which can be used in constructing orbifold models.\
From eq. (\[eq:gauge\_embedding\_hom\_property\]) it is clear that for a given order $N$ of the twist $\theta$, all shifts $V$ of order $M$ are also admissible, as long as $M$ divides $N$. In principle, we could determine the admissible shifts for each $M$ separately, but a more practical approach is to run through the outlined procedure for $N$, dropping the condition on the relative-primeness of the sequence $s = (s_0, \ldots, s_\ell)$, see the remarks preceding eq. (\[eq:order\_N\_intermsof\_s\_i\]). In the cases where $s$ is not relatively prime and the common divisor can be cancelled out from both the numerator and the denominator in eq. (\[eq:V\_in\_Dynkin\_basis\]), the order of the shift is some $M$ which is smaller than $N$.\
We will illustrate the outlined methods using the $\mathbb{Z}_4$ orbifold in section \[subsec:Z4\_classification\].
The $\boldsymbol{\mathbb{Z}_4}$ Orbifold {#subsec:Z4_classification}
----------------------------------------
### Classification {#classification .unnumbered}
We shall use the method presented in section \[subsec:general\_classification\] to compute all admissible shifts for the $\mathbb{Z}_4$ orbifold. For $N=4$, there are 256 different vectors $s=(s_0,\ldots,s_{16})$, which satisfy eq. (\[eq:order\_N\_intermsof\_s\_i\]) with m=1 and the Kač labels $k_i$ given in fig. \[fig:extended\_dynkin\_diagram\_of\_SO32\]. We express the corresponding shift vectors using eq. (\[eq:V\_in\_Dynkin\_basis\]) and a standard choice of roots[@Green:1987sp]. Of these shift vectors, 134 satisfy the first restriction eq. (\[eq:gauge\_embedding\_hom\_property\]) for admissible orbifold shifts and only 30 are left when we impose the modular invariance requirement, given by eq. (\[eq:mod\_inv\_general\]) with the twist $v=\tfrac{1}{4}(1,1,\text{-}2)$. Considering two shifts to be inequivalent if their spectra are different, we find only 16 inequivalent shift vectors in the $\mathbb{Z}_4$ orbifold. These are all possible shifts one can obtain.
### Anomalies {#anomalies .unnumbered}
The 16 inequivalent shift vectors, their corresponding gauge groups and spectra are listed in table \[tab:Z4\_classification\] of appendix \[sec:Z4\_models\]. We have denoted the anomalous $\text{U}_1$ factors by $\text{U}_{1A}$. As a cross-check for our calculations, we have verified the following conditions for anomaly cancellation $$\label{eq:Anomaly_conditions}
\frac{1}{24}\text{Tr}\,Q_i = \frac{1}{6 |t_i|^2}\text{Tr}\,Q_i^3 = \frac{1}{2}\text{Tr}\,lQ_i =
\left\{ \begin{array}{ll}\frac{1}{2|t_j|^2}\text{Tr}\,Q_j^2 Q_A \neq 0 \quad & \text{if } i = A,\,j \neq A \\ 0 &
\text{otherwise}\end{array} \right.$$ where $l$ denotes the index of a given representation. Furthermore, $t_i$ is the generator of the i-th $\text{U}_1$ factor that defines the charge $Q_i$ as: $$Q_i |p_{sh}\rangle_L = (t_i \cdot p_{sh}) |p_{sh}\rangle_L\text{ ,}$$ where $p_{sh}$ is the shifted $\text{Spin}(32)/\mathbb{Z}_2$ lattice vector. In the case when eq. (\[eq:Anomaly\_conditions\]) does not vanish, these conditions guarantee that the anomalous $\text{U}_1$ is cancelled by the generalized Green-Schwarz mechanism [@Green:1984sg; @Witten:1984dg; @Dine:1987xk; @Sagnotti:1992qw; @Berkooz:1996iz], [@Blumenhagen:2005pm].
### Discussion of the Results {#discussion-of-the-results .unnumbered}
A detailed list including the spectra of all $\mathbb{Z}_4$ orbifold models is given in table \[tab:Z4\_classification\] of appendix \[sec:Z4\_models\]. In particular, in the second column of this table we compare our results to those previously obtained in ref. [@Choi:2004wn]. In ref. [@Choi:2004wn] the shift vectors are separated into two classes. The so-called [*vectorial*]{} shifts in the $\mathbb{Z}_4$ orbifold are those whose entries have a maximal denominator of 4, whereas all entries of the [*spinorial*]{} shifts have a denominator of 8 and an odd numerator. Using these definitions, 12 of our shifts are vectorial and 4 are spinorial.
Our result differs in some ways from that of the ref. [@Choi:2004wn]. Some multiplicities of states and $\text{U}(1)$ charges are different from our findings and cannot be related by a change of basis in the $\text{U}(1)$ directions. For the models in question, ref. [@Choi:2004wn] does not fulfill the anomaly cancellation conditions, eq. (\[eq:Anomaly\_conditions\]).
Additionally, we find 16 inequivalent models whereas one can obtain only 10 inequivalent shifts with the general method proposed in ref. [@Choi:2004wn]. This discrepancy is related to two problems. One is that by using the ansatz for a spinorial shift proposed in ref. [@Choi:2004wn] and the weight lattice as given in section \[subsec:general\_classification\], one cannot obtain any of the four spinorial shifts we found in a direct manner for $\mathbb{Z}_4$. In appendix \[sec:ansatz\] we give an alternative ansatz for the form of any shift of $\mathbb{Z}_N$ orbifolds in the SO(32) heterotic string. Yet a classification based on this ansatz is more time consuming than the method presented in section \[subsec:general\_classification\].
The second problem is that the shift vectors V$^{(4)}$ and V$^{(12)}$ of our vectorial shifts are not listed in ref. [@Choi:2004wn]. Here, V$^{(i)}$ denotes the shift vector corresponding to the model number $i$ of table \[tab:Z4\_classification\]. The reason can be traced back to comparing, for instance, the shifts V$^{(3)}$ and V$^{(4)}$ of our list. Since both shifts generate the same unbroken gauge group in four dimensions and the same matter representations in the untwisted and second twisted sectors, one might be tempted to consider them to be equivalent. But the matter content of the first twisted sector is different, therefore, the two shifts lead to different models.
We have deeper reasons to argue that these two shifts are inequivalent. First, as illustrated in fig. \[fig:V3\_V4\_breaking\], the shift vectors come from two different breakings of the original SO(32) gauge group in ten dimensions. Two different breakings may be equivalent if one can transform the corresponding shifts into each other by adding lattice vectors and applying automorphisms of the lattice. We can see that there is no lattice vector relating the shifts V$^{(3)}$ and V$^{(4)}$. There exists an automorphism of $\text{SO}(32)$, which maps V$^{(3)}$ onto V$^{(4)}$ up to a lattice vector, compare fig. \[fig:V3\_V4\_breaking\]. It is not an automorphism of $\text{Spin(32)}/\mathbb{Z}_2$, as it transforms the [*spinor*]{} conjugacy class of the lattice of $\text{Spin(32)}/\mathbb{Z}_2$ into the [*conjugate-spinor*]{} class of Spin(32), which is not part of $\text{Spin(32)}/\mathbb{Z}_2$.
In summary, this shows that the ansatz of ref. [@Choi:2004wn] is incomplete. It leads only to 10 of the 16 shift vectors in the $\mathbb{Z}_4$ orbifold. As we shall show in section \[subsec:Z4\_model\], one of the missing shifts leads to a three-family model.
The $\boldsymbol{\mathbb{Z}_N}$ Orbifold {#subsec:ineq_models}
----------------------------------------
Using the method described in section \[subsec:general\_classification\], we have computed all inequivalent models, which we do not list due to space limitations. All $\mathbb{Z}_N$ shifts, their corresponding gauge groups and spectra are listed on our web page [@SO32:webseite].
### Lists of Models {#lists-of-models .unnumbered}
In summary, there are 5141 $\mathbb{Z}_N$ orbifold models without Wilson lines. We have used the geometry of $\mathbb{Z}_N$ orbifolds as given in ref. [@Kobayashi:1991rp], and for the $\mathbb{Z}_8\text{-I}$ as given in ref. [@Casas:1991ac]. On our web site [@SO32:webseite], we provide for each model the following details:
- the twist $v$ and the 6 dimensional root lattice, which specifies the geometry,\
- the gauge shift $V$ and the corresponding gauge group,\
- the matter content, listed by sectors, including all $\text{U}_1$ charges, where we have denoted the anomalous one by $\text{U}_{1A}$.[^5]
For convenience, we have implemented a search engine, with which one can choose models with a given gauge group. As a side remark, this work can be seen as a contribution to the String Vacuum Project [@SVP] in the context of the heterotic string [@Dienes:2006ut].
### Discussion of the Results {#discussion-of-the-results-1 .unnumbered}
In table \[tab:compareSO32andE8xE8\] and table \[tab:ineq\_models\], we summarize our results. Our classification extends to SO(32) the results of ref. [@Katsuki:1989bf] obtained in the context of $\text{E}_8\times\text{E}_8'$ heterotic orbifolds. Comparing the numbers of inequivalent SO(32) models to those presented in ref. [@Katsuki:1989bf], we find that there are more inequivalent models in the SO(32) heterotic string for $\mathbb{Z}_N$ orbifolds with $N\le7$ and, conversely, the number of inequivalent models for orbifolds with $N>7$ is larger in the case of $\text{E}_8\times\text{E}_8'$. This difference becomes important if Wilson lines are present, since then one can interpret the action of the shift plus the associated Wilson line(s) locally around every fixed point as a new shift. However, this new shift must be one of the set of inequivalent shift vectors we have before Wilson lines are switched on[^6]. In this sense, for $N\le7$ the SO(32) orbifolds lead to a richer variety of models.
---------------------- -------- -------------------------------
$\mathbb{Z}_N$ SO(32) $\text{E}_8\times\text{E}_8'$
$\mathbb{Z}_3$ 6 5
$\mathbb{Z}_4$ 16 12
$\mathbb{Z}_6$-I 80 58
$\mathbb{Z}_6$-II 75 61
$\mathbb{Z}_7$ 56 40
$\mathbb{Z}_8$-I 196 246
$\mathbb{Z}_8$-II 194 248
$\mathbb{Z}_{12}$-I 2295 3026
$\mathbb{Z}_{12}$-II 2223 3013
---------------------- -------- -------------------------------
: Comparison between the number of inequivalent $\mathbb{Z}_N$ orbifold models in the SO(32) heterotic string and in the $\text{E}_8\times\text{E}_8'$ heterotic string [@Katsuki:1989bf].[]{data-label="tab:compareSO32andE8xE8"}
---------------------- ------------------------ ------------------------------- -------------------------------
$\mathbb{Z}_N$ anomalous $\text{U}_1$ ${\boldsymbol{16}}$ of SO(10) ${\boldsymbol{32}}$ of SO(12)
$\mathbb{Z}_3$ 5 0 0
$\mathbb{Z}_4$ 12 2 0
$\mathbb{Z}_6$-I 76 4 4
$\mathbb{Z}_6$-II 65 10 3
$\mathbb{Z}_7$ 55 2 0
$\mathbb{Z}_8$-I 193 12 0
$\mathbb{Z}_8$-II 166 11 7
$\mathbb{Z}_{12}$-I 2269 80 36
$\mathbb{Z}_{12}$-II 2097 116 10
---------------------- ------------------------ ------------------------------- -------------------------------
: Numbers of inequivalent $\mathbb{Z}_N$ orbifold models of the SO(32) heterotic string containing at least one spinor of SO(10) or SO(12). Spinors of bigger groups do not appear in orbifold models of the SO(32) heterotic string. We also present the number of models having an anomalous $\text{U}_1$ factor as part of the gauge group.[]{data-label="tab:ineq_models"}
In the second column of table \[tab:ineq\_models\] we present the number of models having an anomalous $\text{U}_1$. As explained in section \[subsec:Z4\_classification\], all $\text{U}_1$ factors are consistent with the anomaly conditions, eq. (\[eq:Anomaly\_conditions\]). Most of the orbifold models of the SO(32) heterotic string contain an anomalous $\text{U}_1$.
From the phenomenological point of view, the $\text{SO}(32)$ heterotic string has been considered to be a less promising starting point than the $\text{E}_8\times\text{E}_8'$ theory, one of the reasons being that one did not expect spinor representations to be present in the spectrum. As first shown by ref. [@Choi:2004wn], it is possible to obtain spinor representations in orbifold models of the SO(32) heterotic string from the twisted sectors. In the third and fourth columns of table \[tab:ineq\_models\], we list the number of models for each $\mathbb{Z}_N$ orbifold in which there is at least one ${\boldsymbol{16}}$ spinor of SO(10) or one ${\boldsymbol{32}}$ spinor of SO(12), respectively. As we will explain in the next section, the mass formula forbids the appearance of spinors of SO(14) or bigger groups in orbifold models of the SO(32) heterotic string.
Spinors in SO(32) Orbifold Models {#sec:spinors}
=================================
In the light of recent developments, $\text{SO}(10)$ GUTs are attractive candidates for a theory beyond the Standard Model [@Georgi:1974my; @Fritzsch:1974nn; @Nilles:2004ej]. In orbifold constructions, GUTs may be realized in an intermediate picture [@Forste:2004ie; @Kobayashi:2004ud; @Kobayashi:2004ya; @Buchmuller:2004hv; @Buchmuller:2005jr], keeping their successful predictions and avoiding the problems, from which GUTs in 4 dimensions generically suffer. Therefore, we are naturally led to look for orbifold models containing the spinor of $\text{SO}(10)$. The SO(10) gauge group can then be broken to the Standard Model gauge group by the inclusion of Wilson lines.
We investigate here the possibility of having the 16-dimensional spinor representation of SO(10). In the standard basis, the simple roots of SO(10) can be written as
----------------------------------------------------------------------- ----------------------------------------------------------------------------------
$\alpha_1 = \left(1,-1,\phantom{-}0,\phantom{-}0,\phantom{-}0\right)$ $\alpha_4 = \left(0,\phantom{-}0,\phantom{-}0,\phantom{-}1,-1\right)$
$\alpha_2 = \left(0,\phantom{-}1,-1,\phantom{-}0,\phantom{-}0\right)$ $\alpha_5 = \left(0,\phantom{-}0,\phantom{-}0,\phantom{-}1, \phantom{-}1\right)$
$\alpha_3 = \left(0,\phantom{-}0,\phantom{-}1,-1,\phantom{-}0\right)$
----------------------------------------------------------------------- ----------------------------------------------------------------------------------
In this basis, the highest weight of the ${\boldsymbol{16}}$ is given by the 5-dimensional vector $\left(\tfrac{1}{2}^5\right)$. This vector must be part of a 16-dimensional vector $p_{sh}$ as follows $$p_{sh}=p+mV=\left(\tfrac{1}{2} ^5, a_1, a_2,..., a_{11} \right),
\label{eq:gral_p}$$ where $p\in\Lambda_{\text{Spin(32)}/\mathbb{Z}_2}$, V is a shift, $m\in\mathbb{N}$ is the number of the studied sector, and the numbers $a_i$ are selected so that $p_{sh}$ fulfills $N
p_{sh}\in\Lambda_{\text{Spin(32)}/\mathbb{Z}_2}$ and the mass formula for massless states $$p_{sh}^{2}= 2(1-\tilde{N}-\delta c),
\label{eq:mass}$$ where $\delta c$ is the shift of the zero point energy, and $\tilde{N}$ is the number operator, as explained in ref. [@Forste:2004ie]. It is important to notice that there can be more than one combination of different $a_i$’s for which the resulting $p_{sh}$ fulfills all the conditions.
The first consequence of the form of $p_{sh}$ eq. (\[eq:gral\_p\]) is that one cannot get the ${\boldsymbol{16}}$ of SO(10) in the untwisted sector ($m=0$), since it only consists of the roots of SO(32), which can be expressed by the 480 vectors $(\underline{\pm 1,\,\pm 1,\;0^{14}})$.
As a second consequence, one finds that it is not possible to get the ${\boldsymbol{16}}$ of SO(10) in the $\mathbb{Z}_3$ orbifold. The first five entries of $3 p_{sh}$ are half-integer and thus, since $3
p_{sh}\in\Lambda_{\text{Spin(32)}/\mathbb{Z}_2}$, the remaining 11 entries must also be half-integer, i.e. $3a_i\in \mathbb{Z}+\tfrac{1}{2}$. Assuming the smallest value $3a_i=\tfrac{1}{2}$, it follows that $p_{sh}^2\ge\tfrac{5}{4}+\tfrac{11}{36}=\tfrac{14}{9}$. In the case of $\tilde{N}=0$, for any twisted sector the right-hand side of eq. (\[eq:mass\]) is equal to $\tfrac{4}{3}<\tfrac{14}{9}$, which forbids the appearance of $p_{sh}$ in the spectrum of any $\mathbb{Z}_3$ orbifold model. This does not change for $\tilde{N}\neq 0$ since the value of the right-hand side of eq. (\[eq:mass\]) in this case is even smaller.
In particular, one can easily find all shift vectors which produce SO(10) spinors in the first twisted sector. From eq. (\[eq:gral\_p\]), for all $\mathbb{Z}_N$ orbifolds with $N>3$ the shift(s) giving rise to the ${\boldsymbol{16}}$ of SO(10) in the first twisted sector ($m=1$) can be written simply as $$V=p_{sh}-p \stackrel{p = 0}{\longrightarrow}p_{sh},
\label{eq:shift16}$$ where we have chosen $p=0$ because two shifts are equivalent if they differ by an arbitrary lattice vector. This shift is automatically modular invariant.
Finding the highest weight of the ${\boldsymbol{16}}$ of SO(10) is a necessary condition for its existence in the spectrum, but it is not sufficient to guarantee the presence of an SO(10) gauge group. One also needs to compute the gauge group induced by eq. (\[eq:shift16\]), which can be done by simply using the patterns given in appendix \[sec:ansatz\].
As an example, we consider the $\mathbb{Z}_4$ orbifold. The only possible shift consistent with $4V\in\Lambda_{\text{Spin(32)}/\mathbb{Z}_2}$ and eqs. (\[eq:mass\]) and (\[eq:shift16\]) with $\delta c = \tfrac{5}{16}$ is $$V=p_{sh}=\left(\left(\tfrac{1}{2}\right)^5,\,\left(\tfrac{1}{4}\right)^2,\,0^{9} \right).
\label{eq:Z4shiftwith16}$$ This shift is equivalent to V$^{(5)}$ of table \[tab:Z4\_classification\] up to lattice vectors and Weyl reflections. One can also verify that this shift provides indeed several copies of the ${\boldsymbol{16}}$ of SO(10) in the first twisted sector.
It is phenomenologically attractive to have ${\boldsymbol{16}}$’s of $\text{SO}(10)$ in the first twisted sector. Since the ${\boldsymbol{16}}$-plets of the first twisted sector are localized in all six compact dimensions, the inclusion of Wilson lines will break the gauge group and will reduce the degeneracy of the fixed points, but it will not project out parts of a ${\boldsymbol{16}}$-plet. Therefore, models with this feature can lead to potentially realistic string models.
An indirect method to obtain the ${\boldsymbol{16}}$ of SO(10) is to switch on Wilson lines in models having spinor representations of bigger groups, like SO(12). In general, for SO($2n$) groups, the highest weight of the corresponding spinor is a solution of the eq. (\[eq:mass\]) of the form $$p_{sh}=p+mV=\left(\tfrac{1}{2} ^n, a_1, a_2,..., a_{16-n} \right).
\label{eq:gral_p_D_n}$$ By inspecting all possible values that $\delta c$ and $\tilde{N}$ can take in the twisted sectors for all $\mathbb{Z}_N$ orbifolds, one can see that $p_{sh}^2=2(1-\tilde{N}-\delta c)\le \tfrac{31}{18}$ in the mass equation (\[eq:mass\]). This means that the spinor representation of $\text{SO}(2n)$ for $n\ge 7$ given by eq. (\[eq:gral\_p\_D\_n\]) is not allowed, because it is forbidden by the mass equation.
There are indeed some models with the ${\boldsymbol{32}}$ spinor representation of SO(12), as shown in table \[tab:ineq\_models\]. One might switch on Wilson lines on them in search of realistic models.
Three-Family Orbifold Models {#subsec:3_family}
============================
The $\boldsymbol{\mathbb{Z}_4}$ Orbifold {#subsec:Z4_model}
----------------------------------------
As our illustrative example, we consider the $\mathbb{Z}_4$ orbifold. One choice for the 6 dimensional lattice [@Kobayashi:1991rp] is the SO(5)$^2\times$SO(4) root lattice as shown in fig. \[fig:Z4\_geometry\]. The point group $\mathbb{Z}_4$ is generated by $\theta$ which acts as a simultaneous rotation of 90${}^\circ$ in two of the three 2-tori and a rotation of 180${}^\circ$ in the third one; this corresponds to the twist vector $$v = \frac{1}{4}\left(1,\,\,\,1,\,\,-2 \right).
\label{eq:twist_Z4}$$
[**[Fixed point structure.]{}**]{} On the torus, the action of $\theta^1$ has $2\times2\times4 = 16$ fixed points, see fig. \[fig:Z4\_geometry\](a). The twisted sector corresponding to the action of $\theta^3$ gives the anti-particles of the $\theta^1$ sector, so we will not consider it separately.
The element $\theta^2$ of the point group acts non-trivially only in two of the three complex planes, see fig. \[fig:Z4\_geometry\](b). Thus, the strings are localized only in 4 of the 6 compact dimensions and are free to move in the last torus. For convenience, we shall refer to these fixed tori as fixed points. Of the $4\times 4 = 16$ fixed points of the $\theta^2$ sector, only
( ${\scriptstyle \blacksquare}$, ${\scriptstyle \blacksquare}$ ), ( ${\scriptstyle \blacksquare}$, ), ( , ${\scriptstyle \blacksquare}$ ), ( , )
are also invariant under the action of $\theta$. The remaining 12 points are pairwise related by $\theta$ and therefore form pairs
( ${\scriptstyle \blacksquare}$, ) $\leftrightarrow$ ( ${\scriptstyle \blacksquare}$, $\mathbf{\times}$ ), ( , ${\scriptstyle \blacksquare}$ ) $\leftrightarrow$ ( $\mathbf{\times}$, ${\scriptstyle \blacksquare}$ ), ( , ) $\leftrightarrow$ ( $\mathbf{\times}$, $\mathbf{\times}$ ),\
( , ) $\leftrightarrow$ ( $\mathbf{\times}$, ), ( , $\mathbf{\times}$ ) $\leftrightarrow$ ( $\mathbf{\times}$, ), ( , ) $\leftrightarrow$ ( , $\mathbf{\times}$ ).
In this way, these 12 fixed points of the $\theta^2$ sector collapse to 6 by the action of the orbifold. This leaves an effective number of $4 + 6 = 10$ fixed points in the second twisted sector.
[**[Wilson lines.]{}**]{} Of the 16 $\mathbb{Z}_4$ models, none has 3 families of quarks and leptons. In order to reduce the number of families and to further break the gauge symmetry, we need Wilson lines [@Ibanez:1986tp]. The number and the order of Wilson lines one can add in a specific orbifold model is dictated by the geometry of the underlying compactification. In our case, we can have only 4 Wilson lines $A_1$, $A_3$, $A_5$, and $A_6$ of order 2 corresponding to the directions $e_1$, $e_3$, $e_5$, and $e_6$, respectively.
As a toy model, we present a 3-generation SU(5) model. When considering the models presented in table \[tab:Z4\_classification\], the shift vector V$^{(14)}$ seems quite promising. This model has an SU(5) gauge symmetry and, most importantly, the localization of the generations gives some clues. There are two ${\boldsymbol{10}}$’s in the bulk and sixteen ${\boldsymbol{10}}$’s attached to the fixed points of the first twisted sector. We focus our attention on the ${\boldsymbol{10}}$ because the representation ${\boldsymbol{\overline{5}}}$ generically come with the same multiplicity due to anomaly cancellation in orbifold models. By a clever choice of the four Wilson lines, the degeneracy of the fixed points can be lifted, so that the number of families is reduced from 16 in the twisted sectors to 1, and both families of the untwisted sector survive, as depicted in fig. \[fig:Z4\_3gen\_model\]:
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$A_1=\left(\phantom{-}\frac{5}{2}^2,\phantom{-}0^5,-{\frac{1}{2}}^2 -1^2,\phantom{-}3,-1^4 \right)$,
$A_3=\left(-3^2,\phantom{-}0^5,-3^2,-2,-3,-\frac{5}{2},\phantom{-}\frac{3}{2},-\frac{5}{2},-2,\phantom{-}\frac{1}{2}\right)$,
$A_5=\left(\phantom{-}{\frac{1}{2}}^2,\phantom{-}0^5,\phantom{-}{\frac{1}{2}}^2,\phantom{-}2,\phantom{-}\frac{3}{2}^3,\phantom{-}2,\phantom{-}{\frac{1}{2}},\phantom{-}2 \right)$,
$A_6=\left(\phantom{-}3,\phantom{-}\frac{7}{2},\phantom{-}0^5,-1,-\frac{5}{2},-2,-\frac{5}{2}^6
\right)$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The combined action of the shift and the Wilson lines leads to the gauge group $\text{SU}(5)\times\text{SU}(2)^5\times\text{U}(1)^7$, where the first U(1) is anomalous. The complete spectrum of this model is given in table \[tab:spectrum\_of\_SU5\_model\_in\_Z4\]. The main objective of the present publication being the clarification of some outstanding issues in the heterotic $\text{SO}(32)$ theory, we will not explore the phenomenology of this model in detail.
U $\theta^1$ $\theta^2$ sum states
-- ---- ------------ ------------ ----- ---------------------------------------------------------------------------------------------------------------------------------------------------------- --
2 1 3 $({\boldsymbol{10}};\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}})$
3 5 8 $(\phantom{{\boldsymbol{1}}}{\boldsymbol{\overline{5}}};\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}})$
1 4 5 $({\boldsymbol{\phantom{1}5}};\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}})$
10 37 4 51 $({\boldsymbol{\phantom{1}1}};\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}})$
12 4 16 $({\boldsymbol{\phantom{1}1}};\;{\boldsymbol{2}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}})$
12 4 16 $({\boldsymbol{\phantom{1}1}};\;{\boldsymbol{1}},\;{\boldsymbol{2}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}})$
12 4 16 $({\boldsymbol{\phantom{1}1}};\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{2}},\;{\boldsymbol{1}},\;{\boldsymbol{1}})$
12 4 16 $({\boldsymbol{\phantom{1}1}};\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{2}},\;{\boldsymbol{1}})$
12 4 16 $({\boldsymbol{\phantom{1}1}};\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{1}},\;{\boldsymbol{2}})$
: The spectrum of a $\mathbb{Z}_4$ toy model with 3 generations of SU(5).[]{data-label="tab:spectrum_of_SU5_model_in_Z4"}
We would like to stress three features of this model. To the best of our knowledge, this is the first three-generation model in the context of the SO(32) orbifold published in the literature. The shift V$^{(14)}$ that we used for this three-family model does not appear in ref. [@Choi:2004wn]. This model shows clearly the possibility to compute promising models through orbifolds of the SO(32) heterotic string.
Model in the $\boldsymbol{\mathbb{Z}_6}$-II Orbifold {#subsec:Z6II_model}
----------------------------------------------------
In the $\mathbb{Z}_6$-II orbifold, one possible choice of the 6 dimensional lattice is G$_2\times$SU(3)$\times$SO(4). For further details on the geometry and the fixed point structure, see ref. [@Kobayashi:2004ya]. Even without the inclusion of Wilson lines, we find toy models with 3 generations. For instance, using
$V^{(30)} =
\left(\;\frac{1}{2}^{2},\;-\frac{1}{6}^{5},\;-\frac{1}{3}^{6},\;-\frac{1}{2}^{3}\;\right)$
of those $\mathbb{Z}_6$-II shifts listed on our web page [@SO32:webseite], we obtain a model with 3 generations of SO(10). Their localization is illustrated in fig. \[fig:Z6\_II\_3gen\_model\].
The families are localized as follows: there are three ${\boldsymbol{16}}$’s of SO(10) in the second twisted sector, whereas there are six ${\boldsymbol{\overline{16}}}$’s in the fourth twisted sector. Since the families are located in the second and fourth twisted sectors, where two of the six compactified dimensions are left invariant by the orbifold action, the families are free to move in six dimensions.
Even though this model is not realistic, it illustrates how easily one can obtain orbifold models with three families in the context of the SO(32) heterotic string. Therefore, using Wilson lines, potentially realistic models may be derived.
Conclusions and Outlook
=======================
As we have seen, model building with the heterotic SO(32) theory might be as exciting as that with its more famous brother: the $\text{E}_8\times\text{E}_8'$ string, see e.g. [@Faraggi:1989ka; @Kobayashi:2004ud; @Kobayashi:2004ya; @Buchmuller:2004hv; @Braun:2005ux; @Buchmuller:2005jr; @Bouchard:2005ag; @Blumenhagen:2006ux]. It opens new roads for explicit constructions that should be explored as a vital part of the string landscape.
We were somewhat surprised about the frequency of the appearance of spinor representations of SO(2$n$) gauge groups. These spinors might be an important tool to implement the family structure of SU(3) $\times$ SU(2) $\times$ U(1) models. In addition they are an important ingredient for a possible understanding of the SO(32) heterotic type I duality in $d=4$ space time dimensions. We know that these spinors do not appear in the perturbative type I theory. Thus the mentioned duality will need the implementation of nonperturbative effects.
Our classification of the $\mathbb{Z}_N$ orbifolds of the SO(32) theory completes a basic building block for further model constructions. We understand this as a contribution to the study of the string landscape in the spirit of the “String Vacuum Project” [@SVP]. A further step in this program would be the implementation of Wilson lines that leads to enormous complexity and a huge number of models (comparable to that of the $\text{E}_8\times\text{E}_8'$ string). An exploration of this large region of the landscape is currently beyond our capabilities. We therefore gather our present results and make them available to the public on our web page [@SO32:webseite], such that interested people could share our knowledge and contribute to the enterprise.
[**Acknowledgments**]{}
It is a pleasure to thank K. S. Choi, S. Förste and M. Ratz for valuable discussions. This work was partially supported by the European Union 6th Framework Program\
“Quest for Unification” and “ForcesUniverse”. A. W. was supported in part by DOE grant DOE/ER/01545-866.
Table of $\mathbb{Z}_4$ orbifold models {#sec:Z4_models}
=======================================
[|c|c|p[2.85cm]{}|c|p[4.3cm]{}|p[3.7cm]{}|p[4.2cm]{}p[0.5mm]{}|]{}\
\
& & & & & &\
\# & Shift & & 4D gauge group & & & &\
& & & & & &\
\# & Shift & & 4D gauge group & & & &\
1 & $\left( 0^2, \text{-}{\frac{1}{2}}, \text{-}{\frac{3}{4}}^2, 1, 0^{10} \right)$ & & SO$_{26} \times$ SU$_2 \times$ U$_{1A}\times$U$_1$ & & & &\
2 & $\left( 0^2, \text{-}{\frac{1}{2}}^2, {\frac{1}{2}}, {\frac{1}{4}}, \text{-}{\frac{3}{4}}, 1, 0^{8} \right)$ & & SO$_{22} \times$ SU$_4 \times$ SU$_2 \times$ U$_1$ & & & &\
3 & $\left(0^{2},\text{-}\frac{3}{4}^2,\frac{1}{4}^{3},\frac{9}{4},\text{-}2,0^{7}\right)$ & & SO$_{20} \times$ SU$_6 \times$ U$_{1A}$ & & & &\
4 & $\left(0^{2},\text{-}\frac{1}{4},\text{-}\frac{3}{4},\frac{1}{4}^{3},\text{-}\frac{3}{4},1,0^{7}\right)$ & & SO$_{20} \times$ SU$_6 \times$ U$_{1A}$ & & & &\
5 & $\left( 0^2, \text{-}{\frac{1}{2}}^2, {\frac{1}{2}}^3, {\frac{1}{4}}, {\frac{9}{4}}, \text{-}2, 0^{6} \right)$ & & SO$_{18} \times$ SO$_{10} \times$ SU$_2 \times$ U$_{1A}$ & & & &\
6 & $\left( 0^2, \text{-}{\frac{1}{2}}^2, {\frac{1}{4}}^5, \text{-}{\frac{3}{4}}, 1, 0^{5} \right)$ & & SO$_{16} \times$ SU$_2^2
\times$ SU$_6 \times$ U$_{1A}$ & & & &\
7 & $\left( 0^2, \text{-}{\frac{1}{2}}^4, {\frac{1}{2}}^3, {\frac{1}{4}}, {\frac{5}{4}}, \text{-}1, 0^{4} \right)$ & & SO$_{14} \times$ SO$_{14} \times$ SU$_2 \times$ U$_1$ & & & &\
8 & $\left( 0^2, \text{-}{\frac{1}{2}}^4, {\frac{1}{4}}^5, {\frac{9}{4}}, \text{-}2, 0^{3} \right)$ & & SO$_{12} \times$ SO$_{8} \times$ SU$_6 \times$ U$_{1A}$ & & & &\
9 & $\left( 0^2, \text{-}{\frac{1}{2}}, \text{-}{\frac{3}{4}}, {\frac{1}{4}}^8, {\frac{9}{4}},\text{-}2,0^2 \right)$ & & SO$_{10} \times$ SU$_{10} \times$ U$_{1A}\times$U$_1$ & & & &\
10 & $\left( 0^2, \text{-}{\frac{1}{2}}^2, {\frac{1}{2}}, {\frac{1}{4}}^9, {\frac{9}{4}}, 2 \right)$ & & SU$_{4} \times$ SU$_4 \times$ SU$_{10} \times$ U$_1$ & & & &\
11& $\left( {\frac{1}{2}}^2, \text{-}{\frac{1}{4}}^{12}, {\frac{3}{4}}^2 \right)$ & & SU$_{2} \times$ SU$_2 \times$ SU$_{14} \times$ U$_1$ & & & &\
12& $\left( {\frac{1}{2}}^2, {\frac{1}{4}}, \text{-}{\frac{1}{4}}^{11}, {\frac{3}{4}}^2 \right)$ & & SU$_{2} \times$ SU$_2 \times$ SU$_{14} \times$ U$_{1A}$ & & & &\
13 & $\left( \text{-}{\frac{1}{8}}, \text{-}{\frac{7}{8}}, \text{-}{\frac{5}{8}}, {\frac{1}{8}}^{11}, {\frac{17}{8}}^2 \right)$ & & SU$_{15} \times$ U$_{1A}\times$U$_1$ & & & &\
14 & $\left( \text{-}{\frac{9}{8}}, {\frac{1}{8}}, \text{-}{\frac{13}{8}}, \text{-}{\frac{5}{8}}^4, \text{-}{\frac{7}{8}}^9 \right)$ & & SU$_{11} \times$ SU$_5 \times$ U$_{1A}\times$U$_1$ & & & &\
15 & $\left( \text{-}{\frac{1}{8}}, {\frac{1}{8}}, \text{-}{\frac{5}{8}}^4, {\frac{3}{8}}^{5}, {\frac{1}{8}}^5 \right)$ & & SU$_{7} \times$ SU$_9 \times$ U$_{1A}\times$U$_1$ & & & &\
16 & $\left({\frac{3}{8}}, {\frac{5}{8}}, \text{-}{\frac{1}{8}}^{12}, {\frac{15}{8}}, \text{-}{\frac{3}{8}} \right)$ & & SU$_{3} \times$ SU$_{13} \times$ U$_{1A}\times$U$_1$ & & & &\
The General Form of a Shift in $\boldsymbol{\mathbb{Z}_N}$ Orbifolds of the SO(32) Heterotic String {#sec:ansatz}
===================================================================================================
To obtain the general form of a shift, we use the fact that two shifts are equivalent if they are related by lattice vectors or by Weyl reflections, i.e. by any permutation of the entries and pairwise sign flips.
In $\mathbb{Z}_N$ orbifolds with [**[even $\boldsymbol{N}$]{}**]{}, one can prove that the most general form of a [**[vectorial shift]{}**]{} is given by $$V = \frac{1}{N}\left( \left(\pm k\right)^{\alpha}, -(N-k)^{\beta}, 0^{n_0}, 1^{n_1}, \ldots ,
(N-k)^{n_{(N-k)}-\alpha-\beta},\ldots, \left(\frac{N}{2}\right)^{n_{\left(\frac{N}{2}\right)}} \right),$$ where $\alpha, \beta, n_i \in \mathbb{N}$, $\alpha+\beta\in \{0,1\}$, $k\in\{\frac{N}{2}+1,\frac{N}{2}+2,\ldots,N\}$ and $\sum n_i =16$. It leads to a symmetry breaking in four dimensions of the general form $$\textrm{SO(32)} \longrightarrow \textrm{SO(2}n_0\textrm{)}\times \textrm{U(}n_1\textrm{)}\times
\ldots\times\textrm{U}\left(n_{\left(\frac{N}{2}-1\right)}\right)\times \textrm{SO}\left(2n_{\left(\frac{N}{2}\right)}\right).$$ On the other hand, for [**[even $\boldsymbol{N}$]{}**]{} the [**[spinorial shifts]{}**]{} can be written in the standard form $$V = \frac{1}{2N}\left( \left(\pm k\right)^{\alpha}, -(2N-k)^{\beta}, 1^{n_1}, 3^{n_3}, \ldots ,
(2N-k)^{n_{\left(2N-k\right)}-\alpha-\beta},\ldots, (N-1)^{n_{\left(N-1\right)}} \right),$$ with $k\in\{N+1,N+3,\ldots,2N-1\}$, which give rise to the gauge group $$\textrm{SO(32)} \longrightarrow \textrm{U(}n_1\textrm{)}\times \textrm{U(}n_3\textrm{)}\times
\ldots\times\textrm{U}\left(n_{\left(N-3\right)}\right)\times \textrm{U}\left(n_{\left(N-1\right)}\right).$$ For [**[$\boldsymbol{N}$ odd]{}**]{}, the general form of both a vectorial shift and a spinorial one change slightly. As explained in ref. [@Choi:2004wn], in this case it is enough to determine either the vectorial or the spinorial shifts, since one spinorial shift can always be transformed into a vectorial one by the action of Weyl reflections and lattice vectors. Therefore any shift can be written in general as $$V = \frac{1}{N}\left( \left(\pm k\right)^{\alpha}, -(N-k)^{\beta}, 0^{n_0}, 1^{n_1}, \ldots ,
(N-k)^{n_{\left(N-k\right)}-\alpha-\beta},\ldots, \left(\frac{N-1}{2}\right)^{n_{\left(\frac{N-1}{2}\right)}} \right),$$ where $k\in\{\frac{N+1}{2},\frac{N+3}{2},\ldots,N\}$. The resulting four dimensional gauge group is $$\textrm{SO(32)} \longrightarrow \textrm{SO(2}n_0\textrm{)}\times \textrm{U(}n_1\textrm{)}\times
\ldots\times\textrm{U}\left(n_{\left(\frac{N-3}{2}\right)}\right)\times \textrm{U}\left(n_{\left(\frac{N-1}{2}\right)}\right).$$
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L. E. Ibanez, H. P. Nilles and F. Quevedo, Phys. Lett. B [**187**]{} (1987) 25. A. E. Faraggi, D. V. Nanopoulos and K. j. Yuan, Nucl. Phys. B [**335**]{} (1990) 347. V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, Phys. Lett. B [**618**]{}, 252 (2005) \[arXiv:hep-th/0501070\]. V. Bouchard and R. Donagi, Phys. Lett. B [**633**]{} (2006) 783 \[arXiv:hep-th/0512149\]. R. Blumenhagen, S. Moster and T. Weigand, arXiv:hep-th/0603015.
[^1]: For earlier work on SO(32) heterotic string orbifolds, see ref. [@Giedt:2003an; @Choi:2004wn].
[^2]: In a more general context, an orbifold is defined to be the quotient of a manifold by a discrete symmetry.
[^3]: Note that the group $\text{SO}(32)$ and its covering group $\text{Spin}(32)$ share the same algebra.
[^4]: Clearly, $\ell=16$ in our case, but for the time being, we want to keep the discussion general.
[^5]: More details are available from the authors upon request.
[^6]: For more information about the concept of fixed point equivalent models, see ref. [@Gmeiner:2002es].
|
---
abstract: 'We find an exact formula for the thermally averaged cross section times the relative velocity $\langle \sigma v_{\text{rel}} \rangle$ with relativistic Maxwell-Boltzmann statistics. The formula is valid in the effective field theory approach when the masses of the annihilation products can be neglected compared with the dark matter mass and cut-off scale. The expansion at $x=m/T\gg 1$ directly gives the nonrelativistic limit of $\langle \sigma v_{\text{rel}}\rangle$ which is usually used to compute the relic abundance for heavy particles that decouple when they are nonrelativistic. We compare this expansion with the one obtained by expanding the total cross section $\sigma(s)$ in powers of the nonrelativistic relative velocity $v_r$. We show the correct invariant procedure that gives the nonrelativistic average $\langle \sigma_{nr} v_r \rangle_{nr}$ coinciding with the large $x$ expansion of $\langle \sigma v_{\text{rel}}\rangle$ in the comoving frame. We explicitly formulate flux, cross section, thermal average, collision integral of the Boltzmann equation in an invariant way using the true relativistic relative $v_\text{rel}$, showing the uselessness of the Møller velocity and further elucidating the conceptual and numerical inconsistencies related with its use.'
author:
- Mirco Cannoni
title: |
Relativistic and nonrelativistic annihilation of dark matter: a sanity check\
using an effective field theory approach
---
Introduction
============
While there are compelling evidences in astrophysics and cosmology that most of the mass of the Universe is composed by a new form of non baryonic dark matter (DM), there is a lack of evidence for the existence of new physics at LHC and other particle physics experiments. On the theory side, many specific models with new particles and interactions beyond the standard model have been proposed to account for DM.
Under these circumstances where no clear indications in favour of a particular model are at our disposal, the phenomenology of DM as been studied in a model independent way using an effective field theory approach, see for example [@Kurylov:2003ra; @Fan:2010gt; @Fitzpatrick:2012ix; @Beltran:2008xg_vs2; @Cao:2009uw_vs2; @Buckley:2011kk_vs2; @Matsumoto:2014rxa_vs2; @Fedderke:2014wda_vs2_Moll; @Chen:2013gya_vs2_Moll; @Berlin:2014tja_vs2_error; @Chang:2013oia_vs2; @Cheung:2012gi_vs2; @Balazs:2014rsa; @Goodman:2010yf; @Goodman:2010ku; @Buchmueller:2013dya; @Busoni:2013lha; @Busoni:2014sya; @Busoni:2014haa; @Zheng:2010js_vl; @Dreiner:2012xm; @Blumenthal:2014cwa_vl; @Desimone_moller_frame].
Measurements of the parameters of standard model of cosmology [@Bennett:2012zja; @Ade:2013zuv] furnish the present day mass density of DM, the relic abundance, $\Omega h^2 \sim 0.11$ with an uncertainty at the level of 1%. Any model that pretends to account for DM must reproduce this number, which, on the other hand, sets strong constraints on the free parameters of the model.
When the DM particles are weakly interacting massive particles that decouple from the primordial plasma at a temperature when they are nonrelativistic, the relativistic averaged annihilation rate $\langle \sigma v_\text{rel}\rangle$ can be well approximated by taking the nonrelativistic average of the first two terms of the expansion of $\sigma$ in powers of the nonrelativistic relative velocity. With $v_\text{rel}$ we indicate the [*relativistic relative velocity*]{} and with $v_r$ the [*nonrelativistic relative velocity*]{}, as defined in \[app:2\]. To describe collisions in a gas, and in particular in the primordial plasma, the reference frame that matters is the comoving frame (COF) where the observer sees the gas at rest as a whole and the colliding particles have general velocities $\boldsymbol{v}_{1,2}$ without any further specification of the kinematics.
It is thus desirable to formulate cross sections and rates in a relativistic invariant way, such that all the formulas and nonrelativistic expansions are valid automatically in the COF. Obviously, invariant formulas give the same results in the lab frame (LF), the frame where one massive particle is at rest, and in the center of mass frame (CMF) where the total momentum is zero. We will see that the key for the invariant formulation is $v_\text{rel}$.
On the contrary, in DM literature [@GG] instead of $v_\text{rel}$ it is used the so–called Møller velocity $\bar{v}$, see \[app:2\]. That this is incorrect was already discussed in Ref. [@Cannoni:2013bza] but papers using $\bar{v}$ continue to appear. The problem with $\bar{v}$, which is not the relative velocity, is its non invariant and nonphysical nature, for it can take values larger than $c$.
In this paper we first find an exact formula for $\langle \sigma v_{\text{rel}} \rangle$ as a function of $x=m/T$ calculated with the relativistic Maxwell-Boltzmann statistics. The formula is valid in the effective field theory framework such that the masses of the annihilation products can be neglected compared with the DM and the cut-off scale. For concreteness we work with fermion DM. We find the thermal functions corresponding to various interactions and in particular those corresponding to $s$ and $p$ wave scattering in the nonrelativistic limit which is given by the expansion at $x\gg 1$. This is done in Section \[sec2\], and \[app:1\] contains some mathematical results needed for the derivation of the exact formula and its asymptotic expansions.
Then, in Section \[sec3\], we present the correct invariant method for obtaining the same expansion by expanding the total annihilation cross section $\sigma(s)$ in powers of $v_r$.
We then discuss in Section \[sec4\] the problems with the use $\bar{v}$, while the numerical impact on the relic abundance of some incorrect methods employed in literature is evaluated in Section \[sec5\].
\[app:2\] is preparatory for the whole paper: we remind how relativistic flux, cross section, rate, collision term of the Boltzmann equation and thermal averaged rate can be defined in the invariant way in terms of $v_\text{rel}$ showing the uselessness of the Møller velocity.
Exact formula for the thermal average in the effective approach {#sec2}
===============================================================
We consider a DM fermion field $\chi$ that couples to other fermion fields $\psi$ through an effective dimension-6 operator of the type
\_= (| \_a ) (|\_b ). \[operator\]
The DM particles can be of Dirac or Majorana nature and have mass $m$, while $\psi$ are the standard model fermions or new ones. Here $\lambda_{a,b}$ are dimensionless coupling associated with the interactions described by combination of Dirac matrices $\Gamma_{a,b}$. $\Lambda$ is the energy scale below which the effective field theory is valid. In the exact theory $\Lambda$ corresponds to the mass of a heavy scalar or vector boson mediator that appears in the propagators. The $\psi$ masses can be neglected compared to $\Lambda$ and $m$. The exchange of a heavy mediator with mass $\Lambda$ may take place in the $s$-channel and/or in $t$-channel, as depicted in Figure \[Fig1\], depending on the specific model.
Exact formula for $\langle \sigma {v}_{\text{rel}} \rangle$
------------------------------------------------------------
In all generality, for $2 \to 2$ processes, the matrix elements depend only on two independent Mandelstam variables, for example $s$ and $t$, and the squared matrix element is dimensionless. After integrating over the CMF angle, for example, the only remaining dependence is on $s$ and $m$. Any amplitude related to the operator (\[operator\]) gives an integrated squared matrix element $\overline{|\mathcal{M}|^2}$ summed over the final spins and averaged over the initial spins that is a simple polynomial of the type
w= d= p\_2 s\^2 +p\_1 m\^2 s+p\_0 m\^4,
with $p_0,...,p_2$ depending on $\Lambda$ and $\lambda_{a,b}$.
![$s$ and $t$ channel annihilation diagrams reducing to the effective vertex corresponding to the lagrangian Eq. (\[operator\]). []{data-label="Fig1"}](eff_diag_ann1.eps)
To get the formula for $\langle\sigma v_\text{rel}\rangle$ in a useful form, it is convenient to define the [*reduced cross section*]{}
\_0 &= w, \[sigma0\]
and the [*effective cross*]{} section
\_ = , \[sigmalambda\]
which contains all the couplings. In terms of the effective cross section (\[sigmalambda\]), and of the dimensionless variable $y=s/(4m^2)$, the reduced cross section Eq. (\[sigma0\]) becomes
\_0=\_(a\_2 y\^2 + y + ), \[sigma0ann\]
where now $a_2,...,a_0$ are pure numbers. The total unpolarized cross section then is
= \_0. \[sigma\]
We now set $m_1=m_2=m$, $m_3=m_4=0$ in Eq. (\[sigma\]) and in Eq. (\[sigmav\_1\]), and change variable to $y$. Thus Eq. (\[sigmav\_1\]) becomes
\_ = \_[1]{}\^ [dy]{} K\_1 (2x) \_0(y). \[sigmav\_y\]
Using the integrals of \[app:1\], we find
\_ = \_ \[& 8 a\_2 + 2 a\_1 + (5a\_2 +2a\_1 +a\_0)\
&+3a\_2\]. \[sigmav\_final\]
In the case $m_3=m_4=0$ the pure mass terms do not appear in the cross sections, thus $a_0 =0$. Furthermore, we can relate $a_2 $ and $a_1$ each other by an appropriate multiplicative factor,
a\_1=k a\_2, \[a\_2\_a\_1\]
and express the cross sections as a function of $a_2$ only. The general formula (\[sigmav\_final\]) thus finally becomes
\_ = \_a\_2 \_k(x), \[sigmav\_ann\_simple\]
with
\_k(x) = (8 +2k + (5+2k) + 3) \[F\_k\]
the factored out thermal function.
The nonrelativistic thermal average is given by the expansion at $x\gg 1$. Using the asymptotic expansions Eq. (\[Bessel\]) we find
\_[nr]{} v\_r \_[nr]{}=\_a\_2 (1+ - )+(x\^[-2]{}). \[nonrel\]
In the ultrarelativistic limit, $x\ll 1$, using the expansions (\[Bessel2\]), the thermal functions behave as $3/x^2$, thus
\_ \_[ur]{} \~\_a\_2 =3a\_2 T\^2,
which is the expected result for massless particles.
The exact integration is possible because the effective operator removes the momentum dependence in the propagators that are reduced to a multiplicative constant and the assumption $m_3 =m_4 =0$ allows to simplify the square root $\sqrt{\lambda(s,m^2_3,m^2_4)}=s$ in the cross section (\[sigma\]). For example, with $m_3=m_4=m_\psi$, equation (\[sigmav\_y\]) becomes
\_ = \_[y\_0]{}\^ .
with $\rho=m^2_\psi / m^2$ and $y_0 =1$ if $m\geq m_\psi$, $y_0 =\rho$ if $m< m_\psi$. In this case the exact integration is not possible but nonrelativistic expansions exist also in the case $\rho=1$ and $\rho\gg 1$ as we have shown in Ref. [@Cannoni:2013bza].
Applications
------------
In order to show the thermal behaviour of different interactions, we calculate the cross sections for various operators of the type (\[operator\]), both for $s$ and $t$ channel annihilation. We list the quantity $\varpi=\Lambda^4/(\lambda^2_a \lambda^2_b)\,w$ and the resulting average Eq. (\[sigmav\_ann\_simple\]).
For the *$s$-channel annihilation* we find:\
1) Scalar: $(\bar{\chi} \chi) (\bar{\psi} \psi)$, $(\bar{\chi} \chi) (\bar{\psi} \gamma^5 \psi)$.
=2s(s-4m\^2), \_S [v]{}\_ =\_2 \_[-4]{}(x). \[S\_s\]
2\) Pseudo-scalar: $(\bar{\chi}\gamma^5 \chi) (\bar{\psi}\gamma^5 \psi)$, $(\bar{\chi}\gamma^5 \chi) (\bar{\psi} \psi)$:
= 2s\^2, \_[PS]{} [v]{}\_ =\_2 \_[0]{}(x). \[PS\_s\]
3\) Chiral: $(\bar{\chi}P_{L,R} \chi) (\bar{\psi} P_{L,R} \psi)$.
=s(s-2m\^2), \_C [v]{}\_ =\_ \_[-2]{}(x). \[C\_s\]
4\) Pseudo-vector: $(\bar{\chi} \gamma^\mu \gamma_5 \chi) (\bar{\psi} \gamma_\mu \gamma_5\psi)$, $(\bar{\chi} \gamma^\mu \gamma_5 \chi) (\bar{\psi} \gamma_\mu \psi)$.
&=s(s-4m\^2), \_[PV]{} [v]{}\_ =\_ \_[-4]{}(x) \[PV\_s\]
5\) Vector: $(\bar{\chi} \gamma^\mu \chi) (\bar{\psi} \gamma_\mu \psi)$, $(\bar{\chi} \gamma^\mu \chi)(\bar{\psi} \gamma_\mu \gamma^5 \psi)$.
=s(s+2m\^2), \_[V]{} [v]{}\_ =\_ \_[2]{}(x). \[V\_s\]
6\) Vector-chiral: $(\bar{\chi} \gamma^\mu P_{L,R} \chi) (\bar{\psi} \gamma_\mu P_{L,R} \psi)$.
=s(s-m\^2), \_[VC]{} [v]{}\_ =\_ \_[-1]{}(x). \[VC\_s\]
The tensor interaction $\sigma^{\mu \nu}$ gives the same function as the vector case and is not reported. In the case of a Majorana $\chi$ clearly the vector and tensor interactions are absent, and the inclusion of a factor $1/2$ in the operator (\[operator\]) cancels the factor 4 due to the presence of the exchange diagram of the initial identical particles.
Now we consider some examples of *$t$-channel annihilation* for operators common to Dirac and Majorana DM annihilation:\
1) Scalar, pseudo-scalar: $(\bar{\chi} \chi) (\bar{\psi} \psi)$, $(\bar{\chi} \chi) (\bar{\psi} \gamma^5 \psi)$,\
$(\bar{\chi}\gamma^5 \chi) (\bar{\psi}\gamma^5 \psi)$, $(\bar{\chi}\gamma^5 \chi) (\bar{\psi} \psi)$.
\_D&=s(s-m\^2), \^[D,t]{}\_[S,PS]{} [v]{}\_ =\_ \_[-1]{}(x).\
\_M& =s(5s-14m\^2), \^[M,t]{}\_[S,PS]{} [v]{}\_ =\_ \_[-]{}(x).
2\) Chiral: $(\bar{\chi}P_{L,R} \chi) (\bar{\psi} P_{L,R} \psi)$.
\_D& =s(s-m\^2), \^[D,t]{}\_[C]{} [v]{}\_ =\_ \_[-1]{}(x).\
\_M &=s(s-4m\^2), \^[M,t]{}\_[C]{} [v]{}\_ =\_ \_[-4]{}(x).
3\) Pseudo-vector: $(\bar{\chi} \gamma^\mu \gamma_5 \chi) (\bar{\psi} \gamma_\mu \gamma_5\psi)$, $(\bar{\chi} \gamma^\mu \gamma_5 \chi) (\bar{\psi} \gamma_\mu \psi)$.
\_D &= s(4s-7m\^2), \^[D,t]{}\_[PV]{} [v]{}\_ =\_ \_[-]{}(x).\
\_M &=s(7s-16m\^2), \^[M,t]{}\_[PV]{} [v]{}\_ =\_ \_[-]{}(x).
![The thermal function (\[F\_k\]) for the interactions and annihilation cross sections considered in the text.[]{data-label="Fig2"}](Fk.eps)
The thermal functions corresponding to the previous cases are shown in Figure \[Fig2\] where the asymptotic behaviours are clearly seen. In particular we note that
\_0 (x)&= (8 + 5 + 3),\
\_[-4]{}(x)&= (- + ),
behave in the nonrelativistic limit as
\_0 (x) \~1 +(x\^[-2]{}), \_[-4]{}(x) \~ +(x\^[-2]{}).
The function $\mathcal{F}_0(x)$, which appears in the $s$-channel annihilation through a pseudoscalar interaction, is the only case where the term of order $\mathcal{O}(x^{-1})$ is absent, while $\mathcal{F}_{-4}(x)$, which appears in the scalar and axial-vector $s$-channel annihilation and in the chiral $t$-channel Majorana fermion annihilation, is the only case where the constant $\mathcal{O}(x^{0})$ term is zero. These are the exact temperature dependent factors that correspond to the phenomenological interpolating functions proposed in Ref. [@Drees:2009bi] to model the $s$-wave and $p$-wave behaviour in the nonrelativistic limit. For all other interactions both $s$-wave and $p$-wave contribution are present. The function $\mathcal{F}_{-4}(x)$ can be also read off from the formulas of Ref. [@Claudson:1983js] where the $t$-channel annihilation of Majorana fermions with the exchange of a scalar with chiral couplings was considered.
We note that although we have concentrated on the case of fermion DM, the formula is valid for DM scalar and vector candidates as well, with the necessary redefinition of $\sigma_\Lambda$.
Expansion of the cross section in powers of the relative velocity {#sec3}
=================================================================
In the general case $m_3=m_4=m_\psi\neq 0$ the exact integration is not possible. If the relative velocity of the annihilating particles is small compared with the velocity of light we can work directly with nonrelativistic formulas. The exothermic annihilation cross section in the nonrelativistic limit, to the lowest orders in $v_r$, is usually expanded as $\sigma_{nr} \sim a/v_r +b v_r$, and multiplying by $v_r$,
\_[nr]{} v\_r \~a+b v\^2\_r. \[sigmav\_a\_b\]
Then, using Eq. (\[Maxwell\_vrel\]) and (\[nr\_average\_def\]), the nonrelativistic thermal average of Eq. (\[sigmav\_a\_b\]) is
\_[nr]{} v\_r \_[nr]{} \~a + 6 . \[sigmav\_avea\_a\_b\]
In the case of our cross sections, comparing Eq. (\[sigmav\_avea\_a\_b\]) with Eq. (\[nonrel\]), the coefficients are thus [^1]
a =\_a\_2 (1+), b =-\_a\_2 . \[a\_b\_corr\_gen\]
We now ask, given $\sigma(s)$, how to perform the expansion in terms of the relative velocity to find the coefficients $a$ and $b$ that correspond to the large $x$ expansion of the relativistic thermal average in the COF. Combining equations (\[sigma0ann\]), (\[sigma\]), (\[a\_2\_a\_1\]), the general total annihilation cross section reads
=\_ ( +). \[sigmatot\_k\]
The correct way to proceed is to use the invariant relation Eq. (\[vrel\_s\]) with $m_1= m_2=m$ and to solve it for $s$ as a function of $v_{\text{rel}}$:
s=2m\^2 (1+). \[s\_vrel\]
This formula is valid in every frame and substituted in Eq. (\[sigmatot\_k\]) gives the exact dependence of the cross section on the relativistic relative velocity, $\sigma(v_\text{rel})$. Then, if $v_\text{rel}\sim v_r \ll 1$, we can expand the obtained expression to the desired order in $v_r$ and the nonrelativistic average taken using Eq. (\[nr\_average\_def\]) will coincide with the expansion of Eq. (\[sigmav\_ann\_simple\]) for $x\gg 1$, that is the expansion (\[nonrel\]).
Equivalently, in order to find the expansion (\[sigmav\_a\_b\]), we note that the squared roots in the annihilation cross section (\[sigmatot\_k\]) imply that a term of order $v^4_r$ in $s$ will contribute to the order $v^2_r$ in $\sigma$. Thus we need to expand $s$, formula (\[s\_vrel\]), at least to order $v^4_r$,
s \~4m\^2+m\^2 v\^2\_r + m\^2 v\^4\_r. \[s\_expansion\_vrel\]
Substituting Eq. (\[s\_expansion\_vrel\]) in Eq. (\[sigmatot\_k\]) and performing the expansion in powers of $v_r$ it easy to find
\_[nr]{} v\_[r]{} \~\_ (1+-v\^2\_[r]{} ), \[expansion\_\_sigmatot\_k\]
in agreement with (\[a\_b\_corr\_gen\]).
In the case of coannihilations [@Griest:1990kh], for example when a DM particles scatter off another particle with different mass, the Mandelstam invariant takes the form
s=(m\_1 -m\_2)\^2+ 2m\_1 m\_2 (1+), \[s\_vrel\_coan\]
with the expansion
s \~(m\_1-m\_2)\^2+m\_1 m\_2 v\^2\_r + m\_1 m\_2 v\^4\_r. \[s\_expansion\_vrel\_coan\]
This procedure gives the correct expansion in the COF where the velocities $\boldsymbol{v}_{1,2}$ of the colliding particles are specified in this frame. Clearly, the same expansion with the same coefficients is obtained in the LF and in the CMF.
The problems with the Møller velocity {#sec4}
=====================================
The simple outlined procedure has not been recognized in DM literature where, incorrectly, the Møller velocity $\bar{v}$, Eq. (\[v\_moller\]), instead of $v_\text{rel}$ is considered. As reminded in \[app:2\], $\bar{v}$ is a non-invariant, non-physical velocity. The expression of $\bar{v}$ in terms of $s$ is thus different in different frames and the expansion of $\sigma$ takes different values in different frames.
Before discussing the problems with the Møller velocity we note that if we take the limit $m_f\to 0$ in the analogous expansions published many papers [@Beltran:2008xg_vs2; @Cao:2009uw_vs2; @Buckley:2011kk_vs2; @Matsumoto:2014rxa_vs2; @Fedderke:2014wda_vs2_Moll; @Chen:2013gya_vs2_Moll; @Berlin:2014tja_vs2_error; @Chang:2013oia_vs2], we do not reproduce the expansion (\[expansion\_\_sigmatot\_k\]). The reason is that in these papers the expansion of $s$ is truncated to the lowest order in $v^2_{r}$,
s \~4m\^2 + m\^2 v\^2\_[r]{}.
If we substitute this in Eq. (\[sigmatot\_k\]) and expand, we find
\_[nr]{} v\_[r]{} \~\_(1++v\^2\_[r]{} ), \[sv\_incorretta1\]
with an incorrect coefficient $b$. Clearly the same wrong result is obtained truncating (\[s\_expansion\_vrel\]) to order $v^2_r$, whatever the frame in which $v_r$ is specified, CMF, LF or COF.
We now go back to the Møller velocity (\[v\_moller\]). Evaluated in the CMF taking $m_1 =m_2 =m$ reads
|[v]{}\_\* = . \[v\_moller\_cm\]
We indicate the quantities evaluated in the CMF with a “\*”. By inverting Eq. (\[v\_moller\_cm\]) we find
s\_\* =. \[s\_vr\_cmf\]
This relation is different from (\[s\_vrel\]) and is often incorrectly identified as the relation between $s$ and the relative velocity in the CMF, see for example [@Griest:1990kh], [@Berlin:2014tja_vs2_error]. In facts, the expansion to order $\mathcal{O}(v^4_{r,*})$ reads
s\_\* \~4m\^2 + m\^2 v\^2\_[r,\*]{}+ v\^4\_[r,\*]{}. \[s\_vr\_cmf\_1\]
When used in (\[sigmatot\_k\]), it gives the following nonreltivistic expansion of the cross section
\_[nr]{} v\_[r,\*]{} \~\_(1++v\^2\_[r,\*]{} ), \[sv\_incorretta2\]
which is different from the correct expansion (\[expansion\_\_sigmatot\_k\]).
Other authors, [@GG] and [@Zheng:2010js_vl; @Dreiner:2012xm; @Blumenthal:2014cwa_vl; @Desimone_moller_frame], perform the expansion with the Møller velocity evaluated in the rest frame of one particle. Indicating with “$\ell$” the quantities in this frame, Eq. (\[v\_moller\]) becomes
|[v]{}\_=, \[v\_moller\_lab\]
and by inverting Eq. (\[v\_moller\_lab\]) we obtain
s\_=2m\^2 (1+).
This expression is formally identical to Eq. (\[s\_vrel\]), thus when $\bar{v}_\ell \sim v_{r,\ell}$ and $s_\ell$ is expanded up to the order $v^4_{r,\ell}$ we obtain the expansion $\sigma_{nr} v_{r,\ell}$ which formally coincides with Eq. (\[expansion\_\_sigmatot\_k\]), with $v_{r,\ell}$ in place of $v_r$.
It should be clear that this is just a mathematical coincidence due to the fact that $\bar{v}$ reduces to $v_\text{rel}$ only when one of the two velocities $\boldsymbol{v}_{1,2}$ is zero as it is evident from the definitions Eq. (\[v\_rel\_Rel\_def\]) and Eq. (\[v\_moller\]). In other words, the expansion found in Refs. [@Zheng:2010js_vl; @Dreiner:2012xm; @Blumenthal:2014cwa_vl; @Desimone_moller_frame] are correct because the authors have implicitly used the relative velocity, Eq. (\[vrel\_s\]) and (\[s\_expansion\_vrel\]).
We thus emphasize some common statements found in DM literature and why they do not subsist:\
1) *In the relativistic Boltzmann equation the $v$ in $\sigma v$ is $\bar{v}$ and $\langle \sigma v\rangle$ must be calculated in the LF frame.*\
This is not true, as shown in details in Ref. [@Cannoni:2013bza] and in \[app:2\]. Using $v_\text{rel}$ and recognizing the nonphysical nature of $\bar{v}$, one works always with invariant quantities and the consistency of the relativistic and nonrelativistic formulas and expansions is obtained in the comoving frame without any further specification of the kinematics. The LF, also called Møller frame in Ref. [@Desimone_moller_frame], cannot be a privileged frame for the relic abundances calculation also because for massless particles the rest frame does not exist.\
2) *The Møller velocity coincides with relative velocity in a frame where the velocities are collinear.*\
This not true because, for example, in the CMF where the particles have velocities $v_*$, the Møller velocity is $2v_*$ while the relative velocity is $2v_* /(1+v^2_*)$. Note that the true relative velocity is never superluminal.
Impact on the relic abundance {#sec5}
==============================
Only in the case $k=-4$ the incorrect expansions (\[sv\_incorretta1\]) and (\[sv\_incorretta2\]) coincide, incidentally, with the expansion (\[expansion\_\_sigmatot\_k\]). While the lowest order coefficient $a$ turns out to be always the same, the coefficient $b$ is different in any other case. To illustrate the impact of $b$ on the value of the relic abundance we consider the case of the $s$-channel annihilation with vector interaction, Eq. (\[V\_s\]), and the $s$-channel annihilation with a pseudoscalar exchange, Eq. (\[PS\_s\]). In the first case $k=2$, $a_2 =8/3$, and the correct coefficients $a$ and $b$ are
a\_V= 4 \_, b\_V=- , \[a\_b\_V\_corr\]
while the incorrect coefficient $b$ in (\[sv\_incorretta1\]) and (\[sv\_incorretta2\]) is
b\_[V\_1]{}= \_, b\_[V\_2]{}= \_. \[b\_V\_wrong\]
In the second case, $k=0$ and $a_2 =2$, thus
a\_[PS]{}= 2 \_, b\_[PS]{}=0, \[a\_b\_PS\_corr\]
and the wrong $b$ coefficients are
b\_[PS\_1]{}= \_, b\_[PS\_2]{}= \_. \[b\_PS\_wrong\]
We calculate the relic abundance following the exact theory of freeze out presented in Ref. [@Cannoni:2014zqa]. We briefly recall the main points. Let $Y_{0}=45/(4 \pi^4)(g_\chi /g_s)
x^2 K_2 (x)$ be the initial equilibrium abundance (number density over the entropy density), with $g_\chi =2$ for spin 1/2 fermions and $g_s$ the relativistic degrees of freedom associated with the entropy density. The function $Y_1(x)$ that gives the abundance up to the point $x_*$ where $Y_1(x)-Y_0 (x)$ is maximal is
Y\_1(x)&=(1+(x))Y\_0 (x), \[Y1\]\
(x)&= -1, \[delta\]
with $x_\ast$ given by the condition
-= . \[freezeoutcondition\]
The abundance at $x>x_\ast$ is found by integrating numerically the usual equation
= (Y\^2\_[0]{}-Y\^2), \[Beq\]
with the initial condition ($x_*$, $Y(x_*)=Y_1 (x_*)$). The factor $C$ is defined by $C =\sqrt{\frac{\pi}{45}} M_P m_\chi \sqrt{g_*}$, where $M_P$ is the Plank mass and $\sqrt{g_*} ={g_s}/ {\sqrt{g_\rho}} (1+{T}/{3}\, d(\ln g_s )/dT)$ accounts for the temperature dependence of the relativistic degrees of freedom associated with the energy density, $g_\rho$, and $g_s$ [@SWO; @GG]. For WIMP masses larger than 10 GeV we can neglect the temperature dependence of the degrees of freedom [@Steigman:2012nb; @Drees:2015exa] and take $g_s =g_\rho=g=100$, $\sqrt{g_*}=\sqrt{g}$. In solving numerically (\[Beq\]) and (\[freezeoutcondition\]) with the exposed method, we use the exact formula for $\langle \sigma v_\text{rel}\rangle$, Eq. (\[sigmav\_ann\_simple\]).
We compare the previous numerical solution with the one obtained using the nonrelativistic freeze out approximation (FOA) that is commonly employed in literature. The FOA consists in integrating equation (\[Beq\]) with an initial condition ($x_f$, $Y(x_f)$) such that the equilibrium term proportional $Y^2_{0}$ can be neglected. We choose the freeze out point at the point $x_2$ where $Y(x_2)\simeq Y_1(x_2)=2Y_0 (x_2)$. As shown in Ref. [@Cannoni:2014zqa], $Y_1 (x)$ well approximates the true abundance also in the interval $x_* < x < x_2 $. $x_2$ is the optimal point for the FOA and corresponds to the temperature where the extent of the inverse creation reaction $\psi\bar{\psi}\to \chi\chi$ is maximal. The solution in the freeze out approximation is then
Y\_[FOA]{}=. \[Y2inf\]
The freeze out point $x_2$ is given by the condition $-\frac{1}{Y_0} \frac{dY_0}{dx}\\=3 \frac{C}{x^2} \langle \sigma v_\text{rel} \rangle Y_0$, which, in terms of the method of Ref. [@Scherrer:1985zt] corresponds to $c(c+2)=3$, that is $c=1$. Using the nonrelativistic form of $Y_0$,
Y\_[0]{}= x\^[3/2]{} e\^[-x]{}, \[Y\_0\]
$x_2$ is given by the root of
3 [C]{} (a+6 )x\^[-1/2]{}e\^[-x]{}=1. \[x2mio\_general\]
Calling $\alpha=3a C \sqrt{\pi/2} $, an accurate analytical approximate solution of Eq. (\[x2mio\_general\]) is given by
x\_2 = -() +(1+()\^[-1]{}). \[x2\]
![Ratio of the relic abundance obtained by solving numerically equation (\[Beq\]) over the value given by the freeze out approximation, for the pseudoscalar and vector interactions. In the bottom blue curves for the FOA the correct coefficients (\[a\_b\_V\_corr\]) and (\[a\_b\_PS\_corr\]) are used. The red and the black curves show the effect of the wrong coefficients (\[b\_V\_wrong\]) and (\[b\_PS\_wrong\]), respectively. []{data-label="Fig3"}](omega.eps)
The relic abundance normalized over the critical density is $\Omega h^2 = 2.755\times 10^8 (m/\text{GeV}) Y_{(\infty)}$ for a Majorana fermion and two times that quantity for a Dirac fermion with the same density of antiparticles. We now compare the exact relic abundance $\Omega h^2$ with the value $(\Omega h^2)_{FOA}$ furnished by the nonrelativistic FOA calculated using the correct and the wrong expansions. We take the couplings $\lambda_{a,b}=1$ for illustrative purposes and two values of the cut off scale, $\Lambda=1,10$ TeV. The value of the freeze out points $x_\ast$ and $x_2$ varies roughly between 18 and 30 in the parameter space with $m<\Lambda$ where the effective treatment is supposed to be valid. The ratio $\Omega h^2 / (\Omega h^2)_{FOA}$ is shown in Figure \[Fig3\] as a function of the DM mass for the chosen examples. The bottom blue curves show that the FOA with the correct coefficients (\[a\_b\_V\_corr\]) and (\[a\_b\_PS\_corr\]) underestimates the numerical value by less than 2%, and that in most part of the parameter space the error is at the level of 1% or less. This a test of goodness for our FOA, and confirms what shown in Ref. [@Cannoni:2014zqa]. The red and the black curves show the effect of the wrong coefficients (\[b\_V\_wrong\]) and (\[b\_PS\_wrong\]), respectively. The wrong expansions underestimate the relic abundance by a factor between $3\%$ and $12\%$ for both interactions for masses larger than 10 GeV as shown in the plot. The behaviour is similar for the other interactions not shown in figure. The error becomes even larger at smaller masses and we have verified that using for example $c=1/2$ and other values we get even worst approximations. Clearly this kind of error nowadays is not compatible with the precision with which the experimental value is known.
This work was supported by a grant under the MINECO/FEDER project: SOM: Sabor y origen de la Materia (CPI-14-397). The author acknowledges Roberto Ruiz de Austri, Nuria Rius and Pilar Hernandez for hospitality and for useful discussions at the Instituto de Fisica Corpuscolar (IFIC) in Valencia where part of this work was done.
Integrals and expansions {#app:1}
=========================
Equation (\[sigmav\_y\]) can be written as
\_ =\_ (a\_2 \_2 + \_1 + \_0). \[Ap2\_1\]
The integrals are evaluated with methods similar to those described in Ref. [@Cannoni:2013bza] in terms of Bessel functions of the second kind:
\_0&=\_[1]{}\^ [dy]{} K\_1 (2x) = K\^2\_1(x),\
\_1 &=\_[1]{}\^ [dy]{} y K\_1 (2x) = ,\
\_2 &=\_[1]{}\^ [dy]{} y\^2 K\_1 (2x) &= \[5 K\^2\_1(x)+ 8 K\^2\_2(x) +3K\^2\_3(x)\].
The expansions at $x\gg 1$ are
\~ 1-+(x\^[-2]{}), \~ 1++(x\^[-2]{}), \[Bessel\]
while for $x\ll 1$ are
\~ +(x\^[3]{}), \~ +(x\^[2]{}). \[Bessel2\]
Invariant formulation using $v_\textbf{rel}$ {#app:2}
============================================
In this Appendix we remind, based on the results of Ref. [@Cannoni:2013bza], the main points about the relation between the relative velocity, the Møller velocity, flux and thermal average which are used in the main text.
Invariant relative velocity
----------------------------
The relativistic relative velocity that generalizes the nonrelativistic relative velocity
v\_r=|\_[1]{}-\_[2]{}|, \[v\_r\_def\]
is given by
v\_= . \[v\_rel\_Rel\_def\]
We have explicitly written the dependence on the velocity of light $c$ to make manifest that $v_{\text{rel}}$ coincide with $v_r$ in the nonrelativistic limit because the scalar and vector products are of order $(v/c)^2$. In the following we go back to natural units.
The relative velocity $v_{\text{rel}}$ can be written using the Mandelstam invariant $s=(p_1+p_2)^2$, where $p_{1,2}$ are the four-momenta, and $\lambda$, the Mandelstam triangular function,
(s,m\^2\_1,m\^2\_2)=\[s-(m\_1 + m\_2)\^2\]\[s-(m\_1 - m\_2)\^2\], \[Mandelstam\_l\]
in a generic frame,
v\_&= \[vrel\_mom\_rep\]\
&=, \[vrel\_s\]
showing its invariant nature.
Flux factor
-----------
Given two bunches of particles with number densities $n_{1,2}$ and velocities $\boldsymbol{v}_{1,2}$ in a generic inertial frame, in nonrelativistic physics the flux is $F_{nr}=n_1 n_2 v_r$. To obtain the relativistic invariant flux that reduces to $F_{nr}$ in the nonrelativistic limit, the easiest way is to consider the 4-currents $J_i=(n_i, n_i \boldsymbol{v}_i)$, thus $$F=(J_1 \cdot J_2) v_\text{rel}=n_1 n_2 (1-\boldsymbol{v}_1 \cdot \boldsymbol{v}_2) v_{\text{rel}}.
\label{invF}$$ Note that the factor $(1-\boldsymbol{v}_1 \cdot \boldsymbol{v}_2)$ that guarantees the Lorentz invariance of the product of the number densities can also be written as
1-\_1 \_2==, \[fac\]
where $\gamma_{{\text{r}}}=1/\sqrt{1-v^2_{\text{rel}}}$ is the Lorentz factor associated with $v_\text{rel}$ and $\gamma_i$ the Lorentz factors associated with $\boldsymbol{v}_i$.
If the element of Lorentz invariant phase space is defined as usual
d=, \[dlips\]
and one particle states for bosons and fermions are normalized to $2E_i$ such that the density per unit volume is $2E_i$, then, using (\[fac\]), the flux (\[invF\]) simplifies to $$F= 4 (p_1\cdot p_2) v_{\text{rel}}.
\label{invF1}$$ Substituting the expression of $v_{\text{rel}}$ in the momentum representation, formula (\[vrel\_mom\_rep\]), in Eq. (\[invF1\]), the scalar product $p_1 \cdot p_2$ cancels out and the standard explicit form is recovered $$F=4{\sqrt{(p_1 \cdot p_2)^2 -m^2_1 m^2_2}}.$$
Cross section and collision integral
------------------------------------
The integrated collision term of the Boltzmann equation, neglecting quantum effects, can be written as,
\^4\_[i=1]{} d \[f\_3 f\_4 W(3,4|1,2) -f\_1 f\_2 W(1,2|3,4)\],
where $W(ij|kl)=(2\pi)^4\delta^4(P_{ij}-P_{kl})
\sum_{s_i, s_f}{|\mathcal{M}_{ij\to kl}|^2}$, and $f_i$ is the phase space distribution.
Using the unitary condition $\int d\tilde{p_3}d\tilde{p_4} W(3,4|1,2)=\int d\tilde{p_3}d\tilde{p_4} W(1,2|3,4)$ to write the collision integral only in terms of the annihilation rate
&\^4\_[i=1]{} d (f\_3 f\_4 -f\_1 f\_2) W(1,2|3,4),
we keep out a statistical factor accounting for the possibility of identical particles.
By definition, the invariant cross section, using the flux in the form (\[invF1\]), is
=dd , \[cross\]
being $g_i=(2s_i+1)$ the spin degrees of freedom.
Assuming as usual that the annihilation products are described by the equilibrium phase space distribution at zero chemical potential $f_{0,i}$, we have $f_3 f_4 =f_{\text{0},3} f_{\text{0},4}=f_{\text{0},1} f_{\text{0},2}$, the last equality following from energy conservation. Hence
\^2\_[i=1]{} (p\_1p\_2)(f\_[,1]{} f\_[,2]{} -f\_1 f\_2) v\_.
The equilibrium phase-space distribution $f_{0,i}$ is related to the number density $n_0$ and to the momentum distribution $f_{0,p}(\boldsymbol{p})$ by ${g_i}/{(2\pi)^3}\,f_{0,i}=n_{0,i} f_{0,p}(\boldsymbol{p})$. Assuming further that the non-equilibrium phase-space function at finite chemical potential $f_i$ remains proportional to the equilibrium momentum distribution by a factor given by the non-equilibrium number density $n_i$, ${g_i}/{(2\pi)^3}\, f_i =n_i f_{0,p}(\boldsymbol{p})$, we obtain
(n\_[0,1]{} n\_[0,2]{} -n\_1 n\_2)v\_. \[Cf\_final\]
When the species 1 and 2 are the same, it takes the usual form $\langle \sigma v_{\text{rel}}\rangle (n^2_0 -n^2) $ with the factor 1/2 cancelled by stoichiometric coefficient appearing in the left-hand side of the complete kinetic equation, see for example Ref. [@Cannoni:2014zqa].
Averaged thermal rate
---------------------
In Eq. (\[Cf\_final\]) the general definition of relativistic thermal averaged rate is
v\_= \^2\_[i=1]{} (p\_1p\_2) f\_[0,p]{}(\_1) f\_[0,p]{}(\_2) v\_. \[svgeneral\]
In the case of the relativistic Maxwell-Boltzmann-Juttner statistics, the momentum distribution is
f\_[0,p]{} ()=e\^[-/T]{},
and as shown in Ref. [@Cannoni:2013bza], the six-dimensional integral on the right-hand side of Eq. (\[svgeneral\]) reduces to
v\_&=\^1\_0 dv\_ (v\_) v\_, \[rel\_everage\_def\]
where the probability distribution of $v_\text{rel}$, for example for $m_1=m_2=m$, is
(v\_)= K\_1 (x). \[P\_v\_rel\]
This is completely analogous to the nonrelativistic case where the probability distribution of $v_r$, for $m_1=m_2=m$, is
P(v\_[[r]{}]{})=x\^[3/2]{} [v]{}\^2\_[[r]{}]{} e\^[-x]{}, \[Maxwell\_vrel\]
and the thermal average reads
\_[nr]{} v\_r\_[nr]{}&=\^\_0 dv\_r P(v\_r) \_[nr]{} v\_r. \[nr\_average\_def\]
Given the total annihilation cross section $\sigma$ the product $\sigma v_{\text{rel}}$ will reduce to the nonrelativistic limit $\sigma_{nr} v_r$ and $\langle \sigma v_{\text{rel}} \rangle$ to $\langle \sigma_{nr} v_r\rangle_{nr}$ in the COF when $v_\text{rel}\sim v_r \ll 1$. Expressing Eq. (\[rel\_everage\_def\]) in terms of $s$ using (\[vrel\_s\]), we obtain the usual integral [@Edsjo:1997bg; @Cannoni:2013bza] useful for practical calculation
\_ [v]{}\_ &= &\^\_[M\^2]{} ds K\_1 () , \[sigmav\_1\]
with $x_i=m_i/T$ and $M=(m_1+m_2)$.
We have recently become aware of the paper [@weaver] where, probably for the first time, the thermal average of relativistic rates was discussed and it was realized that with the relativistic Maxwell-Boltzmann statistics formula (\[svgeneral\]) reduces to a single integral over the distribution over the relative momentum. With some algebra and change of variables it is easy to verify that for example Eqs. (11b) and (12a) of [@weaver] coincide with Eqs. (29) and (37) of Ref. [@Cannoni:2013bza]. In Ref. [@weaver] the cases of collisions of two massive particles, two massless particles and a massive with a massless particles are treated separately as if different definitions of flux and cross sections were necessary in each case. Clearly this distinction is unnecessary for the formulation we have given is completely general and valid in any case. We finally note that an integral formula similar to (\[sigmav\_1\]) was also given in Ref. [@Claudson:1983js].
No need for the Møller velocity.
--------------------------------
By noting that in Eq. (\[invF\]) the factor $(1-\boldsymbol{v}_1 \cdot \boldsymbol{v}_2)$ can cancel the same factor in the denominator of $v_\text{rel}$, the invariant flux can also be written in the form
F=n\_1 n\_2 . \[flux\_moller\]
In the textbook by Landau and Lifschits [@landau] this form is attributed to Pauli without giving any reference, while its origin is more generally attributed to Møller [@moller].
It is interesting to look at original paper by Møller [@moller]. With our notation, he wants to prove that the flux given (\[flux\_moller\]) is invariant. In order to do that he shows that this can be written as a product of two invariant quantities: the ratio $\frac{n_1
n_2}{E_1 E_2}$ and the quantity $B=\sqrt{(p_1 \cdot p_2)^2 -m^2_1 m^2_2}$ and there he stops.
The flux factor written in the form (\[flux\_moller\]) has the same structure of thee nonrelativistic expression $n_1 n_2 v_r$. Probably for this reason it has been later introduced in the literature the notion of Møller velocity
|[v]{}&= =. \[v\_moller\]
It is worth to stress that neither Møller nor Landau and Lifschits attribute any particular meaning to Eq. (\[v\_moller\]) and do not define it as a particular velocity, even less as relative velocity. Clearly $\bar{v}$ is nothing but the numerator of the formula defining $v_{\text{rel}}$ because $\bar{v}=(1-\boldsymbol{v}_1 \cdot \boldsymbol{v}_2)v_\text{rel}$, where the factor $(1-\boldsymbol{v}_1 \cdot \boldsymbol{v}_2)$ comes from the definition of the invariant flux (\[invF\]). Already this fact indicates that $\bar{v}$ is not a fundamental physical quantity and overall, it is not the relative velocity, nor when the velocities are collinear.
On the contrary, in DM literature and in textbooks, when defining the flux factor for the relativistic invariant cross section, it is incorrectly asserted that in a frame where the velocities are collinear the quantity $|\boldsymbol{v}_{1}-\boldsymbol{v}_{2}|$ is the relative velocity, while in a generic frame is given by (\[v\_moller\]). The form (\[flux\_moller\]) of the flux is a simple consequence of the fundamental quantities (\[v\_rel\_Rel\_def\]) and (\[invF\]), there is no new physics or concept in it. For these reasons, and for its noninvariant and nonphysical nature, $\bar{v}$ should not be used.
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[^1]: This result must coincide with the expansion of Ref. [@SWO; @Cannoni:2013bza]. With our notation the expansion is
\_[nr]{} v\_r \_[nr]{}\~\_0|\_[y=1]{} +(-\_0|\_[y=1]{}+’\_0|\_[y=1]{}),
where the prime indicate derivative respect to the variable $y$. Comparison with the expansion (\[sigmav\_avea\_a\_b\]) requires to identify
a\_0|\_[y=1]{}, b( -\_0|\_[y=1]{}).
Using Eq. (\[sigma0ann\]) with $a_0 =0$ and $a_2 =k a_1$, it is easy to verify that one obtains again Eq. (\[nonrel\]).
|
---
abstract: 'Reynolds-averaged Navier–Stokes (RANS) simulations with turbulence closure models continue to play important roles in industrial flow simulations as high-fidelity simulations are prohibitively expensive for such flows. Commonly used linear eddy viscosity models are intrinsically unable to handle flows with non-equilibrium turbulence (e.g., flows with massive separation). Reynolds stress models, on the other hand, are plagued by their lack of robustness and the poor model conditioning. Recent studies found that even substituting Reynolds stresses from DNS databases (with errors below 0.5%) into RANS equations leads to very inaccurate velocities (e.g., with an error of 35% for plane channel flows at frictional Reynolds number $Re_\tau=5200$). This observation suggests that RANS equations with Reynolds stress closure are *ill-conditioned* for some flows. This observation is not only disturbing for the recently emerging data-driven Reynolds stress models but also relevant for traditional, equation-based models. Our work shows that the ill-conditioning cannot be explained by the global matrix condition number of the discretized RANS equations. As such, we propose a metric based on local condition number function for *a priori* evaluation of the conditioning of RANS equations. Numerical tests on turbulent channel flows at various Reynolds numbers suggest that the proposed metric can adequately explain observations in previous studies, i.e., deteriorated model conditioning with increasing Reynolds number, and better conditioning of the implicit treatment of Reynolds stress compared to the explicit treatment.'
address:
- 'Kevin T. Crofton Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 24060, USA'
- 'Department of Aeronautics and Astronautics, MIT, Cambridge, MA 02139, USA'
author:
- Jinlong Wu
- Heng Xiao
- Rui Sun
- Qiqi Wang
title: 'RANS Equations with Reynolds Stress Closure Can Be Ill-Conditioned'
---
turbulence modeling ,model conditioning ,Reynolds stress transport model ,implicit scheme
Introduction
============
Reynolds-averaged Navier–Stokes (RANS) simulations play an important role in industrial simulations of turbulent flows. The RANS models with linear eddy viscosity assumption (e.g., $k-\varepsilon$, $k-\omega$ and S–A models [@launder74application; @wilcox88reassessment; @spalart92one]) are based on the assumption that the turbulence production and dissipation are in equilibrium, and thus they perform poorly in flows with non-equilibrium turbulence [@speziale96towards; @hamlington08reynolds; @hamlington14modeling], e.g., flows with massive separations or abrupt mean flow changes. On the other hand, Reynolds stress models (RSM) take into account the transport of Reynolds stresses and thus have better performance than eddy viscosity models for flows with non-equilibrium turbulence [@pope00turbulent]. The CFD Vision 2030 white paper of NASA identified advanced turbulence modeling based on Reynolds stresses models as a priority for aeronautic technological advancement in the coming decades [@slotnick14cfd]. However, so far the Reynolds stress models have not been widely used in the engineering applications. A key shortcoming of the Reynolds stress models is the lack of robustness due to their *numerical instabilities* (i.e., difficult to achieve convergence) [@basara03new; @maduta17improved].
Conditioning of RANS Equations with Data-Driven Reynolds Stress Models
----------------------------------------------------------------------
In addition to the relatively well-known stability issue mentioned above, model conditioning is rarely discussed separately. In the context of RANS equations with Reynolds stress closure, *model conditioning* is defined as the sensitivity of the solved quantities (e.g., mean velocity and mean pressure fields) to the modeled terms (Reynolds stresses). Model conditioning provides a measure of amplification of errors in the modeled terms to the solved quantities. Numerical stability and model conditioning are two separate yet closely related issues. Specifically, in solvers that solve RANS equations and turbulence transport equations in a segregated manner, ill-conditioned system would lead to numerical instability due to the sensitivity of solved velocities to residuals in the Reynolds stress from iteration to iteration.
Model conditioning may be even more critical for data-driven Reynolds stress models. Recent works in data-driven turbulence modeling highlighted conditioning as a potential difficulty. This is because some data-driven models do not have explicit expressions for the Reynolds stress [@ling16reynolds; @wang17physics-informed], which make it difficult to treat the Reynolds stresses implicitly in the RANS equations to improve model conditioning. For example, Wang et al. [@wang17physics-informed] used machine learning techniques to predict Reynolds stresses based on existing DNS databases and reported that the solved mean velocity field does not improve over the original RANS simulations, even though the predicted Reynolds stress shows noticeable improvement. Gamahara et al. [@gamahara17searching] proposed a data-driven subgrid-scale stress model in a turbulent channel flow. They reported that the neural network model predicted better subgrid-scale stresses but the less satisfactory mean velocities compared to the predictions of Smagorinsky models. These observations demonstrate the gap between *a priori* and *a posteriori* performances in Reynolds or subgrid-scale stress-based turbulence modeling by using data-driven techniques. That is, an ill-conditioned model can amplify small *a priori* errors in the modeled terms to large *a posteriori* errors in the solved quantities, which would defeating all efforts in improving the closure models. Consequently, efforts must be made to improve the model conditioning in such cases.
Using Reynolds stress obtained from DNS data to solve the RANS equations for velocity can be considered the ideal scenario for data-driven turbulence modeling. Solving for mean velocities with a given Reynolds stress field is referred to as “propagation” in this work, i.e., propagation of Reynolds stresses to mean velocities by solving the RANS equations. Such a methodology represents an upper limit of performances for data-driven Reynolds stress models as pursued in refs. [@ling16reynolds; @wang17physics-informed]. However, even solving RANS equations with Reynolds stresses from DNS data can lead to large errors in the velocities [@thompson16methodology]. Thompson et al. [@thompson16methodology] investigated turbulent channel flows at various Reynolds numbers with such a propagation methodology. These DNS were performed with extreme caution by reputable groups [@lee15direct], and the errors in the reported Reynolds stress are indeed very small (typically less than $0.5\%$; see Table \[tab:summary\]). Thompson et al. [@thompson16methodology] showed that the solved mean velocity has unsatisfactory agreement with the DNS data at high frictional Reynolds numbers (e.g., $Re_{\tau}=5200$). Poroseva et al. [@poroseva16on] also confirmed such observations. As the starting point of our work, we reproduced the two studies of solving for mean velocities by using the DNS Reynolds stresses obtained from Lee and Moser [@lee15direct], and the results are summarized in Table \[tab:summary\]. Although the mean velocities reported here are accurate up to $Re_{\tau}=1000$, researchers have reported that large errors in the propagated velocities can be found at Reynolds number as low as $Re_{\tau}=395$ depending on the DNS database used [@poroseva18personal].
Frictional Reynolds number ($\mathrm{Re}_{\tau}$) 180 550 1000 2000 5200
--------------------------------------------------- ------- ------- ------- ------- -----------
Error in turbulent shear stresses
*volume averaged* 0.17% 0.21% 0.03% 0.15% 0.31%
*maximum* 0.43% 0.38% 0.07% 0.23% 0.41%
Errors in mean velocities
*volume averaged* 0.25% 1.61% 0.17% 2.85% **21.6**%
*maximum* 0.36% 2.70% 0.25% 5.48% **35.1**%
: Summary of the results of the channel flow test, showing errors in the turbulent shear stresses and the propagated mean velocities. The large errors in the high Reynolds number case $\mathrm{Re}_{\tau}=5200$ is highlighted. The true Reynolds stress is obtained by analytical integration of the RANS equation with the DNS mean velocity [@thompson16methodology].
\[tab:summary\]
The results in Table \[tab:summary\] raise several critical questions on Reynolds stress based turbulence modeling:
(1) How to explain the deteriorated conditioning (i.e., increased amplifications of errors in Reynolds stresses to velocities) with increasing Reynolds numbers observed in these studies? Is there a quantitative metric that can characterize the conditioning of a given turbulence model?
(2) What are the implications of this observation to traditional and data-driven turbulence models?
Our present work aims to answer these questions by proposing a quantitative metric and elucidating the relevance of the above-mentioned studies to both data-driven and traditional turbulence modeling. No turbulence models are used. Rather, Reynolds stresses obtained from DNS database are used to represent the ideal performance for any turbulence model of Reynolds stresses for both explicit and implicit treatments.
Relevance of present work to PDE-Based Reynolds stress models
-------------------------------------------------------------
The conditioning issue is an equally important challenge for traditional Reynolds stress models based on Reynolds stress transport equations.[^1] Although solving a monolithic system of Reynolds stress transport equations and RANS equations would allow for implicit treatment and thus enhance model conditioning, it is uncommon due to increased computational costs. Other caveats are further discussed in Section \[sec:discuss-monolithic\]. As of now, most open-source and commercial general-purpose CFD packages (e.g., [@weller98tensorial]) still solve the turbulence transport equations and the RANS mean flow equations in a segregated manner, even in solvers where velocity and pressure are solved concurrently. In such solvers, the modeled Reynolds stress are often updated with the mean velocity field at every iteration step, in the hope that the mean velocity field and the Reynolds stress can consistently adjust to each other during the iterations. However, the conditioning issue within each iteration can lead to error amplification if the Reynolds stresses are treated explicitly as source terms in the RANS equations. Specifically, a small error in the modeled Reynolds stress would lead to large errors in the solved mean velocity field, which is carried over to the Reynolds stresses and further amplified in the next iteration step. Such an error amplification destabilizes the solution procedure and leads to divergence.
For the reasons outlined above, RANS simulations with Reynolds-stress-based turbulence models need to be stabilized to increase the robustness of the solvers. Examples of stabilization include (1) using the velocity solved with eddy viscosity models as an initial condition for iterations in the RSM-based solver and (2) partial implicit treatment of the Reynolds stress, among others. In the latter category, researchers introduced a hybrid scheme of computing Reynolds stress by blending the RSM modeled Reynolds stress $\bm{\tau}_{\textrm{RSM}}$ with the Reynolds stress $\bm{\tau}_{\textrm{Boussinesq}}$ computed from eddy viscosity models, with the later stabilizing the solution, i.e., $\bm{\tau}=\gamma \bm{\tau}_{\textrm{RSM}}+(1-\gamma) \bm{\tau}_{\textrm{Boussinesq}}$ [@basara03new; @maduta17improved]. However, the choice of the blending factor $\gamma$ is ad hoc due to the lack of a method to quantitatively evaluate the model conditioning of using RSM. Large weights ($1-\gamma$) on the eddy viscosity model impair the accuracy of the obtained model, while small weights may not provide adequate stabilization. This shortcoming needs to be addressed.
Summary and Novelty of Present Contribution
-------------------------------------------
In this work, we propose a metric to quantify such error amplification from Reynolds stress to mean velocities, i.e., the model conditioning of RANS equations with Reynolds stress closures. We first demonstrate that the traditional condition number based on matrix norms is incapable of explaining the increased errors in solved mean velocities with the increase of Reynolds number. A local condition number function based on Ref. [@chandrasekaran95on] is derived as a more refined metric to assess the conditioning property for turbulence models. We also demonstrate that the proposed metric can also explain the improved model conditioning by introducing an eddy viscosity to implicitly model the linear part of the Reynolds stress, which is a common practice in traditional RANS modeling to enhance the stability and model conditioning of the simulations. Traditionally, the stability and conditioning of Reynolds stress models are improved by empirically blending the modeled Reynolds stresses from RSM and those obtained from linear eddy viscosity models. Therefore, the metric proposed in this work is also of importance in the analysis of model conditioning for PDE-based turbulence modeling approaches, e.g., guiding the choice of blending factor $\gamma$ in the hybrid scheme of traditional Reynolds stress models to enhance numerical stability and model conditioning [@basara03new; @maduta17improved]. Turbulent channel flows with five Reynolds numbers ranging from $Re_\tau = 180$ to 5200 are investigated. The results show that the proposed metric has clear importance in evaluating and improving the conditioning of Reynolds stress models, for both the traditional and data-driven turbulence modeling approaches.
The rest of this paper is organized as follows. Section \[sec:meth\] introduces the global condition number and shows that it fails to explain the deteriorated conditioning with increasing Reynolds numbers. A local condition number function is derived to achieve such an objective. In Section \[sec:results\], the local condition number is used to evaluate the conditioning of RANS simulations by using Reynolds stresses obtained in the context of both data-driven and traditional RANS modeling. Section \[sec:discussion\] discusses the reasons for different propagated mean velocity fields by using explicit and implicit treatment of Reynolds stresses. Finally, conclusions are presented in Section \[sec:conclusion\] .
Methodology {#sec:meth}
===========
Consider the steady state Reynolds-averaged Navier–Stokes equations for incompressible, constant density fluids:
$$\begin{aligned}
\label{eq:ns}
\bm{u} \cdot \nabla \bm{u} - \nu \nabla^2 \bm{u}
+ \nabla p - \nabla \cdot \boldsymbol{\tau} & = 0 \\
\nabla \cdot \bm{u} & = 0 \end{aligned}$$
where $\bm{u}$ is the mean flow velocity; $\nu$ is molecular viscosity; $p$ is the pressure normalized by the constant density of the fluid; $\boldsymbol{\tau}$ is the Reynolds stress tensor, which needs to be modeled. For simplicity we first consider a Reynolds-stress-based model where $\boldsymbol{\tau}$ is obtained by solving a transport equation in a segregated manner with the RANS equations or by a data-driven function (see e.g. [@ling16reynolds]). The objective of this work is to investigate the sensitivity of the obtained mean velocity with respect to small perturbations on the Reynolds stress.
For notation simplicity, we introduce nonlinear operator $\mathcal{N}$ to include the convection and diffusion terms with
$$\mathcal{N}(\bm{u}) =
\bm{u} \cdot \nabla \bm{u} - \nu \nabla^2 \bm{u}$$
The RANS momentum equation in Eq. (\[eq:ns\]) can be written as
$$\label{eq:ns-concise-N}
\mathcal{N}(\bm{u}) = \nabla \cdot \boldsymbol{\tau}-\nabla p$$
In numerical solvers, the convection term is first linearized around the current velocity $\overline{{\bm{u}}}_{0}$ to obtain the linearized RANS equations:
$$\label{eq:ns-concise-L}
\mathcal{L}(\bm{u}) = \nabla \cdot \boldsymbol{\tau}-\nabla p$$
where $\mathcal{L}$ is the linearized operator of $\mathcal{N}$, i.e.,
$$\label{eq:L-definition}
\mathcal{L}(\bm{u}) =
\bm{u}_0 \cdot \nabla \bm{u} - \nu \nabla^2 \bm{u}$$
The linearized equation (\[eq:ns-concise-L\]) is then discretized on a CFD mesh to obtain a linear system of the following form:
$$\label{eq:ns-matrix}
\mathsf{A} \; {\bm{U}} = \bm{b}$$
where we denoted ${\bm{b}} = {[\nabla\cdot\boldsymbol{\tau} - \nabla p]}$ as the imbalance between the two forces, pressure gradient and Reynolds stress divergence; ${\bm{U}} = {[\bm{u}]}$ is the discretized velocity field to be solved for. Both $\bm{b}$ and ${\bm{U}}$ are $n \times 1$ vectors, where $n$ is the number of cells or grid points in the mesh. The matrix $\mathsf{A}$ with dimension $n \times n$ comes from the implicit discretization of the linearized convection term and the molecular diffusion term.
Matrix-norm as a measure of model conditioning {#sec:global-cn}
----------------------------------------------
We first show the derivation of the traditional matrix-norm-based condition number and explain why it fails to distinguish the sensitivities of solving for mean velocity at different Reynolds numbers as shown in Table \[tab:summary\]. Following the definition of matrix norm, the norm of the error in the velocity is bounded as follows[^2]:
$$\begin{aligned}
& \frac{\| \delta {\bm{U}}\|}{\| {\bm{U}}\|}
\le \mathcal{K}_\mathsf{A} \frac{\|\delta \bm{b} \|}{\| \bm{b} \|}
\label{eq:global-cn}\end{aligned}$$
where
$$\mathcal{K}_\mathsf{A} \equiv \|\mathsf{A}\| \|\mathsf{A}^{-1}\|$$
denotes the condition number of matrix $\mathsf{A}$ (see, e.g., [@strang93introduction]). Considering that the objective is to assess the effects of *Reynolds stress perturbation* $\delta \boldsymbol{\tau}$ on the velocities, the inequality in Eq. (\[eq:global-cn\]) above is formulated as follows:
$$\label{eq:cond-norm-derive}
\frac{\|\delta {\bm{U}}\|}{\|{\bm{U}}\|} \le
\mathcal{K}_\tau
\frac{\|\nabla \cdot \delta \boldsymbol{\tau} \|}{\| \nabla \cdot \boldsymbol{\tau} \|} .$$
where
$$\label{eq:cond-norm}
\mathcal{K}_\tau = \mathcal{K}_\mathsf{A} \frac{\|\nabla \cdot
\boldsymbol{\tau}\|}{\|\bm{b}\|}$$
Detailed derivations are omitted here for brevity and are presented in \[app:global-cn\]. It can be seen that the model condition number $\mathcal{K}_\tau$ consists of the condition number of the matrix $\mathsf{A}$ and the ratio
$$\label{eq:alpha}
\overline{\alpha} = \|\nabla \cdot \boldsymbol{\tau}\| / \|\bm{b}\|.$$
For plane channel flows the convective term disappears, and thus $\bm{b}$ is the force due to the divergence of the viscous stress, i.e., $\bm{b} = \nabla \cdot \nu (\nabla \bm{u}+(\nabla \bm{u})^T) =
\nabla \cdot \boldsymbol{\tau}_{vis}$. Consequently, the ratio $\overline{\alpha}$ indicates the overall relative importance of the forces due to Reynolds stress and viscous stress.
The proposed condition number $\mathcal{K}_\tau$ based on matrix condition number $\mathcal{K}_\mathsf{A}$ is a natural first attempt in explaining the increasing sensitivity of the velocities to the Reynolds stress with increasing Reynolds numbers as shown in Table \[tab:summary\]. However, surprisingly it turns out that the condition number $\mathcal{K}_\tau$ is more or less the same across all Reynolds numbers from $Re_\tau = 180$ to $5200$, which is shown in Fig. \[fig:kappa-tau\]. This observation suggests that the matrix-based condition number $\mathcal{K}_\tau$ cannot explain the ill-conditioning of the $Re_\tau = 5200$ case and the better conditioning of the lower Reynolds number cases as observed in Table. \[tab:summary\].
The following two factors explain why the matrix-based condition number $\mathcal{K}_\tau$ is almost the same at different Reynolds numbers:
(1) the matrix condition number $\mathcal{K}_\mathsf{A}$ is constant for all Reynolds numbers, because the matrix $\mathsf{A}$ itself is independent of the Reynolds number.
(2) The ratios $\overline{\alpha} = \frac{\|\nabla \cdot
\boldsymbol{\tau}\|}{\|\bm{b}\|}$ are very similar at vastly different Reynolds numbers, because both norms (which involve integration or square sums) are dominated by the *viscous wall regions* of each flow. It is well-known that the Reynolds number only determines the thickness of this region in outer coordinates, and the Reynolds number effect is weak here.
![Conditioning measure of Reynolds-stress-based turbulence models based on $\mathcal{K}_\mathsf{A}$ and the ratio $\overline{\alpha}$ as defined in Eq. (\[eq:alpha\]). The Reynolds stress is computed from DNS data to study the ideal scenario of the RANS modeling.[]{data-label="fig:kappa-tau"}](kappaTau-legend.pdf "fig:"){width="35.00000%"}\
![Conditioning measure of Reynolds-stress-based turbulence models based on $\mathcal{K}_\mathsf{A}$ and the ratio $\overline{\alpha}$ as defined in Eq. (\[eq:alpha\]). The Reynolds stress is computed from DNS data to study the ideal scenario of the RANS modeling.[]{data-label="fig:kappa-tau"}](kappaTau.pdf "fig:"){width="48.00000%"}
Each factor above will be detailed below. First, it can be established through simple algebra that for plane channel flows the matrix $\mathsf{A}$ is independent of the Reynolds number but depends on the discretization scheme and the mesh used. Since the convection term disappears for plane channel flows, the matrix $\mathsf{A}$ results solely from the discretization of the diffusion operator $\nu \nabla^2 (\cdot)$. When discretized with central difference on a uniform mesh of $n$ cells, matrix $\mathsf{A}$ can be written as follows [@strang93introduction]:
$$\mathsf{A} = \nu
\begin{bmatrix}
\begin{array}{ccccc}
2 & -1 & & & \\
-1 & 2 & -1 & & \\
& -1 & 2 & \ddots & \\
& & \ddots & \ddots & -1 \\
& & & -1 & 2
\end{array}
\end{bmatrix}.
\label{eq:simple-matrix}$$
The condition number for matrix $\mathsf{A}$ is $\mathcal{K}_\mathsf{A} = 4 n^2 / \pi^2$. Therefore, $\mathcal{K}_\mathsf{A}$ does not explicitly depend on the viscosity or the Reynolds number. This analysis is confirmed by the results shown in Fig. \[fig:kappa-tau\], which shows that $\mathcal{K}_\mathsf{A}$ is strictly constant for all five flows at different Reynolds numbers. Moreover, $\mathcal{K}_\mathsf{A}$ depends on the mesh size $n$, which is a critical shortcoming of the matrix-norm based condition number as a measure of the conditioning property of a turbulence model. To exclude the influences of the mesh, we used the same mesh with 1040 cells in all the flows at different Reynolds numbers presented in Fig. \[fig:kappa-tau\].
Second, the change of Reynolds number has little influence on the ratio $\overline{\alpha}$, as is shown in Fig. \[fig:kappa-tau\]. We examine the profile of turbulent shear ($\nabla \cdot \boldsymbol{\tau}$) and viscous shear ($\nabla \cdot \boldsymbol{\tau}_{vis}$) in the channel in Fig. \[fig:force-profiles\] for the two cases, $Re_\tau = 180$ and 5200. In most of the channel outside the viscous wall region, both forces (and thus the ratio) are fairly uniform. Nevertheless, in the viscous wall region, the two forces are of the same order of magnitude but with opposite signs. In contrast, outside the viscous region, the pressure gradient is the main driving force while the Reynolds shear stress is the resistance, with the viscous shear having negligible effects. The forces in the two distinct regions are illustrated schematically in Fig. \[fig:balance\]. However, when calculating the ratio $\overline{\alpha} = \|\nabla \cdot \boldsymbol{\tau}\|/\|\bm{b}\|$ of the two norms, values within the viscous wall region clearly dominates the calculation of both norms, which involve integration of the functions squared. It can be seen that in both Fig. \[fig:force-profiles\]a ($Re_\tau = 180$) and Fig. \[fig:force-profiles\]b ($Re_\tau=5200$) the areas enclosed by the blue/solid curve and red/dashed curve (with the vertical zero line) are similar. Squaring the function would place even more weights on the regions of larger function values, i.e., the viscous wall region. This observation suggests that the ratio $\|\nabla \cdot \boldsymbol{\tau}\|/\|\bm{b}\|$ is of order $O(1)$ for both cases, as confirmed by examining Fig. \[fig:kappa-tau\]. Consequently, the computed norm mostly reflects the values of the forces in the viscous region and not the outer layer. It is well known that the Reynolds number effects are not pronounced within the viscous wall region. Increasing the Reynolds number merely extends the outer layer in terms of inner coordinates ($y^+ = y/y^*$ where $y^*=\nu/\sqrt{\tau_w/\rho}$ is the viscous unit and $\tau_w$ is the wall shear stress). This explains why the ratio $\overline{\alpha}$ does not vary significantly (much less than proportionally) with the Reynolds number as can be seen in Fig. \[fig:kappa-tau\].
![The balance among forces due to turbulent shear stress ($\nabla \cdot \bm{\tau}$), viscous shear stress ($\nabla \cdot \bm{\tau}_{vis}$), and pressure gradient ($\nabla p$) for two plane channel flows at frictional Reynolds numbers (a) $Re_\tau=180$ and (b) $Re_\tau=5200$. The right vertical axis denotes the inner coordinates ($y^+$).[]{data-label="fig:force-profiles"}](force-legend.pdf "fig:"){width="50.00000%"}\
![Force balance of the plane channel flow in (a) the outer layer and (b) the viscous wall region.[]{data-label="fig:balance"}](force-balance.pdf){width="95.00000%"}
In summary, the matrix-based condition number $\mathcal{K}_\tau$ as derived in Eq. (\[eq:cond-norm\]) is not able to explain the increasing sensitivity of the velocities with respect to Reynolds stresses with increasing Reynolds number. In addition, the matrix condition number $\mathcal{K}_\mathsf{A}$ has another critical drawback of being mesh dependent. The mesh dependency is highly undesirable as the condition number is to measure the conditioning property of *turbulence models* at the PDE level, not any particular numerical discretization thereof. These drawbacks clearly call for a better metric for measuring the conditioning property of Reynolds-stress-based turbulence models.
Proposed metric as a measure of model conditioning {#sec:local-cn}
--------------------------------------------------
In order to address the deficiency of the global condition number as presented in Section \[sec:global-cn\], we derive a metric based on a local condition number function to measure the sensitivity of the solved mean velocity $\bm{u}$ at a given location $\bm{x}$ with respect to perturbation $\delta {\boldsymbol{\tau}}$ on the Reynolds stresses field $\bm{\tau}$. Such a local condition number is formally defined as the following bound:
$$\begin{aligned}
\frac{|\delta \bm{u}({\bm{x}})|}{U_\infty}
& \le
\mathcal{K}({\bm{x}}) \,
\frac{{\left\Vert \nabla \cdot \delta {\boldsymbol{\tau}} \right\Vert_{\Omega}}}
{{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}}
\label{eq:cond-derive}\end{aligned}$$
where $\mathcal{K}({\bm{x}})$ is the local condition number function defined as:
$$\label{eq:cont-cond}
\mathcal{K}({\bm{x}}) =
\frac{{\left\Vert G({{\bm{x}}}, \bm{\xi}) \right\Vert_{\Omega}} \,
{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}}{U_\infty}$$
Function $G$ is the Green’s function corresponding to the linear operator $\mathcal{L}$ (see e.g., [@lanczos96linear]), such that the solution to the linearized RANS equation (\[eq:ns-concise-L\]) can be written formally as:
$$\label{eq:define-green}
\bm{u}({\bm{x}}) = \mathcal{L}^{-1} [{\bm{b}}({\bm{x}})]
\equiv \int_\Omega G({\bm{x}}; {\boldsymbol{\xi}}) \, {\bm{b}}({\boldsymbol{\xi}}) \, d{\boldsymbol{\xi}}$$
where $\mathcal{L}^{-1} [\cdot]$ is the inverse operator of $\mathcal{L}$. Green’s function $G({\bm{x}}; {\boldsymbol{\xi}})$ indicates the contribution of the source ${\bm{b}}({\boldsymbol{\xi}})$ at location ${\boldsymbol{\xi}}$ to the solution $\bm{u}$ at location ${\bm{x}}$. The norm ${\left\Vert f({\boldsymbol{\xi}}) \right\Vert_{\Omega}}$ of function $f({\boldsymbol{\xi}})$ is an integration on domain $\Omega$ defined as [@debnath05hilbert]:
$$\label{eq:function-norm-def}
{\left\Vert f({\boldsymbol{\xi}}) \right\Vert_{\Omega}}=\sqrt{\int_\Omega |f({\boldsymbol{\xi}})|^2 d {\boldsymbol{\xi}}} \; .$$
The detailed derivations to obtain Eq. (\[eq:cont-cond\]) are presented in \[sec:app-local-cn\].
For functions discretized on a CFD mesh of $n$ cells (e.g., those in RANS simulations), the function norm $\| \cdot \|_\Omega$ in Eq. (\[eq:cont-cond\]) can be approximated by the norm of the discretized $n$-vector through numerical quadrature. That is,
$${\left\Vert G(\bm{x}, \bm{\xi}) \right\Vert_{\Omega}} \approx {\left\Vert {{\bm{r}}}_j \right\Vert_{n}}
\qquad
\text{and}
\qquad
{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}} \approx {\left\Vert [\nabla \cdot {\boldsymbol{\tau}}] \right\Vert_{n}}$$
where ${\bm{r}}_{j}$ is the $j$-th row of the matrix $\mathsf{A}^{-1}$. Recall that $[\nabla \cdot {\boldsymbol{\tau}}]$ indicates discretization of field $\nabla \cdot {\boldsymbol{\tau}}$ on the CFD mesh, but the bracket can be omitted inside a vector norm ${\left\Vert \cdot \right\Vert_{n}}$ without ambiguity. The discretized condition number $n$-vector corresponding to $\mathcal{K}(\bm{x})$ in Eq. (\[eq:cont-cond\]) is thus:
$$\label{eq:cont-cond-disc}
\mathcal{K}_j =
\frac{{\left\Vert {\bm{r}}_{j} \right\Vert_{n}} \,
{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{n}}}{U_\infty}
\qquad \text{with} \quad j = 1, 2, \cdots, n$$
which implies that the location ${\bm{x}}$ is the coordinate of the $j$-th cell in the CFD mesh.
The proposed local condition number function $\mathcal{K}(\bm{x})$ has two important merits compared to the global matrix based condition number $\mathcal{K}_\tau$:
(1) First, $\mathcal{K}(\bm{x})$ provides a tighter upper bound than the matrix-norm condition number $\mathcal{K}_{\tau}$. The main reason is that the upper bound of $\mathcal{K}_{\tau}$ can only be achieved when the following conditions are satisfied simultaneously: (1) the discretized mean velocity field vector $\bm{U}$ is aligned with the principal axis of the coefficient matrix $\mathsf{A}$, and (2) that the perturbation vector $\delta \bm{b}$ is aligned with the principal axis of $\mathsf{A}^{-1}$. In contrast, the derivation of $\mathcal{K}(\bm{x})$ does not assume any conditions on the discretized mean velocity field $\bm{U}$. Consequently, the bound provided by $\mathcal{K}(\bm{x})$ is a more precise assessment of the sensitivity $\delta \bm{u}$ with respect to Reynolds stress perturbations.
(2) Moreover, the discretization $\mathcal{K}_j$ of function $\mathcal{K}(\bm{x})$ is mesh indepedent, which is an important property considering that this metric aims to measure the conditioning property of Reynolds stress models. Derivations to obtain Eq. (\[eq:cont-cond-disc\]) and more discussions on the mesh independency of the local condition number $K_j$ are presented in \[sec:discretization\].
As the local condition number $\mathcal{K}(\bm{x})$ is a spatial function and its discretization an $n$-vector, it is desirable to obtain a scalar quantity to provide an integral, more straightforward measure of model conditioning property similar to the global condition number $\mathcal{K}_{\tau}$. To this end, we define a *volume-averaged* local condition number $\overline{\mathcal{K}}_{\bm{x}}$ defined in Eq. (\[eq:avg-kappa\]).
$$\overline{\mathcal{K}}_{\bm{x}} =
\frac{\sum_{j=1}^{n}[\mathcal{K}_j] \, [\Delta V_j]}{V}
\label{eq:avg-kappa}$$
where $\Delta V_j$ denotes the volume of the $j$-th cell in the CFD mesh, and $V$ is the total volume of the computational domain. This volume-averaged local condition number $\overline{\mathcal{K}}_{\bm{x}}$ has a similar interpretation to $\mathcal{K}_{\tau}$ but preserves the merits of $\mathcal{K}_j$ as summarized above. That is, it has a tighter bound and is mesh independent.
In the derivations above the Reynolds stress term is treated explicitly, i.e., $\bm{\tau}$ is substituted directly into the RANS equation. When the Reynolds stress term is treated implicitly as in many practical implementations of Reynolds stress models (e.g. [@basara03new; @maduta17improved]), the corresponding local condition number of the model can be obtained similarly, except that the Green’s function is modified to account for the implicit modeling of the linear part of Reynolds stress with eddy viscosity model. Specifically, the general form of implicit treatment of Reynolds stress can be written as follows:
$$\label{eq:nut-model}
{\boldsymbol{\tau}} = 2\nu_t \mathbf{S} + {\boldsymbol{\tau}}^{\perp}$$
where $\nu_t$ represents the eddy viscosity, $\mathbf{S}=\frac{1}{2}\left( \nabla \bm{u} + (\nabla \bm{u})^T\right)$ denotes the strain rate tensor and ${\boldsymbol{\tau}}^{\perp}$ denotes the nonlinear part. That is, Green’s function $\widetilde{G}$ corresponding to the linear operator
$$\label{eq:L-implicit}
\widetilde{\mathcal{L}}(\bm{u}) =
\mathcal{L}(\bm{u}) - \nu_t^m \nabla^2 \bm{u}
=
\bm{u}_0 \cdot \nabla \bm{u} - (\nu + \nu_t^m) \nabla^2 \bm{u}$$
should be used in Eqs. (\[eq:cont-cond\]) and (\[eq:cont-cond-disc\]), with $\widetilde{G}$ related to $\widetilde{\mathcal{L}}$ in a similar way as $G$ to $\mathcal{L}$ in Eq. (\[eq:define-green\]). The optimal eddy viscosity $\nu_t^m$ is computed by minimizing the discrepancy between the linear eddy viscosity model and the DNS Reynolds stress data, i.e.,
$$\label{eq:nut-definition}
\nu_t^m = \operatorname*{arg\,min}_{\nu_t} || {\boldsymbol{\tau}}^{DNS} - 2\nu_t \mathbf{S}^{DNS}||$$
where $\bm{\tau}^{DNS}$ and $\mathbf{S}^{DNS}$ denote the Reynolds stress and the strain rate tensor from DNS database, respectively. The detailed derivations are presented in \[sec:app-eddy-cn\].
In summary, we proposed a local condition number to assess the sensitivity of local mean velocities in Reynolds stress models. It has the following three forms: the spatial function $\mathcal{K}(\bm{x})$ (i.e., condition number function), an $n$-vector $\mathcal{K}_j$ obtained by discretizing $\mathcal{K}(\bm{x})$ on the CFD mesh, and a scalar $\overline{\mathcal{K}}_{\bm{x}}$ obtained by integration of $\mathcal{K}(\bm{x})$. This metric is applicable to both traditional and data-driven Reynolds stress models with either implicit or explicit treatments.
Results {#sec:results}
=======
The fully developed turbulent plane channel flows are investigated by using the local condition number $\mathcal{K}_j$. Both the explicit treatment and implicit treatment of Reynolds stress models are studied. In this work, we consider an ideal scenario in which the Reynolds stress $\bm{\tau}$ is directly computed from DNS database at various Reynolds numbers $Re_{\tau}=180, 550, 1000, 2000, 5200$. The DNS data were obtained from the University of Texas Austin online database [@lee15direct]. The mean velocity field is then solved by substituting the computed Reynolds stress as the closure term of RANS equations.
We study the data-driven RANS modeling and the traditional RANS modeling in Sections \[sec:data-driven\] and \[sec:tradition-rans\], respectively. In data-driven RANS modeling, the Reynolds stress or eddy viscosity is “frozen” and not updated during the iterations of RANS simulations as in the traditional RANS solvers. By studying these two types of RANS modeling, we show that the proposed local condition number can be used to assess the sensitivities of RANS simulations for both data-driven modeling and traditional modeling.
The RANS simulations are performed in a finite-volume CFD platform OpenFOAM, using a modified flow solver that allows the explicit and implicit treatments of Reynolds stress computed from DNS data. The mesh sizes are $36, 110, 200, 400, 1040$ for the flows at Reynolds numbers $Re_{{\tau}}=180, 550, 1000, 2000$, and $5200$. The $y^+$ of the first cell center is kept below 1. For numerical discretizations, the second-order central difference scheme is chosen for all terms except for the convection term, which is discretized with a second-order upwind scheme. In Section \[sec:data-driven\], the mean velocity is obtained by directly solving Eq. \[eq:ns-concise-L\] since the mean velocity and the pressure are decoupled for the RANS simulation of a fully-developed plane channel flow. The convergence criteria of solving mean velocity is set as $10^{-8}$ in relative error.
Data-driven RANS modeling {#sec:data-driven}
-------------------------
In this section, DNS data is directly used to compute the Reynolds stress term to represent the ideal situation of data-driven RANS modeling. The results show that the magnitude of local condition number $\mathcal{K}_j$ increases rapidly with the Reynolds number by explicit treatment of Reynolds stress. We also demonstrate that the magnitude of local condition number can be reduced by using implicit treatment of Reynolds stress. Such reduction of local condition number is in consistent with the observation that eddy viscosity models are more stable than Reynolds stress transport models.
### Reynolds stress models with explicit treatment
The Reynolds stress term is directly computed from DNS data and substituted into RANS as shown in Algorithm \[alg:reynolds-stress-model\].
Set Reynolds stress from DNS data: $\bm{\tau}=\bm{\tau}^{DNS}$\
The local condition numbers $\mathcal{K}_j$ of the explicit treatment of Reynolds stress are shown in Fig. \[fig:cmptKappaTau\]. With the increase of the Reynolds number, it can be seen that the magnitude of local condition number also increases. Specifically, the local condition number of the flow at $Re_\tau=180$ is of the order $O(1)$, while the local condition number of the flow at $Re_\tau=5200$ is of the order $O(10^2)$. This rapid increase of local condition number agrees well with the increased error of solved mean velocity $U$ as summarized in Table \[tab:summary\]. In addition, the local condition number is greater near the channel center than close to the wall, especially for the high Reynolds number cases. Such pattern of local condition number also agrees with the spatial pattern of the error of the solved mean velocity as illustrated in Fig. \[fig:U-comp-nut\]b.
![The profiles of local condition number $\mathcal{K}_j$ at different Reynolds numbers by using explicit treatment of Reynolds stress, i.e., the computed Reynolds stress is explicitly substituted into RANS equations.[]{data-label="fig:cmptKappaTau"}](cmptKappaTau-legend.pdf "fig:"){width="40.00000%"}\
![The profiles of local condition number $\mathcal{K}_j$ at different Reynolds numbers by using explicit treatment of Reynolds stress, i.e., the computed Reynolds stress is explicitly substituted into RANS equations.[]{data-label="fig:cmptKappaTau"}](cmptKappaTau.pdf "fig:"){width="45.00000%"}
The averaged local condition number $\overline{\mathcal{K}}_{\bm{x}}$ increases with the Reynolds number by using explicit treatment of Reynolds stress, wich is clearly seen in Fig. \[fig:avg-cmptKappaTau-nut\]. Such increase of averaged local condition number with Reynolds number reveals the potential shortcoming of explicit modeling of Reynolds stress, i.e., a relatively accurate but explicit modeling of Reynolds stress does not guarantee the satisfactory mean velocity by solving RANS equations, especially for high Reynolds number flows. This observation has been reported in the work by Thompson et al. [@thompson16methodology], and the proposed averaged local condition number can be used as an integral indicator to estimate the extent of error propagation from the modeled Reynolds stress to the solved mean velocity field.
### Reynolds stress models with implicit treatment {#sec:evm}
The eddy viscosity is directly computed from DNS data and substituted into RANS equations to study the ideal situation of data-driven Reynolds stress models with implicit treatment as shown in Algorithm \[alg:eddy-viscosity-model\].
Compute optimal eddy viscosity $\nu_t^m$ from DNS Reynolds stresses based on Eq. (\[eq:nut-definition\])\
Compared to the Reynolds stress models, it is well known that eddy viscosity models can enhance the stability and conditioning of RANS equations with turbulence closures. In the practice of traditional RSM, the modeled Reynolds stress is empirically blended with the Reynolds stress from eddy viscosity models to achieve better convergence and conditioning [@basara03new; @maduta17improved]. In this work, we demonstrate that the local condition number $\mathcal{K}_j$ can quantitatively explain the improved conditioning of implicit treatment of Reynolds stress by introducing an eddy viscosity. It can be seen in Fig. \[fig:cmptKappaTau-nut\] that the local condition number $\mathcal{K}_j$ is significantly reduced compared with the results of explicit treatment of Reynolds stress as shown in Fig. \[fig:cmptKappaTau\], especially for high Reynolds number flows. Although the local condition number of high Reynolds number is still greater than the one of low Reynolds number, they are at the same order of magnitude for different Reynolds numbers. The volume-averaged local condition number in Fig. \[fig:avg-cmptKappaTau-nut\] is also significantly reduced by using implicit treatment of Reynolds stress, demonstrating the merit of using implicit treatment of Reynolds stress in improving the conditioning when solving RANS equations for mean velocity field.
![The local condition number at different Reynolds numbers by using implicit treatment of Reynolds stress, i.e., the linear part of Reynolds stress is implicitly treated by introducing an optimal eddy viscosity.[]{data-label="fig:cmptKappaTau-nut"}](cmptKappaTau-legend.pdf "fig:"){width="40.00000%"}\
![The local condition number at different Reynolds numbers by using implicit treatment of Reynolds stress, i.e., the linear part of Reynolds stress is implicitly treated by introducing an optimal eddy viscosity.[]{data-label="fig:cmptKappaTau-nut"}](cmptKappaTau-nut.pdf "fig:"){width="45.00000%"}
![The volume-averaged local condition number at different Reynolds numbers for explicit and implicit treatments of Reynolds stress.[]{data-label="fig:avg-cmptKappaTau-nut"}](avg-cmptKappa-legend.pdf "fig:"){width="40.00000%"}\
![The volume-averaged local condition number at different Reynolds numbers for explicit and implicit treatments of Reynolds stress.[]{data-label="fig:avg-cmptKappaTau-nut"}](avg-cmptKappa.pdf "fig:"){width="45.00000%"}
The mean velocity $U$ is solved and presented in Fig. \[fig:U-comp-nut\] at Reynolds numbers $Re_\tau=180$ and $Re_\tau=5200$ by using explicit and implicit treatments of Reynolds stress. At Reynolds number $Re_\tau=180$, it can be seen in Fig. \[fig:U-comp-nut\]a that the solved mean velocity $U$ by using both kinds of treatments has a good agreement with DNS data. It demonstrates that the error propagation from Reynolds stress to mean velocity is not severe at low Reynolds number, and the percentage error of mean velocity is comparable by using either explicit or implicit treatment of Reynolds stress as shown in Fig. \[fig:U-comp-nut\]c. These results have a good agreement with the local condition number presented in Figs. \[fig:cmptKappaTau\] and \[fig:cmptKappaTau-nut\], which shows that the local condition number is of the same order for the flow at Reynolds number $Re_\tau=180$ by using both types of treatments. However, the solved mean velocity fields are noticeably different at high Reynolds number ($Re_\tau=5200$) as shown in Fig. \[fig:U-comp-nut\]b. Specifically, the solved mean velocity by using explicit treatment of Reynolds stress shows a significant difference from the DNS data, while the solved mean velocity by using implicit treatment of Reynolds stress still agrees well with the DNS data at $Re_\tau=5200$. The percentage error of solved mean velocity at $Re_\tau=5200$ in Fig. \[fig:U-comp-nut\]d also confirms that the error in mean velocity by using explicit treatment of Reynolds stress is orders of magnitude higher than the error of using implicit treatment of Reynolds stress. Such comparison of solved mean velocity fields agrees well with the differences in local condition number $\mathcal{K}_j$, demonstrating that the proposed local condition number can be used to quantitatively assess the error propagation from Reynolds stress to mean velocity when solving RANS equations. Wu et al. [@wu18data-driven] further proposed an implicit treatment of Reynolds stress for machine-learning-assisted RANS modeling to improve the conditioning of solving mean velocity field, building upon the comparison of model conditioning between explicit treatment of Reynolds stress in Algorithm \[alg:reynolds-stress-model\] and the implicit treatment in Algorithm \[alg:eddy-viscosity-model\].
![The comparison of solved mean velocity by using explicit and implicit treatments of Reynolds stress, including (a) mean velocity $U$ at $Re_\tau=180$, (b) percentage error of mean velocity $U$ at $Re_\tau=180$, (c) mean velocity $U$ at $Re_\tau=5200$ and (d) percentage error of mean velocity $U$ at $Re_\tau=5200$.[]{data-label="fig:U-comp-nut"}](U-legend.pdf "fig:"){width="60.00000%"}\
\
![The comparison of solved mean velocity by using explicit and implicit treatments of Reynolds stress, including (a) mean velocity $U$ at $Re_\tau=180$, (b) percentage error of mean velocity $U$ at $Re_\tau=180$, (c) mean velocity $U$ at $Re_\tau=5200$ and (d) percentage error of mean velocity $U$ at $Re_\tau=5200$.[]{data-label="fig:U-comp-nut"}](Uerr-legend.pdf "fig:"){width="47.00000%"}\
\
Traditional RANS modeling {#sec:tradition-rans}
-------------------------
The local condition number in Fig. \[fig:cmptKappaTau\] only assess the error propagation from Reynolds stress to the mean velocity when the modeled Reynolds stress is “frozen” during solving for the mean velocity field. In the practice of traditional RANS modeling, iterations are involved in solving RANS equations and the modeling of Reynolds stress is updated by the mean velocity field during the iterations. Therefore, it is possible that the mean velocity field and the Reynolds stress can adjust to each other during the iterations. We employed the ratio $\delta U_{rms}/U^{DNS}_{rms}$ to assess the error of the solved mean flow field at each iteration step. Specifically, the volume-averaged root-mean-squared error of the solved mean velocity is defined as follow:
$$\delta U_{rms} =
\sqrt{\frac{\sum_{j=1}^{n} \left([U]_j-[U^{DNS}]_j\right)^2 \, [\Delta V_j]}{V}}
\label{eq:deltaUrms}$$
The volume-averaged root-mean-squared DNS velocity is defined as follow:
$$U^{DNS}_{rms} =
\sqrt{\frac{\sum_{j=1}^{n}\left([U^{DNS}]_j \right)^2 \, [\Delta V_j]}{V}}
\label{eq:Urms}$$
### Explicit treatment
In this section, we show that the explicit coupling between Reynolds stress and mean velocity during the iterations can gradually amplify the errors and lead to divergence. Such explicit coupling is often used in the Reynolds stress transport models (RSTM). Specifically, the Reynolds stress is obtained by solving its transport equations with the mean velocity field at the previous iteration step. In the following, we use a simplified example with an iterative solver as shown in Algorithm \[alg:exp-model\] to illustrate the convergence issue of RSTM. In addition, we demonstrate that the proposed local condition number can be used to detect and explain the corresponding ill-conditioning issue during iterations.
The Reynolds stress at $i^{\textrm{th}}$ iteration step is explicitly treated by using DNS data according to Algorithm \[alg:exp-model\]. Unlike the data-driven Reynolds stress modeling as shown in Algorithm \[alg:reynolds-stress-model\], this simplified explicit Reynolds stress treatment allows the update of Reynolds stress at each iteration based on the solved mean velocity field at the previous iteration step. Compared to the implicit treatment of Reynolds stress as shown in Algorithm \[alg:eddy-viscosity-model\], the only difference of this explicit treatment is the computing of Reynolds stress with the mean velocity at the previous iteration step, which is indicated by the superscript $i-1$ at line 3 of Algorithm \[alg:exp-model\].
Compute optimal eddy viscosity $\nu_t^m$ from DNS Reynolds stresses based on Eq. (\[eq:nut-definition\])\
The errors of solved mean velocity field by using explicit treatment of Reynolds stress is presented in Fig. \[fig:coupled-Exp-Ux\]a. The DNS mean velocity is used as the initial field in RANS simulations, and thus the initial value of $\delta U_{rms}/U^{DNS}_{rms}$ is small. However, the value of $\delta U_{rms}/U^{DNS}_{rms}$ increases rapidly during the first several iteration steps. It demonstrates that the conditioning issue within each iteration can lead to error amplification, i.e., a small error in the modeled Reynolds stress can lead to noticeable errors in the solved mean velocity field and thus influence the modeled Reynolds stress in the next iteration step. Due to such coupling of error amplification, even a small error of modeled Reynolds stress would lead to divergence of simulation eventually. It can be seen in Fig. \[fig:coupled-Exp-Ux\]b that the volume-averaged local condition number is at $O(10)$ within the first three iteration steps, explaining the rapid growth of error in the solved mean velocity. Therefore, the proposed local condition number is still of importance in traditional RANS modeling since it provides a quantitative assessment of model conditioning at every iteration step.
### Implicit treatment
We further show that the relative error of mean velocity is much smaller by using implicit treatment of Reynolds stress in RANS simulations. In this work, a simplified implicit Reynolds stress treatment is illustrated in Algorithm \[alg:eddy-viscosity-model\], where the Reynolds stress at $i^{\textrm{th}}$ iteration step is computed based on the mean velocity at the same iteration step. It can be seen in Fig. \[fig:coupled-imp-Ux\]a that the relative error of the solved mean velocity is much smaller than the one shown in Fig. \[fig:coupled-Exp-Ux\]a. In addition, the volume-averaged local condition number stays at $O(1)$ as shown in Fig. \[fig:coupled-imp-Ux\]b, which explains the better convergence of solving for mean velocity field by using implicit treatment of Reynolds stress.
Discussion {#sec:discussion}
==========
Monolithic coupling and data-driven turbulence models {#sec:discuss-monolithic}
-----------------------------------------------------
The conditioning analysis above deals with segregated RANS solvers specifically, where RANS momentum equations and turbulence equations are solved in a segregated manner. It is well known that monolithic coupling is the most effective way to improve conditioning. However, there are two caveats worth mentioning here in additional to the possible increase in computational costs mentioned earlier. First, while monolithic coupling generally improves model conditioning and numerical stability, it is by no means a panacea that guarantees well-conditioning and stability. The conditioning and stability ultimately depend on the characteristics of the turbulence model itself. For example, the popularity of S–A model in external aerodynamics is largely attributed to its excellent robustness in terms of both model conditioning and numerical stability, which many other models do not have. Second, a monolithic coupling for data-driven turbulence models is more challenging than for traditional PDE-based models, if possible at all. For example, for a neural-network-based data-driven turbulence model (e.g., [@ling16reynolds]), a monolithic coupling would be possible, because neural networks models are differentiable. However, for non-differentiable models, e.g., those based on random forests or other tree-based models (e.g., [@wang17physics-informed]), a monolithic coupling is not straightforwardly viable.
Discrepancies in velocities obtained with explicit and implicit treatments {#sec:discuss-paradox}
--------------------------------------------------------------------------
It was shown in Fig. \[fig:U-comp-nut\] that the solved mean velocity can be significantly different depending on whether the DNS Reynolds stress used to solve Eq. (\[eq:ns-concise-N\]) is treated explicitly or implicitly. In other words, solving the following two equations
$$\begin{aligned}
\mathcal{L}(\bm{u}^{exp}) & = \nabla \cdot \bm{\tau}^{exp} -\nabla p \qquad
\text{and} \label{eq:exp-u} \\
\mathcal{L}(\bm{u}^{imp}) & = \nabla \cdot \bm{\tau}^{imp} -\nabla p
\label{eq:imp-u}\end{aligned}$$
yields drastically different velocities. The superscript $imp$ indicates the implicit treatment of Reynolds stress and $exp$ denotes the explicit treatment, i.e., $\bm{\tau}^{imp}=\nu_t\left(\nabla\bm{u}^{imp} +(\nabla \bm{u}^{imp})^T\right)+\bm{\tau}^{\perp,DNS}$ and $\bm{\tau}^{exp}=\bm{\tau}^{DNS}$. This finding apparently contradicts the common understanding in traditional CFD practice that the *converged solution* of the mean velocity should be the same regardless of how the Reynolds stress is treated. Indeed, the Reynolds stresses used in the two formulations in Eqs. (\[eq:exp-u\]) and (\[eq:imp-u\]) are approximately equal, since $\nu_t\left(\nabla\bm{u}^{imp} +(\nabla \bm{u}^{imp})^T\right)+\bm{\tau}^{\perp,DNS} \approx \bm{\tau}^{DNS}$. More precisely, the difference between $\bm{u}^{imp}$ and $\bm{u}^{DNS}$ is about $0.1\%$, and the difference between $\bm{\tau}^{imp}$ and $\bm{\tau}^{exp}$ would be at the similar level. However, the condition number with regard to the nonlinear differential operator $\mathcal{L}$ is large for the flows at high Reynolds numbers, and thus a small difference between $\bm{\tau}^{imp}$ and $\bm{\tau}^{exp}$ can lead to a large difference between the solved mean velocities $\bm{u}^{imp}$ and $\bm{u}^{exp}$.
In addition, the better solution of $\bm{u}^{imp}$ with implicit treatment of Reynolds stress can be explained by the improved model conditioning, i.e., the condition number is smaller with regard to the linear differential operator $\widetilde{\mathcal{L}}=\mathcal{L}-\nu_t^m \nabla^2$ for the implicit treatment of Reynolds stress. Specifically, we first define an optimal Reynolds stress $\bm{\tau}^{op}$ that can lead to $\bm{u}^{DNS}$ by solving RANS equations:
$$\mathcal{L}(\bm{u}^{DNS})=\nabla \cdot \bm{\tau}^{op} - \nabla p$$
where $\bm{\tau}^{op}$ denotes the true Reynolds stress that provides $\bm{u}^{DNS}$ by solving RANS equations. The errors $\|\bm{\tau}^{DNS}-\bm{\tau}^{op}\|$ and $\|\bm{\tau}^{\perp,DNS}-\bm{\tau}^{\perp,op}\|$ are of the same order of magnitude. Therefore, $\|\bm{u}^{imp}-\bm{u}^{DNS}\|$ is smaller than $\|\bm{u}^{exp}-\bm{u}^{DNS}\|$ due to the smaller sensitivity of solving mean velocity by using the implicit treatment of Reynolds stress.
Conclusion {#sec:conclusion}
==========
Recently, several researchers [@thompson16methodology; @poroseva16on] employed DNS Reynolds stress data as the closure term and solved the RANS equations for mean velocities on the plane channel flows. They reported unexpected results that the obtained mean velocities deviated significantly (up to 35%) from the DNS data at high Reynolds numbers. In this work, we aim to identify a metric to quantitatively assess the conditioning of the RANS equations with Reynolds stress closure, i.e., how a small error in Reynolds stress can lead to large errors in the mean velocity by solving the RANS equations. The turbulent channel flow is studied to evaluate the candidate metrics. Our analysis shows that the global, matrix-based condition number is not able to distinguish the different sensitivity of solved mean velocities at different Reynolds numbers. A local condition number function is then derived as a more precise indicator of model conditioning. We demonstrate that such a local condition number is mesh-independent and is able to explain the error propagation from the modeled Reynolds stress to the solved mean velocity in RANS simulations. Furthermore, it is also capable of explaining the enhanced conditioning of implicit treatment of Reynolds stress compared to the explicit treatment. The proposed condition number can be a valuable tool for assessing the sensitivity of solved mean velocity field in RANS simulations, providing great potential in facilitating the conditioning-oriented schemes in data-driven turbulence modeling.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank Dr. Gary N. Coleman of NASA Langley and Dr. Florian Menter of ANSYS for the helpful discussions during this research. We gratefully acknowledge Dr. Scott Murman of NASA Ames and Prof. Svetlana Poroseva of The University of New Mexico for their constructive comments on the first draft of the manuscript.
Derivations of Condition Numbers
================================
Derivation of global, matrix-based condition number {#app:global-cn}
---------------------------------------------------
The global matrix-based condition number $\mathcal{K}_{\tau}$ is defined as follow:
$$\label{eq:app-global-define}
\frac{\|\delta {\bm{U}}\|}{\|{\bm{U}}\|} \le
\mathcal{K}_{\tau} \frac{\|\nabla \cdot \delta \bm{\tau} \|}{\|\nabla \cdot \bm{\tau} \|}$$
where $\mathcal{K}_{\tau}$ measures the sensitivity of the solved mean velocity field due to the perturbation of Reynolds stress field， and $|\cdot|$ indicate Euclidean norm of a vector (of all values in the discretized velocity field). To derive the formulation of $\mathcal{K}_{\tau}$, the perturbation $\delta \bm{b}$ in Eq. (\[eq:global-cn\]) is further written as:
$$\delta \bm{b} = \nabla \cdot \delta \boldsymbol{\tau} -
\delta (\nabla p)$$
For the purpose of the sensitivity study here, it is assumed that a constant pressure gradient is imposed to drive the flow, i.e., $\delta (\nabla p) = 0$, and thus we have:
$$\label{eq:app-delta-b}
\delta \bm{b} = \nabla \cdot
\delta \boldsymbol{\tau}.$$
Hence,
$$\label{eq:app-cond-norm-derive}
\frac{\|\delta {\bm{U}}\|}{\|{\bm{U}}\|} \le
\mathcal{K}_\mathsf{A} \frac{\|\delta \bm{b} \|}{\| \bm{b} \|}
= \mathcal{K}_\mathsf{A} \frac{\| \nabla \cdot \delta \boldsymbol{\tau} \|}{\| \bm{b} \|}
= \underbrace{\mathcal{K}_\mathsf{A} \frac{\|\nabla \cdot \boldsymbol{\tau}\|}{\|\bm{b}\|}}_{\textstyle \mathcal{K}_{\tau}}
\frac{\|\nabla \cdot \delta \boldsymbol{\tau} \|}{\| \nabla \cdot \boldsymbol{\tau} \|} .$$
Comparing the forms of Eq. \[eq:app-cond-norm-derive\] with the definition of $\mathcal{K}_\tau$ in Eq. \[eq:app-global-define\], the matrix-norm-based condition number for Reynolds-stress-based turbulence models is thus:
$$\label{eq:app-cond-norm}
\mathcal{K}_\tau = \mathcal{K}_\mathsf{A} \frac{\|\nabla \cdot
\boldsymbol{\tau}\|}{\|\bm{b}\|}$$
Derivation of local condition number function {#sec:app-local-cn}
---------------------------------------------
The continuous local condition number $\mathcal{K}(\mathbf{x})$ is defined as follow:
$$\label{eq:app-define-Kx}
\frac{|\delta \bm{u}(\bm{x})|}{U_{\infty}} \le
\mathcal{K}(\bm{x}) \frac{{\left\Vert \nabla \cdot \delta \bm{\tau} \right\Vert_{\Omega}}}{{\left\Vert \nabla \cdot \bm{\tau} \right\Vert_{\Omega}}}$$
where $\mathcal{K}(\bm{x})$ measures the sensitivity of the solved mean velocity at any given location $\bm{x}$ due to the perturbation of the Reynolds stress field, and $U_\infty$ is a constant representative velocity magnitude for normalization. The function norm ${\left\Vert \cdot \right\Vert_{\Omega}}$ of function $f({\boldsymbol{\xi}})$ on domain $\Omega$ is defined as in Eq. (\[eq:function-norm-def\]).
To derive the formulation of this local condition number, we first consider the solution $\bm{u}$ at a particular location ${\bm{x}}'$:
$$\label{eq:app-u0}
\bm{u}({\bm{x}}') = \int_\Omega G({\bm{x}}'; {\boldsymbol{\xi}}) \, {\bm{b}}({\boldsymbol{\xi}}) \, d{\boldsymbol{\xi}}$$
where $G$ represents the Green’s function of the linear differential operator $\mathcal{L}$ in the linearized RANS equations as defined in Eq. (\[eq:L-definition\]). Denoting $G_{{\bm{x}}'} = G({\bm{x}}'; {\boldsymbol{\xi}})$, the perturbation of the solution is thus:
$$\delta \bm{u}({\bm{x}}') = \int_\Omega G({\bm{x}}'; {\boldsymbol{\xi}}) \, \delta{\bm{b}}({\boldsymbol{\xi}}) \, d{\boldsymbol{\xi}} = {\left< G_{{\bm{x}}'},\delta{\bm{b}} \right>_{\Omega}}$$
where ${\left< \cdot \right>_{\Omega}}$ is the inner product of functions defined on domain $\Omega$.
Using the Schwartz inequality [@steele04cauchy; @debnath05hilbert] leads to:
$$\begin{aligned}
|\delta \bm{u}({\bm{x}}')| & \le
{\left\Vert G_{{\bm{x}}'} \right\Vert_{\Omega}} \,
{\left\Vert \delta {\bm{b}} \right\Vert_{\Omega}} \\
& = {\left\Vert G_{{\bm{x}}'} \right\Vert_{\Omega}} \,
{\left\Vert \nabla \cdot \delta {\boldsymbol{\tau}} \right\Vert_{\Omega}} \end{aligned}$$
As in \[app:global-cn\], the pressure gradient is assumed constant and thus $\delta \bm{b} = \nabla \cdot \delta \boldsymbol{\tau}$. Finally, the sensitivity of mean velocity $\bm{u}$ with respect to the Reynolds stress $\bm{\tau}$ perturbations is derived as follows:
$$\begin{aligned}
\frac{|\delta \bm{u}({\bm{x}}')|}{U_\infty}
& \le \frac{{\left\Vert G_{{\bm{x}}'} \right\Vert_{\Omega}} \,
{\left\Vert \nabla \cdot \delta {\boldsymbol{\tau}} \right\Vert_{\Omega}}}{U_\infty} \\
& =
\frac{{\left\Vert G_{{\bm{x}}'} \right\Vert_{\Omega}} \,
{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}}{U_\infty} \,
\frac{{\left\Vert \nabla \cdot \delta {\boldsymbol{\tau}} \right\Vert_{\Omega}}}
{{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}}
\label{eq:app-cond-derive}\end{aligned}$$
Therefore, by comparing Eqs. (\[eq:app-cond-derive\]) and (\[eq:app-define-Kx\]), we define a **local condition number function** $\mathcal{K}$ of spatial location ${\bm{x}}$ as:
$$\label{eq:app-cont-cond}
\mathcal{K}({\bm{x}}) =
\frac{{\left\Vert G_{{\bm{x}}} \right\Vert_{\Omega}} \,
{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}}{U_\infty}
= \frac{
{\left\Vert G({\bm{x}}, \bm{\xi}) \right\Vert_{\Omega}} \,
{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}
}{U_\infty}$$
Without causing ambiguity, we have dropped the subscript of $\bm{x}'$ in the equation above and in the text for simplicity of notation.
Local condition number for implicit treatment of Reynolds stress {#sec:app-eddy-cn}
----------------------------------------------------------------
In the practice of RANS modeling, eddy viscosity models are widely used, and the modeled eddy viscosity would influence the differential operator $\mathcal{L}$ associated with RANS equations. Therefore, we extend the derivation of Eq. (\[eq:cont-cond\]) to make it compatible with the implicit treatment of Reynolds stress. According to the general form of implicit treatment of Reynolds stress [@pope75more] in Eq. (\[eq:nut-model\]), the linearized RANS equations in Eq. \[eq:ns-concise-L\] can be rearranged as follow:
$$\label{eq:app-ns-concise-nut}
\widetilde{\mathcal{L}}(\bm{u}) = \nabla \cdot \boldsymbol{\tau}^\perp-\nabla p$$
where $\widetilde{\mathcal{L}}=\mathcal{L}-\nu_t^m \nabla^2$ is the modified linear differential operator by using implicit treatment of Reynolds stress. Here we only study the perturbation on the nonlinear term $\bm{\tau}^\perp$ of Reynolds stress $\bm{\tau}$, i.e.,
$$\delta {\boldsymbol{\tau}} = \delta {\boldsymbol{\tau}}^{\perp}$$
Finally, we have the local condition number $\mathcal{K}({\bm{x}}')$ in Eq. (\[eq:cont-cond\]) re-derived as follows for the implicit treatment of Reynolds stress:
$$\begin{aligned}
\frac{|\delta \bm{u}({\bm{x}}')|}{U_\infty}
& \le \frac{{\left\Vert \widetilde{G}_{{\bm{x}}'} \right\Vert_{\Omega}} \,
{\left\Vert \nabla \cdot \delta {\boldsymbol{\tau}}^\perp \right\Vert_{\Omega}}}{U_\infty} \\
& =
\underbrace{\frac{{\left\Vert \widetilde{G}_{{\bm{x}}'} \right\Vert_{\Omega}} \,
{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}}{U_\infty}}_{\textstyle \mathcal{K}_j} \;\;
\frac{{\left\Vert \nabla \cdot \delta {\boldsymbol{\tau}} \right\Vert_{\Omega}}}
{{\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}}\end{aligned}$$
where the kernel function ${\left\Vert \widetilde{G}_{{\bm{x}}'} \right\Vert_{\Omega}}$ represents the Green’s function that corresponds to the linear differential operator $\widetilde{\mathcal{L}}$ defined in Eq. (\[eq:L-implicit\]), taking into account the implicit modeling of linear part of Reynolds stress by introducing an eddy viscosity.
Mesh independency of the local condition number function {#sec:discretization}
========================================================
We present the numerical discretization of the proposed local condition number on a CFD mesh and show that the discretized local condition number is mesh-independent. First, the function norms ${\left\Vert \cdot \right\Vert_{\Omega}}$ are approximated in vector norms ${\left\Vert \cdot \right\Vert_{n}}$ through numerical integration on a CFD mesh of $n$ cells. This is derived as follows:
$$\begin{aligned}
{\left\Vert G_{\bm{x}'} \right\Vert_{\Omega}} & = \sqrt{\int_\Omega | G(\bm{x}'; {\boldsymbol{\xi}})|^2 \,
d {\boldsymbol{\xi}} } \label{eq:fn-defition} \\
& \approx \sqrt{ \sum_{i=1}^n \left( [{\bm{r}}_{j,i}]^2 \, \Delta V_i \right) } \\
& = \sqrt{ \sum_{i=1}^n \left([{\bm{r}}_{j,i}] \, \sqrt{[\Delta V_i]}\right)^2} \\
& = {\left\Vert {\bm{r}}_j \sqrt{[\Delta {\bm{V}}]} \right\Vert_{n}}
= {\left\Vert \tilde{{\bm{r}}}_j \right\Vert_{n}}\end{aligned}$$
with $\tilde{{\bm{r}}}_j = {\bm{r}}_j \sqrt{\Delta {\bm{V}}}$ and “$\approx$” indicating numerical discretization of the integral involved in the function norm in Eq. (\[eq:fn-defition\]); $\Delta V_i$ is the volume for the $i$-th cell; $\Delta {\bm{V}}$ is the $n$-vector consisting of volumes of cells in the mesh; ${\bm{r}}_{j}$ is the $j$-th row of the matrix $\mathsf{A}^{-1}$, with $\mathsf{A}$ being the discretization of the operator $\mathcal{L}$ as seen in Eq. (\[eq:L-definition\]). The numbering implies that the location ${\bm{x}}'$ is the coordinate of the $j$-th cell.
Similarly, the function norm of the Reynolds stress divergence is approximated on the CFD mesh as follows:
$${\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}
\approx {\left\Vert [\nabla \cdot {\boldsymbol{\tau}}] \sqrt{[\Delta {\bm{V}}]} \right\Vert_{n}}$$
It is clear that the function norm ${\left\Vert G_{{\bm{x}}'} \right\Vert_{\Omega}}$ is mesh-independent as its definition does not involve any discretization mesh (quadrature), so its numerical approximation ${\left\Vert {\bm{r}}_j \sqrt{[\Delta {\bm{V}}]} \right\Vert_{n}}$ should also be mesh independent on a sufficiently fine mesh. In the same way, both the function norm ${\left\Vert \nabla \cdot {\boldsymbol{\tau}} \right\Vert_{\Omega}}$ and its numerical approximation ${\left\Vert [\nabla \cdot {\boldsymbol{\tau}}] \sqrt{[\Delta {\bm{V}}]} \right\Vert_{n}}$ are mesh independent. The mesh independency can be further verified in the special case, where the divergence field $\nabla \cdot {\boldsymbol{\tau}}$ is a nonzero constant $\beta$ and the mesh consists of $n$ uniformly sized cells. In this case we have $\Delta V = \frac{|\Omega|}{n}$, where $|\Omega|$ is the total volume of the computational domain $\Omega$ (independent of the discretization mesh). Therefore, the vector norm, which is a numerical approximation of the function form, is as follows:
$$\begin{aligned}
{\left\Vert [\nabla \cdot {\boldsymbol{\tau}}] \sqrt{[\Delta {\bm{V}}]} \right\Vert_{n}} = &
\sqrt{\sum_{i=1}^n \left( \beta \sqrt{\frac{|\Omega|}{n}} \right)^2} \\
&
= \sqrt{n \; \beta^2 \frac{|\Omega|}{n}} \\
&
= \beta \sqrt{|\Omega|},\end{aligned}$$
which is clearly independent of the number of cells in the mesh.
Notation {#notation .unnumbered}
========
We use ${[\phi]}$ to indicate the $n$-vector obtained by discretizing the field $\phi$ on the mesh, where $n$ is number of cells/grid points. $\|{[\phi]}\|$ denotes the norm of vector ${[\phi]}$ resulted from the discretization. Since the norm is always taken on discretized $n$-vector, we abbreviated $\|{[\phi]}\|$ as $\|\phi\|$ without ambiguity. When mentioning function norm and $n$-vector norm simultaneously, we used ${\left\Vert \cdot \right\Vert_{\Omega}}$ and ${\left\Vert \cdot \right\Vert_{n}}$ to distinguish them, with $\Omega$ denoting the domain on which the norm is defined.
XXXX = $\bm{u}$ mean velocity field\
$\mathbf{U}$ discretized mean velocity ($n$-vector)\
$\bm{\tau}$ Reynolds stress tensor\
$\mathbf{S}$ rate-of-strain tensor\
$\bm{b}$ imbalance vector between Reynolds stress divergence and pressure gradient\
$\mathsf{A}$ $n \times n$ coefficients matrix of discretized RANS equations\
$\mathcal{N}$ non-linear differential operator\
$\mathcal{L}$ linear differential operator\
$ G $ Green’s function corresponding to $\mathcal{L}$\
$\mathcal{K}_\mathsf{A}$ condition number of matrix $\mathsf{A}$\
$\overline{\alpha}$ ratio between Reynolds stress divergence and the total source term\
$\mathcal{K}_\tau$ matrix-norm condition number associated with Reynolds stress perturbation\
$ \mathcal{K}_j $ local condition number vector approximated on a CFD mesh ($n$-vector)\
$ \overline{\mathcal{K}}_x $ volume-averaged local condition number (scalar)\
$ \delta U_{rms} $ volume-averaged root-mean-squared error of solved mean velocity\
$ U_{rms}^{DNS} $ volume-averaged root-mean-squared DNS mean velocity\
$ \perp $ superscript indicating the non-linear part of Reynolds stress\
[10]{}
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[^1]: We use “traditional” models to refer to turbulence models based on partial differential equations (PDEs) and/or analytical expressions. This is to distinguish them from recently emerging data-driven turbulence models.
[^2]: As explained in the notation, the norms $\| \overline{{\bm{u}}}\|$, $ \|\bm{b}
\|$ are taken of the discretized vectors ${[\overline{{\bm{u}}}]}$ and ${[\bm{b}]}$, respectively, with the brackets inside the norm omitted for clarity. Such a brief notation would not cause confusion, because norms in this work are always taken for the *discretized vectors or matrices* with dimensions of $n\times 1$ or $n\times n$, respectively, and never for the velocity or force vectors at any particular location.
|
---
abstract: 'The Poisson-Nernst-Planck (PNP) system is a widely accepted model for simulation of ionic channels. In this paper, we design, analyze, and numerically validate a second order unconditional positivity-preserving scheme for solving a reduced PNP system, which can well approximate the three dimensional ion channel problem. Positivity of numerical solutions is proven to hold true independent of the size of time steps and the choice of the Poisson solver. The scheme is easy to implement without resorting to any iteration method. Several numerical examples further confirm the positivity-preserving property, and demonstrate the accuracy, efficiency, and robustness of the proposed scheme, as well as the fast approach to steady states.'
address: 'Iowa State University, Mathematics Department, Ames, IA 50011'
author:
- Hailiang Liu and Wumaier Maimaitiyiming
title: 'Unconditional positivity-preserving and energy stable schemes for a reduced Poisson-Nernst-Planck system'
---
Introduction
============
Biological cells exchange chemicals and electric charge with their environments through ionic channels in the cell membrane walls. Examples include signaling in the nervous system and coordination of muscle contraction, see [@Cu09] for a comprehensive introduction. Mathematically the flow of ions can be modeled by drift-diffusion equations such as the Poisson-Nernst-Planck (PNP) system, see e.g. [@Coalson05; @Eis96; @Eis07; @Eis04].
In this investigation we design, analyze and numerically validate positivity-preserving algorithms to solve time-dependent drift-diffusion equations. As a first step, in this paper we focus on a reduced model derived by Gardner et al [@Eis04] as an approximation to the full three dimensional (3D) PNP system. Let us first recall the full model and its reduction.
Mathematical models
-------------------
The general setup in [@Eis04] is a flow of positive and negative ions in water in a channel plus surrounding baths in an electric field against a background of charged atoms on the channel protein. The distribution of charges is described by continuum particle densities $c_i(\mathbf{x},t)$ for the mobile ions (such as $K^+, N_a^+, C_a^{++},\cdots $). The flow of ions can be modeled by the PNP system of $m$ equations $$\label{PNP0}
\begin{aligned}
\partial_t c_i&= -\nabla\cdot J_i,\quad i=1, \cdots, m; \; \mathbf{x}\in \Omega \subset \mathbb{R}^3, \; t>0, \\
J_i & = -(D_i\nabla c_i+z_i\mu_ic_i\nabla \psi),\\
-\nabla \cdot(\epsilon \nabla \psi)&=\sum_{i=1}^{m}q_ic_i-e\rho,
\end{aligned}$$ where $J_i$ is the flux density, in which $D_i$ is the diffusion coefficient, $\mu_i$ the mobility coefficient which is related to the diffusion coefficient via Einstein’s relation $\mu_i=\frac{D_i}{k_BT_0}$, where $k_B$ is the Boltzmann constant and $T_0$ is the absolute temperature [@Cu09]. In the Poisson equation, $\epsilon$ is the dielectric coefficient, $q_i$ the ionic charge for each ion species $i$, $\rho=\rho(\mathbf{x})$ the permanent fixed charge density, and $e$ the proton charge. The coupling parameter $z_i = q_i/e$. In general, the physical parameters $\epsilon$, $\mu_i$ and $D_i$ are functions of $\mathbf{x}$. Let us mention that the case of no permanent charge does not pertain to biological channels. Even channels without permanent charge (in the form of so called acid and base side chains) have large amounts of fixed charge in their (for example) carbonyl bonds( see, e.g., [@WS15] and references therein).
The derivation of the Nernst-Planck equation typically follows two steps, namely, using the energy variation to obtain the chemical potential and then using Fick’s laws of diffusion to attain the Nernst-Planck equation (see e.g. [@FB97]). In the charge dynamics modeled by the traditional NP equation, mobile ions are treated as volume-less point charges. In order to incorporate more complex effects such as short-range steric effect and long range Coulomb correlation, modifications of the PNP equations were derived ( see, e.g., [@PXZ18] and references therein). Nonetheless, the scheme methodology proposed in this paper can well be adapted to solve such modified PNP systems.
The 3D geometry of the ion channel can be approximated by a reduced problem along the axial direction $x$, with a cross-sectional area $A(x)$ [@SN09; @SGNE08]. Subject to a further rescaling as in [@GLE2018], the corresponding PNP system (\[PNP0\]) reduces to the following equations $$\label{PNP2}
\begin{aligned}
\partial_t c_i&=\frac{1}{A(x)} \partial_x (A(x)D_i(\partial_x c_i+z_ic_i\partial_x \psi)), \quad x \in \Omega=[0,\ 1], \quad t>0,\\
-\frac{1}{A(x)} \partial_x (\epsilon A(x) \partial_x \psi)&=\sum_{i=1}^{m}z_ic_i-\rho(x), \quad x \in \Omega, \quad t>0.
\end{aligned}$$ For ionic channels, an important characteristic is the so-called current-voltage relation, which can characterize permeation and selectivity properties of ionic channels (see [@WS-IV-08] and references therein). For (\[PNP0\]), the electric current density (charge flux) is $J=\sum_{i=1}^m q_i J_i$. Such quantity for (\[PNP2\]) reduces to $$\label{J}
J=-\sum_{i=1}^m z_i D_iA(x) ( \partial_x c_i + z_ic_i \partial_x \psi).$$ System (\[PNP2\]) is a parabolic/elliptic system of partial differential equations, boundary conditions for both $c_i$ and $\psi$ can be Dirichlet or Neumann.
In order to solve the above reduced system, we consider initial data $$c_i(x, 0)=c_i^{\rm in}(x) \geq 0, \quad x\in \Omega.$$
Boundary conditions and model properties
----------------------------------------
We consider two types of boundary conditions. The first is the Dirichlet boundary condition, $$\label{Di}
c_i(0,t)=c_{i,l}, \quad c_i(1,t)=c_{i,r}; \quad \psi(0,t)=0, \psi(1,t)=V,\quad t>0,$$ where $c_{i,l}, c_{i, r}$ are non-negative constants, and $V$ is a given constant. This is the setting adopted in [@Eis04]. One important solution property is $$\label{po}
c_i(x, t) \geq 0, \quad x\in \Omega, \; t>0.$$ Another set of boundary conditions is as follows: $$\label{Zf}
\begin{aligned}
&\partial_x c_i+z_ic_i\partial_x \psi=0,\quad x=0,1, \quad t>0,\\
& (- \eta \partial_x \psi+\psi)|_{x=0}=\psi_{_-},\quad ( \eta \partial_x \psi+\psi)|_{x=1}=\psi_{_+}, \quad t>0,
\end{aligned}$$ where $\psi_{_-}, \psi_{_+}$ are given constants, the size of parameter $\eta $ depends on the properties of the surrounding membrane [@Eis13]. Here the first one is the zero-flux boundary condition for the transport equation, and the second is the Robin boundary condition for the Poisson equation. Such boundary condition is adopted in [@Eis13] to model the effects of partially removing the potential from the ends of the channel. For system (\[PNP2\]) with this boundary condition, solutions have non-negativity, mass conservation, and free energy dissipation properties, i.e., (\[po\]), $$\label{mass}
\int_{\Omega}A(x)c_i(x,t)dx=\int_{\Omega}A(x)c_i^{in}(x)dx, \quad t>0,\quad i=1,\cdots,m, \; \text{and}$$
$$\label{energydiss1}
\frac{dE}{dt}=-\int _{\Omega} \sum_{i=1}^m A(x)D_ic_i|\partial _{x} ( \log c_i+z_i\psi)|^2 dx \leq 0 ,$$
where the total energy $E$ associated to (\[PNP2\]) is defined (see [@Eis13]) by $$\label{energy1}
E=\int_{\Omega}A(x) \bigg(\sum_{i=1}^m c_i\log c_i+\frac{1}{2}(\sum_{i=1}^mz_ic_i-\rho)\psi\bigg) d x+\frac{\epsilon}{2 \eta }(\psi_{_+}A(1)\psi(1)+\psi_{_-}A(0)\psi(0)).$$ The positivity-preserving property is of special importance, since negative values in density would violate the physical meaning of the solution and may destroy the energy dissipation law (\[energydiss1\]). Numerical techniques addressing the positivity preserving property have been introduced in various application problems, see e.g. [@JSY19; @LYY18]. In this paper, we construct second order accurate unconditional positivity-preserving schemes for solving (\[PNP2\]) subject to two types of boundary conditions. For the zero-flux boundary condition, the schemes will be shown to satisfy mass conservation and a discrete energy dissipation law.
Related works
-------------
Numerical methods for solving the PNP system of equations have been studied extensively; see e.g., [@Eis04; @HP16; @HEL11; @LHMZ10; @ZCW11]. We also refer to [@Gwei16] for a review on the PNP model and its generalizations for ion channel charge transport.
For the reduced PNP system (\[PNP2\]), the finite difference scheme with TR–BDF2 time integration was first pursued in [@Eis04] to simulate an ionic channel. For the one dimensional PNP system, the second order implicit finite difference scheme proposed in [@Eis13] can preserve total concentration of ions with the aid of a special boundary discretization, but numerical solutions may not be positive or energy dissipating. An improved scheme, further introduced in [@Flavell7], can preserve a discrete form of energy dissipation law up to $O(\tau^2+h^2)$, where $\tau$ is the time step, and $h$ is the spatial mesh size. In [@CH19] the authors proposed an adaptive conservative finite volume method on a moving mesh that maintains solution positivity. The second oder finite difference scheme in [@LW14] is explicit and shown to preserve positivity, mass conservation, and energy dissipation, while the positivity-preserving property is ensured if $\tau=O(h^2)$. Further extension in [@LW17] is a free energy satisfying discontinuous Galerkin scheme of any high order, where positivity-preserving property is realized by limiting techniques. The finite element scheme obtained by the method of lines approach in [@Me16] preserves positivity of the solutions and a discrete energy dissipation law. Recently in [@Hjingwei19] the authors presented a fully implicit finite difference method where both positivity and energy decay are preserved. In their scheme a fixed point iteration is needed for solving the resulting nonlinear system. These schemes are either explicit or fully implicit in time, the former require a time step restriction for preserving the desired properties while the later preserve desired properties unconditionally but they had to be solved by some iterative solvers.
In this paper we design schemes to preserve all three desired properties of solution: positivity, mass conservation, and energy dissipation, by following [@LiuWu2018], in which a second order finite-volume method was constructed for the class of nonlinear nonlocal equations $$\label{LiuWu}
\partial_t c =\nabla \cdot (\nabla c+c \nabla (V(\mathbf{x})+W*c )).$$ The key ingredients include a reformulation of the equation in its non-logarithmic Landau form and the use of the implicit-explicit time discretization, these together ensure the positivity-preserving property without any restriction on the size of time steps (unconditional!) and do not require iterative solvers.
Contributions and organization of the paper
-------------------------------------------
Our scheme construction is based on the reformulation $$\label{RF}
A(x)\partial_t c_i(x,t) =\partial_x (A(x)D_ie^{-z_i\psi(x,t)}\partial_x( c_i(x,t)e^{z_i\psi(x,t)})),$$ of the transport equation in (\[PNP2\]). Similar formulation has been used in [@LiuWu2018] and in earlier works [@LY12; @LW14]. Following [@LiuWu2018], we adopt a semi-implicit time discretization of (\[RF\]): $$\label{time}
A(x)\frac{c_i^{n+1}(x)-c_i^{n}(x)}{\tau}=\partial_x \left( A(x)D_i e^{-z_i \psi^n(x)} \partial_x (c_i^{n+1}(x) e^{z_i \psi^n(x)}) \right).$$ The feature of such discretization is that it is a linear equation in $c_i^{n+1}(x)$, and easy to solve numerically. For spatial discretization, we use the central finite volume approach. The coefficient matrix of the resulting linear system is an M-matrix and right hand side is a nonnegative vector, thus positivity of the solution is ensured without any time step restriction.
The main contribution in this paper includes the model reformulation, proofs of unconditional positivity-preserving properties for two types of boundary conditions, and of mass conservation and energy dissipation properties for zero flux boundary conditions (\[Zf\]). In addition, the positivity-preserving property is shown to be independent of the choice of Poisson solvers. Our implicit-explicit scheme is easy to implement and efficient in computing numerical solutions over long time.
The paper is organized as follows. In section 2, we derive our numerical scheme for a model equation. Theoretical analysis of unconditional positivity is provided. In section 3, we formulate our scheme to the PNP system and prove positivity, mass conservation and energy dissipation properties of the scheme. Numerical examples are presented in section 4. Finally, concluding remarks are given in section 5.
Numerical methods for a model equation
======================================
In this section, we first demonstrate the key ideas through a model problem. Let $u(x, t)$ be an unknown density, satisfying $$\label{Model1}
\begin{aligned}
A(x)\partial_t u(x,t)&= \partial_x (B(x)(\partial_xu(x,t)-u(x,t)\partial_x \phi (x,t)) ), \quad x \in \Omega=[0,\ 1], \quad t>0,\\
u(x,0)&=u^{in}(x), \quad x \in \Omega, \\
\end{aligned}$$ where $A(x)>0, B(x)>0$ are given functions, and $\phi(x,t)$ is either known or can be obtained from solving another coupled equation. For this model problem, we consider two types of boundary conditions:\
(i) the Dirichlet boundary condition $$\label{Di2}
u(0,t)=u_l, \quad u(1,t)=u_r, \quad t > 0,$$ and (ii) the zero flux boundary condition $$\label{Zf2}
\partial_xu(x,t)-u(x,t)\partial_x \phi (x,t) =0, \quad x=0,1, \quad t>0.$$
Scheme formulation
------------------
Let $N$ be an integer, and the domain $\Omega=[0, \ 1]$ be partitioned into computational cells $I_j=[x_{j-1/2}, \ \ x_{j+1/2}]$ with cell center $x_{j}=x_{j-1/2}+\frac{1}{2}h$, for $j\in \{1,2, \cdots, N\},$ $x_{1/2}=0$ and $x_{N+1/2}=1.$ For simplicity, uniform mesh size $h=\frac{1}{N}$ is adopted. Discretize $t$ uniformly as $t_n=\tau n$, where $\tau$ is time step.
From the reformulation $$\label{Model2}
A(x)\partial_t u(x,t)= \partial_x (B(x)e^{\phi(x,t)}\partial_x( u(x,t)e^{-\phi(x,t)}))$$ of (\[Model1\]), we consider a semi-implicit time discretization as follows: $$\label{time1}
A(x)\frac{u^{n+1}(x)-u^{n}(x)}{\tau}=\partial_x \left( B(x)e^{\phi^n(x)} \partial_x (u^{n+1}(x) e^{-\phi^n(x)}) \right),$$ where $u^{n}(x)\approx u(x,t_n),$ $\phi^n(x)\approx \phi(x,t_n).$ Let $u^n_j\approx \frac{1}{h}\int_{I_j}u^n(x)dx$, and $ A_{j}=\frac{1}{h}\int_{I_j} A(x)dx$, then a fully-discrete scheme of (\[time1\]) can be given by $$\label{fully}
A_j\frac{u^{n+1}_{j}-u^n_{j}}{\tau}=\frac{U_{j+1/2}-U_{j-1/2}}{h},$$ where the flux on interior interfaces are defined by $$\label{UU}
U_{j+1/2}=B_{j+1/2} e^{\phi^n_{j+1/2}}\frac{u^{n+1}_{j+1}e^{-\phi^n_{j+1}}- u^{n+1}_{j}e^{-\phi^n_{j}} }{h}, \quad j=1,2,\cdots, N-1.$$ Here $B_{j+1/2}=B(x_{j+1/2})$; For $\phi^n_{j+1/2}$ we either use $\phi(x_{j+1/2}, t_n)$ if $\phi(x, t)$ is given, or $$\phi^n_{j+1/2}=\frac{\phi_j^n+ \phi^n_{j+1}}{2},$$ where $\phi^n_{j}$ is a numerical approximation of $\phi(x_j, t_n)$.
The boundary fluxes are given as follows:\
(i) for the Dirichlet boundary condition (\[Di2\]) $$\label{U11}
\begin{aligned}
U_{1/2}&=B_{1/2}e^{\phi^n_{1/2}}\frac{2(u^{n+1}_{1}e^{-\phi^n_1}-u_{l}e^{-\phi^n_{1/2}})}{h}, \\
U_{N+1/2}&=B_{N+1/2}e^{\phi^n_{N+1/2}}\frac{2(u_{r}e^{-\phi^n_{N+1/2}}-u^{n+1}_{N}e^{-\phi^n_{N}})}{h};
\end{aligned}$$ (ii) for the zero flux boundary condition (\[Zf2\]), $$\label{Zf3}
U_{1/2}=U_{N+1/2}=0.$$ In either case, the initial data are determined by $$u_{j}^0=\frac{1}{h}\int_{I_j}u^{in}(x)dx, \quad j=1,\cdots, N.$$ Before turning to the analysis of solution properties, we comment on these boundary fluxes.
The factor 2 in the boundary flux (\[U11\]) suffices to ensure the first order accuracy in the approximation of $$B(x)e^{\phi(x,t)}\partial_x( u(x,t)e^{-\phi(x,t)})$$ at the boundary; see [@Eymard]. However, the following flux without the factor 2, i.e. $$\label{U0}
\begin{aligned}
U_{1/2}&=B_{1/2}e^{\phi^n_{1/2}}\frac{(u^{n+1}_{1}e^{-\phi^n_1}-u_{l}e^{-\phi^n_{1/2}})}{h}, \\
U_{N+1/2}&=B_{N+1/2}e^{\phi^n_{N+1/2}}\frac{(u_{r}e^{-\phi^n_{N+1/2}}-u^{n+1}_{N}e^{-\phi^n_{N}})}{h},
\end{aligned}$$ can produce only a zeroth order approximation at the boundary. Order loss of accuracy has been observed in our numerical tests when (\[U0\]) is used.
An alternative boundary flux for (i) is a second order approximation of the form $$\label{U22}
\begin{aligned}
U_{1/2}&=B_{1/2}e^{\phi^n_{1/2}}\frac{-\frac{1}{3}u^{n+1}_{2}e^{-\phi^n_{2}}+3u^{n+1}_{1}e^{-\phi^n_{1}}-\frac{8}{3}u_le^{-\phi^n_{1/2}}}{h}, \\
U_{N+1/2}&=B_{N+1/2}e^{\phi^n_{N+1/2}}\frac{\frac{1}{3}u^{n+1}_{N-1}e^{-\phi^n_{N-1}}-3u^{n+1}_{N}e^{-\phi^n_{N}}+\frac{8}{3}u_re^{-\phi^n_{N+1/2}}}{h}.
\end{aligned}$$ However, it is known that the first order boundary flux does not destroy the second order accuracy of the scheme, we refer to [@LW18] for a such result regarding the Shortley-Weller method. Hence throughout the paper, we will not discuss high order boundary fluxes such as (\[U22\]).
Positivity
----------
It turns out that both schemes, (\[fully\])-(\[UU\])-(\[U11\]) and (\[fully\])-(\[UU\])-(\[Zf3\]), preserve positivity of numerical solutions without any time step restriction.
Scheme (\[fully\])-(\[UU\]) with either (i) (\[U11\]) and $u_l\geq 0, u_r\geq 0$, or (ii) (\[Zf3\]), is positivity-preserving, in the sense that if $u^n_{j}\geq 0$ for all $j=1,\cdots, N$, then $$u^{n+1}_{j}\geq 0 \; \text{for all}\; j=1,\cdots, N.$$
Set mesh ratio $\lambda=\frac{\tau}{h^2}$ and introduce $G_{j}=u^{n+1}_j e^{-\phi^n_{j}}$, so that\
(i) scheme (\[fully\]), (\[UU\]) and (\[U11\]) can be rewritten as $$\label{G}
\begin{aligned}
&(A_1e^{\phi^n_1}+\lambda B_{3/2}e^{\phi^n_{3/2}}+2\lambda B_{1/2}e^{\phi^n_{1/2}})G_1-\lambda B_{3/2}e^{\phi^n_{3/2}}G_2=A_1u^n_1+2\lambda B_{1/2} u_l,\\
&-\lambda B_{j-1/2}e^{\phi^n_{j-1/2}}G_{j-1}+(A_je^{\phi^n_j}+\lambda B_{j+1/2}e^{\phi^n_{j+1/2}}+\lambda B_{j-1/2}e^{\phi^n_{j-1/2}})G_j-\lambda B_{j+1/2}e^{\phi^n_{j+1/2}}G_{j+1}=A_ju^n_j,\\
&-\lambda B_{N-1/2}e^{\phi^n_{N-1/2}}G_{N-1}+(A_N e^{\phi^n_N}+\lambda B_{N-1/2}e^{\phi^n_{N-1/2}}+2\lambda B_{N+1/2}e^{\phi^n_{N+1/2}})G_N=a_Nu^n_N+2\lambda B_{N+1/2} u_r.
\end{aligned}$$ This linear system of $\{G_j\}$ admits a unique solution since its coefficient matrix is strictly diagonally dominant. Since $u^{n+1}_{j}=e^{\phi_j^n} G_j \geq e^{\phi_j^n} G_k$, where $$G_{k}=\min_{1\leq j\leq N} \{ G_{j}\},$$ it suffices to prove $G_k\geq0$. We discuss in cases: if $1<k<N,$ then from the $k$-th equation of (\[G\]) with $A_k>0$ it follows $$\begin{aligned}
A_ku^n_k \leq & -\lambda B_{k-1/2}e^{\phi^n_{k-1/2}}G_{k}+(A_ke^{\phi^n_k} +\lambda B_{k+1/2}e^{\phi^n_{k+1/2}}+\lambda B_{k-1/2}e^{\phi^n_{k-1/2}})G_k\\
&-\lambda B_{k+1/2}e^{\phi^n_{k+1/2}}G_{k}=A_ke^{\phi^n_k}G_k.\end{aligned}$$ Hence $G_{k} \geq u^n_k e^{- \phi^n_k} \geq 0$; if $k=1,$ from the first equation of (\[G\]) we have $$\begin{aligned}
A_1u^n_1+2\lambda B_{1/2}u_l \leq & (A_1e^{\phi^n_1}+\lambda B_{3/2}e^{\phi^n_{3/2}}+2\lambda B_{1/2}e^{\phi^n_{1/2}})G_1-\lambda B_{3/2}e^{\phi^n_{3/2}}G_1= (A_1e^{\phi_1}+2\lambda B_{1/2}e^{\phi_l})G_1.\end{aligned}$$ This implies $G_1\geq 0$; so does the case if $k=N$.
\(ii) Likewise, scheme (\[fully\]), (\[UU\]) and (\[Zf3\]) can be rewritten as $$\begin{aligned}
&(A_1e^{\phi^n_1}+\lambda B_{3/2}e^{\phi^n_{3/2}})G_1-\lambda B_{3/2}e^{\phi^n_{3/2}}G_2=A_1u^n_1,\\
&-\lambda B_{j-1/2}e^{\phi^n_{j-1/2}}G_{j-1}+(A_je^{\phi^n_j}+\lambda B_{j+1/2}e^{\phi^n_{j+1/2}}+\lambda B_{j-1/2}e^{\phi^n_{j-1/2}})G_j-\lambda B_{j+1/2}e^{\phi^n_{j+1/2}}G_{j+1}=A_ju^n_j,\\
&-\lambda B_{N-1/2}e^{\phi^n_{N-1/2}}G_{N-1}+(A_N e^{\phi^n_N}+\lambda B_{N-1/2}e^{\phi^n_{N-1/2}})G_N=A_Nu^n_N.
\end{aligned}$$ Using an entirely same argument, we can show $G_j \geq 0$, hence $u_j^{n+1}\geq0$ for all $j$ involved.
The specific values or choices of $\{\phi^n_j\}$ and $\{\phi^n_{j+1/2}\}$ do not affect the unconditional positivity property of the scheme for $\{u^n_j\}.$ This result thus can be applied to the case when $\phi(x, t)$ is solved by the Poisson equation, see the next section.
Positive schemes for the reduced PNP-system
============================================
The reduced PNP system (\[PNP2\]) is reformulated as $$\label{PNP2+}
\begin{aligned}
A(x) \partial_t c_i &= \partial_x (A(x)D_ie^{-z_i\psi} \partial_x (c_ie^{z_i\psi})),\\
-\partial_x (\epsilon A(x) \partial_x \psi) & = A(x) \left(\sum_{i=1}^{m}z_ic_i-\rho(x)\right).
\end{aligned}$$ Let $c^n_{i,j}$ and $\psi^n_j$ approximate the cell average $\frac{1}{h}\int_{I_j}c_i(x,t_n)dx$ and $\frac{1}{h}\int_{I_j}\psi(x,t_n)$ respectively, then from the discretization strategy in section 2 the fully discrete scheme for system (\[PNP2+\]) follows $$\begin{aligned}
\label{fullyCi}
& A_j\frac{c^{n+1}_{i,j}-c^n_{i,j}}{\tau} =\frac{C_{i,j+1/2}-C_{i,j-1/2}}{h}, \\
\label{fullyPs}
& - \frac{\Psi^n_{j+1/2}-\Psi^n_{j-1/2}}{h}=A_{j}\bigg(\sum_{i=1}^{m}z_ic^n_{i,j}-\rho_j\bigg),\end{aligned}$$ where numerical fluxes on interior interfaces are defined by $$\begin{aligned}
\label{CC}
C_{i,j+1/2} &=A_{j+1/2}D_i e^{-z_i\psi_{j+1/2}^n} \frac{\left( c^{n+1}_{i,j+1}e^{z_i\psi_{j+1}^n} -c^{n+1}_{i,j}e^{z_i\psi_{j}^n}\right)}{h}, \; j=1, \cdots, N-1.\\
\label{Ps2}
\Psi^n_{j+1/2}&=\epsilon A_{j+1/2}\frac{\psi^n_{j+1}-\psi^n_j}{h}, \quad j=1,\cdots, N-1,\end{aligned}$$ where relevant terms are determined by $$\begin{aligned}
& A_j=\frac{1}{h}\int_{I_j}A(x)dx, \quad \rho_j =\frac{1}{h}\int_{I_j}\rho(x)dx \\
& A_{j+1/2}=A(x_{j+1/2}), \; \psi^n_{j+1/2} =(\psi^n_{j}+\psi^n_{j+1})/2.\end{aligned}$$ For non-trivial $A(x), \rho(x)$, numerical integration of high accuracy is used to evaluate $A_j$ and $\rho_j$. The boundary fluxes are defined as follows:
\(i) for Dirichlet boundary condition (\[Di\]), $$\label{BC11}
\begin{aligned}
C_{i,1/2}&=A_{1/2}D_i\frac{2(c^{n+1}_{i,1}e^{z_i\psi^n_1}-c_{i,l})}{h}, \\
C_{i,N+1/2}&=A_{N+1/2}D_ie^{-z_iV}\frac{2(c_{i,r}e^{z_iV}-c^{n+1}_{i,N}e^{z_i\psi^n_{N}})}{h},\\
\Psi^n_{1/2}&=\epsilon A_{1/2}\frac{2\psi^n_{1}}{h}, \\
\Psi^n_{N+1/2}&=\epsilon A_{N+1/2}\frac{2(V-\psi^n_{N})}{h},
\end{aligned}$$
\(ii) for boundary condition (\[Zf\]):\
$$\label{ZfBC}
\begin{aligned}
& C_{i,1/2}=0,\quad C_{i,N+1/2}=0,\\
& \Psi^n_{1/2}=\frac{\epsilon}{\eta} A_{1/2}(\psi^n_{1}-\psi_{_-}),\quad \Psi^n_{N+1/2}=\frac{\epsilon}{\eta} A_{N+1/2}(\psi_{_+}-\psi^n_{N}).
\end{aligned}$$
Scheme properties
-----------------
Scheme (\[fullyCi\])-(\[Ps2\]) with (\[BC11\]) turns out to be unconditionally positivity-preserving.
Let $\psi^n_j$ and $c_{i,j}^{n+1}$ for $i=1,\cdots, m,$ $j=1, \cdots, N$ be obtained from (\[fullyCi\])-(\[Ps2\]) with (\[BC11\]). If $c^n_{i,j}\geq 0$ and $c_{i,l}\geq 0,$ $c_{i,r}\geq 0$ for $i=1,\cdots, m$, $j=1, \cdots, N$, then $c^{n+1}_{i,j}\geq 0$ for all $i=1,\cdots, m,$ $j=1, \cdots, N$.
For fixed $i=1,\cdots,m$, the scheme (\[fullyCi\]), (\[CC\]) and (\[BC11\]) is of the same form as (\[fully\]), (\[UU\]) and (\[U11\]) with $u^n_j=c^n_{i,j},$ $B_{j+1/2}=A_{j+1/2}D_i$, $\phi^n_j=-z_i\psi^n_j$ and $\phi^n_{j+1/2}=-z_i\psi^n_{j+1/2}$. From (i) in Theorem 2.2, we can conclude $c^{n+1}_{i,j}=u^{n+1}_j \geq 0$.
From the above analysis we see that positivity of $c_{i,j}^n$ remains true even when another Poisson solver is used.
For scheme (\[fullyCi\])-(\[Ps2\]) with (\[ZfBC\]), it turns out that the solution $c^n_{i,j}$ is conservative, non-negative, and energy dissipating. In order to state the energy dissipation result, we define a discrete version of the free energy (\[energy1\]) as $$\label{fullyenergy}
E_h^n=\sum_{j=1}^NhA_j\bigg(\sum_{i=1}^mc^n_{i,j}\log c^n_{i,j}+\frac{1}{2}S_j^n \psi^n_j \bigg)+\frac{\epsilon}{2\eta }(\psi_{_+}A_{N+1/2}\psi^n_{N}+\psi_{_-}A_{1/2}\psi^n_{1}),$$ where $$S_j^n=\sum_{i=1}^m z_ic^n_{i,j}-\rho_j.$$
Let $\psi^n_j$ and $c^{n}_{i,j}$ be obtained from (\[fullyCi\])-(\[Ps2\]) and (\[ZfBC\]), then we have:\
(1) Conservation of mass: $$\label{mass2}
\sum_{j=1}^NhA_jc_{i,j}^{n+1}= \sum_{j=1}^Nh A_jc_{i,j}^n \ \ \text{ for } n\geq 0, i=1,\cdots,m;$$ (2) Propagation of positivity: if $c_{i,j}^n\geq0$ for all $j=1,\cdots, N,$ and $i=1,\cdots,m,$ then $$c_{i,j}^{n+1}\geq 0, \quad j=1,\cdots,N, i=1,\cdots,m;$$ (3) Energy dissipation: there exists $C^*>0$ depending on numerical solutions but independent on $\tau$ and $h$, such that if $\tau\leq C^*\epsilon/\eta$, then $$\label{FE}
E_{h}^{n+1}-E_{h}^n\leq-\frac{\tau}{2}I_h^n,$$ where $$I_h^n=\sum_{i=1}^m \sum_{j=1}^{N-1}\frac{1}{h}A_{j+1/2}D_i (c_{i,j+1}^{n+1}e^{z_i\psi_{j+1}^n}-c_{i,j}^{n+1}e^{z_i\psi_j^n})( \log c_{i,j+1}^{n+1}e^{z_i\psi_{j+1}^n}-\log c_{i,j}^{n+1}e^{z_i\psi_j^n})\geq 0.$$
\(1) Mass conservation follows from summing (\[fullyCi\]) over $j=1,\cdots, N$ and using (\[ZfBC\]).
\(2) For each fixed $i=1, \cdots m$, this follows from (ii) in Theorem 2.2, by taking $u^n_j=c^n_{i,j}$, $B_{j+1/2}=A_{j+1/2}D_i$, $\phi^n_j=-z_i\psi^n_j$ and $\phi^n_{j+1/2}=-z_i\psi^n_{j+1/2}.$
\(3) Using (\[fullyenergy\]) we find that $$\begin{aligned}
E^{n+1}_h-E^n_h
=&\sum_{j=1}^NhA_j\bigg( \sum_{i=1}^m(c_{i,j}^{n+1}-c_{i,j}^n)(\log c_{i,j}^{n+1}+z_i\psi_j^n)+ \sum_{i=1}^m c_{i,j}^n\log c_{i,j}^{n+1} -\sum_{i=1}^m c_{i,j}^n\log c_{i,j}^n \\
&+ \frac{1}{2}S_j^{n+1} \psi_j^{n+1}-\frac{1}{2}S_j^n \psi_j^n-\sum_{i=1}^m z_i(c_{i,j}^{n+1}-c_{i,j}^n)\psi_j^n\bigg)\\
& +\frac{\epsilon}{2\eta }(A_{N+1/2}\psi_{_+}\psi_N^{n+1}+A_{1/2}\psi_{_-}\psi_1^{n+1})-\frac{\epsilon}{2\eta}(A_{N+1/2}\psi_{_+}\psi_N^{n}+A_{1/2}\psi_{_-}\psi_1^{n})\\
=: &I+II+III. \end{aligned}$$ We proceed to estimate term by term. For $I$, we use scheme (\[fullyCi\])-(\[CC\]) and (\[ZfBC\]) and summation by parts to obtain $$\begin{aligned}
I=& \sum_{j=1}^NhA_j \sum_{i=1}^m(c_{i,j}^{n+1}-c_{i,j}^n)(\log c_{i,j}^{n+1}+z_i\psi_j^n)\\
=& \tau \sum_{i=1}^m \sum_{j=1}^N (C_{i,j+1/2}-C_{i,j-1/2}) \log (c_{i,j}^{n+1}e^{z_i\psi_j^n})\\
=& -\tau \sum_{i=1}^m \sum_{j=1}^{N-1} C_{i,j+1/2}( \log c_{i,j+1}^{n+1}e^{z_i\psi_{j+1}^n}-\log c_{i,j}^{n+1}e^{z_i\psi_j^n})\\
=& -\tau \sum_{i=1}^m \sum_{j=1}^{N-1}\frac{1}{h}A_{j+1/2}D_i (c_{i,j+1}^{n+1}e^{z_i\psi_{j+1}^n}-c_{i,j}^{n+1}e^{z_i\psi_j^n})( \log c_{i,j+1}^{n+1}e^{z_i\psi_{j+1}^n}-\log c_{i,j}^{n+1}e^{z_i\psi_j^n})\\
=& -\tau I_h^n\leq0.\end{aligned}$$ For $II$, we use $\log(X)\leq X-1$ for $X>0,$ to obtain $$\begin{aligned}
II=&\sum_{j=1}^{N}hA_{j} ( \sum_{i=1}^m c_{i,j}^n\log c_{i,j}^{n+1} -\sum_{i=1}^m c_{i,j}^n\log c_{i,j}^n )\\
=& \sum_{j=1}^{N}hA_{j} \sum_{i=1}^m c_{i,j}^n\log \frac{c_{i,j}^{n+1}}{c_{i,j}^n} \\
\leq & \sum_{i=1}^m \sum_{j=1}^{N}hA_{j} c_{i,j}^n( \frac{c_{i,j}^{n+1}}{c_{i,j}^n}-1) \\
= & \sum_{i=1}^m \sum_{j=1}^{N}hA_{j} ( c_{i,j}^{n+1} -c_{i,j}^n) =0,\end{aligned}$$ where in the last equality we have used conservation of mass.
Rearranging terms in $III$, we find that $$\begin{aligned}
III=&\sum_{j=1}^{N}hA_{j}\bigg( \frac{1}{2}S_j^{n+1} \psi_j^{n+1}+\frac{1}{2}S_j^n\psi_j^n-S_j^{n+1}\psi_j^n\bigg)\\
& +\frac{\epsilon}{2\eta }(A_{N+1/2}\psi_{_+}\psi_N^{n+1}+A_{1/2}\psi_{_-}\psi_1^{n+1})-\frac{\epsilon}{2\eta}(A_{N+1/2}\psi_{_+}\psi_N^{n}+A_{1/2}\psi_{_-}\psi_1^{n})\\
&=\frac{1}{2}\sum_{j=1}^{N}hA_{j}(S_j^{n+1}-S_j^n)(\psi_j^{n+1}-\psi_j^n)+\mathbf{F},\end{aligned}$$ where $$\begin{aligned}
\mathbf{F}=&\frac{1}{2} \sum_{j=1}^{N}hA_{j}( S_j^n \psi_j^{n+1}-S_j^{n+1}\psi_j^n)+\frac{\epsilon}{2\eta }(A_{N+1/2}\psi_{_+}\psi_N^{n+1}+A_{1/2}\psi_{_-}\psi_1^{n+1})\\
&-\frac{\epsilon}{2\eta }(A_{N+1/2}\psi_{_+}\psi_N^{n}+A_{1/2}\psi_{_-}\psi_1^{n}).\end{aligned}$$ Note that $$\begin{aligned}
-\frac{1}{2}(\sum_{i=1}^m z_ic_{i,j}^{n}-\rho_j)\psi_j^{n+1}=& -\frac{1}{2}(\sum_{i=1}^m z_ic_{i,j}^{n+1}-\rho_j)\psi_j^n+\frac{\epsilon}{2}(A_{N+1/2}\psi_{_+}\psi_N^{n+1}+A_{1/2}\psi_{_-}\psi_1^{n+1})\\
&-\frac{\epsilon}{2}(A_{N+1/2}\psi_{_+}\psi_N^{n}+A_{1/2}\psi_{_-}\psi_1^{n}).\end{aligned}$$ Tedious but elementary calculations show that $\mathbf{F} \equiv 0.$ Thus $$\label{C4}
III=\frac{1}{2}\sum_{j=1}^{N}hA_{j}(S_j^{n+1}-S_j^n)(\psi_j^{n+1}-\psi_j^n).$$ Scheme (\[fullyPs\])-(\[Ps2\]) and (\[ZfBC\]) can be written in matrix form $$M\vec{\psi}^n=\vec{b},$$ where $$\label{TrM}
M= \begin{bmatrix}
\frac{h}{\eta} A_{1/2}+A_{3/2} & -A_{3/2} & & & \\
-A_{3/2} & A_{3/2}+A_{5/2} & -A_{5/2} & & \\
&\ddots &\ddots&\ddots & \\
& & -A_{N-3/2} & A_{N-3/2}+A_{N-1/2} & -A_{N-1/2} \\
& & & -A_{N-1/2} & \frac{h}{\eta} A_{N+1/2}+A_{N-1/2}
\end{bmatrix},$$ $$\vec{b}=\frac{h^2}{\epsilon}\bigg(A_1S_1^n +\frac{\epsilon}{h\eta }A_{1/2}\psi_{_-}, A_2S_2^n, \cdots, A_NS_N^n+\frac{\epsilon}{h \eta}A_{N+1/2}\psi_{_+} \bigg)^\top.$$ Hence we have $$\psi_j^{n+1}-\psi_j^n=\frac{\tau h^2}{\epsilon}\sum_{k=1}^N(M^{-1})_{j,k}A_k D_t S_k^n, \quad \tau D_t S_j^n: =S_j^{n+1}-S_j^n,$$ thus (\[C4\]) can be simplified as $$III=\frac{h^3\tau^2 }{2\epsilon} \sum_{j=1}^NA_jD_tS_j^n\sum_{k=1}^N(M^{-1})_{j,k}A_kD_tS_k^n.$$ We claim that for any $\zeta \in \mathbb{R}^N$ $$\begin{aligned}
\label{em}
\zeta \cdot M^{-1} \zeta \leq \frac{N^2 \eta}{ (A_{1/2}+A_{N+1/2})}\|\zeta\|^2,\end{aligned}$$ with which we can bound $III$ as $$\label{C5}
\begin{aligned}
III=& \frac{h^3\tau^2 }{2\epsilon} \sum_{j=1}^NA_jD_tS_j^n\sum_{k=1}^N(M^{-1})_{j,k}A_kD_t S_k^n\\
\leq & \frac{\alpha \eta N^2 h^3\tau^2 }{2\epsilon} \sum_{j=1}^NA^2_j |D_t S_j^n|^2,
\end{aligned}$$ where $\alpha^{-1}=A_{1/2}+A_{N+1/2}$. Note that $hN=1$ and $$|D_t S_j^n|^2\leq m\sum_{i=1}^mz_i^2(D_tc_{i,j}^n)^2,$$ we thus have $$III \leq \sum_{i=1}^m\frac{\alpha \eta z_i^2m\tau^2}{2\epsilon}\sum_{j=1}^NhA_j^2 (D_tc_{i,j}^n)^2.$$ Collecting estimates on $I,II$ and $III$ we arrive at $$E_h^{n+1}-E_h^n\leq \sum_{i=1}^m \Bigg( \tau \sum_{j=1}^NhA_j (D_tc_{i,j}^n) (\log c_{i,j}^{n+1}+z_i\psi_j^n) + \frac{\alpha \eta z_i^2m\tau^2}{2\epsilon}\sum_{j=1}^NhA_j^2 (D_tc_{i,j}^n)^2\Bigg).$$ For (\[FE\]) to hold, it remains to find a sufficient condition on time step $\tau$ so that for all $i=1,\cdots,m$, $$\label{Final}
\frac{\alpha\eta z_i^2m\tau^2}{2\epsilon}\sum_{j=1}^NhA_j^2 (D_tc_{i,j}^n)^2 \leq -\frac{\tau}{2} \sum_{j=1}^NhA_j (D_tc_{i,j}^n)(\log c_{i,j}^{n+1}+z_i\psi_j^n).$$ This is nothing but $$\frac{\alpha\eta z_i^2m\tau^2}{2\epsilon}\|\vec{\xi}\|^2+\frac{\tau}{2} \vec{\xi}\cdot \vec{\mu}\leq 0,$$ where $$\vec{\xi}_j=\sqrt{h}A_jD_t c_{i,j}^{n}, \; \vec{\mu}_j=\sqrt{h}(\log c_{i,j}^{n+1})+z_i\psi^n_j).$$ Note that $I=\tau \vec{\xi}\cdot \vec{\mu} \leq 0$. One can verify using (\[CC\]) and flux (\[fullyCi\]) that $\vec{\xi}\cdot \vec{\mu}=0$ if and only if $ \vec{\xi}=0$. Therefore $$0< c_0 \leq \frac{-\vec{\xi}\cdot \vec{\mu}}{\|\vec{\xi}\|^2}\leq \frac{\|\vec{\mu}\|}{\|\vec{\xi}\|} \quad \text{for} \; \vec{\xi} \not=0,$$ where $c_0$ depends on the numerical solution at $t_n$ and $t_{n+1}$. We thus obtain (\[Final\]) by taking $$\tau\leq C^*\frac{\epsilon}{\eta} , \quad \text{where \ \ } C^*=\min_{1\leq i\leq m} \frac{c_0}{\alpha z_i^2m}>0.$$ Finally, we return to the proof of claim (\[em\]): For any $y\in \mathbb{R}^N$ with $\|y\|=1$, we have the following $$\begin{aligned}
y\cdot M y& =\frac{h}{\eta} A_{1/2}y_1^2+ \sum_{j=1}^{N-1}A_{j+1/2}(y_{j+1}-y_j)^2 +\frac{h}{\eta} A_{N+1/2}y_N^2 \\
& \geq \min_{\|y\|=1} \{ \frac{h}{\eta} A_{1/2}y_1^2+ \sum_{j=1}^{N-1}A_{j+1/2}(y_{j+1}-y_j)^2 +\frac{h}{\eta} A_{N+1/2}y_N^2\}\\
& =\frac{ h}{N \eta} (A_{1/2}+A_{N+1/2}), \end{aligned}$$ where the minimum is achieved at $y=(1, \cdots, 1)/\sqrt{N}$. Replacing $y$ by $y/\|y\|$ and then further set $y=M^{-1/2}\zeta$ leads to (\[em\]).
Though $C^*$ is not explicitly given, it is about $O(1)$ as can be seen from a formal limit $\Delta t \to 0$. The sufficient condition $\tau \leq C^* \epsilon/\eta$ suggests that for smaller $\epsilon/\eta $, one should consider a smaller time step to ensure the scheme stability. This is consistent with our numerical results.
Numerical tests
===============
In this section, we implement the fully discrete scheme (\[fullyCi\])-(\[Ps2\]) with different boundary conditions. Errors are measured in the following discrete $l^{\infty}$ norm: $$e_f=\max_{1\leq j \leq N}| f_j-\bar{f}_{j}|.$$ Here $\bar{f}_{j}$ denotes the average of $f$ on cell $I_j$. In what follows we take $f_j=c_{i,j}^n$ or $\psi_{j}^n$ at time $t=n\tau.$
Accuracy test
-------------
In this example we numerically verify the accuracy and order of schemes (\[fullyCi\])-(\[Ps2\]) with first order boundary flux (\[BC11\]) and second order boundary flux of form (\[U22\]).
\[ex1\] Consider the initial value problem with source term
$$\left \{
\begin{array}{rl}
\hfill \partial_t c_1 =&\frac{1}{A(x)}\partial_x(A(x)D_1(\partial_x c_1 + z_1c_1 \partial_x \psi)+f_1(x,t), \hfill \ \ \ x\in [0, \ 1], \ t>0,\\
\hfill \partial_t c_2 =&\frac{1}{A(x)}\partial_x(A(x)D_2(\partial_x c_2 + z_2c_2 \partial_x \psi)+f_2(x,t), \hfill \ \ \ x\in [0, \ 1], \ t>0,\\
\hfill -\frac{1}{A(x)}&\partial_x(\epsilon A(x) \partial_x \psi) =z_1c_1 + z_2c_2 -\rho(x)+f_3(x,t), \hfill \ \ \ x\in [0, \ 1], \ t>0,\\
\hfill c_1(x,0)=&x^2(1-x) , \quad c_1(0,t)=c_1(1,t)=0, \\
\hfill c_2(x,0)=&x^2(1-x)^2 , \quad c_2(0,t)=c_2(1,t)=0, \\
\hfill \psi(0,t)=&0 , \quad \psi(1,t)=-\frac{1}{60}e^{-t}.
\end{array}
\right.$$
Here we take $A(x)=(5-4x)^2,$ $D_1=D_2=1,$ $z_1=-z_2=1,$ $\epsilon=1$ and $\rho(x)=0,$ source terms are $$\begin{aligned}
f_1(x,t)&=\frac{4x^4 -9x^3+53x^2 -54x+10}{4x-5}e^{-t}+\frac{40x^7-71x^6+30x^5}{20}e^{-2t},\\
f_2(x,t)&=\frac{4x^5-13x^4 +94x^3-161x^2 +84x-10}{5-4x}e^{-t}+\frac{22x^8-60x^7+53x^6-15x^5}{10}e^{-2t},\\
f_3(x,t)&=-\frac{2x^4}{5}e^{-t}.\end{aligned}$$ The exact solution to (\[ex1\]) is $$c_1(x,t)=x^2(1-x)e^{-t}, \quad c_2(x,t)=x^2(1-x)^2e^{-t}, \quad \text{and} \; \psi(x,t)=-\frac{x^5(3-2x)}{60}e^{-t}.$$ We use the time step $\tau=h^2 $ to compute numerical solutions. The errors and orders at $t=1$ are listed in Table 1 and Table 2.
\[ex11\]
-------------- -------------- -------- ------------- -------- -------------- -------- --
N $c_1$ error order $c_2$ error order $\psi$ error order
\[0.5ex\] 40 0.11184E-03 - 0.57759E-04 - 0.83275E-05 -
80 0.28354E-04 1.9798 0.14407E-04 2.0033 0.20810E-05 2.0006
160 0.71370E-05 1.9902 0.36019E-05 1.9999 0.52013E-06 2.0003
320 0.17903E-05 1.9951 0.90047E-06 2.0000 0.13002E-06 2.0001
\[1ex\]
-------------- -------------- -------- ------------- -------- -------------- -------- --
\[ex121\]
-------------- -------------- -------- ------------- -------- -------------- -------- --
N $c_1$ error order $c_2$ error order $\psi$ error order
\[0.5ex\] 40 0.10014E-03 - 0.69633E-04 - 0.37021E-05 -
80 0.25204E-04 1.9903 0.18005E-04 1.9514 0.93954E-06 1.9783
160 0.63218E-05 1.9952 0.45767E-05 1.9760 0.23755E-06 1.9837
320 0.15830E-05 1.9977 0.11536E-05 1.9882 0.59655E-07 1.9935
\[1ex\]
-------------- -------------- -------- ------------- -------- -------------- -------- --
We see from Table 1 and Table 2 that both first and second order boundary fluxes yield second order convergent solutions. The numerical errors with both fluxes are comparable.
In the remaining numerical tests we only use first order boundary flux (\[BC11\]) for the Dirichlet boundary value problem.
Effects of permanent charge and channel geometry
------------------------------------------------
The key structure of an ion channel includes both the channel shape and the permanent charge (see e.g. [@WS15]). We present numerical examples to illustrate the effects from the channel geometry or the permanent change. While we also examine dependence of the total current (\[J\]) on voltage $V$, which is known as the current-voltage (I-V) relation in [@WS15]. Note that (\[J\]) can be reformulated as $$J=-\sum_{i=1}^m z_iD_iA(x) e^{-z_i\psi}\partial_x (c_ie^{z_i\psi}).$$ Let $J^n_{j+1/2}$ be an approximation of $J(x_{j+1/2},t_n),$ then $J^n_{j+1/2}$ can be computed by $$J^n_{j+1/2}=-\sum_{i=1}^m z_iC_{i,j+1/2},$$ where $C_{i,j+1/2}$ is defined in (\[CC\]) with $c^{n+1}_{i, j}$ replaced by $c^n_{i, j}$, that is, $$C_{i,j+1/2} =A_{j+1/2}D_i e^{-z_i\psi_{j+1/2}^n} \frac{\left( c^{n}_{i,j+1}e^{z_i\psi_{j+1}^n} -c^{n}_{i,j}e^{z_i\psi_{j}^n}\right)}{h}, \; j=1, \cdots, N-1.$$
\[ex2\] (Effects of channel geometry with permanent charge) We consider the system $$\label{PNPQ}
\begin{aligned}
A(x) \partial_t c_1 &= \partial_x (A(x)( \partial_x c_1 +c_1 \partial_x \psi)), \quad x\in [0, \ 1], \ t>0,\\
A(x) \partial_t c_2 &= \partial_x (A(x)( \partial_x c_2 -c_2 \partial_x \psi)), \quad x\in [0, \ 1], \ t>0,\\
-\frac{1}{A(x)}\partial_x (\epsilon A(x) \partial_x \psi) & = c_1-c_2 -\rho(x), \quad x\in [0, \ 1], \ t>0,
\end{aligned}$$ where $\epsilon=5\times 10^{-5},$ subject to initial and boundary conditions $$\label{IB}
\begin{aligned}
c_1(x,0)& =c_2(x, 0) =0.5-0.1x, \quad x \in [0, 1], \\
c_i(0,t) &=0.5, \quad c_i(1,t)=0.4; \;\psi(0,t)=0, \psi(1,t)=0.5,\quad t>0.
\end{aligned}$$
![Diagram of 1D computational region for the channel and bath funnels [@Eis04]](pic01){width="50.00000%"}
This corresponds to problem (\[PNP2\]) with $D_1=D_2=1$, $z_1=-z_2=1$, with $c_{i,l}=0.5$, $c_{i,r}=0.4$, and $V=0.5$.
The computational domain diagram is given in Figure 1, while the cross sectional area $A(x)$ is defined as: $$\label{2Ax}
A(x)=\left \{
\begin{array}{rl}
2(r_f+\frac{r_c-r_f}{l_b}x), \quad \hfill & x \in [0,\ l_b], \\\\
2r_c, \quad \hfill & x \in (l_b,\ l_b+l_c), \\\\
2( r_c+\frac{r_f-r_c}{l_b}(x-l_b-l_c)), \hfill & x \in [l_b+l_c,\ 1],
\end{array}
\right.$$ where the shape parameters are allowed to vary in our numerical tests. The permanent charge $\rho(x)$ is taken as $$\label{2Qx}
\rho(x)=\left \{
\begin{array}{rl}
0, \quad \hfill & x \in [0,\ l_b], \\\\
2Q_0 \quad \hfill & x \in (l_b,\ l_b+l_c), \\\\
0, \hfill & x \in [l_b+l_c,\ 1],
\end{array}
\right.$$ with $Q_0$ a fixed constant.
Robert Eisenberg made clear to us the great importance of the tapered representation of the baths in one dimensional versions of PNP models of channels, that became clear in his early work with Wolfgang Nonner [@NE98; @NCE98], followed by many other more formal treatments such as in [@Eis04].
In this numerical test, we take $h=0.01,$ $\tau=5\times 10^{-5}$. The solutions are understood to have reached steady states if $||\psi^{n}-\psi^{n-1}||_{\infty}\leq 10^{-6}$. Table 3 shows times $t_s$ needed for reaching each steady state, number of iterations, and CPU times.
\[ex122\]
--------------------------------- ------------------------------------ ------------ ------------------------- ---------------- -- -- --
channel parameters $||\psi^{n}-\psi^{n-1}||_{\infty}$ time $t_s$ iterations $n=t_s/\tau$ CPU time (sec)
\[0.5ex\] $r_c=l_c=\frac{1}{3}$ 9.9822E-07 0.0744 1488 0.5244
$r_c=l_c=\frac{1}{5}$ 9.9879E-07 0.0992 1984 0.6534
$r_c=l_c=\frac{1}{11}$ 9.9920E-07 0.1116 2232 0.7035
\[1ex\]
--------------------------------- ------------------------------------ ------------ ------------------------- ---------------- -- -- --
From Table 3 we see that $t_s=0.1116$ is the longest time needed for reaching the steady state, so we run the simulation up to $t=0.2$.
In Figure 2 we take $Q_0=0.2$, $r_f=20$, varying $l_c$ and $r_c$ inside the channel, to obtain a series of snapshots. We see that both $c_1$ and $c_2$ coincide outside the channel, but split inside the channel with the shape evolving in terms of the channel geometry. The profile of $\psi$ looks similar.
In Figure 3 we fix the channel shape with $r_f=20,$ $r_c=1/5,$ $l_c=1/5$, varying $Q_0$, we observe that the difference between $c_1$ and $c_2$ inside the channel increases in terms of $Q_0$, roughly we have $c_1-c_2\approx 2Q_0$ inside the channel. We can also observe the effects on $\psi$.
In Figure 4 is the I-V relation for the PNP system with channel shape parameters $l_c=1/5,$ $r_c=1/5,$ $r_f=20$, and $Q_0=0.1$. We see from the figure that the current is linear in the voltage.
\
\
\[ex3\] (No permanent charge in the channel $\rho=0$) We still use problem with (\[PNPQ\]), (\[IB\]), (\[2Ax\]) and (\[2Qx\]) to test the effects of the channel geometry, by taking $\rho=0$, $r_f=20$ and varying $l_c$ and $r_c$. In the simulation we take $h=0.01,$ $\tau=5\times 10^{-5}$.
\[ex13\]
--------------------------------- ------------------------------------ ------------ ------------------------- ---------------- -- -- --
channel parameters $||\psi^{n}-\psi^{n-1}||_{\infty}$ time $t_s$ iterations $n=t_s/\tau$ CPU time (sec)
\[0.5ex\] $r_c=l_c=\frac{1}{3}$ 9.9876E-07 0.0589 1178 0.4647
$r_c=l_c=\frac{1}{5}$ 9.9928E-07 0.0747 1494 0.5529
$r_c=l_c=\frac{1}{11}$ 9.9873E-07 0.0888 1776 0.6116
\[1ex\]
--------------------------------- ------------------------------------ ------------ ------------------------- ---------------- -- -- --
From Table 4 we see that $t_s=0.0888$ is the longest time needed for reaching the steady state, so simulation runs up to $t=0.1$.
In Figure 5 are snap shots of solutions for different channel geometry. In the case of no permanent charge, there does not seem to be any layering phenomenon on $c_1$ and $c_2$: $c_1$ and $c_2$ are rather close both inside the channel (linear) and inside the bath (constant). The profile for $\psi$ is quite similar. This is consistent with the analysis in [@WS15], in which the authors showed that the density in the channel gets steeper as the channel gets narrower. We refer to [@WSzeroQ15] for a study of steady state solutions to (\[PNPQ\]) in the case of $\rho=0.$ They proved that for $\epsilon >0$ small, there is a unique nonnegative steady state to problem (\[PNPQ\]) and (\[IB\]).
\[ex4\] (Variable diffusion coefficient and quadratic area function) We consider the system $$\label{PNP4}
\begin{aligned}
A(x) \partial_t c_1 &= \partial_x (A(x)D_1(x)( \partial_x c_1 +2c_1 \partial_x \psi), \quad x \in [-10, 10],\ t>0\\
A(x) \partial_t c_2 &= \partial_x (A(x)D_2(x)( \partial_x c_2 -3c_2 \partial_x \psi), \quad x \in [-10, 10],\ t>0\\
A(x) \partial_t c_3 &= \partial_x (A(x)D_3(x)( \partial_x c_3 +c_3 \partial_x \psi), \quad x \in [-10, 10],\ t>0\\
-\frac{1}{A(x)}\partial_x (\epsilon A(x) \partial_x \psi) & = 2c_1-3c_2+c_3 -\rho(x), \quad x \in [-10, 10],\ t>0,
\end{aligned}$$ where $\epsilon= 0.1,$ subject to boundary conditions $$\label{IB4}
\begin{aligned}
c_i(\pm 10,t) &=0.5, \quad \psi(\pm 10,t)=0,\quad t>0.
\end{aligned}$$ As in [@GLE2018], we choose $A(x)=1+x^2,$ $D_i(x)=20(1-0.9e^{-x^4})$ and $\rho=Ce^{-x^4}$. This corresponds to problem (\[PNP2\]) with $z_1=2$, $z_2=-3$, $z_3=1$, and $c_{i,l}=c_{i,r}=0.5$, $V=0$. In this numerical test we take $h=0.1$, $\tau=10^{-3}$.
We take two different sets of initial data, first set is given as $$\label{c1}
\begin{aligned}
c^{in}_1(x)&= 0.5-0.5e^{-(x+4)^4},\\
c^{in}_2(x)&= 0.5+2e^{{-x}^4},\\
c^{in}_3(x)&= 0.5+e^{-(x-4)^4}.
\end{aligned}$$ For the second set of initial data we take uniformly distributed random initial data $c^{0}_{i,j}\in (0, 1)$. From Table 5 we see that $t_s=2.7410$ is the longest time needed for reaching the steady state, so simulation runs up to $t=3$. We vary the parameter $C$ to observe effects of the permanent charge.
\[ex14\]
------------------------- ------------------------------------ ------------ ------------------------- ---------------- -- -- --
initial data $||\psi^{n}-\psi^{n-1}||_{\infty}$ time $t_s$ iterations $n=t_s/\tau$ CPU time (sec)
\[0.5ex\] data (\[c1\]) 9.9999E-08 2.7410 2741 0.3874
random data 9.9864E-08 2.0440 2044 0.3167
\[1ex\]
------------------------- ------------------------------------ ------------ ------------------------- ---------------- -- -- --
In Figure 6 (top three) are snap shots of solutions for initial data (\[c1\]). Varying $C$, we can see that $\max c_1-\min c_2$ (or $\max c_3-\min c_2$) increases in terms of $C$.
In Figure 6 (bottom three) are snap shots of solutions for random initial data, we see that the choice of initial data does not affect the steady state densities.
\
Mass conservation and free energy dissipation
---------------------------------------------
In this numerical test we demonstrate mass conservation and free energy dissipation properties.
[(Zero flux + Robin boundary conditions)]{} In this example we consider (\[PNP4\]) with initial condition (\[c1\]) and boundary condition $$\label{Zf5}
\begin{aligned}
&\partial_x c_i+z_ic_i\partial_x \psi =0,\quad x=-10,10, \quad t>0,\\
& (- \eta \partial_x \psi+\psi)|_{x=-10}=-0.1,\quad (\eta \partial_x \psi+\psi)|_{x=10}=0.1, \quad t>0.
\end{aligned}$$ We choose same $A(x), D_i(x), \rho(x)$ and $z_i$ as in Example 4.4 and choose $\eta=0.1$, $\epsilon=0.1$. In this numerical test we use scheme (\[fullyCi\])-(\[Ps2\]) and (\[ZfBC\]), with $h=0.1$, $\tau=10^{-3}$.
In Figure 7 (left) are snap shots of solutions for initial data (\[c1\]), (right) is free energy for the system and total mass for each species, which confirms energy dissipation and mass conservation properties as proved in Theorem 3.2.
Concluding Remarks
==================
In this paper, we have developed an unconditional positivity-preserving finite-volume method for solving initial boundary value problems for the reduced Poisson-Nernst-Planck system. Such a reduced system has been used as a good approximation to the 3D ion channel problem. By writing the underling system in non-logarithmic Landau form and using a semi-implicit time discretization, we constructed a simple, easy-to-implement numerical scheme which proved to satisfy positivity independent of time steps and the choice of Poisson solvers. Our scheme also preserves total mass and satisfies a free energy dissipation property for zero flux boundary conditions. Extensive numerical tests have been presented to simulate ionic channels in different settings.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Robert Eisenberg for stimulating discussions on PNP systems and their role in modeling ion channels. This research was supported by the National Science Foundation under Grant DMS1812666 and by NSF Grant RNMS (KI-Net) 1107291.
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---
abstract: 'We study the thermodynamic properties of horizons using the dynamical description of the gravitational degrees of freedom at a horizon found in a previous work. We use the action of the horizon degrees of freedom to calculate the horizon entropy using the Cardy formula, and obtain the expected Bekenstein-Hawking entropy. We also couple the gravitational degrees of freedom at the horizon to a classical background scalar field, and show that Hawking radiation is produced.'
address: |
Jefferson Physical Laboratory, Harvard University,\
17 Oxford St., Cambridge, MA 02138, USA
author:
- Hyeyoun Chung
title: Hawking Radiation and Entropy from Horizon Degrees of Freedom
---
Hawking radiation ,horizon entropy
Introduction
============
Ever since the work of Bekenstein and Hawking in the 1970s, which established that the laws of thermodynamics can be adapted to describe black holes[@Bekenstein; @Hawking1; @Hawking2], there have been repeated attempts to provide a microscopic description of black hole horizons, and of horizons in general.
There have also been attempts to find an *effective* theory of horizon microstates, that can describe the degrees of freedom of the horizon without reference to the underlying theory of quantum gravity. This suggestion is particularly plausible because of the universal appearance of conformal symmetry in the neighborhood of a horizon[@Carlip; @Dreyer; @Silva; @Koga; @Kang; @Medved1; @Medved2], which indicates that the dynamics of a horizon will be governed by a two-dimensional conformal field theory (CFT). Carlip has suggested that the degrees of freedom of this theory are diffeomorphisms that become dynamical at the horizon due to the presence of boundaries or constraints[@CarlipLec; @Carlip21]. Dynamical actions have been derived for such “would-be gauge” degrees of freedom both at spatial infinity in ${\mathrm{AdS}_3}$[@CarlipAdS1; @CarlipAdS2] and $\mathrm{AdS}_5$[@Aros], and at the horizon of the (2+1)-dimensional BTZ black hole[@Carlip21], and these actions are indeed found to describe conformal theories.
In [@Mine], we derived a dynamical action for the near-horizon gravitational degrees of freedom that arise for a very general class of horizons. This result was obtained by imposing physically motivated boundary conditions that preserved the existence and the essential characteristics of a horizon, and identifying the infinitesimal (i.e. first order) diffeomorphisms that preserved these conditions. We then derived a dynamical action for the gravitational degrees of freedom corresponding to these diffeomorphisms from the Einstein-Hilbert action. The resulting action is similar to that of Liouville theory, and the equation of motion derived from this action approaches that of a free two-dimensional conformal field in the near-horizon region.
In this paper we study the thermodynamic properties of the theory found in [@Mine]. First we use the fact that the gravitational degrees of freedom at a horizon exhibit a two-dimensional conformal symmetry to calculate the entropy of the horizon. Using the Cardy formula, we find that we can reproduce the Bekenstein-Hawking entropy. This work differs in two main ways from earlier results[@Solodukhin; @DiasLemos; @Giacomini] that find a Liouville action for the gravitational degrees of freedom at a horizon, and use this action to derive the Bekenstein-Hawking entropy. Firstly, our result is valid in an arbitrary number of dimensions, and does not require the horizon to be spherically symmetric. Secondly, instead of obtaining the action by an *ad hoc* dimensional reduction of the Einstein-Hilbert action, or by choosing the action so that it will lead to the expected equation of motion for a Liouville field, our work directly relates the gravitational degrees of freedom at the horizon to the diffeomorphisms that preserve horizon boundary conditions. We only integrate over the spatial coordinates as a final step, after determining that the leading order dynamics are in the $r-t$ plane.
A derivation of the Bekenstein-Hawking entropy is a useful criterion for judging the validity of a theory that describes horizon microstates: however, it is far from being a proof that the theory is correct. Another valuable indicator is seeing whether or not the theory predicts the emission of Hawking radiation. Many different methods have been devised for deriving Hawking radiation[@Hawking1; @Hawking2; @Instantons; @Christensen; @RobinsonWilczek]. One common feature of all these works is that they analyze quantum matter fields in a classical black hole background, and derive Hawking radiation as a consequence of quantum fields living in a curved space. In order to have a complete picture of horizon thermodynamics, it is important to do the reverse: couple quantized gravitational degrees of freedom to classical matter, and produce the blackbody spectrum of Hawking radiation.
A few steps have been taken in this direction: in [@Solodukhin2], it was shown that Hawking radiation in the near-horizon region could be modeled using a Liouville conformal field theory. Similarly, in [@RodriguezYildrim], an ansatz Liouville theory was proposed to describe the near-horizon region, and this theory was then used to derive the flux of Hawking radiation from several classes of black holes. An alternative approach was taken in [@EmparanSachs], where the Liouville theory of diffeomorphism degrees of freedom at the spatial infinity of ${\mathrm{AdS}_3}$ was coupled to scalar field matter. It was shown that the decay rate of the BTZ black hole exactly matched the spectrum of Hawking radiation, including greybody factors. In this work we follow a similar approach, but instead of coupling a classical scalar field to a conformal field theory at the boundary, we identify a coupling in a neighborhood of the horizon, and obtain the Hawking radiation spectrum.
This paper uses the method proposed in [@EmparanSachs], by coupling the horizon degrees of freedom found in [@Mine] to scalar field matter and deriving the thermal spectrum of Hawking radiation. Our approach differs from those of [@Solodukhin2; @RodriguezYildrim] as these earlier papers proposed an ansatz Liouville theory to model horizon phenomena. In contrast, this paper uses a theory described by a dynamical action that was directly derived from the Einstein-Hilbert action by relating the gravitational degrees of freedom at the horizon to the diffeomorphisms preserving the existence and characteristics of the horizon.[@Mine]. Our work extends the results of [@EmparanSachs], as our result is valid in an arbitrary number of dimensions, and covers a large class of black holes, not just those black holes whose near-horizon region is ${\mathrm{AdS}_3}$. Moreover, the coupling between the gravitational degrees of freedom and the matter fields occurs in the near-horizon region, not at spatial infinity as in [@EmparanSachs].
This paper is structured as follows. In Sec. \[sec-CFT\] we review the main result found in [@Mine], as well as the formalism necessary to understand the results in this paper. In Sec. \[sec-Entropy\] we derive the Bekenstein-Hawking entropy of a general class of horizons using the conformal field theory of the horizon degrees of freedom. In Sec. \[sec-Hawking\] we show how the spectrum of Hawking radiation from a horizon can be reproduced by coupling classical scalar field matter to these gravitational degrees of freedom. We conclude in Sec. \[sec-conc\].
Conformal Field Theory at the Horizon {#sec-CFT}
=====================================
We first review some of the notation and concepts used to study the diffeomorphism degrees of freedom at the horizon in [@Mine], and then summarize the main result of that paper.
Isolated Horizons
-----------------
Our horizon boundary conditions are based on the notion of **weakly isolated horizons** (WIHs)[@Ashtekar; @Booth]. Isolated horizons are null sub-manifolds $\Delta$ of spacetime, with an intrinsic metric $q_{ab}$ that is the pull-back of the spacetime metric to $\Delta$. A tensor $q^{ab}$ on $\Delta$ is defined to be an inverse of $q_{ab}$ if it satisfies $q_{am}q_{bn}q^{mn}=q_{ab}$. The inverse is not unique, but all of the definitions and constructions in the isolated horizon formalism are independent of the choice of inverse. Given a null normal $l^\mu$ to $\Delta$, the *expansion* $\theta_{(l)}$ of $l^\mu$ is defined to be $$\theta_{(l)} := q^{ab}\nabla_a l_b.$$ A weakly isolated horizon (WIH) is a sub-manifold $\Delta$ of a spacetime that satisfies the following conditions:
1. $\Delta$ is topologically $S^2\times\mathbb{R}$ and null,
2. Any null normal $l^\mu$ of $\Delta$ has vanishing expansion, $\theta_{(l)}=0$, and
3. All equations of motion hold at $\Delta$ and the stress energy tensor $T_{\mu\nu}$ is such that $-T^\mu_\nu l^\nu$ is future-causal for any future directed null normal $l^\mu$.
Note that if Condition 2 holds for one null normal to $\Delta$, then it holds for all. WIHs generalize the definitions of Killing horizons and apparent horizons, with the normal vector $l^\mu$ being analogous to a Killing vector, and the requirement that ${\theta_{(l)}}=0$ clearly being inspired by the notion of trapped surfaces. However, the definition of a WIH is given only *at* the horizon, and does not require a Killing vector to exist even within an infinitesimal neighborhood. Thus, WIHs allow for much greater freedom in the dynamics of the matter and spacetime outside the horizon, while preserving the essential characteristics of the horizon itself.
The surface gravity of an isolated horizon is not uniquely defined. However, given a normal vector $l^\mu$ to $\Delta$, there is a function $\kappa_l$ such that $$\label{eq-sgra}
l^\mu\nabla_\mu l^\nu = \kappa_l l^\nu.$$ It is always possible to choose a normal vector $l^\mu$ such that the corresponding $\kappa_l$ is constant everywhere on $\Delta$. Therefore, $\kappa_l$ may be interpreted as the surface gravity of the horizon corresponding to $l^\mu$.
Gaussian Null Coordinates and Conformal Coordinates
---------------------------------------------------
In this work we use the system of **Gaussian null coordinates** (denoted “GN coordinates”) that are analogous to Eddington-Finkelstein coordinates in Schwarzschild spacetime[@Wald2], and which are well suited for studying horizons as they are adapted to null hypersurfaces. In the neighborhood of any smooth null hypersurface $\Delta$, we can define coordinates $(u,r,x^i)$ such that the metric takes the form: $$\label{eq-GN}
{{\mathop{\mathrm{d}s}}}^2 = rF{{\mathop{\mathrm{d}u}}}^2 + 2{\mathop{\mathrm{d}u}}{\mathop{\mathrm{d}r}}+2rh_i{\mathop{\mathrm{d}u}}{\mathop{\mathrm{d}x^i}}+g_{ij}{\mathop{\mathrm{d}x^i}}{\mathop{\mathrm{d}x^j}}.$$ $g_{ij}$ is positive definite, and $F, h_i,$ and $g_{ij}$ are smooth functions of $(u,r,x^i)$ that can be expanded in powers of $r$. The null hypersurface is defined by $r=0$, and we have chosen a smooth, non-vanishing vector field $l^\mu$ that is normal to $\Delta$, so that the integral curves of $l^\mu$ are the null geodesic generators of $\Delta$ and we have $l^\mu= (\partial/\partial u)^\mu$ on $\Delta$.
Since an isolated horizon is a null hypersurface, we can construct such a coordinate system in a neighborhood of any isolated horizon. In fact, in the neighborhood of the event horizon of a stationary black hole, or a stationary Killing horizon, we can define these coordinates so that all the metric components are independent of $u$. Extremal Killing horizons correspond to the case where $F$ has a simple root at $r=0$, so that it has the form $F=rf(u,r,x^i)$. We will consider only *non-extremal* horizons, such that $F|_{r=0}\neq 0$. We can then define a non-zero surface gravity for the horizon, using the definition (\[eq-sgra\]) and taking the normal vector to the $r=0$ hypersurface to be $l^\mu := g^{r\mu}$. The surface gravity $\kappa$ associated with this normal vector is $-\frac{1}{2}F|_{r=0}$, and is related to the **inverse Hawking temperature** $\beta_H$ of the horizon by $\kappa = \frac{1}{\beta_H}$.
We will also use the coordinate $\tilde{r} := -\ln r$, which puts the metric in the form $$\begin{aligned}
\label{eq-Tort}
{{\mathop{\mathrm{d}s}}}^2 &= e^{-\tilde{r}}\left (F{{\mathop{\mathrm{d}u}}}^2 - 2{\mathop{\mathrm{d}u}}{\mathop{\mathrm{d}\tilde{r}}}+2h_i{\mathop{\mathrm{d}u}}{\mathop{\mathrm{d}x^i}}\right ) +g_{ij}{\mathop{\mathrm{d}x^i}}{\mathop{\mathrm{d}x^j}},\end{aligned}$$ where $F, h_i,$ and $g_{ij}$ are now taken to be functions of $\tilde{r}$. The horizon is now located at $\tilde{r}\to\infty$. We call these coordinates **tortoise Gaussian null coordinates**. Finally, we may define conformal coordinates $({x_+},{x_-})$ in terms of $(u,r)$ such that the metric takes the form $$\label{eq-conf2}
{{\mathop{\mathrm{d}s}}}^2 = 2{g_{+-}}\left ({\mathop{\mathrm{d}{x_+}}}{\mathop{\mathrm{d}{x_-}}}+h_{+i}{\mathop{\mathrm{d}x^+}}{\mathop{\mathrm{d}x^i}}\right)+g_{ij}{\mathop{\mathrm{d}x^i}}{\mathop{\mathrm{d}x^j}},$$ and ${g_{+-}}$ has a simple root at $r=0$, so it can be written in the form ${g_{+-}}= e^{\sigma(u,r,x^i)}$, with: $$\begin{aligned}
\sigma(u,r,x^i) = \ln r + \sigma_0(u,x^i) + O(r).\end{aligned}$$ In order to obtain the results in [@Mine], it is necessary to impose one more restriction on $\sigma(u,r,x^i)$: we require that $\partial_i\partial_+\sigma=O(r)$, so that $\sigma = \ln r + \sigma_0(u) + \sigma_1 (x^i) + O(r)$. As the form of the functions $h_{+i}$ in (\[eq-conf2\]) are left unrestricted, this still allows us to describe a very large class of horizon metrics. For example, all stationary horizons satisfy these conditions.
When defining the conformal coordinates $({x_+},{x_-})$, without loss of generality we can impose $$\begin{aligned}
\label{eq-xuRel}
\partial_- u = 0,\quad\partial_r {x_+}= 0.\end{aligned}$$ We can also determine the useful relations: $$\begin{aligned}
\label{eq-useRel}
\partial_+r &= O(r),\quad\partial_-r = O(r).\end{aligned}$$
An Effective Theory of Gravitational Degrees of Freedom at a Horizon {#sec-Action}
--------------------------------------------------------------------
In this section we review the derivation given in [@Mine] of the dynamical action for the gravitational degrees of freedom at a horizon.
We begin by assuming that we have a weakly isolated horizon $\Delta$ in our spacetime, and that we can define GN coordinates in a neighborhood of $\Delta$ so that the horizon lies at $r=0$ and the metric takes the form (\[eq-conf2\]). This is the background metric, ${g_{\mu\nu}}$, which is taken to be fixed and non-dynamical: the dynamical variables in our setup will be the fluctuations about ${g_{\mu\nu}}$ that preserve the existence and basic properties of the horizon $\Delta$. Our aim is firstly to isolate and identify these fluctuations, and secondly to determine their dynamics in the near-horizon region, by deriving their Lagrangian. Note that we do *not* require ${g_{\mu\nu}}$ to satisfy Einstein’s equations (EEs), but we do require it to satisfy certain constraints that are necessary, but not sufficient, such that it *can* satisfy the EEs as $r\to 0$.
In order to find the relevant excitations about ${g_{\mu\nu}}$, we use the definition of WIHs to formulate physically motivated boundary conditions that preserve the existence and characteristics of a horizon. We then identify the most general diffeomorphisms $\xi$ that preserve these boundary conditions, and apply the diffeomorphism to ${g_{\mu\nu}}$ to obtain a new metric ${g_{\mu\nu}}' := {g_{\mu\nu}}+ \mathcal{L}_{\xi}{g_{\mu\nu}}$. As we wish to study the dynamics of gravitational excitations corresponding to $\xi$, we define the field $\phi$ by ${g_{+-}}' = (1+\phi){g_{+-}}$. Thus $\phi$ becomes the dynamical field in the problem.
We then evaluate the Einstein-Hilbert action for ${g_{\mu\nu}}'$. The transformation ${g_{\mu\nu}}' := {g_{\mu\nu}}+ \mathcal{L}_{\xi}{g_{\mu\nu}}$ is an infinitesimal diffeomorphism, i.e. a diffeomorphism to first order in $\xi$. Therefore, as the E-H action is diffeomorphism invariant, at first order the action changes only by a boundary term. However, to higher order, this transformation is not a diffeomorphism: that is, if we set ${g_{\mu\nu}}' = {g_{\mu\nu}}+ h_{\mu\nu}$ for some perturbation $h_{\mu\nu}$ and expand the Einstein-Hilbert action to second order or higher in $h_{\mu\nu}$, then evaluating the E-H action for ${g_{\mu\nu}}'$ with $h_{\mu\nu} = \mathcal{L}_{\xi}{g_{\mu\nu}}$ will give a non-trivial *bulk* contribution to the action as long as the background metric ${g_{\mu\nu}}$ is not required to satisfy the EEs[@WilczekPorf]. Thus, in order to obtain a non-trivial action for the gravitational degrees of freedom in the near-horizon region, in this work we consider excitations of ${g_{\mu\nu}}$ that are derived from diffeomorphisms (in that they are diffeomorphisms to first order), and compute higher order contributions to the E-H action due to these excitations.
As we are working in the near-horizon region $r\to 0$, and we are considering infinitesimal diffeomorphisms (i.e. small $\phi$), we work to leading order in $(r,\phi)$. Imposing the aforementioned constraints on ${g_{\mu\nu}}$ that are necessary, but not sufficient, for it to be able to satisfy the EEs as $r\to 0$, we identify the leading terms in the action as $r\to 0$, which give a non-trivial action for $\phi$. This action is similar to that of Liouville theory, and the equation of motion of the gravitational degrees of freedom approaches that of a free two-dimensional conformal field in the near-horizon region.
We now give the details of the calculation (note that in the remainder of this paper, the term “diffeomorphism” always refers to a diffeomorphism to first order.) The first step is to define appropriate boundary conditions that preserve a weakly isolated horizon $\Delta$ at $r=0$ in our spacetime, with a background metric of the form (\[eq-conf2\]). In order for $\Delta$ to be a WIH, it should have zero expansion ${\theta_{(l)}}=0$ for all normal vectors $l^\mu$. In GN coordinates, the requirement that the horizon satisfy ${\theta_{(l)}}=0$ is equivalent to saying that $\partial_u g_{ij} = O(r)$. We then impose conditions that preserve the essential characteristics of the horizon by demanding that, after applying a diffeomorphism:
1. There is still a null hypersurface at $r=0$. This is equivalent to saying that $g_{uu}'$ (or, in conformal coordinates, ${g_{+-}}'$) has a simple root at $r=0$. \[cond-1\]
2. In conformal coordinates, the metric remains in the form given by Eq.(\[eq-conf2\]), with ${g_{+-}}' = e^{\sigma'(u,r,x^i)}$ for some $\sigma'(u,r,x^i) = \ln r + \sigma_0'(u) + \sigma_1'(x^i) + O(r)$.\[cond-2\]
3. The induced metric on the $r=0$ hypersurface is preserved, so that $g'_{ij} = g_{ij} + O(r)$.\[cond-3\]. This also ensures that the null hypersurface continues to satisfy ${\theta_{(l)}}=0$.
Each of these boundary conditions is physically motivated. Condition \[cond-1\] is necessary (but not sufficient) for a WIH to still exist at $r=0$ following the diffeomorphism. Condition \[cond-2\] reflects the fact that the class of horizons we are currently considering can all be written in the form (\[eq-conf2\]): thus, if the horizon remains intact after the diffeomorphism then the metric should remain in the same form up to trivial diffeomorphisms. This condition is analogous to the boundary conditions defining asymptotically AdS spaces. The metric of any asymptotically AdS space can be written in a special form, called the Fefferman-Graham form[@FeffermanGraham]. Therefore the diffeomorphisms that keep the metric in Fefferman-Graham form are the dynamical degrees of freedom at the spatial infinity of an asymptotically AdS space[@CarlipAdS1; @CarlipAdS2; @Aros]. Finally, Condition \[cond-3\] is derived from the requirement that ${\theta_{(l)}}=0$ for WIHs. In GN coordinates this requirement is equivalent to $\partial_u g_{ij}=O(r)$. It follows that the induced metric on the $r=0$ hypersurface is a characteristic of the horizon, as it remains constant along $\Delta$. We therefore impose that this characteristic of the horizon is preserved under the diffeomorphism $\xi$, so that $g_{ij}' = g_{ij} + O(r)$. This conditions also ensures that the null hypersurface at $r=0$ continues to satisfy ${\theta_{(l)}}=0$, so that we still have a WIH at $r=0$ following the diffeomorphism.
In conformal coordinates, the diffeomorphisms preserving these conditions have the form: $$\begin{aligned}
\label{eq-diff}
\xi^+ &= \xi^+({x_+}) + O(r)\\
\xi^- &= \xi^-({x_-}) + O(r)\nonumber\\
\xi^i &= O(r)\nonumber\end{aligned}$$ We see from the form of $\xi$ that the $({x_+},{x_-})$ coordinates define a natural two-dimensional submanifold where our CFT will live, with infinitesimal conformal transformations being given by ${x_+}\to{x_+}+\xi^+({x_+})$, ${x_-}\to{x_-}+\xi^+({x_-})$.
After applying a diffeomorphism of the form (\[eq-diff\]), the boundary conditions ensure that the metric remains in the form (\[eq-conf2\]). The non-zero components of the new metric ${g_{\mu\nu}}' := {g_{\mu\nu}}+ \mathcal{L}_{\xi}{g_{\mu\nu}}$ are: $$\begin{aligned}
\label{eq-newMet}
g'_{+-} &= g_{+-}(1+\partial_+\xi^+ +\partial_-\xi^-+\xi^+\partial_+\sigma+\xi^-\partial_-\sigma)\\
g'_{+i} &= \partial_+\xi^+ g_{+i} + \partial_i\xi^-{g_{+-}}+ \xi^-\partial_-g_{+i} + \xi^+\partial_+g_{+i}\nonumber\\
g'_{ij} &= g_{ij} + \xi^+\partial_+g_{ij} +\xi^-\partial_-g_{ij}\nonumber\end{aligned}$$ We now evaluate the Einstein-Hilbert action $$I_{EH} = \frac{1}{16\pi G} \int{\mathop{\mathrm{d}^nx}} \sqrt{-g}\,\,(R-2\Lambda)$$ for the new metric ${g_{\mu\nu}}' := {g_{\mu\nu}}+ \mathcal{L}_{\xi}{g_{\mu\nu}}$ after applying the diffeomorphism, and impose a set of constraints that are necessary, but not sufficient, for ${g_{\mu\nu}}$ to satisfy the EEs asymptotically as $r\to 0$ (see \[app-3\] for details of the computations.) This allows us to isolate the dynamics of the gravitational fluctuations about the background metric.
We define the field $\phi$ by ${g_{+-}}' = (1+\phi){g_{+-}}$, so that $$\label{eq-phiDef}
\phi = \partial_+\xi^+ + \partial_-\xi^- + \xi^+\partial_+\sigma + \xi^-\partial_-\sigma.$$ Although $\sigma$ (and therefore $\phi$) is a function of $x^i$, the fact that $\partial_i\partial_+\sigma=O(r)$ and $\partial_i\partial_-\sigma=O(r)$ means that to leading order $\phi$ is independent of $x^i$ and thus may be considered as a field $\phi({x_+},{x_-})$ on the $({x_+},{x_-})$ submanifold.
We then evaluate the Einstein-Hilbert action for the new metric ${g_{\mu\nu}}'$ to leading order in $(r,\phi)$. We find that all the dynamics in the $x^i$ coordinates disappear to $O(r^2)$, so we may integrate over these coordinates, and the Einstein-Hilbert action takes the form: $$\label{eq-action}
I_{\mathrm{hor}} = \frac{{a_{\Delta}}}{16\pi G} \int{\mathop{\mathrm{d}^2x}} \sqrt{-\hat{g}}\,\, \left (\partial_a\phi\partial^a\phi -\phi\hat{R}+ \lambda (1+\phi)\right )$$ where ${a_{\Delta}}$ is the cross-sectional area of the horizon, $\hat{g}$ is the induced metric on the $({x_+},{x_-})$ submanifold, and the Ricci scalar $\hat{R}$ is computed from $\hat{g}$. The parameter $\lambda$ is given by $$\begin{aligned}
\lambda := \frac{1}{a_{\Delta}} \int \mathrm{d}^{n-2}x^i\,\,\, \tilde{\lambda}\sqrt{\tilde{g}},\end{aligned}$$ where $\tilde{\lambda} := \frac{4\Lambda e^{\sigma_1(x^i)}}{n-2}$, and $\tilde{g}_{ij} := g_{ij}|_{r=0}$ is the induced metric on the $r=0$ hypersurface.
The equation of motion for the field $\phi$ has the form: $$\begin{aligned}
\partial_+\partial_-\phi + O(r) = 0,\end{aligned}$$ so that $\phi$ becomes a free two-dimensional conformal field in an infinitesimal neighborhood of the horizon.
Note: the above result requires some conditions on the fall-off behavior of the matter energy-momentum (EM) tensor as $r\to 0$ (for details, see \[app-3\].) If these conditions are not met, then the potential term for $\phi$ in the action becomes modified, and we need to know the precise form of $T_{\mu\nu}$ in order to find an explicit form for the action. However, the dynamics of the field $\phi$ are unchanged by the addition of matter, as the non-kinetic terms in the action are multiplied by $\sqrt{-\hat{g}}$, which is $O(r)$. Thus these terms become irrelevant as $r\to 0$, so that $\phi$ still becomes a free two-dimensional conformal field in an infinitesimal neighborhood of the horizon. This observation will be important in Sec. \[sec-Entropy\] and \[sec-Hawking\], when we study the thermodynamic properties of the theory described by the action (\[eq-action\]).
Horizon Entropy {#sec-Entropy}
===============
We now calculate the entropy of a large class of horizons using the Cardy formula, a remarkable result that allows us to determine the entropy of any system with a two-dimensional conformal symmetry. The Cardy formula states that, given any unitary two-dimensional conformal field theory with generators $L_n^{\pm}$ of conformal transformations, the asymptotic density of states at eigenvalues $\Delta^{\pm}$ of $L_0^{\pm}$ is given by: $$\ln \rho(\Delta^+, \Delta^-) \sim 2\pi\sqrt{\frac{c^+\Delta^+}{6}}+2\pi\sqrt{\frac{c^-\Delta^-}{6}}$$ This formula has often been used to calculate the horizon entropy of black holes, simply by using a knowledge of the symmetries that govern the gravitational degrees of freedom in a spacetime. Brown and Henneaux[@BrownHenneaux] showed that the generators of the asymptotic symmetry group of $AdS_3$ formed a Virasoro algebra with a nontrivial central charge. Strominger then used this result to show that the entropy of $AdS_3$ black holes could be reproduced by using this central charge in the Cardy formula[@Strominger].
This approach has been extended by others [@Carlip; @Dreyer; @Silva; @Koga; @Kang], who have imposed physically motivated boundary conditions at a horizon or at spatial infinity to identify the symmetries that preserve these boundary conditions and thus govern the gravitational degrees of freedom in the spacetime. Then, computing the algebra of charges corresponding to these symmetries, and a corresponding central extension (if one exists) has allowed the Bekenstein-Hawking entropy to be reproduced for several types of black holes. The striking feature of this method is that a *classical* conformal symmetry can be enough to determine the entropy, without knowing anything about the underlying quantum theory, lending support to the notion that there is an effective description of the horizon degrees of freedom that is independent of the true theory of quantum gravity. Moreover, the appearance of the Virasoro algebra suggests that this effective theory is a two-dimensional conformal field theory (CFT).
As we have found that the gravitational degrees of freedom at a horizon have an effective description as a 2D CFT, we can also apply the Cardy formula to calculate the entropy of the horizon. As in the works cited above, we have identified the diffeomorphisms $\xi^+$ and $\xi^-$ given in (\[eq-diff\]) that preserve the existence and characteristics of a horizon, and we have derived a dynamical action for the gravitational fluctuations in the near-horizon region that correspond to these diffeomorphism degrees of freedom. We can therefore compute the algebra of charges corresponding to the symmetries $\xi^+, \xi^-$ of the theory. Our work is close in spirit to that of Solodukhin[@Solodukhin] and Giacomini and Pinamonti[@Giacomini].
The energy-momentum tensor derived from $I_{\mathrm{hor}}$ is $$\begin{aligned}
T_{ab} &= \frac{{a_{\Delta}}}{16\pi G} \Bigl [ \partial_a\phi\partial_b\phi - \hat{g}_{ab} \left ( \frac{1}{2}\partial_a\phi\partial^a\phi + \frac{\lambda}{2}(1+\phi)\right )\nonumber\\
&\hspace{1.7cm}-(\hat{g}_{ab}\Box\phi - \nabla_a\nabla_b\phi)\Bigr ]\end{aligned}$$ The first step in calculating the horizon entropy is to compute the algebra of charges corresponding to the symmetries of the CFT. The symmetries are given by the diffeomorphisms $\xi^+(x^+)$ and $\xi^-({x_-})$. In order to obtain a countable set of charges, we impose a cutoff scale $l$ and define $$\zeta^\pm_n = \frac{l}{2\pi}e^{\pm\frac{2\pi i}{l} nx_\pm}$$ The generators of the corresponding conformal transformations are: $$\begin{aligned}
L_n^{\pm} &= \int_{-l/2}^{l/2} {\mathop{\mathrm{d}x_{\pm}}}\,\zeta_n^{\pm} T_{\pm\pm}\nonumber\\
&= \frac{{a_{\Delta}}}{16\pi G} \frac{l}{2\pi}\int_{-l/2}^{l/2} {\mathop{\mathrm{d}x_{\pm}}}e^{\pm\frac{2\pi i}{l} nx_\pm} \left [(\partial_{\pm}\phi)^2 +\partial_{\pm}^2\phi\right ]\end{aligned}$$ To evaluate the algebra of charges, we define coordinates $(t,\rho)$ such that ${x_+}= t + \rho$, ${x_-}= t - \rho$, and define $t$ to be the time coordinate. We see from the action in (\[eq-action\]) that the canonical momentum $\Pi$ conjugate to the field $\phi$ is $\Pi := \delta\mathcal{L}/\delta(\partial_t \phi) = \frac{{a_{\Delta}}}{16\pi G} \partial_t\phi$. We thus obtain the Poisson bracket $$\{\phi(t, \rho), \partial_t\phi(t, \rho')\} = \frac{16\pi G}{{a_{\Delta}}}\delta(\rho - \rho')$$ Using the Poisson bracket to evaluate the algebra of charges, we find that $$\{ L_n^{\pm}, L_m^{\pm}\} = i(n-m)L_{n+m}^{\pm} + \frac{ic^{\pm}}{12}n^3\delta_{n+m,0}$$ with central charges $$c^{\pm} = \frac{3{a_{\Delta}}}{4G}$$ Now we simply need to determine the eigenvalues of $L_0^{\pm}$ for a given horizon in order to calculate its entropy.
Note that our calculations are completely independent of the form of the potential for $\phi$ in the action. Thus, even if the fall-off conditions on the matter energy-momentum tensor required to obtain the action (\[eq-action\]) are not met, and the action becomes modified by an additional potential term for $\phi$, the results in this section remain unchanged.
We would now like to evaluate $L_0^{\pm}$ for the classical horizon configuration: this corresponds precisely to the case when there are no fluctuations about the horizon, so that $\phi = 0$, giving $L_0^{\pm}=0$. This is not a serious problem: it is merely due to the fact that the overall normalization of $L_0^{\pm}$ has not been fixed.
In order to determine the normalization, recall the definition of $\phi$ given in (\[eq-phiDef\]). We see from (\[eq-newMet\]) that the metric $g_{\mu\nu}'$ after the diffeomorphism has $g_{+-}' = (1+\phi)g_{+-}$. Since we are considering infinitesimal diffeomorphisms, we have ${g_{\mu\nu}}' \approx e^{\phi}{g_{\mu\nu}}$. Thus $\phi$ can be interpreted as a fluctuation of the conformal factor of the metric on the $({x_+},{x_-})$ submanifold. Recall that we have ${g_{\mu\nu}}= e^{\sigma(u,r,x^i)}$ with $\sigma(u,r,x^i) = \ln r + \sigma_0(u)+\sigma_1(x^i)+O(r)$, and our boundary conditions require ${g_{\mu\nu}}' = e^{\sigma'(u,r,x^i)}$ with $\sigma'(u,r,x^i) = \ln r + \sigma_0'(u)+\sigma_1'(x^i)+O(r)$. Thus the $\ln r$ term in $\sigma$ is common to all the horizons we are considering, and we can think of it as a fixed, non-dynamical background around which $\phi$ fluctuates. When $\phi = 0$, the conformal factor of the metric on the $({x_+},{x_-})$ submanifold is simply $\ln r$. The ground state value of $L_0^{\pm}$ is therefore given by evaluating $L_0^{\pm}$ for $\phi$ shifted by this background value.
We can calculate $L_0^{\pm}$ for $\phi = \ln r$ by writing $\ln r$ in terms of the coordinates $({x_+},{x_-})$. Consider a general metric of the form (\[eq-GN\]). As previously stated, such a metric has a well-defined inverse Hawking temperature $\beta_H = \frac{1}{\kappa}$, where $\kappa = -\frac{1}{2}F_{r=0}$. By defining ${x_+}= u$ and ${x_-}= u - \beta_H\ln r$, we can put this metric in the conformal form (\[eq-conf2\]). It follows that $\ln r = \frac{1}{\beta_H}({x_+}- {x_-})$. Evaluating $L_0^{\pm}$ for the solution $\phi = \ln r = \frac{1}{\beta_H}({x_+}- {x_-})$ gives: $$L_0^{\pm} = \frac{{a_{\Delta}}l^2 }{8\pi^2 G\beta_H^2}$$ We can now calculate the horizon entropy using the Cardy formula, and find: $$S_H = \frac{{a_{\Delta}}l}{4G\beta_H}$$ We obtain the desired result of $S_H = \frac{{a_{\Delta}}}{4G}$ with $l = \beta_H$, which is a natural choice for $l$ as $\beta_H$ is the period for a thermal ensemble after analytically continuing to imaginary time.
Hawking Radiation {#sec-Hawking}
=================
We now investigate another aspect of horizon thermodynamics: Hawking radiation. We couple the gravitational degrees of freedom in the near-horizon region to a classical scalar field, and show that we can produce Hawking radiation from this coupling. Unlike most derivations of Hawking radiation, we will quantize the gravitational theory, and treat the scalar field as a classical background, as in [@EmparanSachs]. Our calculations extend and modify those given in [@EmparanSachs], as we work in an arbitrary number of dimensions and couple the scalar field to the gravitational degrees of freedom at the horizon, rather than spatial infinity.
We know from the results described in Sec. \[sec-CFT\] that the gravitational degrees of freedom in a neighborhood of a horizon can be described by a 2D CFT. We also know how the metric components change under the infinitesimal conformal transformations $x_+ \to x_+'(x_+)$ and $x_- \to x_-'(x_-)$ of the CFT. The fields and operators of the CFT are the components of the metric and objects constructed from the metric, since the metric is the only dynamical field in our framework. Note that this result remains unchanged even if the matter energy-momentum tensor does not satisfy the fall-off conditions that lead to the precise form of the action in (\[eq-action\]), as $\phi$ still becomes a free two-dimensional conformal field in an infinitesimal neighborhood of the horizon.
There is one subtle point that must be taken into account when quantizing this CFT. When calculating the classical charges $L_0^{\pm}$ in Sec. \[sec-Entropy\], we evaluated the charges by shifting $\phi$ by a fixed background configuration corresponding to a horizon. However, when we quantize the field $\phi$, we are quantizing its fluctuations about this background, which have the form given in (\[eq-phiDef\]). We can see from the relations (\[eq-xuRel\]) and (\[eq-useRel\]) that $\partial_-\phi = O(r)$ for these fluctuations. It follows that $L_n^{-} = 0$ for $n \neq 0$, and thus when we quantize the 2D CFT, we end up with only a chiral half of the original theory, with infinitesimal conformal transformations $x_+ \to x_+'(x_+)$.
To determine the conformal weight[@DiFrancesco] of the fields and operators in the chiral CFT, we note that as $x_+ = x_+(u)$ and $u = u(x_+)$, infinitesimal transformations of $u$ are equivalent to transformations of $x_+$. Conversely, since $\partial_{r} x_+ = 0$, we find that the coordinate $r$ does not transform under transformations of $x_+$. It follows that the conformal weight of a metric component is equal to the number of lower $u$ indices it has. For example, $g_{ur}$ has conformal weight 1. This result still applies when we switch from using GN coordinates to the tortoise GN coordinates given in (\[eq-Tort\]).
When we consider adding matter to the system, the total action is the sum of the matter action and the Einstein-Hilbert action. Since the metric appears in the matter action, and $\phi$ is a gravitational degree of freedom, we see that the matter action gives rise to a coupling between $\phi$ and the matter fields. Since we are taking the matter fields to be classical background fields, this means that the matter action will take the form of an operator constructed from $\phi$ that perturbs the theory described by the original action(\[eq-action\]). Moreover, this perturbation does not disappear in the limit $r\to 0$, which means that it continues to have an effect even in an infinitesimal neighborhood of the horizon. We can determine the conformal weight of this operator, and thus we find that this perturbation results in Hawking radiation.
In order to determine the form of the coupling between a scalar field $\psi$ and the conformal field $\phi$ that arises in the matter action, we consider the scalar field as a classical background, and quantize the CFT. This means that we take the scalar field on-shell in the bulk of the spacetime. Thus the scalar field action will result in a perturbation of the CFT by adding an operator to the CFT action. In tortoise GN coordinates, the scalar field action in the near-horizon region is $$\begin{aligned}
I_s &= \int_{\mathcal{M}} \sqrt{-g}g^{\mu\nu}\partial_\mu\psi\partial_\nu\psi + \int_{\Delta} \sqrt{-g}g^{\tilde{r}\mu}\psi\partial_\mu\psi\end{aligned}$$ The boundary term is evaluated at the horizon. Inspecting the action, we find that all the terms either go to zero in the near-horizon region as $\tilde{r}\to\infty$, or exhibit a coupling of the classical scalar field to an operator of conformal weight 1. These couplings are given by: $$\begin{aligned}
I_{\mathrm{int}} &= \int_{\mathcal{M}} \sqrt{-g}\left ( g^{\tilde{r}\tilde{r}}\partial_{\tilde{r}}\psi\partial_{\tilde{r}}\psi\right ) + \int_{\Delta} \sqrt{-g}\left ( g^{\tilde{r}\tilde{r}}\psi\partial_{\tilde{r}}\psi\right )\nonumber\\
&\sim \int_{\mathcal{M}} g_{u\tilde{r}}\left ( g^{\tilde{r}\tilde{r}}\partial_{\tilde{r}}\psi\partial_{\tilde{r}}\psi\right )+ \int_{\Delta} g_{u\tilde{r}}\left ( g^{\tilde{r}\tilde{r}}\psi\partial_{\tilde{r}}\psi\right )\label{eq-coupling}\end{aligned}$$ When we quantize the near-horizon CFT, these couplings add a perturbation to the CFT action in the form of an operator $\mathcal{O}(u)$ of conformal weight 1. This perturbation will induce transitions between closely spaced states of the CFT, resulting in the emission of radiation. From the form of the metric in (\[eq-Tort\]) we see that this perturbation remains finite even in an infinitesimal neighborhood of the horizon, so that our calculation remains valid as $r\to 0$.
We assume that the form of the scalar field is not affected by small deformations of the metric. It is not necessary to know the specific form of $\psi$, but it is instructive to work out a simple example. If the background metric is a spherically symmetric, static metric with a Killing horizon that has the form $${{\mathop{\mathrm{d}s}}}^2 = -f(x){{\mathop{\mathrm{d}t}}}^2+\frac{1}{f(x)}{{\mathop{\mathrm{d}x}}}^2 + x^2{{\mathop{\mathrm{d}\Omega}}}^2,$$ with $f(x) = \frac{2}{\beta_H}(x-x_h) + O((x-x_h)^2)$, then we can write this metric in GN coordinates by defining $r:= x-x_h$ and $u:=t+\frac{\beta_H}{2}\ln r$. We can describe $\psi$ as an infinite collection of 2-dimensional scalar fields $\psi_{l,m}$ of the form $$\psi_{l,m} = e^{i(t-\tilde{r})} + e^{i(t+\tilde{r})}$$ where $\tilde{r}:=\frac{\beta_H}{2}\ln r$ is the radial tortoise coordinate[@RobinsonWilczek]. In this simple case of a static metric with a Killing horizon, substituting the modes $\psi_{l,m}$ into (\[eq-coupling\]) gives a coupling that remains finite even infinitesimally close to the horizon.
This coupling will lead to transitions between the states of the CFT, so that the horizon produces Hawking radiation. We can compute the macroscopic decay rate using standard conformal field theory methods, following the approaches of [@EmparanSachs] and [@MaldacenaStrominger]. The operator $\mathcal{O}(u)$ introduced by the coupling to the scalar field will lead to a transition amplitude between initial and final states of the horizon in the presence of an external flux with frequency $\omega$ of the form: $$\mathcal{M} \sim \int {\mathop{\mathrm{d}u}} \langle f | \mathcal{O}(u) | i \rangle e^{-i\omega u}.$$ Squaring and summing over final states, we get: $$\label{eq-M}
\sum_f |\mathcal{M}|^2 \sim \int {\mathop{\mathrm{d}u}}{\mathop{\mathrm{d}u'}} \langle i | \mathcal{O}(u)\mathcal{O}(u') | i \rangle e^{-i\omega (u-u')}$$ for the decay rate. As we have already stated, we can define the surface gravity $\kappa$ (and therefore, the temperature) of any horizon that satisfies our boundary conditions as $\kappa := -\frac{1}{2}F|_{r=0}$. Thus, the class of horizons we are studying are in thermal equilibrium and can therefore be considered as thermal states with a well-defined temperature. We therefore average over the intial states assuming that the distribution is given by a Boltzmann spectrum. If the temperature of the horizon is $T_H$, then the decay rate is given by finite temperature two-point functions, which have the form $$\langle \mathcal{O}^\dagger(0)\mathcal{O}(u)\rangle_{T_H} \sim \left [\frac{\pi T_H}{\sinh(\pi T_H u)} \right ]^2$$ In order to evaluate the integrals in (\[eq-M\]), we use standard techniques of contour integration and assume that we are calculating emission rates. We find that the emission rate is given by $$\Gamma \sim \frac{\pi\omega}{e^{\frac{\omega}{T_H}}-1},$$ where we have divided by a factor of $\omega$ to account for the normalization of the outgoing scalar. Thus we obtain the familiar blackbody spectrum of Hawking radiation.
Conclusion {#sec-conc}
==========
We have investigated the thermodynamic properties of horizons by using the dynamical description of the diffeomorphism degrees of freedom obtained in [@Mine]. Using the Cardy formula, we computed the entropy of the horizon and reproduced the expected Bekenstein-Hawking entropy. This result suggests that the classical conformal symmetry imposed by boundary conditions at a horizon is enough to determine the entropy of the horizon, without reference to the underlying theory of quantum gravity. The wide applicability of the result to many kinds of horizons, including cosmological and acceleration horizons[@Unruh; @GibbonsHawking], indicates that the universality of the Bekenstein-Hawking entropy formula is results from the fact that a two-dimensional conformal symmetry is always induced near a horizon. We have also provided evidence for the validity of the effective description of horizon degrees of freedom as a 2D CFT by coupling the effective theory to a classical scalar background and showing that this produces Hawking radiation. Our result shows that although the effective theory is not the “true” theory of quantum gravity, it can provide a way of quantizing the gravitational degrees of freedom at a horizon.
Computing the Dynamical Action {#app-3}
==============================
In this section we present the details of the calculations used to derive the dynamical action in Sec. \[sec-Action\].
Evaluating the Ricci tensor for metrics of the form (\[eq-conf2\]), we find: $$\begin{aligned}
R_{+-} &= -\partial_+\partial_-\sigma + O(r)\\
R_{-i} &= \frac{1}{2}(\partial_-^2\sigma h_{+i} + \partial_-^2h_{+i} + \partial_-\sigma\partial_-h_{+i}) + O(r)\\
R_{+i} &= \frac{1}{2}(\partial_-\sigma \partial_+h_{+i} + \partial_+\partial_-h_{+i}) + O(r)\end{aligned}$$ We will assume that the matter energy-momentum tensor in the near-horizon region satisfies: $$\begin{aligned}
T_{+-} \,\,\,\hat{=}\,\,\, O(r^2)\label{eq-TAssumpGen1}\\
T_{-i} \,\,\,\hat{=}\,\,\, O(r)\label{eq-TAssumpGen2}\\
T_{+i} \,\,\,\hat{=}\,\,\, O(r)\label{eq-TAssumpGen3}\\
T_{ij} \,\,\,\hat{=}\,\,\, O(r)\label{eq-TAssumpGen4}\end{aligned}$$ where “$\hat{=}$” indicates that the constraint holds as $r\to0$ (The reasonableness of these constraints is discussed at greater length in [@Mine]. If they are not met, then the effective action for the gravitational degrees of freedom in the near-horizon region become modified by a potential term. However, the dynamics described by the action remain unchanged, even in this case.)
Looking at the Einstein equation: $$\label{eq-EE}
R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G T_{\mu\nu},$$ this gives the following constraints on the metric: $$\begin{aligned}
\partial_+\partial_-\sigma \,\,\,\hat{=}\,\,\, O(r)\label{eq-GenConst1}\\
\partial_-^2\sigma h_{+i} + \partial_-^2 h_{+i} + \partial_-\sigma\partial_- h_{+i} \,\,\,\hat{=}\,\,\, O(r)\label{eq-GenConst2}\\
\partial_-\sigma \partial_+ h_{+i} + \partial_+\partial_- h_{+i} \,\,\,\hat{=}\,\,\, O(r)\label{eq-GenConst3}\\
R - \frac{2n\Lambda}{n-2} \,\,\,\hat{=}\,\,\, O(r)\label{eq-GenConst4}\end{aligned}$$ These conditions are necessary (but not sufficient) for ${g_{\mu\nu}}$ to satisfy the Einstein equations as $r\to 0$.
We apply a diffeomorphism $\xi$ of the form (\[eq-diff\]) to the background metric ${g_{\mu\nu}}$, obtaining a new metric ${g_{\mu\nu}}' := {g_{\mu\nu}}+ {\mathcal{L}_{\xi}}{g_{\mu\nu}}$. We then evaluate the Einstein-Hilbert action for the new metric ${g_{\mu\nu}}'$ with the constraints (\[eq-GenConst1\])-(\[eq-GenConst4\]) applied to ${g_{\mu\nu}}$ in the near-horizon region, thus isolating the gravitational fluctuations about this background that preserve the horizon. When evaluating the Einstein-Hilbert action, everything is calculated from the new metric, including the inverse metric and the metric determinant, as the metric itself is the only dynamical field in the problem. The final form of the action determines the dynamics of the horizon degrees of freedom in an infinitesimal neighborhood of the horizon. As we do not require ${g_{\mu\nu}}$ to satisfy the Einstein equations, we get a non-trivial form for the action.
We begin with the Einstein-Hilbert action $$I_{EH} = \frac{1}{16\pi G} \int{\mathop{\mathrm{d}^nx}} \sqrt{-g}\,\,(R-2\Lambda),$$ and evaluate the action for the new metric ${g_{\mu\nu}}'$. We see from (\[eq-newMet\]) that this metric has the form ${g_{+-}}' = (1+\phi){g_{+-}}$ and $g_{ij}' = g_{ij} + O(r)$, with $\phi$ given by $$\phi = \partial_+\xi^+ +\partial_-\xi^-+\xi^+\partial_+\sigma+\xi^-\partial_-\sigma.$$ Although $\sigma$ (and therefore $\phi$) is now a function of $x^i$, the requirement that $\partial_i\partial_+\sigma=O(r)$ and $\partial_i\partial_-\sigma=O(r)$ means that to leading order $\phi$ is independent of $x^i$ and may be considered as a field on the $({x_+},{x_-})$ submanifold. This $\phi$ will end up being the dynamical degree of freedom in the near-horizon region. As we are interested in the near-horizon region $r\to 0$, and we are considering infinitesimal diffeomorphisms, we work to leading order in $(r,\phi)$. To second order in $\phi$, the inverse metric ${g'}^{\mu\nu}$ has the form $$\begin{aligned}
{g'}^{+-} &= (1-\phi + \phi^2)g^{+-}\\
{g'}^{ij} &= g^{ij} + O(r)\end{aligned}$$ We can now begin evaluating each of the terms necessary to compute the Einstein-Hilbert action. First we find that $$\begin{aligned}
\label{eq-AppMetDet}
\sqrt{-g'} &= {g_{+-}}'\sqrt{\tilde{g}} + O(r^2)\end{aligned}$$ In order to evaluate the Einstein-Hilbert action for ${g_{\mu\nu}}'$, we need to compute $$\begin{aligned}
R' &= 2(g^{'+-}R'_{+-} + g^{'i-}R'_{i-}) + g'^{ij}R'_{ij}\end{aligned}$$ to leading order in $(r,\phi)$. We first find $$\begin{aligned}
R_{+-} &= -\partial_-\Gamma_{++}^+\nonumber\\
&\qquad+ \frac{1}{2} \biggl [g^{ij}(\partial_-\partial_j g_{+i} - \partial_i\partial_j{g_{+-}}- \partial_-\partial_+g_{ij})\nonumber\\
&\qquad\qquad\qquad+ (\partial_-g_{+j} - \partial_j{g_{+-}})g^{ij}(\tilde{\Gamma}^k_{ki} + \frac{1}{2}g^{k-}\partial_-g_{ik})\\
&\qquad\qquad\qquad+ (\partial_-g_{+j} - \partial_j{g_{+-}})(\partial_-g^{j-} + \partial_i g^{ij})\nonumber\\
&\qquad\qquad\qquad - g^{i-}(\partial_i\partial_-{g_{+-}}- \partial_-^2 g_{+i}) \biggr ] + O(r^2)\\
R_{-i} &= \frac{1}{2}g^{+-}(\partial_-^2 g_{+i} - \partial_-\partial_i {g_{+-}}) + \frac{1}{2}(g^{+-})^2(\partial_-{g_{+-}})(\partial_i{g_{+-}}- \partial_- g_{+i})\nonumber\\
&\hspace{0.7cm} + O(r)\end{aligned}$$ where $\tilde{A}$ denotes a quantity $A$ computed with respect to the induced metric $\tilde{g}_{ij} := g_{ij}|_{r=0}$ on the $r=0$ hypersurface. This gives: $$\begin{aligned}
2(g^{i-}R_{i-} + g^{+-}R_{+-})&= g^{i-}g^{+-}(\partial_-^2g_{+i} - \partial_i\partial_-{g_{+-}})\nonumber\\
&\qquad+ g^{i-}(g^{+-})^2(\partial_-{g_{+-}}\partial_i{g_{+-}}- \partial_-{g_{+-}}\partial_i g_{+i})\nonumber\\
&\qquad -g^{+-}2\partial_-\Gamma_{++}^+\nonumber\\
&\qquad + g^{ij}g^{+-}(\partial_-\partial_j g_{+i} - \partial_i \partial_j {g_{+-}}- \partial_- \partial_+ g_{ij})\nonumber\\
&\qquad + g^{+-}(\partial_- g_{+j} - \partial_-{g_{+-}})g^{ij}(\tilde{\Gamma}^k_{ki} + \frac{1}{2}g^{k-}\partial_- g_{ik})\nonumber\\
&\qquad + g^{+-}(\partial_-g_{+j} - \partial_-{g_{+-}})(\partial_i g^{ij} + \partial_- g^{j-})\nonumber\\
&\qquad + O(r)\label{eq-Gen1}\end{aligned}$$ Similarly, we can compute: $$\begin{aligned}
g^{ij}R_{ij} &= -g^{ij}g^{+-}(\partial_+\partial_-g_{ij} + \partial_i\partial_j{g_{+-}})\nonumber\\
&\qquad + \frac{1}{2}g^{ij}g^{+-}(\partial_-\partial_ig_{+j} +\partial_-\partial_j g_{+i})\nonumber\\
&\qquad + \frac{1}{2}g^{ij}(\partial_j{g_{+-}}\partial_i{g_{+-}}- \partial_-g_{+j}\partial_-g_{+i})\nonumber\\
&\qquad + g^{ij}g^{+-}\partial_k{g_{+-}}\tilde{\Gamma}_{ij}^k + g^{ij}\tilde{R}_{ij} + O(r)\label{eq-Gen2}\end{aligned}$$ We can now compute ${g_{+-}}'R'$ by varying all the quantities in (\[eq-Gen1\]) and (\[eq-Gen2\]) under a diffeomorphism $\xi$ of the form (\[eq-diff\]). We write: $$\begin{aligned}
{g_{+-}}'R' &= -2{g_{+-}}'g^{'+-}\partial_-\Gamma_{++}^{'+} + B(\phi, {g_{\mu\nu}})\end{aligned}$$ for some function $B(\phi, {g_{\mu\nu}})$. If we can show that $$\begin{aligned}
\label{eq-Key}
B(\phi, {g_{\mu\nu}}) - {g_{+-}}'(R + 2g^{+-}\partial_+\partial_-\sigma) = O(r^2),\end{aligned}$$ then we can write: $$\begin{aligned}
{g_{+-}}'R' &= -2{g_{+-}}'g^{+-}\partial_-\Gamma_{++}^{'+} + {g_{+-}}' (R + 2g^{+-}\partial_+\partial_-\sigma) + O(r^2)\nonumber\\
&= 2{g_{+-}}'(-g^{+-}\partial_-\Gamma_{++}^{'+} + g^{+-}\partial_+\partial_-\sigma)\nonumber\\
&\qquad + {g_{+-}}'\frac{2n\Lambda}{n-2} + O(r^2)\label{eq-GenFinalR}\end{aligned}$$ Calculating the first term in the above expression, we find: $$\begin{aligned}
\label{eq-AppAction1}
&2{g_{+-}}'(-g^{+-}\partial_-\Gamma_{++}^{'+} + g^{+-}\partial_+\partial_-\sigma)\nonumber\\
&\qquad= 2{g_{+-}}'g^{+-}(-{g'}^{+-}\partial_+\partial_-{g_{+-}}' + ({g'}^{+-})^2\partial_-{g_{+-}}'\partial_+{g_{+-}}')\nonumber\\
&\qquad\qquad + 2{g_{+-}}'g^{+-}\partial_+\partial_-\sigma\end{aligned}$$ To evaluate the first two terms in (\[eq-AppAction1\]) in terms of $\phi$ and the background metric ${g_{\mu\nu}}$, we compute: $$\begin{aligned}
-{g'}^{+-}\partial_+\partial_-{g_{+-}}' + ({g'}^{+-})^2\partial_-{g_{+-}}'\partial_+{g_{+-}}' &= \partial_+\phi\partial_-\phi\nonumber\\
&\qquad+ (1+\partial_-\xi^- + \partial_+\xi^+)\partial_+\partial_-\sigma\nonumber\\
&\qquad+ \xi^+\partial_+^2\partial_-\sigma + \xi^-\partial_+\partial_-^2\sigma\nonumber\\
&\qquad+ O(\phi^3, r\phi^2)\label{eq-RicciNewApp}\end{aligned}$$ We are ignoring terms of $O(\phi^3)$ and $O(r\phi^2)$ as we are working to leading order in $(r,\phi)$ and the leading terms in the action will be at most $O(\phi^2)$ or $O(r\phi)$.
We can now write (\[eq-AppAction1\]) as: $$\begin{aligned}
&2{g'}^{+-}\partial_+\phi\partial_-\phi\label{eq-Kin}\\
&\qquad-2{g'}^{+-}\partial_+\partial_-\sigma(1+\partial_+\xi^+ + \partial_-\xi^-)\label{eq-R1}\\
&\qquad-2{g'}^{+-}(\xi^+\partial_+^2\partial_-\sigma + \xi^-\partial_+\partial_-^2\sigma)\label{eq-R2}\\
&\qquad+ 2g^{+-}\partial_+\partial_-\sigma + O(r)\label{eq-R3}\end{aligned}$$ Multiplying the first term (\[eq-Kin\]) of (\[eq-AppAction1\]) with ${g_{+-}}'$ then gives: $$\begin{aligned}
\sqrt{-\hat{g}} \left (2g^{+-}\partial_+\phi\partial_-\phi\right ) &= \sqrt{-\hat{g}} \left (g^{ab}\partial_a\phi\partial_b\phi\right )\nonumber\\
&= \sqrt{-\hat{g}} \left (\partial_a\phi\partial^a\phi\right )\label{eq-Final1}\end{aligned}$$ where the index $a \in ({x_+},{x_-})$ and $\hat{g}_{ab}$ is the induced metric on the $({x_+},{x_-})$ submanifold.
Similarly, multiplying the terms (\[eq-R1\])-(\[eq-R3\]) with ${g_{+-}}'$ gives: $$\begin{aligned}
&-2\left (\xi^+\partial_+^2\partial_-\sigma + \xi^-\partial_+\partial_-^2\sigma \right )-2\partial_+\partial_-\sigma\left ( 1+\partial_+\xi^+ + \partial_-\xi^-\right ) \nonumber\\
&\qquad+ 2\partial_+\partial_-\sigma(1+\phi)\end{aligned}$$ This expression simplifies to $$\begin{aligned}
&2\left [\partial_+\partial_-\sigma (\xi^+\partial_+\sigma + \xi^-\partial_-\sigma) -\xi^+\partial_+^2\partial_-\sigma - \xi^-\partial_+\partial_-^2\sigma\right ]\label{eq-Final2a}\end{aligned}$$ We now use integration by parts to rewrite (\[eq-Final2a\]) as $$\begin{aligned}
&= 2\partial_+\partial_-\sigma (\xi^+\partial_+\sigma + \xi^-\partial_-\sigma + \partial_+\xi^+ + \partial_-\xi^-)\nonumber\\
&= -\left ({g_{+-}}\phi\hat{R}\right)\nonumber\\
&= -\sqrt{\hat{g}}\phi\hat{R}\label{eq-Final2b},\end{aligned}$$ where $\hat{R}$ is the Ricci scalar corresponding to the induced metric $\hat{g}_{ab}$ on the $({x_+},{x_-})$ submanifold, given by $$\hat{R} = -2g^{+-}\partial_+\partial_-\sigma\label{eq-hatR}$$
Thus we can see that (\[eq-AppAction1\]) is equivalent to $$\begin{aligned}
\label{eq-GenFinal1}
\sqrt{-\hat{g}}(\partial_a\phi\partial^a \phi - \phi\hat{R})\end{aligned}$$ Looking at the form of the metric in (\[eq-conf2\]), we might be concerned that $x^i$-dependence will enter through $\hat{g}$ and $\hat{R}$, when we are trying to define an action on the $(x_+, x_-)$ submanifold. However, recall that $g_{+-} = e^{\sigma(u,r,x^i)}$ with $\sigma = \ln r + \sigma_0(u) + \sigma_1(x^i) + O(r)$. Thus the $x^i$-dependence in $\sqrt{-\hat{g}} = g_{+-}$ cancels with the $x^i$ dependence in the kinetic term $\partial_a\phi\partial^a\phi = 2g^{+-}\partial_+\phi\partial_-\phi$. We have already established that $\phi$ can be interpreted as a field on the $(x_+, x_-)$ submanifold to leading order, so the kinetic term can be defined on the $(x_+, x_-)$ submanifold. A similar cancellation of the $x^i$-dependence occurs for the $\hat{R}$ term, as can be seen from the definition (\[eq-hatR\]) of $\hat{R}$. As a result, we can interpret the expression (\[eq-GenFinal1\]) as a quantity defined on the $(x_+, x_-)$ submanifold, by redefining: $$\begin{aligned}
\hat{g}_{+-} := e^{\ln r + \sigma_0(u) + O(r)}\end{aligned}$$ and $$\begin{aligned}
\hat{R} = -2\hat{g}^{+-}\partial_+\partial_-\sigma\end{aligned}$$ Substituting (\[eq-GenFinal1\]) back into (\[eq-GenFinalR\]) we find: $$\begin{aligned}
{g_{+-}}'(R' - 2\Lambda) &= \sqrt{-\hat{g}}(\partial_a\phi\partial^a \phi - \phi\hat{R} + \tilde{\lambda}(1+\phi))\end{aligned}$$ where the $x^i$-dependence of $\sqrt{-\hat{g}}$ only appears in the last term, and has been incorporated into $\tilde{\lambda} := \frac{4\Lambda e^{\sigma_1(x^i)}}{n-2}$. This gives: $$\begin{aligned}
\sqrt{-g'}(R' - 2\Lambda) &= \sqrt{\tilde{g}}\sqrt{-\hat{g}}(\partial_a\phi\partial^a \phi - \phi\hat{R} + \tilde{\lambda}(1+\phi))\end{aligned}$$ to leading order in $(r,\phi)$. We can integrate over the $x^i$ coordinates in the Einstein-Hilbert action to obtain the final dynamical action: $$\begin{aligned}
I_{\mathrm{hor}} = \frac{{a_{\Delta}}}{16\pi G} \int{\mathop{\mathrm{d}^2x}} \sqrt{-\hat{g}}\,\, \left (\partial_a\phi\partial^a\phi -\phi\hat{R}+ \lambda (1+\phi)\right )\nonumber\end{aligned}$$ where ${a_{\Delta}}$ is the cross-sectional area of the horizon, the variables of integration are $({x_+},{x_-})$, and the parameter $\lambda$ is given by: $$\begin{aligned}
\lambda := \frac{1}{a_{\Delta}} \int \mathrm{d}^{n-2}x^i\,\,\, \tilde{\lambda}\sqrt{\tilde{g}},\end{aligned}$$ where the integral is carried out over the $(n-2)$ coordinates $x^i$.
Now all we have to do is derive (\[eq-Key\]) in order to obtain our final result. By direct computation of ${g_{+-}}'R'$, we find: $$\begin{aligned}
&B(\phi, {g_{\mu\nu}}) - {g_{+-}}'(R + 2g^{+-}\partial_+\partial_-\sigma)\nonumber\\
&\quad= -\phi g^{ij}\tilde{\Gamma}^k_{ij}\partial_k{g_{+-}}+ g^{ij}\tilde{\Gamma}_{ij}^k\partial_k(\delta{g_{+-}})\label{eq-Term3}\\
&\quad-\phi\partial_i g^{ij}(\partial_- (\delta g_{+j}) - \partial_j(\delta {g_{+-}}))+ \partial_i g^{ij}(\partial_i (\delta g_{+j}) - \partial_j(\delta {g_{+-}}))\label{eq-Term9b}\\
&\quad-\phi g^{ij}\tilde{\Gamma}^k_{ki}(\partial_- g_{+j} - \partial_j{g_{+-}})+ g^{ij}\tilde{\Gamma}^k_{ki}(\partial_i (\delta g_{+j}) - \partial_j(\delta {g_{+-}}))\label{eq-Term4b}\\
&\quad-2\phi g^{ij}(\partial_-\partial_j g_{+i} - \partial_i\partial_j{g_{+-}}- \partial_-\partial_+ g_{ij})\label{eq-Term2a}\\
&\quad+ 2g^{ij}(\partial_-\partial_j (\delta g_{+i}) - \partial_i\partial_j(\delta {g_{+-}}) - \partial_-\partial_+ (\delta g_{ij}))\label{eq-Term2b}\\
&\quad+ \frac{1}{2}g^{ij}g^{+-}(1-\phi)(\partial_j {g_{+-}}'\partial_i{g_{+-}}' - \partial_- g_{+j}'\partial_- g_{+i}')\label{eq-Term5}\\
&\quad- \frac{1}{2}g^{ij}g^{+-}(1+\phi)(\partial_j {g_{+-}}\partial_i{g_{+-}}- \partial_- g_{+j}\partial_- g_{+i})\label{eq-Term5b}\\
&\quad+ 2g^{'i-}(\partial_-^2 g_{+i}' - \partial_-\partial_i {g_{+-}}')- 2g^{i-}(1+\phi)(\partial_-^2 g_{+i} - \partial_-\partial_i {g_{+-}})\label{eq-Term6b}\\
&\quad+ g^{'i-}g^{+-}(1-\phi)\partial_-{g_{+-}}'(\partial_i{g_{+-}}' - \partial_-g_{+i}')\label{eq-Term7}\\
&\quad- g^{i-}g^{+-}(1+\phi)\partial_-{g_{+-}}(\partial_i{g_{+-}}- \partial_-g_{+i})\label{eq-Term7b}\\
&\quad+ \partial_-g^{'j-}(\partial_-g_{+j}' - \partial_j{g_{+-}}')- \partial_-g^{j-}(1+\phi)(\partial_-g_{+j} - \partial_j{g_{+-}})\label{eq-Term8b}\end{aligned}$$ We simplify the above expression by applying the constraints (\[eq-GenConst1\])-(\[eq-GenConst3\]) to the background metric ${g_{\mu\nu}}$. To simplify the terms (\[eq-Term3\]): $$\begin{aligned}
-\phi\partial_k{g_{+-}}+ \partial_k(\delta{g_{+-}}) &= -\phi \partial_k{g_{+-}}+ \partial_k(\phi{g_{+-}})\nonumber\\
&= {g_{+-}}\partial_k\phi = O(r^2),\nonumber\end{aligned}$$ as ${g_{+-}}= O(r)$ and $\partial_k\phi = O(r)$ due to the conditions $\partial_{\pm}\partial_i\sigma = O(r)$. It follows that the terms (\[eq-Term3\]) combine to give a quantity that is $O(r^2)$.
In order to simplify the terms (\[eq-Term9b\]) as well as (\[eq-Term4b\]), consider: $$\begin{aligned}
&-\phi(\partial_- (\delta g_{+j}) - \partial_j(\delta {g_{+-}}))+ (\partial_i (\delta g_{+j}) - \partial_j(\delta {g_{+-}}))\nonumber\\
&\qquad\qquad= \xi^+{g_{+-}}(\partial_-\sigma\partial_+h_{+i} + \partial_-\partial_+ h_{+i})\nonumber\\
&\qquad\qquad\qquad+ \xi^-{g_{+-}}(\partial_-^2\sigma h_{+i} + \partial_-\sigma\partial_- h_{+i} + \partial_-^2 h_{+i})\nonumber\\
&\qquad\qquad= O(r^2)\end{aligned}$$ by the constraints (\[eq-GenConst2\])-(\[eq-GenConst3\]) on ${g_{\mu\nu}}$. It follows that the terms (\[eq-Term9b\])-(\[eq-Term4b\]) combine to give a quantity that is $O(r^2)$. The terms (\[eq-Term2a\])-(\[eq-Term2b\]) simplify in the same way to give a quantity that is $O(r^2)$.
We are left with the terms (\[eq-Term5\])-(\[eq-Term8b\]). In order to simplify these terms, we use the fact that $$\begin{aligned}
{g'}^{-i} &= -g^{+-}(1-\phi)g'_{+j}g^{ij} + O(r)\\
&= (1-\phi)g^{-i} - (1-\phi)g^{+-}g^{ij}\delta g_{+j} + O(r)\end{aligned}$$ Direct computation and the application of the constraints (\[eq-GenConst2\])-(\[eq-GenConst3\]) shows that the terms (\[eq-Term5\])-(\[eq-Term8b\]) also combine to give a quantity that is $O(r^2)$. So finally, we find that (\[eq-Key\]) holds. Note that constraint (\[eq-GenConst4\]) was not required for these computations.
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¶ c ł Ł §
CERN-TH/99-270\
hep–th/9909041\
.6in
[**States and Curves of Five-Dimensional Gauged Supergravity**]{}
0.5in
[**Ioannis Bakas${}^1$** ]{}and 0.1in [*${}^1\!$Department of Physics, University of Patras\
GR-26500 Patras, Greece\
[[email protected], ajax.physics.upatras.gr]{}*]{}\
.2in [*${}^2\!$Theory Division, CERN\
CH-1211 Geneva 23, Switzerland\
[[email protected]]{}*]{}\
.5in
**Abstract**
We consider the sector of ${\cal N}=8$ five-dimensional gauged supergravity with non-trivial scalar fields in the coset space $SL(6,\IR)/SO(6)$, plus the metric. We find that the most general supersymmetric solution is parametrized by six real moduli and analyze its properties using the theory of algebraic curves. In the generic case, where no continuous subgroup of the original $SO(6)$ symmetry remains unbroken, the algebraic curve of the corresponding solution is a Riemann surface of genus seven. When some cycles shrink to zero size the symmetry group is enhanced, whereas the genus of the Riemann surface is lowered accordingly. The uniformization of the curves is carried out explicitly and yields various supersymmetric configurations in terms of elliptic functions. We also analyze the ten-dimensional type-IIB supergravity origin of our solutions and show that they represent the gravitational field of a large number of D3-branes continuously distributed on hyper-surfaces embedded in the six-dimensional space transverse to the branes. The spectra of massless scalar and graviton excitations are also studied on these backgrounds by casting the associated differential equations into Schrödinger equations with non-trivial potentials. The potentials are found to be of Calogero type, rational or elliptic, depending on the background configuration that is used.
.5cm CERN-TH/99-270\
September 1999
16 pt
Introduction
============
Ungauged and gauged ${\cal N}=8$ supergravities in five dimensions were constructed several years ago in [@cremmer] and [@PPN; @GRW], following the analogous construction made in four dimensions in [@CJ] and [@WN1]. More recently it has become clear that solutions of five-dimensional gauged supergravity play an important rôle in the context of the AdS/CFT correspondence [@Maldacena; @Witten; @GKP]. In particular the maximum supersymmetric vacuum state in five-dimensional gauged supergravity with $AdS_5$ geometry, originates from the $AdS_5\times S^5$ solution in ten-dimensional type-IIB supergravity. The latter solution arises as the near horizon geometry of the solution representing the gravitational field of a large number of coincident D3-branes and has been conjectured to provide the correct framework for analyzing ${\cal N}=4$ supersymmetric $SU(N)$ Yang–Mills for large $N$ and ’t Hooft coupling constant at the conformal point of the Coulomb branch.
The supergravity approach to gauge theories at strong coupling is applicable not only at conformality, but also away from it. In particular, when the six scalar fields of the ${\cal N}=4$ supersymmetric Yang–Mills theory acquire Higgs expectation values we move away from the origin of the Coulomb branch and the appropriate supergravity solution corresponds to a multicenter distribution of D3-branes with the centers, where the branes are located, associated with the scalar Higgs expectation values in the gauge theory side. A prototype example of such D3-brane distributions is the two-center solution that has been studied in [@Maldacena; @MW; @KW], whereas examples of continuous D3-brane distributions arise naturally in the supersymmetric limit of rotating D3-brane solutions [@KLT; @sfe1]. Concentrating on the case of continuous distributions, note that from a ten-dimensional type-IIB supergravity view point the $SO(6)$ symmetry, associated with the round $S^5$-sphere, is broken because this sphere is deformed. On the other hand, from the point of view of five-dimensional gauged supergravity the deformation of the sphere is associated with the fact that some of the scalar fields in the theory are turned on. Hence, finding solutions of five-dimensional gauged supersgravity might shed more light into the AdS/CFT correspondence as far as the Coulomb branch is concerned. Using such solutions, investigations of the spectrum of massless scalars excitations and of the quark-antiquark potential have already been carried out with sometimes suprising results [@FGPW2; @BS1; @CR-GR]. Solutions of the five-dimensional theory are also important in a non-perturbative treatment of the renormalization group flow in gauge theories at strong coupling [@GPPZ1; @DZ; @KPW; @FGPW1].
An additional motivation for studying solutions of five-dimensional gauged supergravity is the fact that for a class of such configurations, four-dimensional Poincaré invariance is preserved. It turns out that our four-dimensional space-time can be viewed as being embedded non-trivially in the five-dimensional solution with a warp factor. This particular idea of our space-time as a membrane in higher dimensions is quite old [@GW] and has been recently revived with interesting phenomenological consequences on the mass hierarchy problem [@rasu]. In that work, in particular, our four-dimensional world was embedded into the $AdS_5$ space from which a slice was cut out; it results into a normalizable graviton zero mode, but also to a continuum spectrum of massive ones above it with no mass gap separating them. The use of more general solutions of five-dimensional gauged supergravity certainly creates more possibilities and in fact there are solutions with a mass gap that separates the massless mode from the massive ones [@BS2].
This paper is organized as follows: In section 2 we present a brief summary of some basic facts about ${\cal N}=8$ five-dimensional gauged supergavity with gauge group $SO(6)$. In particular, we restrict our attention to the sector of the theory where only the metric and the scalar fields associated with the coset space $SL(6,\IR)/SO(6)$ are turned one. In section 3 we find the most general supersymmetric configuration in this sector, which as it turns out, depends on six real moduli. Our solutions have a ten-dimensional origin within type-IIB supergravity and represent the gravitational field of continuous distributions of D3-branes in hyper-surfaces embedded in the transverse space to the branes. In section 4 we further analyze our solution using some concepts from the theory of algebraic curves and in particular Riemann surfaces. We find that our states correspond to Riemann surfaces with genus up to seven, depending on their symmetry groups, which are all subgroups of $SO(6)$. In section 5 we provide details concerning the geometrical origin of the supersymmetric states in five dimensions from a ten-dimensional point of view using various distributions of D3-branes in type-IIB supergravity. This approach yields explicit expressions for the metric and the scalar fields, and it can be viewed as complementary to the algebro-geometric classification of section 4 in terms of Riemann surfaces. In section 6 we consider massless scalar and graviton fluctuations propagating on our backgrounds. We formulate the problem equivalently as a Schrödinger equation in one dimension and compute the potential in some cases of particular interest. We also note intriguing connections of these potentials to Calogero models and various elliptic generalizations thereof. Finally, we end the paper with section 7 where we present our conclusions and some directions for future work.
Elements of five-dimensional gauged supergravity
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${\cal N}=8$ supergravity in five dimensions involves 42 scalar fields parametrizing the non-compact coset space $E_{6(6)}/USp(8)$ that describes their couplings in the form of a non-linear $\s$-model [@cremmer]. In five-dimensional gauged supergravity the global symmetry group $E_{6(6)}$ breaks into an $SO(6)$ subgroup which corresponds to the gauge symmetry group of the resulting theory, and a non-trivial potential develops [@PPN; @GRW]. In the framework of the AdS/CFT correspondence [@Maldacena; @Witten; @GKP] the supergravity scalars represent the couplings of the marginal and relevant chiral primary operators of the ${\cal N}=4$ supersymmetric Yang–Mills theory in four dimensions. The invariance of the theory with respect to the gauge group, as well as the $SL(2,\IR)$ symmetry inherited from type-IIB supergravity in ten dimensions, restricts the scalar potential to depend on $42-15-3=24$ invariants of the above groups. However, it seems still practically impossible to deal with such a general potential. In this paper we restrict attention to the scalar subsector corresponding to the symmetric traceless representation of $SO(6)$, which parametrizes the coset $SL(6,\IR)/SO(6)$, and set all other fields (except the metric) equal to zero. In this sector we will be able to find explicitly the general solution of the classical equations of motion that preserves supersymmetry.
The Lagrangian for this particular coupled gravity-scalar sector includes the usual Einstein–Hilbert term, the usual kinetic term for the scalars as well as their potential = [14]{} [R]{} - \_[i=1]{}\^[5]{} (\_i)\^2 - P . \[lag1\] A few explanations concerning the scalar-field part of this action are in order. It has been shown that in this subsector the scalar potential $P$ depends on the symmetric matrix $SS^T$ only, where $S$ is an element of $SL(6,\IR)$ [@GRW] (for a recent discussion see also [@DZ; @FGPW2]). Diagonalization of this matrix yields a form that depends only on five scalar fields. It is convenient, nevertheless, to represent this sector in terms of six scalar fields $\beta_i$, $i=1,2,\dots , 6$ as [@FGPW2] P= -[18 R\^2]{} ( (\_[i=1]{}\^6 e\^[2 \_i]{})\^2 - 2 \_[i=1]{}\^6 e\^[4 \_i]{} ) , \[ppotenti\] where = . \[beaa\] Note that the $6\times 5$ matrix that relates the auxiliary scalars $\b_i$ with the $\a_i$’s is not unique; it only has to satisfy the condition $\sum_i\b_i =0$. The choice in [(\[beaa\])]{} is particularly useful for certain computational purposes. It also has the property that if the fields $\b_i$ are canonically normalized, the five independent scalar fields $\a_i$ will be canonically normalized as well, i.e. $\sum_{i=1}^6 (\del \b_i)^2 = 2 \sum_{i=1}^5 (\del \a_i)^2$.
The form of the kinetic term for the scalars in [(\[lag1\])]{} suggests that the metric in the corresponding coset space is taken to be $\d^{ij}$. This was explicitly shown for the case of only one scalar field in [@DZ] and the general result was quoted without detailed explanation in [@FGPW2]. One can generally prove this statement by first realizing that the kinetic term of these scalars can depend on two type of terms, namely ${\rm Tr}(\del_\m S S\inv) {\rm Tr}(\del^\m S S\inv)$ and ${\rm Tr}(\del_\m S S\inv \del^\m S S\inv)$. Since $\del_\m S S\inv$ belongs to the algebra of $SL(6,\IR)$ the first term is zero because of the traceless condition. The second term gives a result proportional to $\sum_{i=1}^6 (\del \b_i)^2 = 2 \sum_{i=1}^5 (\del \a_i)^2$, thus showing that the scalar kinetic term in [(\[lag1\])]{} has indeed the above form. The equations of motion follow by varying the action [(\[lag1\])]{} with respect to the five-dimensional metric and the scalar fields. Using the metric $G_{MN}$, we have R\_[MN]{} & =& \_[i=1]{}\^5 \_M \_i \_N \_i + [13]{} G\_[MN]{} P ,\
D\^2 \_i & =& [P \_i]{} . \[eqs32\] There is a maximally supersymmetric solution of the above equations that preserves all 32 supercharges, in which all scalar fields are set zero and the metric is that of $AdS_5$ space. Then, the potential in [(\[ppotenti\])]{} becomes $P=-3/R^2$ and equals by definition to the negative cosmological constant of the theory. This defines the length scale $R$ that will be used in the following.
The coupled system of non-linear differential equations [(\[eqs32\])]{} is in general difficult to solve. In this paper we will be interested in solutions preserving four-dimensional Poincaré invariance $ISO(1,3)$. Hence, we make the following ansatz for the five-dimensional metric ds\^2 = e\^[2 A]{}(\_ dx\^dx\^+ dz\^2) , \[aans\] where $\eta_{\m\n}={\rm diag}(-1,1,1,1)$ is the four-dimensional Minkowski metric and the conformal factor $e^{2A}$, as well as the scalar fields $\a_i$, depend only on the variable $z$. In addition, we demand that our solutions preserve supersymmetry. The corresponding Killing spinor equations, arising from the supersymmetry transformation rules for the 8 gravitinos and the 42 spin-$1/2$ fields, give rise to the first order equations [@FGPW2] A\^= [23 R]{} e\^A W ; \^\_i = -[1R]{} e\^A [W\_i]{} ,i =1,2,…, 5 , \[hjd1\] where W=-[14]{}\_[i=1]{}\^6 e\^[2 \_i]{} , \[h2d1\] and the derivative is taken with respect to the coordinate $z$. It is straightforward to check that all supersymmetric solutions satisfying the first order equations [(\[hjd1\])]{} also satisfy the second order equations [(\[eqs32\])]{}. In doing so, it is convenient to use the alternative expression for the potential, instead of [(\[ppotenti\])]{}, P = [12 R\^2]{} \_[i=1]{}\^5 (W\_i)\^2 - [43 R\^2]{} W\^2 . \[alt1\]
The general supersymmetric solution
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We begin this section with the construction of the most general solution of the non-linear system of equations [(\[hjd1\])]{} and discuss some of the general properties of the corresponding supersymmetric configurations. We also show how our solution can be lifted to ten dimensions in the context of type-IIB supergravity.
Five-dimensional solutions
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It might still seem difficult to find solutions of the coupled system of equations [(\[hjd1\])]{} at first sight, due to non-linearity. It turns out, however, that this is not the case, but instead it is possible to find the most general solution. In order to proceed further, we first compute the evolution of the auxiliary scalar fields $\b_i$. Using [(\[beaa\])]{} and [(\[hjd1\])]{} we find \_i\^= [e\^AR]{}([23]{} W + e\^[2 \_i]{} ) = A\^+ [1R]{} e\^[2 \_i + A]{} ,i=1,2,…, 6 , \[eh1\] where for the last equality we have used the first equation in [(\[hjd1\])]{}. This substitution results into six decoupled first order equations for the $\b_i$’s which can be easily integrated, as we will soon demonstrate. Of course, after deriving the explicit solution for the $\b_i$, we also have to check the self-consistency of this substitution.
Let us reparametrize the function $A(z)$ in terms of an auxiliary function $F(z/R^2)$ as follows e\^A = [1R]{} (-F\^/2)\^[1/3]{} . \[ai1\] We have included a minus sign in this definition since, according to the boundary conditions that we will later choose, $F$ will be a decreasing function of $z$. Then, according to this ansatz, the general solution of [(\[eh1\])]{} is given by e\^[2 \_i]{} = [(-F\^/2)\^[2/3]{}F-b\_i]{} , i=1,2,…, 6 , \[hja1\] where the prime denotes here the derivative with respect to the argument $z/R^2$. The $b_i$’s are six constants of integration, which, sometimes is convenient to order as b\_1b\_2…b\_6 , \[hord1\] without loss of generality. Note that we may fix one combination of them to an arbitrary constant value because [(\[ai1\])]{} determines the function $F$ up to an additive constant. Also, since the sum of the $\b_i$’s is zero, we find that the function $F$ has to satisfy the differential equation (F\^)\^2 =4 \_[i=1]{}\^6 (F-b\_i)\^[1/2]{} , \[jds1\] which thus contains all the information about the supersymmetric configurations and provides a non-trivial algebraic constraint. Using [(\[ai1\])]{}, [(\[hja1\])]{} and [(\[jds1\])]{} one may easily check that the first equation in [(\[hjd1\])]{} is also satisfied. If we insist on presenting the solution in the conformally flat form [(\[aans\])]{} the differential equation [(\[jds1\])]{} needs to be solved to obtain $F(z/R^2)$. This will be studied in detail in section 4, as it is a necessary step for investigating the massless scalar and graviton fluctuations in section 6.
At the moment we present our general solution in an alternative coordinate system, where $F$ is viewed as the independent variable. Indeed, using [(\[jds1\])]{}, we obtain for the metric ds\^2 = [f\^[1/6]{}R\^2]{} \_dx\^dx\^ + [R\^2 4 f\^[1/3]{}]{} dF\^2 ; f = \_[i=1]{}\^6 (F-b\_i) , \[fh3\] whereas the expression for the scalar fields in [(\[hja1\])]{} becomes e\^[2 \_i]{} = [f\^[1/6]{}F-b\_i]{} , i=1,2,…, 6 . \[hja2\] When the constants $b_i$ are all equal, our solution becomes nothing but $AdS_5$ with all scalar fields turned off to zero. In the opposite case, when all constants $b_i$ are unequal from one another, there is no continuous subgroup of $SO(6)$ preserved by our solution. If we let some of the $b_i$’s to coincide we restore various continuous subgroups of $SO(6)$ accordingly. As for the five scalar fields $\a_i$, they can be found using [(\[beaa\])]{} \_1 & = & [12]{} ( \_1 + \_2 - \_3 -\_4) ,\
\_2 & = & [12]{} ( \_1 + \_4 - \_2 -\_3) ,\
\_3 & = & [12]{} ( \_1 + \_3 - \_2 -\_4) , \[anti1\]\
\_4 & = & (\_5-\_6)\
\_5 & = & - (\_5+\_6) .
Note that imposing the reality condition on the scalars in [(\[hja2\])]{} restricts the values of $F$ to be larger that the maximum of the constants $b_i$, which according to the ordering in [(\[hord1\])]{} means that $F\ge b_1$. For $F\gg b_1$ the scalars tend to zero and $f\simeq F^6$, in which case the metric in [(\[fh3\])]{} approaches $AdS_5$ as expected; put differently, in this limit $F\simeq 1/z$ close to $z=0$ that is taken as the origin of the $z$-coordinate. For intermediate values of $F$ we have a flow in the five-dimensional space spanned by all scalar fields $\b_i$. In general we may have $b_1=b_2=\dots = b_n$, with $n\leq 6$, when $b_1$ is $n$-fold degenerate. In this case, the solution preserves an $SO(n)$ subgroup of $SO(6)$ and the flow is actually taking place in $6-n$ dimensions. On the other hand, let us consider the case when $F$ approaches its lower value $b_1$. Then, the scalars in [(\[hja2\])]{} are approaching e\^[2\_i]{}{
[ccc]{} f\_0\^[1/6]{} (F-b\_1)\^[(n-6)/6]{} , & [for]{} & i=1,2,…, n\
\
[f\_0\^[1/6]{}b\_1-b\_i]{} (F-b\_1)\^[n/6]{} ,& [for]{} & i=n+1,…, 6\
} , \[jkef1\] where $f_0=\prod_{i=n+1}^6 (b_1-b_i)$. Consequently, we have a one-dimensional flow in this limit since the scalar fields $\b_i$ can be expressed in terms of a single (canonically normalized) scalar $\a$, as & & [1]{} (n-6,…, n-6,n,…,n) ,\
& & [4 ]{} (F-b\_1) . \[fjk23\] It is also useful to find the limiting form of the metric [(\[fh3\])]{} when $F\to b_1$. Changing the variable to $\r$ as F= b\_1 + ([(6-n) f\_0\^[1/6]{} 3 R]{})\^[66-n]{} , \[jdr23\] the metric [(\[fh3\])]{} becomes for $\r\to 0^+$ ds\^2 d\^2 + ((6-n3)\^n [f\_0 R\^[12-n]{}]{})\^[16-n]{} \^[n6-n]{} \_ dx\^dx\^ . \[jksd1\] Hence, at $\r=0$ (or equivalently at $F=b_1$) there is a naked singularity which has an interpetation, as we will see later in the ten-dimensional context, as the location of a distribution of D3-branes. It is instructive to compare this with the singular behaviour of non-conformal non-supersymmetric solutions found in [@KS1]. A similar naked singularity was found there, but the corresponding metric near the singularity had a power law behaviour in $\r$ with exponent equal to $1/2$, which coincides with the result in [(\[jksd1\])]{} only for $n=2$.
Type-IIB supergravity origin
----------------------------
It is possible to lift our solution with metric and scalars given by [(\[fh3\])]{} and [(\[hja2\])]{} to a supersymmetric solution of type-IIB supergravity, where only the metric and the self-dual five-form are turned on. This proves that our five-dimensional solution is a true compactification of type-IIB supergravity on $S^5$. This is not a priori obvious because unlike the case of the $S^7$ compactification of eleven-dimensional supergravity to four dimensions [@witnik], there is no general proof that the full non-linear five-dimensional gauged supergravity action can be fully encoded into the action or equations of motion of the type-IIB supergravity for the $S^5$ compactification. However, there is a lot of evidence that this is indeed the case and our result gives further support in its favour.
We will show that the ten-dimensional metric corresponds to the gravitational field of a large number of D3-branes in the field theory limit with a special continuous distribution of branes in the transverse six-dimensional space. Namely, the metric has the form ds\^2 = H\_0\^[-1/2]{} \_ dx\^dx\^+ H\_0\^[1/2]{} (dy\_1\^2+dy\_2\^2+…+ dy\_6\^2) , \[d33\] where $H_0$ is a harmonic function (yet to be determined) in the six-dimensional space transverse to the brane parametrized by the $y_i$ coordinates. However, instead of being asymptotically flat, the metric [(\[d33\])]{} will become asymptotically $AdS_5\times S^5$ for large radial distances (or equivalently in the UV region using the terminology of the AdS/CFT correspondence). The ten-dimensional dilaton field is constant, i.e. $e^\Phi = g_s =
{\rm const.}$ and, as usual, the self-dual five-form is turned on. Under these conditions, the ten-dimensional solution breaks half of the maximum number of supersymmetries (see, for instance, [@KaKou]).
We proceed further by first performing the coordinate change in [(\[d33\])]{} y\_i = R e\^[A-\_i]{} x\_i = (F-b\_i)\^[1/2]{} x\_i ,i=1,2,…, 6 , \[ejh1\] where the $\hat x_i$’s define a unit five-sphere, i.e. they obey $\sum_{i=1}^6 \hat x_i^2 =1$. Various convenient bases for these unit vectors can be chosen, depending on the particular applications that will be presented later. It can be shown that the flat six-dimensional metric in the transverse part of the brane metric [(\[d33\])]{} can be written as \_[i=1]{}\^6 dy\_i\^2 = R\^2 e\^[2 A]{} d\^2 + [e\^[-2 A]{}4 R\^2]{} \_[i=1]{}\^6 e\^[2 \_i]{} x\_i\^2 dF\^2 , \[d34\] where the line element $d\hat \s^2$ defines the metric of a deformed five-sphere given by d\^2 = \_[i=1]{}\^6 e\^[-2 \_i]{} (dx\_i)\^2 ,g = [Vol]{}([S\^5]{}) \_[i=1]{}\^6 e\^[2 \_i]{} x\_i\^2 . \[kjf1\] For later use, we have also written the expression for the determinant of the deformed five-sphere in [(\[kjf1\])]{}. In computing this determinant we have used the fact that the sum of the $\b_i$’s is zero. Note that a similar expression also holds for a general $n$-sphere.
The harmonic function $H_0$ is determined by comparing the massless scalar equation $\Box_{10}\Phi=0$ for the ten-dimensional metric [(\[d33\])]{} with the equation arising using the five-dimensional metric [(\[aans\])]{}, i.e. $\Box_{5}\Phi=0$. In both cases one makes the ansatz that the solution does not depend on the sphere coordinates, i.e. $\Phi= e^{i k \cdot x}
\phi(z)$. Since the solutions for the scalar $\Phi$ should be the same in any consistent trancation of theory, the resulting second order ordinary differential equations should be identical. A comparison of terms proportional to $\phi(z)$ determines the function $H_0$ as follows, H\_0\^[-1]{} = [1 R\^4]{} f\^[1/2]{} \_[i=1]{}\^6 [x\_i\^2F-b\_i]{} = [1 R\^4]{} f\^[1/2]{} \_[i=1]{}\^6 [y\_i\^2(F-b\_i)\^2]{} , \[dhj1\] where in the second equality the harmonic function $H_0$ has been expressed in terms of the transverse coordinates $y_i$. Comparison of the terms proportional to the first and second derivative of $\phi(z)$ yields, using the expression for $\det \hat g$ in [(\[kjf1\])]{}, an identity and provides no further information. The coordinate $F$ is determined in terms of the transverse coordinates $y_i$ as a solution of the algebraic equation \_[i=1]{}\^6 [y\_i\^2F-b\_i]{} =1 . \[jk4\] This is a sixth order algebraic equation for general choices of the constants $b_i$, and its solution cannot be written in closed form. However, this becomes possible when some of the $b_i$’s coincide in such a way that the degree of [(\[jk4\])]{} is reduced to four or less. Even then, the resulting expressions are not very illuminating and we will refrain from presenting them except in the simplest case in section 5 below.
The corresponding D3-brane solution that is asymptotically flat is obtained by replacing $H_0$ in [(\[d33\])]{} by $H=1+H_0$. Then, in this context, the length parameter $R$ has a microscopic interpretation using the string scale $\a'$, the string coupling $g_s$, and the (large) number of D3-branes $N$, as $R^4 = 4 \pi g_s N \a'^2$.
In the rest of this section we demonstrate for completeness the proof that the function $H_0$, as defined in [(\[dhj1\])]{}, is indeed harmonic in the six-dimensional transverse space spanned by $y_i$, $i=1,2,\dots , 6$. This is not a trivial check since $F$ that appears in [(\[dhj1\])]{} is itself a function of the transverse space coordinates $y_i$ due to the condition [(\[jk4\])]{}. For notational convenience we define the functions A\_m= \_[i=1]{}\^6 [y\_i\^2(F-b\_i)\^m]{} ,B\_m=\_[i=1]{}\^6 [1 (F-b\_i)\^m]{} . \[jd23\] Then, using [(\[jk4\])]{} we determine the derivative of the function $F(y)$ \_i F = 2 [y\_i A\_2 (F-b\_i)]{} . \[lll3\] Also, the first derivative of $H_0$ with respect to $y_i$ turns out to be \_i H\_0 = - f\^[-1/2]{} [B\_1 y\_iA\_2\^2 (F-b\_i)]{} - 2 f\^[-1/2]{} [y\_i A\_2\^2 (F-b\_i)\^2]{} + 4 f\^[-1/2]{} [A\_3 y\_iA\_2\^3 (F-b\_i)]{} . \[hjg4\] Taking the derivative with respect to $y_i$ once more, summing over the free indices and after some algebraic manipulations, we obtain the desired result \_[i=1]{}\^6 \_i\^2 H\_0 & = & 2 f\^[-1/2]{} ([B\_2A\_2\^2]{} - [B\_1 A\_3A\_2\^3]{})\
&& - 2 f\^[-1/2]{} ([B\_2A\_2\^2]{} - [B\_1 A\_3A\_2\^3]{}) + 16 f\^[-1/2]{} ([A\_4A\_2\^3]{} - [A\_3\^2A\_2\^4]{})\
&& - 16 f\^[-1/2]{} ([A\_4A\_2\^3]{} - [A\_3\^2A\_2\^4]{}) = 0 , \[jfh56\]\
where the terms appearing in the three different lines above arise from the three distinct terms of [(\[hjg4\])]{} respectively.
Riemann surfaces in gauged supergravity
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In this section we will present the basic mathematical aspects of our general ansatz for the supersymmetric conditions of five-dimensional gauged supergravity and find the means to obtain explicit solutions in several cases by appealing to methods of algebraic geometry. In fact, we will classify all possible solutions according to symmetry groups (subgroups of $SO(6)$) and use the uniformization of algebraic curves that result in this approach for deriving the corresponding expressions. To simplify matters the parameter $R$ will be set equal to 1, but it can be easily reinstated by appropriate scaling in $z$.
Schwarz–Christoffel transform
-----------------------------
A useful way to think about the differential equation for the unknown function $F(z)$ is in the context of complex analysis. Suppose that $z$ and $F$ are extended in the complex domain and let us consider a closed polygon in the $z$-plane, including its interior, and map it via a Schwarz–Christoffel transformation onto the upper half $F$-plane. This provides a one-to-one conformal transformation and it is assumed that $F(z)$ is analytic in the polygon and is continuous in the closed region consisting of the polygon together with its interior. Considering the behaviour of $dz$ and $dF$ as the polygon is transversed in the counter-clockwise direction, we know that the transformation is described as = A (F-b\_1)\^[-\_1 /]{} (F-b\_2)\^[-\_2 /]{} (F-b\_n)\^[-\_n /]{} , where $A$ is some constant that changes by rescaling $F$. The vertices of the polygon are mapped to the points $b_1, b_2, \cdots ,b_n$ on the real axis of the upper complex $F$-plane and the exponents ${\varphi}_i$ that appear in the transformation are the exterior (deflection) angles of the polygon at the corresponding vertices. When the polygon is closed their sum is ${\varphi}_1 + {\varphi}_2 + \dots + {\varphi}_n = 2\pi$. Of course, without loss of generality, we may take one point (say $b_n$) to infinity. Letting $A = B/(-b_n)^{-{\varphi}_n /\pi}$ we see that as $b_n \rightarrow \infty$ the Schwarz–Christoffel transformation becomes = B (F-b\_1)\^[-\_1 /]{}(F-b\_2)\^[-\_2 /]{} (F-b\_[n-1]{})\^[-\_[n-1]{} /]{} , where $B$ is another constant factor. To make contact with our problem we choose $n=7$ and let the angles ${\varphi}_1 = {\varphi}_2 = \cdots =
{\varphi}_6 = \pi/4$. Then, we arrive at the differential equation ([dz dF]{})\^4 = B\^4(F-b\_1)\^[-1]{}(F-b\_2)\^[-1]{} (F-b\_6)\^[-1]{} , which is the same as the one implied by our ansatz for the general solution of gauged supergravity (with $B=1/2$).
The solutions of this equation are difficult to obtain in practice for generic values of the moduli $b_i$. We will investigate this problem in connection with the theory of algebraic curves in $C^2$ and we will see that in many cases explicit solutions can be given using the theory of elliptic functions. Before proceeding further we note that in our formulation we are looking for the map from the interior of the polygon onto the upper half-plane, $F(z)$, and not for the inverse transformation.
Symmetries and algebraic curves
-------------------------------
If we extend the variable $z$ to the complex domain, as before, and set x= 4F(z) , y=4F\^(z) , ł\_i = 4b\_i , the Schwarz–Christoffel differential equation will become an algebraic curve in $C^2$, y\^4 = (x-ł\_1)(x-ł\_2) …(x-ł\_6) . This is a convenient formulation for finding solutions of the supersymmetry equations, but at the end we have to restrict to real values of $z$ and demand that the resulting supergravity fields ${\beta}_i$ are also real. For generic values of the parameters $b_i$, so that they are all unequal and hence there is no symmetry in the solution of five-dimensional gauged supergravity, the genus of the curve can be easily determined (like in any other case) via the Riemann–Hurwitz relation. Recall that for any curve of the form y\^m = (x-ł\_1)\^[\_1]{}(x-ł\_2)\^[\_2]{} …(x-ł\_n)\^[\_n]{} , which is reduced, i.e. the integers $m$ and $\a_i$ have no common factors, and all $\l_i$’s are unequal, the genus $g$ can be found by first writing the ratios = [d\_1 c\_1]{} , , [\_n m]{} = [d\_n c\_n]{} ; [\_1 + + \_n m]{} = [d\_0 c\_0]{} in terms of relatively prime numbers and then using the relation g = 1 - m +[m 2]{} \_[i=0]{}\^n (1-[1 c\_i]{}) . According to this the genus of our surface turns out to be $g=7$ when all $b_i$ are unequal, and so it is difficult to determine explicitly the solution in the general case. However, by imposing some isometries in the solution of gauged supergravity the genus becomes smaller and hence the problem becomes more tractable. The presence of isometries manifests by allowing for multiple branch points in the general form of the algebraic curve, which in turn degenerates along certain cycles that effectively reduce its genus.
Note for completeness that if we had not taken $b_7$ to infinity in our discussion of the Schwarz–Christoffel transformation, we would have had an additional factor $(x-\l_7)^2$ in the equation of the algebraic curve because ${\varphi}_7 = \pi/2$ instead of $\pi/4$ that was chosen for the remaining ${\varphi}_i$’s. It can be easily verified that this does not affect the genus of the curve, as expected on general grounds.
Next, we enumerate all possible cases with a certain degree of symmetry that correspond to various subgroups of $SO(6)$; this amounts to various deformations of the round five-sphere, $S^5$, which is used for the compactification of the theory from 10 to 5 dimensions. Consequently, this will in principle determine the solution for the scalar fields in the remaining 5 dimensions as we will see later in detail. If all the branch points are different, the $SO(6)$ isometry of $S^5$ will be completely broken, whereas if all of them coalesce to the same point the maximal isometry $SO(6)$ will be manifestly present. The classification is presented below in an order of increasing symmetry or else in decreasing values of $g$.
\(1) : It corresponds to setting two of the $\l_i$ equal to each other and the remaining are all unequal. The curve becomes y\^4 = (x-ł\_1)(x-ł\_2)(x-ł\_3)(x-ł\_4)(x-ł\_5)\^2 and its genus turns out to be $g=5$.
\(2) : It corresponds to setting three of the $\l_i$ equal and all other remain unequal. The curve becomes y\^4 = (x-ł\_1)(x-ł\_2)(x-ł\_3)(x-ł\_4)\^3 and the genus turns out to be $g=4$.
\(3) : In this case two pairs of $\l_i$ are mutually equal and the remaining two parameters are unequal. The curve becomes y\^4 = (x-ł\_1)(x-ł\_2)(x-ł\_3)\^2(x-ł\_4)\^2 and its genus is $g=3$.
\(4) : In this case three $\l_i$ are equal and another two are also equal to each other. The curve becomes y\^4 = (x-ł\_1)(x-ł\_2)\^2(x-ł\_3)\^3 and its genus is $g=2$. Therefore we know that it can be cast into a manifest hyper-elliptic form by introducing appropriate bi-rational transformations of the complex variables.
\(5) : It corresponds to setting four $\l_i$ equal to each other and the other two remain unequal. The curve becomes y\^4 = (x-ł\_1)(x-ł\_2)(x-ł\_3)\^4 and its genus is $g=1$. It can also be cast into a manifest (hyper)-elliptic form as we will see shortly.
\(6) : It corresponds to three different pairs of mutually equal $\l_i$, but in this case the curve is not irreducible, since $y^4 = (x-\l_1)^2(x-\l_2)^2(x-\l_3)^2$. The reduced form is y\^2 = (x-ł\_1)(x-ł\_2)(x-ł\_3) and clearly has genus $g=1$ as it is written directly in (hyper)-elliptic form.
\(7) : In this case we have two groups of triplets with equal values of $\l_i$. The curve becomes y\^4 = (x-ł\_1)\^3(x-ł\_2)\^3 and its genus is $g=1$. It can also be cast into a manifest (hyper)-elliptic form.
\(8) : In this case five $\l_i$ are equal to each other and the last remains different. The curve becomes y\^4 = (x-ł\_1)(x-ł\_2)\^5 and its genus is also $g=1$ as before.
\(9) : It corresponds to separating the $\l_i$ into four equal and another two equal parameters. The curve becomes $y^4 = (x-\l_1)^2(x-\l_2)^4$, but it is not irreducible. The reduced form is y\^2 = (x-ł\_1)(x-ł\_2)\^2 and has genus $g=0$, as it can also be obtained by degenerating a genus 1 surface along its cycles. Therefore, we expect the solution to be given in terms of elementary functions.
\(10) : This is the case of maximal symmetry in which all $\l_i$ are set equal to each other. The curve becomes $y^4 = (x-\l_1)^6$, whose reduced form is y\^2 = (x-ł\_1)\^3 and has genus $g=0$ as before.
Of course, when certain cycles contract by letting various branch points to coalesce, the higher genus surfaces reduce to lower genus and a bigger symmetry group emerges in the solutions corresponding to gauged supergravity. For genus $g \leq 2$ one can always transform to a manifest hyper-elliptic form so that two sheets (instead of four) are needed for picturing the Riemann surface by gluing sheets together along their branch cuts. We will investigate in detail the cases corresponding to genus 0 and 1 surfaces since the solutions can be given explicitly in terms of elementary and elliptic functions respectively. Some results about the genus 2 case will also be presented. The other cases are more difficult to handle in detail even though the general form of the solution is known implicitly for all $g$ according to our ansatz.
Genus 0 surfaces
----------------
There are two genus 0 surfaces according to the previous discussion, namely the curve $y^2 = (x-\l_1)^3$ for the isometry group $SO(6)$ and the curve $y^2 = (x-\l_1)(x-\l_2)^2$ for the isometry group $SO(4) \times SO(2)$. According to algebraic geometry every irreducible curve $f(x,y) = 0$ with genus 0 is representable as a unicursal curve (straight line) w = v by means of a bi-rational transformation $x(v,w)$, $y(v,w)$ and conversely $v(x,y)$, $w(x,y)$. In our two examples the underlying transformations are summarized as follows:
\(a) : We have x = vw + ł\_1 , y = vw\^2 and conversely v = [(x-ł\_1)\^2 y]{} , w = [y x-ł\_1]{} .
\(b) : We have x = vw + ł\_1 , y = w(vw + ł\_1 - ł\_2) and conversely v = [(x-ł\_1)(x-ł\_2) y]{} , w = [y x-ł\_2]{} . Of course, the first curve arises as special case of the second for $\l_2 \rightarrow \l_1$.
For general $\l_1$ and $\l_2$ we may use $u$ as a (trivial) uniformizing complex parameter for the unicursal curve, i.e. $v=u=w$. Then, the expressions for $x$ and $y$ yield 4F(z) = u\^2 + 4b\_1 , 4[dF(z) dz]{} = u(u\^2 + 4(b\_1 - b\_2) ) , where we have taken into account the rescaling $x=4F(z)$, $y=4F^{\prime}(z)$, $\l_i = 4b_i$ that was introduced earlier. So we can determine $u$ as a function of $z$ by simple integration since = [1 2]{}(u\^2 + 4(b\_1-b\_2)) . In fact there are three different cases for generic values of $b_1$ and $b_2$. Choosing appropriately the integration constant, so that the resulting conformal factor $e^{2A(z)}$ will behave like $1/z^2$ as $z \rightarrow 0$, we have: ([i]{}) u & = & - 2( z), [for]{} b\_1>b\_2 ,\
([ii]{}) u & = & -2( z), [for]{} b\_2>b\_1 ,\
([iii]{}) u & = & -[2 z]{} , [for]{} b\_1=b\_2 . The first two cases correspond to the $SO(4) \times SO(2)$ isometry and they are obtained by analytic continuation from one other, depending on the size of the $b_i$’s, whereas the last case has $SO(6)$ isometry. Here, we do not assume any given ordering among the $b_i$’s. As for the functions $F(z)$ we have respectively F(z) = (b\_1-b\_2) [cot]{}\^2 (z) + b\_1 , (b\_2-b\_1) [coth]{}\^2 (z) + b\_1 , [1 z\^2]{} + b\_1 .
Then, the expression for the conformal factor of the metric is ([i]{}) e\^[2A(z)]{} &=& (b\_1 -b\_2) [[cos]{}\^[2/3]{}( z) \^2( z)]{} , [for]{} b\_1>b\_2 , \[a42\]\
([ii]{}) e\^[2A(z)]{}&=& (b\_2 -b\_1) [[cosh]{}\^[2/3]{}( z) \^2 ( z)]{} , [for]{} b\_2>b\_1 , \[a42a\]\
([iii]{}) e\^[2A(z)]{}&= &[1 z\^2]{} , [for]{} b\_1 =b\_2 , which indeed behaves as $1/z^2$ in all three cases for $z \rightarrow 0$.
The solution for the scalar fields ${\beta}_i(z)$ of five-dimensional gauged supergravity follows by simple substitution into our ansatz. We have explicitly in each case &([i]{}) & e\^[2\_1(z)]{} = e\^[2\_2(z)]{} = [1 \^[4/3]{}(z)]{} ,\
&& e\^[2\_3(z)]{} = e\^[2\_4(z)]{} = e\^[2\_5(z)]{} = e\^[2\_6(z)]{} = [cos]{}\^[2/3]{}(z) , \[b42\]\
&([ii]{})& e\^[2\_1(z)]{} = e\^[2\_2(z)]{} = [1 \^[4/3]{}(z)]{} ,\
&& e\^[2\_3(z)]{} = e\^[2\_4(z)]{} = e\^[2\_5(z)]{} = e\^[2\_6(z)]{} = [cosh]{}\^[2/3]{}(z) , \[b42b\]\
&([iii]{})& e\^[2\_i(z)]{} = 1 , i=1,…, 6 .
Equally well we could have transformed the genus 0 curves into the quadratic form $Y^2 = 1 - X^2$ using the following transformation for the curve $y^2 = (x-\l_1)(x-\l_2)^2$ x = [1 + X 1-X]{} + ł\_1 , y = [Y 1-X]{} ([1+X 1-X]{} + ł\_1 - ł\_2 ) and conversely X = [x - ł\_1 - 1 x - ł\_1 + 1]{} , Y = [2y (x-ł\_1 +1)(x-ł\_2)]{} . In this case we can use another uniformizing complex parameter $u$, so that $X={\rm sin}u$ and $Y={\rm cos}u$, and proceed as above. Either way, the uniformization problem is solved in terms of elementary functions, which in turn determine the function $F(z)$ every time and hence the particular supersymmetric solutions of five-dimensional gauged supergravity.
Genus 1 surfaces
----------------
Recall first that given a genus 1 algebraic curve in its Weierstrass form w\^2 = 4v\^3 - g\_2v - g\_3 the uniformization problem is solved by introducing the Weierstrass function ${\cal P}(u)$ and its derivative ${\cal P}^{\prime}(u)$ with respect to a complex parameter $u$. Then, $v={\cal P}(u)$ and $w={\cal P}^{\prime}(u)$ in which case the Weierstrass function satisfies the time independent KdV equation ${\cal P}^{\prime \prime \prime}(u) -12 {\cal P}(u){\cal P}^{\prime}(u) =0$. The two periods of the elliptic curve are denoted by $2{\omega}_1$ and $2{\omega}_2$ and the Weierstrass function is double periodic with respect to them. Also the values of the Weierstrass function at the half-periods coincide with the roots of the algebraic equation $4v^3 -g_2v -g_3 = 0$, namely $e_1 = {\cal P}({\omega}_1)$, $e_2 = {\cal P}({\omega}_1 + {\omega}_2)$ and $e_3 = {\cal P}({\omega}_2)$. Conversely, given the differential equation ( [dG(z) dz]{})\^2 = 4G\^3(z) - g\_2 G(z) - g\_3 the general solution is described in terms of the Weierstrass function $G(z) = {\cal P}(\pm z + a)$, where $a$ is the constant of integration and $g_2$, $g_3$ are related as usual to the periods of the elliptic curve. This may be seen by taking a new dependent variable $u$ defined by the equation $G = {\cal P}(u)$, when the differential equation reduces to $(du/dz)^2 =1$; for this recall that the number of roots of the equation ${\cal P}(u) = c$ that lie in any cell depend only on ${\cal P}(u)$ and not on $c$, which can assume arbitrary values, like for any other elliptic function.
We have four different curves with genus 1 that follow from the classification that we described above. It is known from algebraic geometry that any genus 1 surface is hyper-elliptic (in particular elliptic since $g=1$), but we see that only the one that corresponds to the case of $SO(2) \times SO(2) \times SO(2)$ isometry is essentially written in such form with roots $e_1 = \l_1$, $e_2 = \l_2$ and $e_3 = \l_3$ (when $\l_1 + \l_2 + \l_3 = 0$). The other three curves can be transformed to $w^2 = 4v^3 -g_2v -g_3$ for appropriately chosen coefficients $g_2$ and $g_3$ provided that one performs the necessary bi-rational transformations of the complex variables $v(x, y)$, $w(x, y)$ and conversely $x(v,w)$, $y(v, w)$. Only then the solution can be easily deduced from the resulting genus 1 curve in its Weierstrass form using elliptic functions. This is precisely what we are about to describe in the sequel.
Note first that all three curves that correspond to the symmetry groups $SO(4)$, $SO(3) \times SO(3)$ and $SO(5)$ can be transformed into the same curve Y\^4 = (X - ł\_1) (X - ł\_2) according to the following bi-rational transformations && X = x , Y = [y x - ł\_3]{} [for]{} y\^4 = (x-ł\_1)(x-ł\_2)(x-ł\_3)\^4 ,\
&& X = x , Y = [(x- ł\_1)(x-ł\_2) y]{} [for]{} y\^4 = (x-ł\_1)\^3(x-ł\_2)\^3 ,\
&& X = x , Y = [y x-ł\_2]{} [for]{} y\^4 = (x-ł\_1)(x-ł\_2)\^5 , respectively. Then, defining X- ł\_1 = [\^2 ]{} , Y = we arrive at the curve ${\eta}^2 (1- {\zeta}^2) = (\l_1 - \l_2){\zeta}^3$ in all three cases. This simplifies further by defining new variables $V$, $W$ so that = [W V]{} , + 1 = [1 V]{} , in which case the curve becomes $W^2 (2V-1) = (\l_1 - \l_2) V(1-V)^3$. Finally, letting V = [2v ł\_2 - ł\_1]{} + [1 2]{} , W = [1 ł\_1 - ł\_2]{}[w v]{} (v + [1 4]{}(ł\_1 - ł\_2)) , we obtain the genus 1 curve in its standard Weierstrass form w\^2 = 4v\^3 - g\_2v -g\_3 [with]{} g\_2 = [14]{} (ł\_1 - ł\_2)\^2 , g\_3 = 0 for all three cases of interest. This is a non-degenerate Riemann surface with $g=1$, but it is more special than the $SO(2) \times SO(2) \times SO(2)$ surface since the latter depends on three parameters $\l_1$, $\l_2$ and $\l_3$ instead of the two $\l_1$ and $\l_2$ that appear in the Weierstrass form for higher (non-abelian) symmetry. Actually, in the present case we have ${\omega}_2/{\omega}_1 = i$, and so by introducing the modulus of elliptic integrals, $k$, and its complementary value $k^{\prime}$, one finds $k = k^{\prime} = 1/\sqrt{2}$.
We summarize the bi-rational transformations that are needed to transform each one of the genus 1 surfaces into their Weierstrass forms according to the symmetry groups of the solutions that they represent:
\(a) : The curve $y^2 = (x-\l_1)(x-\l_2)(x-\l_3)$ can be brought into the standard Weierstrass form $w^2 = 4v^3 - g_2 v - g_3$ with g\_2& = & [1 36]{}((ł\_1 + ł\_2 -2ł\_3)\^2 - (ł\_2 + ł\_3 -2 ł\_1)(ł\_1 + ł\_3 -2 ł\_2)) ,\
g\_3 & = & -[1 432]{}(ł\_1 + ł\_2 -2ł\_3) (ł\_2 + ł\_3 -2 ł\_1)(ł\_1 + ł\_3 -2 ł\_2) , using the simple transformation y=4w , x=4v +[1 3]{}(ł\_1 + ł\_2 + ł\_3) .
\(b) : The curve $y^4 = (x-\l_1)^3(x-\l_2)^3$ also transforms into the Weierstrass form $w^2 = 4v^3 - g_2v$ with $g_2 = (\l_1 - \l_2)^2 /4$ using x = ł\_1 - [1 v]{} (v+[1 4]{}(ł\_1 -ł\_2))\^2 , y = [w\^3 8v\^3]{} , and conversely v & = & [1 4]{}(ł\_1-ł\_2)[y\^2 - (x-ł\_1)(x-ł\_2)\^2 y\^2 +(x-ł\_1)(x-ł\_2)\^2]{} ,\
w & = & [1 2]{}(ł\_1 -ł\_2)[(x-ł\_1)(x-ł\_2) y]{} [y\^2 -(x-ł\_1)(x-ł\_2)\^2 y\^2 + (x-ł\_1)(x-ł\_2)\^2]{} .
\(c) : The curve $y^4 = (x-\l_1)(x-\l_2)(x-\l_3)^4$ transforms into the Weierstrass form $w^2 = 4v^3 -g_2v$ with $g_2 = (\l_1 -\l_2)^2 /4$ using x & = & ł\_1 - [1 v]{} (v +[1 4]{} (ł\_1 -ł\_2))\^2 ,\
y & = & [w 2v]{} (ł\_1 - ł\_3 - [1 v]{} (v + [1 4]{}(ł\_1 - ł\_2) )\^2 ) , and conversely v & = & [1 4]{} (ł\_2 - ł\_1) [y\^2 -(x-ł\_1)(x-ł\_3)\^2 y\^2 +(x-ł\_1)(x-ł\_3)\^2]{} ,\
w & = & [1 2]{} (ł\_2 -ł\_1) [y x-ł\_3]{} [y\^2 -(x-ł\_1) (x-ł\_3)\^2 y\^2 + (x-ł\_1)(x-ł\_3)\^2]{} .
\(d) : This case arises from $SO(4)$ when $\l_3 \rightarrow \l_2$, and so x = ł\_1 -[1 v]{}(v+[1 4]{}(ł\_1 -ł\_2))\^2 , y = -[w 2v\^2]{}(v-[1 4]{}(ł\_1 -ł\_2))\^2 .
Note that in the three last cases (b)–(d) one may choose $v = {\cal P}(u)$ and $w = {\cal P}^{\prime}(u)$, where $u$ is the uniformizing parameter of the same Riemann surface. Thus, the $x$’s ($=4F(z)$) are the same functions of $u$ in these three cases, given in terms of Weierstrass functions and their derivatives, but the $y$’s ($=4F^{\prime}(z)$) are all different as can be readily seen. This simply means that the variable $z$ is not equal to the uniformizing parameter $u$ of the genus 1 curve in its Weierstrass form, but rather a more complicated function $u(z)$ that has to be found in each case separately by integration (in analogy with what we did in the $g=0$ cases). This complication does not arise for the case (a), since there we can take $z = u$ (more generally $(du/dz)^2 = 1$, as we have already seen). For (a) the solution has already been very simply expressed in terms of the Weierstrass functions $x = 4{\cal P}(u) + (\l_1 + \l_2 + \l_3)/3$ and $y = 4{\cal P}^{\prime}(u)$, though of another Riemann surface with different coefficients $g_2$ and $g_3$.
Next, we take into account the field redefinitions $x = 4F(z)$, $y = 4F^{\prime}(z)$, $\l_i = 4b_i$ and solve for $z(u)$ and its inverse $u(z)$, when this is possible in closed form, thus determining $F(z)$ in each case of interest. Of course, the elliptic functions that appear, refer to the corresponding curves with $g_2$ and $g_3$ determined as above. Summarizing the results, including some technical details, we have:
$\bullet$ : We have already seen that the uniformizing parameter $u$ equals to $z$ and hence F(z) = [P]{}(z) + [1 3]{}(b\_1 + b\_2 + b\_3) \[f222\] . According to this we find e\^[2A(z)]{} = ([1 2]{}[P]{}\^(z))\^[2/3]{} , \[a222\] and so the conformal factor of the metric behaves as $1/z^2$ when $z$ approaches 0. The solution for the scalar fields of gauged supergravity follows by substitution into our general ansatz. We have in fact && e\^[2\_1(z)]{} = e\^[2\_2(z)]{} = [([P]{}\^(z)/2 )\^[2/3]{} (z) - e\_1]{} ,\
&& e\^[2\_3(z)]{} = e\^[2\_4(z)]{} = [([P]{}\^(z)/2 )\^[2/3]{} (z) - e\_2]{} , \[s222\]\
&& e\^[2\_5(z)]{} = e\^[2\_6(z)]{} = [([P]{}\^(z)/2 )\^[2/3]{} (z) - e\_3]{} , where e\_i = b\_i - [1 3]{}(b\_1 + b\_2 + b\_3) , i=1,2,3. \[e222\]
$\bullet$ : This is the next simple case to consider. The relation between the differentials $dz$ and $du$ can be found by first computing $4dF/du$ as a function of $u$; it turns out to be $-{{\cal P}^{\prime}}^3(u)/4{\cal P}^3(u)$. Then, using the expression for $y = 4dF(z)/dz$ we arrive at the simple relation = -[1 2]{} and so $u= -z/2$, up to an integration constant that is taken zero. Consequently, F(z) = b\_1 - [1 4[P]{}(z/2)]{}([P]{}(z/2) + b\_1 - b\_2)\^2, which in turn implies the following result for the conformal factor of the metric e\^[2A(z)]{} = ([[P]{}\^(z/2) 4 [P]{}(z/2)]{})\^2 = [1 4]{} ([P]{}(z/2) - [(b\_1-b\_2)\^2(z/2)]{}) . \[a33\] The derivative of the Weierstrass function is taken with respect to its argument $z/2$. The conformal factor clearly approaches $1/z^2$ as $z \rightarrow 0$, which justifies our choice of the integration constant above.
As for the solution corresponding to the scalar fields of gauged supergravity, we obtain by substitution into our general ansatz the result && e\^[2\_1(z)]{} =e\^[2\_2(z)]{} =e\^[2\_3(z)]{} = -[[P]{}(z/2) -b\_1+b\_2 (z/2) +b\_1-b\_2]{} ,\
&& e\^[2\_4(z)]{} =e\^[2\_5(z)]{} =e\^[2\_6(z)]{} = -[[P]{}(z/2) +b\_1-b\_2 (z/2) -b\_1+b\_2]{} \[b33\] , which completes the task. At this point we add a clarifying remark, which takes into account the discrete symmetry $x \rightarrow -x$ and $b_i \rightarrow -b_i$ of the underlying algebraic curves. The uniformization that gave rise to eq. (4.64) implies that as $z$ ranges from 0 to $2 \omega_1$, $F(z)$ ranges from $- \infty$ to $b_2$ (provided that $b_1 > b_2$ so that ${\cal P}(\omega_1) \equiv e_1 = b_1 - b_2$). If one applies the discrete symmetry mentioned above, eq. (4.64) will change accordingly so that $F(z)$ will range from $+ \infty$ to $b_1$ (taken as the maximum of the two moduli parameters). This particular symmetry implies in turn that the expressions for the scalar fields ${\rm exp}(2 \beta_i)$ get modified by simply changing the overall sign according to the defining relation (3.3). Hence, despite appearances, the fields $\beta_i(z)$ are real provided that $z$ is real with values in the range where $F(z)$ is bigger than the maximum of $b_1$ and $b_2$, as it is usually taken.
$\bullet$ : This case is computationally more difficult to handle. Since $4dF/du = -{{\cal P}^{\prime}}^3 / 4{\cal P}^3(u)$ again as a function of $u$, we find the following relation between the differentials $dz$ and $du$, = [1 2]{}[[P]{}(u) -b\_1 +b\_2 (u) +b\_1 -b\_2]{} . Then, integrating over $u$ we arrive at the formula z = (u) + [1 2]{} [[P]{}\^(u) (u) -b\_1 +b\_2]{} , up to an integration constant that should be determined by the asymptotic behaviour $e^{2A(z)} \rightarrow 1/z^2$ as $z$ approaches 0. Here $\zeta(u)$ is the Weierstrass zeta-function. Note that the above expression will somewhat simplify if one uses the identity (u +\_1) - (\_1) &= & (u) + [1 2]{} [[P]{}\^(u) (u) -b\_1 +b\_2]{} , [for]{} b\_1>b\_2 ,\
(u +\_2) - (\_2) &= & (u) + [1 2]{} [[P]{}\^(u) (u) -b\_1 +b\_2]{} , [for]{} b\_2>b\_1 , where $\omega_1$ and $\omega_2$ are the half-periods of the curve. In either case, it is not possible to invert the relation and explicitly find $u(z)$ in closed form.
We give the result for the conformal factor of the metric as a function of $u$, e\^[2A]{} = [1 4[P]{}(u)]{} ([P]{}(u) +b\_1-b\_2)\^[1/3]{} ([P]{}(u) -b\_1 +b\_2)\^[5/3]{} . Similar expressions are obtained for the scalar fields of gauged supergravity, e\^[2\_1]{} &=& - ([[P]{}(u) -b\_1+b\_2 (u) +b\_1-b\_2]{})\^[5/3]{} ,\
e\^[2\_2]{} &= &= e\^[2\_6]{} = - ( [[P]{}(u) + b\_1-b\_2(u)-b\_1+b\_2]{})\^[1/3]{} . Similar remarks apply here for the overall sign appearing in eq. (4.72), as for the scalar fields of the model $SO(3) \times SO(3)$, using the discrete symmetry $x \rightarrow -x$, $b_i \rightarrow -b_i$ of the underlying algebraic curve.
We mention for completeness that as $b_1 \rightarrow b_2$ the Riemann surface degenerates and one recovers the $SO(6)$ model that was already discussed. It might seem that this contradicts the relation between $u$ and $z$ at first sight, since the left hand side becomes zero irrespective of $z$. However, for elliptic functions in the degeneration limit $g_2 = g_3 = 0$ we have ${\cal P}(u) = 1/u^2$ and $\zeta (u) = 1/u$ for all $u$, and so the right hand side also becomes zero irrespective of $u$; hence there is no problem in taking this limit.
$\bullet$ : In this situation the calculation becomes even more involved. We find that = [1 2]{} [([P]{}(u) -b\_1 + b\_2)\^2 -4(b\_2-b\_3)[P]{}(u) ([P]{}(u) +b\_1-b\_2)([P]{}(u)-b\_1+b\_2)]{} , but as in the $SO(5)$ case it is still not possible to find explicitly $u(z)$ in closed form. Besides, the integrals are more difficult to perform when $b_2 \neq b_3$ and so the resulting expressions are not very illuminating in terms of algebraic geometry. We postpone the presentation of the corresponding configuration for the next section, where a more geometrical approach is employed for it.
Genus 2 surface
---------------
Here we have only one such curve corresponding to the isometry group $SO(3) \times SO(2)$, which is described by the algebraic equation $y^4 = (x-\l_1)(x-\l_2)^2(x-\l_3)^3$. According to algebraic geometry it can be brought into a manifest hyper-elliptic form by performing appropriate bi-rational transformations. To achieve this explicitly we consider the following sequence of transformations: First, let x= X , y = [(X-ł\_3)(X-ł\_2) Y]{} , that brings the curve into the form (X-ł\_1)Y\^4 = (X-ł\_2)\^2(X-ł\_3) . The second step consists in performing the transformation X-ł\_2 = [\^2 ]{} , Y = , that transforms it further into the form \^2(1-\^2) = (ł\_1 -ł\_2)+ (ł\_2 -ł\_3)\^3 . Next, we introduce $V$ and $W$ so that = [W V]{} , + 1 = [1 V]{} , and the algebraic curve simplifies to W\^2(2V-1) = (ł\_1-ł\_2)V\^3(1-V) + (ł\_2-ł\_3)V(1-V)\^3 . Finally, as last step let us consider V = v , W=[w 2v-1]{} , which turns the curve into the desired hyper-elliptic form of genus two w\^2 = v(v-1)(2v-1)((ł\_3-ł\_1)v\^2 -2(ł\_3-ł\_2)v + ł\_3 -ł\_2 ) , with five distinct roots when all $\l_i$ are different from each other.
Summarizing the sequence of the above operations, which are similar to the genus 1 examples, we have for the transformation $x(v, w)$, $y(v,w)$ the final result x = ł\_2 - [(ł\_3-ł\_1)v\^2 -2(ł\_3-ł\_2)v+ł\_3-ł\_2 2v-1]{} , y = (ł\_1-ł\_3)[vw (2v-1)\^2]{} , whereas for its inverse $v(x,y)$, $w(x,y)$ we have v = [(x-ł\_2)(x-ł\_3)\^2 (x-ł\_2)(x-ł\_3)\^2 + y]{} , w= [y x-ł\_3]{}[(x-ł\_2)(x-ł\_3)\^2 -y\^2 (x-ł\_2)(x-ł\_3)\^2 + y\^2]{} , and so it is bi-rational, as required. These formulae are useful for addressing the uniformization problem of the original form of the curve in terms of theta functions. However, the resulting solution for gauged supergravity is rather complicated in this algebro-geometric context and we postpone its presentation for the next section using a different approach.
Before concluding this section note that the $SO(3) \times SO(3)$ model, which arises as $\l_1 \rightarrow \l_2$, corresponds in this context to the curve $w^2 =(\l_3-\l_2)v(v-1)^3(2v-1)$, which according to the Riemann–Hurwitz relation has genus 1 as required; letting $w \rightarrow w(v-1)$, we see that the cubic form $w^2 = (\l_3 -\l_2)v(v-1)(2v-1)$ results in this case. Also, the $SO(5)$ model arises as $\l_2 \rightarrow \l_3$ and it corresponds in this context to the curve $w^2 = (\l_2-\l_1)v^3(v-1)(2v-1)$, which again has genus 1, as required; it transforms, in turn, into the cubic form $w^2 = (\l_2-\l_1)v(v-1)(2v-1)$ under the transformation $w \rightarrow wv$. Last, the $SO(4) \times SO(2)$ model is described by $w^2 = (\l_2 -\l_1) v(v-1)(2v-1)^2$ as $\l_1 \rightarrow \l_3$. This has genus 0 and it can be brought into a manifest quadratic form $w^2 = (\l_2-\l_1)v(v-1)$ using the transformation $w \rightarrow w(2v-1)$. However, the bi-rational transformation for $y$ is appearing singular now, and the same is also true for the fully symmetric $SO(6)$ model; notice that for both of them the original form of the curve is not irreducible. Hence, we assume that the $SO(3) \times SO(2)$ model has $\l_1$, $\l_2$, $\l_3$ all different from each other (in particular $\l_1 \neq \l_3$). In any event, all previous models with genus 0 and 1 arise as special cases of $SO(3) \times SO(2)$ apart from the $SO(2) \times SO(2) \times SO(2)$ and the $SO(4)$ models that have already been described.
Examples
========
The five- as well as the ten-dimensional forms of our solutions preserve four-dimensional Poincaré invariance $ISO(1,3)$ along the three-brane, but for general values of the constants $b_i$, they have no other continuous isometries. In order to obtain some continuous group of isometries we have to choose some of the $b_i$’s equal. By means of [(\[hja2\])]{} the corresponding scalars $\b_i$ are also equal to one another. In this section we work out explicitly the expression for the metric and the scalar fields for some cases of particular interest using the ten-dimensional geometric frame where $F$ is more conveniently regarded as a coordinate instead of using $z$. We will present the models with isometry groups $SO(3)\times
SO(2)$ and its limiting cases $SO(3)\times SO(3)$ and $SO(5)$, as well as the cases with isometry groups $SO(2)\times SO(2)\times SO(2)$, $SO(4)$ and their limiting model $SO(4)\times SO(2)$. They describe all solutions with genus $\leq 2$ from the point of view of the previous section. The examples are ordered by starting from the more general configurations and then specializing to models with higher symmetry.
The variable $z$ is more natural to use for addressing the uniformization problem of the algebraic curves underlying in our solutions. In here, we adapt our presentation to the ten-dimensional type-IIB supergravity description for two reasons: first as an alternative method for constructing explicit forms of our supersymmetric configurations, and second for providing a higher dimensional view point for the compactification to five space-time dimensions, and naturally for questions regarding the AdS/CFT correspondence. To avoid confusion note that certain choices of the moduli $b_i$ made in the sequel differ slightly from those made in the previous section, but this should cause no problem.
Solutions with $SO(3)\times SO(2)$ symmetry
-------------------------------------------
In this case it is convenient to use a basis for the unit vectors $\hat x_i$ that define the five-sphere in such a way that it is in one to one correspondence with the decomposition of the vector representation $\bf 6$ of $SO(6)$ with respect to the subgroup $SO(3)\times SO(2)$, as ${\bf 6}\to ({\bf 3},{\bf 1})
\oplus ({\bf 1},{\bf 2}) \oplus ({\bf 1},{\bf 1})$. Hence, we choose x\_1 & =& , = ,\
&= & ,x\_6 = . \[jwoi1\] It is also convenient to choose the constants $b_i$ as follows b\_1=b\_2=b\_3=0 ,b\_4=b\_5=-l\_1\^2 ,b\_6=-l\_2\^2 , \[fdj1\] where $l_1$ and $l_2$ are real constants, thus ordering now the moduli $b_i$ in an increasing order according to [(\[hord1\])]{}. We moreover adopt the change of variable $F=r^2$ with $r\ge 0$, which is legitimate as $b_{\rm max}=0$. Then, the corresponding ten-dimensional metric takes the form ds\^2& =& H\^[-1/2]{} \_ dx\^dx\^+ H\^[1/2]{}[\_1 \_2]{} dr\^2\
&& + r\^2 H\^[1/2]{} , where the various functions appearing in it are \_1 & =& 1+ [l\_1\^2r\^2]{} ,\_2 = 1+ [l\_2\^2r\^2]{} ,\
& =& \_1 \_2 \^2+ \^2(\_1 \^2+ \_2 \^2) , \[dh21\]\
H & = & 1 + [R\^4 \_2\^[1/2]{} r\^4 ]{} , and $d \Om_2^2$ is the two-sphere metric d\_2\^2 = d\^2 + \^2d\_1\^2 . \[d2j1\]
In terms of five-dimensional gauged supergravity, the five-dimensional metric [(\[fh3\])]{} is described by the form ds\^2 = [\_1\^[1/3]{} \_2\^[1/6]{} r\^2R\^2]{} \_ dx\^dx\^ + [R\^2r\^2 \_1\^[2/3]{} \_2\^[1/3]{}]{} dr\^2 \[kao2\] and the expressions for the scalars [(\[hja2\])]{} become &&e\^[2\_1]{} = e\^[2\_2]{}= e\^[2\_3]{}=\_1\^[1/3]{} \_2\^[1/6]{} ,\
&& e\^[2\_4]{} = e\^[2\_5]{}= \_1\^[-2/3]{} \_2\^[1/6]{} , \[esc1\]\
&& e\^[2\_6]{}= \_1\^[1/3]{} \_2\^[-5/6]{} .
The metric [(\[mee1\])]{} has a singularity at $r=0$ where the harmonic function $H$ diverges. However, this is not a point-like singularity as it occurs for all possible values of the angular variables $\th$, $\om$ and $\varphi_2$. Hence, [(\[mee1\])]{} may be interpreted as representing the distribution of a large number of D3-branes inside a solid three-dimensional ellipsoid defined by the equation + [y\_6\^2l\_2\^2]{} = 1 , \[ell1\] and the three-dimensional hyper-plane $y_1=y_2=y_3=0$. We note that, by analytic continuation on the $l_i$’s we may obtain brane distributions other than [(\[ell1\])]{}, but we will not elaborate more on this point.
Solutions with $SO(3)\times SO(3)$ symmetry
-------------------------------------------
In this case we may obtain the metric by just setting $l_1=l_2\equiv l$ in [(\[mee1\])]{}, since then the symmetry is enhanced, from $SO(3)\times SO(2)$ to $SO(3)\times SO(3)$. The metric becomes ds\^2 & =& H\^[-1/2]{} \_ dx\^dx\^+ H\^[1/2]{} [r\^2 + ł\^2\^2r\^2+ l\^2]{} dr\^2\
&& + H\^[1/2]{} , \[fe2\] where the harmonic function $H$ that follows from the corresponding expression in [(\[dh21\])]{} is H = 1 + [R\^4r (r\^2+l\^2\^2)(r\^2+l\^2)\^[1/2]{}]{} \[dfj3\] and the two different line elements for the two-dimensional sphere appearing in [(\[fe2\])]{} are d\_2\^2 = d\^2 + \^2d\_1\^2 ,d\_2\^2 = d\^2 + \^2d\_2\^2 . \[d2d1\] The field theory limit form of the metric [(\[fe2\])]{} (with the 1 omitted in [(\[dfj3\])]{}) has appeared before in [@FGPW2].
In the description in terms of gauged supergravity, the five-dimensional metric [(\[fh3\])]{} takes the form ds\^2 = [r (r\^2+l\^2)\^[1/2]{}R\^2]{} \_ dx\^dx\^ + [R\^2r\^2 + l\^2 ]{} dr\^2 \[kao3\] and the expressions for the scalars [(\[esc1\])]{} simplify to && e\^[2\_1]{}= e\^[2\_2]{}= e\^[2\_3]{}= (1+[l\^2r\^2]{})\^[1/2]{} ,\
&& e\^[2\_4]{}= e\^[2\_5]{}= e\^[2\_6]{}= (1+[l\^2r\^2]{})\^[-1/2]{} . \[esc51\] Note that the five-dimensional metric, as well as the corresponding scalar fields, take the equivalent form [(\[a33\])]{} and [(\[b33\])]{}, respectively, when written in terms of the variable $z$.
Specializing [(\[ell1\])]{} to the case at hand with $l_1=l_2=l$, we deduce that the metric [(\[fe2\])]{} represents the distribution of a large number of D3-branes inside the solid three-dimensional ball y\_4\^2+y\_5\^2+ y\_6\^2= l\^2 , \[spph1\] in the three-dimensional hyper-plane defined by $y_1=y_2=y_3=0$.
Solutions with $SO(5)$ symmetry
-------------------------------
In this case we may obtain the metric by just setting $l_1=0$ (and also redefining $l_2\equiv l$) in [(\[mee1\])]{}, since then the symmetry is enhanced from $SO(3)\times SO(2)$ to $SO(5)$. However, in order to present a metric with manifest $SO(5)$ symmetry, the basis [(\[jwoi1\])]{} is not appropriate. A convenient basis for the unit vectors $\hat x_i$ should be such that it is in one to one correspondence with the decomposition of the vector representation $\bf 6$ of $SO(6)$, with respect to $SO(5)$, as ${\bf 6}\to {\bf 5} \oplus {\bf 1}$. Hence we choose & = & ,\
& = & ,\
x\_5 & =& ,\
x\_6 & =& . \[jdj3\] The metric becomes ds\^2 & =& H\^[-1/2]{} \_ dx\^dx\^+ H\^[1/2]{} [r\^2 + l\^2\^2r\^2+ l\^2]{} dr\^2\
&& + H\^[1/2]{} , \[fe24\] where the harmonic function is H = 1 + [R\^4 (r\^2+l\^2)\^[1/2]{}r\^3 (r\^2+l\^2\^2)]{} , \[df2\] and the line element for the four-sphere is defined as d\_4\^2= d\^2 + \^2d\_1\^2 + \^2(d\^2 +\^2d\_2\^2) . \[jkg2\] The field theory limit form of the metric [(\[fe24\])]{} (with the 1 omitted in [(\[df2\])]{}) has also appeared before in [@FGPW2].
The five-dimensional metric [(\[fh3\])]{} takes the form ds\^2 = [r\^[5/3]{} (r\^2+l\^2)\^[1/6]{}R\^2]{} \_dx\^dx\^+ [R\^2r\^[4/3]{} (r\^2+l\^2)\^[1/3]{}]{} dr\^2 , \[jfk4\] whereas the expressions for the scalars [(\[esc1\])]{} become && e\^[2\_1]{}= e\^[2\_2]{}= e\^[2\_3]{}=e\^[2\_4]{}= e\^[2\_5]{}= (1+[l\^2r\^2]{})\^[1/6]{} ,\
&& e\^[2\_6]{}= (1+[l\^2r\^2]{})\^[-5/6]{} . \[esc17\]
The singularity of the metric [(\[fe24\])]{} for $r=0$ may be interpreted as due to the presence of D3-branes distributed along the $y_6$ axis. This can be also obtained from [(\[ell1\])]{} in the limit $l_1\to 0$ (and $l_2\equiv l$). In this limit, $y_4$ and $y_5$ are forced to be zero and therefore imposing [(\[ell1\])]{} leads to $y_6=l$. Hence, the distribution of D3-branes is taken over a segment of length $l$.
Solutions with $SO(2)\times SO(2)\times SO(2)$ symmetry
-------------------------------------------------------
In this case it is convenient to use a basis for the unit vectors $\hat x_i$ that define the five-sphere in such a way that it corresponds to the decomposition of the vector representation $\bf 6$ of $SO(6)$ with respect to the full Cartan subgroup $SO(2)\times SO(2)\times SO(2)$, as ${\bf 6}\to ({\bf 2},{\bf 1},{\bf 1})
\oplus ({\bf 1},{\bf 2},{\bf 1}) \oplus ({\bf 1},{\bf 1},{\bf 2})$. Hence, we choose &= & ,\
&= & , \[jwoi\]\
& = & . We also make the choice b\_1=b\_2 a\_1\^2 ,b\_3=b\_4 a\_2\^2 ,b\_5=b\_6 a\_3\^2 , \[fh1\] where $a_i$, $i=1,2,3$ are some real constants. Using the change of variable $F=r^2$ (with $r\ge a_1$), the metric is written as && ds\^2 = H\^[-1/2]{} \_ dx\^dx\^+ H\^[1/2]{}[r\^4f]{} dr\^2\
&&+ r\^2 H\^[1/2]{} (\_1 d\^2 +\_2 \^2d\^2 + 2 [a\_2\^2-a\_3\^2r\^2]{}dd\
&&+ (1-[a\_1\^2r\^2]{})\^2d\_1\^2 + (1-[a\_2\^2r\^2]{}) \^2\^2d\_2\^2 + (1-[a\_3\^2r\^2]{})\^2\^2d\_3\^2 ) \[dsiib\] where the various functions are defined as H & = & 1 + [R\^4r\^4 ]{} ,\
f & = & (r\^2-a\_1\^2)(r\^2-a\_2\^2)(r\^2-a\_3\^2) ,\
&=& 1 -[a\_1\^2r\^2]{} \^2-[a\_2\^2r\^2]{} (\^2\^2+\^2) - [a\_3\^2r\^2]{}(\^2\^2+\^2)\
&+& [a\_2\^2a\_3\^2r\^4]{}\^2+[a\_1\^2 a\_3\^2r\^4]{} \^2\^2+[a\_1\^2a\_2\^2r\^4]{}\^2\^2 , \[d12\]\
\_1 &=& 1-[a\_1\^2r\^2]{}\^2- [a\_2\^2r\^2]{}\^2\^2- [a\_3\^2r\^2]{}\^2\^2 ,\
\_2 &=& 1-[a\_2\^2r\^2]{}\^2-[a\_3\^2r\^2]{}\^2 . The metric [(\[dsiib\])]{}, together with the defining relations [(\[d12\])]{}, corresponds to the supersymmetric limit of the most general non-extremal rotating D3-brane solution [@RS1]. Using this interpretation, it turns out that $a_1,a_2$ and $a_3$ correspond to the three rotational parameters of the solution, after a suitable Euclidean continuation. We also note that the metric [(\[dsiib\])]{} corresponds to the extremal limit of the three-charge black hole solution found in [@BCS], in ansaetze for solutions to $N=8$, $D=5$ gauged supergravity preserving an $U(1)^3$ subgroup of $SO(6)$ [@Cetall].
The five-dimensional metric [(\[aans\])]{} takes the form ds\^2 = [\_[i=1]{}\^3 (r\^2-a\_i\^2)\^[1/3]{}R\^2]{} \_ dx\^dx\^+ [R\^2 r\^2\_[i=1]{}\^3 (r\^2-a\_i\^2)\^[2/3]{}]{} dr\^2 , \[hh3h\] whereas the expressions for the scalar fields are && e\^[2 \_1]{} = e\^[2 \_2]{} = (r\^2-a\_1\^2)\^[-2/3]{} (r\^2-a\_2\^2)\^[1/3]{} (r\^2-a\_3\^2)\^[1/3]{} ,\
&& e\^[2 \_3]{} = e\^[2 \_3]{} = (r\^2-a\_1\^2)\^[1/3]{} (r\^2-a\_2\^2)\^[-2/3]{} (r\^2-a\_3\^2)\^[1/3]{} , \[bbb\]\
&& e\^[2 \_5]{} = e\^[2 \_6]{} = (r\^2-a\_1\^2)\^[1/3]{} (r\^2-a\_2\^2)\^[1/3]{} (r\^2-a\_3\^2)\^[-2/3]{} ,
The relationship to elliptic functions is made explicit by first using the definition [(\[e222\])]{}, which is rewritten here in terms of three parameters $a_i$ as e\_i = a\_i\^2 -[13]{} (a\_1\^2 + a\_2\^2 + a\_3\^2) ,i=1,2,3 . \[wkp1\] Then, the differential equation [(\[jds1\])]{} has as solution the Weierstrass elliptic function ${\cal P}$ F(z)= [P]{}(z/R\^2) ,\[soo1\] which is the same as [(\[f222\])]{} after ignoring the irrelevant additive constant. The invariants of the curve that define the Weierstrass elliptic function ${\cal P}$ are g\_2= -4(e\_1 e\_2 + e\_2 e\_3 + e\_3 e\_1) ,g\_3 = 4 e\_1 e\_2 e\_3 . \[jr2\] Since the Weierstrass function ${\cal P}$ is double periodic with half-periods $\om_1$ and $\om_2$ given by \_1 = [[ K]{}(k)]{} ,\_2 = [i [ K]{}(k’)]{} , \[dawh1\] where ${ K}$ is the complete elliptic integral of the first kind with modulus $k$ and complementary modulus $k'$, we arrive at the following identification in terms of the rotational parameters && k\^2=[e\_2-e\_3e\_1-e\_3]{}=[a\_2\^2-a\_3\^2a\_1\^2-a\_3\^2]{} ,\
&& k’\^2= 1-k\^2= [e\_1-e\_2e\_1-e\_3]{}= [a\_1\^2-a\_2\^2a\_1\^2-a\_3\^2]{} . \[dh1\] Finally, after changing variable r = [ u]{} , u z , \[ej1\] where ${\rm sn} u$ is the Jacobi function, the metric [(\[hh3h\])]{} assumes the conformally flat form [(\[aans\])]{} with e\^A = [R]{} [[cn]{}\^[1/3]{}u [dn]{}\^[1/3]{}u u]{} = [1R]{} ([P]{}\^(z/R\^2)2)\^[1/3]{} . \[fj3\] The last equality describes precisely the result found in [(\[s222\])]{} using the algebro-geometric method of uniformization. Also, in terms of the variable $z$, the scalar fields [(\[bbb\])]{} coincide with [(\[s222\])]{}.
Solutions with $SO(4)\times SO(2)$ symmetry
-------------------------------------------
These solutions can be obtained by letting $e_2=e_3$ (equivalently $a_2=a_3$) into the various expressions of the previous subsection. In this limit, by taking into account the change of radial variable as $r^2\to r^2 + a_2^2$, the metric [(\[dsiib\])]{} becomes && ds\^2 = H\^[-1/2]{} \_ dx\^dx\^+ H\^[1/2]{} [r\^2-r\_0\^2\^2r\^2-r\_0\^2]{} dr\^2\
&& + H\^[1/2]{} ( (r\^2-r\_0\^2\^2) ([dr\^2r\^2-r\_0\^2]{}+ d\^2 ) + (r\^2-r\_0\^2) \^2d\_1\^2 +r\^2 \^2d\_3\^2) \[ruu1\] where $r_0^2 \equiv a_1^2-a_2^2$ and the harmonic function is H & =& 1 + [R\^4r\^2 (r\^2-r\_0\^2 \^2)]{}\
& = & 1+ [2 R\^4 ( r\_6\^2 + r\_0\^2 + ) ]{} , \[dj32\] where $r_6^2=y_1^2+\dots + y_6^2$ and $r_2^2=y_1^2+y_2^2$. In the second line of [(\[dj32\])]{} we have written for completeness the harmonic function $H$ in terms of the Cartesian coordinates by explicitly substituting the function $F$ as a solution of the condition [(\[jk4\])]{}. The result agrees with what was obtained previously in [@KLT; @sfe1]. The three-sphere line element that appears in [(\[ruu1\])]{} is given by d\_3\^2 = d\^2 + \^2d\_2\^2 + \^2d\_3\^2 . \[lpj2\]
The five-dimensional metric [(\[aans\])]{} takes the form [@BS2] ds\^2 = [r\^[4/3]{} (r\^2-r\_0\^2)\^[1/3]{}R\^2]{} \_ dx\^dx\^+ [R\^2 r\^[2/3]{} (r\^2-r\_0\^2)\^[2/3]{}]{} dr\^2 , \[h3h\] whereas the expressions for the scalar fields are given by && e\^[2 \_1]{} = e\^[2 \_2]{} = (1-[r\_0\^2r\^2]{})\^[-2/3]{} ,\
&& e\^[2 \_3]{} = e\^[2 \_4]{} = e\^[2 \_5]{} = e\^[2 \_6]{}= (1-[r\_0\^2r\^2]{})\^[1/3]{} . \[bb4\] Assuming that $r_0^2> 0$, we find that the metric [(\[ruu1\])]{} has a singularity at $r=r_0$ and $\th=0$. This is not a point-like singularity as it occurs for general values of $\psi,\varphi_2$ and $\varphi_3$. It describes the situation where the horizon of the non-extremal metric coincides with the singularity as one approaches the extremal limit. The singularity of the metric [(\[ruu1\])]{} can be interpreted as arising from the presence of D3-branes distributed on a spherical shell [@KLT; @sfe1] defined in the $y_1=y_2=0$ hyper-plane by the equation y\_3\^2+y\_4\^2+y\_5\^2+y\_6\^2=r\_0\^2 . \[ppo1\]
In the case that $e_1=e_2$ (equivalently $a_2=a_1$) it turns out that the previous results apply with $r_0^2= a_3^2-a_1^2<0$. It is then appropriate to define a new positive parameter by just letting $r_0^2\to -r_0^2$. Then, the singularity of the metric [(\[ruu1\])]{} occurs at $r=0$ and may be interpreted as coming from the presence of D3-branes distributed over a disc [@KLT; @sfe1], whose boundary is defined in the $y_3=y_4=y_5=y_6=0$ hyper-plane by the circle y\_1\^2+y\_2\^2=r\_0\^2 . \[fjh4\]
It is instructive to recover the metric of five-dimensional gauged supergravity corresponding to our solution as a limiting case of [(\[fj3\])]{}, in analogy with the limiting description of the ten-dimensional metric [(\[ruu1\])]{}. To comment on this, let us first consider the limiting case $e_3=e_2$ (or equivalently $a_3=a_2$), where the modulus $k\simeq 0$ and the elliptic curve degenerates along the $a$-cycle. Then, using the well known properties of the Jacobi functions ${\rm cn u}\simeq \cos u$, ${\rm sn u}\simeq \sin u$ and ${\rm dn u}\simeq 1$, that are valid for $k\simeq 0$, we obtain from [(\[fj3\])]{} that the conformal factor in the corresponding five-dimensional metric [(\[aans\])]{} is given by e\^[2A]{} = [r\_0\^2R\^2]{} [\^[2/3]{} (r\_0 z/R\^2)\^2(r\_0 z/R\^2)]{} , \[dj9\] where the variable $u$ in [(\[ej1\])]{} becomes $u=r_0/R^2 z$, with $r_0^2=a_1^2-a_2^2$, when $a_3=a_2$. Another limiting case arises when $e_2=e_1$ (or equivalently $a_2=a_1$), in which the complementary modulus $k'\simeq 0$ and the elliptic curve degenerates along the $b$-cycle. Then, using the properties of the Jacobi functions for $k'\simeq 0$ we have ${\rm cn u}\simeq 1/\cosh u$, ${\rm sn u}\simeq \tanh u$ and ${\rm dn u}\simeq 1/\cosh u$. >From [(\[fj3\])]{} we obtain that the conformal factor of the corresponding five-dimensional metric [(\[aans\])]{} becomes e\^[2A]{} = [r\^2\_0R\^2]{} [\^[2/3]{} (r\_0 z/R\^2)\^2(r\_0 z/R\^2)]{} , \[dj89\] where we have used the fact that the variable $u$ in [(\[ej1\])]{} becomes $u=r_0/R^2 z$, with $r_0$ now defined as $r_0^2= a_1^2-a_3^2$, when $a_2=a_1$. We note that the conformal factors appearing in [(\[dj9\])]{} and [(\[dj89\])]{} are the same as those found before in [@BS2]. They are also precisely the same factors as those appearing in [(\[a42\])]{} and [(\[a42a\])]{} by appropriate identification of the parameters and after reinstating the scale factor $R$ into the equations.
Solutions with $SO(4)$ symmetry
-------------------------------
In this case we choose four of the constants $b_i$ equal to each other as follows b\_1=-l\_1\^2 ,b\_2= - l\_2\^2 ,b\_3=b\_4=b\_5=b\_6 = 0 . \[jwef3\] Using the basis [(\[jwoi\])]{} for the $\hat x_i$’s we find that the metric takes the form ds\^2 & = & H\^[-1/2]{} \_ dx\^dx\^+ H\^[1/2]{}[\_1 \_2]{} dr\^2\
&& + r\^2 H\^[1/2]{} , where \_1 & = & 1+[l\_1\^2r\^2]{} ,\_2 = 1+[l\_2\^2r\^2]{} ,\
& = & \_1 \_2 \^2+\^2 (\_1 \^2\_1+\_2 \^2\_1) , \[jkasf1\]\
H & = & 1+ [ R\^4 \_1\^[1/2]{} \_2\^[1/2]{}r\^4 ]{} .
The five-dimensional gauged supergravity metric [(\[aans\])]{} becomes ds\^2 = [r\^2 \_1\^[1/6]{} \_2\^[1/6]{}R\^2]{} \_ dx\^dx\^+ [R\^2r\^2 \_1\^[1/3]{} \_2\^[1/3]{}]{} dr\^2 , \[jfh6\] and the scalar fields are given by && e\^[2 \_1]{} = \_1\^[-5/6]{} \_2\^[1/6]{} ,\
&& e\^[2 \_2]{} = \_1\^[1/6]{} \_2\^[-5/6]{} , \[bb47\]\
&& e\^[2 \_3]{} = e\^[2 \_4]{} = e\^[2 \_5]{} = e\^[2 \_6]{}= \_1\^[1/6]{} \_2\^[1/6]{} .
The metric [(\[merhj\])]{} has a singularity at $r=0$, where the harmonic function $H$ in [(\[jkasf1\])]{} blows up. It can be interpreted as being due to a continuous distribution of D3-branes in the ellipsoidal disc defined by + [y\_2\^2l\_2\^2]{} = 1 , \[fj33\] lying in the $y_3=y_4=y_5=y_6=0$ hyper-plane. Note also that in the case when $l_1=l_2$, the symmetry is enhanced from $SO(4)$ to $SO(4)\times SO(2)$. Then, the expressions for the metric [(\[merhj\])]{} and the scalars fields [(\[bb47\])]{} coincide with those found in [(\[ruu1\])]{} and [(\[bb4\])]{} respectively using the identification $r_0^2= -l_1^2=-l_2^2$. Also, when one of the $l_i$’s becomes zero, the symmetry is enhanced from $SO(4)$ to $SO(5)$ and by a suitable change of coordinates one recovers the results of subsection 4.3.
Spectrum for scalar and spin-two fields
=======================================
In this section we investigate the problem of solving the differential equations for the massless scalar field as well as for the graviton fluctuations in our general five-dimensional background metric [(\[aans\])]{}. We formulate the problem in terms of an equivalent Schrödinger equation in a potential that depends on the particular background. Later in this section we will discuss explicitly some cases of particular interest and determine the exact form of the corresponding potentials.
Generalities
------------
We begin with the massless scalar field equation $\Box_5 \Phi =0$ in the background geometry [(\[aans\])]{}. In the context of the AdS/CFT correspondence, the solutions and eigenvalues of this equation have been associated with the spectrum of the operator ${\rm Tr} F^2$ [@GKT; @Witten; @FFZ]. On the other hand, the fluctuations of the graviton polarized in the directions parallel to the brane are associated with the energy momentum tensor $T_{\m\n}$ on the gauge theory side [@GKT; @Witten; @FFZ]. A priori, one expects that the spectra of the two operators ${\rm Tr} F^2$ and $T_{\m\n}$ are different. However, as was shown in [@BS2], when graviton fluctuations on a three-brane embedded in a five-dimensional metric as in [(\[aans\])]{} are considered, the two spectra and the corresponding eigenfunctions coincide. In particular, in order to study the graviton fluctuations, the Minkowski metric $\eta_{\m\n}$ along the three-brane is replaced in [(\[aans\])]{} by $\eta_{\m\n} + h_{\m\n}$ and then the equations of motion [(\[eqs32\])]{} are linearized in $h_{\m\n}$. Reparametrization invariance allows to gauge-fix five functions. In the gauge $\del_\m h^{\m}{}_\n=h^\m{}_\m =0$, where indices are raised and lowered using $\eta_{\m\n}$ and its inverse, the graviton fluctuations obey the equation $\Box_5 h_{\m\n}=0$, which is the same equation as that for a scalar field [@BS2] (the same observation has been made in a slightly different context in [@CM-BMT]). Hence, the spectra for the operators ${\rm Tr} F^2$ and $T_{\m\n}$ indeed coincide. In what follows, $\Phi$ will denote either a massless scalar field or any component of the graviton tensor field.
We proceed further by making the following ansatz for the solution (x,z) = (i kx) (z) , \[hd22\] which represents plane waves propagating along the three-brane with an amplitude function that is $z$-dependent. The mass-square is defined as $M^2 = - k\cdot k$. Using the expression for the metric in [(\[aans\])]{}, we find that the equation for $\phi(z)$ is \^ + 3 A\^\^+ M\^2 = 0 . \[acc2\] This equation can be cast into a Schrödinger equation for a wavefunction $\Psi(z)$ defined as $\Psi= e^{3 A/2} \phi$. We find -\^ + V = M\^2 , \[ss2\] with potential given by V = [94]{} [A\^]{}\^2 + [32]{} A\^ . \[jwd\] So far our discussion is quite general and applies to all solutions of the system of equations [(\[eqs32\])]{}. When the solutions are supersymmetric we may use [(\[ai1\])]{}, [(\[hja1\])]{} and [(\[jds1\])]{} to find alternative forms for the potential, namely V & = & [e\^[2 A]{}16 R\^2]{}\
& = & [f\^[1/2]{}16 R\^4]{} . \[hf8\] This expression for the potential depends, of course, on the variable $z$ through the function $F(z)$. Even without having knowledge of the explicit $z$-dependence of the potential, we may deduce some general properties about the spectrum in the various cases of interest. Further details will be worked out in the following subsection using the results of section 4.
In general, $F$ takes values between the maximum of the constants $b_i$ (which according to the ordering made in [(\[hord1\])]{} is taken to be $b_1$) and $+\infty$. When $F\to \infty$, the five-dimensional space approaches $AdS_5$ and the potential becomes V , F , \[hjf4\] and hence it is unbounded from above. Let us now consider the behaviour of the potential close to the other end, namely when $F\to b_1$. Consider the general case where $b_1$ appears $n$ times, as in the corresponding discussion made at the end of subsection 3.1. Using [(\[hf8\])]{} we find that the potential behaves (including the subscript $n$ to distinguish the various cases) as V\_n n (3 n -8) (F-b\_1)\^[[n2]{}-2]{} , Fb\_1 , \[hjfg3\] with $f_0$ being a constant given, as before, by $f_0=\prod_i(b_1-b_i)$. Hence, for the value $n=6$, corresponding to $AdS_5$ the potential goes to zero and the spectrum is continuous. The same is true for the value $n=5$ corresponding to the $SO(5)$ symmetric model. For the case $n=4$ the potential approaches a constant value with the metric given by [(\[merhj\])]{}. Using the definitions [(\[jwef3\])]{}, the general expression for the the minimum value of the potential is in this case V\_[4,[min]{}]{}=[l\_1 l\_2R\^4 ]{} . \[hf56\] Therefore, although the spectrum is continuous, it does not start from zero, but there is a mass gap whose value squared is given by the minimum of the potential in [(\[hf56\])]{}. For the $SO(4) \times SO(2)$ model, where the metric is given by [(\[ruu1\])]{} with $r_0^2<0$, the existence of a mass gap has already been proven in [@FGPW2; @BS1]. For $n=3$ the potential goes to $+\infty$ as $F\to b_1$ and therefore the spectrum is not continuous but discrete. Quite generally we may show, using simple scaling arguments, that the typical unit of mass square is $f_0^{1\ov 6-n}/R^4$. Hence, for $n=3$, we expect that $M^2$ will be quantized in units of $f_0^{1/3}/R^4$. For $n=2$ the potential goes to $-\infty$ and there is the danger that it is unbounded from below. Nevertheless, at least for the $SO(4) \times SO(2)$ symmetric model with metric given by [(\[ruu1\])]{} with $r_0^2>0$, it was shown before that $M^2$ is discrete and positive [@FGPW2; @BS1].
Examples of potentials
----------------------
###
Let us consider first the massless scalar equation for the most symmetric case, namely when the background is given by the $AdS_5$ metric itself. In this case we have $e^{A} =R/z$ and the potential becomes V(z) = [154 z\^2]{} , 0 z < , \[hj11\] which obviously has a continuous spectrum for $M^2$. The corresponding Schrödinger equation can be transformed into a Bessel equation and the result for the amplitude of the fluctuations [(\[hd22\])]{} is given by \_M = (MR)\^[-3/2]{} (Mz)\^2 J\_2(M z) , \[bell2\] where $J_2$ is the Bessel function of index 2, which is regular at the origin $z=0$. The arbitrary overall constant is chosen so that the Dirac-type normalization condition is satisfied \_0\^dz e\^[3A]{} \_M \_[M’]{} = (M-M’) . \[noormm\] The measure factor $e^{3A}$ in the integrand of the equation above is such that the Schrödinger wave function $\Psi = e^{3 A/2} \phi$ obeys a normalization condition similar to [(\[noormm\])]{}, but with measure 1.
###
Consider now the first non-trivial case with metric given by [(\[ruu1\])]{} with $r_0^2>0$. The spectrum for massless scalar and graviton fluctuations has already been analyzed in [@FGPW2; @BS1] and [@BS2], respectively. We include this case here not only for completeness, but also because we will make connections with Calogero-type models later. The explicit form for the potential turns out to be V(z) = [r\_0\^2R\^4]{} (-1- [14 \^2(r\_0z/R\^2) ]{} +[154 \^2(r\_0 z/R\^2)]{} ) ,0z , \[j1d\] and clearly possesses the features we have discussed at the begining of this section. In fact [(\[j1d\])]{} belongs to a family of potentials called Pöschl–Teller potential of type I in the literature of elementary quantum mechanics. The solution for the massless scalar or graviton fluctuations is given by [(\[hd22\])]{} with the quantized amplitude modes being given by [@BS1] \_n = (1-x)\^2 P\_[n]{}\^[(2,0)]{}(x) ,x=1-2 [r\_0\^2r\^2]{}= (2 r\_0 z/R\^2) ,n=0,1,… , \[jdkl3\] where in general $P^{(\a,\b)}_n$ denote the classical Jacobi polynomials. Note that the arbitrary overall constant in [(\[jdkl3\])]{} has been chosen so that the $\phi_n$’s are normalized to 1 with measure $e^{3A}$, similar to [(\[noormm\])]{}, where $A$ is now given by [(\[dj9\])]{}. The associated mass spectrum is M\^2\_n= [4r\_0\^2R\^4]{} (n+1)(n+2) ,n=0,1,… . \[jdf9\]
Let us now turn to the case of the metric [(\[ruu1\])]{} with $r_0^2<0$ and replace $r_0^2$ by $-r_0^2$. Then, the potential takes the form (also given in [@BS2]) V(z) = [r\_0\^2R\^4]{} (1+ [14 \^2(r\_0 z/R\^2) ]{} +[154 \^2(r\_0 z/R\^2)]{} ) ,0z < , \[jd\] which is the so called Pöschl–Teller potential of type II. Note that this potential is related to the one appearing in [(\[j1d\])]{} by analytic continuation $r_0\to i r_0$, as expected. This potential approaches the value $r_0^2/R^4$ as $z\to \infty$ and therefore its spectrum is continuous, but with a mass gap given by [@FGPW2; @BS1] M\^2\_[gap]{} = [r\_0\^2R\^4]{} . \[hgef3\] In this case the solution for the massless scalar and the graviton fluctuations is given by [(\[hd22\])]{} with amplitude \_q \~x\^[(q-1)/2]{} F\_q(x) - x\^[-(q+1)/2]{} F\_[-q]{}(x) , 0x= [r\^2r\^2+r\_0\^2]{}= [1\^2(r\_0 z/R\^2)]{} 1 . \[duw\] The constant $q$ and the function $F_q(x)$ are related to the mass $M$ via hypergeometric functions as F\_q(x) = F([q-12]{},[q-12]{},1+q; x) ,q = . \[gqe\] Note that the constant $q$ is purely imaginary due to the mass gap of the model.
###
Since this case has not been discussed in the literature, we will explain the derivation of the potential $V(z)$ in some detail. Using the variable $u = z/R^2$, for simplicity, we find according to the definition that the potential becomes V(u) &=& [1 16 R\^4]{}[[P]{}\^]{}\^2(u)(3( [1 (u) -e\_1]{} + [1 (u) - e\_2]{} + [1 (u) -e\_3]{})\^2\
& & -[4 ([P]{}(u) -e\_1)\^2]{} -[4 ([P]{}(u) -e\^2)\^2]{} -[4 ([P]{}(u) -e\^3)\^2]{} ) , where $e_1 + e_2 + e_3 =0$ for the roots of the corresponding elliptic curve. Then, using the addition theorem for the Weierstrass function we have & &[1 4]{}([[[P]{}\^]{}\^2(u) ([P]{}(u) -e\_1)\^2]{} + [[[P]{}\^]{}\^2(u) ([P]{}(u) -e\_2)\^2]{} + [[[P]{}\^]{}\^2(u) ([P]{}(u) -e\_3)\^2]{} ) =\
& &3[P]{}(u) +[P]{}(u+\_1) +[P]{}(u+\_2) +[P]{}(u+\_1 + \_2) and so a straightforward calculation yields the final result V(z) = [1 4R\^4]{}(15[P]{}([z R\^2]{}) - [P]{}([z R\^2]{} + \_1) - [P]{}([z R\^2]{} + \_2) -[P]{}([z R\^2]{} + \_1 + \_2) ) with the dependence on $R$ appearing now explicitly.
It is easy to see how the degeneration of the curve leads to the rational potential of the $SO(4) \times SO(2)$ model. Recall that for $e_1 \neq e_2 = e_3$, i.e. for elliptic modulus $k =0$, the $a$-cycle of the Riemann surface shrinks to zero size and the Weierstrass function simplifies to (u) =-[3g\_3 2g\_2]{} +[9g\_3 2g\_2]{} [1 \^2(u )]{} . In this limiting case we have $9g_3 / 2g_2 = a_1^2 - a_2^2$, whereas $a_2 = a_3$, using the rotational parameters of our ten-dimensional solution. Since this combination equals to $r_0^2$, we obtain the trigonometric function ${\rm sin}(r_0z/R^2)$. In this limit we also have $\omega_1 = \pi /2r_0$ and $\omega_2 = i \infty$, and so ${\cal P}(u + \omega_1)$ will involve the function ${\rm cos}(r_0z/R^2)$, while the terms originating from ${\cal P}(u +\omega_2)$ and ${\cal P}(u +\omega_1 +\omega_2)$ contribute only to the constant. In this fashion we recover the potential of the $SO(4) \times SO(2)$ model. Its hyperbolic counterpart appears when $r_0^2 <0$ and so the new potential can be obtained by suitable analytic continuation.
Note finally that in the general case the potential becomes infinite at the Weierstrass points 0, $\omega_1$, $\omega_2$, $\omega_1 + \omega_2$, because the Weierstrass function blows up at 0 modulo the periods; put differently, some term of the potential becomes infinite at each one of these points. Unfortunately, we do not have complete grasp of the spectrum for the Schrödinger equation in this potential. We hope that its computation will be discussed elsewhere.
###
This case also leads to a new form for the potential that has not been investigated before. Again, using for simplicity the parameter $u=z/(2 R^2)$, since for $R=1$ the uniformizing parameter is $u=-z/2$, and the minus sign plays no rôle in $V$, we have for our solution (with general $R$) V(u) = [3 256 R\^4]{}[[[P]{}\^]{}\^2(u) \^2 (u)]{} ( ([[P]{}(u) -b\_1 +b\_2 (u) +b\_1-b\_2]{} )\^2 +([[P]{}(u) + b\_1 -b\_2 (u) -b\_1 +b\_2]{})\^2 + 18 ) . This follows by substitution of our algebro-geometric solution into the defining relation of the potential, after reinstating the $R$-dependence. The elliptic curve has presently $g_3 = 0$ and so $e_2 = 0$, $e_1 = - e_3$. By employing the identities, special to this surface, \^(u+\_1) = -2e\_1\^2 [[P]{}\^(u) ([P]{}(u) - e\_1)\^2]{} , [P]{}\^(u+\_2) = -2e\_3\^2 [[P]{}\^(u) ([P]{}(u) - e\_3)\^2]{} , where $e_1^2 = e_3^2 = (b_1-b_2)^2$, we arrive after some calculation at the final result for the potential V(z) = [3 (b\_1 -b\_2)\^4 4R\^4]{} ([1 \^2([z 2 R\^2]{} + \_1)]{} + [1 \^2([z 2 R\^2]{} + \_2)]{} + [18 \^([z2 R\^2]{} + \_1) [P]{}\^([z 2 R\^2]{} + \_2)]{} ) .
Note that this potential also becomes infinite at the four Weierstrass points, but its structural dependence on elliptic functions seems to be different from the previous example. However, making use of some further identities (special to the curves with $g_3 =0$), it can be cast into a form proportional to ([z 2R\^2]{} ) + [3 4]{} [P]{} ([z 2R\^2]{} + \_1 ) + [15 4]{} [P]{} ([z 2R\^2]{} + \_1 + \_2 ) + [3 4]{} [P]{} ([z 2R\^2]{} + \_2 ) , which is invariant under shifts with respect to $\omega_1 + \omega_2$. We leave the computation of the spectrum for the corresponding Schrödinger equation to future investigation, as before.
For the other examples we have been unable to derive the form of the potential in closed form, because there are no closed formulae for the solutions in terms of the variable $z$ that appears naturally in the corresponding Schrödinger equation, or else in the algebro-geometric description of the various models as Riemann surfaces. We close this section with some general remarks concerning the rational and elliptic variations of Calogero-like models.
Comments
--------
It is rather amusing that the Schrödinger problem one has to solve in $z$ is of Calogero type. This is a characteristic feature of $AdS_5$ and possibly of more general $AdS$ spaces. For the maximally symmetric $SO(6)$ model, in particular, the potential is precisely $V(z) = 1/z^2$ (up to an overall scale) [(\[hj11\])]{}. It can be seen that the solutions of the less symmetric $SO(4) \times SO(2)$ model are also related to a Calogero problem, namely the three-body model in one dimension. Recall that the quantum states of the general three-body problem can be found by solving the time independent Schrödinger equation (- \_[i=1]{}\^3 [\^2 x\_i\^2]{} + 2g \_[i, j = 1]{}\^3 [1 (x\_i - x\_j)\^2]{} + 6f \_[i, j, k = 1]{}\^3 [1 (x\_i + x\_j -2 x\_k)\^2]{} - E) (x\_1, x\_2, x\_3) = 0 , for ${i \neq j \neq k}$, where the $x_i$’s describe the coordinates of the three particles and $g$, $f$ denote the strength of the two-body and three-body Calogero interactions respectively. In fact, from a group theory point of view, this potential describes couplings between the particles according to the root system of the simple Lie algebra $G_2$. Introducing the center of mass coordinates r [sin]{}= x\_1 - x\_2 , r [cos]{}= x\_1 + x\_2 -2x\_3 , 3R\_[cm]{} = x\_1 + x\_2 +x\_3 , one obtains, after moding out the $R_{\rm cm}$-dependence, a differential equation for the wavefunction ${\Psi}(r, \varphi)$. It can be separated, as usual, into two independent equations for the radial and angular dependence of the wavefunctions (- [\^2 r\^2]{} - [1 r]{}[r]{} + [\^2 r\^2]{} - E) X(r) = 0 , (-[\^2 \^2]{} + [9g \^2 (3)]{} + [9f \^2 (3)]{} - \^2 ) () = 0 , where ${\lambda}^2$ is the separation constant, ${\Psi}(r, \varphi) = X(r) \Phi (\varphi)$, and $E$ is now the energy in the center of mass frame [@Ref1]. Of course, the Hamiltonian is Hermitian provided that the coupling constants $g \geq -1/4$ and $f \geq -1/4$. For $f = 0$ only two-body interactions are present in the problem.
The integrability of the classical Calogero model persists quantum mechanically and helps us to determine its spectrum and wave eigenfunctions. In particular, the differential equation for the angular dependence is solved in general as follows, && \_n() = [sin]{}\^[\_1]{} (3) [cos]{}\^[\_2]{} (3) P\_n\^[\_1 -1/2 , \_2 -1/2]{}([cos]{}(6)) ,\
&& \_n\^2 = 9(2n + \_1 + \_2)\^2 , n = 0, 1, 2, … , where the ${\mu}_i$’s are introduced as g = \_1 (\_1 -1) , f = \_2(\_2 -1) . The angle $\varphi$ assumes the values between 0 and $\pi /6$; because of its dependence on the Cartesian coordinates $x_i$, a particular value of $\varphi$ gives a specific ordering of the three particles and hence the problem can be divided into sectors depending on the range of $\varphi$. For a general overview of these issues, see for instance [@Ref2] and references therein. Interestingly enough, the Schrödinger problem that arose in studying the spectrum of quantum fluctuations for scalar and spin-two fields in the background of the $SO(4) \times SO(2)$ model of five-dimensional gauged supergravity fits precisely into the integrable class of such Calogero potentials with ${\mu}_1 = 5/2$ and ${\mu}_2 = 1/2$, which thus attains its minimum value required by hermiticity. Note, however, that presently ${\mu}_1 \neq {\mu}_2$. To make exact contact with our problem for the $SO(4)\times SO(2)$ model, first introduce the necessary rescaling with respect to $R$, setting $3\varphi = r_0 z/R^2$, and then conclude that the mass spectrum is given in general by M\_n\^2 = [r\_0\^2 R\^4]{}([\_n\^2 9]{} - 1) , n= 0, 1, 2, … , \[jf5\] as there is also a constant term which is present now that shifts the energy levels. For the values ${\mu}_1 = 5/2$ and ${\mu}_2 = 1/2$ the spectrum [(\[jf5\])]{} coincides with that in [(\[jdf9\])]{}.
On the other hand, the elliptic generalization of the $1 /z^2$ potential arose historically more than a century ago in connection with the problem of finding ellipsoidal harmonics for the 3-dim Laplace equation. When one deals with physical problems connected to ellipsoids, like having sources with a general ellipsoidal distribution, the mathematical structure of spheres, cones and planes usually associated to polar coordinates gets replaced by the structure of confocal quadrics. Since the transformation from Cartesian coordinates is not singled valued, elliptic functions are employed for its proper description. Introducing uniformizing parameters associated with confocal coordinates, it turns out that the solutions of Laplace’s equation are obtained by separation of variables, in which case one arrives at the Lamé equation (-[d\^2 dz\^2]{} + n(n+1) [P]{}(z) - E) (z) = 0 for harmonics of degree $n$; $E$ is a separation constant that appears in the mathematical analysis of the problem (for details see [@Ref3]). It is interesting to note that this particular Schrödinger problem was fully investigated much later in connection with finite zone potentials, Riemann surfaces and the KdV hierarchies (see for instance [@Ref4] and references therein), since the Weierstrass function satisfies the time independent KdV equation. It comes as no surprize that potentials consisting of Weierstrass functions also arise in our study, because the relevant configurations can be obtained from distributions of D3-branes on ellipsoids in ten dimensions, as it has already been noted in the geometrical setting of our solutions. In this sense, all potentials that occur in the supergravity models with genus $g>0$ should be considered as appropriate generalizations of the original derivation of Lamé’s equation in a ten-dimensional IIB framework.
Multi-particle systems with two-body interactions described by ${\cal P}(z_i - z_j)$ have also been studied extensively as integrable systems [@Ref2]. However, the trigonometric identities used earlier for expressing ${\rm sin} 3\varphi$ and ${\rm cos} 3\varphi$ in terms of ${\rm sin}\varphi$ and ${\rm cos}\varphi$ for the rational three-body Calogero model, thus arriving at a separation of the angular $\varphi$ dependence, are not generalizable to elliptic functions. Hence, there is no analogous understanding of the Schrödinger equation that determines the spectrum of scalar and spin-two fields in the five-dimensional background of our elliptic configurations using many-body elliptic Calogero systems. To the best of our knowledge, the specific quantum problems that arise here have not been investigated and pose a set of interesting questions for future work.
We mention for completeness that the only problem which has been studied in detail among the class of potentials given by a sum of Weierstrass functions concerns the Schrödinger equation with V(z) = 2\_[i=1]{}\^n [P]{}(z-z\_i(t)) when $z_i(t)$ are moduli that evolve in time as elliptic Calogero particles with two-body interactions only, namely = 4\_[i j]{} [P]{}\^(z\_i - z\_j) . Such systems are naturally encountered in the description of elliptic solutions of the KP equation, in analogy with the rational solutions of the KP equation where the ordinary $1/z^2$ Calogero models make their appearance (see for instance [@Ref5]). A static solution is easily obtained by considering four such particles located at the corners of a parallelogram inside the fundamental domain of elliptic functions described by the points $z_1 = 0$, $z_2 = {\omega}_1$, $z_3 = {\omega}_2$ and $z_4 = {\omega}_1 + {\omega}_2$. In this case, their differences $z_i -z_j$ equal to half-periods (modulo the periods) for all $i \neq j$, so the derivative of the Weierstrass function vanishes there and the elliptic Calogero equations are trivially satisfied. Then, the potential for the Schrödinger equation becomes V(z) = 2([P]{}(z) + [P]{}(z+\_1) + [P]{}(z+\_2) +[P]{}(z+\_1 + \_2)) , which by the way equals to $8 {\cal P}(2z)$ and reduces to the usual Lamé equation with $n=1$ after rescaling $z$. The generalization to potentials consisting of similar Weierstrass terms but with more arbitrary relative coefficients, as in the $SO(2)\times
SO(2)\times SO(2)$ model, or as in the $SO(3)\times SO(3)$ model, remain open for study and we hope to return elsewhere in view of their relevance in five-dimensional gauged supergravity.
Conclusions
===========
In this paper we have analyzed the conditions for having supersymmetric configurations in five-dimensional gauged supergravity in the sector where only five scalar fields, associated with the coset space $SL(6,\IR)/SO(6)$, are turned on apart from the metric. These conditions were integrated using an ansatz for the conformal factor of the five-dimensional metric in terms of a function $F(z)$, and the scalar fields were subsequently determined provided that a certain non-linear differential equation for the function $F(z)$ could be solved. This approach provides a natural algebro-geometric framework in which Riemann surfaces and their uniformization play a prominent role. A key ingredient was the interpretation of the non-linear differential equation for $F(z)$ as a Schwatz–Christoffel transformation by extending the range of parameters to the complex domain. In fact, the general solution depends on six real moduli, which when they start coalescing lead to configurations with various symmetry groups. The case with maximal symmetry $SO(6)$ corresponds to the maximally supersymmetric solution of $AdS_5$ with all scalar fields set equal to zero. More generally, we have classified all such algebraic curves according to their genus, and associated symmetry groups (all being subgroups of $SO(6)$). We also made use of their uniformization for finding explicit forms of the supersymmetric states in terms of elliptic functions. The calculations have been carried out in detail for the models of low genus, but they can be extended to all other cases with higher genus (or else smaller symmetry groups).
There is an alternative description of our solutions in terms of type-IIB supergravity in ten dimensions, which is a natural place for discussing solutions of five-dimensional gauged supergravity via consistent truncations. This higher dimensional point of view is also interesting for addressing various questions related to the AdS/CFT correspondence and supersymmetric Yang-Mills theory in four space-time dimensions. We found that the algebraic classes of our five-dimensional configurations could be understood as representing the gravitational field of a large number of D3-branes continuously distributed on hypersurfaces embedded in the six-dimensional space that is transverse to the branes. The geometry of these hypersurfaces is closely related to the Riemann surfaces underlying in the algebro-geometric approach, as the distribution of D3-branes is taken to be in the interior of certain ellipsoids for the corresponding elliptic solutions. Also, as more and more scalar fields are turned on, the geometry of the five-dimensional sphere that appears in the ten-dimensional description of our states (together with the remaining five dimensions which are asymptotic to $AdS_5$ space) becomes deformed and respects less and less symmetry from the original $SO(6)$ symmetry group of the round $S^5$. In this geometrical approach, there is no need to perform the uniformization of Riemann surfaces, as the metric is formulated in another frame with $F(z)$ being the coordinate variable instead of $z$. The Schwarz–Christoffel transform describes precisely this particular change of coordinates, when it is restricted to real values. Then, the calculation reduces to finding appropriate harmonic functions that correspond to the continuous distribution of D3-branes. In any event, both approaches are equivalent to each other and complement nicely the classification of the supersymmetric states that has been considered.
Finally, we have examined the spectra of the massless scalar and graviton fields on these backgrounds and found that they can be determined by a Schrödinger equation in one dimension, which is $z$, with a potential that depends on the conformal factor of the five-dimensional metric. It is rather curious that all these potentials are essentially of Calogero type. In the fully symmetric $SO(6)$ model, whose solution represents $AdS_5$, the potential is $1/z^2$, which is a characteristic feature of Calogero systems. For other models with less symmetry, the potential turns out to be either a rational form of Calogero interactions or elliptic generalizations thereof depending on each case. Such generalized potentials were not investigated in the literature before and there are many questions that are left open concerning their integrability properties and the exact determination of the spectrum. We think that supersymmetric quantum mechanics could help to make progress in this direction.
It will be also interesting to consider in future study the precise characterization of all these states in connection with the representation theory of the complete supersymmetry algebra. Shrinking cycles that lower the genus of our algebraic curves and lead to enhancement of the symmetry group of the various models should have an interesting interpretation in more traditional terms, using the representations of supersymmetry and the associated multiplets. Moreover, the extention of our techniques to other theories of gauged supergravity, in particular in higher dimensions, seems possible and we hope to return to all these elsewhere.
**Acknowledgements**
One of the authors (I.B.) wishes to thank CERN/TH for hospitality and support during the course of this work. He is also grateful to the organizers of the summer institute at Ecole Normale Superieure for their kind invitation to present a preliminary version of these results and for stimulating conversations.
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---
abstract: 'We present an ab-initio study of photocarrier dynamics in graphene due to electron-phonon (EP) interactions. Using the Boltzmann relaxation-time approximation with parameters determined from density functional theory (DFT) and a complementary, explicitly solvable model we show that the photocarrier thermalization time changes by orders of magnitude, when the excitation energy is reduced from 1 eV to the 100 meV range. In detail, the ultrafast thermalization at low temperatures takes place on a femtosecond timescale via optical phonon emission, but slows down to picoseconds once excitation energies become comparable with these optical phonon energy quanta. In the latter regime, thermalization times exhibit a pronounced dependence on temperature. Our DFT model includes all the inter- and intraband transitions due to EP scattering. Thanks to the high melting point of graphene we extend our studies up to 2000 K and show that such high temperatures reduce the photocarrier thermalization time through phonon absorption.'
author:
- Dinesh Yadav
- Maxim Trushin
- Fabian Pauly
bibliography:
- 'hot-carrier-graphene.bib'
title: Photocarrier thermalization bottleneck in graphene
---
Introduction
============
Recent progress in nanotechnology has made it possible to fabricate high-quality materials that are only one atom thick and hence reach the fundamental two-dimensional (2D) limit for solid crystals [@cao2015quality]. Due to their ultimate thinness these materials demonstrate various properties that are qualitatively different from those of the three-dimensional parent crystals and, at the same time, are found to be useful in photodetection and photovoltaic applications [@Nanoscale2015roadmap]. Indeed, the central phenomenon employed in photodetection and photovoltaics is the conversion of light energy into electricity. It is a quantum conversion process, employing absorption of photons to deliver photoexcited carriers to an external circuit, where they do electrical work [@Nelson2004]. There are two obvious strategies for increasing the amount of energy transferred by photocarriers. One can try to speed up the photocarrier extraction such that the carriers are collected, while they are still hot or even out of thermal equilibrium. Alternatively, one can try to slow down the cooling or photocarrier thermalization for the same purpose.
Graphene in a combination with other 2D semiconductors offers an interesting opportunity to employ both strategies. Thanks to the extremely small thickness of the junctions between 2D materials (also known as van der Waals heterostructures [@Science2016novo]), interlayer photocarrier transport may occur faster than the intralayer relaxation processes [@NatPhys2016ma]. At the same time the optical phonon emission is strongly suppressed for low-energy excitations in graphene due to unusually high energy quanta of optical phonons [@mihnev2016microscopic; @cooling_in_graphene_PhysRevLett.117.087401]. As a consequence, the photocarriers can be extracted well before they thermalize and dissipate useful energy by means of phonon emission. By incorporating graphene into a heterostructure, we can combine the two strategies in one optoelectronic device. In this way the photoresponse can be substantially increased simultaneously to the device performance. In this paper, we focus on the photocarrier evolution in graphene, providing conclusive evidence for the existence of a thermalization bottleneck that makes such applications possible.
The photocarrier dynamics in graphene has been studied experimentally by means of pump-probe spectroscopy as well as time- and angle-resolved photoemission spectroscopy [@Daniela_Nature_communication_Brida2013; @gr_pp1_PhysRevB.83.153410; @gr_pp2_PhysRevLett.105.127404; @gr_pp3_doi:10.1021/nn200419z; @gr_pp4; @Direct_view_of_hot_carrier_in_Graphene_PhysRevLett.111.027403; @gierz2017probing; @aeschlimann2017ultrafast; @Trushin_pseudospin]. In the experiments the photoexcited carriers lie far above the Dirac point (by more than $1$ eV), and the ultrafast relaxation of hot carriers is mainly attributed to optical phonon emission and carrier-carrier scattering, taking place within 150-170 fs [@Direct_view_of_hot_carrier_in_Graphene_PhysRevLett.111.027403; @Daniela_Nature_communication_Brida2013]. Excitations below the highest optical phonon energy (of around $200$ meV in graphene) have been studied in Refs. [@Acoustic_graphene_PhysRevLett.107.237401; @cooling_in_graphene_PhysRevLett.117.087401; @mihnev2016microscopic], where it has been observed that the relaxation time is drastically enhanced from the femtosecond to the picosecond timescale. Despite multiple theoretical contributions in the field of photocarrier thermalization and cooling in graphene [@mihnev2016microscopic; @PRB2009kubakaddi; @PRL2009bistritzer; @PRB2009tse; @PRB2012low; @PRB2011Malic; @PRB2011kim; @PRL2012song; @Malic2012efficient; @tomadin2013nonequilibrium; @menabde2017interface], the leading role of phonons in this enhancement still requires conclusive evidence from a parameter-free ab-initio point of view.
In what follows, we present an ab-initio approach to calculate the relaxation time of photoexcited carriers in graphene, relying on EP scattering. We use DFT to calculate EP scattering rates. Inclusion of contributions arising from all the optical and acoustical phonon branches in the whole Brillouin zone (BZ) makes it possible to calculate the energy-dependent relaxation time without adjustable parameters. Moreover, we include inter- and intraband processes, arising from the EP scattering. We investigate the relaxation time for different excitation energies from 0.05 to 0.8 eV and, due to the high melting point of graphene at around 5000 K [@melting_graphene], over a wide range of temperatures from 0 to 2000 K. Finally, we develop an explicitly solvable model to understand the energy dependence of the photocarrier thermalization.
Our paper is organized as follows. In Sec. \[sec:theory\] we present the theoretical approaches used in this work. Next, we discuss the results obtained within the models in Sec. \[sec:results\] before we end with a summary and outlook in Sec. \[sec:summary-outlook\].
Theoretical approaches {#sec:theory}
======================
In this section, we describe the theoretical approaches that we apply. In subsection \[subsec:method-ab-initio\] these are the details of our DFT calculations to determine electronic and phononic properties. Subsequently, we present in subsection \[subsec:method-time-evolution\] the Boltzmann equations in the relaxation-time approximation, as employed to determine the photocarrier dynamics. In subsection \[subsec:method-analytical\] we finally discuss simplifications to the relaxation-time approximation in order to obtain an explicitly solvable model.
Ab-initio theory for electronic and phononic properties {#subsec:method-ab-initio}
-------------------------------------------------------
We use DFT within the local density approximation (LDA) to calculate the ground-state electronic properties of graphene with <span style="font-variant:small-caps;">Quantum Espresso</span> [@QUANTUMESPRESSO_0953-8984-21-39-395502]. We employ a plane-wave basis set with a kinetic energy cutoff of 110 Ry, a charge density cutoff of 440 Ry and a Troullier-Martins pseudopotential for carbon with a $2s^{2}2p^{2}$ valence configuration [@TM_pseudopotential_PhysRevB.43.1993]. The unit cell of graphene is relaxed with the help of the Broyden-Fletcher-Goldfarb-Shanno algorithm until the net force on atoms is less than $10^{-6}$ Ry/a.u., and total energy changes are below $10^{-8}$ Ry. A vacuum of 20 along the out-of-plane direction is used to avoid artificial interactions with periodic images of the graphene sheet, and the BZ is sampled with a $45\times45\times1$ $\Gamma$-centered $\mathbf{k}$-grid. We construct Wannier functions to get localized orbitals from plane-wave eigenfunctions. By interpolating wavefunctions, we finally obtain electronic eigenenergies, dynamical matrices and EP couplings on fine grids in the BZ [@Wannierorbitals]. We calculate the phonon dispersion spectrum of graphene through density functional perturbation theory (DFPT) [@DFPT_RevModPhys.73.515], employing a $12\times12\times1$ $\mathbf{q}$-grid to evaluate phonon dynamical matrices.
By performing the DFT procedures, we obtain an optimized in-plane lattice constant of graphene of $a=|\mathbf{a}_1|$ = $|\mathbf{a}_2|$ = 2.436 , see Fig. \[fig:bandstruct\](a), which is in good agreement with previous reports of 2.458 [@graphene_phonon_DFT_PhysRevB.77.125401]. We calculate electronic and phononic band structures along high symmetry lines of the first BZ, as plotted in Fig. \[fig:bandstruct\](b). Fig. \[fig:bandstruct\](c) shows the electronic band structure, as computed from DFT with plane waves. The excellent agreement with those determined through the Wannier function method demonstrates the high quality of the interpolated localized orbitals. The phonon dispersion is finally displayed in Fig. \[fig:bandstruct\](d). Longitudinal optical and transverse optical phonon modes of graphene at the $\Gamma$-point are degenerate at an energy of 198.37 meV, which matches well with a previously reported value of 197.75 meV [@graphene_phonon_DFT_PhysRevB.77.125401].
![(a) Lattice structure of graphene with in-plane lattice vectors $\mathbf{a}_1$ and $\mathbf{a}_2$. (b) Reciprocal lattice of graphene with high symmetry points in the first BZ. (c) Electronic band structure of graphene, as obtained directly from the calculations with the plane-wave basis set and the corresponding curve from the Wannier-function formalism. (d) Phonon band structure of graphene.[]{data-label="fig:bandstruct"}](fig1.pdf){width="1.0\columnwidth"}
Having determined electronic and phononic band structures, we calculate the electronic self-energy $\Sigma_{n\mathbf{k}}(T)$ due to the EP interaction for the electronic eigenstate $|n\mathbf{k}\rangle$ with the <span style="font-variant:small-caps;">EPW</span> code. It is defined as follows [@EPW_package_PONCE2016116]
$$\begin{gathered}
\Sigma_{n\mathbf{k}}(T)=\sum_{m,p}\int_{\text{BZ}}\frac{d^3q}{\Omega_{\text{BZ}}}\vert
g_{mn,p}(\mathbf{k},\mathbf{q})\vert^{2}\times\left[\frac{N_{\hbar\omega_{p\mathbf{q}}}(T)+f^{(0)}_{\varepsilon_{m\mathbf{k+q}}}(T)}{{{\varepsilon_{n\textbf{k}}}}-(\varepsilon_{m\mathbf{k+q}}-\varepsilon_{\text{F}})+\hbar\omega_{p\mathbf{q}}+\text{i}\eta}+\frac{N_{\hbar\omega_{p\mathbf{q}}}(T)+1-f^{(0)}_{\varepsilon_{m\mathbf{k+q}}}(T)}{\varepsilon_{n\mathbf{k}}-(\varepsilon_{m\mathbf{k+q}}-\varepsilon_{\text{F}})-\hbar\omega_{p\mathbf{q}}+\text{i}\eta}\right],\label{eq:self-energy}
\end{gathered}$$
where $n$ is the band index, $\mathbf{k}$ is an electronic wave vector in the BZ, $\hbar\omega_{p\mathbf{q}}$ is the energy of the phonon of branch $p$ at wave vector $\mathbf{q}$, $\varepsilon_{\text{F}}=0$ is the Fermi energy, $f^{(0)}_{\varepsilon_{n\mathbf{k}}}(T)=1/[\exp(\frac{\varepsilon_{n\textbf{k}}-\varepsilon_{\text{F}}}{k_{\text{B}}T})+1]$ is the Fermi-Dirac distribution, $N_{\hbar\omega_{p\mathbf{q}}}(T)=1/[\exp(\frac{\hbar\omega_{p\mathbf{q}}}{k_{\text{B}}T})-1]$ is the Bose function, $\Omega_{\text{BZ}}$ is the volume of the BZ, and $\eta=10~\text{meV}$ is the small broadening parameter. The EP matrix elements are defined as [@EPW_package_PONCE2016116] $$g_{mn,p}(\mathbf{k,q})=\frac{1}{\sqrt{2\omega_{p\mathbf{q}}}}\left<m\mathbf{k+q}|\partial_{p\mathbf{q}}V|n\mathbf{k}\right>$$ and provide information about the scattering processes happening between the Kohn-Sham states $|m\textbf{k+q}\rangle$ and $|n\textbf{k}\rangle$, as mediated by the derivative $\partial_{p\mathbf{q}}V$ of the self-consistent Kohn-Sham potential with respect to the phonon wavevector **q** in branch $p$. Note that we assume that electron and phonon baths are at same temperature $T$. The first term in the brackets of Eq. (\[eq:self-energy\]) can be seen as arising from absorption of phonons and the second one from their emission.
To obtain converged results for Eq. (\[eq:self-energy\]), we first calculate the electronic and vibrational states on a $36\times36\times1$ **k**-grid and a $12\times12\times1$ **q**-grid using DFT and DFPT with plane-wave basis functions [@QUANTUMESPRESSO_0953-8984-21-39-395502], respectively. Finally, the electron eigenenergies, wavefunctions and phonon dynamical matrices are interpolated on fine grids using Wannier functions [@Wannier90_MOSTOFI20142309]. We use a $1200\times1200\times1$ **k**-grid and a $300\times300\times1$ **q**-grid, which we find necessary to accurately map out the whole BZ and to converge the integral over **q** in Eq. (\[eq:self-energy\]).
We assume that electronic wavefunctions and phonon dynamical matrixes do not change with EP interactions [@EPW_package_PONCE2016116]. The EP scattering time, resolved according to the electronic band and momentum, is calculated as $$\tau_{n\mathbf{k}}(T)=\frac{\hbar}{2\text{Im}[\Sigma_{n\textbf{k}}(T)]}\label{eq:tau-nk}$$ from Eq. (\[eq:self-energy\]) by using the imaginary part of the self-energy.
Time-evolution of excited charge carriers {#subsec:method-time-evolution}
-----------------------------------------
The time evolution of the electronic occupation $\tilde{f}_{n\textbf{k}}(t,T)$ is calculated using the Boltzmann equation in the relaxation-time approximation $$\frac{d\tilde{f}_{n\textbf{k}}(t,T)}{dt}=-\frac{\tilde{f}_{n\textbf{k}}(t,T)-f^{(0)}_{\varepsilon_{n\textbf{k}}}(T)}{\tau_{n\textbf{k}}}\label{eq:Boltzmann}$$ with the solution $$\tilde{f}_{n\textbf{k}}(t,T)=f^{(0)}_{\varepsilon_{n\textbf{k}}}(T)+e^{-\frac{t}{\tau_{n\textbf{k}}}}[\tilde{f}_{n\textbf{k}}(0,T)-f^{(0)}_{\varepsilon_n\textbf{k}}(T)],\label{eq:Boltzmann-solution}$$ if the excitation is assumed to happen at time $t=0$. Eq. (\[eq:Boltzmann-solution\]) states that when the system is weakly perturbed, the perturbation decays exponentially with the scattering time $\tau_{n\textbf{k}}$ to restore the equilibrium Fermi-Dirac distribution $f^{(0)}_{\varepsilon_{n\textbf{k}}}(T)$ [@Lundstrom_Book_lundstrom_2000]. The tilde sign indicates the time dependence of the occupation function.
We generate the initial hot-carrier occupation $\tilde{f}_{n\textbf{k}}(0,T)$ as a combination of a Fermi-Dirac distribution $f^{(0)}_{\varepsilon_{n\textbf{k}}}(T)$ at the temperature $T$ and a Gaussian peak at energy $+\zeta$ for electrons in the conduction band ($\varepsilon_{n\textbf{k}}>\varepsilon_{\text{F}}$) and $-\zeta$ for the holes in the valence band ($\varepsilon_{n\textbf{k}}<\varepsilon_{\text{F}}$) as $$\tilde{f}_{n\textbf{k}}(0,T)=f^{(0)}_{\varepsilon_{n\textbf{k}}}(T)\begin{cases}+\frac{\lambda_{\text{e}}}{\sqrt{2\pi\sigma^{2}}}e^{\frac{(\varepsilon_{n\textbf{k}}-\zeta)^{2}}{2\sigma^{2}}},
& \varepsilon_{n\textbf{k}}\geq\varepsilon_{\text{F}},\\
-\frac{\lambda_{\text{h}}}{\sqrt{2\pi\sigma^{2}}}e^{\frac{(\varepsilon_{n\textbf{k}}+\zeta)^{2}}{2\sigma^{2}}},
& \varepsilon_{n\textbf{k}}<\varepsilon_{\text{F}}.\end{cases}\label{eq:initial-distribution}$$ Throughout this work, we choose a small energy smearing $\sigma=8.47$ meV and small perturbation $\lambda_{\text{e}}=2.4 \times 10^{-3}$ . The parameter $\lambda_{\text{h}}$ is selected such that the initially excited number of electrons and holes is the same. Since the density of states (DOS) of graphene is rather symmetric in the range of excitation energies $-0.8~\text{eV}\leq\zeta\leq0.8~\text{eV}$ studied by us \[see Fig. \[fig:bandstruct\](c)\], it turns out to be an excellent approximation to set $\lambda=\lambda_{\text{e}}=\lambda_{\text{h}}$.
While we use $\lambda$ here as a free parameter to adjust the initial occupation, it can be related to measurements through $\lambda=4\pi^{2}\alpha\hbar^{2}\Phi v^2_{\text{F}}/\zeta^2$. In the expression, $\pi\alpha$ is the linear absorption of graphene, $\Phi$ is the pump-fluence and $v_{\text{F}}$ is the Fermi velocity of electrons in graphene [@Maxim_Paper_PhysRevB.94.205306].
We determine the time $\tau_{\text{th}}$, when hot carriers have relaxed through the relation $P(\zeta,0,T)-P(\zeta,t,T)<P(\zeta,0,T)/\mathrm{e}$. In the expression we have defined the population $$P(E,t,T)=\sum_{n\mathbf{k}}\delta(E-\varepsilon_{n\mathbf{k}})
\begin{cases}\times[1-\tilde{f}_{n\mathbf{k}}(t,T)],&
E<\varepsilon_{\text{F}},\\\times\tilde{f}_{n\mathbf{k}}(t,T),&
E\geq\varepsilon_{\text{F}}.\end{cases} \label{eq:population}$$ Our definition ensures that the population is symmetric with regard to electrons and holes, as long as the DOS is symmetric.
Analytical model {#subsec:method-analytical}
----------------
Before performing ab-initio calculations of charge carrier dynamics, we estimate the photocarrier thermalization time of intrinsic graphene within an explicitly solvable model. For simplicity we assume only optical phonon modes $p$ that are dispersionless, i.e., exhibit the fixed energy $\hbar\omega_p$. For this reason phonon wave vectors will be omitted. Furthermore, we consider only the two linear electronic bands of the Dirac cone with $\varepsilon_{n\mathbf{k}}=n\hbar v_{\text{F}} k$, $n=\pm$ and $k=|\mathbf{k}|$. Additionally, we will suppress all time and temperature arguments of the occupation functions in this subsection, while the tilde sign will still be indicative of a time dependence of the electronic occupation function.
The EP collisions in the given optical phonon mode $p$ are governed by the following integral $$\begin{aligned}
I_{p}[\tilde{f}_{n\mathbf{k}}] & = & \sum\limits
_{n'\mathbf{k}'}\left[\tilde{f}_{n'\mathbf{k}'}\left(1-\tilde{f}_{n\mathbf{k}}\right)W_{n'\mathbf{k}'\to
n\mathbf{k}}\right.\nonumber \\
& & \left.-\tilde{f}_{n\mathbf{k}}\left(1-\tilde{f}_{n'\mathbf{k}'}\right)W_{n\mathbf{k}\to
n'\mathbf{k}'}\right], \end{aligned}$$ where $\tilde{f}_{n\mathbf{k}}=f_{\varepsilon_{n\mathbf{k}}}^{(0)}+\tilde{f}_{n\mathbf{k}}^{(1)}$ denotes the carrier occupation with the time-independent Fermi-Dirac distribution $f_{\varepsilon_{n\mathbf{k}}}^{(0)}$ and the non-equilibrium addition $\tilde{f}_{n\mathbf{k}}^{(1)}$, representing the second term in Eq. (\[eq:Boltzmann-solution\]). The transition probability is given by Fermi’s golden rule $$\begin{aligned}
W_{n\mathbf{k}\to n'\mathbf{k}'} & = &
\frac{2\pi}{\hbar}W_{p}\left[\left(N_{p}+1\right)\delta\left({{\varepsilon_{n\textbf{k}}}}-{{\varepsilon_{n'\textbf{k}'}}}-\hbar\omega_{p}\right)\right.\nonumber
\\ & &
\left.+N_{p}\delta\left({{\varepsilon_{n\textbf{k}}}}-{{\varepsilon_{n'\textbf{k}'}}}+\hbar\omega_{p}\right)\right]\label{w1}\end{aligned}$$ for carriers outgoing from the state $|n\mathbf{k}\rangle$, and $$\begin{aligned}
W_{n'\mathbf{k}'\to n\mathbf{k}} & = &
\frac{2\pi}{\hbar}W_{p}\left[\left(N_{p}+1\right)\delta\left({{\varepsilon_{n'\textbf{k}'}}}-{{\varepsilon_{n\textbf{k}}}}-\hbar\omega_{p}\right)\right.\nonumber
\\
& & \left.+N_{p}\delta\left({{\varepsilon_{n'\textbf{k}'}}}-{{\varepsilon_{n\textbf{k}}}}+\hbar\omega_{p}\right)\right]\label{w}\end{aligned}$$ for carriers incoming to the state $|n\mathbf{k}\rangle$. Making use of nearly dispersionless optical phonon modes, the EP interaction matrix element $W_p$ is assumed to be independent of momentum. The first term in both Eqs. (\[w1\]) and (\[w\]) corresponds to the phonon emission, while the second one describes the phonon absorption. The phonons are treated as a non-interacting gas, characterized by the Bose-Einstein distribution $N_{p}= N_{\hbar\omega_{p}}(T)$. Due to the strong carbon-carbon bonding in graphene the optical phonon energy is higher than $100$ meV \[see Fig. \[fig:bandstruct\](c)\] and, hence, we assume $\hbar\omega_{p}\gg k_{\text{B}}T$ for typical temperatures or, in other words, $N_{p}\ll1$. The collision integral can then be simplified to
$$I_{p}[\tilde{f}_{n\mathbf{k}}] = \frac{2\pi}{\hbar}W_{p}\sum\limits
_{n'\mathbf{k}'}\left[\tilde{f}_{n'\mathbf{k}'}\left(1-\tilde{f}_{n\mathbf{k}}\right)\delta\left({{\varepsilon_{n'\textbf{k}'}}}-{{\varepsilon_{n\textbf{k}}}}-\hbar\omega_{p}\right)
-\tilde{f}_{n\mathbf{k}}\left(1-\tilde{f}_{n'\mathbf{k}'}\right)\delta\left({{\varepsilon_{n\textbf{k}}}}-{{\varepsilon_{n'\textbf{k}'}}}-\hbar\omega_{p}\right)\right].\label{Ieph1}$$
Let us now assume $\tilde{f}_{n\mathbf{k}}$ to be a function of ${{\varepsilon_{n\textbf{k}}}}$ and integrate in momentum space. Making use of the $\delta$-function and $\varepsilon_{nk}=\varepsilon_{n\mathbf{k}}$, we obtain $$I_{p}[\tilde{f}_{\varepsilon_{nk}}] =
\frac{W_{p}}{\hbar^{3}v^{2}_{\text{F}}}\left[|\varepsilon_{nk}+\hbar\omega_{p}|\tilde{f}_{\varepsilon_{nk}+\hbar\omega_{p}}\left(1-\tilde{f}_{\varepsilon_{nk}}\right)-|\varepsilon_{nk}-\hbar\omega_{p}|\tilde{f}_{\varepsilon_{nk}}\left(1-\tilde{f}_{\varepsilon_{nk}-\hbar\omega_{p}}\right)\right].\label{Ieph2}$$ Finally, we employ a linear response approximation and the property of intrinsic graphene $1-f_{\varepsilon_{nk}}^{(0)}=f_{-\varepsilon_{nk}}^{(0)}$ so that $$\begin{aligned}
\tilde{f}_{\varepsilon_{nk}+\hbar\omega_{p}}\left(1-\tilde{f}_{\varepsilon_{nk}}\right)&\approx&
f_{\varepsilon_{nk}+\hbar\omega_{p}}^{(0)}f_{-\varepsilon_{nk}}^{(0)}-\tilde{f}_{\varepsilon_{nk}}^{(1)}f_{\varepsilon_{nk}+\hbar\omega_{p}}^{(0)}+\tilde{f}_{\varepsilon_{nk}+\hbar\omega_{p}}^{(1)}f_{-\varepsilon_{nk}}^{(0)}, \\
\tilde{f}_{\varepsilon_{nk}}\left(1-\tilde{f}_{\varepsilon_{nk}-\hbar\omega_{p}}\right)&\approx& f_{-\varepsilon_{nk}+\hbar\omega_{p}}^{(0)}f_{\varepsilon_{nk}}^{(0)}+\tilde{f}_{\varepsilon_{nk}}^{(1)}f_{-\varepsilon_{nk}+\hbar\omega_{p}}^{(0)}-\tilde{f}_{\varepsilon_{nk}-\hbar\omega_{p}}^{(1)}f_{\varepsilon_{nk}}^{(0)}.\end{aligned}$$ Hence, Eq. (\[Ieph2\]) can be written as a sum of two terms $I_{p}[\tilde{f}_{\varepsilon_{nk}}]=I_{p}[f_{\varepsilon_{nk}}^{(0)}]+\hat{I}_{p}[f_{\varepsilon_{nk}}^{(0)},\tilde{f}_{\varepsilon_{nk}}^{(1)}]$, where $$I_{p}[f_{\varepsilon_{nk}}^{(0)}]=\frac{W_{p}}{\hbar^{3}v^{2}_{\text{F}}}\left[|\varepsilon_{nk}+\hbar\omega_{p}|f_{\varepsilon_{nk}+\hbar\omega_{p}}^{(0)}f_{-\varepsilon_{nk}}^{(0)}-|\varepsilon_{nk}-\hbar\omega_{p}|f_{-\varepsilon_{nk}+\hbar\omega_{p}}^{(0)}f_{\varepsilon_{nk}}^{(0)}\right],\label{Ieph3gen}$$ $$\hat{I}_{p}[f_{\varepsilon_{nk}}^{(0)},\tilde{f}_{\varepsilon_{nk}}^{(1)}]=\frac{W_{p}}{\hbar^{3}v^{2}_{\text{F}}}\left[|\varepsilon_{nk}+\hbar\omega_{p}|\left(\tilde{f}_{\varepsilon_{nk}+\hbar\omega_{p}}^{(1)}f_{-\varepsilon_{nk}}^{(0)}-\tilde{f}_{\varepsilon_{nk}}^{(1)}f_{\varepsilon_{nk}+\hbar\omega_{p}}^{(0)}\right)-|\varepsilon_{nk}-\hbar\omega_{p}|\left(\tilde{f}_{\varepsilon_{nk}}^{(1)}f_{-\varepsilon_{nk}+\hbar\omega_{p}}^{(0)}-\tilde{f}_{\varepsilon_{nk}-\hbar\omega_{p}}^{(1)}f_{\varepsilon_{nk}}^{(0)}\right)\right].\label{Ieph4gen}$$ Eqs. (\[Ieph3gen\]) and (\[Ieph4gen\]) are valid for any ratio between $\varepsilon_{nk}$ and $\hbar\omega_{p}$ so that we can investigate the thermalization behavior for photocarriers excited below and above the phonon frequency. Note that only Eq. (\[Ieph4gen\]) is responsible for thermalization, because Eq. (\[Ieph3gen\]) does not contain $\tilde{f}_{\varepsilon_{nk}}^{(1)}$.
In what follows we consider the thermalization of electrons (i.e., $\varepsilon_k=\varepsilon_{+k}=\hbar v_{\text{F}}k$), as the thermalization of holes is equivalent in the case of intrinsic graphene at not too high excitation energies \[see Fig. \[fig:bandstruct\](c)\]. Assuming the initial non-equilibrium distribution to be $\delta$-shaped, $f_{\varepsilon_k}^{(1)}\propto\delta\left(\varepsilon_k-\hbar\omega/2\right)$, we find $$I_{p}[\tilde{f}_{\varepsilon_k}]=\frac{\omega
W_{p}}{2\hbar^{2}v^{2}_{\text{F}}}\left(\tilde{f}_{\varepsilon_k+\hbar\omega_{p}}^{(1)}f_{-\frac{\hbar\omega}{2}+\hbar\omega_{p}}^{(0)}+\tilde{f}_{\varepsilon_k-\hbar\omega_{p}}^{(1)}f_{\frac{\hbar\omega}{2}+\hbar\omega_{p}}^{(0)}\right)
-\frac{W_{p}}{\hbar^{2}v^{2}_{\text{F}}}\tilde{f}_{\varepsilon_k}^{(1)}\left(\left|\frac{\omega}{2}+\omega_{p}\right|f_{\frac{\hbar\omega}{2}+\hbar\omega_{p}}^{(0)}+\left|\frac{\omega}{2}-\omega_{p}\right|f_{-\frac{\hbar\omega}{2}+\hbar\omega_{p}}^{(0)}\right). \label{eq:Iph5}$$ Eq. (\[eq:Iph5\]) contains cascade terms, generated each time, when a phonon is emitted or absorbed [@PRB2011Malic]. We use the relaxation-time approximation, i.e., we truncate the cascade to a single term proportional to $\tilde{f}_{\varepsilon}^{(1)}$. This results in the thermalization time given by $$\frac{1}{\tau_{\mathrm{th}}}=\sum\limits_{p}\frac{W_{p}}{\hbar^{2}v^{2}_{\text{F}}}\left(\left|\frac{\omega}{2}+\omega_{p}\right|f_{\frac{\hbar\omega}{2}+\hbar\omega_{p}}^{(0)}+\left|\frac{\omega}{2}-\omega_{p}\right|f_{-\frac{\hbar\omega}{2}+\hbar\omega_{p}}^{(0)}\right).\label{tauth}$$
This analytical model is of course not able to give quantitative predictions, but it suggests that the thermalization time at $\omega_{p}\gg\omega/2$ is much longer than at $\omega_{p}\ll\omega/2$. Indeed, in the latter limit we have $$\frac{1}{\tau_{\mathrm{th}}}=\sum\limits _{p}\frac{\omega W_{p}}{2\hbar^{2}v^{2}_{\text{F}}}, \quad \omega_{p} \ll \omega/2,
\label{eq:optical}$$ whereas in the former case the rate contains an exponentially small multiplier, resulting in the following expression $$\frac{1}{\tau_{\mathrm{th}}}=\sum\limits_{p}\frac{2\omega_{p}W_{p}}{\hbar^{2}v^{2}_{\text{F}}}{\mathrm{e}}^{-\frac{\hbar\omega_{p}}{k_{\text{B}}T}},
\quad \omega_{p} \gg \omega/2.
\label{eq:farinfra}$$
We will confirm the predictions of Eqs. (\[eq:optical\]) and (\[eq:farinfra\]) in the next section using the [*ab-initio*]{} approach. Note, however, that while the approximation $\hbar\omega_{p}\gg
k_{\text{B}}T$ or $N_{p}\ll1$, made for their derivation, is excellent for most temperatures studied, we will consider temperatures of up to 2000 K with our ab-initio approach, where this approximation becomes questionable.
Results {#sec:results}
=======
We will now use the ab-initio parameters for electrons, phonons and their couplings, determined as described in subsection \[subsec:method-ab-initio\], and combine them with the Boltzmann formalism of subsection \[subsec:method-time-evolution\] to study photocarrier thermalization. At the end, we will compare to the results of the analytical equations as derived in subsection \[subsec:method-analytical\].
Since we determine scattering times $\tau_{n\mathbf{k}}$ of the Boltzmann formalism \[see Eq. (\[eq:Boltzmann\])\] from the imaginary part of the EP self-energy \[see Eq. (\[eq:self-energy\])\], we investigate this quantity first. Fig. \[fig:ImSigma-tau\](a) plots $\text{Im}[\Sigma_{n\textbf{k}}(T)]$ as a function of energy for different temperatures. For a given temperature it shows a pronounced energy dependence. Increasing initially monotonically and rather symmetrically in the vicinity of the Dirac point at $E=\varepsilon_{\text{F}}=0$, it follows the same behavior as the electronic DOS \[see Eqs. (\[eq:self-energy\])\]. This results from the fact that the electronic DOS represents the phase space for EP scattering events to take place. As can be inferred from Fig. \[fig:ImSigma-tau\](a) and \[fig:ImSigma-tau\](b), $\text{Im}[\Sigma_{n\textbf{k}}(T)]$ is very sensitive to temperature close to $E=0$. In contrast it shows a much weaker temperature dependence at energies above around 200 meV, coinciding with the highest optical phonon energies. Indeed, we see for low temperatures (0-300 K) that Im$(\Sigma_{n\textbf{k}})$ increases roughly exponentially until the highest optical phonon energy is reached, while the energy dependence is comparatively weak for elevated temperatures (600-2000 K). The behavior shows that scattering below the optical phonon threshold takes place rather inefficiently via acoustical phonons. With increasing temperature there are more phonons available for the carriers to interact with, leading to the increase of $\text{Im}[\Sigma_{n\textbf{k}}(T)]$. Analogously, the available phase space for optical phonon emission grows with increasing energy.
In the inset of Fig. \[fig:ImSigma-tau\](b), we consider the scattering times $\tau_{n\textbf{k}}(T)$, which are inversely proportional to the self-energy \[see Eq. (\[eq:tau-nk\])\]. We observe that around the Dirac point the scattering time becomes very sensitive to temperature and can be on the order of a few picoseconds for low $T$. In contrast, at energies above 200 meV the scattering times exhibit only weak energy and temperature dependencies. As argued before, this behavior can be rationalized by the fact that for low $T$ at $E<200$ meV excited carriers can relax via acoustical phonon scattering only, while they thermalize efficiently via optical phonons above 200 meV.
The behavior of $\text{Im}[\Sigma_{n\textbf{k}}(T)]$ in Fig. \[fig:ImSigma-tau\] can also be analyzed in terms of Eq. (\[eq:self-energy\]). Lets consider low temperatures and electrons with $\varepsilon_{n\mathbf{k}}\geq0$. In this case both $f^{(0)}_{\varepsilon_{n\mathbf{k}}}(T)$ and $N_{\hbar\omega_{p\mathbf{q}}}(T)$ are vanishingly small, and thus only the second term of the Eq. (\[eq:self-energy\]) contributes. For this reason, excited electrons relax via emission of phonons. But as temperature increases, we get $0\leq f^{(0)}_{\varepsilon_{n\mathbf{k}}}(T)\leq1$ and $N_{\hbar\omega_{p\mathbf{q}}}(T)>0$, and both terms in Eq. (\[eq:self-energy\]) start contributing. For this reason $\text{Im}[\Sigma_{n\textbf{k}}(T)]$ increases with increasing temperature in Fig. \[fig:ImSigma-tau\] for $E>0$. An analogous argumentation can be carried out for holes.
![(a) Imaginary part of the EP self-energy as a function of energy, evaluated at different temperatures, and the electronic DOS of graphene. (b) Zoom in on the energy and temperature dependence of $\text{Im}[\Sigma_{n\textbf{k}}(T)]$. We consider only positive energies close to the Dirac point. The inset represents the corresponding scattering](fig2.pdf){width="1.0\columnwidth"}
times. \[fig:ImSigma-tau\]
To simulate the temporal dynamics, we use Eq. (\[eq:Boltzmann-solution\]), starting with the initial distribution of Eq. (\[eq:initial-distribution\]) at time $t=0$. Choosing the parameters $\lambda$ and $\sigma$ as described above, we calculate time evolutions of occupations for different temperatures $T$ and excitation energies $\zeta$. We are particularly interested in the behavior of thermalization times for excitations below and above the optical phonon threshold.
Fig. \[fig:thermalization-tevol-lowT\] shows the hot carrier population $P(E,t,T)$ \[see Eq. (\[eq:population\])\] for excitation energies $\zeta=0.05, 0.5$ eV and temperatures $T=0,10,100$ K. Below the optical phonon threshold for $\zeta=0.05$ eV in Fig. \[fig:thermalization-tevol-lowT\](a)-(c), thermalization of the hot carriers takes place on the ps timescale via low-energy acoustical phonons. In this excitation range the relaxation time decreases with increasing temperature, because the background equilibrium electron distribution allows excited carriers to scatter increasingly efficiently with the optical phonons [@Acoustic_graphene_PhysRevLett.107.237401]. Our thermalization time $\tau_{\text{th}}$ at $T=10$ K, as extracted from Fig. \[fig:thermalization-tevol-lowT\](b), is around 175 ps. This is lower than the 300 ps reported in Ref. [@Acoustic_graphene_PhysRevLett.107.237401] for an excitation energy of 51 meV on an epitaxially grown graphene sample containing around $\sim70$ layers and arranged over a SiC substrate. Above the optical phonon threshold, our results in Fig. \[fig:thermalization-tevol-lowT\](d)-(f) predict a weak or almost no temperature dependence of the relaxation time. With $\tau_{\text{th}}\approx60$ fs it takes a value of similar size as the photocarrier isotropization time from Ref. [@Trushin_pseudospin], originating from scattering by optical phonons. Our qualitative findings of a strong temperature dependence of $\tau_{\text{th}}$ below the optical phonon threshold and none above are consistent with the experimental observations in Ref. [@Acoustic_graphene_PhysRevLett.107.237401]. The plots in Fig. \[fig:thermalization-tevol-lowT\] also demonstrate that the populations of electrons and holes evolve with time quite symmetrically around the Dirac point, confirming that the dynamics of holes are similar as those of electrons.
![Time-dependent thermalization of photocarriers at an excitation energy of (a)-(c) 0.05 eV and (d)-(f) 0.5 eV for different temperatures.[]{data-label="fig:thermalization-tevol-lowT"}](fig3.pdf){width="1.0\columnwidth" height="50.00000%"}
Due to the extraordinarily high melting temperature of nearly $5000$ K predicted theoretically for graphene [@melting_graphene], we extend our analysis of time evolutions to high temperatures $T=300,600,1200,2000$ K. We find carriers to relax at $T=300$ or $600$ K on a 100 fs time scale. At 1200 K this reduces to around 34 fs and is even below 26 fs at 2000 K.
![Same as Fig. \[fig:thermalization-tevol-lowT\] at an excitation energy of 0.5 eV for elevated temperatures.[]{data-label="fig:thermalization-tevol-highT"}](fig4.pdf){width="1.0\columnwidth"}
In Fig. \[fig:t-th\] we summarize the relaxation times $\tau_{\text{th}}$, which we have extracted from our ab-initio modeling at different excitation energies and temperatures. For $\zeta=0.05$ eV the thermalization time decreases with increasing temperature from $T=0$ to $1200$ K by more than 3 orders of magnitude. In contrast, there is only little change in the relaxation time with temperature for a fixed excitation with $\zeta=0.4,0.6,0.8$ eV above the optical phonon threshold. A slight decrease is seen at the temperatures, where thermal energies are similar to those of optical phonon quanta, i.e., $k_\text{B}T\approx\hbar\omega_p$. In addition, for a fixed temperature, relaxation times depend only little on $\zeta$, if the excitation energy is above the optical phonon threshold. To summarize, taking into account only EP scattering events, we thus observe intriguingly that relaxation times in graphene can span an extraordinary range from 170 ps down to 60 fs, if the temperature is varied and carriers are excited below the optical phonon threshold.
Our ab-initio predictions can be qualitatively understood by using the concept of a thermalization bottleneck in graphene. Thanks to the high optical phonon energy quanta of about 200 meV \[see Fig. \[fig:bandstruct\](d)\], the low-energy (THz) electrons cannot relax as fast as the optically excited photocarriers, because at low temperatures (i) the phonon absorption is a very rare process and (ii) the phonon emission requires an empty electron state below the Fermi level, but states below $\varepsilon_{\text{F}}$ are almost fully occupied. The relevant thermalization times can be estimated by using our analytical model. We assume an explicit form for the EP interaction matrix element given by [@PRB2012low] $$W_{p}=\frac{\hbar \Delta_{p}^{2}F_{p}}{2\rho\omega_{p}},$$ where $\Delta_{p}$ is the deformation potential for a mode $p$, $F_{p}$ is a dimensionless geometric factor, and $\rho=7.6\times10^{-8}$ g/cm$^{2}$ is the mass density. In what follows, we take into account the two most important phonon modes [@PRB2012low], $p=\Gamma,K$, where $F_{\Gamma}=1$, $F_{K}=1/2$, $\hbar\omega_{\Gamma}=197$ meV, $\hbar\omega_{K}=157$ meV, and $\Delta_{\Gamma}=\Delta_{0}$, $\Delta_{K}=\sqrt{2}\Delta_{0}$ with $\Delta_{0}=11$ eV/Å [@PRB2012low]. At $\omega\gg\omega_{p}$ the thermalization time can be found from Eq. (\[eq:optical\]) as
$$\tau_{\mathrm{th}}=\frac{4\omega_{0}}{\omega}\frac{\hbar v^{2}_{\text{F}}\rho}{
\Delta_{0}^{2}},\quad\omega\gg\omega_{\Gamma,K},$$
where $1/\omega_{0}=1/\omega_{\Gamma}+1/\omega_{K}$. Assuming an excitation energy of $\hbar\omega=1.55$ eV (i.e., a radiation wavelength of 800 nm), we estimate $\tau_{\text{th}}\approx 58$ fs. In the opposite limit $\omega\ll\omega_{p}$ we get from Eq. (\[eq:farinfra\]) $$\tau_{\mathrm{th}}=\frac{\hbar
v^{2}_{\text{F}}\rho}{\Delta_{0}^{2}}\frac{1}{\exp(-\frac{\hbar\omega_{\Gamma}}{k_{\text{B}}T})+\exp(-\frac{\hbar\omega_{K}}{k_{\text{B}}T})},\quad\omega\ll\omega_{\Gamma,K}.$$ Assuming the most relevant temperature of $300$ K, we estimate $\tau_{\text{th}}\approx 92$ ps.
Our considerations confirm that (i) the thermalization timescales differ at $\omega\ll\omega_{p}$ and $\omega\gg\omega_{p}$ by three orders of magnitude at room temperature, (ii) the photocarrier thermalization time strongly depends on temperature at $\omega\ll\omega_{p}$, whereas at $\omega\gg\omega_{p}$ it does not, (iii) in the former case, the thermalization time decreases rapidly with increasing temperature. This is exactly what we see in the summary of the relaxation times $\tau_{\text{th}}$ shown in Fig. \[fig:t-th\], as determined through our first principles approach.
![Thermalization time of the excited carriers, as determined with our ab-initio approach, as a function of temperature for different excitation energies.[]{data-label="fig:t-th"}](fig5){width="0.8\columnwidth"}
Summary and Outlook {#sec:summary-outlook}
===================
In summary, we have studied the relaxation dynamics of hot carriers in single-layer graphene near and away from the Dirac point subject to the EP interaction. By determining electron and phonon dispersions as well as EP couplings from DFT, our model based on the Boltzmann equation in the relaxation-time approximation contains no free parameters and takes into account contributions from all of the optical as well as acoustical branches in the whole BZ. In excellent agreement with analytical predictions we find that relaxation times computed with our ab-initio model are strongly enhanced, if carriers are excited below the optical phonon energies. In addition, we have shown that the carrier relaxation times depend strongly on temperature for such low excitation energies, while being rather temperature-independent for excitation energies above optical phonon energy quanta.
These effects could be employed to facilitate the photoexcited electron transport from graphene to a semiconductor across a Schottky barrier [@SciRep2014zhang; @ACSNano2016defazio; @massicotte2016photo]. Thanks to the longer relaxation time at lower excitation energies, the photocarriers can contribute to the interlayer transport before thermalization is completed, thus improving the photoresponsivity [@trushin2017theory]. From the device engineering point of view, the most important assumption made in this work is the absence of a substrate. It might provide additional dielectric screening and unintentional doping, which overall influence the electron-electron scattering contribution neglected here. Moreover, the photocarriers might experience interactions with remote polar surface phonons [@PRB2012low]. Since the precise effects caused by a substrate strongly depend on the chosen material and its interface properties, the model should be tailored for each device to make quantitative predictions. Such a fine tuning is out of scope here.
acknowledgment {#acknowledgment .unnumbered}
==============
D.Y. and F.P. acknowledge financial support from the Carl Zeiss Foundation as well as the Collaborative Research Center (SFB) 767 of the German Research Foundation (DFG). M.T. is supported by the Director’s Senior Research Fellowship from the Centre for Advanced 2D Materials at the National University of Singapore (NRF Medium Sized Centre Programme R-723-000-001-281). Part of the numerical modeling was performed using the computational resources of the bwHPC program, namely the bwUniCluster and the JUSTUS HPC facility.
|
---
abstract: 'Neutron diffraction and muon spin relaxation measurements are used to obtain a detailed phase diagram of PrFe$_{1-x}$Ru$_{x}$AsO. The isoelectronic substitution of Ru for Fe acts effectively as spin dilution, suppressing both the structural and magnetic phase transitions. The temperature, $T_S$, of the tetragonal-orthorhombic structural phase transition decreases gradually as a function of x. Slightly below $T_S$ coherent precessions of the muon spin are observed corresponding to static magnetism, possibly reflecting a significant magneto-elastic coupling in the FeAs layers. Short range order in both the Fe and Pr moments persists for higher levels of x. The static magnetic moments disappear at a concentration coincident with that expected for percolation of the J$_1$ - J$_2$ square lattice model.'
author:
- 'Yuen Yiu$^a$, Pietro Bonfà$^b$, Samuele Sanna$^{c}$, Roberto De Renzi$^b$, Pietro Carretta$^c$, Michael A. McGuire$^d$, Ashfia Huq$^e$, and Stephen E. Nagler$^{a,f,g}$'
bibliography:
- 'paper.bib'
title: 'Tuning the magnetic and structural phase transitions of PrFeAsO via Fe/Ru spin dilution'
---
\[sec:level1\]Introduction
==========================
The precise role of magnetism and its coupling to the lattice is a central problem in the physics of unconventional iron based superconductors and related materials [@Lumsdenreview; @Johnston2010; @Tranquada2014]. In general, the undoped parent compounds of the 1111 family iron-pnictide superconductors are tetragonal paramagnets at high temperatures. Upon cooling they display a tetragonal-orthorhombic structural transition at $T_S$, followed or accompanied by a spin density wave (SDW) transition at $T_{SDW}$. [@Johnston2010; @Stewart2011; @Johrendt2011]. Superconductivity can usually be induced by suppressing these transitions and inevitably results when this is done by using dopants that introduce charge carriers. The use of isovalent dopants, for example the substitution of Ru for Fe, allows for investigations of the physics without the complications induced by changing the electron count. In the 122 family compound BaFe$_{2-x}$Ru$_x$As$_2$ the suppression of the structural and magnetic transitions via Ru substitution indeed results in a superconducting ground state, but at a much larger Ru content than has been observed with non-isovalent dopants [@Thaler2010; @Kim2011]. Investigations of the 1111 compounds including PrFe$_{1-x}$Ru$_{x}$AsO and LaFe$_{1-x}$Ru$_{x}$AsO also showed that Ru/Fe substitution suppresses the structural and magnetic phase transitions but in contrast to the 122s, there is no observation of a superconducting ground state for any concentration of Ru.[@mcguire09a; @Bonfa2012; @yiu12; @Martinelli2013].
Arguably, when Ru is substituted for Fe in the 1111 compounds, the main effect on the magnetism can be understood by considering the substitution as simply equivalent to spin dilution. This is consistent with local density approximation calculations on LaFe$_{1-x}$Ru$_{x}$AsO [@tropeano10], which illustrates that Ru atoms do not show any tendency to sustain a magnetic moment regardless of their concentration. This is also compatible with previous experimental data on PrFe$_{1-x}$Ru$_{x}$AsO [@mcguire09a]. In this paper, we present a systematic study of the evolution of the magnetic and structural transitions in the isovalently doped PrFe$_{1-x}$Ru$_{x}$AsO system. The neutron diffraction measurements of Ref. have been extended and complemented by new muon spin relaxation measurements. The previous neutron work [@yiu12] showed no evidence for the structural transition in PrFe$_{1-x}$Ru$_{x}$AsO above $x = 0.4$ as determined by Rietveld refinements. The magnetic transition in the FeAs layers was not detected beyond $x = 0.1$. The x = 0.1 sample was previously measured using elastic scattering at the HB1A triple axis spectrometer[@yiu12], and was not sensitive to ordered moment sizes less than $0.02 \mu_B$. The other neutron diffraction measurements had significantly lower sensitivities. The sensitivity of $\mu$SR ($\approx 0.001 \mu_B$) is therefore more than an order of magnitude better than the neutron measurements, enabling a more complete determination of the phase diagram. Remarkably, it is observed that all signatures of magnetic order disappear at the percolation concentration of the $J_1-J_2$ square-lattice model. We also note that the negative thermal expansion (NTE) reported earlier in PrFe$_{1-x}$Ru$_{x}$AsO [@Kimber08; @yiu12] persists across the entire Ru doping range even for pure PrRuAsO.
This paper is organized as follows: Sample synthesis is described in section II, bulk characterization and neutron diffraction results in section III, and $\mu$SR results in section IV. Discussion and conclusions follow in sections V and VI respectively.
\[sec:level1\]Sample Synthesis
==============================
Methods reported earlier [@mcguire09a; @mcguire12] were used to synthesize the samples. PrFe$_{1-x}$Ru$_{x}$AsO samples were made from powders of PrAs, Fe$_2$O$_3$, RuO$_2$, Fe and Ru. The starting materials were crushed and mixed inside a He glovebox, then pressed into a 1/2" diameter pellets ($\sim$ 2g each) and placed in covered alumina crucibles inside silica tubes. The tube was evacuated, backfilled with ultra-high-purity Ar and flame sealed. Each individual sample was heated at 1200$^\circ$C for $12 - 36$h several times, and was thoroughly ground and pressed into pellets between the heating cycles.
\[sec:level1\] Bulk characterization and Neutron diffraction results
====================================================================
Heat capacity and dc magnetic susceptibility measurements were performed using the MPMS SQUID and PPMS system by Quantum Design. Fig.1(a) shows the reciprocal magnetic susceptibility temperature dependence for x = 1, i.e. PrRuAsO, with no indication of superconductivity down to 2K. The Curie-Weiss law describes the data well down to 14K, coinciding with the Neél temperature for Pr ordering in PrFeAsO. Data points for $T > 50K$ were fitted to the Curie-Weiss law, with the resultant fit intersecting the temperature axis at T$_{CW}$ = -33(5)K with a Curie constant of 1.4(1), close to the expected value of 1.6 for Pr$^{3+}$. Fig.1(b) shows the field dependence of the magnetization at T = 2K, with no sign of saturation up to 6T. Fig.1(c) shows the temperature dependence of the heat capacity. A broad hump was observed around 14K, the same temperature where the anomaly in the $1/\chi$ due to the Pr ordering is detected (Fig.1(a)).
![(a): Temperature dependences of the reciprocal susceptibility for PrRuAsO. The anomaly around 14K is possibly related to magnetic ordering of the Pr sublattice. The straight line is the Curie-Weiss law fitted to data above 50K. (a, inset): d$\chi$/dT vs T, better illustrating the anomaly at 14K. (b): Field dependence of the magnetic susceptibility at T = 2K, with no sign of moment saturation up to 6T; (c): Temperature dependence of heat capacity. Similar to that from the $x = 0.1 - 0.75$ samples, the sharp peak in PrRuAsO at T$_{N,Pr}$ = 14K is suppressed, but a broad hump remains [@yiu12].](susceptibilitynew){width="88mm"}
Neutron powder diffraction was performed using POWGEN at the Spallation Neutron Source of Oak Ridge National Laboratory. Rietveld refinement of the data confirmed that PrRuAsO is isostructural to PrFeAsO at room temperature, and remains in the tetragonal *P4/nmm* structure down to the base temperature of 10K, similar to other PrFe$_{1-x}$Ru$_{x}$AsO samples with $x \ge 0.33$[@yiu12]. Some of the figures below include data previously reported[@yiu12]. In Ref. the neutron diffraction was analyzed via Rietveld refinement, and the transition temperature was determined by whether or not the quality of fit was better for the orthorhombic or tetragonal structure. For x=0.4 the difference in quality of fit was undetectable down to base temperature, and it was concluded that the structure was tetragonal. As described in the next paragraph, here we have re-analyzed that data using a different criterion for identifying the transition.
Fig.2 shows the temperature dependence of the orthorhombicity, defined as $(a-b)/(a+b)$. At high temperature all of the samples are tetragonal and the orthorhombicity is zero by definition. The orthorhombicity values plotted in Fig.2 were determined as follows: lattice parameters a and b were extracted by imposing an orthorhombic structure on the Rietveld refinement over the entire temperature range for all samples. For each doping concentration the fitted value $(a-b)/(a+b)$ determined by the refinement to an orthorhombic structure at T = 200K was subtracted from the corresponding values at other temperatures. This analytical method is useful for detecting structural transitions that are too subtle to be observed via the splitting or broadening of a single nuclear Bragg peak. The data shows clear evidence for the structural transition temperature $T_S$ in samples up to $x = 0.4$. (Here $T_S$ is defined operationally as the temperature at which the orthorhombiciy reaches 1/2 of the asymptotic low temperature value.) For samples with $x \ge 0.5$, no deviation from zero orthorhombicity can be detected at any temperature.
{width="77mm"}
Fig. 3(a) shows the doping dependence of lattice parameters a, b and c (including some data previously published in Ref.). As Fe is substituted by Ru, the in-plane (a,b)-axis elongates, and the out-of-plane c-axis shrinks [@mcguire09a]. The difference of the lattice parameters between x = 0 and x = 1 is of the order of a few %. Fig.3(b) shows the temperature dependence of a and c for PrRuAsO. As reported previously [@yiu12; @Kimber08], for $x \le 0.75$ PrFe$_{1-x}$Ru$_x$AsO exhibits NTE in the c-axis for temperatures below approximately 50K. The NTE is also observed clearly in stoichiometric PrRuAsO. The magnitude of NTE in the c-axis is about 0.02% relative to the minimum at 50K. The a-axis shrinks more than that predicted by the Debye-Grüneisen model, and compensates somewhat for the NTE in the c-axis, resulting in a smaller NTE as determined by the unit cell volume shown in Fig. 3(c). This compensating behavior can be explained by considering that an expansion in the a-b plane forces the unit cell to shorten along c in order to satisfy Fe/Ru-As bonding requirements [@mcguire09a]. The opposite signs of the x-dependence of the a-b and c lattice parameters can similarly be understood. The in-plane expansion as a function of x has been attributed to the substitution of larger Ru atoms for Fe atoms, which stretches along the a-b plane [@mcguire09a].
{width="77mm"}
\[sec:level1\] $\mu$SR
======================
![Time dependence of ZF-$\mu$SR asymmetry for x = 0.33 between 110K and 5K, the lines represent the best fit using Eq.\[Eq:ZFmusrfit\].](33difftemp){width="77mm"}
{width="66mm"}
Zero field (ZF) and longitudinal field (LF) $\mu$SR experiments were performed on the GPS spectrometer at the Laboratory for Muon Spin Spectroscopy of Paul Sherrer Institut. Here the findings are presented in two parts, one for the SDW ordering involving FeAs layers, and the other for the magnetic ordering of the Pr moment.
\[sec:SDWordering\] SDW ordering
--------------------------------
The following methods were used to interpret the data. For the ZF-data, the time dependence of the spin-polarization function for a positive, 100% spin-polarized muon in a magnetic sample can be described as:
$$\begin{aligned}
\label{Eq:ZFmusrfit}
\frac{A_{ZF}(t)}{A_{0}} & = & \left(1 - V_{m}\right)
e^{-\frac{\sigma_{N}^{2}
t^{2}}{2}} {} \nonumber\\ & + & \sum_{i = 1}^{N}
f_{i} \left[w_{i}^{\perp} F_{i}(t) e^{-\frac{\sigma_{i}^{2} t^{2}}{2}}+
w_{i}^{\parallel} e^{-\lambda_{i} t}\right]\,\,\,\end{aligned}$$
$A_{ZF}$ is the asymmetry of the muon decay and $A_0$ is the initial muon asymmetry (i.e. t = 0). $V_m$ represents the fraction of muons probing a static local field **B**$_i$, i.e. the sample’s magnetic volume fraction. The index *i* represents each of N crystallographically-inequivalent muon stopping sites and each stopping site is characterized by a stopping probability $f_i$, with $\sum_{i = 1}^{N} f_{i} = V_m$. The two terms in the square brackets reflect the orientation of the internal field with respect to the initial muon spin $S_\mu$ direction: transverse for $\bm{B}_i\perp \bm{S}_\mu$ and longitudinal for $\bm{B}_i\!\parallel\! \bm{S}_\mu$. For powder samples, the ratio of the two terms is related and normalized by $w_{i}^{\perp} = 2/3$ and $w_{i}^{\parallel} = 1/3$.
The longitudinal component ($\parallel$) can be described by a Lorenztian decay function with relaxation rate $\lambda$. For the transverse component ($\perp$), $F_i(t)$ represents the time dependence, and $\sigma_i$ is the depolarization rate which reflects the second moment of the field distribution $\Delta B_i\!\!\equiv\!\!(\overline{B_i^2}- \overline{B_i}^2)^{1/2} \!\! = \!\! \sigma_i/\gamma_\mu $, where $\gamma_\mu/2\pi = 136$ MHz/T is the muon gyromagnetic ratio. When the muon goes through a local field $B_i$, for example inside a long range ordered sample, the muon asymmetry displays Larmor oscillations described with $F_i= cos(\gamma_\mu B_i t)$, with $B_i$ proportional to the mean magnetic order parameter $<S(T)>$. In case of a short range ordered sample, the width of the field distribution at the muon site broadens and as a result, the transverse muon fraction yields to a fast decay rate ($\sigma_i\gtrsim 1/\gamma_\mu B_i$), with overdamped oscillations and $F_i = 1$.
For the undoped parent PrFeAsO the ZF-$\mu$SR time spectra are well fitted with $N = 2$ and occupancy $f_1 \approx 75\%$ and $f_2 \approx 25\%$, which reflects the presence of two inequivalent muon sites[@Maeter2009; @Derenzi2012; @Prando2013]. In the x = 0.33 and 0.4 samples, the two frequencies are still detectable, but with $f_2$ reduced in half. The complementary missed amplitude gives rise to overdamped oscillations and can be easily fitted as a third additional non-oscillating component i = 3, with $f_3 \approx f_2$, $F_3 = 1$ and $\sigma_3 \sim 5 \mu s^{-1}$. This change might be simply due to the increase of disorder by Ru.
These three components provide a good fit of the time evolution of the ZF muon asymmetry, as seen in Fig.4, which shows data for $x = 0.33$ at different temperatures, fitted with eq.\[Eq:ZFmusrfit\]. At higher temperatures the oscillations become overdamped and the transverse amplitude $(\propto V_m)$ reduces and vanishes at $T \geq 100$K. Fig.5 displays the low temperature ZF-$\mu$SR time spectra for all our samples. For x = 0.5 and 0.6 the coherent oscillations are absent and the fit uses only 2 components but with $F_1 = F_2 = 1$, suggesting that the increase of Ru/Fe substitution induces field inhomogeneity. Accordingly, the presence of the magnetic phase is reflected by the sizeable decay rate detected corresponding to $\Delta B_1 \approx 40$ mT and $\Delta B_2 \approx 5$ mT. The same behavior has also been reported in Ru/Fe substituted LaFeAsO [@Bonfa2012] at a similar doping level.
Fig.5 shows the components with fast decay rates in samples from x = 0 up to 0.6. The lack of a fast decay component in the x = 0.75 and 1 samples indicates that Fe moments do not order in those samples. However, for T below $\approx 14$K, the fit of the LF muon asymmetry requires two non-oscillating amplitudes with distinct relaxation rates, as shown in Fig.6(a) and 6(b). This behavior can be attributed to activities in the Pr sublattice around $T_{Pr} \approx 14$K, where there is noticeable features in both the susceptibility and specific heat measurements (Fig.1 and Ref. ). We will discuss this point again later in section IV-B.
{width="70mm"} {width="70mm"}
{width="75mm"}
To summarize, the temperature dependence of the magnetic volume fraction for all samples is shown in Fig.7. We are able to detect ordering in the FeAs layer up to $x = 0.6$. For the construction of a phase diagram later in this paper, we define $T_{SDW}$ as the temperature at which the magnetic volume fraction is 80%. For $x \leq 0.40$, the magnetic transition temperatures, $T_{SDW}$, can be directly determined from the evolution of the mean magnetic order parameter $<S(T)>\propto B_i (T)$ as a function of temperature, shown in Fig.7(b). The magnetic order parameter $<S(T)>$ has been successfully fit to the phenomenological function $<S(T)>=S(0) [1-(T/T_{SDW})^{2.4}]^{0.24}$, which is found to hold generally for REFeAsO compounds [@Maeter2009]. The values of $T_{SDW}$ determined using the two criteria are consistent to within about 2K.
\[sec:Prordering\] Pr magnetic ordering
---------------------------------------
Previous neutron diffraction measurements did not detect long range ordering (LRO) of Pr in samples with $x \ge 0.1$ [@yiu12]. However, the $\mu SR$ data shows that the muon relaxation rate increases below $T \sim 14$ K in PrFe$_{0.25}$Ru$_{0.75}$AsO (see Fig.6(a)), hinting at a possible short range ordered (SRO) state involving the Pr sublattice. Fig.6(b) shows the LF-$\mu SR$ spectra for PrRuAsO. For x = 0.75, the muon relaxation function ($A_{ZF}$) in zero magnetic field consists of two separate components: $A_f e^{-\lambda_f t} + A_s e^{-\lambda_s t}$, where $\lambda_f = 1.4 \mu$s$^{-1}$ represents the fast decay rate and $\lambda_s = 0.25\mu$s$^{-1}$ represents the slower one.
This empirical fitting function mimics the trend expected for a quasi-static Kubo-Toyabe relaxation[@Kubodyn] with a Lorentzian distribution of internal fields having HWHM $\Delta B = \lambda _f / \gamma_\mu \sim 2$ mT and characterized by a slow dynamics with correlation time $\tau \sim 1/ \lambda_s$. In the static case ($\tau \rightarrow \infty$) $A_f = 2A_s$, but $A_s$ is expected to grow as dynamics sets in.
The onset of longitudinal fields decreases the fast decay rate and increases the slow decay rate in both samples. For x = 0.75, a field of $H = 300$G completely suppresses the faster (static) component of the muon relaxation and only leaves the slower dynamical component. This suggests that the ordering is quasi static. The observed values of $\Delta B$ are consistent with the dipolar field from Pr moments ($\sim 3.6 \mu_B$). Fig.6(b) shows the same for $x = 1$. In this case a longitudinal field $H = 50$G completely suppresses the faster (static) component of the zero field relaxation and the residual dynamics is a little bit faster.
Discussion
==========
{width="82mm"}
Fig.8(a) displays the T vs x phase diagram for PrFe$_{1-x}$Ru$_x$AsO as determined by neutron diffraction and $\mu$SR, with some additional points determined from anomalies detected in heat capacity and resistivity measurements. Using the orthorhombicity criterion described above, the tetragonal to orthorhombic structural transition is detected by neutron scattering for $x$ up to 0.4. Long range magnetic order as detected by neutron diffraction is observed only for $ x \le 0.1$. Conversely, from the $\mu$SR data one detects signs of static moments in the FeAs layers to approximately $x=0.6$, and evidence for Pr moments up to x = 1. To reconcile these observations it must be noted that the neutron measurements of magnetic Bragg peaks are sensitive to spatially dependent long range order. Conversely, muons provide a sensitive probe of local magnetic fields, and therefore may detect local fields associated with static short range order, i.e. with magnetic moments that fluctuate slowly (correlation times less than 1 $\mu$s) and with a short magnetic coherence length (even less than 10 lattice spacings) [@Routier]. Such SRO does not contribute to the magnetic Bragg peaks detected via neutron diffraction.
The muon data for samples with $0.4 \le x \le 0.6$ exhibits overdamped oscillations and the $\mu$SR asymmetry displays a component with a fast decay rate ($\sigma_1 \sim 50\mu s^{-1}$), indicating that the system is still magnetically ordered but the muon spin precessions have become incoherent. Conventionally, this implies that the correlation length of the Fe ordered domains has become shorter than about 10 unit cells [@Routier]. This suggests a transition from LRO to SRO as detected by muons. This is indicated by a dashed vertical line in Fig.8(a) around x = 0.4, which interestingly coincides with the suppression of the structural transition as detected by neutrons. The fact that the progressive reduction of $T_{SDW}$ is closely accompanied by the reduction of $T_S$ hints at significant magneto-elastic coupling in the FeAs layers.
The persistence of static moments as observed by $\mu$SR despite the absence of LRO implied by the neutron measurements may provide a clue to the reason for the failure of superconductivity to appear in PrFe$_{1-x}$Ru$_{x}$AsO.
Notwithstanding the fact that the coexistence of magnetic order and superconductivity is possible in Fe pnictide compounds, superconductivity almost always emerges in these materials when both the structural and magnetic transitions are suppressed. However there is evidence that a necessary condition for the emergence of superconductivity is the persistence of magnetic fluctuations. Some indirect evidence for the latter statement is the complete absence of magnetic fluctuations in the collapsed tetragonal phase observed in 122 family materials[@Soh2013]. One can speculate that mutually uncorrelated but effectively frozen Fe spin clusters might exist in randomly diluted PrFe$_{1-x}$Ru$_{x}$AsO at low temperatures. As a consequence a static moment might be detected by a local probe such as $\mu$SR, and the absence of significant magnetic fluctuations would impede the emergence of a superconducting ground state.
Fig.8(b) shows the evolution of the magnetic order parameter $<S(T\rightarrow 0)>$ vs x as determined by $\mu$SR. The spin dilution caused by Ru/Fe substitution reduces both the ordering temperature $T_{SDW}$ and the moment size gradually. The complete suppression of the SDW ordering is determined to be around x = 0.6 which is similar to the concentration inferred from previous resistivity measurements[@mcguire09a]. Perhaps coincidentally, this is very close to the disruption of superconductivity by Ru/Fe substitution in F-optimally doped 1111 [@Satomi2010; @Sato2010; @Sanna2011; @Sanna2013]. This value may be very significant as discussed below.
A proper description of the magnetism in the Fe-pnictides must account for the fact that the systems are itinerant, however despite this many of the main features can be understood in terms of Hamiltonian models related to local spins. The two dimensional $J_{1}-J_{2}$ model with Heisenberg nearest neighbor ($J_{1}$) and next nearest neighbor ($J_{2}$) interactions on a square lattice[@Chandra1990] exhibits a striped phase for $J_2/J_1 \ge 1/2 $. Moreover, any non-zero coupling to the lattice results in an Ising-nematic transition associated with a rectangular lattice. It can be argued on the basis of symmetry that spin driven Ising nematic order must be accompanied by both a structural phase transition and orbital order [@Fernandes2014], and that nematic order may arise from a correlation driven electronic instability. In any case, the close association of the orthorhombic structural transition and striped antiferromagnetic order in the iron pnictides inspired many applications of the $J_{1}-J_{2}$ model to explain the underlying physics[@Fang2008; @Fernandes2010; @Abrahams2011; @Si2008; @Xu2008; @Fernandes2012]. Caution must be exercised in applying the $J_{1}-J_{2}$ model to iron pnictide systems. Inelastic neutron scattering experiments[@Wysocki2011; @Lynn2009; @Zhao2009; @Diallo2009] and first-principles calculations[@Yin2008] found that fitting observed magnetic excitations using the $J_{1}-J_{2}$ Hamiltonian led to parameters that were physically incompatible with the known ordering scenarios and incapable of giving an acceptable explanation of the response functions. Since then there has been much work[@Hu2012; @Yu2012; @Wysocki2011; @Glasbrenner2014] showing that a minimal effective model must also include a biquadratic exchange term $K(S_1\cdot S_2)^2$ and a small interplane coupling $J_c$.
The biquadratic exchange term must exist in the system and also accounts at least partially for the expected effects of itinerancy. With this Hamiltonian the observed magnetic excitations can be explained with physically reasonable fitted parameters. Within the context of this expanded model, the scenario for magnetic and structural order remain the same as that expected for the $J_{1}-J_{2}$, with some minor renormalizations of the parameters[@Hu2012].
As the moment size is reduced the significance of the biquadratic term is also diminished and one expects that the $J_{1}-J_{2}$ model can provide an even better description of the system. In this context, it is very interesting to consider the vanishing of detectable magnetic order near $x=0.6$. In the simplest scenario for magnetic dilution, magnetic order is expected if the concentration of magnetic ions (here $1-x$) is greater than or equal to the percolation concentration of the lattice. When the interactions are of the same magnitude, the percolation concentration of the $J_{1}-J_{2}$ model should be essentially the same as the square lattice with nearest neighbor and next nearest neighbor connectivity. This model leads to a percolation concentration for magnetic ions almost exactly at the value $1-x=0.4$ [@malarz05]. The fact that this coincides with the disappearance of static magnetism in PrFe$_{1-x}$Ru$_x$AsO is a strong indicator that the core physics of the $J_{1}-J_{2}$ model is at play. We note that the $J_{1}$ only model exhbits percolation at $1-x=0.59$[@malarz05]. The inset of Fig.8(b) compares the doping dependence of $T_{SDW}$ in PrFe$_{1-x}$Ru$_x$AsO to two other systems that also cited the $J_1-J_2$ model for magneto-elastic coupling, namely LaFe$_{1-x}$Ru$_x$AsO[@Bonfa2012] and Li$_2$V$_{1-x}$Ti$_x$SiO$_5$[@Papinutto2005], the latter being an archetype of the $S=1/2$ $J_1-J_2$ square lattice model. The close association of the structural transition with the magnetic order is also explained naturally by the Ising-nematic scenario predicted for localized spins in the $J_1-J_2$ model. If one takes into account for the presence of a magneto-elastic coupling in the system, the structural transition is closely linked to the occurrence of a spin nematic phase at $T_S$ which anticipates the breaking of the rotational symmetry that is associated with the magnetic transition at $T_{SDW}$.
As discussed earlier, $\mu$SR shows that SRO of the Pr moments in PrFe$_{1-x}$Ru$_{x}$AsO persists up to $x = 1$. This is consistent with the anomalies observed in the susceptibility and specific heat measurements around $T_{Pr} \sim 14 K$ (see Fig.1 for the PrRuAsO data ). The ZF-$\mu$SR results indicates a moderately fast depolarization rate in the muon asymmetry around the same temperature. This implies the presence of a broad field distribution generated by a non-collinear arrangement of the Pr moments. LF-$\mu$SR spectra shown in Fig.6 suggests that the magnetic phase is mainly quasi static. Notably, the NTE seen in the c-axis also persists over the entire range of Ru concentrations. Although $T^{Pr}_N$ and $T_{NTE}$ are markedly different, the continuous presence of both throughout the entire doping range leads one to speculate that there is a relation between the ordering of the Pr sublattice and the NTE, and if the latter is driven by magneto-elastic coupling [@Kimber08], the Pr moments are relevant. Indeed, as shown in Fig. 8(b), the doping dependence of the quantities $T_{SDW}$ and $<S(0)>$ in LaFe$_{1-x}$Ru$_{x}$AsO is similar to that seen in PrFe$_{1-x}$Ru$_{x}$AsO, yet the NTE effect has not been observed in LaFe$_{1-x}$Ru$_{x}$AsO [@Martinelli2013].
\[sec:conclusions\]Conclusion
=============================
In summary, we have combined neutron powder diffraction and muon spin relaxation data for the PrFe$_{1-x}$Ru$_{x}$AsO series, completing the study up to $x = 1$. The substitution of diamagnetic Ru for magnetic Fe generates a spin dilution process which gradually suppresses the ordering in the FeAs layers, in which evidence for static moments persists until around $x = 0.6$, the magnetic percolation threshold expected under a localized $J_1-J_2$ model [@Bonfa2012; @malarz05]. The gradual suppression of the magnetic phase is closely followed by the reduction of the structural tetragonal-orthorhombic phase transition temperature. The lattice distortion and the magnetic ordering are found to be strongly coupled, as predicted for pnictides by many theoretical works [@Si2008; @Fang2008; @Xu2008; @Fernandes2010; @Abrahams2011; @Fernandes2012]. The persistence of static moments and possible suppression of magnetic fluctuations may be related to the absence of superconductivity in the system.
In addition, we found that both the magnetic ordering of the Pr sublattice and the negative thermal expansion of the c-axis phenomena persist up to $x = 1$. We speculate that the abnormal thermal expansion behavior can be linked to the magneto-elastic coupling within the Pr sublattice, which survives despite of the disruption of the ordering of Fe moments.
\[sec:level1\]Acknowledgements
==============================
The research reported here utilized the $\mu$SR facilities at the Paul Scherrer Institute, Villigen, Switzerland, and neutron scattering facilities at the Spallation Neutron Source, Oak Ridge National Laboratory (ORNL), which is sponsored by the Scientific User Facilities Division of the Office of Science, Basic Energy Sciences, US Department of Energy (BES DOE). We are grateful to A. Amato and H. Luetkens of the Swiss Muon Source group for technical support. We would like to thank A. A. Aczel and T. J. Williams of ORNL for valuable conversations. A.H. and S.E.N. were supported by the Scientific User Facilities Division of BES DOE. M.A.M. was supported by the Materials Sciences and Engineering Division of BES DOE. S.S. and P.C. acknowledge the financial support of Fondazione Cariplo (Research Grant n.2011-0266). R.D.R., P.B., P.C. and S.S. acknowledge partial support of PRIN2012 project 2012X3YFZ2. Y.Y. was supported by the BES DOE, through the EPSCoR, Grant No. DE-FG02-08ER46528.
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abstract: 'A detailed description of an isoperibol calorimeter for temperatures between 0.05 and 4 K is presented. The proposed setup can provide absolute values of the heat capacity $C$ of small samples (typically 1 mg). The extremely simple design of the sample platform, based on a sapphire substrate, and the experimental setup, which makes use only of a lock-in amplifier and a temperature controller, make the construction of such a calorimeter easy and inexpensive. The thermal-relaxation method is employed, which utilizes a permanent thermal link $k$ between the sample platform and the low-temperature bath. The temperature dependence of $k(T)$ is shown for several platforms throughout the entire temperature range: $k(T)/T$ is nearly constant down to 1 K, where it starts to decrease smoothly. The observed behavior is thoroughly explained by considering the thermal resistances of the platform constituents. A comparison between the values of $k(T)/T$ for platforms based on sapphire and on silver is presented where no significant difference has been observed. Each platform can be assembled to have a particular value of $k/T$ at 1 K. Since the sample relaxation time $\tau \sim C/k$, $k(T)$ can be adjusted to $C(T)$ to give a reasonably fast measuring time: Here, it is demonstrated how this calorimeter can be used in so-called single-shot refrigerators ($^{3}$He or demagnetization cryostats), where the time for a single measurement is limited. In addition, it can be used in moderate magnetic fields $B \leq 10$ T, because the platform constituents are weakly field dependent.'
author:
- 'M. Brando'
title: 'Development of a relaxation calorimeter for temperatures between 0.05 and 4 K'
---
\[sec:introduction\]Introduction
================================
It is often necessary to work with small samples of only few milligrams when studying the thermal properties of solid materials at very low temperatures, as they are easier to cool down. The first modern low-temperature semi-adiabatic calorimeter was developed by Eucken and Nernst (1910), but this technique is restricted to temperatures $T > 2$ K and a minimum sample mass of about $100$ mg.[@eucken-1909; @nernst-1910] For temperatures $T < 2$ K, calorimeters which employ adiabatic techniques, such as pulse or continuous warming methods,[@hemminger-1984; @cochran-1966; @pinel-1972] require heat switches to allow the sample to be cooled to the lowest temperature and to provide adequate thermal isolation during the measurement. Mechanical heat switches are not suitable when working with small samples at temperatures below 1 K because of the large heating effects induced by friction, while superconducting heat switches cannot guarantee adequate thermal isolation above 1 K. Thus, improvements have been made to refine experimental techniques which utilize a permanent weak thermal link $k$ between the sample and the low-temperature bath (isoperbol calorimeters), eliminating the need for a heat switch, while providing a reasonably short cooling time.[@bachmann-1972; @schutz-1974; @schwall-1975; @gmelin-1979; @stewart-1983; @ogata-1984]\
With the development of lock-in-amplifiers, such non-adiabatic microcalorimetry techniques allow measuring the heat capacity of bulk crystals of a mass of typically 1 mg. Experimentalists favour two principal methods: the steady-state ac heating method (AC) [@sullivan-1968; @eichler-1979; @schmiedeshoff-1987] and the thermal-relaxation (TR) method.[@bachmann-1972; @schutz-1974; @schwall-1975; @regelsberger-1986] Recently, a semi-adiabatic compensated heat-pulse (CHP) calorimeter has been developed, which requires an even weaker thermal link to the bath, compared to the TR method, and allows high precision data within a short measuring time.[@fisher-1995; @wilhelm-2004] The AC method has the advantage of providing heat capacity data as a continuous function of temperature, and it is therefore suitable, e.g., for studying phase transitions. Moreover, it can be used to carry out measurements at constant $T$, while continuously varying other external parameters, such as magnetic field and pressure. This technique does not, however, provide absolute values of the sample heat capacity $C$. This has to be determined in a different way. The TR method, on the other hand, provides heat-capacity data of high accuracy as it measures sample relaxation time $\tau \sim C/k$ while maintaining both, bath temperature and external parameters, constant. The only drawback of this method is that it gets inappropriate close to $1^{st}$ order phase transitions. The CHP method is fast, but restricted to measurements with increasing temperature and does, therefore, not permit measurements at constant $T$ while continuously varying other parameters. A TR isoperibol calorimeter has been chosen because of its versatility and the necessity of making precise measurements of the specific heat capacity under the influence of external parameters at constant $T$.\
To allow measurements in the range $0.05 \leq T \leq 4$ K on a large variety of small samples of insulating or metallic materials (as metal-oxides, 3d-electron metals, heavy-fermions, spin glasses etc.), the sample platform was constructed with a simple design, developed by G. R. Stewart’s group,[@sievers-1994] which makes it possible to employ the TR method with high precision, while substantially cutting down costs on the necessary construction materials. In this paper, the construction of such simple and rather inexpensive sample platforms, based on a sapphire substrate, is described, along with the measurement setup, which makes use of only a lock-in amplifier and a temperature controller. Although similar platform designs have already been used by other groups, the temperature dependence of their thermal conductance $k(T)$ is only known for $T > 2$ K.[@klemens-1962; @greene-1972] In section \[t2:s-therm-conductance\], measurements of $k(T)$ vs. $T$ are shown between 0.05 and 4 K for several platforms. These data can be well fitted by a theoretical model based on the thermal resistances of the platform constituents and on simple geometrical considerations. Experiments were carried out with platforms based on a sapphire substrate (insulating) and on silver (metallic) to investigate the effect of different substrate materials on $k(T)$. No significant difference between the two performances can be observed.\
Since $\tau \sim C/k$, a large sample mass, or a large $C$, could imply a measuring time of the order of many hours. For this reason, the TR technique is usually mounted in dilution refrigerators, where the temperature can be kept constant permanently. It is demonstrated here that it is also possible to use this kind of technique in so-called single-shot refrigerators ($^{3}$He or demagnetization cryostats), where the measuring time is limited, if the platform material is properly chosen.\
\[sec:principle-of-operation\]Principle of operation
====================================================
In the TR method, the sample is placed on a platform which contains a heater and a thermometer. The thermal link between sample and platform is $k_{2}$. The platform is connected through a small heat link $k_{1}$ to a reservoir at temperature $T_{0}$.
![(Color online) The RT experiment. : One-dimensional heat-flow model with a poor thermal contact between sample and platform ($\tau_{2}$ effect): $C_{p}$ and $C_{a}$ are the heat capacities of the sample and platform, respectively. : Corresponding thermal behavior of the platform thermometer $T_{a}(t)$ during and after the heating pulse.[]{data-label="fig:model"}](fig1){width="48.00000%"}
After the bath and the sample have reached a constant temperature $T_{0}$, a well defined constant heating pulse with power $P_{0}$ and duration $t_{1}-t_{0}$ is applied to the platform, until a steady-state temperature $T_{1}$ is reached. The heater power is then turned off and the temperature decays to $T_{0}$ with a time constant $\tau_{1}\approx (C_{p}+C_{a})/k_{1}$. The principle of the experiment is shown in Fig. \[fig:model\], where $C_{p}$ is the heat capacity of the sample and $C_{a}$ is the heat capacity of the platform.\
To analyse this process in detail, it will be assumed assume that the thermal contact between the sample and the platform is good, but not ideal (finite values for $k_{2}$), while the internal thermal conductivity of the platform will be considered ideal. With these assumptions the following one-dimensional heat-flow equations can be solved: $$\label{eq:heat-balance-0}
\left\{ \begin{array}{l}
P(t)=C_{a}\dot{T}_{a}(t)+k_{1}[T_{a}(t)-T_{0}]+k_{2}[T_{a}(t)-T_{p}(t)]\\
\\
C_{p}\dot{T}_{p}(t)=k_{2}[T_{a}(t)-T_{p}(t)]\end{array} \right.$$ where $P(t)$ is the power applied on the platform, $T_{p}(t)$, $T_{a}(t)$ and $T_{0}$ are the sample, platform and bath temperatures; $k_{1}$ and $k_{2}$ are the thermal conductances between platform and bath, and between sample and platform respectively. The heat is flowing in one direction only. If the thermal conductivity of either the sample or the link between sample and substrate is small, compared to that of the link to the bath, the relaxation curves are characterized by a second relaxation time $\tau_{2}$ between sample and platform:[@shepherd-1985] These cooling curves show an abnormally high initial slope compared to the rest of the decay (see Fig. \[fig:model\] for $t_{1} \leq t \leq t_{2}$). If the sample is a good thermal conductor, but the thermal contact with the substrate is poor, the $\tau_{2}$ effect is called “lumped”.[@shepherd-1985; @brando-2000]
![Measurement exhibiting a “lumped” $\tau_{2}$ effect: The relaxation time $\tau_{2}$ decreases with increasing temperature.[@brando-2000][]{data-label="t2:f-lumpedtau2"}](fig2.eps){width="48.00000%"}
![Measurement exhibiting a “distributed” $\tau_{2}$ effect: The relaxation time $\tau_{2}$ increases with increasing temperature.[@brando-2000][]{data-label="t2:f-distrtau2"}](fig3.eps){width="48.00000%"}
In this case the relaxation time $\tau_{2}$ decreases with increasing temperature, as shown in Fig. \[t2:f-lumpedtau2\] for a single crystal of Pr$_{0.65}$Ca$_{0.28}$Sr$_{0.07}$MnO$_{3}$.\
By applying a constant power $P(t)=P_{0}$, and maintaining a maximum temperature rise $\Delta T=T_{1}-T_{0}$ below $2-3\%$, the heat capacity of the sample can be calculated exactly by solving the differential equations \[eq:heat-balance-0\] (see Appendix), where the decay in temperature can be represented by a curve consisting of the sum of two exponentials with different time constants $\tau_{1}$ and $\tau_{2}$: $$T_{a}(t)=T_{1}-A_{1}e^{-t/\tau_{1}}-A_{2}e^{-t/\tau_{2}}~.\\$$ The solution yields: $$\label{eq:solution}
C_{p}=k_{1}\tau_{1}\left (1-\frac{k_{1}\tau_{2}}{C_{a}}\right ) +k_{1}\tau_{2}-C_{a}~.$$ The complete calculation is given in the Appendix.\
From the experimental value of the heat capacity we have to subtract the heat capacity of the platform $C_{a}$ to obtain $C_{p}$. Since $C_{a}$ can be deduced from preliminary heat-capacity measurements without sample, and since the thermal relaxation time between the sample and the substrate can be calculated from $$\label{eq:tau2}
\tau_{2}=\frac{A_{2}\tau_{1}C_{a}}{(A_{1}+A_{2})\tau_{1}k_{1}-A_{1}C_{a}}~,$$ a measure of $A_{1}$, $A_{2}$, $k_{1}$ and $\tau_{1}$ constitutes a measure of $C_{p}$. The thermal conductance $k_{1}$ does not have to be measured every time, but only once, by calibrating the platform, using $$\label{eq:k1}
k_{1}=P_{0}/(T_{1}-T_{0})=P_{0}/\Delta T~.$$ Details of this calibration process are given in Sec. \[t2:s-therm-conductance\]. The parameters $A_{1}$, $A_{2}$ and $\tau_{1}$ can be determined experimentally by analysing the $log(T_{a}(t))$ vs. $t$ plots of the exponential decay. The main slope of the curve corresponds to $\tau_{1}$, and the initial drop measures the $\tau_{2}$ contribution: With decreasing $\tau_{2}$ effect the drop vanishes.\
In case the sample has a poor thermal conductivity (as in insulators), different regions of the sample will be at significantly different temperatures. Such a phenomenon is called “distributed” $\tau_{2}$ effect. [@shepherd-1985] The solution of the problem is a sum of exponential decays. The time constants of these are given by the solution of a transcendental equation. The calculation of the heat capacity is difficult, but for small effects this contribution can be corrected with the same method used for the “lumped” $\tau_{2}$ effect. As shown in Fig. \[t2:f-distrtau2\] for a single crystal of La$_{0.8}$Sr$_{0.2}$MnO$_{3}$, the values of $\tau_{2}$ increase with increasing temperature.[@brando-2000] If the effect is significant, i.e. if the rate $\tau_{1}/\tau_{2}\rightarrow 1$, this method cannot be used.\
Assuming an ideal thermal contact between sample and substrate ($k_{2}\rightarrow \infty$, $\tau_{2}=0$), the model leads to: $$\label{eq:solutions}
\left\{ \begin{array}{l}
T_{a}(t)=T_{1}-A_{1}e^{-t/\tau_{1}}\\
\\
C_{p}=k_{1}\tau_{1}-C_{a}\\
\\
P_{0}=k_{1}(T_{1}-T_{0})=k_{1}A_{1}.\\
\end{array} \right.$$ One of the principal advantages of the TR method, e.g. when comparing it with the AC method, is that the corrections for the $\tau_{2}$ effect can be calculated exactly. This is usually valid for $T > 1$, since the “lumped” $\tau_{2}$ effect vanishes with increasing temperature (cf. Fig. \[t2:f-lumpedtau2\]). For samples with masses lower than 1 mg, the platform heat capacity $C_{a}$ provides the dominant source of error for $C_{p}$. This explains why some effort must be applied to building platforms with small $C_{a}$ values at low temperature, e.g. using sapphire single crystals as platform substrates.
\[experimental-setup\]Experimental setup
========================================
The apparatus for all heat-capacity experiments is shown in Fig. \[fig:design\]: The illustrated design consists of a copper ring-shaped platform holder in which four electrically isolated copper pins are inserted and fixed by a thermally-conducting epoxy cement. The ring is permanently screwed into the low-temperature stage of the cryostat which represents the thermal bath at $T_{0}$. The sample platform consists of a substrate, a heater, a thermometer and bonding silver epoxy.
![(Color online) General design of the experimental apparatus. The platform is suspended and held mechanically by four Pt wires, which are bonded to electrically isolated Cu-pins inserted in a Cu-ring. The ring is permanently screwed into the low-temperature stage of the cryostat.[]{data-label="fig:design"}](fig4.eps){width="35.00000%"}
It is held mechanically in horizontal position by four platinum wires, which are bonded to the copper pins. The wires provide a well defined thermal connection (essentially $k_{1}$) to the isothermal ring, as well as electrical connections for the heater and the temperature sensor attached to the lower side of the platform. In some cases, thin gold wires are used to electrically connect the thermometer with the Pt wires.\
The data that will be shown hereafter have been collected from measurements carried out with five sample platforms, labelled PLS-1 to 5, mounted in two different systems: a $^{3}$He *Oxford Instruments* Heliox $2^{VL}$ cryostat for $0.3 \leq T \leq 4$ K and a *Cambridge Magnetic Refrigeration* mFridge for $0.05 \leq T \leq 4$ K.
The measurement platforms
-------------------------
The construction materials are slightly different for each platform; they are listed in Tab. \[tab:materials\]. A collection of selected images for three of the platforms are shown in Fig. \[fig:pls0\]. The substrate for PLS-1 is a high quality silver (5N) substrate of $100~\mu$m thickness, while for the other platforms sapphire single-crystals of 6 mm diameter and $200~\mu$m thickness were used.\
![(Color online) Pictures of three of the five sample platforms utilized for this report. The black chips are the RuO$_{2}$ sensors, the green one is the heater chip. On PLS-3, the gray large surface is the sputtered Cr-film heater, which is a squared path on the PLS-5 sapphire disc. On PLS-3 a very small (less than 1 mm$^{3}$) Cernox type sensor is attached to the sapphire surface and connected with tiny gold wires (used also in PLS-5).[]{data-label="fig:pls0"}](fig5.eps){width="45.00000%"}
*Lake Shore* Cernox type sensors or polished *Bourns* $2$ k$\Omega$ ruthenium-oxide (RuO$_{2}$) resistors were utilized as thermometers (cf. Fig. \[fig:pls0\]). The platform temperature $T_{a}$ can be estimated by measuring their resistance. They guarantee high sensitivity below 4 K: Decreasing the temperature down to 0.05 K, the resistance of the RuO$_{2}$ sensors increases from $2$ k$\Omega$ up to values higher than 40 k$\Omega$. In addition, the heat capacity contribution to $C_{a}$ of these thermometers is very small (cf. section \[t2:s-addenda-calibration\]).\
Polished commercial thin film chips (mass $\approx 1$ mg, R $\approx 10$ k$\Omega$) were used as heaters for the first two platforms, while chromium film resistors (with different geometry and resistance of the order of k$\Omega$) were sputtered on one side of the other sapphire discs as heater.
[lllrr]{} Platform & Substrate & Wires & Wire-Diameter & Heater\
\
PLS-1 & Ag & Pt$_{0.9}$Ir$_{0.1}$ & 50$\mu$m & chip\
PLS-2 & Al$_{2}$O$_{3}$ & Pt$_{0.9}$Ir$_{0.1}$ & 50$\mu$m & chip\
PLS-3 & Al$_{2}$O$_{3}$ & Pt$_{0.9}$Rh$_{0.1}$ & 50$\mu$m & Cr-film\
PLS-4 & Al$_{2}$O$_{3}$ & Pt$_{0.9}$Rh$_{0.1}$ & 125$\mu$m & Cr-film\
PLS-5 & Al$_{2}$O$_{3}$ & Pt$_{0.9}$Ir$_{0.1}$ & 50$\mu$m & Cr-film\
\[tab:materials\]
The resistance of both kinds of heaters has a very weak temperature dependence and is practically constant below 4 K.\
The Pt lead wires are made of Ir- or Rh-doped platinum with 50$\mu$m diameter (125$\mu$m for the PLS-4), and provide the well defined thermal link $k_{1}$ to the isothermal ring, as well as electrical connections for the heater and sensor. The Cu-pins in PLS-5 were bent slightly towards the center of the ring to decrease the length of the Pt$_{0.9}$Ir$_{0.1}$ wires, and to thus increase $k_{1}$ (cf. Fig. \[fig:KTvsT\]). By varying the wire length, their thickness or their dopant concentration, the value of $k_{1}$ can obviously be tuned, and, considering $\tau_{1} \sim C_{p}/k_{1}$, be adjusted to what time is available for the experiment.\
On some of the sapphire substrates, tiny gold wires (25$\mu$m diameter) were spot-welded to the sensor as well as directly to the Pt wires, to reduce $C_{a}$. All constituents were fixed with conducting silver-epoxy cement (*Polytec* Epo-Tex HB1LV). This arrangement ensures an excellent thermal connection between thermometer, heater and substrate. The sample is attached to the back of the sapphire substrate with thermally conductive grease (Apiezon N).[@swenson-1999] In the silver platform PLS-1, the wires were directly attached with the same epoxy to the heater and sensor contacts. Soldering was avoided because of the risk of having thin superconducting contacts, which could dramatically reduce the thermal conductance between platform and Cu-ring.
The measurement systems
-----------------------
The measurement system is rather simple and requires just two instruments: A temperature controller to measure and stabilize $T_{0}$ at a thermometer positioned at the bottom of the low-temperature stage and a lock-in amplifier to measure $T_{a}(t)$ and supply the heat pulses.\
In the Heliox $2^{VL}$ cryostat, $T_{0}$ was measured and controlled with a *RV-Elektroniikka OY* AVS-47 ac resistance bridge and a TS-530 temperature controller. In the mFridge, $T_{0}$ was read by a *Lakeshore* model 340. The direct-temperature-control (DTC) routine, which drives the demagnatisation magnet, seemed the most elegant solution of stabilizing it. In both cases it was possible to stabilize the temperature within 0.2%.\
On account of low $\tau_{1}$ and $\tau_{2}$ (e.g. 1 s and 0.05 s, respectively), the voltage signal on the platform sensor has to be read quickly, which is why a lock-in amplifier has been used for measuring the platform thermometer. $T_{a}(t)$ was measured by a *Signal Recovery* 7265 lock-in amplifier connected to the previously calibrated platform sensor through a 10 M$\Omega$ resistance. The lock-in frequency $f$ was set at values close to 100 Hz. It is clear that the lock-in time constant has to be set at a lower value than our $\tau_{1}$ and be comparable to $\tau_{2}$, in order to permit measurement. It can be verified that $\tau_{lock-in}\approx \tau_{1}/40$ provides a good response of the lock-in amplifier. The applied voltage varied from 0.1 V at 50 mK to about 2 V at 3 K to prevent the sensor from self heating and to garantee a good signal reading. The heat pulse was given by the same lock-in amplifier, connecting the DAC output to the platform heater through a 200 k$\Omega$ resistance. High-speed data acquisition was achieved by directly monitoring the output signal with the lock-in buffer option, set at 10 ms. The error due to data noise was successfully reduced to a considerably lower level than the one due to thermal fluctuations.\
Although the signal offset was large, compared to its changing due to the heat pulse, the quality of the data has been improved by adjusting the lock-in parameters before every single measurement. It was also possible to reduce the signal offset to values close to zero by inserting another RuO$_2$ thermometer on the Cu ring and, with both sensors, building a standard ac Wheatstone bridge, which is driven directly by the lock-in amplifier reference oscillator. Doing that, the resolution of the voltage reading increased by a factor of ten, allowing the bridge to operate now at higher frequency, optimally at $f \approx 1500$ Hz. Values for $\tau_{2}$ lower than 1 ms have thus been detected.\
As the platform thermometer is calibrated, a reading of $C_{p}$ can always be obtained, along with a measurement of $k_{1}$, with the configuration described above. A major advantages of utilizing the TR method, is that the thermometer on the sample platform is only necessary for measuring relaxation time constants and, therefore, it does not always have to be calibrated. In systems with various measurement platforms assembled together, or with two different thermometers (for separate temperature ranges) on the same platform, this can be a very convenient feature. The thermal conductance $k_{1}$ for each platform has then to be measured separately once (see section \[t2:s-therm-conductance\]) and the base temperature $T_{0}$ may be detected by the low-temperature stage thermometer of the cryostat.\
\[experimental-results\]Experimental results
============================================
Measurement of the platforms thermal conductance {#t2:s-therm-conductance}
------------------------------------------------
The thermal conductance $k_{1}(T)$ was determined by measuring the supplied power $P_{0}$ and the temperature difference $\Delta T = (T_{1}-T_{0})$, as indicated in Eq. \[eq:k1\] . $\Delta T$ was kept below 3 %. As the thermometers of all platforms had previously been calibrated, $k_{1}(T)$ was measured each time, along with the specific heat.\
With a suitable choice of lead wires, the heat link can be controlled and varied: The upper frame of Fig. \[fig:KTvsT\] shows the thermal conductance for the PLS-3 with Pt$_{0.9}$Rh$_{0.1}$ lead wires of $50~\mu$m diameter, and for the PLS-4 with $125~\mu$m of diameter. A relaxation time of these platforms (without a sample) was measured at less than 400 ms even for the lowest temperatures. It is useful to note that, according to Eq. \[eq:solution\], the first platform allows measuring smaller samples, because of its smaller $k_{1}$.\
Instead of very pure platinum wires, Rh- and Ir-doped ones were chosen, mainly for two reasons: Their robustness and the lower thermal conductivity values.
![(Color online) Upper frame: Comparison of the thermal conductance for PLS-3 and PLS-4 with two different sizes for the diameter of the lead wires: $50~\mu$m and $125~\mu$m, respectively. The results for PLS-2 in a magnetic field of 1 T are also shown. Lower frame: Measurement of the thermal conductance for platform PLS-1,-2,-3 and PLS-5. The black lines represent the fit curves resulting from equation \[t2:s-thermal-cond-model\]. The fit parameters are quoted in Tab. \[tab:parameters\].[]{data-label="fig:KTvsT"}](fig6.eps){width="48.00000%"}
In fact, the thermal conductivity of a metal at low temperature is strongly influenced by impurities in the material, which tend to decrease $k_{1}$. In addition to that, the thermal conductivity of these wires is not very sensitive to magnetic fields of a relative magnitude ($B\leq 10$ T). This allows measurements of $Cp(T)$ vs. $T$ in magnetic field with the same platforms. A measurement of $k_{1}(T)$ vs. $T$ for the PLS-2 in $B = 1$ T leads to the same results as in zero field. To avoid the overlapping with the data in zero field, shown in the lower frame of Fig. \[fig:KTvsT\], this measurement is plotted in the upper frame of the same figure.\
The lower frame of Fig. \[fig:KTvsT\] shows the thermal conductance for PLS-1,-2, and -5, down to 50 mK and, for PLS-3 down to 300 mK. In all curves, $k_{1}(T)/T$ begins to decrease below 1 K; in PLS-5, this drop is very pronounced. With a simple theoretical model it is possible to explain this effect and to calculate the thermal conductance of the Pt and Au wires, as well as that of the platform substrate. Not only the sapphire disc (described here as substrate) is relevant for this model, but also the insulating substrates of the sensors and all parts where there is a thermal boundary resistance between metal and insulator. We need to consider that the low-temperature thermal conductivity of a metal is primarily due to conduction electrons, and is known to follow a linear-in-$T$ dependence for $T\rightarrow 0$. It is larger than the thermal conductance of an insulator, which is solely due to lattice phonons and follows a $T^{3}$-power law for $T\leq \Theta_{D}$ ($\Theta_{D}$ is the Debye temperature).
![Schematic representation of the thermal resistances on the sample platform: R$_{S}$, R$_{Au}$ and R$_{Pt}$ are the resistances of the substrate, of the gold and platinum wires, respectively.[]{data-label="t2:k-model"}](fig7.eps){width="45.00000%"}
If we suppose that the heat originates from the center of the platform and flows simultaneously through the Au wires and the substrate of the platform, and after that through the platinum wires, a simple schematic representation of all thermal resistances can be drawn as in Fig. \[t2:k-model\]. The numbers in the picture indicate the four wires which are considered identical. The thermal resistance along every wire is: $$R_{tot}=R_{Pt}+\frac{R_{S}R_{Au}}{R_{S}+R_{Au}}$$ and the relative thermal conductance is: $$k=\frac{k_{Pt}(k_{S}+k_{Au})}{k_{Pt}+k_{S}+k_{Au}}~.$$ Considering the following temperature dependence for the gold and platinum wires and for the substrate contribution: $$\left\{ \begin{array}{l}
k_{Au}=aT+bT^{3}\\
k_{Pt}=a_{1}T+b_{1}T^{3}\\
k_{S}=b_{2}T^{3}
\end{array}\right.$$ the temperature dependence of $K$ can be written as: $$\label{t2:s-thermal-cond-model}
\frac{k(T)}{T}=\frac{aa_{1}+(a_{1}b_{2}+a_{1}b+ab_{1})T^{2}+(b_{1}b_{2}+bb_{1})T^{4}}{(a+a_{1})+(b+b_{1}+b_{2})T^{2}}~.$$ Since we have four wires, $k(T)=4k_{wire}(T)$. The derived function fits well the experimental data for PLS-1 to -4 (cf. Fig. \[fig:KTvsT\]): The resulting parameters are shown in Tab. \[tab:parameters\].\
Taking the fit parameters for PLS-5 at 1 K, and supposing that the total length of the four platinum wires is 2 cm, the corresponding thermal conductivity of every wire is about 53 mW/Kcm. For the sapphire thermal conductivity $K_{Sh}$, we can perform a similar calculation, assuming that $b\approx 20~\mu$W/K$^{4}$ at 1 K, and that the length of four conducting paths from the center of the platform to its edges is $l=0.3\times 4$ cm, with a flow area $A=200~\mu$m$\times 1$ mm: We obtain a reasonable value of $K_{Sh}=bl/A=12$ mW/Kcm.[@pobell-1996]\
This model cannot explain the experimental data above 2.4 K for PLS-5. Since the sapphire substrate and the heater design of this platform are different from the others, it is possible that the measured data above 2.4 K are subject to a systematic error. One possibility is that the well polished surface of the sapphire does not allow the epoxy to stick well and creates large thermal boundary resistances. It might also be possible to bond the wires using the technique described in Ref. [@varmazis-1978] to avoid this problem, and to afterwards fix them with epoxy. If we look at the heater shape, on the other hand, we can see that it is closer to the sapphire substrate border than to the thermometer. This design was meant to create a better steady state on the entire platform, but above 2.4 K it may cause the thermometer to be at a lower temperature than the heater. Specific-heat measurements on gold samples also confirm that the data for $k_{1}(T)$ above 2.4 K are not correct.
[cccccccc]{} Platform & $a$ & $b$ & $a_1$ & $b_1$ & $b_2$\
& ($\mu$W/K$^2$) & ($\mu$W/K$^4$) & ($\mu$W/K$^2$)& ($n$W/K$^4$) & ($\mu$W/K$^4$)\
\
PLS-1 & 4.75 & 0 & 0.80 & 4.19 & 20.24\
PLS-2 & 1.01 & 0 & 0.26 & 1.69 & 13.30\
PLS-3 & 2.28 & 0 & 0.40 & 1.87 & 7.75\
PLS-4 & 4.54 & 0 & 1.54 & 20.36 & 28.80\
PLS-5 & 0.98 & 0 & 0.52 & 0 & 18.08\
Although the reason for the behavior of PLS-5 has not yet been understood, the fact that the silver platform PLS-1 behaves exactly like PLS-2, -3 and -4 indicates that such behavior might derive from the sapphire substrate.\
Comparing the temperature behavior and the performance of the silver platform with that of the other sapphire platforms, it becomes clear that a sapphire substrate can be used perfectly well instead of a metallic silver one for this kind of heat-capacity measurements (at least down to 50 mK).
Calibration {#t2:s-addenda-calibration}
-----------
The advantage of our platform design is that its contribution $C_{a}$ to the total heat capacity is small. It consists only of the substrate (sapphire was chosen because of its low specific heat at low temperatures), a small amount of grease, tiny polished thermometer and heater chips, and $1/3$ of the lead wires (cf. Ref.[@bachmann-1972]).
Accurate values for the specific heat of every single platform constituent can be determined separately or found in literature. [@swenson-1999; @pobell-1996] Moreover, it is possible to use very small samples and still keep the platform heat capacity well below the sample’s one: e.g. the platform PLS-3 consists of a 34 mg sapphire substrate, a 4 mg Cernox thermometer, 0.678 mg silver-epoxy cement, 0.711 mg gold wires and 6.2 mg Pt$_{0.9}$Rh$_{0.1}$ wires for an amount of heat capacity at 1 K of about $1.7231\times 10^{-7}$ J/K. This value is about 3.5 times lower than the heat capacity of 100 mg gold at 1 K, which is about $5.84\times 10^{-7}$ J/K.\
Since the specific heat of the Pt$_{0.9}$Ir$_{0.1}$ wires was unknown, two calibration measurements were carried out with two gold samples (purity of 99.99%) of different weight, 59.1 mg and 72.6 mg. After having subtracted the literature data of Ref. [@martin-1973] for the measured 59.1 mg gold sample, the total heat capacity of the platform is obtained: Its behavior for PLS-2 and PLS-5 can be seen in Fig. \[t2:f-addenda\]. For $1.5 \leq T \leq 4$ K, $C_{a}/T$ versus $T^{2}$ is linear, as expected, but below 1.5 K it increases sharply. This effect is very common and due to magnetic impurities in the platform constituents (e.g. in the alumina substrate of the RO$_{2}$ chips) and in the Pt wires. [@pobell-1996; @ho-1965] This contribution can be fitted well by adding a $T^{-2}$ Schottky factor to the expected fit function, corrected by a constant $d$, which ideally should be zero: $$\label{t2:e-addenda-function}
\frac{C_{a}}{T}=\gamma+\beta T^{2} + \frac{\delta}{d+T^{3}}~.$$ The results of the fits are shown as black lines in Fig. \[t2:f-addenda\]. There is a big difference between the heat capacities of the two platforms PLS-2 and PLS-5, as in the PLS-2 a chip has been attached as heater instead of a Cr-film,
and a larger amount of silver epoxy has been used. However, even on PLS-2, the total heat capacity of the platform remains very small when compared to the sample.\
Fig. \[t2:f-gold\] shows the measurement of the second gold sample on PLS-2, after having substracted the platform heat capacity $C_{a}$, in $C_{p}/T$ versus $T^{2}$. The linear fit results in a Sommerfeld coefficient $\gamma = 0.70$ mJ/K$^{2}$mol, very close to the one measured in Ref. [@martin-1973] (0.69 mJ/K$^{2}$mol). The total error is displayed in the inset; it is less than 2% within the whole temperature range. It is mainly due to the instability of the temperature at all points. Applications of the technique described in this article can be seen, e.g. in Ref. [@knebel-1999; @brando-2008], where samples with masses between 0.2 and 10 mg have been used.
\[conclusions\]Conclusions
==========================
A large number of techniques have been developed to measure the heat capacity of small samples at low temperatures. This paper presented the progress made in improving the TR method. The proposed inexpensive platform design and the simple measurement procedure, which makes use of only two instruments, offer a new prospective for the low-temperature laboratories which suffer of limited funding. The distinct advantage of the method described here, is that it simultaneously provides high-precision measurements, absolute heat capacity values, a certain flexibility in varying temperature or external parameters, while performing the measurements at constant $T$.\
There are limitations of the TR method, including the fact that the temperature must be kept constant during measurement time, which is why this method has not been used in single-shot refrigerators. In this article it has been demonstrated that this kind of method can be used down to 0.05 K in single-shot refrigerators, too, i.e. $^{3}$He and demagnetization cryostats, provided that materials for the construction of the sample platforms are properly chosen. Detailed information on how to build sample platforms utilizing inexpensive and effective materials has been given, along with the calibration results of the platform heat capacities.\
Platforms with sapphire and silver substrates were used. For the first time, measurements of thermal conductance $k_{1}(T)$ vs. $T$ for such platforms have been shown below 2 K. A simple theoretical understanding of its behavior has been proposed. Our results indicate that the behavior of the platforms with different substrates are comparable across the entire range of temperature investigated.\
Finally, measurements of the platform thermal conductance $k_{1}(T)$ were carried out in magnetic field and it could been observed that the magnetic field has no influence on the $k_{1}(T)$ vs. $T$ behavior. This is due to the fact that the thermal conductivity of the lead wires is almost field independent.
I would like to acknowledge F. M. Grosche, J. Hemberger, G. Knebel, F. Mayr, M. Nicklas, E.-W. Scheidt, W. Trinkl and R. Wehn for their precious help and suggestions, C. Klingner, T. Lühmann, M. Sugrue and N. Rothacher for having examined the manuscript.
Solution of the one-dimentional model
=====================================
The thermal equations for the model depicted in Fig. \[fig:model\] can be written as:[@shepherd-1985] $$\label{eq:heat-balance}
\left\{ \begin{array}{l}
P(t)=C_{a}\dot{T}_{a}(t)+k_{1}[T_{a}(t)-T_{0}]+k_{2}[T_{a}(t)-T_{p}(t)]\\
\\
C_{p}\dot{T}_{p}(t)=k_{2}[T_{a}(t)-T_{p}(t)]\end{array} \right.$$ Rearranging yields: $$\left \{ \begin{array}{l}
k_{2}T_{p}(t)=-[P(t)+k_{1}T_{0}]+(k_{1}+k_{2})T_{a}(t)+C_{a}\dot{T}_{a}(t)\\
\\
C_{p}\dot{T}_{p}(t)=k_{2}[T_{a}(t)-T_{p}(t)]\end{array} \right.$$ Considering $P(t)=P_{0}=const$ for $t_{0}\leq t < t_{1}$ and $P(t)=0$ for $t_{1} \leq t < t_{2}$, we derive the first equation: $$\left\{ \begin{array}{l}
k_{2}\dot{T}_{p}(t)=(k_{1}+k_{2})\dot{T}_{a}(t)+C_{a}\ddot{T}_{a}(t)\\
\\
C_{p}\dot{T}_{p}(t)=k_{2}T_{a}(t)-k_{2}T_{p}(t)\end{array} \right.$$ Substituting $T_{p}(t)$ and $\dot{T}_{p}(t)$ into the second equation we obtain: $$\label{eq:main-equation}
\frac{C_{p}C_{a}}{k_{1}k_{2}}\ddot{T}_{a}(t)+\left [\frac{C_{p}+C_{a}}{k_{1}}+\frac{C_{p}}{k_{2}}\right ]\dot{T}_{a}(t)+T_{a}(t)=T_{0}+\frac{P(t)}{k_{1}}$$ If a power $P_{0}$ is applied between $t_{0}$ and $t_{1}$, the platform temperature $T_{a}$ will rise to $T_{1}$ (see Fig. \[fig:model\]), according to the following relation: $$\label{eq:solution-0}
\left\{ \begin{array}{l}
T_{a}(t)=T_{1}-A_{1}e^{-t/\tau_{1}}-A_{2}e^{-t/\tau_{2}}\\
\\
\dot{T}_{a}(t)=\frac{A_{1}}{\tau_{1}}e^{-t/\tau_{1}}+\frac{A_{2}}{\tau_{2}}e^{-t/\tau_{2}}\\
\\
\ddot{T}_{a}(t)=-\frac{A_{1}}{\tau_{1}^{2}}e^{-t/\tau_{1}}-\frac{A_{2}}{\tau_{2}^{2}}e^{-t/\tau_{2}}\\
\end{array} \right.$$ For $t_{0}=0$ this leads to: $$\begin{array}{l}
T_{a}(0)=T_{0}=T_{1}-(A_{1}+A_{2})\\
\\
\Delta T=T_{1}-T_{0}=A_{1}+A_{2}
\end{array}$$ Inserting solution (\[eq:solution-0\]) into (\[eq:main-equation\]) we obtain: $$\begin{aligned}
-\frac{A_{1}}{\tau_{1}^{2}}e^{-t/\tau_{1}}-\frac{A_{2}}{\tau_{2}^{2}}e^{-t/\tau_{2}}-\left [\frac{k_{1}+k_{2}}{C_{a}}+\frac{k_{2}}{C_{p}}\right ]\left (\frac{A_{1}}{\tau_{1}}e^{-t/\tau_{1}}+\frac{A_{2}}{\tau_{2}}e^{-t/\tau_{2}}\right ) \\
\\
+\frac{k_{1}k_{2}}{C_{p}C_{a}}\left (\Delta T -A_{1}e^{-t/\tau_{1}}-A_{2}e^{-t/\tau_{2}}\right )=\frac{k_{2}P_{0}}{C_{p}C_{a}}\\\end{aligned}$$ Isolating the two exponential terms the equation is fulfilled if $$\label{eq:solution-1}
\left\{ \begin{array}{lr}
e^{-t/\tau_{1}}\left [\frac{A_{1}}{\tau_{1}}\left (\frac{k_{2}}{C_{p}}+\frac{k_{1}+k_{2}}{C_{a}}\right )-\frac{A_{1}}{\tau_{1}^{2}}-\frac{A_{1}k_{1}k_{2}}{C_{p}C_{a}}\right ]=0 & \rm (i)\\
\\
e^{-t/\tau_{2}}\left [\frac{A_{2}}{\tau_{2}}\left (\frac{k_{2}}{C_{p}}+\frac{k_{1}+k_{2}}{C_{a}}\right )-\frac{A_{2}}{\tau_{2}^{2}}-\frac{A_{2}k_{1}k_{2}}{C_{p}C_{a}}\right ]=0 & \rm (ii)\\
\\
\frac{k_{1}k_{2}}{C_{p}C_{a}}(T_{1}-T_{0})-\frac{k_{2}P_{0}}{C_{p}C_{a}}=0 & \rm (iii)
\end{array} \right.
\nonumber$$ From the (iii) we obtain: $$\label{eq:power}
P_{0}=k_{1}(T_{1}-T_{0})=k_{1}\Delta T$$ and from (i) and (ii) we have the system of equations $$\label{eq:solution-2}
\left\{ \begin{array}{lr}
\frac{k_{2}}{C_{p}}+\frac{k_{1}+k_{2}}{C_{a}}=\frac{1}{\tau_{1}}+\frac{\tau_{1}k_{1}k_{2}}{C_{p}C_{a}}\\
\\
\frac{k_{2}}{C_{p}}+\frac{k_{1}+k_{2}}{C_{a}}=\frac{1}{\tau_{2}}+\frac{\tau_{2}k_{1}k_{2}}{C_{p}C_{a}}
\end{array} \right.
\nonumber$$ with solutions $$\frac{1}{\tau_{1}}+\frac{\tau_{1}k_{1}k_{2}}{C_{p}C_{a}}=\frac{1}{\tau_{2}}+\frac{\tau_{2}k_{1}k_{2}}{C_{p}C_{a}}$$ $$\label{eq:final-1}
C_{p}C_{a}=k_{1}k_{2}\tau_{1}\tau_{2}$$ and (substituting $k_{2}$ into the first equation of the system): $$\frac{C_{a}}{k_{1}\tau_{1}\tau_{2}}+\frac{C_{p}}{k_{1}\tau_{1}\tau_{2}}+\frac{k_{1}}{C_{a}}=\frac{1}{\tau_{2}}+\frac{1}{\tau_{1}}$$ $$\label{eq:final-2}
C_{p}=k_{1}\tau_{1}\left (1-\frac{k_{1}\tau_{2}}{C_{a}}\right ) +k_{1}\tau_{2}-C_{a}$$ To obtain the $\tau_{2}$ constant we take the first of the equations (\[eq:heat-balance\]) with the initial condition $t_{0}=0$ and $P(0)=P_{0}$ $$P_{0}=C_{a}\dot{T}_{a}(0)+k_{1}[T_{a}(0)-T_{0}]+k_{2}[T_{a}(0)-T_{p}(0)]~,$$ where $T_{a}(0)=T_{p}(0)=T_{0}$. Considering Eq. (\[eq:power\]) we have: $$k_{1}(A_{1}+A_{2})=C_{a}\left (\frac{A_{1}}{\tau_{1}}+\frac{A_{2}}{\tau_{2}}\right )$$ and therefore: $$\label{eq:tau2-0}
\tau_{2}=\frac{A_{2}\tau_{1}C_{a}}{(A_{1}+A_{2})\tau_{1}k_{1}-A_{1}C_{a}}$$ Considering now the simple case when $\tau_{2}=0$ (with, of course, $A_{2}=0$), we deduce: $$\label{eq:solution-3}
\left\{ \begin{array}{l}
T_{a}(t)=T_{1}-A_{1}e^{-t/\tau_{1}}\\
\\
C_{p}=k_{1}\tau_{1}-C_{a}\\
\\
P_{0}=k_{1}(T_{1}-T_{0})=k_{1}A_{1}\\
\end{array} \right.$$
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---
abstract: 'The discovery of collectivity in proton–proton collisions, is one of the most puzzling outcomes from the first two runs at LHC, as it points to the possibility of creation of a Quark–Gluon Plasma, earlier believed to only be created in heavy ion collisions. One key observation from heavy ion collisions is still not observed in proton–proton, namely jet-quenching. In this letter it is shown how a model capable of describing soft collective features of proton–proton collisions, also predicts modifications to jet fragmentation properties. With this starting point, several new observables suited for the present and future hunt for jet quenching in small collision systems are proposed.'
address:
- 'Niels Bohr Institute, University of Copenhagen, Blegdamsvej 19, 21000 København Ø, Denmark.'
- 'Department of Astronomy and Theoretical Physics, Lund University, S[ö]{}lvegatan 14A, S 223 62 Lund, Sweden.'
author:
- Christian Bierlich
bibliography:
- 'jetq.bib'
title: 'Soft modifications to jet fragmentation in high energy proton–proton collisions'
---
Quark–Gluon Plasma ,QCD ,Collectivity ,Jet quenching ,Hadronization ,Monte Carlo generators
Introduction
============
One of the key open questions from Run 1 and Run 2 at the LHC, has been prompted by the observation of collective features in collisions of protons, namely the observation of a near–side ridge [@Khachatryan:2010gv], as well as strangeness enhancement with multiplicity [@ALICE:2017jyt]. Similar features are, in collisions of heavy nuclei, taken as evidence for the emergence of a Quark–Gluon Plasma (QGP) phase, few fm after the collision.
The theoretical picture of collective effects in heavy ion collisions is vastly different from the picture known from proton–proton (pp). Due to the very different geometry of the two system types, interactions in the final state of the collision become dominant in heavy ion collisions, while nearly absent in pp collisions. The geometry of heavy ion collisions is so different from pp collision that in fact even highly energetic jets suffer an energy loss traversing the medium, a phenomenon known as jet quenching.
The ATLAS experiment has recently shown that the ridge remains in events tagged with a $Z$-boson [@ATLAS:2017nkt]. While maybe unsurprising by itself, the implication of this measurement is a solid proof that *some* collective behaviour exists in events where a high-$p_\perp$ boson is produced, possibly with an accompanying jet. In this letter this observation is taken as a starting point to investigate how the same dynamics producing the ridge in $Z$-tagged collisions, may also affect jet fragmentation. To that end, the microscopic model for collectivity, based on interacting strings [@Bierlich:2017vhg; @Bierlich:2016vgw; @Bierlich:2014xba] is used. The model has been shown to reproduce the near side ridge in minimum bias pp, and has been implemented in the [P8]{}event generator [@Sjostrand:2014zea], allowing one to study its influence also on events containing a $Z$ and a hard jet.
The non-observation of jet quenching in pp and pPb collisions is, though maybe not surprising due to the vastly different geometry, one of the most puzzling features of small system collectivity. If collectivity in small systems is due to final state interactions, it should be possible to also measure its effect on jets. If, on the other hand, collectivity in small collision systems is *not* due to final state interactions, but mostly due to saturation effects in the initial state – as predicted by Color Glass Condensate calculations [@Schenke:2016lrs] – the non-observation of jet quenching will follow by construction. The continued search for jet quenching in small systems is therefore expected to be a highly prioritized venue for the upcoming high luminosity phase of LHC [@Citron:2018lsq].
{width="35.00000%"} {width="40.00000%"}
The microscopic model for collectivity
======================================
Most general features of pp collisions, such as particle multiplicities and jets, can be described by models based on string fragmentation [@Andersson:1983jt; @Andersson:1979ij]. In the original model, such strings have no transverse extension, and hadronize independently. The longitudinal kinematics of the $i$’th breaking is given by the Lund symmetric fragmentation function: $$\label{eq:lufrac}
f(z) = Nz^{-1}(1-z)^a\exp\left(\frac{-bm_\perp}{z}\right),$$ where $z$ is the fraction of the *remaining* available momentum taken away by the hadron. $N$ is a normalization constant, and $a$ and $b$ are tunable parameters, relating the fragmentation kinematics to the breakup space-time points of the string, which are located around a hyperbola with a proper time of: $$\label{eq:stime}
\langle \tau^2 \rangle = \frac{1+a}{b\kappa^2},$$ where $\kappa \sim 1$ GeV/fm is the string tension. The transverse dynamics is determined by the Schwinger result: $$\label{eq:schwinger}
\frac{d\mathcal{P}}{d^2p_\perp} \propto \kappa \exp\left(\frac{\pi m_\perp^2}{\kappa}\right),$$ where $m_\perp$ is the transverse mass of the *quark* or *diquark* produced in the string breaking[^1].
When a $q\bar{q}$ pair moves apart, spanning a string between them, the string length is zero at time $\tau = 0$. To obey causality, its transverse size must also be zero, allowing no interactions between strings for the first short time ($<1$ fm/c) after the initial interaction. After this initial transverse expansion, strings may interact with each other, by exerting small transverse shoves on each other. In refs. [@Bierlich:2017vhg; @Bierlich:2016vgw] a model for this interaction was outlined, based on early considerations by Abramowski [@Abramovsky:1988zh]. Assuming that the energy in a string is dominated by a longitudinal colour–electric field, the transverse interaction force, per unit string length is, for two parallel strings, given by: $$\label{eq:shoving}
f(d_\perp) = \frac{g\kappa d_\perp}{\rho^2}\exp\left(-\frac{d_\perp^2}{4\rho^2}\right),$$ where both $d_\perp$ (the transverse separation of the two strings), and $\rho$ (the string transverse width) are time dependent quantities. The parameter $g$ is a free parameter, which should not deviate too far from unity. Equation (\[eq:stime\]) gives an (average) upper limit for how long time the strings should be allowed to shove each other around, as the strings will eventually hadronize[^2]. String hadronization and the shoving model has been implemented in the [P8]{} event generator, and all predictions in the following are generated using this implementation.
Effects on jet hadronization {#sec:jet-had}
============================
We consider now a reasonably hard $Z$-boson, produced back–to–back with a jet. Due to the large $p_\perp$ of the jet, its core will have escaped the transverse region in which shoving takes place well before it is affected. See figure \[fig:pp-sketch\] (a, left) in black for a sketch. In the following, a toy geometry where the jet is prevented from escaping before shoving, will also be studied, see figure \[fig:pp-sketch\] (a, right) in red for a sketch. The toy geometry is motivated by studies of jet fragmentation in Pb–Pb collisions, where the jet must still traverse through a densely populated region before hadronizing, due to the much larger geometry. Indeed in Pb–Pb, the observed effect by CMS [@Sirunyan:2017jic], is that the jet-$p_\perp$ relative to the $Z$-$p_\perp$ is reduced, moving the $z_j = p_{\perp, j}/p_{\perp, Z}$ distribution to the left.
Both geometries are constructed by picking the transverse position of each MPI according to the convolution of the two proton mass distributions, which are assumed to be 2D Gaussians. The jet is placed in origo. In the first, more realistic, geometry, all string pieces – including that corresponding to the jet core – are allowed to propagate for a finite time, indicating the time it takes for the strings to from infinitesimal transverse size, to their equilibrium size. In the toy geometry no such propagation is allowing, and all strings are treated *as if* expanded to full transverse size at $\tau = 0$. As such, strings from the underlying event, are allowing to shove even the hardest fragment of the jet. This clearly violates causality, and is not meant to be a realistic picture of a pp interaction. It is implemented in order to give an effect similar to what one should expect from a heavy-ion collision, where the event geometry allows strings from other nucleon-nucleon sub-collisions to interact with the jet core.
A set-up similar to that of the CMS study [@Sirunyan:2017jic], just for pp collisions at $\sqrt{s} = 7$ TeV, is studied in the following. A $Z$-boson reconstructed from leptons with $80$ GeV$ < M_Z< 100$ GeV, $p_\perp > 40$ GeV is required, and the leptons are required each to have $p_\perp > 10$ GeV. The leading anti-$k_\perp$ [@Cacciari:2008gp] jet (using FastJet [@Cacciari:2011ma] in Rivet [@Buckley:2010ar]) is required to have $p_\perp > 80$ GeV and $\Delta \phi_{zj} > 3\pi/4$. We study three different situations, with the result given in figure \[fig:pp-sketch\] (b).
In red, default [P8]{} is shown, where geometry has no impact on the result. In blue [P8]{} + shoving, with the normal event geometry, with the jet escaping. In green [P8]{} + shoving with the toy geometry, where the jet core interacts with the underlying event.
![\[fig:zridge\]The ridge in $Z$-tagged, high multiplicity pp collisions at 8 TeV, with default [P8]{} (dashed line, no ridge), and [P8]{} + shoving (full line, ridge). Simulation is compared to preliminary ATLAS data [@ATLAS:2017nkt].](zridge.pdf){width="45.00000%"}
While shoving in a toy geometry (green) produces an effect qualitatively similar to what one would expect from jet quenching, the effect in real events (blue) is far too geometrically suppressed to be seen (comparing blue to red in figure \[fig:pp-sketch\] (b)). Several suggestions exist for accommodating this problem, prominently using jet substructure observables [@Andrews:2018jcm], or using a delayed signal from top decays [@Apolinario:2017sob] (in AA collisions). In the remaining paper another approach will be described. Instead of looking for deviations in the spectrum of a narrow jet compared to a “vacuum” expectation, we start from the wide-$R$ ($R^2 = \Delta \eta^2 + \Delta \phi^2$) part where collectivity in the form of a ridge is known to exist even in pp collisions. The same observable is then calculated as function of $R$, all the way to the core, where the soft modification is expected to vanish.
Near side ridge in $Z$-tagged events
====================================
The ridge, as recently measured by ATLAS in events with a $Z$ boson present [@ATLAS:2017nkt], provides an opportunity. The requirement of a $Z$ boson makes the events in question very similar to the events studied above. The $Z$ does not influence the effect of the shoving model, and in figure \[fig:zridge\] high multiplicity events, with and without shoving, are shown, with the appearance of a ridge in the latter – in accordance with the experimental results[^3].
It is instructive to discuss the result of figure \[fig:zridge\] with the sketch in figure \[fig:pp-sketch\] (a) in mind. Since the ridge analysis requires a $|\Delta \eta|$ gap of 2.5, the jet region is, by construction, cut away. (Keeping in mind that in this case there is no required jet trigger.) The underlying event does, however, continue through the central rapidity range, and if only one could perform a true separation of jet particles from the underlying event in an experiment, the ridge should be visible. Since that is not possible, it is reasonable to naively ask if the presence of a ridge in the underlying event will by itself give rise to a shift in $z_j$. The result presented in figure \[fig:pp-sketch\] (b) (blue line) suggests that it does not. It is therefore necessary to explore more exclusive observables to isolate the effect of the soft modification of the jet.
![\[fig:jetmass\] The jet mass of anti-$k_\perp$ jets with $R = 0.7$, in events with a $Z-$boson with $p_\perp > 120$ GeV, in bins of jet-$p_\perp$. Data compared to default [P8]{} (red) and [P8]{} + shoving (black).](jetmass.pdf){width="45.00000%"}
The comparison in figure \[fig:zridge\] also serves the purpose of fixing the parameters of the shoving model before studying jet-related quantities. The only free parameter of the model is the $g$-parameter in equation (\[eq:shoving\]), the rest are fixed to default values [@Skands:2014pea]. As shown in ref. [@Bierlich:2017vhg], the free parameter determines the height of the ridge. The value $g=4$ is chosen in this paper, which also gives a good description of the ridge in minimum bias events.
Influence on jet observables
============================
As the ATLAS measurement has established, there is indeed collectivity present in (high multiplicity) events with a $Z$ present. In the previous section it was shown that the measured signature can be adequately described by the shoving model. Now the situation will be extended to include also a high-$p_\perp$ jet trigger in the same way as in section \[sec:jet-had\], and the effect of the collective behaviour on the jet will be discussed.
Hard measures: Jet mass and jet cross section
---------------------------------------------
The jet masses, binned in jet-$p_\perp$, is a key calorimetric observable for comparing observed jet properties to predictions from models. With the advent of jet grooming techniques, the precision of such comparisons have increased, and any model seeking to predict new phenomena, must be required to not destroy any previous agreement with this observable. The mass of hard jets produced in events with a $Z$-boson present has been measured by the CMS experiment [@Chatrchyan:2013vbb], and in figure \[fig:jetmass\] the results are compared to [P8]{}with and without shoving, in red and black respectively. Shoving increases the jet mass slightly, bringing the prediction closer to data, though not at a significant level. In the analysis by CMS [@Chatrchyan:2013vbb], various grooming techniques are also explored. These are not shown in the figure, but all remove most of the effect from shoving. This is the expected result, as the grooming techniques are in fact introduced to remove soft QCD radiation from jets.
![\[fig:jetr\]The $R$ dependence of $\sigma_j$ for four configurations of the leading jet in $Z+$jet in pp collisions at 7 TeV. Special attention is given to the difference between [P8]{} default and [P8]{} + shoving in the large-$R$ limit.](jetr.pdf){width="45.00000%"}
An effect of shoving at the level is seen for low jet masses. While also the most difficult region to assess experimentally, this effect could be worthwhile to explore further. A prediction for the jet-$p_\perp$ bin 40-125 GeV is also shown, as one could imagine that a larger effect could be observed if the jet threshold could be experimentally lowered. The effect on a level persists, but does not increase.
The jet-$p_\perp$ is also a well studied quantity. As there is little effect on the raw jet-$p_\perp$ spectra, the jet cross section is used: $$\sigma_j = \int_{p_{\perp,0}}^\infty dp_{\perp,j} \frac{d\sigma}{dp_{\perp,j}},$$ where $p_{\perp,j}$ is the $p_\perp$ of the leading jet in the event, and $p_{\perp, 0}$ is the imposed phase space cut–off. It was pointed out by Ellis [@Ellis:1992qq], that the $R$-dependence of $\sigma_j$ under the influence of MPIs in a pp collision, can be parametrized as $A + B\log(R) + CR^2$. Later Dasgupta [@Dasgupta:2007wa] noted that hadronization effects contributes like $-1/R$. This gives a total parametrization: $$\label{eq:sigmaj}
\sigma_j(R) = A + B\log(R) + CR^2 - DR^{-1}.$$ By construction, the ridge effect from the previous chapter is far away from the jet in $\eta$, and therefore also in $R$. Any contribution from shoving can be reasonably expected to be most pronounced for large $R$. Equation (\[eq:shoving\]) gives a contribution of $\langle dp_\perp / d\eta \rangle \propto f\left(\langle d_\perp \rangle \right)$, where $\langle d_\perp \rangle$ is density dependent. In the previously introduced semi-realistic geometry, a contribution to $\sigma_j$, which is $\propto R^2$, is expected, a correction to the parameter $C$ in equation (\[eq:sigmaj\]).
In figure \[fig:jetr\], $\sigma_j(R)$ is shown without MPIs and hadronization (red), with MPI, no hadronization (blue), [P8]{} default (green) and [P8]{} + shoving (black). The analysis setup is the same as in section \[sec:jet-had\]. Results from the Monte Carlo is shown as crosses, and the resulting fits as dashed lines, with parameters given in table \[partable\].
\[pb/GeV\] No MPI, no had. No had. Default Shoving
------------ ----------------- ----------------- ----------------- -----------------
A $1.46 \pm 0.03$ $1.31 \pm 0.01$ $1.28 \pm 0.04$ $1.29 \pm 0.05$
B $8.44 \pm 0.03$ $8.22 \pm 0.01$ $8.18 \pm 0.02$ $8.19 \pm 0.03$
C - $1.16 \pm 0.01$ $1.35 \pm 0.03$ $1.49 \pm 0.03$
D - - $0.05 \pm 0.01$ $0.05 \pm 0.01$
: \[partable\]Parameters obtained by fitting equation (\[eq:sigmaj\]) to [P8]{}. Errors are fit errors ($1\sigma$), fits shown in figure \[fig:jetr\].
From the fits it is visible that shoving contributes to the $R^2$ dependence as expected. Directly from figure \[fig:jetr\] it is visible that shoving contributes to the jet cross section at a level comparable to hadronization effects. As it is also seen from the figure and table, MPI effects contributes much more than the additional effects from hadronization or shoving. This means that the usual type of centrality measure (number of charged particles measured in some fiducial region) is not quite applicable for such observables, as the large bias imposed on MPI selection, would overcome any bias imposed on selection of the much smaller collective effects[^4]
![\[fig:avgmass\] The average hadron mass in the leading anti-$k_\perp$ jet with $R = 0.1$ (dashed) and $R = 0.7$ (full) in $Z$+jet, using default [P8]{} (red) and [P8]{} + shoving (black). The deviation imposed by shoving grows larger with increasing $R$.](massdist.pdf){width="45.00000%"}
Soft measures: Average hadron mass and charge
---------------------------------------------
The hadrochemistry of the jet is here quantified in a quite inclusive manner by the average hadron mass: $$\langle m_h \rangle = \frac{1}{N_p} \sum_i^{N_p} m_{h,i},$$ where $N_p$ is the number of hadrons in the jet, and $m_h$ are the individual hadron masses. Furthermore the total jet charge is studied: $$Q_j = \sum_i^{N_p} q_{h,i},$$ where $q_i$ are the individual hadron electric charges. As shoving only affects these quantities indirectly, the predicted effect is not as straight forward as was the case for jet cross section, but requires a full simulation to provide predictions. In figure \[fig:avgmass\] the average hadron mass in the leading jet (still in $Z$+jet collisions as above) is shown for two exemplary values of $R$. For small $R$, $\langle m_h \rangle$ is unchanged, but as $R$ grows, a significant change, on the order of is visible. The average hadron mass in jets has to this authors’ knowledge not been measured inclusively, but related quantities (ratios of particle species) has been preliminarily shown by ALICE [@Hess:2014xba] to be adequately described by [P8]{}.
The $Q_j$ distribution for $R=0.3$ jets is shown in figure \[fig:jetchargedist\]. It is seen directly, that for this particular value of $R$, shoving widens the distribution, and also the mean is further shifted in the positive direction. The $R$-dependency of this behaviour is shown in figure \[fig:jetcharger\]. Here both the mean and the width of the jet distribution at different values of $R$ is shown (note the different scales on the axes). It is seen that this observables shows deviations up to in the large-$R$ limit. Jet identification techniques to reveal whether the seed parton is a gluon or a quark [@Gras:2017jty; @Bright-Thonney:2018mxq] might be able to increase the discriminatory power even further.
![\[fig:jetchargedist\] An example of a jet charge distribution for the leading anti-$k_\perp$ jet in $Z$+jet with $R = 0.3$. Shoving has the effect of making the distribution wider.](jcdist.pdf){width="40.00000%"}
[P8]{} provides a good description of jet charge in di-jet events [@Sirunyan:2017tyr], giving further significance to any deviation introduced by shoving in this special configuration. It should, however, be noted that the jet charge has been a challenge for fragmentation models since the days of $e^+e^-$ collisions at LEP [@Abdallah:2006ve]. The renewed interest in fragmentation properties from the observation of collectivity in small systems provides a good opportunity to also go back and revisit older observations.
The jet hadrochemistry can be studied in a more exclusive manner, by means of particle identification, similar to what is done in nuclear collisions. Such observables will also be largely affected by formation of colour multiplets, increasing the string tension [@Biro:1984cf; @Bierlich:2014xba]. In the context of this letter, it is noted that rope formation contributes negligibly to the observables studied above. Some studies of rope effect in jets in pp collisions have been performed [@Mangano:2017plv], but could require further attention to the important space–time structure, as described in section \[sec:jet-had\].
Conclusions
===========
The non-observation of jet quenching in small systems is one of the key open questions to understand collective behaviour in collisions of protons. For the coming high luminosity era at LHC, the search for new observables to either observe jet quenching, or provide quantitative exclusion limits is necessary. In this letter we have shown that the microscopic model for collectivity implemented in [P8]{}, can reproduce one observed collective feature already observed in pp collisions with a hard probe, namely the ridge in $Z$ tagged events. Basic features like $z_j$ are, however, unaffected, but highly sensitive to the collision geometry. For a toy event geometry, the model produces features similar to those observed in Pb–Pb collisions. The toy geometry study highlights the need for a better motivated theoretical description of the space–time structure of the initial state. The realization that the complicated interplay between fragmentation time and spatial structure is significant for precision predictions, dates back to the 1980’s for collisions of nuclei [@Bialas:1986cf]. With the discovery of small system collectivity, several approaches have been developed also for pp collisions (*e.g.* [@Avsar:2010rf; @dEnterria:2010xip; @Albacete:2016pmp; @Ferreres-Sole:2018vgo]), most (but not all) aiming for a description of flow effects. It is crucial for the future efforts that such space–time models attempt at describing both soft and hard observables at once, in order to avoid “over tuning” of sensitive parameters. In this letter it was done by first describing the ridge in $Z$-tagged events, and then proceed to investigate jet observables with the same parameters, while the models remains able to describe key observables like jet mass. An effect from shoving up to for low jet masses was shown, but is within the current experimental uncertainty.
![\[fig:jetcharger\] $R$-dependency of the average jet charge and the distribution (see fig. \[fig:jetchargedist\]) width with and without shoving. Note the different scales for the two quantities.](jetcharge.pdf){width="45.00000%"}
The major contribution of this letter is the proposal of several new observables to understand the effects on jet fragmentation from the shoving model in $Z$+jet events. The main idea behind these observables is to go from the wide-$R$ region (wide jets), where collective effects, in form of the ridge, is already observed, to the very core of the jet, where only little effect is expected. The jet-$p_\perp$ is only affected little, and the observed effect on the integrated quantity $\sigma_j$, will be difficult to observe when also taking into account uncertainties from PDFs and NLO corrections, but nevertheless provides a crucial challenge for the upcoming high luminosity experiments at LHC, where larger statistics can help constraining the theoretical uncertainties better. More promising are the effects observed on hadron properties inside the jet, where the average hadron mass shows a deviation and jet charge even larger. Even if an effect this large is not observed in experiment, its non-observation will aide the understanding of soft collective effects better, as the shoving model predicting the effect, adequately describes the ridge in $Z$-tagged collisions.
Acknowledgements
================
I thank Johannes Bellm for valuable discussions, and Peter Christiansen, Leif Lönnblad and Gösta Gustafson for critical comments on the manuscript. I am grateful for the hospitality extended to me by the ALICE group at the Niels Bohr Institute during the preparation of this work. This work was funded in part by the Swedish Research Council, contract number 2017-0034, and in part by the MCnetITN3 H2020 Marie Curie Initial Training Network, contract 722104.
[^1]: The formalism does not dictate whether to use current or constituent quark masses. In [P8]{} the suppression factors s/u and diquark/quark are therefore determined from data, with resulting quark masses providing a consistency check.
[^2]: Eq. (\[eq:stime\]) is written up with a string in vacuum in mind. It might be possible that the string life time is modified in the dense environment of a heavy ion collision.
[^3]: The simulation is compared to preliminary ATLAS data [@ATLAS:2017nkt], with the caveat that the analysis procedure is very simplistic compared to the experimental one. Instead of mixing signal events with a background sample, distributions are instead divided each with their minimum to obtain comparable scales.
[^4]: In order to use this procedure to set limits on jet quenching in small systems, comparison must be made to predictions. In figure \[fig:jetr\] only LO predictions are given, but while NLO corrections are sizeable enough that figure \[fig:jetr\] cannot be taken as a numerically accurate prediction, such corrections will not affect the relative change in $\sigma_j$ from shoving, and will not affect the result. More crucial is the effect of parton density uncertainties, which may affect $\sigma_j$ up to 10% for this process [@Bendavid:2018nar]. This points to the necessity of more precise determinations of PDFs, if microscopic non-perturbative effects on hard probes in pp collisions are to be fully understood.
|
---
abstract: 'We explore the tidal excitation of stellar modes in binary systems using [*Kepler*]{} observations of the remarkable eccentric binary KOI-54 (HD 187091; KIC 8112039), which displays strong ellipsoidal variation as well as a variety of linear and nonlinear pulsations. We report the amplitude and phase of over 120 harmonic and anharmonic pulsations in the system. We use pulsation phases to determine that the two largest-amplitude pulsations, the 90th and 91st harmonics, most likely correspond to axisymmetric $m=0$ modes in both stars, and thus cannot be responsible for resonance locks as had been recently proposed. We find evidence that the amplitude of at least one of these two pulsations is decreasing with a characteristic timescale of $\sim 100\,$yr. We also use the pulsations’ phases to confirm the onset of the traveling wave regime for harmonic pulsations with frequencies $\lesssim 50\; {\ensuremath{\Omega_{\rm orb}}}$, in agreement with theoretical expectations. We present evidence that many pulsations that are not harmonics of the orbital frequency correspond to modes undergoing simultaneous nonlinear coupling to multiple linearly driven parent modes. Since coupling among multiple modes can lower the threshold for nonlinear interactions, nonlinear phenomena may be easier to observe in highly eccentric systems, where broader arrays of driving frequencies are available. This may help to explain why the observed amplitudes of the linear pulsations are much smaller than the theoretical threshold for decay via three-mode coupling.'
author:
- |
Ryan M. O’Leary$^{1}$[^1][^2] and Joshua Burkart$^{1,2}$\
$^{1}$Department of Astronomy and Theoretical Astrophysics Center, University of California, Berkeley, CA 94720, USA\
$^2$Department of Physics, 366 LeConte Hall, University of California, Berkeley, CA 94720, USA
bibliography:
- 'p.bib'
title: 'It takes a village to raise a tide: nonlinear multiple-mode coupling and mode identification in KOI-54'
---
binaries: close – asteroseismology – stars: oscillations – waves – instabilities
Introduction {#sec:intro}
============
Stars and planets in eccentric orbits exchange energy and angular momentum through tidal interactions. The net tidal fluid response can be conceptually divided into two components. The equilibrium tide is the large-scale prolate distortion caused by nonresonantly excited stellar modes . The dynamical tide, which is our focus in this work, corresponds to low-frequency internal waves (gravity and inertial) that are resonantly excited by the time-varying tidal potential. Such waves have much shorter damping times, and are expected to play an important role in the circularization of orbits and spin up of stars . Although most prior work on tides has been calculated in the linear regime, nonlinear effects may play an important role in redistributing energy and angular momentum in binary systems on much shorter timescales [e.g., @2010MNRAS.404.1849B; @2012ApJ...751..136W].
In this work, we explore the role of linear and nonlinear dynamical tides in the recently discovered *Kepler* system KOI-54 (HD 187091; KIC 8112039; @W11, hereafter W11). In particular, we aim to understand the nature of the largest-amplitude tidally excited pulsations to ascertain if they are responsible for resonance locks, a phenomenon that allows for efficient spin-orbit coupling in binary systems . More generally, we explore the harmonic and anharmonic pulsations that we observe in this particular binary system, and address how their amplitudes and frequencies are determined by a complex set of nonlinear interactions amongst many different modes.
Recently, @B12 [hereafter B12] developed a quantitative framework for analyzing such pulsations, which they termed tidal asteroseismology (see also @2011arXiv1107.4594F). This work used theoretical stellar models to investigate the linear, nonadiabatic response of stars to linearly excited resonant oscillations. This analysis naturally explained many qualitative features of KOI-54’s harmonic pulsations. Nonetheless, several puzzles remained.
One unresolved question was the nature of the two most prominent pulsations in the system, which have frequencies that are the 90th & 91st harmonics of the orbital frequency. The amplitudes of the pulsations ($294\,\mu$mag and $228\,\mu$mag, or parts-per-million, respectively) are considerably larger than any of the other pulsations; indeed, B12 estimated that the probability of their occurrence through purely linear excitation to be only $\sim 1\%$. Both @2011arXiv1107.4594F and B12 considered the possibility that the two pulsations could be occurring due to two $l=2$, $|m|=2$ g-modes responsible for resonance locks. @2011arXiv1107.4594F showed that the frequencies of the 90th and 91st harmonics are within a few percent of the natural, *a priori* prediction of the most likely frequency at which resonance locks should occur, given KOI-54’s observed system parameters. However, B12 found that the maximum torque possible from likely modes was much too small to effect a resonance lock, and also pointed out that it is unlikely for two resonance locks to occur simultaneously.
B12 and @2011arXiv1107.4594F also reported that the two largest anharmonic pulsations appear to be driven by the parametric instability of the $91$st harmonic. However, B12 found that the estimated amplitude for a mode to become overstable and drive the parametric instability is $\sim 100\times$ larger than the observed amplitude of the $91$st harmonic if it is an $m=0$ mode.
We address the nature of the harmonic pulsations by analyzing their amplitudes and frequencies together with their phases. B12 showed that the observed phase of the pulsations depends on the excited mode’s azimuthal order $m$, its damping rate $\gamma$, and the difference $\delta \omega$ between the eigenmode frequency and the driving frequency (see eq. \[eq:phase\]).
In this work, we extend the work of W11 using additional publicly available [*Kepler*]{} data to determine all of the observable properties of the pulsations, in particular their phases, which were not originally reported. This allows us to determine more information about the modes responsible for the individual pulsations, in particular the azimuthal order $m$. Furthermore, with the addition of five more quarters of data, and consequently greater signal-to-noise ratios, we are able to search for pulsations with amplitudes lower than W11 were able to detect.
The structure of our paper is as follows. We describe *Kepler* photometry and our data reduction routine in § \[sec:data\], and give an overview of the observed pulsations in § \[sec:sin\]. In § \[sec:lin\], we analyze the tidally driven, linear oscillations of KOI-54. In § \[sec:nonlin\], we analyze the nonlinear oscillations. Finally, we summarize our results and discuss their implications in § \[sec:disc\].
{width="\textwidth"} \[fig:reduced\]
*Kepler* photometry and detrending {#sec:data}
==================================
We use 785.4d of nearly continuous photometric observations of KOI-54 that have been publicly released by the *Kepler* mission [@2010Sci...327..977B; @2010ApJ...713L..79K]. During these observations, the satellite underwent multiple safe-mode shutdowns, as well as scheduled rollings that introduce large artifacts into the raw data. We detrend the data and remove these artifacts in the following way. We visually inspect the released raw data, and remove, by eye, obvious outliers due to cosmic rays. After inspection, we fit cubic polynomials (and lines) to each series of contiguous data simultaneously along with a photometric model for the periastron brightening events as well as 30 sinusoids for the largest-amplitude pulsations. We then remove all outliers that deviate from the remaining data by more than three standard deviations, and refit our models and detrending curves. After we detrend the data, we subsequently subtract the photometric model from the normalized lightcurve and remove low-frequency pulsations with a Hann window function of width $8\,$d.
W11 employed the proprietary [ELC]{} modeling code to simultaneously fit the [*Kepler*]{} photometry together with complementary radial velocity observations of KOI-54. We instead use the much faster and simpler photometric model detailed in B12 (Appendix B), which is sufficient to determine the amplitudes, frequencies, and phases (and any variations) of the high-frequency ($f \gtrsim 20 {\ensuremath{\Omega_{\rm orb}}}$) dynamical tide with minimal contamination from the equilibrium tide and stellar irradiation. This linear model decomposes the stellar flux perturbation induced by the equilibrium tide into spherical harmonics, making use of von Zeipel’s theorem [@1924MNRAS..84..665V]. The effects of irradiation of each star by its companion are also included, and are modeled as absorption and immediate reemission at the photosphere. We evaluate changes in luminosity due to the equilibrium tide and irradiation up to spherical harmonic degree $l=3$; higher-degree harmonics do not contribute significantly to the signal.
To model limb darkening, we use the four-coefficient nonlinear fit found in @2011MNRAS.413.1515H determined for the *Kepler* bandpass for an $8,800 $K star with a surface gravity $\log g = 4$, consistent with best-fit parameters of both stars in W11. We then find the least-squares best fit for the orbital period, epoch of periastron, and bandpass correction coefficients $\beta_1$ & $\beta_2$. For the remaining parameters, we use the best-fit values in W11. In principle, it should be possible to determine $\beta_1$ and $\beta_2$ *a priori*, as was done in W11. We find that for the objectives of this paper, by fitting for $\beta_1$ and $\beta_2$ and subtracting the ellipsoidal variation and irradiation from the lightcurve, we do not introduce any significant errors in our assessment of the high-frequency pulsations. The parameters of KOI-54 used in this study are listed in Table \[tab:params\].
[llrrc]{} & parameter & value& error & unit\
\
& $T_{1}$ & 8500 & 200 & K\
&$T_{2}$ & 8800 & 200 & K\
&$L_2/L_1$ & 1.22 & 0.04 &\
&$v_{{\mathrm{rot}},1} \sin{i_1}$ & 7.5 & 4.5 & ${\mathrm{km}}/{\mathrm{s}}$\
&$v_{{\mathrm{rot}},2} \sin{i_2}$ & 7.5 & 4.5 & ${\mathrm{km}}/{\mathrm{s}}$\
&$[\text{Fe}/\text{H}]_1$ & 0.4 & 0.2 &\
&$[\text{Fe}/\text{H}]_2$ & 0.4 & 0.2 &\
\
& ${\ensuremath{P_{\rm orb}}}$ & 41.8050 & 0.0003 & days\
&$T_{\rm p}$ & 2455103.5490 & 0.0010 & BJD\
&$e$ & 0.8335 & 0.0005 &\
&$\omega$ & 36.70 & 0.90 & degrees\
&${\ensuremath{i}}$ & 5.50 & 0.10 & degrees\
&$a$ & 0.3956 & 0.008 & AU\
&$M_{2}/M_{1}$ & 1.024 & 0.013 &\
&$R_{1}$ & 2.20 & 0.03 & ${\ensuremath{{\rm R}_{\odot}}}$\
&$R_{2}$ & 2.33 & 0.03 & ${\ensuremath{{\rm R}_{\odot}}}$\
\
& ${\ensuremath{P_{\rm orb}}}$ & 41.8050 & 0.0001 & days\
&$T_{\rm p}$ & 2455061.73814 & 0.006 & BJD\
&$\beta(T_1)$ & 0.489 & &\
&$\beta(T_2)$ & 0.818 & &\
\
& $b_0$ & 1.0000 & &\
&$b_1$ & 0.7076& &\
&$b_2$ & 0.3230& &\
&$b_3$ & 0.0596& &\
&$c_0$ & 0.0000& &\
&$c_1$ & 1.4152& &\
&$c_2$ & 1.9380& &\
&$c_3$ & 0.7154& &\
------------------------------------------------------------------------
![Frequency spectrum of residual data after removing the ellipsoidal variation as well as 182 of the largest-amplitude pulsations (see § \[sec:osc\]). No pulsations have amplitude $\gtrsim 1.2 \mu$mag. In the figure we have marked two of the largest-amplitude pulsations that we did not include in our global fits, as they have frequencies that are too close to other frequencies to be independently resolved in the data. The frequency marked (a) is close to F5, the largest-amplitude anharmonic pulsation. The peak marked (b) is close to F1, the largest-amplitude harmonic pulsation $f = 90 {\ensuremath{\Omega_{\rm orb}}}$. []{data-label="fig:spectrum"}](residualspectrum182.pdf){width="\columnwidth"}
Compared to W11, who detrended the data by completely masking the brightening events and then similarly fitting cubic polynomials, our method gives smaller residuals near the brightening events and smaller formal errors in the derived properties of the pulsations.
Sinusoidal pulsations in KOI-54 {#sec:sin}
===============================
Observed frequencies, amplitudes, and phases {#sec:osc}
--------------------------------------------
We systematically search for the $200$ largest-amplitude[^3] pulsations in the lightcurve by removing our best-fit ellipsoidal variation model and then iteratively removing the largest-amplitude pulsation found in the power spectrum of the residual data. Between steps, we simultaneously refit the data for the frequency, phase, and amplitude of all known pulsations. In practice, the process of refitting the data via least-squares can cause the frequencies of two or more initially distinct pulsations to converge. As a result, we remove 18 cases where the frequency spacing between two pulsations is less than the reciprocal of the total duration of observations (i.e., pairs that have not undergone one full beat in the window of observations), and simultaneously refit all the remaining $182$ pulsations.
In Table \[tab:fitsh\], we present the $70$ largest-amplitude [ *harmonic*]{} pulsations with frequency $f >20{\ensuremath{\Omega_{\rm orb}}}$ and $|f - k {\ensuremath{\Omega_{\rm orb}}}| < 0.02{\ensuremath{\Omega_{\rm orb}}}$, where $k$ is an integer,[^4] along with their measured phase (from 0 to 1) with respect to the epoch of periastron as well as BJD$-2454833$. In Table \[tab:fitsa\], we similarly present the $59$ largest-amplitude [*anharmonic*]{} pulsations with $|f - k {\ensuremath{\Omega_{\rm orb}}}|
\ge 0.02{\ensuremath{\Omega_{\rm orb}}}$. The derived properties of low-frequency pulsations ($f < 20{\ensuremath{\Omega_{\rm orb}}}$) have much greater uncertainties because of systematic trends in the raw data that we were unable to remove completely, so we do not report these pulsations in our results. After removing the $182$ largest-amplitude pulsations, we are left with no pulsations with amplitude $\gtrsim 1.2\,\mu$mag. To be consistent with W11 and much of the observing literature, we fit for the amplitude $A_i > 0$, frequency $f_i$ and phase offset $\delta_i$, using the form $A_i \sin{2\pi (f_it+\delta_i)}$, where the time $t$ is measured relative to the epoch of periastron, $t_p$, as well as BJD-$2454833$. The variance in the residual lightcurve is $\approx
16\, \mu$mag, approximately 1.6 times larger than the photometric uncertainty of each observation. The frequency spectrum of the residuals is presented in Figure \[fig:spectrum\].
We estimate the formal uncertainties of our fits by a Monte Carlo method (for systematic uncertainties, see § \[sec:unc\]). We create mock light curves using the best-fit parameters of the pulsations, using the same observing window as the real data. We then add Gaussian white noise to each point with an amplitude $10^{-5}$, consistent with the photometric uncertainty in each data point. Finally, we add $1/f$ noise to the data by passing it through the same high-pass Hann filter as the data, and then rescaling the amplitude of the $1/f$ noise so the final noise amplitude of the mock light curve equals the observed amplitude of the residuals. The power spectrum of the noise we generate matches both the amplitude and shape of the residual power spectrum much better than white noise alone, which is commonly used in other analyses. We refit each mock data set 100 times using the previously known best-fit parameters as our initial values, and report the variance of the best-fit parameters as the formal uncertainties in Tables \[tab:fitsh\] & \[tab:fitsa\].
For some pulsations that have similar frequencies, we find that fitting routine can be pathological, resulting in the two pulsations to be fit with the same frequency. Even when this happens in only one of the noise realizations, the reported formal uncertainty can be significantly larger than the intrinsic amplitude of the observed pulsation. In Tables \[tab:fitsh\] & \[tab:fitsa\] we report all observed pulsations with amplitudes larger than $\approx 0.7 \mu$mag, which typically are $\approx 3\sigma$ local detections, and have not been corrected for searching over the entire parameter space. Some of these pulsations, especially the low-frequency anharmonic pulsations, may be noise. We choose to report all of these pulsations because any spurious pulsations do not contaminate our analysis.
Systematic uncertainties {#sec:unc}
------------------------
{width="\columnwidth"} {width="\columnwidth"}
Our work here, and work on similar systems, inevitably possesses important systematic errors. The greatest uncertainty in our calculations is the systematic uncertainty in fitting for the epoch of periastron, which impacts the derived phase offsets of the harmonic pulsations. Indeed, in W11, the authors’ best-fit epoch of periastron derived from radial velocity data alone is $6\sigma$ away from the epoch found using radial velocities together with photometry. Visually, the systematic errors in the fitting of the RV data appear more significant. The quoted uncertainties in the W11 RV data were also probably too low due to heterogeneity of the spectra (G. Marcy, private communication). For this reason, we exclusively use the photometrically determined epoch of periastron. In addition, fitting only $15$ of the largest-amplitude pulsations, as done in W11, is likely insufficient for determining the epoch of periastron. As we will show in § \[sec:lin\] (see Fig. \[fig:scatterphase\]), many of the low-amplitude pulsations are $m=0$ oscillations that reach their maximum near periastron and bias the fitting of the ellipsoidal variation and irradiation of the binary. Indeed we found that the peak amplitudes of the observed brightening events were $80\,\mu$mag larger than the best-fit model if the pulsations are not taken into consideration.
In W11, the authors compensated for the undetected pulsations by increasing the uncertainty of each data point by a factor of $10$, thereby obtaining a normalized $\chi^2$ less than unity. Nevertheless, systematic errors from the undetected pulsations may remain. We attempt to ascertain how the pulsations bias the epoch of periastron by refitting the detrended data simultaneously with 15 or 30 of the largest-amplitude pulsations. We find systematic changes in the epoch of periastron approximately three times larger than the uncertainty reported in W11. We conservatively estimate that the true uncertainty is six times larger than reported in W11 (Table 2).
Unfortunately, many of the low-frequency pulsations are systematic errors from our fitting routine using cubic polynomials. We therefore fit the pulsations by both including all low-frequency pulsations and compare this to the fit excluding pulsations with $f < 20 {\ensuremath{\Omega_{\rm orb}}}\approx 0.5\,$day$^{-1}$. We find that the difference in the fits of the high-frequency pulsations are less than the formal uncertainties listed in Tables \[tab:fitsh\] & \[tab:fitsa\]. We also note that we do not include the impact of Doppler boosting in our light curve models. We find that including the impact of boosting does not change the amplitude of any pulsations by more than $0.5\,\mu$mag, although the Doppler boosting signal is important in other eccentric binaries reported by [*Kepler*]{} that show strong tidal distortions [@2012ApJ...753...86T; @2013arXiv1306.1819H].
Incompletely subtracting the ellipsoidal variation can also systematically contaminate our fits of the harmonic pulsations. Fortunately, the Fourier decomposition of the ellipsoidal variation is continuous with frequency. We estimate that the total contamination is $\lesssim 0.3\mu$mag by inspecting the limits we placed on the amplitude of undetected harmonic pulsations. A similar technique may be useful when constraining the contamination of lower harmonics in less eccentric binaries.
Linearly driven harmonic pulsations {#sec:lin}
===================================
Each star is expected to have tidally driven pulsations at perfect harmonics of the orbital frequency ${\ensuremath{\Omega_{\rm orb}}}$. In § \[sec:linintro\] we briefly outline the general theory of dynamical tides and describe how the pulsations reveal detailed properties of the star (B12, @2011arXiv1107.4594F). In § \[sec:harm\] we analyze the phases of the harmonic pulsations relative to periastron in order to determine the azimuthal order, $m$, of the oscillations. In § \[sec:time\], we search for time variability in the amplitudes, phases, and frequencies of KOI-54’s pulsations.
Introduction {#sec:linintro}
------------
For a nonrotating star, the time-varying tidal potential excites $l
\geq 2$ mode spherical harmonics in the star. The oscillations within the star are observed as sinusoidal pulsations,[^5] which are averaged over the disk of the entire star. The steady-state, equilibrium solution is simply a sinusoidal pulsation with an observed frequency $k{\ensuremath{\Omega_{\rm orb}}}$ that is a harmonic of the orbital frequency, where $k$ is an integer. The amplitude and phase of the pulsation are determined by how well tuned the oscillator is relative to an eigenfrequency as measured by the detuning $\delta \omega = \omega_{nl} -
k{\ensuremath{\Omega_{\rm orb}}}$, by the damping rate of the excited mode $\gamma_{nl}$, as well as by the spatial coupling between the driving force and the mode [@1977ApJ...213..183P]. By directly measuring the amplitude and phase of a harmonic pulsation, it is possible to determine the detuning-to-damping ratio $\delta \omega/
\gamma_{nl}$, as well as the azimuthal order of the oscillation, $m$.
The phases of observed pulsations of a star are determined by disk averages of the flux perturbations on the stellar surface. In this work we measure the phase of the pulsation from the light curve using the equation (see § \[sec:osc\]) $$\label{eq:fit}
A_i \sin{2\pi (f_it+\delta_i)},$$ where $A_i >0$, and $t$ is measured relative to the epoch of periastron, $t_p$. The observed phase of the pulsation directly depends on the order of the harmonic, $m$. B12 derived the observed pulsation phase relative to the epoch of periastron to be[^6] $$\label{eq:phase}
\delta = \left(\frac{1}{4}+\Psi_{nlmk} + m \phi_0 \right) \mod \frac{1}{2},$$ where $\phi_0 = 1/4 - \omega_p/(2\pi) \mod 1$, $\omega_p$ is the argument of periastron of the orbit, $$\Psi_{nlmk} = -\frac{1}{2\pi}
\arctan{\left(\frac{\omega_{nl}^2-\sigma_{km}^2}{2 \gamma_{nl}\sigma_{km}}\right)} \mod \frac{1}{2},
\label{eq:PSI}$$ and $\sigma_{km} = k{\ensuremath{\Omega_{\rm orb}}}- m \Omega_*$ is the driving frequency of the tide in the corotating frame of the star with rotation frequency $\Omega_*$. We expect that most excited oscillations are poorly tuned, i.e., $|\delta \omega| \gg \gamma_{nl}$, since the frequency spacing of the eigenmodes is much larger than the damping rate of high-frequency g-modes. In this limit, equations \[eq:phase\] & \[eq:PSI\] reduce to $$\label{eq:finalphase}
\delta = \left(\frac{1}{4} + m \phi_0\right) \mod \frac{1}{2}.$$ Since $\phi_0$ is a quantity derived from modeling the equilibrium tide as well as the radial velocity data, we can directly determine the order of an oscillation $m$ in this limit.
In this work, we report the absolute phases of the pulsations $\in
[0,1)$ relative to the epoch of periastron using the sine function to be consistent with the observing literature (see § \[sec:osc\]). A pulsation is observed at its maximum at the epoch of periastron when $\delta = 1/4$. For $m=0$ modes, based on equation \[eq:finalphase\] we would expect the pulsation to be near maximum ($\delta \approx 1/4$) or near minimum ($\delta \approx 3/4$) at the epoch of periastron. B12 show that if the largest-amplitude harmonics of KOI-54 are $m=0$ modes, then their detuning-to-damping ratio is $|\delta \omega|/\gamma_{nl} \sim 20$ and so should be offset from $1/4$ or $3/4$ by no more than a few percent. Given the face-on orientation of KOI-54, most observable pulsations should be $m=0$, $l=2$ modes. Large amplitude $m=2$, $l=2$ pulsations may also be present, but the amplitudes of these pulsations are suppressed by a factor of $\sin^{|m|}{i} \approx 1/200$.
There are two other regimes where equation \[eq:finalphase\] does not apply. First, when oscillations are no longer standing waves but are instead in the traveling regime, the phases begin to deviate from equation \[eq:finalphase\], which we discuss in more detail in § \[sec:travel\]. Second, when an excited mode is in a resonance lock, it is possible that $\delta
\omega / \gamma_{nl} \sim 1$, allowing for a potentially arbitrary phase. Such a resonance lock can only occur when $m \neq
0$. For stars similar to KOI-54, each star is expected to be in a resonance lock only $\sim 10\,$percent of the time, and as discussed in B12, it is much rarer for both stars to be in a simultaneous lock.
As we have discussed, for most of the large-amplitude pulsations, only one eigenmode of a single star will contribute to the corresponding pulsations. However, *Kepler* photometry is so precise that we are able to detect pulsations with amplitudes comparable to $1\,\mu$mag. A fraction of these low-amplitude pulsations will have contributions from both stars or from multiple modes in a single star.
{width="\textwidth"}
Harmonic pulsation phases {#sec:harm}
-------------------------
### Standing waves
In Figure \[fig:scatterphase\], we plot the phases of the harmonic pulsations as well as the phases in a theoretical model from B12. The model was derived to qualitatively match the amplitudes and frequencies of observed pulsations in KOI-54. In both plots, the area of each point is in proportion to the observed amplitude of the pulsation, the error bars show the uncertainties in the phase, and the filled vertical regions of the plot show the phases expected for $m=0$ and $m=\pm2$ modes (§ \[sec:linintro\], equation \[eq:finalphase\]). For standing waves, with $k \equiv
f/{\ensuremath{\Omega_{\rm orb}}}\gtrsim 50$ (see § \[sec:travel\] for traveling waves), we expect the $m=0$ oscillations to have phases near $1/4$ and $3/4$ (§ \[sec:linintro\]). Many of the observed pulsations are consistent with this result (e.g., F1, F2, F7, F11, and F16 among others). In particular we find that the phases of the largest-amplitude 90th and 91st harmonics, $.752 \pm .0001\,$(statistical)$\,\pm 0.013\,$(systematic) and $.759 \pm .0001\,$(statistical)$\,\pm
0.013\,$(systematic[^7]) are consistent with being $m=0$ modes. If either of these pulsations were in a resonance lock, its phase could be arbitrary (when $\delta \omega / \gamma_{nl} \sim 1$; § \[sec:linintro\]).
There are a number of pulsations that appear to be $m \ne 0$ modes, e.g., F18, F32, F67, & F68. Not shown in the figure is the pulsation with $k = 171$ (F68) which has a phase near $0.5$. This is consistent with what we expect for a typical $m=2$ oscillation, especially given its low observed amplitude, which excludes it as originating from a resonance lock. The phase of the 105th harmonic (F67) is closer to the phase expected for an $|m| = 3$ oscillation. The phase of the 76th (F18) and 127th (F57) harmonics are consistent with $m=1$ modes.
There are at least four possible explanations for why some pulsations do not have phases that correspond to $m=0$ modes in Figure \[fig:scatterphase\]. 1) The oscillation is an $m\ne 0$, $l\ge2$ mode. 2) Given the large number of excited pulsations, we might expect a few of the pulsations from each star to interfere. This causes the observed phase to shift between the intrinsic phase of each star when $\delta_A \ne \delta_B$. The observed phase of the pulsation, $\delta_{AB}$, is $$\delta_{AB} = \frac{1}{2\pi} \cos^{-1}{\left(\frac{A_A \cos{2 \pi \delta_A}+A_B\cos{2 \pi \delta_B}}{A_{AB}}\right)}$$ where the amplitude of the resulting pulsation is $$A_{AB} = \sqrt{A_A^2+A_B^2+2A_AA_B\cos{2\pi(\delta_A-\delta_B})}.$$ 3) If the oscillation is primarily driven by nonlinear terms via three or higher mode coupling its phase will no longer reflect the expected linear offset (see § \[sec:nonlin\]). 4) Lastly, if it is an $m \ne 0$ oscillation that was shifted from the traveling wave regime because of the star’s rotation, then it could also have an arbitrary phase. For our outliers, this is possible only if $\Omega_* \gtrsim 13 {\ensuremath{\Omega_{\rm orb}}}$, which is satisfied by the expected rotation rate of the star (B12).
It is unlikely that 90th and 91st harmonic pulsations originate in the same star. For $k\sim 90$, the typical frequency spacing between the eigenfrequencies of $m=0$, $l=2$ g-modes is significantly larger than $\Delta k \sim 1$. Indeed, the g-mode eigenfrequencies of a star are expected to behave asymptotically as [@oscillations] $$\label{eq:asymp}
\omega_{ln} \approx \omega_0 \frac{l+1/2}{(n+\alpha)\pi},$$ where we assume that $l=2$ and $n \gg 1$. Assuming that $m=0$ for both the $k=90$ and $k=91$ pulsations, we can estimate where other large amplitude $m=0$ fluctuations should exist using stellar models or by tuning equation \[eq:asymp\]. Just as importantly, when a pulsation appears with the frequency between two consecutive eigenmodes, we can conclude $|m| > 0$, since it must have been shifted from stellar rotation by $m\Omega_*$. We can calibrate equation \[eq:asymp\] by finding the eigenfrequencies of stellar models with parameters near the best-fit found by W11. We use an untuned MESA stellar model [@2011ApJS..192....3P] for a star similar to those in KOI-54 to calibrate equation \[eq:asymp\] at $n\approx 10$–$20$. We find the best-fit parameters are $\alpha
\approx 1.07$ and $\omega_0 = 40.5\,$d$^{-1}$.
Assuming that $n=14$ for $k = 90$, we roughly estimate that the next few eigenfrequencies are near $83.5$, $79$, $74.5$, $70.6$, $63.8$, $60.8$, $58.1 {\ensuremath{\Omega_{\rm orb}}}$, where the uncertainty is of order $\Delta k
\approx 1$. Interestingly there are two large-amplitude pulsations in KOI-54 with $k=71$ and $k=72$, both with phases consistent with $m=0$, which may correspond to the $n=18$ modes of each star. There are no pulsations with amplitudes $\gtrsim 1.7\,\mu$mag between where we expect the $n=14$ and $n=15$ pulsations. Identifying the eigenfrequencies of the stars in this manner can greatly reduce the large computational overhead of finding stellar models that reproduce the properties of the two stars. The frequency spacing between the high-frequency pulsations, with $f \gtrsim 100{\ensuremath{\Omega_{\rm orb}}}$, is also consistent with the MESA models of B12.
Figure \[fig:scatterphase\] shows how the phases of the pulsations are a powerful tool to determine the nature of the pulsations (e.g., the azimuthal order, $m$, or whether a mode is in a resonance) as well as to verify the epoch of periastron and $\phi_0$.
### Traveling waves {#sec:travel}
![Fractional amplitude change ($\delta A/A$) of the $k = 90$ and $k = 91$ pulsations over time. The error bars show the estimated standard deviation of the fits for each individual time bin. The $k=91$ data points are offset from the center of the bin in order to increase legibility.[]{data-label="fig:ampchange"}](ampchangef.pdf){width="\columnwidth"}
![Distribution of anharmonic pulsation frequencies. Plotted is the fractional part of $f/{\ensuremath{\Omega_{\rm orb}}}$ as a function of $f/{\ensuremath{\Omega_{\rm orb}}}$ for the anharmonic pulsations of KOI-54. The area of each circle corresponds to the amplitude of the pulsation. Assuming that the nonlinear coupling is with the 91st harmonic, the complementary daughter modes to the anharmonic pulsations are shown as pink squares, where $f_{\rm complement}\equiv 91 {\ensuremath{\Omega_{\rm orb}}}- f$. Four pairs of the observed anharmonic pulsations add to $91 {\ensuremath{\Omega_{\rm orb}}}$ within their uncertainties: F5 & F6, F8 & F100, F25 & 76, and F39 & F42. The other pulsations either have complementary pairs that are not visible, or are the daughter modes of other pulsations. This figure only shows the pulsations that have frequencies $f/{\ensuremath{\Omega_{\rm orb}}}>
20$ and amplitudes $> 1.0\,\mu$mag.[]{data-label="fig:anharmonic"}](nonharmsgood2.pdf){width="\columnwidth"}
At lower frequencies, the group travel timescale of the wave becomes comparable to the damping timescale near the surface at the outer turning point of the mode. This results in a phase offset of the wave of order $\sim \sqrt{t_{\rm group} / t_{\rm damp}}$. As such, pulsation phases need not obey the relation in equation \[eq:phase\]. B12 estimated that this should occur for $k
\lesssim 50$ for stars similar to KOI-54 when $m = 0$. These estimates were made under the approximation of solid-body rotation. For differentially rotating bodies, the shear between layers could dramatically impact the onset of the traveling wave regime, resulting in much larger phase offsets for frequencies $\omega \lesssim 2 \Omega_*$ [@1989ApJ...342.1079G]. Only rapidly rotating stars, with much faster spins than expected for KOI-54 (B12), would impact the phases for frequencies much larger than $50{\ensuremath{\Omega_{\rm orb}}}$.
The horizontal dashed line in Figure \[fig:scatterphase\] marks where $k = 50$, approximately when the onset of the traveling regime is expected (B12). We see that below this frequency there are more observed pulsations with phases that are not consistent with $m=0$ standing waves. The phases of the two large-amplitude harmonics, $k=44$ (F3) and $k=40$ (F4), in particular, are significantly offset from the phases of the $k=90$ and $k=91$ pulsations. The phase offsets of traveling waves in KOI-54 appear comparable to the phase offsets in the best-fit semi-quantitative model of B12 (right panel of Fig. \[fig:scatterphase\]). For modes with $m \ne 0$, the traveling wave regime is determined by the frequency of the tide in the corotating frame, which is Doppler shifted from the observers frame by $k {\ensuremath{\Omega_{\rm orb}}}- m \Omega_*$. For $m > 0$, modes with $k\gtrsim 50$ can still be in the traveling wave regime, especially when it is rotating rapidly. In principle, the rotation frequency of the star may be inferred by identifying where the $m > 0$ modes enter the traveling wave regime.
We note again that as the pulsation amplitude decreases, it is more likely that modes from both stars will have comparable contributions to the pulsations. This can occur at any frequency, but is more likely at lower frequencies where the spacing between eigenfrequencies is smaller (eq. \[eq:asymp\]). When this happens, the observed phase will be determined by a combination of the pulsation amplitudes and phases.
Time variation of the pulsations {#sec:time}
--------------------------------
The estimated timescale for KOI-54 to circularize is very long, $\sim
10^{10}\,$yr, compared to the duration of the observations, so we do not expect variations in the orbital elements to be directly observable with the current data set. However, the amplitude and phases of the pulsations in KOI-54 can vary on much shorter timescales, especially if the pulsations are in a nonlinear limit cycle [@2001ApJ...546..469W] or are close to resonance ($\delta
\omega \sim \gamma$). We therefore search for time variation in the amplitudes, frequencies, and phases of the pulsations. To do so we split the data into 6 bins each with equal duration of $131\,$d, and refit each with the $60$ largest-amplitude pulsations. We omit 7 pulsations from these fits because their frequencies are spaced too closely together to resolve in the shorter duration of the observations. We measure the uncertainty in each fit in the same way as for the overall uncertainties: via a Monte Carlo method with a $1/f$ plus white noise spectrum (see § \[sec:osc\]).
Figure \[fig:ampchange\] shows how the amplitudes of the 90th and 91st harmonic pulsations vary with time. We find that their amplitudes declined by $3$ and $2$ percent, respectively, over the duration of the observations. The characteristic timescales associated with the changes are $A/{\dot A} \approx 60$ and $90$ years for the 90th and 91st harmonics, respectively. If the entire change in amplitude was systematic, we would expect the change to be the same in both pulsations, but they are different at the 3$\sigma$ level. We find similar variations in the amplitude of the two pulsations when using only four or five bins, or when looking at individual quarters of the *Kepler* data.
To check that our estimated errors are correct, we compare the distribution of $\delta A/A$ to the estimated error for all of the observed pulsations. Excluding the outliers beyond four standard deviations, we find the estimated errors are only 25 percent smaller than the variance in $\delta A/A$. Finally, if the changes are indeed systematic, we would expect the brightening events at pericenter to show a similar evolution. A $3\,$percent error in the absolute calibration of the source would reduce the peak amplitude of the tidally induced static tide and irradiation by $\approx 240 \,\mu$mag over the observed duration of the brightening events. We do not see signatures of this deviation in the residuals near the brightening events even if we only subtract the 20 brightest pulsations.
Such a large change in the amplitude of the pulsations is not expected. The timescale for the changes is comparable to the damping time for $n=14$ g-modes. Even if the oscillations are close to resonance, which we have already argued the $k=90$ and $k=91$ modes are likely not, the maximum change in amplitude expected is of order $$\label{eq:da}
\frac{\delta A}{A} \sim \frac{\Delta t}{t_{\rm tidal}} \frac{\omega_{nl}}{\gamma_{nl}},$$ where $\Delta t$ is the duration of the observations, $t_{\rm tidal}$ is the characteristic timescale for tides to change the orbital parameters, $\gamma_{nl} \approx 1 (n)^4\,$Myr$^{-1}$, with $n = 14$ for $k
\approx 90$ in KOI-54 (W11, B12). Alternatively we can invert equation \[eq:da\] to get $t_{\rm tidal}$, the timescale of the change in the orbital parameters. For $n= 14$ and $ \gamma \approx 4 \times
10^{-2}\,$yr$^{-1}$, $\Delta t \approx 3\,$yr, we derive $t_{\rm tidal}
\lesssim 10^{6}\,$yr. This is a strong upper limit, and is shorter than the expected synchronization timescale by a factor of $\sim 80$ (B12).
As we alluded to earlier, one possible explanation of our observations is that the modes in question are undergoing limit cycles as a result of parametric decay (§ \[sec:nonlin\]). This is expected to occur when the frequency offsets of the daughter modes are less than their damping rates [@2001ApJ...546..469W]. The timescale associated with the limit cycle, which is $\sim \gamma^{-1}$, is comparable to the inferred timescale for the amplitude changes.
However, in § \[sec:threemode\], we find no evidence to suggest that the $k=90$ pulsation is undergoing parametric decay. However, nonlinear second-order coupling between groups of oscillators can result in chaotic behavior in many circumstances. This can occur even when the oscillators have the same resonant frequency [e.g., @chimera].
More data or a more sophisticated detrending routine that explicitly preserves long term trends may be able to determine whether some the variation of the pulsation amplitudes is systematic or if the variations are intrinsic to KOI-54. Systematic variations of approximately one percent have been observed in the transit depths of the Hot Jupiter HAT-P-7b [@2013arXiv1307.6959V]. The systematic effects were found to be seasonal, relating to the rotation of the [*Kepler*]{} spacecraft. However, @2013arXiv1307.6959V found no evidence of systematic trends when comparing data between the same season. We find that the amplitudes of the 90th and 91st harmonic pulsations decreased with time, even when compared to the same season of data. Finally, we note that nodal regression owing to a third body could change the amplitudes of $m\ne 0$ pulsations by altering the inclination of the system. This would additionally cause changes to the amplitude of the equilibrium tide, which we do not observe.
Nonlinearly excited tides {#sec:nonlin}
=========================
Introduction {#sec:nonlinintro}
------------
If a mode achieves a large enough amplitude, the linearized fluid equations are no longer sufficient to describe its evolution. Instead, it may decay into daughter modes via the parametric instability. Traditionally, this has been calculated as a form of resonant three-mode coupling that minimizes the threshold for the instability to develop (e.g., @1982AcA....32..147D, @2001ApJ...546..469W, B12, @2011arXiv1107.4594F). These daughter modes increase the rate at which energy is damped in the star, potentially accelerating the impact of the tides. KOI-54 exhibits a variety of pulsations that have frequencies which are not integer harmonics of the orbital period. In §\[sec:threemode\], we will explore the distribution of anharmonic pulsations in KOI-54 in the context of isolated three-mode coupling. We will show in § \[sec:multimode\] that these anharmonic pulsations are best explained as the nonlinearly excited daughter modes of several different parent modes.
Modes can efficiently couple to each other in the second order when they are near resonance, i.e., when $$\label{eq:restrict0}
|\omega_A-(\omega_a+\omega_b)| = |\delta \omega_{Aab}| \ll |\omega_A|,$$ where we denote the primary parent mode with the capital subscript $A$ and the the pair of daughter modes as lowercase $a$ and $b$. They must also satisfy certain selection rules[^8]: $$\label{eq:restrict2}
m_A = m_a+m_b ,$$ $$\label{eq:restrict1}
\mod (l_A + l_a + l_b, 2) = 0,$$ and $$\label{eq:restrict3}
|l_a-l_b| \le l_{A} < l_a+l_b.$$
Not only do the intrinsic frequencies of the excited oscillations sum to the parent frequency, but equation \[eq:restrict2\] also implies that the frequencies of the observed daughter pulsations also sum to the parent frequency: $$\label{eq:obsthree}
f_a+f_b = f_A$$ For sufficiently small-amplitude oscillations, the daughter modes do not grow, and the instability is suppressed. The daughter modes will only grow in amplitude when the parent mode is sufficiently excited for the parametric instability to develop (see, e.g., B12).
All modes with $l_a=l_b$ have nonzero coupling coefficients. However, for KOI-54 the coupling becomes much less efficient when $l_{a,b} \gg
2$ (B12), so we restrict our attention to $l_{a,b}\le 3$. For parent modes with $m_A = 0$, equation \[eq:restrict2\] implies $m_a = -
m_b$. Although the onset of parametric decay occurs at relatively large amplitudes, the complete triplet of modes should not necessarily be visible in the [*Kepler*]{} observations. In many cases the parent mode or one of the daughter modes can have an amplitude that is much smaller than any of the other oscillations (see, e.g, B12). Additionally, because KOI-54 is nearly face on, the visibility of $m \ne
0 $ oscillations is suppressed by $\sin^{|m|}{i} \approx (1/10)^{|m|}$.
![Diagram of traditional, isolated three-mode coupling versus multiple-mode coupling. The [*top*]{} panel shows two sets of three coupled modes. In previous nonlinear analyses, these sets are treated independently of the other. The [*bottom*]{} panel shows a coupled system of five modes. It is the same as the top panel—both are coupled at second-order—except the two parent modes share the coupled daughter mode $b$. KOI-54 shows evidence for at least three parent modes coupled to each daughter mode. Because the coupling still occurs at second-order, the selection rules for strong coupling still apply to multiple-mode coupling. Namely, for the [ *bottom*]{} panel, $ f_a+f_b = f_A$, $m_a+m_b = m_A$, and additionally $f_b+f_d = f_B$ and $m_b+m_d = m_B$. \[fig:modecouple\]](modecouple-f.pdf){width="7cm"}
Three-mode coupling and the observed distribution of daughter pairs {#sec:threemode}
-------------------------------------------------------------------
The anharmonic pulsations in KOI-54 are best explained as daughter modes that are nonlinearly excited by higher-frequency tidally driven oscillations. As expected from equation \[eq:obsthree\], many of the anharmonic pulsations have frequencies that, when summed, equal the frequencies of some of the largest-amplitude harmonic pulsations within the formal precision of the measurements. The two most prominent are the F5 ($f_5 = 22.419{\ensuremath{\Omega_{\rm orb}}}$) and F6 ($f_6 =
68.582{\ensuremath{\Omega_{\rm orb}}}$) pulsations, which appear to be daughter modes of the 91st harmonic of the orbital period. These were the only two anharmonic pulsations reported in W11 that were consistent with being a complete daughter mode pair of the $91$st harmonic (B12, @2011arXiv1107.4594F). Here we report a total of four pairs of anharmonic pulsations which have frequencies that all add to $91{\ensuremath{\Omega_{\rm orb}}}$. We additionally find that the frequencies of many anharmonic pulsations sum to other harmonic parents, specifically $f_7 = 72{\ensuremath{\Omega_{\rm orb}}}$ and $f_{11} = 53{\ensuremath{\Omega_{\rm orb}}}$. These three parent modes in particular have frequencies that are equally spaced with $\Delta f/{\ensuremath{\Omega_{\rm orb}}}= \Delta k = 19$. We address this in more detail in § \[sec:multimode\].
In Figure \[fig:anharmonic\], we plot the frequency ($f / {\ensuremath{\Omega_{\rm orb}}}$) and fractional part of the frequency ($f/{\ensuremath{\Omega_{\rm orb}}}\mod {1.0}$) for each anharmonic pulsation with measured amplitude $> 1.2\,\mu$mag. The area of each circle is proportional to the amplitude of the pulsation. For each point in the graph, we also highlight the complementary “sister” frequency of the pulsation with a pink square, assuming that it is a daughter of the $k = 91$ harmonic. In other words, the pink squares represent frequencies that satisfy $f_{\rm
complement}\equiv 91 {\ensuremath{\Omega_{\rm orb}}}- f$. There are eight points that are highlighted in pink, corresponding to four complete daughter pairs of the 91st harmonic. The vast majority of the observed anharmonic pulsations, however, do not have complementary pairs with amplitudes above $1.2\,\mu$mag. As discussed in § \[sec:nonlinintro\], this is consistent with theoretical expectations. No two observed anharmonic pulsations add to $k = 90$.
There are two striking features of the distribution of frequencies plotted in Figure \[fig:anharmonic\]. First, many of the pulsations have regular frequency spacing. For all the completely observed daughter pulsation pairs, there are other observed anharmonic pulsations with frequencies separated by $\Delta f = \Delta k {\ensuremath{\Omega_{\rm orb}}}=
19 {\ensuremath{\Omega_{\rm orb}}}$. For example, $f_5+f_6 = f_1 = 91{\ensuremath{\Omega_{\rm orb}}}$, but another complementary mode sums to the fifth largest-amplitude harmonic pulsation, $f_5+f_{42} = (91-19){\ensuremath{\Omega_{\rm orb}}}= f_{7} = 72{\ensuremath{\Omega_{\rm orb}}}$, and to the sixth largest-amplitude harmonic pulsation $f_5+f_{76} = f_{11} = (91
- 2\times 19){\ensuremath{\Omega_{\rm orb}}}= 53{\ensuremath{\Omega_{\rm orb}}}$. A similar pattern emerges for the daughter pair of F8 and F100: $f_8+f_{100} = f_1$, $f_{49} + f_{100} =
f_7$ and $f_{47}+f_{100} = f_{11}$. Although it is possible that this is just a coincidence, we are compelled to suggest that all of these nonlinearly driven daughter pairs share multiple parent modes. In the case of KOI-54, three of most prominent parent modes are separated by $\Delta f = 19 {\ensuremath{\Omega_{\rm orb}}}$. This particular value of $19$ is probably set randomly by the eigenfrequency spacing unique to one of the stars. Additionally, each parent mode clearly has multiple daughter pairs. As we will discuss in § \[sec:multimode\], the coupling between multiple parents and daughters is one possible explanation for why the amplitude of the 91st harmonic is so much lower than the threshold estimated using only isolated three-mode coupling (B12).
The second salient feature of the anharmonic pulsations is that many share common fractional (noninteger) parts in units of the orbital frequency, as was first noted by W11. There are eight pulsations that have frequencies with fractional parts near $0.42$ (F5, F25, F39, F80, F87, F110, F124, & F128) , and nine observed pulsations that have frequencies with their fraction part $0.58$ (F6, F9, F42, F61, F63, F76, F102, F123, & F130). That is, if you sum the frequencies of one pulsation from each group you will get a harmonic (integer) frequency. There are also six pulsations with the fractional part of their frequency near $0.08$ (F8, F47, F49, F58, F59, & F75). There are eight near $0.84$ (F17, F19, F48, F71, F72, F86, F108, & F125). What would cause all the pulsations to share similar fractional parts? Again, we are led to the situation where each daughter is being nonlinearly excited by multiple parents. We explore this further in § \[sec:multimode\].
The anharmonic pulsations can also be susceptible to the parametric instability. In this scenario, the daughter modes will themselves couple to granddaughter modes, which must obey the same restrictions (eqs. \[eq:restrict2\]-\[eq:restrict3\]). We do not find evidence for granddaughter modes in KOI-54. However, it would be more difficult to observe daughters of the already low-frequency anharmonic pulsations in KOI-54. The frequency of the largest-amplitude anharmonic pulsation is close to the frequency cutoff we have imposed in our search for pulsations ($\approx 20{\ensuremath{\Omega_{\rm orb}}}$). Only a pulsation with amplitude $\gtrsim
10\,\mu$mag would be observable above all the noise at low frequencies.
Multiple-mode coupling in KOI-54 {#sec:multimode}
--------------------------------
{width="\columnwidth"} {width="\columnwidth"}
As described in § \[sec:threemode\], we find that the anharmonic pulsations have two features that are not explained in the traditional picture of three-mode coupling between individual overstable parent modes and their daughter modes. First, many anharmonic pulsations have frequency offsets from a perfect harmonic (residual fractions) that are nearly identical, and second, many observed pairs have common frequency differences of $\Delta f = 19 {\ensuremath{\Omega_{\rm orb}}}$. We propose that the distribution of frequencies in the anharmonic pulsations is naturally explained by coupling between [*groups*]{} of daughter pairs and parents. Figure \[fig:modecouple\] contrasts the case of isolated three-mode coupling, as we described in § \[sec:nonlinintro\], and multiple-mode coupling, as we describe further here.[^9] Throughout this section, we use the same terminology as B12.
A common simplifying assumption employed in deriving the nonlinear parametric instability is to consider coupling only within isolated mode triplets. We show this in the top diagram of Figure \[fig:modecouple\]. By minimizing over all possible sets of three modes, constrained by equations \[eq:restrict2\] – \[eq:restrict3\], one can estimate when the parametric instability should develop (e.g., @1982AcA....32..147D, @2001ApJ...546..469W). Recently, @2012ApJ...751..136W extended the traditional isolated three-mode coupling calculations as described in § \[sec:nonlinintro\] to include the case of a single parent with $N$ distinct daughter pairs. The authors found this can reduce the threshold for the parent to become unstable by $\approx 1/N$. More generally, however, each eigenmode of the star can additionally couple to multiple parent modes in second-order perturbation theory, as we show in the bottom diagram of Figure \[fig:modecouple\]. We explore this scenario by analyzing the system of equations for five coupled oscillators: two linearly driven parent modes and two daughter pairs that share a common eigenmode. We ask 1) does multiple-mode coupling result in the same quantitative peculiarities that we observed in § \[sec:threemode\]? and 2) does multiple-mode coupling lower the threshold for the parametric instability to develop?
Under the assumption of an equilibrium solution, i.e., that each eigenmode is a sinusoid with a fixed frequency $f_i$ and amplitude $A_i$, the observed frequencies will naturally reproduce the trends in the system. For a network of five oscillators coupled at second-order, the frequencies of the daughter pairs obey the relations[^10] $f_a + f_b = f_A$ and $f_b+f_d = f_B$ in order for the presumptive sinusoidal time dependence to cancel out. If the daughters are coupled to linearly driven parents, then both $f_A$ and $f_B$ will be harmonics of the orbital frequency, $k_{A,B} {\ensuremath{\Omega_{\rm orb}}}$. It then holds that $f_a-f_d = f_B - f_A$ must also be an integer multiple of ${\ensuremath{\Omega_{\rm orb}}}$. Thus, nonlinear multiple-mode coupling in eccentric binaries manifests itself with harmonic frequency differences between the anharmonic pulsations, just as we observe in KOI-54. This property is specific to dynamically excited tides in eccentric binaries. Parent oscillations that are driven by instabilities will not have integer spacing. As such, it becomes less likely that a daughter mode will couple to many parents.
Although it is a simple matter to derive the relationship between the frequencies of a network of five oscillators that are coupled to second-order, it is not clear if there is a closed-form analytic equilibrium solution for a network of five (or more) coupled oscillators, as the steady-state solution of three-mode coupling is already nontrivial. Instead, we integrate the coupled equations numerically to look at the consequences of multi-mode coupling on the steady state properties of the pulsations. In this calculation, we use realistic damping and driving rates as well as coupling coefficients computed in B12. The two parents are linearly driven by the tidal potential at $91{\ensuremath{\Omega_{\rm orb}}}$ and $72{\ensuremath{\Omega_{\rm orb}}}$. In this calculation, we use realistic damping and driving rates as well as coupling coefficients computed in B12. The two parents are linearly driven by the tidal potential at $91{\ensuremath{\Omega_{\rm orb}}}$ and $72{\ensuremath{\Omega_{\rm orb}}}$. Rather than trying to represent KOI-54 exactly, we arbitrarily choose the frequencies of the parents and daughter pairs to be close to resonance to demonstrate the impact of multi-mode coupling. We show the results of two calculations in Figure \[fig:pickle\]. In the two simulations, the intrinsic parent mode frequencies are ($91.295{\ensuremath{\Omega_{\rm orb}}}$, $72.044{\ensuremath{\Omega_{\rm orb}}}$) and ($90.615{\ensuremath{\Omega_{\rm orb}}}$, $71.996{\ensuremath{\Omega_{\rm orb}}}$). In one case, shown in the left panel, the system undergoes the parametric instability [ *only*]{} when the set of five coupled oscillators are solved. When we treat the oscillators as two independent triplets, the parents are stable, and no daughter modes are excited. In the right panel, we show a case where the amplitude of the parent that undergoes the parametric instability actually ends up pumping energy into an initially low-amplitude linearly driven oscillation (which was stable in the isolated three-mode calculations).
To summarize, some consequences of nonlinear multi-mode coupling are: 1) the frequencies of many anharmonic pulsations have the same residual fraction in units of the orbital frequency; 2) many anharmonic pulsations may share the same frequency separation; 3) the largest-amplitude harmonic pulsation is not necessarily the pulsation undergoing decay (i.e., the energy can be transported via daughter pairs to another linearly driven parent.); 4) if one parent $l=2$, $m=0$ is overstable, then all $l=2$, $m=0$ oscillations have some growth from nonlinear interactions even if all of their amplitudes are below the threshold for parametric decay. Furthermore, networks of coupled oscillators are more likely to enter limit cycles, where the modes do not reach a steady state, but rather transfer energy back and forth, sometimes chaotically [see, e.g., @2005PhRvD..71f4029B].
Since the nonlinear coupling is constrained by equations \[eq:restrict2\]-\[eq:restrict3\], we can use the observed anharmonic frequencies to group the pulsations into their respective modes by degree $l$ and azimuthal order $m$. For example, we have identified the 91st harmonic as an $l=2, m=0$ oscillation (§ \[sec:harm\]). Hence, the daughters pairs that couple to the 91st harmonic in KOI-54 must have $m_a=-m_b$, and most likely have $l_{a,b} \le 3$. Those pulsations with frequencies that have noninteger parts near $0.42{\ensuremath{\Omega_{\rm orb}}}$ (e.g., F5) most likely have the same degree, and must have the opposite azimuthal order of those with noninteger parts near $0.58{\ensuremath{\Omega_{\rm orb}}}$ (such as F6) since they have similar observed amplitudes. The pulsations with noninteger parts near $0.08$ (F8) must belong to a different chain of non-linear interactions from F5. Because the F5 and F6 pulsations are so prominent, we can speculate that these two pulsations have the smallest degree, $l=1$ or $l=2$, as low degree pulsations are more easily observable. There are very few pulsations observed with noninteger parts near the “sister” pair to F8, F100. One possible reason is because they are $l=3$ pulsations, which have higher damping rates and have lower observed amplitudes since they are averaged out over the disk of the star.
The complicated nature of a network of coupled oscillators makes analyzing nonlinear systems such as KOI-54 challenging. In particular, if there was a mode creating a resonance lock, then it may only be visible through the nonlinear interactions it has with its daughter modes, because its observed amplitude may be reduced by projection effects (§ \[sec:linintro\]) and nonlinear interactions (§ \[sec:nonlinintro\]). The two anharmonic pulsations with frequencies larger than $91{\ensuremath{\Omega_{\rm orb}}}$, F51 and F97, are likely further evidence of multiple mode-coupling. In isolated three-mode coupling the resonant parametric instability can only develop for daughter modes with frequencies less than the parent mode.
Additional nonlinear effects
----------------------------
In linear theory, the observed frequency of a tidally driven oscillation is always a perfect harmonic of the orbital period. However, nonlinear effects can cause the observed frequency of the oscillation to deviate from a perfect harmonic. Many of the pulsation frequencies in Tables \[tab:fitsh\] & \[tab:fitsa\] have uncertainties smaller than one part per million. Although we have not determined the orbital frequency with a comparable level of precision, we can still compare two harmonic pulsations to each other and assess whether their frequencies deviate from perfect harmonics. For the two largest-amplitude pulsations, we see that $f_{91}-91/90
f_{90}\approx 3.31 \pm 0.06\times 10^{-5}\,$day$^{-1}$. This means that they are inconsistent with both being perfect harmonics of the orbital frequency by $\sim 50 \sigma$. Similarly, the 76th and 57th harmonics deviate from perfect harmonics with respect to the $90$th harmonic by $\sim 15\sigma$ ($10\,$ppm) and $3\sigma$ ($7\,$ppm) respectively. These deviations are further evidence of nonlinear interactions in KOI-54.
Multiple-mode coupling may also be an important factor in some of the observed amplitudes and phases of the harmonic pulsations. For example, at harmonics that lie between two eigenmodes, $|\delta
\omega|$ is at a maximum, which means that the linear prediction for the amplitude of the corresponding tidally driven pulsation is small (see Fig 9. of B12). In this case, the second-order coupling between such an oscillation and other tidally driven oscillations (as well as nonlinearly driven oscillations) may be its dominant source of energy. That is, second-order effects are likely important for all oscillations with low intrinsic amplitudes. This not only impacts the observed amplitude of the pulsation, but also its phase.
In addition, if a parent mode undergoes a significant amount of nonlinear interaction, with a nonlinear amplitude threshold much less than the expected linear amplitude, then the observed phase of the pulsation may not reflect its intrinsic, linear phase (as we assumed in § \[sec:harm\]). On the other hand, if we find that the phase offset is near the expected linear phase, as with the 90th and 91st harmonic pulsations, then it is likely that its dominant source of energy is the linear tide. We still cannot be certain that the 91st harmonic is the dominant parent that is driving all the nonlinear effects in KOI-54. It has the largest observed amplitude, but as we see in Figure \[fig:pickle\], the mode that is undergoing parametric decay can have an intrinsic amplitude less than the other pulsations in the system, especially for large $m$ or $l$ oscillations.
Finally, even higher-order nonlinear effects may be present in KOI-54. As we saw in the introduction of this section, the second-order coupling between three modes is restricted. In § \[sec:multimode\], we saw how these selection rules combined with nonlinear multiple-mode couplings naturally led to the frequency spacing of the anharmonic pulsations. Likewise, third-order coupling between four or more modes may also impact the observed frequencies of the anharmonic pulsations. For example, there are linear combinations of three pulsations which add to harmonic pulsations, $f_{17}+f_{6}-f_{5} = f_7$ ($25.846+68.582-22.419 = 72.009$), as well as many anharmonic pairs that have frequencies that add to half harmonics. Theoretical analysis of higher-order nonlinear mode coupling may provide deeper insight into these relations. Furthermore, future observations of other similar eccentric binaries [e.g., @2012ApJ...753...86T; @2013arXiv1306.1819H] may show similar features.
Summary and discussion {#sec:disc}
======================
In this paper we have explored the nature and origin of the harmonic and anharmonic pulsations of the eccentric binary KOI-54. We used 785 days of nearly continuous [*Kepler*]{} observations of KOI-54 to find and report over 120 sinusoidal pulsations with frequencies $f \ge 20
{\ensuremath{\Omega_{\rm orb}}}$ and amplitudes as small as $\sim 1\,\mu$mag. We then used the phase offset of each harmonic pulsation with respect to the epoch of periastron to identify the azimuthal order, $m$, of the the corresponding modes of oscillation. We also explored the properties of the anharmonic pulsations; in particular, we focused on the unique distribution of the pulsation frequencies in KOI-54.
One open question regarding KOI-54 was the nature of the two most prominent pulsations that have frequencies that are the 90th and 91st harmonics of the orbital period. It had been suggested that the large amplitudes of these two pulsations may be explained if they were responsible for a resonance lock between the stars’ spins and their orbital motion (@2011arXiv1107.4594F; B12). In § \[sec:harm\], we used the phase offset of these pulsations relative to the epoch of periastron to show that they are most likely $m=0$ modes. Such oscillations are axisymmetric and thus cannot be responsible for a resonance lock. Furthermore, we found that most of the high frequency $f \gtrsim 50\,{\ensuremath{\Omega_{\rm orb}}}$ harmonic pulsations in KOI-54 are $m=0$ oscillations. This is consistent with the prior theoretical expectation that, since KOI-54 is nearly face-on, only the largest-amplitude $m=\pm 2$ oscillations are observable.
Oscillations with frequencies $f\lesssim 50\,{\ensuremath{\Omega_{\rm orb}}}$ have characteristic damping timescales comparable to their group travel time. They are thus expected to be traveling waves rather than standing waves (B12). In § \[sec:travel\], we compared the phase offset of the low-frequency pulsations in KOI-54 to a theoretical model derived in B12. We indeed found that the observed phase offsets were consistent with the pulsations being traveling waves, precisely where B12 predicted the oscillations to transition to the traveling wave regime. Because differential rotation can dramatically impact on the onset of the traveling wave regime, we conclude that the approximation of solid-body rotation made in B12 was correct.
In § \[sec:time\] we systematically searched for time variations in the pulsations’ amplitudes and frequencies. We found that the amplitudes of the 90th and 91st harmonics have decreased by $3\,\%$ and $2\,\%$, respectively, over the duration of the observations, corresponding to timescales of $\approx 60$ and $90$ yr, respectively. This is a much larger change than is expected from linear tidal theory alone. The 91st harmonic is clearly nonlinearly coupled to many daughter modes (§ \[sec:nonlin\]), and may be in a limit cycle [@2001ApJ...546..469W], which can cause such rapid variation. However, the 90th harmonic shows no evidence for nonlinear coupling.
KOI-54 has a rich population of anharmonic pulsations, which are naturally explained as being driven by the nonlinear parametric instability. Whenever a parent mode exceeds the amplitude threshold for the parametric instability to develop, it excites daughter mode pairs, which have frequencies that sum to the parent mode frequency. Indeed, we observed four complete daughter pairs of the 91st harmonic and several more for the 72nd and 53rd harmonics (§ \[sec:threemode\]). However, there are no two anharmonic pulsations with frequencies that sum to the 90th harmonic. This is consistent with the 90th and 91st harmonic pulsations as originating in different stars in the binary, and only one star undergoing the parametric instability, although it is unclear why this would be true since they have similar observed amplitudes.
Although it was expected that nonlinear interactions can result in multiple sets of daughter modes, we found strong evidence that the daughter modes simultaneously couple to multiple parent modes at second order (§ \[sec:multimode\]). Many of the anharmonic pulsation frequencies are spaced by harmonics of the orbital frequency. We showed that a network of oscillators coupled at the second-order naturally reproduce the characteristic distribution of frequencies we observe in KOI-54.
Furthermore, we found through numerical experiments that coupling among multiple modes can lower the threshold for the onset of nonlinear interactions, and thereby accelerate the impact of tides, similar to what happens when a single parent couples to multiple daughters [see @2012ApJ...751..136W]. Because tidally driven oscillations are naturally spaced as perfect harmonics of the orbital frequency, we expect multiple-mode coupling to be more common and more important in highly eccentric systems, such as KOI-54.
Nonlinear effects in eccentric binaries are expected to play an important role in redistributing the energy and angular momentum between the orbit and stars [@1996ApJ...466..946K; @1998ApJ...507..938G; @2010MNRAS.404.1849B; @2012ApJ...751..136W]. Since we have identified the 90th and 91st harmonic pulsations in KOI-54 as $m=0$ modes, we would like to estimate how much energy they dissipate compared to the anharmonic pulsations in the system. Unfortunately, we do not know the azimuthal order of the anharmonic pulsations, and so we only know their minimum, projected amplitude, which is $\propto
\sin^{|m|}i \approx (1/10)^{|m|}$. Even with this uncertainty, we can compare the relative damping rates of the modes if we use the scaling relation that $\gamma \propto \omega^{-4}$ (confirmed for adiabatic modes in B12), and we also assume that the observed pulsation amplitude is a constant times the intrinsic amplitude of the oscillation. Using the units of B12, the energy dissipation rate in each mode is $\propto A^2\gamma \propto A^2\omega^{-4}$. With these rather uncertain assumptions, the largest nonlinear mode [*may*]{} dissipate $(A_5/A_1)^2 (f_5/f_1)^{-4} \sim 20$ times as much energy as the largest amplitude linearly excited mode. Even if the asymptotic relationship for the damping rate of the modes was much shallower, e.g., $\gamma \propto \omega^{-2}$, the two modes would have dissipation rates that were comparable.
We have explored the nature of the observed pulsations in KOI-54 without using well tuned stellar models of the stars themselves. Instead we focused on using qualitatively similar stellar models first developed in B12, and the asymptotic relations of high-order g-modes in order to understand the eigenmodes of KOI-54. We have identified many to be $m=0$ modes of the system using the phase information of the pulsations. We have also been able to determine that the two largest amplitude pulsations at the $91$st and $90$th harmonic of the orbital frequency come from different stars. In particular we have identified eight individual anharmonic pulsations that belongs to the star with the large-amplitude $m=0$ mode at the 91st harmonic of the orbital frequency. Most, if not all, of the other anharmonic pulsations belong to this star as well. In future work, it may be possible to perform much more precise modeling of KOI-54 using these constraints, as well as in other identified systems [@2012ApJ...753...86T; @2013arXiv1306.1819H] with similar analyses.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank K. Burns, E. Petigura, and E. Quataert for useful discussions. This paper includes data collected by the *Kepler* mission. Funding for the *Kepler* mission is provided by the NASA Science Mission directorate. All of the *Kepler* data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. R.O. is supported by the National Aeronautics and Space Administration through Einstein Postdoctoral Fellowship Award Number PF0-110078 issued by the Chandra X-ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. J.B. is an NSF Graduate Research Fellow.
[@lcrrcrr@]{} ID & Frequency & $f / {\ensuremath{\Omega_{\rm orb}}}$ & Amplitude & Phase & Phase & Phase\
& (day$^{-1}$) & & ($\mu$mag) & $(t=0)$ & (BF) & (W11)\
[@lcrrcrr@]{} ID & Frequency & $f / {\ensuremath{\Omega_{\rm orb}}}$ & Amplitude & Phase & Phase & Phase\
& (day$^{-1}$) & & ($\mu$mag) & $(t=0)$ & (BF) & (W11)\
[@lcrrcrr@]{} ID & Frequency & $f / {\ensuremath{\Omega_{\rm orb}}}$ & Amplitude & Phase & Phase & Phase\
& (day$^{-1}$) & & ($\mu$mag) & $(t=0)$ & (BF) & (W11)\
[^1]: Einstein Fellow
[^2]: E-mail: [email protected]
[^3]: In practice our search routine looks for the pulsations with the most power in a fast [@1989ApJ...338..277P] Lomb–Scargle periodogram .
[^4]: We choose this specific value of the frequency offset because no observed pulsations were observed with an offset between $0.019{\ensuremath{\Omega_{\rm orb}}}$ and $0.035{\ensuremath{\Omega_{\rm orb}}}$.
[^5]: We distinguish the intrinsic changes within the star, i.e., [*oscillations*]{}, from the extrinsically observed [*pulsations*]{}.
[^6]: This equation is derived from Eq. 33 of B12. B12 incorrectly included a $\pm$ on the right hand side of Eq. 33, which we have omitted here. In addition, B12 defined the phase in radians using the cosine function.
[^7]: The systematic uncertainty includes the uncertainty in the epoch of periastron, and is proportional to the pulsation frequency.
[^8]: In this work, we assume that all the frequencies are positive. In B12 the daughter frequencies are considered to be negative, such that $\omega_A+\omega_a+\omega_b = \delta_{Aab}$. In this convention, equation \[eq:restrict2\] would instead read $m_A+m_a+m_b = 0$.
[^9]: Note that the multiple-mode coupling scenario we are describing is still second-order perturbation theory, and involves only three-mode coupling coefficients (see Fig. \[fig:modecouple\]).
[^10]: The derivation is similar to the calculation of the equilibrium solution to three-mode coupling in Appendix D of @2012ApJ...751..136W.
|
---
abstract: 'We obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class of Metropolis-Hastings algorithms which satisfy a central limit theorem and invariance principle when the variance is not linear in $n$.'
author:
- 'George Deligiannidis [^1]'
- 'Magda Peligrad[^2] [^3]'
- 'Sergey Utev [^4]'
title: Asymptotic variance of stationary reversible and normal Markov processes
---
Introduction
============
Let $({\xi}_{n})_{n\in\mathbb{Z}}$ be a stationary Markov chain defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with values in a general state space $(S,\mathcal{A})$ and let the marginal distribution be denoted by $\pi(A)=\mathbb{P}({\xi}_{0}\in A)$. We assume that there is a regular conditional distribution denoted by $Q(x,A)=\mathbb{P}({\xi}_{1}\in A|\,{\xi
}_{0}=x)$. Let $Q$ also denote the Markov transition operator acting via $(Qg)(x)=\int_{S}g(s)Q(x,\mathrm{d}s)$, on $\mathbb{L}_{0}^{2}(\pi)$, the set of measurable functions on $S$ such that $\int_{S}g^{2}(s)\pi(\mathrm{d}s)<\infty$ and $\int_{S}g(s)\pi(\mathrm{d}s)=0.$ If $g, h\in \mathbb{L}
_{0}^{2}(\pi)$, the integral $\int_{S}g(s)h(s)\pi(\mathrm{d}s)$ will sometimes be denoted by $\langle g,h\rangle$.
For some function $g \in \mathbb{L}_{0}^{2}(\pi)$, let $$X{_{i}=g(\xi}_{i}{),\quad S_{n}(}X{)=\sum\limits_{i=1}^{n}}X_{i},
\quad\sigma _{n}({g)}=(\mathbb{E}S_{n}^{2}(X{)})^{1/2}. \label{defcsi}$$ [ Denote by $\mathcal{F}_{k}$ the $\sigma$–field generated by $\xi_{i}$ with $i\leq k$. ]{}
For any integrable random variable $X$ we denote by $\mathbb{E}_{k}X=\mathbb{E}(X|\mathcal{F}_{k}).$ With this notation, $\mathbb{E}_{0}X_{1}=Qg($[$\xi$]{}$_{0})=\mathbb{E}(X_{1}|$[$\xi$]{}$_{0}).$ We denote by ${{\|X\|}_{p}}$ the norm in [$\mathbb{L}^{p}$]{}$(\Omega,\mathcal{F},\mathbb{P}).$
The Markov chain is called *normal* when the transition operator $Q$ is normal, that is it commutes with its adjoint $Q^{\ast},$ namely $QQ^{\ast}=Q^{\ast}Q$.
From the spectral theory of normal operators on Hilbert spaces (see for instance [@R]), it is well known that for every $g\in L_{0}^{2}(\pi )$ there is a unique *transition spectral measure* $\nu $ supported on the spectrum of the operator $D:=\{z\in \mathbb{C}:|z|\leq 1\}$, such that $$\mathrm{cov}(X_{0},X_{n})=\mathrm{cov}((g({\xi }_{0}),Q^{n}g({\xi }_{0}))=\langle g,Q^{n}g\rangle =\int_{D}z^{n}\nu (\mathrm{d}z). \label{cov}$$and $$\mathrm{cov}(\mathbb{E}_{0}(X_{i}),\mathbb{E}_{0}(X_{j}))=\langle
Q^{i}g,Q^{j}g\rangle =\langle g,Q^{i}(Q^{\ast })^{j}g\rangle =\int_{D}z^{i}\bar{z}^{j}\nu (\mathrm{d}z).$$
In particular, the Markov chain is reversible if $Q=Q^{\ast}.$ The condition of reversibility is equivalent to requiring that $($[$\xi$]{}$_{0},$[$\xi$]{}$_{1})$ and $({\xi}_{1},{\xi}_{0}) $ have the same distribution. Furthermore, in the reversible case $\nu$ is concentrated on $[-1,1]$.
Limit theorems for additive functionals of reversible Markov chains have received considerable attention in the literature not only for their intrinsic interest, but also for their great array of applications which range from interacting particle systems (see the seminal paper by Kipnis and Varadhan [@KV]) and random walks in random environments (see for example [@Toth]), to the relatively recent applications in computational statistics with the advent of Markov Chain Monte Carlo algorithms (e.g. [@HaggRos; @RR]). Limit theorems have appeared under a great array of conditions, notably geometric ergodicity (for an overview see [@KM]), conditions on the growth of the conditional expectations $\mathbb{E}(S_{n}|X_{1})$ (see e.g. [@MW], [@PU]), under spectral conditions (see [@KV; @GL; @DL]) or under conditions on the resolvent of the transition operator (see [@Toth]), a method which is also applicable in the non-normal case where spectral calculus may not apply.
The variance of the partial sums plays a major role in limit theorems where it acts as a normalizer, and also in computational statistics where the asymptotic variance is used as a measure of the efficiency of an algorithm (see e.g. [@Tierney]). It is not surprising then, that in certain cases, conditions for the central limit theorem have been imposed directly on the growth of the variance. In fact in 1986 Kipnis and Varadhan [@KV] proved the functional form of the central limit theorem for functionals of stationary reversible ergodic Markov chains under the assumption that $$\lim_{n\rightarrow \infty }\frac{\mathrm{var}(S_{n})}{n}=\sigma _{g}^{2},
\label{condvar}$$and further established necessary and sufficient conditions for the variance of the partial sums to behave linearly in $n$ in terms of the transition spectral measure $\nu $. In particular they showed that for any reversible ergodic Markov chain the convergence in is sufficient for the functional central limit theorem $S_{[nt]}/\sqrt{n}\Rightarrow |\sigma
_{g}|W(t)$ (where $W(t)$ is the standard Brownian motion, $\Rightarrow $ denotes weak convergence and $[x]$ denotes the integer part of $x$). Moreover, is equivalent to the fact that the finite limiting variance is then given by $$\sigma _{g}^{2}=\int\nolimits_{-1}^{1}\frac{1+t}{1-t}\nu (\mathrm{d}t)<\infty . \label{SR}$$Furthermore, according to Remark 4 on page 514 in [@DL] if in addition to (\[SR\]) we assume $\rho (-1)=0$ then, we also have $$\lim_{n\rightarrow \infty }{\sum\limits_{i=0}^{n}}\mathrm{cov}(X_{0},X_{i})=\int\nolimits_{-1}^{1}\frac{1}{1-t}\nu (\mathrm{d}t).$$See also [@HaggRos] for a discussion of when and are equivalent.
It is remarkable that in the reversible case, conditions and are equivalent, both sufficient for the central limit theorem and invariance principle, and conjectured to be sufficient for the almost sure conditional central limit theorem.It is an open problem, whether is also necessary for the central limit theorem.
On the other hand, notice that any invertible transformation $T$ generates a unitary, and thus normal, transition operator $Qf(x)=f(T(x))$, since $Q^{\ast }f(x)=f(T^{-1}(x))$ whence $QQ^{\ast }=Q^{\ast }Q=I$ is the identity operator. In particular, any stationary sequence $\xi _{i}$, can be treated as a functional of a normal Markov chain. Therefore for normal, non-reversible Markov chains, and the central limit theorem and invariance principle are no longer equivalent without further assumptions (see e.g. Bradley [@Bra] and Giraudo and Volný [@GiVo] for counterexamples).
For the non-reversible case, Gordin and Lifšic [@GL] applied martingale methods and stated, among other results, the central limit theorem for functionals of stationary ergodic Markov chains with normal transition operator, under the spectral condition $$\int_{D}\frac{1}{|1-z|}\nu (\mathrm{d}z)<\infty . \label{eq:GLcond}$$If condition (\[eq:GLcond\]) holds then (\[condvar\]) also holds with $$\sigma ^{2}:=\int_{D}\frac{1-|z|^{2}}{|1-z|^{2}}\nu (\mathrm{d}z)<\infty .
\label{sigma}$$One of our main results, Theorem \[pr:NSC\], gives necessary and sufficient conditions for the existence of the limit $\mathrm{var}(S_{n})/n\rightarrow K<\infty $. We shall see that $\mathrm{var}(S_{n})/n\rightarrow K$ if and only if $\sigma ^{2}<\infty
$ and $\nu (U_{x})/x\rightarrow C$ as $x\rightarrow 0^{+}$, where $$U_{x}=\{z=(1-r)e^{iu}\;:|z|\leq 1,\;0\leq r\leq |u|\leq x\}.$$In this case $K=\sigma ^{2}+{\pi C}$. Furthermore if ([eq:GLcond]{}) holds then $C=0.$
Recently Zhao et al. [@ZWV] and Longla et al. [@LPP], in the context of reversible Markov chains, studied the asymptotic behavior of $S_{n}$ for the more general case when the variance of partial sums behaves as a regularly varying function $\sigma _{n}^{2}=\mathrm{var}(S_{n})=nh(n)$ where $h(x)$ is slowly varying, i.e. $h:(0,\infty )\rightarrow (0,\infty ),$ continuous, and $h(st)/h(t)\rightarrow 1$ for all $s>0$. For this case the situation is different and in [@ZWV; @LPP] examples are given of stationary, reversible, and ergodic Markov chains that satisfy the CLT under a normalization different of $\sigma _{n},$ namely $S_{n}/\sigma
_{n}\Rightarrow N(0,c^{2})$ for a $c\neq 1$ and $c\neq 0$.
On the other hand, in a recent paper, Deligiannidis and Utev [@DU] have studied the relationship between the variance of the partial sums of weakly stationary processes and the spectral measure induced by the unitary shift operator. To be more precise, by the Birghoff-Herglotz Theorem (see e.g.Brockwell and Davis [@BD]), there exists a unique measure on the unit circle, or equivalently a non-decreasing function $F,$ called the *spectral distribution function* on $[0,2\pi ]$, such that $$\mathrm{cov}(X_{0},X_{n})=\int_{0}^{2\pi }\mathrm{e}^{\mathrm{i}n\theta }F(\mathrm{d}\theta ),\quad \text{for all}\,\,n\in \mathbb{Z}\,. \label{SpM}$$If $F$ is absolutely continuous with respect to the normalized Lebesgue measure $\lambda $ on $[0,2\pi ]$, then the Radon-Nikodym derivative $f$ of $F$ with respect to the Lebesgue measure is called the *spectral density;* in other words $F(\mathrm{d}\theta )=f(\theta )\mathrm{d}\theta $,. The main result of [@DU] is given below. In the sequel, the notation $a_{n}\sim b_{n}$ as $n\rightarrow \infty $ means that $\lim_{n\rightarrow
\infty }a_{n}/b_{n}=1.$
**Theorem A**. \[Deligiannidis and Utev [@DU]\]\[thm:DU\] *Let $S_{n}:=X_{1}+\cdots+X_{n}$ where $(X_{i})_{i\in\mathbb{Z}}$ is a [real]{} weakly stationary sequence. For $\alpha\in(0,2)$, define $C(\alpha):=\Gamma(1+\alpha)\sin(\tfrac{\alpha\pi}{2})/[\pi(2-\alpha )]$, and let $h$ be slowly varying at infinity. Then $\mathrm{var}(S_{n})\sim n^{\alpha}h(n)$ as $n\rightarrow\infty$ if and only if $F(x)\sim {\frac{1}{2}}C(\alpha)x^{2-\alpha}h(1/x)$ as $x\rightarrow0$.*
In this paper we obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of stationary Markov chains with normal operators. The necessary and sufficient conditions are based on several different representations in terms of:
1. the spectral distribution function in the sense of the Birghoff-Herglotz theorem,
2. the transition spectral measure of the associated transition operator,
3. the harmonic measure of Brownian motion in the disk,
4. a martingale decomposition.
In the case of stationary reversible Markov Chains we also construct a class of Metropolis-Hastings algorithms with non-linear growth of variance, for which we establish the invariance principle and conditional central limit theorem with normalization $\sqrt{nh(n)}$. *Continuous-time processes.* In the continuous time setting, let $\{\xi
_{t}\}_{t\geq 0}$ be a stationary Markov process with values in the general state space $(S,\mathcal{A})$, defined on a probability space $(\Omega ,\mathcal{F},\mathrm{P})$, with stationary measure $\pi $. We assume that the contraction semigroup $$T_{t}g(x):=\mathbb{E}[g(\xi _{t})|\xi _{0}=x],\quad g\in L^{2}(\pi ),\quad
t\geq 0,$$is strongly continuous on $L^{2}(\pi )$, and we let $\{\mathcal{F}_{t}\}_{t\geq 0}$ be a filtration on $(\Omega ,\mathcal{F},\mathrm{P})$ with respect to which $\{\xi _{t}\}_{t}$ is progressively measurable and satisfies the Markov property $\mathrm{E}(g(\xi _{t})|\mathcal{F}_{u})=T_{t-u}g(\xi _{u})$, for any $g\in L^{2}(\pi )$ and $0\leq u<t$. Furthermore we can write $T_{t}=\mathrm{e}^{Lt}$, where $L$ is the infinitesimal generator of the process $\{\xi _{t}\}_{t}$, and $\mathcal{D}(L)$ its domain in $L^{2}(\pi )$. We assume $T_{t}$ to be normal, that is $T_{t}^{\ast }=T_{t}$, which then implies that $L$ is a normal, possibly unbounded operator, with spectrum supported in the left half-plane $\{z\in
\mathbb{C}:\Re (z)\leq 0\}$ (see [@R Theorem 13.38]). In the reversible case the spectrum of $L$ is supported on the left real half-axis(see [@KV Remark 1.7]).
Similarly to the discrete case, with any $f\in L^2(\mu)$ we can associate a unique spectral measure $\nu(\mathrm{d} z) = \nu_f (\mathrm{d} z)$ supported on the spectrum of $L$ such that $$\langle f, T_t f\rangle = \mathrm{cov}( f(\xi_0), f(\xi_t)) =
\int_{\Re(z)\leq 0} \mathrm{e} ^{z t} \nu (\mathrm{d} z).$$
In the reversible case Kipnis and Varadhan [@KV] proved an invariance principle under the condition that $f\in \mathcal{D}\big( (-L)^{-1/2}\big)$, which in spectral form is equivalent to $$\int_{\alpha=-\infty}^{0}\frac{-1}{\alpha}\nu(\mathrm{d}\alpha)<\infty .
\label{eq:1/a}$$ Building on the techniques in [@GL; @KV], Holzmann [@HH; @HH2] established the central limit theorem for processes with normal transition semi-groups (see also [@O]), under the condition $$\int_{\Re(z)\leq0}|z|^{-1}\nu(\mathrm{d}z)<\infty. \label{eq:1/z}$$ In this case $$\lim_{T\rightarrow\infty}\frac{\mathrm{var}(S_{T})}{T}= -2\int_{\mathbb{H}^{-}}\Re(1/z)\nu(\mathrm{d}z)=:\varsigma^{2}.$$ On the other hand, using resolvent calculus, Toth [@Toth2; @Toth] treated general discrete and continuous-time Markov processes and obtained a martingale approximation, central limit theorem and convergence of finite-dimensional distributions to those of Brownian motion, under conditions on the resolvent operator which may hold even in the non-normal case. Similar conditions, albeit in the normal case, also appeared later in [@HH; @HH2].
Under any of the above conditions, it is clear that the variance of $S_{T}$ is asymptotically linear in $T$. Similarly to the discrete case, we show in Theorem \[pr:NSCcts\], that $\mathrm{var}(S_n)/n \to K = \varsigma^2 + \pi
C$ if and only if $\varsigma^2<\infty$ and $\nu(U_x)/x\to C$, where $$U_{x} = \{ a+ \mathrm{i} b: 0\leq-a\leq|b|\leq x\}.$$
The rest of the paper is structured as follows. We provide our results for discrete time processes in Section 2 and for continuous time in Section 3. Section 4 contains the proofs, while the Appendix contains two standard Tauberian theorems to make the text self-contained, and technical lemmas used in Section 3.
Results for Markov chains
=========================
Relation between the transition spectral measure and spectral distribution function
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Our first result gives a representation of the spectral distribution function in terms of the transition spectral measure. This link makes possible to use the results in [@DU] to analyze the variance of partial sums. Quite remarkably, if the transition spectral measure is supported on the open unit disk, the spectral distribution function is absolutely continuous with spectral density given by , and in this case the sequence $\mathrm{cov}(X_{0},X_{n})$ converges to $0$.
\[thm:normaldensity\] Let $({\xi}_{n})_{n\in \mathbb{Z}}$ be a stationary Markov chain, with normal transition operator $Q$. Let $g\in
L_{0}^{2}(\pi)$, $X_{i}:=g(\xi_{i})$ and write $\nu=\nu_{g}$ for the operator spectral measure with respect to $g$. Also denote the unit circle $\Gamma:=\{z:|z|=1\}$ and by $D_{0}:=\{z:|z|<1\}.$ Denote by $\nu_{\Gamma}$ the restriction of measure $\nu$ to $\Gamma$ and by $\nu_{0}$ denote the restriction of measure $\nu$ to $D_{0}.$ Then $$\mathrm{cov}(X_{0},X_{n})=\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}tn}[\nu
_{\Gamma}(\mathrm{d}t)+f(t)\mathrm{d}t],$$ where $$f(t)={\frac{1}{2\pi}}\int_{D_{0}}\frac{1-|z|^{2}}{|1-z\mathrm{e}^{\mathrm{i}t}|^{2}}\nu_{0}(\mathrm{d}z). \label{eq:normaldensity}$$ Furthermore the spectral distribution function has the representation $$F(\mathrm{d}t)=\nu_{\Gamma}(\mathrm{d}t)+f(t)\mathrm{d}t. \label{SMR}$$
By integrating relation (\[SMR\]) we obtain$$F(x)=x\int_{D_{0}}T_{x}(z)\nu_{0}(\mathrm{d}z)+\nu_{\Gamma}([0,x])
\label{SpMe}$$ where $$T_{x}(z):=\frac{1}{2\pi}(1-|z|^{2})\int_{0}^{1}\frac{dt}{|1-z\mathrm{e}^{\mathrm{i}tx}|^{2}}\text{.}$$
By combining Representation Lemma \[thm:normaldensity\] with Theorem A we obtain the following corollary.
\[cor5\]Let $({\xi}_{n})_{n\in\mathbb{Z}}$ be as in Lemma [thm:normaldensity]{} and let $\alpha\in(0,2).$ Then $\mathrm{var}(S_{n})=n^{\alpha}h(n)$ as $n\rightarrow\infty$ if and only if $F(x)={\frac{1}{2}}{C(\alpha
)}x^{2-\alpha}h(1/x)$ as $x\rightarrow0^{+}.$
It should be obvious from the statement of Representation Lemma [thm:normaldensity]{} that Theorem A is directly applicable to the measure $\mathrm{d} F$. The conditions on $F,$ mentioned in Corollary \[cor5\], when expressed in terms of the operator spectral measure, become technical conditions on the growth of integrals of the Poisson kernel over the unit disk. To get further insight into this lemma we shall apply it to reversible Markov chains.
Our next result, a corollary of Representation Lemma \[thm:normaldensity\] combined with Theorem A, provides this link and points out a set of equivalent conditions for regular variation of the variance for reversible Markov chains. In fact, as it turns out, if the spectral measure has no atoms at $\pm1$, it follows that the spectral distribution function is absolutely continuous and we obtain an expression for the spectral density. Related ideas, under more restrictive assumptions have appeared in [@JB], while in [@DL] a spectral density representation was obtained for positive self-adjoint transition operators, in other words $\nu$ supported on $[0,1)$.
\[prop:SpDens\] Assume that $Q$ is self-adjoint and that the transition spectral measure $\nu$ does not have atoms at $\pm1.$ Then, the spectral distribution function $F$ defined by (\[SpM\]) is absolutely continuous with spectral density given by $$f(t)={\frac{1}{2\pi}}\int_{-1}^{1}\frac{1-\lambda^{2}}{1+\lambda^{2}-2\lambda\cos t}\mathrm{d}\nu(\lambda), \label{eq:revdensity}$$ and for $\alpha\in\lbrack1,2),$ the following are equivalent:
(i)
: $\mathrm{var}(S_{n})=n^{\alpha}h(n)\text{ as }n\rightarrow\infty,
$
(ii)
: $F(x)={\frac{1}{2}}{C(\alpha)}x^{2-\alpha}h(1/x)\text{ as }x\rightarrow0^{+}. $
Moreover, if $h(x)\rightarrow\infty$ as $x\rightarrow0^{+}$, then (i), (ii) are equivalent to
$\displaystyle \int_{0}^{1}\frac{r(\mathrm{d}y)}{x^{2}+y^{2}}={\frac{\pi}{{2}} C(\alpha )}x^{1-\alpha}h(1/x)+O(1)\text{ as }
x\rightarrow0^{+}$; where $r(0,y]=\nu(1-y,1)$.
Relation between spectral measure and planar Brownian motion
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Our next result makes essential use of the Poisson kernel which appears in to provide a fascinating interpretation of the spectral distribution function $F$ in terms of the harmonic measure of planar Brownian motion started at a random point in the open unit disk.
\[thm:harmonicmeasure\] Let $\nu$ be the transition spectral measure, and let $(B_{t}^{z})_{t\geq0}$ be standard planar Brownian motion in $\mathbb{C}$, started at the point $z\in D$. Also let $Z$ be a random point in $D$ distributed according to $\nu$ and let $\tau_{D}^{Z}:=\inf\{t {\geq}
0:B_{t}^{Z}\notin D\}$. Let $\Gamma_{x}:= \{z: z=\mathrm{e}^{\mathrm{i}t y},
|y|<x\}$ and $\alpha\in(0,2).$ Then, the following statements are equivalent:
(i)
: $\mathrm{var}(S_{n})\sim n^{\alpha}h(n)$ as $n\rightarrow \infty$;
(ii)
: $\mathrm{P}\big\{B_{\tau_{D}^{Z}}^{Z}\in\Gamma _{x}\big\}\sim
C(\alpha)x^{2-\alpha}h(1/x)/\nu(D)$ as $x\rightarrow0$.
Linear growth of variance for partial sums for normal Markov Chains
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By applying martingale techniques we establish necessary and sufficient conditions for the asymptotic linear variance behavior for general normal Markov chains.
\[pr:NSC\] With the notation of Lemma \[thm:normaldensity\]
(a)
: The limit, $\mathrm{var}(S_{n})/n\rightarrow K<\infty$ exists if and only if $$\begin{aligned}
\sigma^{2}&:=\int_{D}\frac{1-|z|^{2}}{|1-z|^{2}}\nu(\mathrm{d}z)<\infty,
\qquad \text{and} \label{normalsigma} \\
I_{n}&:=\frac{1}{n}\int_{D}\frac{|1-z^{n}|^{2}}{|1-z|^{2}}\nu(\mathrm{d}z)\rightarrow L, \label{new_lin}\end{aligned}$$ where $K=\sigma^{2}+L$ $.$
(b)
: Moreover, under (\[normalsigma\]) the following are equivalent:
(i)
: (\[new\_lin\]) holds with $L={\pi C}$*.*
(ii)
: $\nu (U_{x})/x\rightarrow C$* as* $x\rightarrow 0^{+}$*, where* $$U_{x}=\{z=(1-r)e^{\mathrm{i}u}\in D\;:\;0\leq r\leq |u|\leq x\}.$$
(iii)
: $n\nu (D_{n})\rightarrow C$* as* $x\rightarrow 0^{+}$*, where* $$D_{n}=\{z=re^{2\mathrm{i}\pi \theta };\text{ }1-\frac{1}{n}\leq r\leq 1,\text{ }-\frac{1}{n}\leq \theta \leq \frac{1}{n}\}.$$
It should be noted that there are many sufficient conditions for the convergence $$\mathrm{var}(S_{n})/n\rightarrow\sigma^{2}<\infty. \label{popular}$$
\(1) It is immediate from the proof of Theorem \[pr:NSC\], that (\[popular\]) is equivalent to $\sigma^{2}<\infty$ and $$\frac{1}{n}\int_{D}\frac{|1-z^{n}|^{2}}{|1-z|^{2}}\nu(\mathrm{d}z)\rightarrow0. \label{cl_linear:LG}$$ (2) In Corollary 7.1 in [@BI] it was shown that if we assume $$\int_{D}\frac{1}{|1-z|}\nu(\mathrm{d}z)<\infty,$$ then both (\[normalsigma\]) and (\[cl\_linear:LG\]) are satisfied and so convergence (\[popular\]) holds, the result attributed to Gordin and Lifšic [@GL] (see Theorem 7.1 in [@BI]; see also [@DL]).
\(3) From Representation Lemma \[thm:normaldensity\] and Ibragimov version of Hardy-Littlewood theorem $$\mathrm{var}(S_{n})/n\rightarrow2\pi f(0)=\sigma^{2}.$$
\(4) On the other hand, from the Representation Lemma [thm:normaldensity]{} and Theorem A, convergence (\[popular\]) is equivalent to the uniform integrability of $T_{x}(z)$ with respect to $\nu _{0}$ as $x\rightarrow0^{+}$.
\(5) Motivated by the complex Darboux-Wiener-Tauberian approach (e.g. as in [@DU11]), by analyzing $$V(\lambda)=\sum_{n=1}^{\infty}\mathrm{var}(S_{n})\lambda^{n}\;=\frac{\lambda
}{(1-\lambda)^{2}}\int_{D}\frac{1+\lambda z}{1-\lambda z}\nu(dz),$$ a sufficient condition for (\[popular\]) is $$\int_{D}\frac{1+\lambda z}{1-\lambda z}\nu(dz)\rightarrow\sigma^{2}=\int _{D}\frac{1+z}{1-z}\nu(dz)\text{ as }\lambda\rightarrow1\text{ with }|\lambda|<1.$$ Here, since $\nu(\mathrm{d}z)=\nu(\mathrm{d}\bar{z})$, the integral is understood in the Cauchy sense i.e. $$\int_{D}f(z)\nu(dz)=\frac{1}{2}\int_{D}[f(z)+f(\bar{z})]\nu(dz).$$
\(6) From Theorem \[pr:NSC\], it follows that is equivalent to $\sigma ^{2}<\infty $ and $\nu (U_{x})/x\rightarrow 0$ as $x\rightarrow 0^{+}$.
\(7) Finally, again from Theorem \[pr:NSC\], it follows that is equivalent to $\sigma^{2}<\infty$ and $n\nu(D_{n})\rightarrow0$ as $n\rightarrow\infty$. This result is also a corollary of the following inequality motivated by Cuny and Lin [@CL] $$\frac{n\nu(D_{n})}{36}\leq\frac{1}{n}\mathbb{E}(\mathbb{E}_{1}(S_{n}))^{2}\leq\frac{4}{n}{\displaystyle\sum\limits_{j=1}^{n-1}}j\nu(D_{j}).$$
Notice that when the transition spectral measure $\nu$ is concentrated on $\Gamma$ (dynamical system), then $\sigma^{2}=0$ and so $\sigma^{2}$ cannot be the limiting variance, in general.
By inspecting the proof of the Theorem \[pr:NSC\], under we are able to characterize regular variation of $\mathrm{var}(S_{n})$ when $\liminf_{n\rightarrow \infty }\mathrm{var}(S_{n})/n>0$. More exactly, we have the following proposition.
Assume $\liminf_{n\rightarrow \infty }\mathrm{var}(S_{n})/n>0$ and that holds. Then, for $\alpha \in \lbrack 1,2)$, and a positive function $h$, slowly varying at infinity, $\mathrm{var}(S_{n})\sim
n^{\alpha }h(n)$ as $n\rightarrow \infty $ if and only if $\nu (U_{x})\sim
C(\alpha )x^{2-\alpha }h(1/x)$ as $x\rightarrow 0^{+}$, with $C(\alpha )$ as defined in Theorem *A*. In particular, $\mathrm{var}(S_{n})/n$ is slowly varying at $n\rightarrow \infty $ if and only if $\nu (U_{x})/x$ is slowly varying as $x\rightarrow 0^{+}$.
Relation between the variance of partial sums and transition spectral measure of reversible Markov chains
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We continue the study of stationary reversible Markov chains and provide further necessary and sufficient conditions for its variance to be regularly varying, in terms of the operator spectral measure by a direct approach, without the link with the spectral distribution function.
\[thR(x)\]Assume $Q$ is self-adjoint, $\alpha \geq 1$, $\mathrm{var}(S_{n})/n\rightarrow \infty $, and let $c_{\alpha }:=\alpha (2-\alpha
)/2\Gamma (3-\alpha )$. Then $$V(x)=\int\nolimits_{-1}^{1-x}\frac{1}{1-t}\nu (\mathrm{d}t)\sim c_{\alpha
}x^{1-\alpha }h(\frac{1}{x})\text{ as }x\rightarrow 0^{+}$$if and only if $$\mathrm{var}(S_{n})=n^{\alpha }h(n)\text{ as }n\rightarrow \infty .$$Furthermore if $\alpha >1$ then $\mathrm{var}(S_{n})=n^{\alpha }h(n)$ as $n\rightarrow \infty $ iff $$\nu (1-x,1]\sim d_{\alpha }x^{2-\alpha }h(\frac{1}{x})\text{ as }x\rightarrow 0^{+},$$where $d_{\alpha }:=\alpha (\alpha -1)/2\Gamma (3-\alpha )$.
It should be obvious from the statement in the above theorem, that regular variation of the variance is equivalent to regular variation of the transition spectral measure only in the case $\alpha>1$. As the following example demonstrates, in the case $\alpha=1$, there are reversible Markov chains whose variance of partial sums varies regularly with exponent $1$ even though $\nu(1-x,1]$ is not a regularly varying function.
Take a probability measure $\upsilon$ on $[-1,1]$ defined for $0<a<1/2$ by $$\mathrm{d}\upsilon=\frac{1}{c}(1-|x|) \Big(1+a\sin[\ln(1-|x|)]+a\cos[\ln(1-|x|)]\Big)\mathrm{d}x. \label{def-niu}$$ where $c$ is the normalizing constant. Then, the unique invariant measure is $$\mathrm{d}\pi=\frac{\mathrm{d}\upsilon}{\theta(1-|x|)}=\frac{1}{2} \Big(1+a\sin[\ln(1-|x|)]+a\cos [\ln(1-|x|)]\Big)\mathrm{d}x.$$ We first compute the following integral $$\begin{aligned}
\int_{0}^{1-x}\frac{\mathrm{d}\pi}{1-y} & =\frac{1}{2}\int_{x}^{1}\frac{1}{t}\Big(1+a\sin[\ln(t)]+a\cos[\ln(t)]\Big)\mathrm{d}x \\
& =-\frac{1}{2}\ln x+O(1)\text{ as }x\rightarrow0,\end{aligned}$$ whence, by Theorem \[thR(x)\]$$\lim_{n\rightarrow\infty}\frac{\mathrm{var}(S_{n})}{n\log n}=c.$$ However, the covariances are not regularly varying because the spectral measure is not. To see why it is enough to show that $r(x)$ is not regularly varying at $0$. Indeed, if we take $y_{k}=e^{-2\pi k}\rightarrow0^{+},$ and $y_{k}=e^{\pi/2-2\pi k}\rightarrow0^{+},$ then $r(y_{k})=y_{k}$ and $r(y_{k})/y_{k}\rightarrow1.$ However, for the choice $r(z_{k})=z_{k}(1+2\alpha),$ we have $r(z_{k})/z_{k}\rightarrow1+2\alpha,$ and hence the spectral measure is not regularly varying.
Often in the literature, conditions for the linear growth of the variance are given in terms of the covariances (see for example [@HaggRos]). As it turns out, one can construct positive covariance sequences such that $\sum_{k=0}^{n}\mathrm{cov}(X_{0},X_{k})=h(n)$ is slowly varying, and hence the variance is regularly varying, but $a_{n}=\mathrm{cov}(X_{0},X_{n})>0,$ is not slowly varying. To construct such a chain, suppose that $\varepsilon
_{n}$ is an oscillating positive sequence such that $\varepsilon
_{n}\rightarrow 0$ and $\sum_{k}a_{k}=\infty $ where $a_{k}:=\varepsilon
_{k}/k$. Then $g_{n}=\sum_{k=1}^{n}a_{k}$ is slowly varying since $$g_{[bn]}=g_{n}+\sum_{i=n+1}^{[bn]}\frac{\varepsilon _{i}}{i}=g_{n}+O\big(\varepsilon _{n+1}\log b\big).$$So, a priori we have many situations when $\mathrm{var}(S_{n})=nh(n)$ even though the covariances (and hence the operator spectral measure) are not regularly varying.
The above proof was direct in the sense, that it relied only on the use of classical Tauberian theory without linking the transition spectral measure with the spectral distribution function, and thus without invoking the results of [@DU].
Examples of limit theorems with non-linear normalizer
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As an application we construct a class of stationary irreducible and aperiodic Markov Chains, based on the Metropolis-Hastings algorithm, with $\mathrm{var}(S_{n})\sim n h(n)$. Markov chains of this type are often studied in the literature from different points of view, in Doukhan et al. [@Douk], Rio ([@Rio1] and [@Rio2]), Merlevède and Peligrad [@mp], Zhao et al. [@ZWV]) and Longla et al. [@LPP].
Let $E=\{|x|\leq1\}$ and define the transition kernel of a Markov chain by $$Q(x,A)=|x|\delta_{x}(A)+(1-|x|)\upsilon(A),$$ where $\delta_{x}$ denotes the Dirac measure and $\upsilon$ is a symmetric probability measure on $[-1,1]$ in the sense that for any $A\subset
\lbrack0,1]$ we have $\upsilon(A)=\upsilon(-A).$ We shall assume that $$\label{eq:theta}
\theta=\int_{-1}^{1}\frac{1}{(1-|x|)}\upsilon(\mathrm{d}x)<\infty.$$ We mention that $Q$ is a stationary transition function with the invariant distribution $$\mu(\mathrm{d}x)=\frac{1}{\theta(1-|x|)}\upsilon(\mathrm{d}x).$$ Then, the stationary Markov chain $(\xi_{i})_{i}$ with values in $E$ and transition probability $Q(x,A)$ and with marginal distribution $\mu$ is reversible and positively recurrent. Moreover, for any odd function $g$ we have $Q^{k}(g)(x)=|x|^{k}g(x)$ and therefore $$Q^{k}(g)(\xi_{0})=\mathbb{E}(g(\xi_{k})|\xi_{0})=|\xi_{0}|^{k}g(\xi_{0})\text{ a.s.} \label{operator}$$ For the odd function $g(x)=\mathrm{sgn}(x)$, define $X_{i}:=\mathrm{sgn}(\xi_{i})$. Then for any positive integer $k$ $$\langle g,Q^{k}(g)\rangle={\int\nolimits_{-1}^{1}}|x|^{k}\mu(\mathrm{d}x)=2{\int\nolimits_{0}^{1}}x^{k}\mu(\mathrm{d}x),$$ and so on $[0,1]$, $2\mu$ coincides with the transition spectral measure $\nu,$ associated to $Q$ and $g$. Furthermore, the operator $Q$ is of positive type $\nu\lbrack-1,0)=0$. In other words $$\nu= \left\{
\begin{array}{c}
2\mu\text{ on }[0,1] \\
0\text{ on }[-1,0)\end{array}
\right. .$$ Therefore, by Theorem \[thR(x)\] applied with $\alpha=1$, $\mathrm{var}(S_{n})=nh(n)$ with $h(n)\rightarrow\infty$ slowly varying at $n\rightarrow\infty$ if and only if $$V(x)=\int_{0}^{1-x}\frac{\nu(\mathrm{d}y)}{1-y} =\int_{0}^{1-x}\frac {2\mu(\mathrm{d}y)}{1-y}\sim\frac{1}{2} h\big(\frac{1}{x}\big),$$ is slowly varying as $x\rightarrow0^{+}$.
Our next result presents a large class of transition spectral measures for the model above which leads to functional central limit theorem.
\[thm:MH\] Let $V(x)$ be slowly varying as $x\rightarrow0^{+}$. Then, the central limit theorem, the functional central limit theorem and the conditional central limit theorem hold for partial sums of $X_{i}$ defined above.
Next, we give a particular example of a Metropolis-Hastings algorithm in which a non-degenerate central limit theorem holds under a certain normalization. However, when normalized by the standard deviation we have degenerate limiting distribution.
**Example.** For $0<x<1$, we take the slowly varying function $V(x)=\exp (\sqrt{\ln (1/x)})$. By Theorem \[thR(x)\], as $n\rightarrow \infty $ $$\mathrm{var}(S_{n})=2nV(1/n)(1+o(1)).$$On the other hand, let us choose $b_{n}$ such that $nE[\tau _{1}^{2}I(\tau
_{1}\leq b_{n})]/b_{n}^{2}\sim 1$ as $n\rightarrow \infty $. By Lemma [lem:aux]{} it follows that $2\theta nV(1/b_{n})\sim b_{n}^{2}$, with $\theta $ as defined in Eq. . Note now that $$\begin{aligned}
2\theta nV(1/b_{n})& =2\theta n\exp (\sqrt{\ln (b_{n})})=2\theta n\exp (\sqrt{(1/2)\ln (4nV(1/b_{n})}) \\
& =2\theta n\exp (o(1)+\sqrt{(1/2)\ln n}+(1/2)^{3/2}),\end{aligned}$$which implies that $b_{n}^{2}\sim 2n\theta \exp (\sqrt{(1/2)\ln n})$, giving the following CLT: $$S_{n}/[2n\exp (\sqrt{(1/2)\ln n})^{1/2}\Rightarrow N(0,1).$$However $$\frac{b_{n}^{2}}{nV(1/n)}\rightarrow 0\quad \text{and therefore\ }\frac{S_{n}}{\sqrt{\mathrm{var}(S_{n})}}\rightarrow ^{P}0.$$
Continuous-time Markov processes
================================
Suppose we have a stationary Markov process $\{\xi_{t}\}_{t\geq0}$, with values in the general state space $(S, \mathcal{A})$, and for $g \in
L_{0}^{2}(\pi)$ let $T_{t} g(x) := \mathbb{E} [ g(\xi_{t})| \xi_{0}=x]$. Further $T_{t} = \mathrm{e}^{Lt}$, where $L$ is the infinitesimal generator which we assume to be normal, which then implies that its spectrum is supported on $\{z\in\mathbb{C}: \Re(z) \leq0\}$, such that $$\mathrm{cov}(f(\xi_{t}), f(\xi_{0})) = \int_{\Re(z)\leq0} \mathrm{e}^{z t}
\nu(\mathrm{d} z).$$ Finally define $S_{T}(g):= \int_{s=0}^{T} g(\xi_{s}) \mathrm{d} s$.
The following result is a continuous time analogue of Theorem [thm:harmonicmeasure]{}, linking the spectral distribution function with the harmonic measure of planar Brownian motion.
\[pr:harmoniccts\] Let $\{\xi_t\}_t$ be a stationary Markov process with normal generator $L$, invariant measure $\pi$, and let $g\in L^2(\pi)$ and $\nu=\nu_f$ be the transition spectral measure associated with $L$ and $g$. Write $(B_{t}^{z})_{t\geq0}$ for a standard planar Brownian motion in $\mathbb{C}$, started at the point $z\in\mathbb{H}^{-}:= \{z\in\mathbb{C}: \Re(z)\leq 0\}$. Also let $Z$ be a random point in $\mathbb{H}^{-}$ distributed according to $\nu$ and let $\tau_{\mathbb{H}^{-}}^{Z}:=\inf\{t {\geq} 0:B_{t}^{Z}\notin\mathbb{H}^{-}\}$. For $\alpha\in(0,2)$, the following statements are equivalent:
*(i)*
: $\mathrm{var}(S_{T}(g))\sim T^{\alpha}h(T)$ as $n\rightarrow \infty$;
*(ii)*
: $\mathrm{P}\big\{B_{\tau _{D}^{Z}}^{Z}\in (-\mathrm{i}x,\mathrm{i}x)\big\}\sim C(\alpha )x^{2-\alpha }h(1/x)/\nu (\mathbb{H}^{-})$ as $x\rightarrow 0^{+}$.
The following theorem gives a necessary and sufficient condition in terms of the transition spectral measure $\nu$. Define for $x>0$, $$U_{x} := \{ a+ \mathrm{i} b: 0\leq-a\leq|b|\leq x\}.$$ and let $$\varsigma^{2}:= -2\int_{\mathbb{H}^{-}}\Re(1/z)\nu(\mathrm{d}z).$$
\[pr:NSCcts\]With the notation of Theorem \[pr:harmoniccts\] the following are equivalent:
(i)
: $\mathrm{var}(S_{T}(g))/T \to L = \varsigma^{2} +K$, where $K>0$;
(ii)
: $\varsigma ^{2}<\infty $ and $\nu (U_{x})/x\rightarrow K/\pi $ as $x\rightarrow 0^{+}$.
In addition, if $\varsigma^2<\infty$ and $\liminf_{T\to\infty}\mathrm{var}(S_{T})/T = \infty$, then $\mathrm{var}(S_{T})\sim T^{\alpha}h(T)$, for $\alpha\geq1$ and $h$ slowly varying at infinity, if and only if $\nu(U_{x})\sim C(\alpha) x^{2-\alpha }h(1/x)$.
Proofs
======
**Proof of Representation Lemma \[thm:normaldensity\].** For $t\in\lbrack-\pi,\pi]$ and $z\in D_{0}$ define the function $$f(t,z):={\frac{1}{2\pi}}\Bigg[1+\sum_{k=1}^{\infty}\Big(z^{k}\mathrm{e}^{\mathrm{i}tk}+\bar{z}^{k}\mathrm{e}^{-\mathrm{i}tk}\Big)\Bigg]={\frac {1}{2\pi}}\frac{1-|z|^{2}}{|1-z\mathrm{e}^{\mathrm{i}t}|^{2}},$$ the Poisson kernel for the unit disk. Our approach is to integrate on $D_{0}$ with respect to $\nu_{0}(\mathrm{d}z)$, obtaining in this way a function defined on $[0,2\pi]$ as follows $$f(t)={\frac{1}{2\pi}}\int_{D_{0}}\left( 1+\sum_{k=1}^{\infty}\Big(z^{k}\mathrm{e}^{\mathrm{i}tk}+\bar{z}^{k}\mathrm{e}^{-\mathrm{i}tk}\Big)\right)
\nu_{0}(\mathrm{d}z)={\frac{1}{2\pi}}\int_{D_{0}}\frac{1-|z|^{2}}{|1-z\mathrm{e}^{\mathrm{i}t}|^{2}}\nu_{0}(\mathrm{d}z).$$ The function is well defined since we are integrating the positive Poisson kernel over the open disk, and in fact, by using polar coordinates, we also have $$\begin{aligned}
\int_{0}^{2\pi}f(t)\mathrm{d}t & ={\frac{1}{2\pi}}\int_{s=0}^{2\pi}\int
_{r=0}^{1^{-}}\int_{t=0}^{2\pi}\frac{1-r^{2}}{1-2r\cos(s+t)+r^{2}}\mathrm{d}t\,\nu_{0}(\mathrm{d}r,\mathrm{d}s) \\
& \leq{\frac{1}{2\pi}}\int_{s=0}^{2\pi}\int_{r=0}^{1^{-}}\frac{2\pi(1-r^{2})}{1-r^{2}}\,\nu_{0}(\mathrm{d}r,\mathrm{d}s)={2\pi}<\infty.\end{aligned}$$ Therefore, it is obvious that $f\in L^{1}(0,2\pi)$, and it makes sense to calculate $$\begin{aligned}
\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}tn}f(t)\mathrm{d}t & ={\frac{1}{2\pi}}\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}tn}\int_{D_{0}}\Big[1+\sum_{k=1}^{\infty}\Big(z^{k}\mathrm{e}^{\mathrm{i}tk}+\bar{z}^{k}\mathrm{e}^{-\mathrm{i}tk}\Big)\Big]\nu_{0}(\mathrm{d}z)\mathrm{d}t \label{comp} \\
& =\int_{D_{0}}z^{n}\nu_{0}(\mathrm{d}z). \notag\end{aligned}$$ Because of the decomposition $$\mathrm{cov}(X_{0},X_{n})=\int_{D}z^{n}\mathrm{d}\nu(z)=\int_{D_{0}}z^{n}\mathrm{d}\nu_{0}(z)+\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}tn}\mathrm{\ }\nu_{\Gamma}(\mathrm{d}t),$$ by (\[comp\]) we obtain that$$\mathrm{cov}(X_{0},X_{k})=\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}tn}f(t)\mathrm{d}t+\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}tn}\nu_{\Gamma }(\mathrm{d}t).$$ Now, by (\[SpM\]) the spectral distribution function $F$ associated with the stationary sequence $(X_{i})_{i}$, is then given by (\[SMR\]). $\square$
**Proof of Corollary \[prop:SpDens\].** The result follows from Representation Lemma \[thm:normaldensity\] and Theorem A. To obtain the last point in the theorem, by standard analysis, the spectral measure has the following useful asymptotic representation $$\begin{aligned}
F(x) & =\int_{0}^{x}f(t)\mathrm{d}t={\frac{1}{2\pi}}\int_{0}^{x}\int_{-1}^{1}\frac{1-\lambda^{2}}{1+\lambda^{2}-2\lambda\cos t}\nu(\mathrm{d}\lambda)dt \\
& =O(x)+{\frac{x}{\pi}}\int_{0}^{1}r(y)\frac{\mathrm{d}y}{x^{2}+y^{2}}\text{
\ \ as }x\rightarrow0^{+}.\end{aligned}$$ So, we derive $F(x)\sim {\frac{1}{2}}C(\alpha)x^{2-\alpha}h(1/x)$ if and only if $$\int_{0}^{1}r(y)\frac{\mathrm{d}y}{x^{2}+y^{2}}\sim{\frac{\pi}{{2}} C(\alpha)}x^{1-\alpha }h(1/x)+O(1)\text{ \ \ as }x\rightarrow0^{+}.$$
**Proof of Theorem \[thm:harmonicmeasure\].** As usual, let $D$ be the closed unit disk, and $D_{0}$ its interior. From Representation Lemma \[thm:normaldensity\] the spectral density $f\in L_{1}([-\pi ,\pi ])
$ is given by the formula $$f(t)={\frac{1}{2\pi }}\int_{D_{0}}\frac{1-|z|^{2}}{|1-z\mathrm{e}^{\mathrm{i}t}|^{2}}\nu (\mathrm{d}z).$$Notice that $D$ is regular for Brownian motion, in the sense that all points in $\Gamma =\partial D$ are regular, i.e. for all $z\in \partial D$ and for $\tilde{\tau}_{D}^{z}:=\inf \{t>0:B_{t}^{z}\notin D\}$ we have $\mathrm{P}^{z}\{\tilde{\tau}_{D}^{z}=0\}=1$. The harmonic measure in $D$ from $z$ is the probability measure on $\partial D$, $\mathrm{hm}(z,D;\cdot )$ given by $$\mathrm{hm}(z,D;V)=\mathrm{P}^{z}\{B(\tau _{D}^{z})\in V\},$$where $\mathrm{P}^{z}$ denotes the probability measure of Brownian motion started at the point $z$, and $V$ is any Borel subset of $\partial D$.
Since $\partial D$ is piecewise analytic, $\mathrm{hm}(z,D;\cdot )$ is absolutely continuous with respect to Lebesgue measure (length) on $\partial
D$ and the density is the Poisson kernel (see for example [@Law]). In the case of the unit disk $D$ the density for $\mathrm{hm}(z,D;\cdot )$ for $z\in D$, $w\in \partial D$ or $t\in \lbrack 0,2\pi ]$, is given by $$H_{D}(z,w)=\frac{1}{2\pi }\frac{1-|z|^{2}}{|w-z|^{2}}=\frac{1}{2\pi }\frac{1-|z|^{2}}{|1-e^{it}z|^{2}}.$$Let $Z$ be a $D$-valued random variable with probability measure $\nu $ properly normalized, independent of the Brownian motion. Then $$\begin{aligned}
\int_{-x}^{x}f(t)\mathrm{d}t& ={\frac{1}{2\pi }}\int_{D_{0}}\int_{-x}^{x}\frac{1-|z|^{2}}{|1-z\mathrm{e}^{\mathrm{i}t}|^{2}}\mathrm{d}t\,\nu _{0}(\mathrm{d}z) \\
& =\int_{D_{0}}\mathrm{P}^{z}\{B(\tau _{D}^{z})\in (-x,x)\}\nu _{0}(\mathrm{d}z).\end{aligned}$$On the other hand, since $\Gamma =\partial D$ is regular for Brownian motion, we have for all $z\in \Gamma $, $\mathrm{P}^{z}(B(\tau _{D}^{z})\in
\Gamma _{x})=1$ if $z\in \Gamma _{x}$ and $0$ otherwise. Thus $$\nu (\Gamma _{x})=\int_{\Gamma _{x}}\nu _{\Gamma }(\mathrm{d}z)=\int_{\Gamma
}\mathrm{P}^{z}(B(\tau _{D}^{z})\in \Gamma _{x})\nu _{\Gamma }(\mathrm{d}z).$$Therefore from Representation Lemma \[thm:normaldensity\] we have $$\begin{aligned}
G(x):=\int_{-x}^{x}F(\mathrm{d}x)& =\int_{D_{0}}\mathrm{P}^{z}\{B(\tau
_{D}^{z})\in \Gamma _{x}\}\nu _{0}(\mathrm{d}z)+\int_{\Gamma }\mathrm{P}^{z}(B(\tau _{D}^{z})\in \Gamma _{x})\nu _{\Gamma }(\mathrm{d}z) \\
& =\nu (D)\mathrm{P}\Big(B^{Z}(\tau _{D}^{Z})\in \Gamma _{x}\Big)=:\nu
(D)H_{\nu ,D}(\Gamma _{x}).\end{aligned}$$The measure $H_{\nu ,D}(\cdot )$, is essentially the harmonic measure when Brownian motion starts at a random point and stops when it hits $\partial D$. Finally, from Theorem A we conclude that $\mathrm{var}(S_{n})$ is regularly varying if and only if the measure $H_{\nu ,D}$ is regularly varying at the origin. $\square $
**Proof of Theorem \[pr:NSC\].** The first part is motivated by Gordin and Lifšic [@GL] (see Theorem 7.1 in [@BI]; see also [@DL]). We write the martingale type orthogonal decomposition:$$S_{n}=\mathbb{E}_{0}(S_{n})+\sum_{i=1}^{n}\mathbb{E}_{i}(S_{n}-S_{i-1})-\mathbb{E}_{i-1}(S_{n}-S_{i-1}).$$ So $$\begin{aligned}
\mathrm{var}(S_{n}) & =\mathbb{E}(\mathbb{E}_{0}(S_{n}))^{2}+\sum_{i=1}^{n}\mathbb{E(E}_{i}(S_{n}-S_{i-1})-\mathbb{E}_{i-1}(S_{n}-S_{i-1}))^{2} \\
& =\mathbb{E}(\mathbb{E}_{0}(S_{n}))^{2}+\sum_{i=1}^{n}\mathbb{E(E}_{1}(S_{i})-\mathbb{E}_{0}(S_{i}))^{2}.\end{aligned}$$ By applying spectral calculus, $$\begin{aligned}
& =\mathbb{E}(\mathbb{E}_{0}(S_{n}))^{2}+\sum_{j=1}^{n}\int_{D}|1+z+...+z^{j-1}|^{2}(1-|z|^{2})\nu(\mathrm{d}z) \\
& =\mathbb{E}(\mathbb{E}_{0}(S_{n}))^{2}+\sum_{j=1}^{n}\int_{D_{0}}\frac{|1-z^{j}|^{2}(1-|z|^{2})}{|1-z|^{2}}\nu(\mathrm{d}z) \\
& =\mathbb{E}(\mathbb{E}_{0}(S_{n}))^{2}+n\int_{D_{0}}\delta_{n}(z)\frac{(1-|z|^{2})}{|1-z|^{2}}\nu(\mathrm{d}z).\end{aligned}$$ Note that $\delta_{n}(z)\leq4$ and for all $z\in D_{0}$ $$\delta_{n}(z)=\frac{1}{n}\sum_{j=1}^{n}|1-z^{j}|^{2}\rightarrow1\text{ as }n\rightarrow\infty.$$ Thus, by the Lebesgue dominated theorem and our conditions $$\lim_{n\rightarrow\infty}\int_{D_{0}}\delta_{n}(z)\frac{(1-|z|^{2})}{|1-z|^{2}}\nu(\mathrm{d}z)=\int_{D_{0}}\frac{1-|z|^{2}}{|1-z|^{2}}\nu(\mathrm{d}z)=\sigma^{2}.$$ This, along with Fatou’s lemma, proves that $\mathrm{var}(S_{n})/n\rightarrow K$ exists if and only if $\sigma^{2}<\infty$ and $$\frac{1}{n}\mathbb{E}(\mathbb{E}_{0}(S_{n}))^{2}\rightarrow L=K-\sigma^{2}.$$ Now, let us introduce a new measure on $D_{0}$ $$\mu(z)=\frac{1-|z|}{|1-z|^{2}}\nu(\mathrm{d}z),$$ which is finite when $\sigma^{2}<\infty$.
To complete the proof of the first part of the theorem, we notice that by the spectral calculus $$\begin{aligned}
\mathbb{E}(\mathbb{E}_{0}(S_{n}))^{2} & =\int_{D}|z+...+z^{n}|^{2}\nu(\mathrm{d}z)\;=\int_{D}|z|^{2}\frac{|1-z^{n}|^{2}}{|1-z|^{2}}\nu(\mathrm{d}z)
\\
& =\int_{D}\frac{|1-z^{n}|^{2}}{|1-z|^{2}}\nu(\mathrm{d}z)-\int_{D}|1-z^{n}|^{2}(1+|z|)\mu(\mathrm{d}z) \\
& =\int_{D}\frac{|1-z^{n}|^{2}}{|1-z|^{2}}\nu(\mathrm{d}z)+O(1),\end{aligned}$$ since $\mu$ is a finite measure.
To prove the second part of this theorem, we show equivalence of (i) and (ii) and then of (ii) and (iii). In addition, throughout we use notation: $$z=|z|e^{\mathrm{i}\mathrm{Arg}(z)},|z|=1-y,\theta=\mathrm{Arg}(z).$$
We note first that $$|1-z^{n}|^{2}=(1-|z|^{n})^{2}+|z|^{n}\sin^{2}(n\theta/2).$$ The proof strategy consists in showing, several successive approximation steps, that $$\frac{1}{n}\mathbb{E}(\mathbb{E}_{0}(S_{n}))^{2} = \int_0^{\pi} \frac{\sin^2(n\theta/2)}{\sin^2 (\theta/2)} G(\mathrm{d} \theta) + o(1),$$ for some appropriate measure $G$, and then to apply Theorem A. With this in mind we write $$\begin{aligned}
I_{n} & =\frac{1}{n}\int_{D}\frac{|1-z^{n}|^{2}}{|1-z|^{2}}\nu(\mathrm{d}z)
\\
& =\frac{1}{n}\int_{D}\frac{|z|^{n}\sin^{2}(n\mathrm{Arg}(z)/2)}{|1-z|^{2}}\nu(\mathrm{d}z)+\frac{1}{n}\int_{D}\frac{(1-|z|^{n})^2}{|1-z|^{2}}\nu(\mathrm{d}z)=:I_{n}^{\prime}+\Delta_{n}^{\prime}.\end{aligned}$$ Note that $$\begin{aligned}
\Delta_{n}^{\prime} & =\int_{D_{0}}\Big(\frac{(1-|z|^{n})}{n(1-|z|)}\Big)(1-|z|^{n})\frac{(1-|z|)}{|1-z|^{2}}\nu(\mathrm{d}z)\; \\
& =\int_{D}\frac{1}{n}\Big(\sum_{j=0}^{n-1}|z|^{j}\Big)(1-|z|^{n})\mu(\mathrm{d}z).\end{aligned}$$ By Lebesgue dominated convergence theorem, since the bounded integral argument goes to $0$ for each $|z|\leq1,$ we have $\Delta_{n}^{\prime}\rightarrow0$ as $n\rightarrow\infty$.
Then, write $$\begin{aligned}
I_{n}^{\prime} & =\frac{1}{n}\int_{D}\frac{\sin^{2}(n\mathrm{Arg}(z)/2)}{|1-z|^{2}}\nu(\mathrm{d}z)\;-\;\frac{1}{n}\int_{D}(1-|z|^{n})\frac{\sin^{2}(n\mathrm{Arg}(z)/2)}{|1-z|^{2}}\nu(\mathrm{d}z) \\
& =:I_{n}^{\prime\prime}+\Delta_{n}^{\prime\prime},\end{aligned}$$ and again $$\begin{aligned}
\Delta_{n}^{\prime\prime} & =\int_{D_{0}}\Big(\frac{(1-|z|^{n})}{n(1-|z|)}\Big)\sin^{2}(n\mathrm{Arg}(z)/2)\frac{(1-|z|)}{|1-z|^{2}}\nu(\mathrm{d}z)\;
\\
& =\int_{D_{0}}\Big(\frac{(1-|z|^{n})}{n(1-|z|)}\Big)\sin^{2}(n\mathrm{Arg}(z)/2)\mu(\mathrm{d}z).\end{aligned}$$ Note that by Lebesgue dominated theorem $\Delta_{n}^{\prime\prime}\rightarrow0$ as $n\rightarrow\infty$, since the bounded integral argument goes to $0$ for each $|z|<1$.
Fix now a small positive $a>0$, recall that $z=(1-y)e^{i\theta }$ and define an auxiliary subset of $D$ $$D_{a}=\{z=(1-y)e^{i\theta }\;:\;0<|\theta |\leq a\;,\;0\leq y\leq a\;\}.$$Further, notice that by the dominated convergence theorem $$\varepsilon _{n}=\int_{D_{a}}\Big|\frac{\sin (n\mathrm{Arg}(z)/2)}{n\mathrm{Arg}(z)}\Big|\mu (\mathrm{d}z)\rightarrow 0\text{ as }n\rightarrow \infty ,$$since the bounded integral argument goes to $0$ for each $|z|<1$.Let $N$ be large enough, so that $|\epsilon _{n}|<1$ for all $n\geq N$, and take [$\delta_{n}=\max(\mathrm{e}^{-n},\sqrt{\varepsilon_{n}})$]{} so that $\delta
_{n}>0$ for all $n$. In this way $\epsilon _{n}/\delta _{n}=0$ is well-defined if $\epsilon _{n}=0$, and $\epsilon _{n}/\delta _{n}\leq \sqrt{\epsilon _{n}}\rightarrow 0$ as $n\rightarrow \infty $. Further define two auxiliary sequences of subsets of $D_{a}$ $$\begin{aligned}
D_{a,n}& =\{z=(1-y)e^{i\theta }\;:\;0<\delta _{n}|\theta |\leq y\leq
a,|\theta |\leq a\;\}\;, \\
U_{a,n}& =U_{a}\setminus {D_{a,n}}=\{z=(1-y)e^{i\theta }\;:\;0\leq y<\delta
_{n}|\theta |,\;|\theta |\leq a\;\}.\end{aligned}$$Next, let $$g_{n}(z)=\frac{\sin ^{2}(n\mathrm{Arg}(z)/2)}{n|1-z|^{2}}$$and write $$I_{n}^{\prime \prime }=\frac{1}{n}\int_{D}g_{n}(z)\nu (\mathrm{d}z)=\frac{O(1)}{n}+\int_{D_{a}}g_{n}(z)\nu (\mathrm{d}z)$$and $$\int_{D_{a}}g_{n}(z)\nu (\mathrm{d}z)=\int_{D_{a,n}}g_{n}(z)\nu (\mathrm{d}z)+\int_{U_{a,n}}g_{n}(z)\nu (\mathrm{d}z)=:\Delta _{n}^{\prime \prime
\prime }+I_{n}^{\prime \prime \prime }\text{ .}$$Notice that by construction ($\theta =\mathrm{Arg}(z)$) $$\begin{aligned}
\Delta _{n}^{\prime \prime \prime }& \leq \int_{D_{a,n}}\Big|\frac{\sin
(n\theta /2)}{n\theta }\Big||\sin (n\theta /2)|\frac{|\theta |}{|1-z|^{2}}\nu (\mathrm{d}z)\; \\
& \leq \frac{1}{\delta _{n}}\int_{D_{a,n}}\Big|\frac{\sin (n\theta /2)}{n\theta }\Big|\frac{1-|z|}{|1-z|^{2}}\nu (\mathrm{d}z)\; \\
& =\frac{1}{\delta _{n}}\int_{D_{a,n}}\Big|\frac{\sin (n\theta /2)}{n\theta }\Big|\mu (\mathrm{d}z)\leq \frac{1}{\delta _{n}}\int_{D_{a}}\Big|\frac{\sin
(n\theta /2)}{n\theta }\Big|\mu (\mathrm{d}z)\;=\frac{\varepsilon _{n}}{\delta _{n}}\rightarrow 0.\end{aligned}$$In addition, on $U_{a,n}$ $$|1-z|^{2}=y^{2}+(1-y)\sin ^{2}(\theta /2)=\sin ^{2}(\theta /2)(1+\delta
_{z,n}),$$where $|\delta _{z,n}|\leq c\delta _{n}$ for some positive $c$ and hence $$\begin{aligned}
I_{n}^{\prime \prime \prime }& =\frac{1}{n}\int_{U_{a,n}}\frac{\sin
^{2}(n\theta /2)}{|1-z|^{2}}\nu (\mathrm{d}z) \\
& =\frac{1}{n}\int_{U_{a,n}}\frac{\sin ^{2}(n\theta /2)}{\sin ^{2}(\theta /2)}\nu (\mathrm{d}z)\;+\;\frac{1}{n}\int_{U_{a,n}}\frac{\sin ^{2}(n\theta /2)}{\sin ^{2}(\theta /2)}\Big(1-\frac{1}{1+\delta _{z,n}}\Big)\nu (\mathrm{d}z)
\\
& =:I_{n}^{(4)}+\Delta _{n}^{(4)}.\end{aligned}$$By construction $\Delta _{n}^{(4)}=o(1)I_{n}^{iv}$ and then $I_{n}^{\prime
\prime \prime }=I_{n}^{(4)}(1+o(1))$. Write $$\begin{aligned}
I_{n}^{(4)}& =\frac{1}{n}\int_{U_{a}}\frac{\sin ^{2}(n\theta /2)}{\sin
^{2}(\theta /2)}\nu (\mathrm{d}z)\;-\;\frac{1}{n}\int_{U_{a}\cap D_{a,n}}\frac{\sin ^{2}(n\theta /2)}{\sin ^{2}(\theta /2)}\nu (\mathrm{d}z). \\
& =I_{n}^{(5)}-\Delta _{n}^{(5)}.\end{aligned}$$In a similar way as $\Delta _{n}^{\prime \prime \prime }$ has been estimated in Step 3, by construction ($\theta =\mathrm{Arg}(z)$) we obtain $$\begin{aligned}
\Delta _{n}^{(5)}& \leq \int_{U_{a}\cap D_{a,n}}\Big|\frac{\sin (n\theta /2)}{n\theta }\Big||\sin (n\theta /2)||(1+\delta _{z,n})|\frac{|\theta |}{|1-z|^{2}}\nu (\mathrm{d}z)\; \\
& \leq \frac{1+c\delta _{n}}{\delta _{n}}\int_{D_{a,n}}\Big|\frac{\sin
(n\theta /2)}{n\theta }\Big|\frac{1-|z|}{|1-z|^{2}}\nu (\mathrm{d}z)\; \\
& \leq \frac{1+c\delta _{n}}{\delta _{n}}\int_{D_{a}}\Big|\frac{\sin
(n\theta /2)}{n\theta }\Big|\mu (\mathrm{d}z)=\frac{(1+c\delta
_{n})\varepsilon _{n}}{\delta _{n}}\rightarrow 0\text{ as }n\rightarrow
\infty .\end{aligned}$$Finally, let us define the set $$D_{+}=\{z=(1-y)e^{i\theta }\;:\;0\leq y\leq |\theta |<\pi \}\;.$$Then, it follows that $I_{n}/n\rightarrow L$ if and only if $$J_{n}=\int_{D_{+}}I_{n}(\mathrm{Arg}(z))\nu (\mathrm{d}z)\rightarrow 0,$$where $I_{n}$ is the Fejer kernel, $$I_{n}(x):=\frac{\sin ^{2}(nx/2)}{n\sin ^{2}(x/2)}.$$Define $G(x)=\nu (U_{x})$ and notice that it is a non-negative, non-decreasing bounded function. In addition, for any step function $g(\theta )=I(u<|\theta |\leq v)=I(|\theta |\leq v)-I(|\theta |\leq w)$ with $u,w$ being continuity points of $G(x)$ we have $$\begin{aligned}
\int_{D_{+}}g(\mathrm{Arg}(z))\nu (\mathrm{d}z)& =\int_{D_{+}}I(g(\theta
))\nu (\mathrm{d}z)=\int_{D_{+}}I(|\theta |\leq v)\nu (\mathrm{d}z)-\int_{D_{+}}I(|\theta |\leq w)\nu (\mathrm{d}z) \\
& =\nu (U_{v})-\nu (U_{w})=\int g(x)dG(x)\end{aligned}$$and then, by Caratheodory and Lebesgue theorem we have $$J_{n}=\int_{0}^{\pi }I_{n}(x)dG(x).$$By Theorem A, $J_{n}/n\rightarrow L$ if and only if $L={\pi
C}$ where $G(x)/x=\nu (U_{x})/x\rightarrow C$ as $x\rightarrow 0^{+}$, completing the proof of equivalence of (i) and (ii) in Part (b).
To prove the equivalence of (ii) and (iii) in Part (b) under the finiteness of integral , let us define $$\begin{aligned}
W_{x}& =\{z=(1-r)e^{iu}\in D\;:\;0\leq |u|\leq r\leq x\}\;, \\
D_{1/x}& =\{z=(1-r)e^{iu}\in D\;:\;0\leq |u|,r\leq x\}\;.\end{aligned}$$Since $\nu ({1})=0$, it follows that $\nu (W_{x})\rightarrow 0$ as $x\rightarrow 0^{+}$.
On the other hand, on $W_{x}$ $$\frac{1-|z|}{|1-z|^{2}}\geq \frac{1}{2x}$$and hence by , $$0\leftarrow \int_{W_{x}}\frac{1-|z|}{|1-z|^{2}}\nu (\mathrm{d}z)\geq \frac{1}{2x}\nu (W_{x}),$$which implies that $\nu (D_{1/x})/x\rightarrow C$ if and only if (ii) holds as $x\rightarrow 0^{+}$ and this completes the proof. $\square $
The sufficient part in Theorem \[pr:NSC\] can be derived directly by performing coordinate transformation mapping the open unit disk to the upper half-plane and further careful analysis. Briefly, we change coordinates to the upper half-plane $\mathbb{H}:=\{(a,b):a\in\mathbb{R},b>0\}$, via the inverse Cayley transform $\phi(w):\mathbb{H}\rightarrow D_{0}$, where $\phi
(w):=(1+\mathrm{i}w)/(1-\mathrm{i}w)$. The finite measure $\nu$ is transformed to a finite measure $\rho$ on $\mathbb{H}$, which for simplicity we can assume to be a probability measure. Then, for $w:=a+\mathrm{i}b$ we have $$\sigma^{2}=\int_{\mathbb{H}}\frac{1-|\phi(a+\mathrm{i}b)|^{2}}{|1-\phi (a+\mathrm{i}b)|^{2}}\mathrm{d}v(\phi^{-1}(z))=\int_{a=-\infty}^{\infty}\int_{b=0}^{\infty}\frac{b}{a^{2}+b^{2}}\mathrm{d}\rho(a,b).$$ and a further change of variables $z=\tan(t/2)$ gives $$\int_{t=0}^{x}f(t)\mathrm{d}t=\frac{1}{2\pi}\int_{t=0}^{x}\iint_{\mathbb{H}}\frac{b}{(a+t/2)^{2}+b^{2}}\rho(\mathrm{d}a,\mathrm{d}b)\mathrm{d}t+o(x).$$
**Proof of Theorem \[thR(x)\].** Let $1\leq \alpha \leq 2$ and denote $C(n):=\sum_{i=0}^{n-1}\mathrm{cov}(X_{0},X_{i}).$ We start by the well known representation $$\mathrm{var}(S_{n})=n[\frac{2}{n}\sum_{k=1}^{n}C(k)-E(X_{0}^{2})].$$It is clear then, since $\mathrm{var}(S_{n})/n\rightarrow \infty $, that $\mathrm{var}(S_{n})$ has the same asymptotic behavior as $2\sum_{k=1}^{n}C(k).$ Implementing the notations$$a_{k}=\int\nolimits_{0}^{1}x^{j}\nu _{1}(\mathrm{d}x)\text{ and }b_{k}=\int\nolimits_{-1}^{0}x^{j}\nu (\mathrm{d}x),$$where $\nu _{1}$ coincides with $\nu $ on $(0,1]$, and $\nu _{1}(\{0\})=0,$ we have the representation $$C(k)=\sum_{j=0}^{k-1}a_{j}+\sum_{j=0}^{k-1}b_{j}=C_{1}(k)+C_{2}(k).$$We shall show that the terms $C_{1}(k)$ have a dominant contribution to the variance of partial sum. To analyze $C_{2}(k)$ it is convenient to make blocks of size $2$, a trick that has also appeared in [@HaggRos] where it is attributed to [@Geyer]. We notice that $$c_{l}=b_{2l}+b_{2l+1}=\int\nolimits_{-1}^{0}(x^{2l}+x^{2l+1})d\nu >0.$$Furthermore, for all $m$$$\sum_{l=0}^{m-1}c_{l}=\sum_{l=0}^{m-1}\int\nolimits_{-1}^{0}x^{2l}(1+x)\nu (\mathrm{d}x)=\int\nolimits_{-1}^{0}\frac{1-x^{2m}}{1-x}\nu (\mathrm{d}x)\leq E(X_{0}^{2}).$$Therefore $|C_{2}(k)|\leq 2E(X_{0}^{2})$ and so, $\sum_{k=1}^{n}C_{2}(k)\leq
2nE(X_{0}^{2})$. Because $\mathrm{var}(S_{n})/n\rightarrow \infty $ we note that $\mathrm{var}(S_{n})$ has the same asymptotic behavior as $2\sum_{k=1}^{n}C_{1}(k).$ Now, each $C_{1}(k)=\sum_{j=0}^{k-1}a_{k}$ with $a_{k}>0.$ So, because the sequence $(C_{1}(k))_{k}$ is increasing, by the monotone Tauberian theorem (see Theorem \[thm:tm\] in [[Bing]{}]{}) for all $\alpha \geq 1\mathrm{\ }$we have
$$\mathrm{var}(S_{n})\sim 2\sum_{k=1}^{n}C_{1}(k)\sim n^{\alpha }h(n)\quad
\text{ if and only if }\quad C_{1}(n)\sim \alpha n^{\alpha -1}h(n)/2.\
\label{rel1}$$
Note now that $$C_{1}(n)=\int\nolimits_{0}^{1}\frac{1-x^{n}}{1-x}\nu _{1}(\mathrm{d}x).$$It is convenient to consider the transformation $T:[0,1]\rightarrow \lbrack
0,1]$ defined by $T(x)=1-x$. For a Borelian $A$ of $[0,1]$ define the measure$$r(A)=\nu _{1}(T(A)). \label{def r}$$Then $$C_{1}(n)=\int\nolimits_{0}^{1}\frac{1-(1-y)^{n}}{y}\ r(\mathrm{d}y).$$We shall integrate by parts. Denote$$R(u)=\int\nolimits_{u}^{1}\frac{1}{y}r(\mathrm{d}y)\text{ and }U_{n}(u)=[1-(1-u)^{n}].$$Let $0<b<1.$ By the definition of $\nu _{1}$ we have $r(\{1\})=0.$ Since $U_{n}$ is continuous, by [@Bill95 Theorem 18.4],$$\int\nolimits_{b}^{1}\frac{1-(1-y)^{n}}{y}r(\mathrm{d}y)=[1-(1-b)^{n}]R(b)+n\int\nolimits_{b}^{1}(1-u)^{n-1}R(u)\mathrm{d}u.$$Note that $$\lim \sup_{b\rightarrow 0^{+}}[1-(1-b)^{n}]R(b)\leq nbR(b)\leq n\mathbb{E}(X_{0}^{2})=o(\sigma _{n}^{2}).$$Therefore$$C_{1}(n)=o(\sigma _{n}^{2})+n\int\nolimits_{0}^{1}(1-u)^{n-1}R(u)\mathrm{d}u.$$By the change of variables $1-u=\mathrm{e}^{-y}$ we have $$\int\nolimits_{0}^{1}(1-u)^{n-1}R(u)du=\int\nolimits_{0}^{\infty }R(1-\mathrm{e}^{-y})\mathrm{e}^{-yn}\mathrm{d}y.$$It follows that$$\begin{gathered}
C_{1}(n)\sim \alpha n^{\alpha -1}h(n)/2\quad \text{ if and only if}\quad
n\int\nolimits_{0}^{\infty }R(1-\mathrm{e}^{-y})\mathrm{e}^{-yn}dx\sim
\alpha n^{\alpha -1}h(n)/2 \label{rel2} \\
\int\nolimits_{0}^{\infty }\quad \text{ if and only if}\quad {\displaystyle\int\nolimits_{0}^{\infty }}R(1-\mathrm{e}^{-y})\mathrm{e}^{-yn}\mathrm{d}x\sim \alpha n^{\alpha -2}h(n)/2\text{.} \notag\end{gathered}$$From here we shall apply Karamata’s Tauberian Theorem \[t1\]. Since $\alpha \leq 2,$$$\begin{gathered}
\int\nolimits_{0}^{\infty }R(1-\mathrm{e}^{-y})\mathrm{e}^{-yn}\mathrm{d}x\sim \alpha n^{\alpha -2}h(n)/2\text{ as }n\rightarrow \infty
\label{rel 3} \\
\text{if and only if }\,\,\int\nolimits_{0}^{x}R(1-\mathrm{e}^{-y})\mathrm{d}y\sim c_{\alpha }^{\prime }x^{2-\alpha }h(1/x)\text{ as }x\rightarrow 0^{+},
\notag\end{gathered}$$where $c_{\alpha }^{\prime }=\alpha /[2\Gamma (3-\alpha )].$ Again by the monotone Karamata Theorem \[thm:tm\] this happens if and only if $$R(1-\mathrm{e}^{-y})\sim (2-\alpha )c_{\alpha }^{\prime }x^{1-\alpha }h(1/x)\text{ as }x\rightarrow 0^{+}. \label{rel4}$$Changing variables $x=1-\mathrm{e}^{-y}$ and taking into account Karamata’s representation for slowly varying functions we get$$\begin{aligned}
R(x)& \sim c_{\alpha }\Big[{\ln (1/(1-x))}\Big]^{{1}-\alpha }h({-1/\ln (1-x)})\ \label{rel 5} \\
& \sim c_{\alpha }x^{{1}-\alpha }h(1/x)\ \text{as }x\rightarrow 0^{+},
\notag\end{aligned}$$where $c_{\alpha }=(2-\alpha )c_{\alpha }^{\prime }$. By combining the results in relations (\[rel1\])-(\[rel 5\]) we have that$$\mathrm{var}(S_{n})\sim n^{\alpha }h(n)\quad \text{ if and only if }\quad
R(x)\sim c_{\alpha }x^{{1}-\alpha }h(1/x)\ \text{as }x\rightarrow 0^{+}.$$It remains to note that $$R(x)=\int\nolimits_{x}^{1}\frac{1}{y}r(\mathrm{d}y)=\int\nolimits_{0}^{1-x}\frac{1}{1-y}\upsilon _{1}(\mathrm{d}y)\sim V(x)\text{ as }x\rightarrow
0^{+}.$$
When $1<\alpha<2,$ one can say more: the distribution function induced by the spectral measure is regularly varying.
Note that again by Theorem \[thm:tm\], since the sequence $a_{k}$ is a monotone sequence of positive numbers and $\alpha-1>0$ we have $$C_{1}(n)=\sum_{k=0}^{n-1}a_{k}=\alpha n^{\alpha-1}h(n)/2\quad\text{ if and
only if }\quad a_{n}\sim\alpha(\alpha-1)n^{\alpha-2}h(n)/2.$$ Now, by considering the mapping $T^{\prime}:[0,1]\rightarrow\lbrack0,\infty),
$ given by $T^{\prime -x},$ we obtain $$a_{n}=\int_{0}^{1}t^{n}\ \nu_{1}(\mathrm{d}t)=\int_{0}^{\infty}e^{-nx}\psi(\mathrm{d}x),$$ where $\psi(A)=r(T^{\prime}(A)),$ for $A$ Borelian in $[0,\infty)$. Letting $d_{\alpha}:=\alpha(\alpha-1)/2\Gamma(3-\alpha),$ it follows by Theorem [t1]{} ([[@Bing Thm 1.7.1’]]{}), that$$a_{n}\sim\alpha(\alpha-1)n^{\alpha-2}h(n)/2\text{ as }n\rightarrow\infty\quad\text{ iff }\quad\psi\lbrack0,x]\sim d_{\alpha}x^{2-\alpha}h(1/x)\text{ as }x\rightarrow0^{+}.$$ Then, we obtain as before, by the properties of slowly varying functions $$\psi\lbrack0,x]\sim d_{\alpha}x^{2-\alpha}h(1/x)\text{ as }x\rightarrow 0^{+}\text{ iff }\nu_{1}(1-x,1]\sim d_{\alpha}x^{2-\alpha}h(1/x)\text{ as }x\rightarrow0^{+}.$$ This last relation combined with (\[rel2\]) gives the last part of the theorem.$\square$
We only prove the central limit theorem, with the other results following in a similar manner. Our approach is based on the regeneration process. Define $$\begin{aligned}
T_{0} & =\inf\{i>0:\xi_{i}\neq\xi_{0}\}\\
T_{k+1} & =\inf\{i>T_{k}:\xi_{i}\neq\xi_{i-1}\},\end{aligned}$$ and let $\tau_{k}:=T_{k+1}-T_{k}$. It is well known that $(\xi_{\tau_{k}},\tau_{k})_{k\geq1}$ are i.i.d. random variables with $\xi_{\tau_{k}}$ having the distribution $\upsilon.$ Furthermore, $$\mathbb{P}(\tau_{1}>n|\xi_{\tau_{1}}=x)=|x|^{n}\text{.}$$ Then, it follows that$$\mathbb{E}(\tau_{1}|\xi_{\tau_{1}}=x)=\frac{1}{1-|x|}\text{ }\ \ \text{and
\ \ }\mathbb{E}(\tau_{1})={\displaystyle\int\nolimits_{-1}^{1}}
\frac{1}{1-|x|}\upsilon(\mathrm{d}x)=\theta.$$ So, by the law of large numbers $T_{n}/n\rightarrow\theta$ a.s. Let us study the tail distribution of $\tau_{1}.$ Since $$\mathbb{P}(\tau_{1}|X_{\tau_{1}}|>u|\xi_{\tau_{1}}=x)=\mathbb{P}(\tau
_{1}>u|\xi_{\tau_{1}}=x)=|x|^{u},$$ by integration we obtain$$\mathbb{P}(\tau_{1}>u)=\int_{-1}^{1}|x|^{u}\upsilon(\mathrm{d}x)=2\int_{0}^{1}x^{u}\upsilon(\mathrm{d}x).\label{tail tau}$$ Using now the relation between $\upsilon(\mathrm{d}x)$ and $\mu(\mathrm{d}x)$ and symmetry we get $$\mathbb{P}(\tau_{1}>u)=2\theta\int_{0}^{1}x^{u}(1-x)\mu(\mathrm{d}x)=\theta\int_{0}^{1}x^{u}(1-x)\nu(\mathrm{d}x),$$ where $\nu$ is spectral measure. By using the fact that $V(x)$ is slowly varying and Lemma \[lem:aux\] in Section “Technicalities” it follows that $$H(u):=E[\tau_{1}^{2}I(\tau_{1}\leq u)]\text{ is slowly varying as
}u\rightarrow\infty.\label{Feller}$$ For each $n$, let $m_{n}$ be such that $T_{m_{n}}\leq n<T_{m_{n}+1}.$ Note that we have the representation $$\sum_{k=1}^{n}X_{k}-\sum_{k=1}^{[n/\theta]}Y_{k}=(T_{0}-1)X_{0}+(\sum
_{k=1}^{m_{n}}\tau_{k}X_{\tau_{k}}-\sum_{k=1}^{[n/\theta]}\tau_{k}X_{\tau_{k}})+\sum_{k=T_{m_{n}+1}}^{n}X_{k}\text{,}\label{Rep}$$ where $Y_{k}=\tau_{k}X_{\tau_{k}}$ is a centered i.i.d. sequence, and by (\[Feller\]) is in the domain of attraction of a normal law (see Feller [@Feller66]). Therefore, $$\frac{\sum_{k=1}^{[n/\theta]}Y_{k}}{b_{[n/\theta]}}\Rightarrow N(0,1)\text{.}\label{Fell}$$ where $b_{n}^{2}\sim nH(b_{n}).$ The rest of the proof is completed on the same lines as in the proof of Example 12 in [@LPP], the final result being that$$\frac{\sum_{k=1}^{n}X_{k}}{b_{[n/\theta]}}\Rightarrow N(0,1)\text{.}$$
The proof is similar to that of Theorem \[thm:harmonicmeasure\] once one observes that for $z=a+\mathrm{i} b$, with $a\leq0$ and $b\in\mathbb{R}$, $t>0$ and $x\in\mathbb{R}$ we have $$\mathrm{e}^{zt} = -\frac{1}{\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}
t x} \Re\Big[ \frac{1}{z-\mathrm{i}x}\Big] \mathrm{d} x =\frac{1}{\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} t x} \frac{-a}{a^{2} +
(b-x)^{2}} \mathrm{d} x.$$ Therefore, by Fubini’s theorem $$\mathrm{cov}(f(\xi_{0}),f(\xi_{t})) = \int_{\Re(z)\leq0} \mathrm{e}^{z t}
\nu(\mathrm{d} z) = \int_{x=-\infty}^{\infty}\mathrm{e}^{\mathrm{i} t x}
\Big[ \frac{-1}{\pi} \int_{\Re(z)\leq0} \Re\big(\frac{1}{z-\mathrm{i}x}\big) \nu(\mathrm{d} z) \Big] \mathrm{d}x.$$ By letting $z=w\mathrm{i}$, where $w\in\mathbb{H}$, and the conformal invariance of Brownian motion, one can immediately deduce that $$-\frac{1}{\pi} \Re\big(\frac{1}{z-\mathrm{i}x}\big) \mathrm{d}x,$$ is the harmonic measure of Brownian motion in the left half-plane started at the point $z$.
First observe that $$\begin{aligned}
\mathrm{var}(S_{T}) & = 2\int_{s=0}^{T} (T-s) \int_{\Re(z)\leq0}
\mathrm{e}^{zx} \nu(\mathrm{d} z) \mathrm{d} s = 2 \int_{\Re(z)\leq0}
\Re\Big[ \frac{\mathrm{e}^{zT}-zT-1}{z^{2}}\Big] \nu(\mathrm{d} z).\end{aligned}$$ By splitting $S_{T}= \mathbb{E}_{0}(S_{T})+ S_{T}-\mathbb{E}_{0}(S_{T})$ we obtain $$\begin{aligned}
\mathrm{var}(S_{T}) & = \mathbb{E}\big[(S_{T}-\mathbb{E}_{0}(S_{T}))^{2}\Big]+ \mathbb{E}\big[ \mathbb{E}_{0}(S_{T})^{2}\big]\\
& = \int_{\Re(z)\leq0} \Re\Big[ 2\frac{\mathrm{e}^{zT}-zT-1}{z^{2}} -
\frac{|1-\mathrm{e}^{zT}|^{2}}{|z|^{2}}\Big] \nu(\mathrm{d} z) + \int
_{\Re(z)\leq0} \frac{|1-\mathrm{e}^{zT}|^{2}}{|z|^{2}} \nu(\mathrm{d} z)=:
I_{1} +I_{2}.\end{aligned}$$ Assume $-\Re(1/z)\in L_{1}(\mathrm{d} \nu)$. Careful calculation shows that $$-\Re\big(\frac{1}{z}\big)\leftarrow\Re\Big[ 2\frac{\mathrm{e}^{zT}-zT-1}{z^{2}} - \frac{|1-\mathrm{e}^{zT}|^{2}}{|z|^{2}}\Big]
\leq\frac{C|x|}{x^{2}+y^{2}}\in L_{1}(\mathrm{d} \nu),$$ and thus $I_{1}/T \to-\int_{\Re(z)\leq0} \Re(1/z)\nu(\mathrm{d} z)$. Fatou’s lemma also shows that if $\mathrm{var}(S_{T})/T$ converges then $\Re(1/z)\in
L_{1}(\mathrm{d} \nu)$.
Next we analyze $I_{2}/T$. Notice that $$|1-\mathrm{e}^{zT}|^{2} = (1-\mathrm{e}^{Tx})^{2}+ 4 \mathrm{e}^{Tx} \sin^{2}
(Ty/2),$$ and since $(1-\mathrm{e}^{Tx})^{2}\leq T|x|$ for $x<0$, we have that $$\begin{aligned}
\frac{1}{T}\int_{\Re(z) \leq0} \frac{|1-\mathrm{e}^{zT}|^{2}}{|z|^{2}}
\nu(\mathrm{d} z) & = \frac{1}{T}\int_{\Re(z)\leq0} \frac{4 \mathrm{e}^{Tx}\sin^{2}(Ty/2)}{x^{2} + y^{2}} \nu(\mathrm{d} x, \mathrm{d} y) + o(1)\\
& =\frac{1}{T}\int_{\Re(z) \leq0} \frac{4\sin^{2}(Ty/2)}{x^{2} + y^{2}}
\nu(\mathrm{d} x, \mathrm{d} y) + o(1).\end{aligned}$$ For $a>0$ write $$D_{a}:=\{x+\mathrm{i}y:0\leq-x\leq a,0\leq|y|\leq a\}.$$ Notice that on $D_{a}^{(c)}$ the integrand is less than $1/2a^{2}$ and therefore $$\int_{\Re(z)\leq0}\frac{4\sin^{2}(Ty/2)}{x^{2}+y^{2}}\nu(\mathrm{d}x,\mathrm{d}y)=\int_{D_{a}}\frac{4\sin^{2}(Ty/2)}{x^{2}+y^{2}}\nu
(\mathrm{d}x,\mathrm{d}y)+o(1).$$ Let $$\epsilon_{T}:=\int_{D_{a}}\Big|\frac{\sin(Ty/2)}{Ty}\Big|\times\frac
{|x|\nu(\mathrm{d}x,\mathrm{d}y)}{x^{2}+y^{2}}\rightarrow0,$$ since the bounded integrand vanishes and $\mu(\mathrm{d}z):=|x|\nu
(\mathrm{d}z)/(x^{2}+y^{2})$ is a finite measure. Let [$\delta_{T}
:=\max(\mathrm{e}^{-T},\sqrt{\epsilon_{T}})$]{}, so that $\delta_T>0$ and $\epsilon_{T}/\delta_{T}\rightarrow0$, and define $$\begin{gathered}
U_{a}:=\{x+\mathrm{i}y:0\leq-x\leq|y|\leq a\},\quad D_{a,T}:=\{x+\mathrm{i}y:0\leq\delta_{T}|y|\leq-x,|y|\leq a\}\\
U_{a,T}:=\{x+\mathrm{i}y:0\leq-x\leq\delta_{T}|y|,|y|\leq a\}\end{gathered}$$ Since on $D_{a,T}$ we have $|y|\leq|x|/\delta_{T}$ $$\frac{1}{T}\int_{D_{a,T}}\frac{\sin^{2}(Ty/2)}{x^{2}+y^{2}}\mathrm{d}\nu
\leq\frac{1}{\delta_{T}}\int_{D_{a,T}}\frac{|\sin(Ty/2)|}{Ty}\frac{|x|}{x^{2}+y^{2}}\mathrm{d}\nu\leq\frac{\epsilon_{T}}{\delta_{T}}\rightarrow0,$$ and thus since $U_{a,T}=D_{a}-D_{a,T}$ it follows that $y^{2}+x^{2}=y^{2}(1+O(\delta_{T}^{2}))$ and $$\frac{1}{T}\int_{D_{a}}\frac{4\sin^{2}(Ty/2)}{x^{2}+y^{2}}\nu(\mathrm{d}x,\mathrm{d}y)=\frac{1}{T}\int_{U_{a,T}}\frac{\sin^{2}(Ty/2)}{(y/2)^{2}}\nu(\mathrm{d}x,\mathrm{d}y)\times(1+o(1))+o(1).$$ Notice that on $U_{a}\cap D_{a,T}$ we have $0\leq\delta_{T}|y|\leq|x|\leq|y|\leq a$ and thus $$\frac{|x|}{x^{2}+y^{2}}\geq\frac{|x|}{x^{2}+y^{2}}\geq\frac{\delta_{T}|y|}{2y^{2}}=\frac{\delta_{T}}{2|y|}\Longrightarrow\frac{1}{|y|}\leq\frac
{1}{\delta_{T}}\frac{2|x|}{x^{2}+y^{2}}.$$ Since $U_{a,T}=U_{a}-U_{a}\cap D_{a,T}$, from the above $$\frac{1}{T}\int_{U_{a,T}}\frac{\sin^{2}(Ty/2)}{(y/2)^{2}}\nu(\mathrm{d}x,\mathrm{d}y)=\frac{1}{T}\int_{U_{a}}\frac{\sin^{2}(Ty/2)}{(y/2)^{2}}\nu(\mathrm{d}x,\mathrm{d}y)+o(1).$$ Therefore $$\begin{gathered}
\frac{1}{T}\int_{U_{a}\cap D_{a,T}}\!\!\!\!\frac{\sin^{2}(Ty/2)}{(y/2)^{2}}\nu(\mathrm{d}x,\mathrm{d}y)
\leq C\int_{U_{a}\cap D_{a,T}}\!\!\!\!\frac{|\sin
(Ty/2)|}{T|y|}\times\frac{1}{|y|}\nu(\mathrm{d}x,\mathrm{d}y)\\
\leq\frac{C}{\delta_{T}}\int_{U_{a}\cap D_{a,T}}\!\!\!\!
\frac{|\sin(Ty/2)|}{T|y|}\times\frac{|x|}{x^{2}+y^{2}}\nu(\mathrm{d}x,\mathrm{d}y)\leq
C\frac{\epsilon_{T}}{\delta_{T}}\rightarrow0.\end{gathered}$$ Finally let $$U:=\{x+\mathrm{i}y:0\leq-x\leq|y|\}.$$ From the above computation, since $a>0$ was arbitrary and all error terms depending on $a$ vanish as $a\rightarrow\infty$ we have $$\begin{gathered}
\lim_{T\rightarrow\infty}\frac{1}{T}\int_{\Re(z)\leq0}\frac{|1-\mathrm{e}^{zT}|^{2}}{|z|^{2}}\nu(\mathrm{d}z)=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{U_{a}}\frac{\sin^{2}(Ty/2)}{(y/2)^{2}}\nu(\mathrm{d}x,\mathrm{d}y)\\
=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{U}\frac{\sin^{2}(Ty/2)}{(y/2)^{2}}\nu(\mathrm{d}x,\mathrm{d}y)=\lim_{T\rightarrow\infty}\frac{1}{T}\int
_{x=0}^{\infty}\frac{\sin^{2}(Tx/2)}{(x/2)^{2}}G(\mathrm{d}x),\end{gathered}$$ where $G(x)=\nu(U_{x})$, using similar arguments to the proof of Theorem \[pr:NSC\]. Now we have $$\begin{aligned}
\frac{1}{T}\int_{x=0}^{\infty}\frac{\sin^{2}(Tx/2)}{(x/2)^{2}}G(\mathrm{d}x)
& =\frac{1}{T}\int_{x=0}^{\pi}\frac{\sin^{2}(Tx/2)}{\sin^2(x/2)}G(\mathrm{d}x)+O(1/T),
$$ since $$\frac{1}{T}\int_{x=0}^{\pi}\sin^{2}(Tx/2)\Big|\frac{1}{(\tfrac{x}{2})^{2}}-\frac{1}{\sin^{2}(\tfrac{x}{2})}\Big|G(\mathrm{d}x)\leq\frac{1}{T}\int
_{x=0}^{\pi}\frac{(x/2)^{4}}{\sin^{2}(x/2)(x/2)^{2}}G(\mathrm{d}x)=
O(1/T).$$ The result then follows from Theorem A.
Technical lemmas
================
Standard Tauberian Theorems
---------------------------
In order to make this paper more self-contained we state the following classical Tauberian theorems (Theorem 2.3 In [@Seneta] or Theorem 1.7. in [@Bing], due to Feller).
\[t1\] Let $U(x)$ be a monotone non-decreasing function on $[0$,$\infty)$ such that$$w(x)=\int_{0^{-}}^{\infty}e^{-xu}\mathrm{d}U(u)\text{ is finite for all }x>0.$$ Then if $\rho\geq0$ and $L$ is a slowly varying function, then $$w(x)=x^{-\rho}L(x)\text{ as }x\rightarrow\infty\text{ \ iff \ }U(x)=x^{\rho
}L(1/x)/\Gamma(\rho+1)\text{ as }x\rightarrow0^{+}.$$
We give the monotone Tauberian theorem (Theorem 2.4 in [@Seneta] or Theorem 2.4 in [@Bing])
\[thm:tm\] Let $U(x)$ defined and positive on $[A,\infty)$ for some $A>0$ given by $$U(x)=\int_{A}^{x}u(y)\mathrm{d}y,\text{ }$$ where $u(y)$ is ultimately monotone. Then if $\rho\geq0$ and $L$ is a slowly varying function, then$$U(x)=x^{\rho}L(x)\text{ as }x\rightarrow\infty\text{ implies }u(x)\sim\rho
x^{\rho-1}L(x)\text{ as }x\rightarrow\infty.$$ If $\rho>0$ then $u(x)$ is regularly varying.
Auxiliary Lemma for Theorem \[thm:MH\]
--------------------------------------
\[lem:aux\] For $V$ and $H$ as defined in Theorem \[thm:MH\], we have $$\frac{H(1/x)}{2\theta V(x)}\rightarrow1\text{ as }x\rightarrow0^{+}.$$ In particular if $V(x)$ is slowly varying at $0$, $H(1/x)$ is slowly varying at $\infty$.
**Proof.** Let $r$ be the measure defined in (\[def r\]). By Definition (\[tail tau\]) $$\mathbb{P}(\tau_{1}>u)\mathbb{=}2\theta\int_{0}^{1}x^{u}(1-x)\mu (\mathrm{d}x)=\theta\int_{0}^{1}(1-y)^{u}yr(\mathrm{d}y).$$ We show first for any $\delta\in(0,1)$ $$\int_{0}^{1}(1-y)^{u}r(\mathrm{d}y)=O(u^{\delta-3})+\int_{0}^{1}e^{-uy}yr(\mathrm{d}y). \label{tail2}$$ To prove this, notice that for $u\geq0,$ $0\leq y\leq1,$ for positive $t,m$ and some $C_{m}$ $$|(1-y)^{u}-\mathrm{e}^{-u y}|\leq|1-y-\mathrm{e}^{-y}|u\mathrm{e}^{-(u-1)y}\text{ and }\mathrm{e}^{t}\geq 1+t^{m}/C_{m}.\text{ }$$ Then for any $\delta\in(0,1)$ $$\int_{0}^{1}\mathrm{e}^{-uy}uy^{3}r(\mathrm{d}y)
\leq C\int_{0}^{1}\frac{r(\mathrm{d}y)}{u^{3-\delta}y^{1-\delta}}
\leq Cu^{\delta-3}\int_{0}^{1}\frac{r(\mathrm{d}y)}{y^{1-\delta}},$$ and after some rearrangement$$\int_{0}^{1}\frac{r(\mathrm{d}y)}{y^{1-\delta}}=C\int_{y=1}^{\infty}\frac{V(1/y)}{y^{1+\delta}}\mathrm{d}y<\infty,$$ since $V(u):=\int_{u}^{1}r(\mathrm{d}y)/y$, is slowly varying as $u\rightarrow0^{+}$. Since for $u\geq0$ $$H(u)=\mathbb{E}\tau_{1}^{2}I(\tau_{1}<u)=2\int_{0}^{u}x\mathbb{P}(\tau
_{1}>x)\mathrm{d}x-u^{2}\mathbb{P}(\tau_{1}>u),$$ by Fubini’s theorem and (\[tail2\]) we derive $$\begin{aligned}
H(u) & =O(u^{\delta-1})-\theta u^{2}\int_{0}^{1}e^{-uy}yr(\mathrm{d}y)+2\theta\int_{0}^{1}\frac{1-e^{-yu}}{y}r(\mathrm{d}y)-2\theta
u\int_{0}^{1}e^{-yu}r(\mathrm{d}y) \\
& =:O(u^{\delta-1})-\theta I_{1}(u)+2\theta I_{2}(u)-4I_{3}(u).\end{aligned}$$ Also note that by integration by parts, $$r(z)=\int_{0}^{z}r(\mathrm{d}s)=-\int_{0}^{z}yV(\mathrm{d}y)=\int_{0}^{z}[V(y)-V(z)]\mathrm{d}y.$$ Then $R$ can also be written as $$r(z)=\int_{1/z}^{\infty}\frac{\mathrm{d}V(1/y)}{y},$$ and since $V(1/y)$ is slowly varying as $y\rightarrow\infty$ we have by Theorem 1.6.5 in [@Bing] that $$\frac{r(z)}{zV(z)}\rightarrow0,\quad\text{as $z\rightarrow0^{+}$.}$$
Now let $K$ be an arbitrary positive number. Since we will first take limits as $x\rightarrow 0^{+}$, we can assume that $x$ is small enough so that $Kx<1
$. Therefore splitting the integral $\int_{0}^{1}=\int_{0}^{Kx}+\int_{Kx}^{1}
$ and applying standard analysis, we derive $$\begin{aligned}
I_{2}(u)& =\int_{y=0}^{Kx}\frac{1-e^{-yu}}{y}r(\mathrm{d}y)+\int_{y=Kx}^{1}\frac{1-e^{-yu}}{y}r(\mathrm{d}y) \\
& =K\frac{r(Kx)}{Kx}+V(Kx)(1+O(\mathrm{e}^{-Kux})).\end{aligned}$$Then for $u=1/x$, arbitrary $K>0$, since for each fixed $K$ $$K\frac{r(Kx)}{KxV(x)}\rightarrow 0\quad \mbox{and}\quad \frac{V(Kx)}{V(x)}\rightarrow 1,\text{ as }x\rightarrow 0^{+}$$we have $$\limsup_{x\rightarrow 0^{+}}\Big|\frac{I_{2}(1/x)}{V(x)}-1\Big|\leq C\mathrm{e}^{-K},$$and since $K>0$ is arbitrary $I_{2}(1/x)/V(x)\rightarrow 1$, as $x\rightarrow 0^{+}$ and $u=1/x$.
Finally let again $K>0$ be arbitrary. Then $$\sup_{t>K}e^{-t}[1+t+t^{2}]=W_{K},\quad W=\sup_{K>0}W_{K}$$ and notice that $W_{K}\rightarrow0$ as $K\rightarrow\infty$. Then we have for fixed arbitrary $K>0$, and $x$ small enough so that $Kx<1$, $$\begin{aligned}
I_{1}+I_{3} & =\int_{y=0}^{1}e^{-yu}(uy+(uy)^{2})\frac{r(\mathrm{d}y)}{y}\leq(1+W)u\int_{y=0}^{Kx}dr(y)+W_{K}\int_{y=Kx}^{1}\frac{r(\mathrm{d}y)}{y}
\\
& =(1+W)\frac{r(Kx)}{x}+W_{K}V(Kx).\end{aligned}$$ Therefore for $u=1/x$, as above we have $$\limsup_{x\rightarrow0^{+}}\Big|\frac{I_{1}(x^{-1})+I_{3}(x^{-1})}{V(x)}\Big|\leq W_{K},$$ and since $K$ is arbitrary the claim follows from the fact that $$\frac{I_{1}(x^{-1})+I_{3}(x^{-1})}{V(x)}\rightarrow0,\text{ as }x\rightarrow0^{+}.$$ $\square$
[99]{}
[ ]{}
[ ]{}
[ ]{}
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[^1]: Department of Statistics, University of Oxford, OX1 3TG, UK; `[email protected]`
[^2]: Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, Oh 45221-0025, USA. `[email protected], [email protected]`
[^3]: Supported in part by a Charles Phelps Taft Memorial Fund grant, and the NSF grant DMS-1208237.
[^4]: Department of Mathematics, University of Leicester, University Road, LE1 7RH UK, ` [email protected]`
|
---
abstract: 'We present an approach that combines appearance and semantic information for 2D image-based localization (2D-VL) across large perceptual changes and time lags. Compared to appearance features, the semantic layout of a scene is generally more invariant to appearance variations. We use this intuition and propose a novel end-to-end deep attention-based framework that utilizes multimodal cues to generate robust embeddings for 2D-VL. The proposed attention module predicts a shared channel attention and modality-specific spatial attentions to guide the embeddings to focus on more reliable image regions. We evaluate our model against state-of-the-art (SOTA) methods on three challenging localization datasets. We report an average (absolute) improvement of $19\%$ over current SOTA for 2D-VL. Furthermore, we present an extensive study demonstrating the contribution of each component of our model, showing $8$–$15\%$ and $4\%$ improvement from adding semantic information and our proposed attention module. We finally show the predicted attention maps to offer useful insights into our model.'
bibliography:
- 'biblio.bib'
title: 'Semantically-Aware Attentive Neural Embeddings for Long-Term 2D Visual Localization'
---
Introduction
============
Visual localization (VL) is the problem of estimating the precise location of a captured image and is crucial to applications in autonomous navigation [@se2002mobile; @kendall2015posenet; @lim2012real; @sattler17:are-3d-models-necessary; @naseer17:semantics-aware-visual-localization; @maddern20171; @chowdhary2013gps]. Here, we target the problem of *long-term VL* [@stenbord18:localization-semantically-segmented; @toft2017long; @naseer17:semantics-aware-visual-localization; @maddern20171] in real-world environments, which is required to operate under (1) extreme perceptual changes such as weather and illumination, and (2) dynamic scene changes and occlusions such as moving vehicles. We also evaluate this task to focus on a key requirement of long-term VL, which is to localize within a narrow radius of ${<}10$m instead of $25$m– the localization radius for evaluating the related task of *place recognition* [@arandjelovic18:netvlad; @gomez-ojeda15:appearance-invariant].
Prior works on VL generally utilize two broad classes of methods: 3D structure-based localization (3D-VL) and 2D image-based localization (2D-VL) [@sattler17:are-3d-models-necessary; @sattler2018benchmarking], both of which utilize local or global low-level appearance descriptors. 3D-VL methods typically associate a local descriptor with each point in a 3D model of the scene [@irschara2009structure; @li2012worldwide; @sattler2015hyperpoints], while 2D-VL methods extract a holistic descriptor or multiple local descriptors for query-based matching [@arandjelovic14:disloc; @torii15:repetitive-structures; @arandjelovic18:netvlad; @torii18:dense-vlad]. A primary drawback with methods using low-level features is that they are not robust to changes in viewing conditions [@schonberger2018semantic; @naseer17:semantics-aware-visual-localization]; while 3D models are less scalable and present greater computational complexity. To this end, we focus on improving 2D-VL methods specifically for operating under severe changes in viewing conditions in large-scale urban environments and over large time lags [@maddern20171].
![[]{data-label="fig:intro"}](flow_diag.pdf){width="0.7\linewidth"}
Recently 2D-VL methods have used deep convolutional neural networks (DCNN) based frameworks that learn to generate view-invariant image representations (or embeddings) [@gomez-ojeda15:appearance-invariant; @arandjelovic18:netvlad; @zhu18:apa]. Despite advancing the state-of-the-art, DCNN still suffer from two key issues in long-term VL. First, due to the reliance on low or mid-level appearance features, DCNN methods suffer a loss in accuracy for large changes in viewing conditions [@naseer17:semantics-aware-visual-localization; @schonberger2018semantic]. Second, since DCNN methods extract holistic representations from an entire image without any explicit understanding of scene composition, the resulting image representation is degraded by non-discriminative visual elements (vehicles) [@naseer17:semantics-aware-visual-localization; @zhu18:apa; @piasco2018survey]. To alleviate these issues, we propose a novel DCNN framework to improve the robustness of learned visual embeddings by incorporating
high-level semantic information inside the neural network, and
an attention-based framework that uses both appearance and semantic information to guide the model to focus on informative and stable image regions.
Our method, referred to as Semantically-Aware Attentive Neural Embeddings (SAANE) (see [first@reffig:intro,@]{}), is an end-to-end trainable deep neural network, that effectively combines semantic[^1] and mid-level appearance features with spatial attention to learn semantically-aware embeddings. The design of SAANE is inspired by recent advances in deep multimodal and attention guided learning [@ahuja2018understanding; @woo2018cbam; @xu2016ask; @yang2016stacked; @afouras2018deep] and is composed of three key modules:
the *modality fusion module:* to fuse information from the two modalities,
the *multimodal attention module:* to use the fused representation to predict a shared channel attention and separate modality-specific spatial attentions, and
the *spatial pooling module:* to pool the attended multimodal features to generate embeddings, which are then used to retrieve matches for a query image.
We use convolutional feature maps from models pre-trained for image classification and semantic segmentation as appearance and semantic representations respectively. Our model is trained end-to-end by using a ranking based objective.
The key motivation behind our approach is that compared to low/mid-level appearance descriptors, the spatial layout of semantic classes in the image yields scene descriptions that have a higher invariance to large changes in viewing conditions for long-term VL. The semantic understanding of the image content along with spatial attention helps to determine which regions of the scene may be unreliable for localization across large time scales. A key advantage of SAANE over prior works, that have attempted to improve the robusness of low-level features, is that it offers a principled end-to-end module to learn semantically-aware representations with spatial attention- guided by multimodal features.
The key contributions of this paper are as follows:
1. We incorporate higher-level semantic information along with commonly-used mid-level appearance features to enhance the quality of the *learned* image embeddings for visual localization under large appearance changes.
2. We propose an attention-based neural module to allow the model to focus on stable and discriminative regions. To the best of our knowledge, ours is the first work to propose a DCNN-based pipeline that combines multimodal representations (appearance and semantic), with spatial attention for visual localization.
3. We perform extensive ablation studies across three datasets to show the contributions of each part of our model. We particularly investigate and establish the necessity of the proposed separable spatial with shared channel attention for current problem.
Related Work
============
Methods for image-based visual localization generally fall into two classes: 3D structure-based localization (3D-VL) [@sattler2015hyperpoints; @zeisl2015camera; @svarm2017city; @sattler17:are-3d-models-necessary; @schonberger2018semantic] and 2D image-based localization (2D-VL) [@cummins2008fab; @cummins11:fab-map; @torii18:dense-vlad; @arandjelovic18:netvlad; @zhu18:apa]. 3D-VL methods create a 3D model of the scene by either using Structure-From-Motion (SfM) or associating local patches to 3D point clouds. On the other hand, 2D-VL methods formulate visual localization as an image retrieval problem by matching a query image with geo-tagged database images for approximate localization. In this paper we focus on improving 2D-VL methods in large-scale scenes, which can then be used in combination with 3D methods for more precise localization as shown in [@sattler17:are-3d-models-necessary].
Most initial works for visual localization relied on matching Bag of Visual Word (BoVW)-type features [@glover10:fabmapratslam; @arandjelovic14:disloc] or using global image descriptors in addition to sequential search [@milford12:seqslam]. Building on the success of deep convolutional neural networks (DCNN) in other areas [@girshick2015fast; @krizhevsky2012imagenet], recent works have extensively demonstrated the effectiveness of off-the-shelf DCNN features for visual place recognition and VL [@sunderhauf15:cnn-placerecog; @chen14:cnn-placerecog; @garg18:lost]. Several works have also focused on improving feature pooling methods for off-the-shelf features [@babenko2015aggregating; @yue2015exploiting; @torii18:dense-vlad] or those learned in end-to-end pipelines [@arandjelovic18:netvlad; @zhu18:apa].
In general, (deep) learned global descriptors are more robust than hand-crafted features but are still susceptible to perceptual aliasing from repeated patterns such as road markings and buildings that introduce indistinguishable global matches; , only specific, meaningful regions are useful for localization [@chen17:only-look-once; @tolias2015particular; @finn2016deep]. In prior works, the contribution of combining mid-level image features with higher-level semantic information has been entangled with the addition of 3D information [@schonberger2018semantic; @radwan2018vlocnet++]. Semantics were also used to focus on manually-selected regions [@naseer17:semantics-aware-visual-localization] or to enhance off-the-shelf appearance features [@arandjelovic2014visual; @garg18:lost]. Our method’s novel multimodal attention module learns not only to combine features across modalities but also to focus on the most stable regions by discovering region types in a data-driven fashion.
We also situate our work within the space of making image features semantically-grounded and more robust to appearance changes. A typical approach to matching image features across large perceptual changes is to learn a transformation from one type of appearance to another—such as across seasons [@naseer14:robust-visual-localization] or across different times of day [@lowry14:morning-to-afternoon; @anoosheh2018night]—or to train a DCNN for appearance invariance using paired images [@gomez-ojeda15:appearance-invariant; @chen17:deep-learning-place-recog; @chen2018learning]. Our method seeks to integrate this innovation with stable semantic scene features to further improve the invariance of these features.
Our work is also related to recent works on combining vision and language for the Visual Question Answering (VQA) task [@antol2015vqa]. The underlying fusion networks vary from pooling methods such as sum/bilinear pooling [@fukui2016multimodal] to more complex attention based methods [@teney2018tips; @anderson2018bottom; @yang2016stacked; @lu2016hierarchical]. Although the proposed multimodal attenion module is motivated by these works, it is both different and better suited to the requirements of the long-term VL task, which is fundamentally different from the VQA task. For example, the appearance and semantic modalities in VL are more closely-aligned as compared to vision and language in VQA, due to which additional blocks are required for aligning the modalities prior to fusion.
Approach: SAANE
===============
We now describe our approach, refered to as **SAANE** (Semantically-Aware Attentive Neural Embeddings), in detail. Our model passes an input RGB through two DCNNs pre-trained for image classification and semantic segmentation to obtain mid-level appearance features (denoted as ) and high-level semantic features (denoted as ) respectively. We denote feature maps from layer $l$ of the image-classification and semantic-segmentation CNNs as $\mF^{A}_{l} \in \R^{C_A \times H \times W}$ and $\mF^{S}_{l} \in \R^{C_S \times H \times W}$ respectively, where $C_A$ and $C_S$ are the number of channels. We drop the notation $l$ in the rest of the paper for brevity. We work in a supervised setting with a database of geo-tagged images captured under different viewing conditions.
SAANE is an end-to-end trainable DCNN that learns to generate robust image embeddings that, in addition to being invariant to changes in viewing conditions, are aware of semantic composition of the scene and focus explicitly on informative visual elements due to the use of spatial attention. Our model () consists of three NN-based modules:
the modality fusion module,
the multimodal attention (MM-Att) module, and
the spatial pooling module.
SAANE operates by first transforming and fusing features from the appearance and the semantic input streams by using the modality fusion module. The output is a semantically-informed multimodal representation of input, which is then used to estimate per-modality spatial attentions using the proposed MM-Att module. We encourage sharing of information between the two modalities by building upon a prior work for unimodal attention [@woo2018cbam] and computing a shared channel attention which is then used to generate separate spatial attentions as described below. The output from this module is used to refine the feature maps from both modalities, which are then fused together with another modality fusion module. Finally, we use a spatial pooling to output the embeddings. We train our model in an end-to-end fashion to learn each of these modules for visual localization. We now describe these modules along with the loss function and the training procedure.
#### Modality Fusion Module.
The modality fusion module aligns the feature maps— and —by first projecting them in a common space and then adding them together [@yang2016stacked; @ahuja2018understanding]. We use $1 \times 1$ convolutions, denoted by $\mW_{A}^{1} \in \R^{C \times C_A \times 1 \times 1}$ and $\mW_{S}^{1} \in \R^{C \times C_S \times 1 \times 1}$ for the appearance and semantic streams respectively, to project the feature maps in a $C$-dimensional common space. $$\begin{aligned}
\mF^{M} &= \mW_{A}^{1} \conv \mF^A +\mW_{S}^{1} \conv \mF^S \\
{} &= \mF^M_A + \mF^M_S,
\end{aligned}
\label{eq:initial_projection}$$
where $\mF^{M}$ is the fused multimodal representation of the image, $F^{M}_{A}$ and $F^{M}_{S}$ are the aligned features maps from and respectively, and $\conv$ is the convolutional operator. The output is a semantically-informed multimodal representation of the input and is used as input to both MM-Att and later to the spatial pooling module. Although recent methods have used sophisticated pooling approaches [@fukui2016multimodal], we opted for projected sum pooling, as it uses few trainable parameters and maintains the spatial configuration of the feature maps, as required for the attention step.
#### Multimodal Attention Module (MM-Att).
The multimodal attention module is responsible for predicting attention at different spatial locations independently for appearance and semantic input streams. The spatial attention allows our network to selectively focus on discriminative and stable visual elements such as buildings instead of confusing/dynamic elements such as cars/pedestrians [@zhu18:apa]. This results in embeddings that are more robust to perceptual changes especially in urban environments. We use the combined multimodal representation, computed by the fusion module, to sequentially predict a shared channel attention (denoted by $\mM_c \in \R^{C}$) and individual spatial attentions for the two modalities (denoted by $\mM_{xy}^A \in \R^{H \times W}$ and $\mM_{xy}^S \in \R^{H \times W}$ for appearance and semantic channels respectively). We believe that a tied channel attention allows sharing of information between the two modalities leading to a better spatial attention (as also evident later in our results in ). The channel attention is computed by summarizing the feature maps across the spatial dimensions by *average* ($\mF^{M}_\text{avg}$) and *max* ($\mF^{M}_\text{max}$) pooling, and passing them through a multi-layer perceptron (MLP) followed by an addition and a non-linearity: $$\mM_c = \sigma(\phi(\mF^{M}_\text{avg}) + \phi(\mF^{M}_\text{max})),$$ where $\sigma$ denotes the sigmoid function, $\phi$ denotes a two-layer MLP shared across the two pooled inputs. The refined multimodal representation with attended channels is computed as $\hat{\mF}^{M} = \mF \odot \mM_c$, where $\odot$ denotes element-wise multiplication with appropriate broadcasting (copying) of attention values along the spatial dimension.
The refined image representation is then used to predict per modality spatial attentions by using two $7 \times 7$ convolutional filters—${\mW}_A^{2}$ and $\mW_S^{2}$—for appearance and semantic input streams, respectively. $\hat{\mF}^{M}$ is pooled across the channel dimension by both *average* ($\hat{\mF}^{M}_\text{avg}$) and *max* ($\hat{\mF}^{M}_\text{max}$) pooling and concatenated across the channel dimension and convolved with the corresponding filters. The spatial attention maps are then used with the common channel attention to attend to the transformed maps from and generate refined features denoted as $\hat{\mF}^A$ and $\hat{\mF}^S$ for and , respectively: $$\begin{gathered}
M_{xy}^{Z} = \mM_c \odot \sigma(\hat{\mW_Z^{2}} \conv
([\hat{\mF}^{M}_\text{avg}\,;\hat{\mF}^{M}_\text{max}])) \quad \forall Z \in \{A, S\}
\end{gathered}$$ The final attended features $\hat{\mF}^{A}$ and $\hat{\mF}^{S}$ from the appearance and semantic input streams are given by $\hat{\mF}^A = \mF^M_A \odot M_{xy}^{A}$ and $\hat{\mF}^S = \mF^M_S \odot M_{xy}^{S}$ respectively. We use another modality fusion module to fuse these refined features and then input them to the spatial pooling module.
#### Spatial Pooling Module.
This module is responsible for pooling the information from the attended and fused features from the previous modules. In this work we use spatial pyramid pooling (SPP) [@lazebnik06:spatial-pyramid-matching] since it has been previously shown to be effective, and does not include any trainable parameters. Other equally effective alternatives, such as NetVLAD [@arandjelovic18:netvlad], would work in our framework. Finally, following the intuition of Ranjan [@ranjan2017l2], we $L_2$-normalize these embeddings and scale them by a factor of $\alpha = 10$.
#### Loss.
We use a max-margin-based triplet ranking loss function to learn our model [@schroff2015facenet]. This loss optimizes the network such that images from similar locations should be located closer in the embedding space than images from different locations. For computational efficiency, we form triplets in an online manner by sampling them from each minibatch [@schroff2015facenet].
Experimental Results {#sec:exp}
====================
We first describe our train/test datasets along with the evaluation metric (). We then compare our model against a strong DCNN baseline while carefully demonstrating the contribution of each component of our model (). Thereafter, we discuss qualitative results to provide important insights into the proposed attention module (). We finally compare our approach with state-of-the-art (SOTA) 2D-VL methods on the three test datasets ().
Datasets and Evaluation {#sec:datasets}
-----------------------
#### Datasets.
We utilize a version of the Specific Places Dataset (SPED) [@chen17:deep-learning-place-recog] to train our model. We randomly sample $\sim 2600$ cameras from the Archive of Many Outdoor Scenes [@jacobs2007consistent] and download images collected every half hour from Feb-Aug 2014. We remove all images where (i) the camera feed was corrupted, obscured, or too dark for visibility, and (ii) capture location of the camera was not fixed. The final dataset comprises $1.3$ million images drawn from $2079$ cameras featuring significant scene diversity, ranging from urban roads to unpopulated landscapes, and appearance changes due to seasonal and day-night cycles. We train our model on this dataset and evaluate the performance on 2D-VL on three challenging datasets as shown in ). Since our focus is long-term VL and not place recognition as mentioned previously, we use the benchmarks as described in [@chen17:deep-learning-place-recog; @sunderhauf15:cnn-placerecog] . For St. Lucia and Nordland, we use the procedure in [@sunderhauf15:cnn-placerecog] due to insufficient experimental details in [@chen17:deep-learning-place-recog]. Since the remaining datasets used in [@chen17:deep-learning-place-recog] were either less challenging (Eynsham) or small (Gardens Point), we adapt RobotCar [@maddern20171] to provide a more challenging test-bed. We refer readers to for additional details regarding the datasets.
#### Evaluation Metric.
We report Area Under the Curve (AUC) by constructing precision-recall curves using “the ratio test” [@sunderhauf15:cnn-placerecog; @chen17:deep-learning-place-recog]. In brief, a match between a query image and the database is considered positive if the ratio of the Euclidean distances of the best match and the second-best match by nearest neighbors is above some threshold. A positive match is a true positive if it is within $5$ frames of the ground truth.[^2] A precision-recall curve is then constructed by varying the threshold on the ratio test. The use of the ratio test enables the measurement of the frame-by-frame matches without relying on the specific pose being made available, which is the case for our test datasets (collected as synchronized traversals of the same route). More concretely, as the ratio test requires the feature distance between the best and second-best match to be above some threshold, it actually provides a stricter requirement for correct localization than Recall@1 metric, which is used to evaluate place recognition.
#### Prior Methods.
We refer readers to for implementation details regarding our model. For a fair comparison with prior methods, we use the same backbone DCNN networks for all methods. We implement AMOSNet [@chen17:deep-learning-place-recog] by fine-tuning all layers of the DCNN on the SPED dataset. We also implement a recent SOTA method using both attention and pyramid pooling: Attention-based Pyramid Aggregation Network (APANet) [@zhu18:apa]. We first implement their proposed cascade pyramid attention block using the appearance features. Although their method did not utilize semantic information, we also experiment with a stronger counterpart that uses two cascade pyramid attention blocks over the multimodal feature maps as used in our work (plus our multimodal projection as in ). We use the learned attention to sum pool the spatial pyramid features across both modalities to achieve the final embedding. We also compare with implementations of DenseVLAD [@torii18:dense-vlad] and NetVLAD [@arandjelovic18:netvlad], trained on the Pitts30k visual place recognition dataset [@torii15:repetitive-structures], provided by the respective authors.
Quantitative Results {#sec:quant}
--------------------
The performance of the proposed method along with different baselines and prior SOTA approaches is presented in . We first compare our model with different baselines to highlight the benefits of the proposed ideas in using
*semantic* information, and
novel *multimodal attention module* to focus on discriminative regions for visual localization.
We validate the benefits of semantic information by comparing the performance of the baseline model using only appearance information (App \[Baseline\]) with that using both appearance and semantic information (App+Sem). Across our test datasets, there is an average absolute improvement of $9\%$, with the largest gain ($25\%$) on the RobotCar AM$\to$PM by using semantic information.
The remaining variants (App-Att, App-Att+Sem-Att, and SAANE) serve to demonstrate the benefits of the proposed *attention* module. Between App \[Baseline\] and App-Att, we see an average performance hike of $8\%$ (averaged on all datasets) from the attention module that learns to focus on informative visual regions. The improvements on Nordland ($16\%$) from attention alone demonstrate the ability of the module to suppresses confusing regions. These consistent gains demonstrate the benefits of using spatial attention with appearance information. However, this does not seem to be the case if we naively compute attention for both modalities with separate attention modules and then combine the resulting features. For example, while comparing App-Att+Sem-Att—the model variant that predicts separate attention over each modality—with App+Sem, we observe only minor improvements on the RobotCar AM$\to$PM dataset ($2\%$), likely because there is no sharing of information between the two modalities to encourage semantically informed and more consistent attention maps.
Our model addresses this by using the fused multimodal image representation to predict spatial attention for each modality by first predicting an intermediate *shared channel attention*. SAANE yields the best performance across all variants on each dataset ($12\%$ improvement over the baseline and $4\%$ over App+Sem and $5\%$ over App-Att+Sem-Att). Both Nordland ($9\%$) and St. Lucia ($11\%$ in the worst case) are further refined by sharing channel attention across modalities, while the most perceptually-challenging test, RobotCar AM$\to$PM, sees a further performance increase of $4\%$ over App-Att+Sem-Att and of $31\%$ over App \[Baseline\]. The proposed model is thus able to show consistent improvements over different baselines across all datasets for the task of visual localization.
Qualitative Results {#sec:qual}
-------------------
![[]{data-label="fig:rbc"}](retrieval_rbc.pdf){width="0.8\linewidth"}
We show top retrievals in for a case with significant variations in viewing conditions. We visualize the attended regions by showing the spatial attention maps from both modalities along with the semantic segmentation maps. We show a query image from the Night Autumn matched against retrieved database images from the Overcast Autumn set of RobotCar (, AM$\to$PM in ). For methods relying only on appearance information (, AMOSNet and the baseline), the retrieved images in the rightmost column are incorrect but present similar scene layouts, while our model retrieves a match within two meters of the query location. We see that across time of day, the maps from both attention modalities remain consistent and focus on stable features, with the appearance attention focusing mostly on fine details such as lane markings or architectural elements and semantic attention giving emphasis to scene layout and skyline shape. Interestingly, we note the bus present in the matched database image, obscuring a significant part of the scene. While the appearance modality attends highly to the bus’s features, as if it were any other structure, we can see that the semantic attention module has learned to disregard this region as a dynamic object and gives more emphasis to the remaining scene layout. These results show that the proposed attention module guides the features to look at consistent regions even across extreme changes in viewing conditions. We refer readers to for further analysis on the semantic-classes of regions attended across both modalities.
Comparison with state-of-the-art {#sec:sota}
--------------------------------
shows the comparison of our model with several SOTA techniques. Our model shows consistent improvements on the test datasets in comparison to DenseVLAD ($28\%$ average absolute) and NetVLAD ($19\%$), both SOTA 2D-VL methods. We note that NetVLAD performs comparably to our model on RobotCar Sum.$\to$Win. where the appearance changes are relatively minor; however, its performance is much lower on the test datasets with more extreme changes, which is consistent with prior work [@schonberger2018semantic; @garg18:lost]. We also observe an average absolute improvement of $12\%$ (across all datasets) over AMOSNet, which was the previous SOTA method on both the Nordland and St. Lucia datasets. We note that for datasets which present minor appearance variation (, St. Lucia), nearly the same result is achieved from fine-tuning on SPED (AMOSNet) as from adding additional semantic features (App+Sem). However, the capacity of our complete (SAANE) attention module to further refine the localization is shown again, with $3\%$ improvement over AMOSNet. For cases with more severe appearance variation, we see a much larger improvement from training the proposed modules combining semantics, modality-specific spatial attention, and shared channel attention; , results on the Nordland dataset show an absolute improvement of $32\%$ and on RobotCar AM$\to$PM an improvement of $22\%$. Similarly, SAANE also shows improvements between $37\%$ and $38\%$ over both implementations of APANet. Adding the semantic modality to APANet-MM is insufficient to significantly boost the performance of this pooling method in this scenario.
Conclusion
==========
We present an attention-based, semantically-aware deep embedding model for image-based visual localization: SAANE. Our model targets the sensitivity of appearance-based visual localization to both perceptual aliasing from repetitive structures and extreme differences in viewing conditions. SAANE uses a novel multimodal attention module that fuses both mid-level appearance and high-level semantic features by predicting attention maps for both modalities with a shared channel attention. The attended maps are then fused to generate semantically informed image embedding. We evaluate the performance benefits of our model on three challenging public visual localization datasets, while showing significant gains compared to the baseline ($12\%$) and prior state-of-the-art methods ($19\%$). We also show that SAANE learns to produce stable attention maps that focus on consistent image regions across changing views, which is crucial for autonomous navigation.
Acknowledgement
===============
We would like to thank Ajay Divakaran for proof-reading the manuscript and providing helpful comments.
Implementation Details {#appsec:impl}
======================
The backbone of SAANE is two parallel DCNNs. We use a ResNet50 [@he2016deep] pre-trained for the Imagenet classification task for mid-level feature extraction and a Pyramid Scene Parsing Network (PSPNet) [@zhao17:pspnet] pre-trained on the ADE20K [@zhou2017scene] semantic segmentation task for extracting semantic features. We also experimented with a version of PSPNet pre-trained on Cityscapes [@Cordts2016Cityscapes]; however, we found the ADE20K version to be more robust to viewpoint changes and the 150 classes of ADE20K to be more useful in the presence of diverse scene types. We use the output of the third residual block from ResNet50 as mid-level appearance features ($\mF^{A}$). For high-level semantic features ($\mF^{S}$), we use the output before the final convolutional layer of PSPNet. The resulting number of channels in appearance and semantic features are $C_R = 1024$ and $C_S = 512$, respectively. We set the number of channels of the common embedding space in the modality fusion module ($C$), both before and after MM-Att, to $256$. We use spatial pyramid pooling with pooling sizes of $[4, 3, 2, 1]$ and concatenate the feature maps from all layers to produce the final embedding. The dimensionality of the final embeddings after spatial pooling is $7680$. For our experiments, we the two pre-trained DCNNs and fine-tune the two modality fusion modules and the MM-Att module. We use the Adam optimizer [@kingma2014adam] with a learning rate of $5 \times
10^{-5}$ and weight decay of $5 \times 10^{-4}$ for training. We use online triplet sampling with batches comprised of 16 different classes with 4 examples per class. Within a batch, we utilize distance-weighted triplet sampling [@manmatha17:sampling-matters] to increase the stability of the training process. We use a margin of $m=0.5$, selected based on giving the best performance on a small validation set. Due to our assumption that our test data come from a dissimilar distribution as our training data, we did not by default experiment with any form of whitening as used in [@arandjelovic18:netvlad; @zhu18:apa].
Finally, to explore the effect of the model capacity on performance on the test datasets, we experiment with varying the dimensionality $C$ of the multimodal fusion network. As shown in , the performance across all of the datasets plateaus between $128$ and $256$ channels and shows evidence of overfitting, particularly in the case of Nordland, above $256$ channels. The best dimensionality of the multimodal fusion module also appears to be a function of the dataset difficulty. Our model’s performance on RobotCar Sum.$\to$Win., in the presence of minor seasonal variations, is relatively stable, even down to $16$ channels, while the tasks with more extreme variation, such as RobotCar AM$\to$PM, sharply decline below $128$.
Test Datasets {#appsec:ds}
=============
#### Nordland
[@neubert2015superpixel] is derived from continuous video footage of a train journey recorded for a Norwegian television program, recorded from the front car of the train across four different seasons. We extract one frame per second from the first hour of each traversal, excluding images where the train is either stopped or in tunnels. This results in $1403$ frames per traversal. We perform our experiments by constructing a database with the summer traversal and querying it with the winter traversal (Sum.$\to$Win.). The images feature no viewpoint variation, due to travel on fixed rails; however, the seasonal appearance changes are quite severe.
#### St. Lucia
[@glover10:fabmapratslam] comprises ten different traversals recorded by a forward-facing webcam affixed to the roof of a car, following a single route through the suburb of St. Lucia, Queensland, Australia. This dataset was captured at five different times of day on different days across two weeks. We use the first traversal (‘100909\_0845’) as the database and query with the remaining nine, reporting the average as well as the worst case result over the nine trials. We sample images at one frame per second, which results in each traversal containing on an average $1350$ frames. The dataset features slight viewpoint variations due to differences in the route taken by the vehicle. There are mild to moderate appearance changes due to differences in time of day and the presence of dynamic objects in the scene.
#### Oxford RobotCar
[@maddern20171] comprises several different traversals of the city of Oxford by a vehicle. It was collected across varying weather conditions, seasons, and times of day over a period of a year. We select two pairs of traversals, referred to as Overcast Autumn/Night Autumn and Overcast Summer/Snow Winter.[^3] We perform an experiment by building a database with either Overcast Summer or Overcast Autumn and querying it with Snow Winter (Sum.$\to$Win.) or Night Autumn (AM$\to$PM), respectively. We make use of the center image from the front-facing camera and extract one frame per second. On average, each traversal covers nearly $9$ km and $2000$ frames. There are mild viewpoint variations present, again due to slight differences in the starting point and road position of the traversals. The appearance change in the day-night pair is quite drastic, largely from the difference in illumination quality in the transition from sunlight to street lights; while it is more moderate in the summer-winter pair, with minor variation from seasonal vegetation and ground cover.
[^1]: By *semantic* we refer to representations that provide high-level information about the input; , per-pixel depth or semantic segmentation maps.
[^2]: For Nordland, we use the synchronized frame correspondence. On the other test datasets, $5$ frames covers the ground-truth frame variance ($0$–$25$ m), a stricter requirement for positive localization than is typically used for place recognition ($25$–$40$ m). See Appendix A for a detailed discussion.
[^3]: The first three were introduced in [@garg18:lost], and the traversals were originally referred to in [@maddern20171] as 2014-12-09-13-21-02, 2014-12-10-18-10-50, 2015-05-19-14-06-38, and 2015-02-03-08-45-10, respectively.
|
---
abstract: 'Morphologies of a vesicle confined in a spherical vesicle were explored experimentally by fast confocal laser microscopy and numerically by a dynamically-triangulated membrane model with area-difference elasticity. The confinement was found to induce several novel shapes of the inner vesicles, that had been never observed in unilamellar vesicles: double and quadruple stomatocytes, slit vesicle, and vesicles of two or three compartments with various shapes. The simulations reproduced the experimental results very well and some of the shape transitions can be understood by a simple theoretical model for axisymmetric shapes.'
author:
- Ai Sakashita
- Masayuki Imai
- Hiroshi Noguchi
title: 'Confinement-induced shape transitions in multilamellar vesicles'
---
Cells and cell organelles have various shapes depending on their functions. For example, red blood cells (RBC) have a biconcave disk shape; this allows large deformations with a fixed area and volume so that RBCs can flow in microvessels narrower than themselves [@fung04; @fedo13]. These discocyte shapes can be observed in unilamellar liposomes and reproduced by minimizing the membrane bending energy with area and volume constraints [@canh70; @helf73]. In living cells, organelles such as Golgi apparatus, endoplasmic reticulum, and mitochondria have much more complicated shapes; their local curvatures are considered to be regulated by BAR and other proteins [@shib09; @baum11]. Among these organelles, mitochondria have a specific feature, [*i.e.*]{}, it consists of two bilayer membranes [@frey00; @mann06; @sche08]. The inner membrane has a much larger surface area than the outer one and forms numerous invaginations called cristae. The cristae have tubular and planar structures, and their narrow junctions regulate diffusion between different compartments. Although the confinement by the outer membrane is expected to play a role in determining the shape of the inner membrane, it is not well understood so far. In this letter, we reveal the effects of the confinement using multilamellar liposomes as a simple model system.
Unilamellar liposomes form various morphologies such as stomatocyte, pear, pearl-necklace, and branched starfish-like shapes. All of these shapes can be reproduced by the area-difference-elasticity (ADE) model [@lipo95; @svet89; @seif97; @khal08; @saka12]. In contrast, the shapes of multilamellar vesicles have not received attention. The main reasons are experimental difficulties with distinguishing inner and outer membranes and controlling the volume and area of inner vesicles. We used a fast confocal laser microscope to extract three-dimensional (3D) images of multilamellar liposomes and observed several confinement-induced shapes of the inner vesicles.
Kahraman [*et al.*]{} [@kahr12a; @kahr12b] recently simulated a vesicle in spherical or ellipsoidal confinement and found that weak confinement produces a stomatocyte with an open (circular or elliptic) neck; further confinement results in the formation of a double stomatocyte where the inner sphere of a typical stomatotye is filled by additional spherical invagination. Hereafter, we call a normal stomatocyte a single stomatocyte to distinguish it from a double stomatocyte. Kahraman [*et al.*]{} considered the bending energy with weak spontaneous curvatures but not the ADE energy. We simulated the confined vesicles by using a dynamically-triangulated membrane with the ADE model and obtained many more varieties of vesicle shapes, including experimentally observed shapes. We also analyzed the mechanism for shape determination using a simple theoretical model of axisymmetric shapes.
We prepared single-component vesicles from DOPC (1,2-dioleoyl-sn-glycero-3-phosphocholine, Avanti Polar Lipids) using the gentle hydration method with pure water [@saka12]. TR-DHPE (Texas Red, 1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine, Molecular Probes) was used as the dye. We kept vesicle suspensions at room temperature ($24$-$25^\circ$C) and observed them by a fast confocal laser microscope (Carl Zeiss, LSM 5Live). At this stage, most vesicles formed either a spherical or tubular shape spontaneously, and some had one or more vesicles inside. We then focused on multilamellar vesicles.
Various shapes of the inner liposomes were observed (see Figs. \[fig:sto2db\](a) and \[fig:phase\_v\](a) and the 3D image of a double stomatocyte in Supplemental Material [@epaps1]). Among them, the double stomatocytes \[Fig. \[fig:phase\_v\](a)iv\] and open single stomatocytes \[Fig. \[fig:sto2db\](a)i\] agree with the shapes predicted in the previous simulations [ [@kahr12a; @kahr12b]]{}. We also observed other liposome shapes that have not been previously reported: a vesicle with a deep slit (Fig. \[fig:sto2db\](a)iii,iv), two hemispheres (or buds) connected by a small neck \[called doublet, see Figs. \[fig:sto2db\](a)v and \[fig:phase\_v\](a)ii\], and a single stomatocyte of discoidal invagiation \[see Fig. \[fig:phase\_v\](a)iii\].
![ Shape changes of liposome confined in spherical liposome. (a) Time-sequential microscopy images of liposomes. From left to right: $t=0$, $223$, $225$, $227$, and $230$ s. Scale bar: $10$ $\mu$m. (b) Sequential snapshots of triangulated-membrane simulation at $v_{\rm r}=0.6$ and $v_{\rm {con}}= 0.727$. Intrinsic area difference $\Delta a_0$ gradually increased. From left to right: $\Delta a_0=0.93$, $1.30$, $1.44$, $1.55$, and $1.80$. Inner and outer vesicles are shown in green and light gray, respectively. (c) Free energy $F$ of simple axisymmetric shapes. Solid and dashed lines represent energy-minimum and metastable states, respectively. Typical shapes are shown in the inset. (d) Area difference $\Delta a$ of two leaflets depending on $\Delta a_0$. Black and gray lines represent simulation results with increasing or decreasing $\Delta a_0$, respectively. Other solid and dashed lines represent same data in (c) for simple axisymmetric shapes. []{data-label="fig:sto2db"}](fig1a.eps "fig:") ![ Shape changes of liposome confined in spherical liposome. (a) Time-sequential microscopy images of liposomes. From left to right: $t=0$, $223$, $225$, $227$, and $230$ s. Scale bar: $10$ $\mu$m. (b) Sequential snapshots of triangulated-membrane simulation at $v_{\rm r}=0.6$ and $v_{\rm {con}}= 0.727$. Intrinsic area difference $\Delta a_0$ gradually increased. From left to right: $\Delta a_0=0.93$, $1.30$, $1.44$, $1.55$, and $1.80$. Inner and outer vesicles are shown in green and light gray, respectively. (c) Free energy $F$ of simple axisymmetric shapes. Solid and dashed lines represent energy-minimum and metastable states, respectively. Typical shapes are shown in the inset. (d) Area difference $\Delta a$ of two leaflets depending on $\Delta a_0$. Black and gray lines represent simulation results with increasing or decreasing $\Delta a_0$, respectively. Other solid and dashed lines represent same data in (c) for simple axisymmetric shapes. []{data-label="fig:sto2db"}](fig1b.eps "fig:")
In order to understand these distinctive vesicle shapes, we describe the inner vesicle as a fluid membrane of volume $V_{\rm {ves}}$ and outer vesicle as a hard sphere of volume $V_{\rm {sp}}=(4\pi/3) R_{\rm {sp}}^3$. In the ADE model, the free energy of a single-component vesicle with a fixed topology is given by $$F = \int \Bigl[ \frac{\kappa}{2}(C_1+C_2)^2 \Bigr] dA
+ \frac{\pi k_{\rm {ade}}}{2Ah^2}(\Delta A - \Delta A_0)^2
\label{eq:ade}$$ where $C_1$ and $C_2$ are the principal curvatures at each point in the membrane. The coefficients $\kappa$ and $k_{\rm {ade}}$ are the bending rigidity and ADE coefficient, respectively. The areas of the outer and inner leaflets of a bilayer vesicle differ with $\Delta A= h \oint (C_1+C_2) dA$, where $h$ is the distance between the two leaflets. Since the flip-flop of lipids (traverse motion between leaflets) is very slow, the area difference $\Delta A_0=(N_{\rm {out}}-N_{\rm {in}})a_{\rm {lip}}$ preferred by lipids is typically different from $\Delta A$, where $N_{\rm {out}}$ and $N_{\rm {in}}$ are the number of lipids in the outer and inner leaflets, respectively, and $a_{\rm {lip}}$ is the area per lipid. We used typical values for the lipid membranes $\kappa=k_{\rm {ade}}=20k_{\rm B}T$, where $k_{\rm B}T$ is the thermal energy [@seif97; @saka12]. We present our results using the reduced volume $v_{\rm r}= V_{\rm {ves}}/(4\pi R_{\rm A}^3/3)$, confinement volume ratio $v_{\rm {con}}= V_{\rm {ves}}/V_{\rm {sp}}$, reduced area differences $\Delta a =\Delta A/8\pi h R_{\rm A}$, and $\Delta a_0 = \Delta A_0/8\pi h R_{\rm A}$, where $R_{\rm A}= \sqrt{A/4\pi}$.
![ (a) Microscopy images of liposomes with various shapes of inner liposomes. i) Single stomatocyte of prolate invagiation. ii) Doublet. iii) Single stomatocyte of discoidal invagiation. iv) Double stomatocyte. Scale bar: $10$ $\mu$m. (b) Phase diagram at the most confined state in simulations on ($v_{\rm r}, \Delta a_{0}$) plane. Shapes are categorized by types of compartmentalization: (blue) single stomatocytes of one inner bud, (violet) single stomatocytes of two inner buds, (green) double stomatocytes, (yellow) slit vesicles, (orange) doublets, and (pink) triplets. Typical simulation snapshots are shown in the inset of (b). Their front halves are removed and cut-off sections are shown in red. []{data-label="fig:phase_v"}](fig2.eps)
The double stomatocyte can be modeled as three spheres (radii $R_1 \ge R_2\ge R_3$) connected by two small necks. The volume, area, and free energy $F$ are estimated as $V_{\rm {ves}}=(4\pi/3)(R_1^3- R_2^3+ R_3^3)$, $A=4\pi(R_1^2+ R_2^2+ R_3^2)$, and $F/8\pi\kappa=3 + \pi\{ (R_1- R_2+ R_3)/R_{\rm A} - \Delta a_0\}^2$, respectively. At $R_{\rm {sp}}=R_1$ and $R_2=R_3$, the internal space of the outer sphere is completely filled by the inner vesicle, [*i.e.*]{}, $v_{\rm {con}}=1$, and $F/8\pi\kappa= 3 + \pi( {v_{\rm r}}^{1/3} - \Delta a_0)^2$ for $v_{\rm r} \geq 3^{-3/2} \simeq 0.19$.
Dynamically-triangulated membrane models are widely used to study the shape deformation of membranes and vesicles [@gomp04c; @nogu09]. We used the molecular dynamics of this membrane model with a Langevin thermostat [@nogu09]. The membrane is described by vertices of $N_{\rm v}=4000$, and the mean distance of the bonds connecting neighboring vertices is $\sigma= 0.06R_{\rm A}$. Free energy $F$ in Eq. (\[eq:ade\]) is discretized using dual lattices of triangulation and the area and volume are maintained by harmonic constraint potentials [@nogu05]. The vertices have short-range excluded volume interactions with each other and with the outer sphere. To obtain the most confined state, the radius of outer sphere is varied as $R_{\rm {sp}}=R_{\rm {max}}+0.05\sigma$; the repulsion with the outer sphere efficiently pushes the inner membranes into compact states, where $R_{\rm {max}}$ is the maximum distance of the vertices from the center of the sphere. Since the excluded volumes yield the finite minimum distance $l_{\rm {mb}}\sim \sigma$ between the membranes, $v_{\rm {con}}$ does not reach unity. The minimum radius $R_{\rm {sp}}^{\rm {min}}$ obtained in the simulation agrees with the estimation by the three-sphere model using the finite minimum distance [@epaps1]. The double stomatocyte is only obtained at $v_{\rm r} \lesssim 0.6$. The details on the model are described in Supplemental Material [@epaps1].
The simulations of the ADE model reproduced all of the experimentally observed shapes \[see Figs. \[fig:sto2db\](b) and \[fig:phase\_v\](b)\]. In particular, sequential shape transitions of a liposome from an open single stomatocyte to a doublet via a slit vesicle were reproduced by gradual changes to the intrinsic area difference $\Delta a_0$ (compare Figs. \[fig:sto2db\](a) and (b) and see a movie in Supplemental Material [@epaps1]). Previously, we observed similar $\Delta a_0$ changes in unilamellar liposomes under the same experimental conditions [@saka12]. Small reservoirs of lipid are likely present on the membrane, and the laser illumination of the microscopy induces fusion into either leaflet, which leads to changes in $\Delta a_0$. In the unilamellar (unconfined) liposomes, as $\Delta a_0$ increases, a single stomatocyte changes into a discocyte and then to starfish or pearl-necklace shapes. The confinement destabilizes the discocyte shape; instead, transient open single stomatocytes become stable. The doublets obtained in the confinement are understood as compressed pearl-necklace shapes. The slit vesicle shape is not seen in unilamellar liposomes. The enlargement of an ellipsoidal neck of the open stomatocyte leads to the formation of a tongue-like invagination, that resembles a crista shape in the old view of the classic baffle model [@frey00].
![(color online) Snapshots of vesicles in metastable states. (i) Quadruple stomatocyte at $(v_{\rm r}, \Delta a_{0})=(0.15, 0.80)$. (ii) Double stomatocyte with outer bud at $(v_{\rm r}, \Delta a_{0})=(0.40, 1.40)$. (iii) Triplet at $(v_{\rm r}, \Delta a_{0})=(0.5, 2.3)$. (iv) Tubular (green) compartment wraps the other spherical (red) compartment at $(v_{\rm r}, \Delta a_{0})=(0.54, 3.8)$. (i–iii) Half-cut snapshots are displayed. []{data-label="fig:others"}](fig3.eps)
![Phase diagram of ($v_{\rm {con}}, \Delta a_{0}$) plane at $v_{\rm r}=0.6$. Solid black lines represent phase boundaries estimated by the simple model for axisymmetric shapes. Half-cut snapshots are displayed in the inset. []{data-label="fig:phase_r"}](fig4.eps)
Using the minimum radius $R_{\rm {sp}}^{\rm {min}}$ at $\Delta a_0=0.4$, the phase diagram is constructed as shown in Fig. \[fig:phase\_v\](b). Note that two or more shapes coexist (in particular, several shapes at $v_{\rm r}=0.4$), so that the shape with the lowest energy is chosen. Some of the metastable shapes are shown in Fig. \[fig:others\]. The obtained shapes are categorized by the number of compartments separated by necks. As $\Delta a_0$ increases, the vesicle shapes change from inner-budded shapes (single stomatocyte) to outer-budded shapes (doublets and triplets). In the middle range, slit vesicles and double stomatocytes appear.
Although we consider five categories for simplicity, as shown in Fig. \[fig:phase\_v\](b), the shapes in each category have detailed variations. At $v_{\rm r}=0.9$, the doublet consists of one small spherical compartment and one large compartment. With decreasing $v_{\rm r}$, the size of the smaller compartment increases and reaches the same size of the other at $v_{\rm r}\simeq 0.6$. At lower $v_{\rm r}$, two compartments become interdigitated like the seam of a baseball \[see the second-lowest snapshot on the right of Fig. \[fig:phase\_v\](b)\]. As $v_{\rm r}$ decreases, the inner bud of the single stomatocyte changes its shape: a sphere, prolate, discocyte, and bent tube \[see four snapshots in the blue region of Fig. \[fig:phase\_v\](b)\]. Since the radius of the single stomatocyte is restricted by the outer sphere, the excess area to the confined volume does not always allow a spherical shape of the inner bud. Thus, the inner bud exhibits similar shape transitions to a unilamellar vesicle with the reduced volume $v_{\rm r}^{\rm {in}}=v_{\rm r}(1-v_{\rm {com}})/(v_{\rm {con}}^{2/3}-v_{\rm {r}}^{2/3})^{3/2}$ and area difference $\Delta a_{\rm {in}}= \{(v_{\rm r}/v_{\rm {con}})^{1/3}-\Delta a\}/\sqrt{1-(v_{\rm r}/v_{\rm {con}})^{2/3}}$. The discoidal invagination connected to spherical membranes via a narrow neck and tubular invagination resemble the crista structures in the modern cristae model based on 3D electron microscopy observations [@frey00; @mann06; @sche08]. The large surface area of the inner mitochondrial membrane and small volume between the inner and outer membranes are key factors to determine the crista structures.
Around the right (left) boundary of the double-stomatocyte region in Fig. \[fig:phase\_v\](b), the neck connecting the outer (inner) and middle membranes opens. As the vesicles cross the left phase boundary, the inner membranes exhibit a shape transition to a discocyte \[snapshots (iv) to (iii) in Fig. \[fig:phase\_v\](b)\]. A typical triplet shape is a spherical compartment surrounded by two compartments \[see bottom-right snapshot in Fig. \[fig:phase\_v\](b)\]. Three symmetric compartments (like sections of oranges) are obtained as a metastable state \[see Fig. \[fig:others\](iii)\]. Quadruple stomatocytes can be formed at $v_{\rm r} \leq 0.15$ \[see Fig. \[fig:others\](i)\].
The effects of the confinement strength were investigated through simulations that varied the outer radius $R_{\rm {sp}}$ (see Fig. \[fig:phase\_r\]). With increasing confinement (increasing $v_{\rm {con}}$), the number of compartments decreases at large values of $\Delta a_0$; the triplets transform into doublets. This is caused by larger deformation under greater confinement. More deformed doublets can have larger values of $\Delta a_0$. The double stomatocyte appears when the confinement is sufficiently strong at $v_{\rm {con}} > 0.85$.
To understand the confinement effects more deeply, we analytically calculated the membrane free energy of several simple geometries: single and double stomatocytes with open or closed necks and doublets. Only axisymetirc shapes are considered, and the cross-sectional shape is represented as a combination of arcs \[see cross-section shapes drawn in the insets of Fig. \[fig:sto2db\](c)\]. Details on the model and calculation are described in Supplemental Material [@epaps1]. This model can well explain the shape transition from the single stomatocyte to the doublet (see Fig. \[fig:sto2db\]). Although the slit vesicle is not considered because of its lack of axisymmetry, $\Delta a$ dependence on $\Delta a_0$ is also well reproduced \[see Fig. \[fig:sto2db\](d)\]. The doublet consisting of two hemispheres of the same size is obtained not at $v_{\rm r}=0.6$ but at $v_{\rm r}=0.65$ in this simple model. This difference from the experimental and simulation results would be due to the limitation of this simple shape representation.
At the limit of the full confinement, [*i.e.*]{}, $v_{\rm {con}}\to 1$, the double stomatocyte has the lowest energy at any $\Delta a_0$ (see the solid line in Fig. \[fig:phase\_r\]). The other possible shapes have sharply-bent edges, which bending energy diverges at $v_{\rm {con}}\to 1$. Thus, the double stomatocytes are in thermodynamically stable state at $v_{\rm {con}}\to 1$ for $v_{\rm r} \geq 3^{-3/2}$. For $v_{\rm r} < 3^{-3/2}$, two inner buds of the double stomatocyte buckle together to form open or closed quadruple stomatocytes.
In summary, we revealed various vesicle shapes induced by the confinement in a spherical vesicle. Some of the shapes, such as double stomatocytes and slit vesicles, were observed experimentally for the first time. All of the shapes were reproduced by the computer simulation when the ADE energy is taken into account. Interestingly, tubular and discoidal invaginations obtained at low values of $\Delta a_0$ are very similar to the crista structures in mitochondria. This may suggest that the ADE energy is one of the key quantities to determine the crista structures. The outer vesicles were modeled as a sphere here, but the outer membranes of liposomes and mitochondria are also flexible and can exhibit tubular or more complicated shapes. The effects of deformation of the outer membrane and local spontaneous curvatures induced by proteins are an important unsolved problem for future studies.
We thank P. Ziherl for informative discussions. This work was partially supported by the Japan Society for the Promotion of Science and the female leadership program at Ochanomizu University.
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---
abstract: |
The paper provides new examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau.
More precisely, let $X$ be the standard resolution of a transversal slice to an adjoint nilpotent orbit of a simple Lie algebra over ${{\mathbb C}}$. An action of the affine braid group on the derived category $D^b(Coh(X))$ and a collection of $t$-structures on this category permuted by the action have been constructed in [@BR] and [@BM] respectively. In this note we show that the $t$-structures come from points in a certain connected submanifold in the space of Bridgeland stability conditions. The submanifold is a covering of a submanifold in the dual space to the Grothendieck group, and the affine braid group acts by deck transformations. In the special case when $\dim (X)=2$ a similar (in fact, stronger) result was obtained in [@Br]. The dimension of our subset equals (in most cases) that of the second cohomology of $X$, so it may deserve the name of stringy moduli space; it is in a sense smaller than one may want, hence the attribute “thin”.
We also propose a new variant of definition of stabilities on a triangulated category, which we call a “real variation of stability conditions” and discuss its relation to Bridgeland’s definition. The main theorem provides an illustration of such a relation.
address:
- 'Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA '
- 'Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts ave., Cambridge, MA 02139, USA '
- 'Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003, USA '
author:
- Rina Anno
- Roman Bezrukavnikov
- Ivan Mirković
title: A thin stringy moduli space for Slodowy slices
---
Introduction and statement of result
====================================
Let ${{\mathfrak g}}$ be a semi-simple Lie algebra over ${{\mathbb C}}$, let $e\in {{\mathfrak g}}$ be a nilpotent element, $S\subset {{\mathfrak g}}$ be a transversal (Slodowy) slice to the $G$-orbit of $e$. Let ${{\mathcal B}}=G/B$ be the flag variety of $G$, and $\pi:T^*({{\mathcal B}})\to {{\mathfrak g}}$ be the Springer (moment) map. Set ${{\mathcal B}}_e=\pi^{-1}(e)$ and $X=\pi^{-1}(S)$. Set ${{\mathcal C}}=D^b(Coh_{{{\mathcal B}}_e}(X))$ where $Coh_{{{\mathcal B}}_e}(X)$ is the category of coherent sheaves on $X$ supported on ${{\mathcal B}}_e$.
Certain $t$-structures on ${{\mathcal C}}$ were constructed in [@BM] (announced in [@ICM]). In this note we show that they arise from a certain explicit connected subset of the space $Stab({{\mathcal C}})$ of Bridgeland stability conditions on ${{\mathcal C}}$. To state the result we need more notations.
Let ${{\mathfrak h}}$ denote the (abstract) Cartan algebra of ${{\mathfrak g}}$.
We have ${{\mathfrak h}}^*=\Lambda\otimes _{{\mathbb Z}}{{\mathbb C}}$ where $\Lambda$ is the weight lattice. Let ${{\mathfrak h}}^*_{{\mathbb R}}=\Lambda \otimes_{{\mathbb Z}}{{\mathbb R}}\subset {{\mathfrak h}}^*$ be the real dual Cartan. The affine Weyl group $W_{aff}$ acts on ${{\mathfrak h}}^*$ and on ${{\mathfrak h}}^*_{{\mathbb R}}$ by affine-linear transformations. Let ${{\mathfrak h}}^*_{reg}$ be the union of free orbits of $W_{aff}$ on ${{\mathfrak h}}^*$, thus ${{\mathfrak h}}^*_{reg}$ is the complement to the affine coroot hyperplanes $H_{\check{{\alpha}}, n}=\{{\lambda}\in {{\mathfrak h}}^*\ | \ \langle {\lambda}, \check{{\alpha}} \rangle =n\}$, where $n\in {{\mathbb Z}}$ and $\check{{\alpha}}$ is a coroot.
Action of the affine braid group and $t$-structures assigned to alcoves.
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Recall that an [*alcove*]{} is a connected component of ${{\mathfrak h}}^*_{{{\mathbb R}},reg}={{\mathfrak h}}^*_{reg}\cap {{\mathfrak h}}^*_{{\mathbb R}}$. The natural action of the affine Weyl group on ${{\mathfrak h}}^*$ induces a simply transitive action of $W_{aff}$ on the set of alcoves. We denote this set by $Alc$.
The argument below is based on the construction of [@BM] which assigns a $t$-structure $\tau_A$ on ${{\mathcal C}}$ to an alcove $A\in Alc$. The $t$-structure $\tau_A$ can be described using derived localization over a field of characteristic $p>0$ [@BMR]. Roughly speaking, modules for the sheaf $D_\lambda({{\mathcal B}})$ of twisted differential operators on ${{\mathcal B}}$ are closely related to coherent sheaves on $T^*({{\mathcal B}})$; on the other hand, the derived category $D^b(D_\lambda({{\mathcal B}}))$ is identified with the derived category of an appropriate quotient of the enveloping algebra $U{{\mathfrak g}}$. Thus one can get a $t$-structure on $D^b(Coh(T^*({{\mathcal B}})))$ which is compatible with the tautological $t$-structure on $D^b(U{{\mathfrak g}}-mod)$. The $t$-structure $\tau_A$ arises this way when the twisting parameter $\lambda$ satisfies the condition $\frac{\lambda+\rho}{p}\in A$. There exists also a more direct construction of the $t$-structure $\tau_A$ over a characteristic zero field, though available proof of its properties relies on positive characteristic picture.
Let $B_{aff}=\pi_1({{\mathfrak h}}^*_{reg}/W_{aff})$ be the affine braid group (this is the affine braid group of Langlands dual group in the standard terminology). An action of $B_{aff}$ on ${{\mathcal C}}$ was defined in [@BR]. This action permutes the $t$-structures $\tau_A$. More precisely, to each pair of alcoves $A,\, A'\in Alc$ one can assign an element $b_{A,A'}\in B_{aff}$; it is then shown in [@BM] that $b_{A,A'}$ sends $\tau_A$ to $\tau_{A'}$. To define $b_{A,A'}$ notice that an element in $B_{aff}$ is determined by a homotopy class of a path connecting two alcoves in ${{\mathfrak h}}^*_{reg}$. The element $b_{A,A'}$ corresponds to a path $\phi: [0,1]\to {{\mathfrak h}}^*_{reg}$ such that $\phi(0)\in A$, $\phi(1)\in A'$ and $\phi(t)\in {{\mathfrak h}}^*_{{\mathbb R}}+ i ({{\mathfrak h}}^*_{{\mathbb R}})^+ $ for $t\in (0,1)$; here $({{\mathfrak h}}^*_{{\mathbb R}})^+\subset {{\mathfrak h}}^*_{{\mathbb R}}$ is the dominant Weyl chamber. This requirement characterizes the homotopy class of $\phi$ uniquely.
For future reference we fix a universal covering ${\widetilde}{{{\mathfrak h}}^*_{reg} }$. We also fix a continuous lifting of each alcove $A\in Alc$ to a subset ${\widetilde}{A}$ in ${\widetilde}{{{\mathfrak h}}^*_{reg} }$, so that for each two alcoves $A$, $A'$ a path representing $b_{A,A'}$ lifts to a continuous path connecting ${\widetilde}{A}$ to ${\widetilde}{A'}$.
Embedding ${{\mathfrak h}}^*\to K^0({{\mathcal C}})^*$ and the “quasi-exponential” map
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We identify $H^*(G/B,{{\mathbb C}})$ with $K^0(Coh(G/B))\otimes {{\mathbb C}}$ by means of the Chern character map. Notice that the class of the line bundle ${{\mathcal O}}({\lambda})$ attached to ${\lambda}\in \Lambda$ corresponds to $\exp({\lambda})\in H^*(G/B)$ where ${\lambda}\in {\Lambda}$ is considered as an element in ${{\mathfrak h}}^*=H^2(G/B)$; it is a nilpotent element in the commutative algebra $H^*(G/B)$, so its exponent is well defined.
We have a bilinear pairing $K^0(G/B)\times K^0({{\mathcal C}})\to {{\mathbb Z}}$ given by $([{{\mathcal F}}], [{{\mathcal G}}])= \chi(pr^*({{\mathcal F}})\otimes {{\mathcal G}})$. Here $\chi$ stands for Euler characteristic and $pr$ for the projection $T^*(G/B)\to G/B$. This gives a map $H^*(G/B) \to (K^0({{\mathcal C}})\otimes {{\mathbb C}})^*$. We will omit complexification from notation where it is not likely to lead to a confusion, and identify an element in $H^*(G/B)$ with its image in $K^0({{\mathcal C}})^*$.
We extend the map ${\Lambda}\to H^*(G/B)\cong K^0(Coh(G/B))\otimes {{\mathbb C}}$, ${\lambda}\mapsto \exp({\lambda})$ to ${{\mathfrak h}}^*$ as follows. Define the “quasi-exponential” map $E:{{\mathfrak h}}^*={{\mathfrak h}}^*_{{\mathbb R}}\times ({\sqrt{-1}}{{\mathfrak h}}^*_{{\mathbb R}})\to H^*(G/B)$ by: $$E:x+{\sqrt{-1}}y\mapsto \exp(x)(1+{\sqrt{-1}}\exp(y)).$$
In fact, the map $x+{\sqrt{-1}}y\mapsto (x,x+y)$ is a $W_{aff}$ equivariant isomorphism between ${{\mathfrak h}}^*$ and $({{\mathfrak h}}^*_{{\mathbb R}})^2$, where $W_{aff}$ acts on $({{\mathfrak h}}^*_{{\mathbb R}})^2$ diagonally. Written as a map from $({{\mathfrak h}}^*_{{\mathbb R}})^2$, the map $E$ takes the form $({\lambda},\mu)\mapsto \exp({\lambda}) +{\sqrt{-1}}(\exp(\mu))$.
A variation of the argument below also works for the map $E(z)=\exp(z)$ (with a less explicit and not necessarily open, though still connected neighborhood $V$ of $({{\mathfrak h}}^*_{{\mathbb R}})^{ar}$). The proof of the statement involving the above quasi-exponential map is a bit shorter, so we opted for presenting that version of the result.
The map $E$ is compatible with the $W_{aff}$ action where the action on the source is the standard affine linear action on ${{\mathfrak h}}^*$, and the one on the target is induced by the $B_{aff}$ action on $D^b(Coh(T^*({{\mathcal B}}))$ from [@BM].
[*Proof.*]{} Translations act on the target by twisting with a line bundle and on the source by shifting the real part, thus it is easy to deduce that the map is compatible with the action of the lattice of translations. Compatibility with the action of the finite Weyl group $W$ follows from [@BM Theorem 1.3.2].
The main result
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Define the “almost regular” part $({{\mathfrak h}}^*_{{\mathbb R}})^{ar}$ of the real Cartan as the set of points in ${{\mathfrak h}}^*_{{\mathbb R}}$ whose stabilizer in $W_{aff}$ has at most two elements. For ${\lambda},\mu \in {{\mathfrak h}}^*_{{\mathbb R}}$ we will write ${\lambda}\preceq \mu$ if ${\lambda}$ lies in the closure of the face which contains $\mu$. Here by a face we mean a stratum of the stratification of ${{\mathfrak h}}^*_{{\mathbb R}}$ cut out by the coroot hyperplanes (thus alcoves are faces of maximal dimension). Define a neighborhood $V$ of $({{\mathfrak h}}^*_{{\mathbb R}})^{ar}$ in ${{\mathfrak h}}^*=({{\mathfrak h}}^*_{{\mathbb R}})^2$ by:
$$V=\{({\lambda}, \mu)\in ({{\mathfrak h}}^*_{{\mathbb R}})^{ar}\times {{\mathfrak h}}^*_{{\mathbb R}}\ |\ {\lambda}\preceq \mu \bigvee {\lambda}\in ({{\mathfrak h}}^*_{{\mathbb R}})^{reg} \}.$$ Thus $V$ is an open neighborhood of $({{\mathfrak h}}^*_{{\mathbb R}})^{ar}$ in ${{\mathfrak h}}^*$. Let $V^{reg}=V\cap {{\mathfrak h}}^*_{reg}$; we have:
$V^{reg}=\{({\lambda}, \mu)\in ({{\mathfrak h}}^*_{{\mathbb R}})^{ar}\times {{\mathfrak h}}^*_{{\mathbb R}}\ |({\lambda}\in ({{\mathfrak h}}^*_{{\mathbb R}})^{reg}) \bigvee ({\lambda}\in \bar{A}, \mu \in A$ for some $A\in Alc) \}$.
Let ${\widetilde}{V^{reg}}$ be the preimage of $V^{reg}$ in ${\widetilde}{{{\mathfrak h}}^*_{reg}}$.
There exists a unique map $\iota:{\widetilde}{V^{reg}}\to Stab({{\mathcal C}})$ such that
1\) The composed map $Z\circ \iota$, where $Z$ is the projection $Stab\to K^0({{\mathcal C}})^*$, coincides with the map ${\sqrt{-1}}E\circ \pi$ where $\pi$ is the projection ${\widetilde}{{{\mathfrak h}}^*_{reg}}\to {{\mathfrak h}}^*_{reg}$.
2\) For some (equivalently, for any) $A\in Alc$ and $z\in {\widetilde}{A}$ the underlying $t$-structure of the stability $\iota(z)$ coincides with $\tau_A$.
[*Proof.*]{} Uniqueness of a map $\iota$ satisfying (1) and (2) for some fixed alcove $A$ and $z\in {\widetilde}{A}$ follows from a Theorem of Bridgeland [@Br1] which asserts that the map $Z$ is a local homeomorphism. It remains to show existence of a map $\iota$ which satisfies (1) and (2) for all $A$, $z\in {\widetilde}{A}$. This will be done in section \[sect3\].
If $\dim (X)=2$, i.e. $e$ is sub-regular, $X$ is well known to be the minimal resolution of a Kleinian singularity. In this case a component of the space $Stab({{\mathcal C}})$ was described in [@Br]. It is easy to see that our submanifold is contained in the one described in [*loc. cit.*]{}
It is easy to show that $\pi_1(V^{reg}/W_{aff})$ is a free group with $rank(G)$ generators. This group surjects to $B_{aff}=\pi_1({{\mathfrak h}}^*_{reg}/W_{aff})$. (The same remains true if $V$ is replaced by any sufficiently small convex $W_{aff}$ invariant neighborhood of $({{\mathfrak h}}^*_{{\mathbb R}})^{ar}$ in ${{\mathfrak h}}^*$). Thus the covering ${\widetilde}{V^{reg}}\to V^{reg}$ is connected but is far from being universal. It would be interesting to construct an explicit subset in $Stab({{\mathcal C}})$ which is a universal covering of a domain whose fundamental group is isomorphic to the affine braid group. The difficulty seems to come from the fact that for an irreducible object $L$ in the heart of $\tau_A$ the corresponding functional $d_L$ can vanish on several faces of $A$ (see below).
Real variations of stabilities
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In this subsection we discuss the motivation for the main result and suggest a new definition.
The real dimension of the manifold ${\widetilde}{V^{reg}}$ is twice the second Betti number of $G/B$; in almost all cases (in particular, in all cases when $G$ is simply-laced except for the degenerate case when $e$ is regular and $X$ is a point; see [@list Theorem 1.3] for the list of exceptional cases) this is equal to twice the second Betti number of $X$. It is our understanding that physicists expect a canonical submanifold in the stability space $Stab(D^b(Coh(M)))$ for a Calabi-Yau manifold $M$ of real dimension $2b_2(M)$, which they call stringy moduli space (under mirror duality it should correspond to a covering of the moduli space of deformations of the dual Calabi-Yau manifold). We hope that the submanifold ${\widetilde}{V^{reg}} \subset
Stab({{\mathcal C}})$ is related to the stringy moduli space of $X$ (hence the title of the note).
More precisely, we conjecture that the following structure is relevant, at least in some examples, for understanding some aspects of the structure of Calabi-Yau categories which have been studied in the literature via the concept of Bridgeland stabilities.
Let ${{\mathcal C}}$ be a finite type triangulated category and $V$ a real vector space. Suppose that a discrete collection $\Sigma$ of affine hyperplanes in $V$ is fixed, let $V^0$ denote their complement. For each hyperplane in $\Sigma$ consider the parallel hyperplane passing through zero, let $\Sigma_{lin}$ be the set of those linear hyperplanes. Fix a component $V^+$ of the complement to the union of hyperplanes in $\Sigma_{lin}$. The choice of $V^+$ determines for each $H\in \Sigma$ the choice of the positive half-space $(V\setminus H)^+\subset V\setminus H$, where $(V\setminus H)^+=H+V^+$. By an [*alcove*]{} we mean a connected component of the complement to hyperplanes in $\Sigma$ and we let ${{{\mathrm{Alc}} }}$ denote the set of alcoves. For two alcoves $A$, $A'\in {{{\mathrm{Alc}} }}$ sharing a codimension one face which is contained in a hyperplane $H\in \Sigma$ we will say that $A'$ is [*above*]{} $A$ and $A$ is below $A'$ if $A'\in (V\setminus H)^+$.
\[def1\] A [*real variation of stability conditions*]{} on ${{\mathcal C}}$ parametrized by $V^0$ and directed to $V^+$ is the data $(Z,\tau)$, where $Z$ (the central charge) is a polynomial map $Z:V\to (K^0({{\mathcal C}})\otimes {{\mathbb R}})^*$, and $\tau$ is a map from ${{{\mathrm{Alc}} }}$ to the set of bounded $t$-structures on ${{\mathcal C}}$, subject to the following conditions.
1. If $M$ is a nonzero object in the heart of $\tau(A)$, $A\in {{{\mathrm{Alc}} }}$, then ${\langle}Z(x), [M]{\rangle}>0$ for $x\in A$.
2. Suppose $A$, $A'\in {{{\mathrm{Alc}} }}$ share a codimension one face $H$ and $A'$ is above $A$. Let ${{\mathcal A}}$ be the heart of $\tau(A)$, and let ${{\mathcal A}}_n\subset {{\mathcal A}}$ be the full subcategory in ${{\mathcal A}}$ given by: $M\in {{\mathcal A}}_n$ if the polynomial function on $V$, $x\mapsto {\langle}Z(x), [M] {\rangle}$ has zero of order at least $n$ on $H$. One can check that ${{\mathcal A}}_n$ is a Serre subcategory in ${{\mathcal A}}$, thus ${{\mathcal C}}_n=\{
C\in {{\mathcal C}}\ |\ H^i_{\tau(A)}(C)\in {{\mathcal A}}_n\}$ is a thick subcategory in ${{\mathcal C}}$. We require that
1. The $t$-structure $\tau(A')$ is compatible with the filtration by thick subcategories ${{\mathcal C}}_n$.
2. The functor of shift by $n$ sends the $t$-structure on $gr_n({{\mathcal C}})={{\mathcal C}}_n/{{\mathcal C}}_{n+1}$ induced by $\tau(A)$ to that induced by $\tau(A')$. In other words, $$gr_n({{\mathcal A}}')=gr_n({{\mathcal A}})[n]$$ where ${{\mathcal A}}'$ is the heart of $\tau(A')$, $gr_n={{\mathcal A}}'_n/{{\mathcal A}}'_{n+1}$, ${{\mathcal A}}'_n={{\mathcal A}}'\cap {{\mathcal C}}_n$.
In many cases (including the examples considered in this paper) one has natural equivalences $D^b({{\mathcal A}}')\cong {{\mathcal C}}\cong D^b({{\mathcal A}})$. The resulting equivalence $D^b({{\mathcal A}}')\cong D^b({{\mathcal A}})$ belongs to a class of equivalences which appeared in the work of Chuang – Rouquier and Craven – Rouquier under the name of [*perverse equivalences*]{} see [@CR] and references therein (our setting may be slightly more general, but the generalization is straightforward).
\[sliceex\] Let ${{\mathcal C}}=D^b(Coh_{{{\mathcal B}}_e}(X))$ as above, $V={{\mathfrak h}}_{{\mathbb R}}^*$ and let $\Sigma$ consist of the affine coroot hyperplanes. Let $V^+$ be the positive Weyl chamber. Let $\tau:A\to \tau_A$ be the map described in [@BM 1.8]. The polynomial map $Z:{{\mathfrak h}}^*_{{\mathbb R}}\to K^0({{\mathcal C}})_{{\mathbb R}}^*$ is characterized uniquely by its values at the points of the lattice $\Lambda\subset {{\mathfrak h}}^*$; these values are given by $$\label{Zatla}
{\langle}Z(\lambda), [{{\mathcal F}}]{\rangle}=\chi({{\mathcal F}}\otimes {{\mathcal O}}(\lambda) ) ,$$ where $\chi$ denotes the Euler characteristic and ${{\mathcal O}}(\lambda)$ is the line bundle attached to $\lambda$.
Using Proposition \[prop1\] below one can show that this data provides an example of a real variation of stability conditions. Notice that in this case for every pair of neighboring alcoves as in part 2 of the Definition the filtration ${{\mathcal C}}_n$ is just a two term filtration, i.e. ${{\mathcal C}}_2=\{0\}$. Another special feature of this example is that all the $t$-structures $\tau(A)$ lie in one orbit of the group of automorphisms of ${{\mathcal C}}$ (in fact, of the group $B_{aff}$ acting on ${{\mathcal C}}$).
\[rem\_symplres\] In a forthcoming work we plan to present more examples of real variations of stability conditions. All our examples are of the following sort: ${{\mathcal C}}=D^b(Coh_Z(X))$ where $X$ is a symplectic resolution of singularities and $Z\subset X$ is a closed projective subvariety; $V=N(X)\otimes {{\mathbb R}}$ where $N(X)$ is the group of numerical classes of divisors on $X$; $V^+$ is spanned by the ample cone, and the central charge map $Z$ is given by a formula similar to . The $t$-structures appearing in the definition of a real variation of stability conditions can in this case be constructed using quantization in positive characteristic, cf. e.g. [@BK], [@BFG], [@Kal].
Requirement (1) of the Definition implies that $({\sqrt{-1}}Z,\tau)$ define a map from $V^0$ to the space of Bridgeland stabilities $Stab({{\mathcal C}})$. Since $V^0$ is disconnected, this structure by itself does not provide any relation between the different $t$-structures, thus it is too weak to yield interesting results. Axiom (2) connects the $t$-structures assigned to different connected components of $V^0$; it is based on the same intuition as Bridgeland’s definition (as we understand it): as $x$ travels from $A$ to $A'$ in the complexification $V_{{\mathbb C}}\setminus H_{{\mathbb C}}$ the phase of a stable objects in ${{\mathcal C}}_n\setminus {{\mathcal C}}_{n+1}$ is shifted by $n\pi$, hence the homological shift by $n$ in requirement (2). This heuristics suggests that given a real variation of stability conditions one might expect a map from a connected covering of the complexification $V^0_{{\mathbb C}}=V_{{\mathbb C}}\setminus \bigcup\limits_{H\in \Sigma} H_{{\mathbb C}}$ to $Stab({{\mathcal C}})$. The main Theorem of this note is a partial result in that direction. However, the fact that we get a map from a covering of a proper subset in $V^0_{{\mathbb C}}$ which is not even homotopy equivalent to the whole space, and have to use a somewhat unnatural quasi-exponential map is an indication of technical difficulties in connecting the two definitions. We expect even more serious difficulties in the cases when filtrations $({{\mathcal C}}_n)$ do not reduce to a two step filtration.
Instead of trying to establish a direct relation between the two structures, it may be more fruitful to view them as different implementations of the same intuition of “physical” origin and possibly try to find a common generalization of the two.
Real variation of stabilities and automorphisms of derived categories
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In some examples in the literature (see e.g. [@Br], [@BrK3], [@BrP2]) (a component of) the space $Stab({{\mathcal C}})$ is realized as a covering of a domain in $K^0({{\mathcal C}})_{{\mathbb C}}^*$ where the group of automorphisms of ${{\mathcal C}}$ acts by deck transformations. We suggest the following counterpart of this picture in the framework of real variations of stability conditions.
A real variation of stability conditions is [*symmetric*]{} if the following holds.
1. For any alcoves $A$, $A'$ as in part 2 of Definition \[def1\] there exists an auto-equivalence $m_{A,A'}$ of ${{\mathcal C}}$ preserving the subcategories ${{\mathcal C}}_n\subset {{\mathcal C}}$, so that the induced auto-equivalence of ${{\mathcal C}}_n/{{\mathcal C}}_{n+1}$ is isomorphic to the shift functor $M\mapsto M [2n]$.
2. The auto-equivalences $m_{A,A'}$ can be chosen so that the following holds. Consider the groupoid $P(V^0_{{\mathbb C}})$ whose objects are alcoves and morphisms from $A$ to $A'$ are homotopy classes of paths in $V^0_{{\mathbb C}}$ starting at $A$ and ending at $A'$. Then there exists a functor from $F:P(V^0_{{\mathbb C}})\to Cat$ where $ Cat$ is the category of categories with morphisms being the isomorphism classes of functors,[^1] such that:
1. $F(A)={{\mathcal C}}$ for all $A\in {{{\mathrm{Alc}} }}$.
2. For $A$, $A'\in {{{\mathrm{Alc}} }}$ as above $F$ sends the class of a path going from $A$ to $A'$ around $H$ in the positive direction to the identity functor.
3. For $A$, $A'$ as above $F$ sends the class of a path going from $A'$ to $A$ in the positive direction to $m_{A,A'}$.
It is easy to see that a symmetric real variation of stability leads to a more symmetric collection of data, which does not use the choice of the “positive cone” $V^+$. Namely, for each alcove $A$ there corresponds a triangulated category ${{\mathcal C}}_A$ with a $t$-structure $\tau_A$. To each homotopy class $\varpi$ of a path in $V^0_{{\mathbb C}}$ connecting an alcove $A$ to an alcove $A'$ there corresponds an equivalence $\varphi_\varpi:{{\mathcal C}}_A\to {{\mathcal C}}_{A'}$. Moreover, if $A$ and $A'$ share a codimension one face, and $\varpi$ runs from $A$ to $A'$ in the counterclockwise direction, then $\varphi_\varpi$ is compatible with $t$-structures “up to a shift on the associated graded pieces” as described above.
Having fixed $V^+$ we can introduce the additional data of an equivalence between ${{\mathcal C}}_A$ and a fixed triangulated category ${{\mathcal C}}$, which is compatible with $\varphi_\pi$ as above when $A'$ lies above $A$.
Using the action of $B_{aff}$ it is easy to see that the real variation of stability conditions in Example \[sliceex\] is symmetric.
More generally, we also expect to obtain symmetric real variation of stability conditions in many cases of the type described in Remark \[rem\_symplres\]. In this situation the heart ${{\mathcal A}}_A$ of the $t$-structures $\tau_A$ is a characteristic zero lifting of the category of finite dimensional modules for an algebra $R_{{\mathcal A}}^k$, where $k$ is a field of positive characteristic and $R_A^k$ is the algebra of global sections of a quantization of a symplectic variety over $k$. The parameter of the quantization should belong to a subset determined by the alcove $A$. Then ${{\mathcal C}}_A\cong D^b({{\mathcal A}}_A-mod^{fd})$ by definition, and the equivalence ${{\mathcal C}}_A\cong {{\mathcal C}}=D^b(Coh_Z(X))$ comes from a derived localization theorem in positive characteristic, cf. [@BMR], [@ICM]. Finally, the equivalence $m_{A,A'}$ can be obtained as a composition of the derived localization equivalences and equivalences induced by the isomorphisms $(R_{-A}^k)\cong R_A^k$, which come from an automorphism of $X$ sending the symplectic form $\omega$ to $-\omega$.
Acknowledgements
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We thank Tom Bridgeland for providing the proof of Lemma \[LBr\]. A part of the work on the paper was done while the last two authors enjoyed the hospitality of Institute for Advanced Study at Hebrew University of Jerusalem, they would like to thank that Institution for excellent work conditions. R.B. and I.M. were supported by NSF grants.
Positivity property
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\[prop1\] Let $A\in Alc$ and let $M\ne 0$ be an object in the heart of $\tau_A$.
a\) The function $d_M:x\mapsto \langle E(x), [M] \rangle$ is a polynomial taking positive real values on $x\in A$.
b\) Let $F$ be a codimension one face of $A$. Then either $d_M$ takes positive values on $F$, or $d_M|_F=0$. In the latter case the object $M[\pm 1]$ lies in the heart of $A'$ where $A'$ is the alcove separated from $A$ by $F$ and the $+$ (respectively, $-$) sign should be taken if $A'$ lies above (respectively, below) $A$.
[*Proof.*]{} The construction of $t$-structures $\tau_A$, $A\in Alc$ on ${{\mathcal C}}=D^b(Coh_{{{\mathcal B}}_e}(X))$ is carried out in [@BM] for $e$, $X$ defined over an algebraically closed field $k$ of arbitrary characteristic, except for positive characteristic $p \leq h$, where $h$ is the Coxeter number of $G$. Furthermore, for each irreducible object $L$ in the heart of ${{\mathcal C}}$, for almost all values of $p$ there exists an object $L_k$ in the heart of $\tau_A^k$ with the same class in the Grothendieck group; here we use a standard identification $K^0({{\mathcal B}}_{e_{{\mathbb C}}})=K^0({{\mathcal B}}_{e_k})$ for matching nilpotent elements $e_{{\mathbb C}}$, $e_k$ defined over ${{\mathbb C}}$ and $k$ respectively, see e.g. [@BMR §7].
Fix such a prime $p$, and let ${\lambda}\in {\Lambda}$ be such that $\frac{{\lambda}+\rho}{p}\in A$, where $\rho$ is the sum of fundamental weights. Then it is shown in [@BMR §6] that $p^{\dim {{\mathcal B}}} d_M(\frac{{\lambda}+\rho}{p})= \dim (\Gamma_{\lambda}(M))$, where $\Gamma_{\lambda}$ is a certain exact conservative functor (depending on ${\lambda}$) from the heart of $\tau_A^k$ to the category of finite dimensional vector spaces over $k$ (in fact, to the category of modules over the Lie algebra ${{\mathfrak g}}_k$). Thus for a large prime number $p$, the polynomial $d_M$ takes positive values at points ${\lambda}\in A$ such that $p{\lambda}\in {\Lambda}$. Since the set of such points, for varying $p$, is dense, we see that $d_M({\lambda})\geq 0$ for any ${\lambda}\in A$. It remains to see that the inequality is strict.
Suppose that $d_M({\lambda}_0)=0$ for some ${\lambda}_0\in A$. Consider the lowest nonzero term $P$ in the Taylor expansion of the polynomial $d_M$ at ${\lambda}_0$. This is a homogeneous polynomial on ${{\mathfrak h}}^*_{{\mathbb R}}$ taking non-negative values only. On the other hand, we claim that $d_M$, and hence $P$ is a harmonic polynomial, i.e. it is annihilated by any $W$ invariant differential operator with constant coefficients and zero constant term. This follows from the differential equation for the exponential function and the fact that the map $Sym({{\mathfrak h}}^*)\to H^*(G/B)$ factors through the quotient by the ideal generated by $Sym({{\mathfrak h}}^*)^W_+$. Now, we claim that a harmonic polynomial taking non-negative values only is necessarily zero. Indeed, for a harmonic polynomial $P$ and ${\lambda}\in {{\mathfrak h}}^*$ we have $\sum_w P(w({\lambda}))=0$; if $P(w({\lambda}))\geq 0$, this implies $P\equiv 0$. This proves (a).
The proof of (b) is based on “singular localization” Theorem of [@BMR2]. Namely, let ${\lambda}\in {\Lambda}$ be such that $\frac{{\lambda}+\rho}{p}$ lies on a codimension one face of an alcove $A$. Then again we have $p^{\dim {{\mathcal B}}} d_M(\frac{{\lambda}+\rho}{p})=\dim {\Gamma}_{\lambda}(M)$ where the functor $\Gamma_{\lambda}$ is exact but not necessarily conservative (i.e. it may kill some non-zero objects).
Moreover, the functor $b_{A,A'}$ sending $\tau_A$ to $\tau_{A'}$ satisfies $b_{A,A'}(L)\cong L[\pm 1]$ for any irreducible object killed by $\Gamma_{\lambda}$, where the sign is chosen as in the statement of the Lemma. We have ${\Gamma}_{\lambda}(M)=0$ if and only if ${\Gamma}_\mu(M)=0$ for every $\mu$ with $\frac{\mu +\rho}{p}\in F$. Assuming that $p$ is large enough, we see that if $\Gamma_{\lambda}(M)=0$ for some $\frac{{\lambda}+\rho}{p}\in F$, then the polynomial $d_M$ vanishes at all points of $F$ with sufficiently large prime denominator, hence $d_M|_F\equiv 0$. In this case we see that $M[\pm 1]\cong b_{A,A'}(M)$ is in the heart of $\tau_{A'}$.
Otherwise if $d_M$ takes positive values at all points of $F$ with a large prime denominator, then an argument involving harmonic polynomials as in the proof of part (a) shows it takes positive values at all points of $F$.
Proof of the Theorem. {#sect3}
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We now construct the map as follows. Recall that $A_0$ denotes the fundamental alcove. It is easy to see that the set $$S=\{({\lambda},\mu) \ | \ ({\lambda}\in A_0) \bigvee ({\lambda}\in \bar{A_0}, \, \mu \in A_0) \}$$ is a fundamental domain for the action of $W_{aff}$ on $V^{reg}$; it is the intersection of a contractible fundamental domain for the action of $W_{aff}$ on ${{\mathfrak h}}^*$ with $V$.
Thus a point in ${\widetilde}{V^{reg}}$ can be represented by a pair $(b,x)$ where $x\in S$ and $b$ is a homotopy class of path from $A_0$ to some alcove $A\in Alc$ (the projection to $V^{reg}$ is then given by $(b,x)\mapsto \bar{b}(x)$ where $\bar{b}$ is the element of $W_{aff}$ corresponding to $b$). We define the map $\iota$ by: $$\iota: (b,x)\mapsto (b(\tau_{A_0}), {\sqrt{-1}}E(\bar{b}(x))),$$ where we use the same notation for $b$ and the corresponding element of $B_{aff}$. It remains to check that $\iota$ is continuous. This is done using the next
\[LBr\] Let ${{\mathcal F}}\in {{\mathcal C}}$ be an object which is stable with respect to a stability $\sigma\in {\widetilde}{V^{reg}}$. Then there exists a neighborhood $U$ of $\sigma$ in $Stab({{\mathcal C}})$ such ${{\mathcal F}}$ is stable with respect to any $\sigma'\in U$.
Assuming the Lemma we finish the argument as follows. We have to check continuity at the boundary of the region corresponding to a given $b\in B_{aff}$. Without loss of generality we can assume $b=1$. Let ${\lambda}$ be a point in the boundary of $S$. By Bridgeland openness Theorem [@Br1] there exists a neighborhood of the point $(\tau_{A_0},{\lambda})$ in $Stab({{\mathcal C}})$ mapping isomorphically to a neighborhood of ${\sqrt{-1}}E({\lambda})$ in $K^0({{\mathcal C}})^*$. It suffices to see that for a neighboring alcove $A'=s_{\alpha}(A_0)$ and a point $\tilde z$ in the neighborhood mapping to $z\in s_{\alpha}(S)$, the $t$-structure underlying $\tilde z$ is $\tau_{A'}$.
Let $L$ be an irreducible object. It suffices to show that $b_{A,A'}(L)$ lies in the heart of the $t$-structure of $\tilde z$. The argument is similar to [@Br p. 10]. There are two cases. Either $b_{A,A'}(L)=L[\pm 1]$, then using the Lemma we see that $L$ is stable for $\tilde z$ (if the neighborhood is chosen small enough), then $L[\pm 1]$ is in the heart of the $t$-structure since its phase is in $[0,1)$. Or $b_{A,A'}(L)$ lies in the heart of $\tau_A$, its Harder-Narasimhan filtration has length two, and both stable subquotients remain stable and have phases in $[0,1)$ in stability $\tilde z$.
It remains to prove Lemma \[LBr\]. The proof below was suggested to us by Tom Bridgeland.
It is easy to see that any $\sigma\in \iota({\widetilde}{V^{reg}})$ is locally finite. If ${{\mathcal F}}$ is stable in $\sigma=(P,Z)$ of phase $t$ then using local-finiteness we can find a finite length quasi-abelian category ${{\mathcal B}}= P(t-a,t+a)$ (notations of [@Br1]). The object ${{\mathcal F}}$ has a finite Jordan-Hoelder series in ${{\mathcal B}}$. Thus there are only finitely many elements in $K^0({{\mathcal C}})$ which can be represented by a subobject of ${{\mathcal F}}$ in ${{\mathcal B}}$, since such a class is a sum of classes of some of the simple subquotients. Since ${{\mathcal F}}$ is stable, the class of a subobject ${{\mathcal F}}'\ne {{\mathcal F}}$ has phase strictly less than $t$. Thus there exists $s<t$ such that the class of any such subobject has phase less than $s$. If $U$ is a sufficiently small neighborhood of $\sigma$, then the phase of any subobject of ${{\mathcal F}}$ with respect to $\sigma'\in U$ differs from its phase with respect to $\sigma$ by less than $\frac{t-s}{2}$. Thus the phase of subobject ${{\mathcal F}}'\ne {{\mathcal F}}$ with respect to $\sigma'$ is less than the phase of ${{\mathcal F}}$ with respect to $\sigma'$, which shows that ${{\mathcal F}}$ is stable with respect to $\sigma'$.
[1]{}
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[^1]: We present the weak version of the definition for brevity, one can upgrade it to a definition of a finer structure involving the 2-category of categories, or a version of the infinity category of $DG$-categories.
|
---
abstract: |
We argue that geometrical scaling (GS) proposed originally in the context of Deep Inelastic Scattering (DIS) at HERA works also in pp collisions at the LHC energies and in NA61/SHINE experiment. We show that in DIS GS is working up to relatively large Bjorken $x \sim 0.1$. As a consequence negative pion multiplicity $p_{\rm T}$ distributions at NA61/SHINE energies exhibit GS in mid rapidity region. For $y \ne 0$ clear sign of scaling violations can be seen when one of the colliding partons has Bjorken $x \ge 0.1$. Finally, we argue that in the case of identified particles GS scaling is still present but the scaling variable is a function of transverse mass rather than $p_{\rm T}$.\
\
PACS number(s):[ 13.85.Ni,12.38.Lg]{}
author:
- |
Michal Praszalowicz[^1]\
[M. Smoluchowski Institute of Physics, Jagellonian University,]{}\
[Reymonta 4, 30-059 Krakow, Poland]{}\
nocite: '[@Praszalowicz:2011tc; @Praszalowicz:2011rm; @Praszalowicz:2012zh; @Praszalowicz:2013uu]'
title: |
Emergence and violation of geometrical scaling\
in pp collisions [^2]
---
Introduction {#intro}
============
This talk based on Refs. [@McLerran:2010ex]–[@Praszalowicz:2013fsa] (where also an extensive list of references can be found) follows closely an earlier report of Ref. [@Praszalowicz:2013swa]. We shall discuss the scaling law, called geometrical scaling (GS), which has been introduced in the context of DIS [@Stasto:2000er]. It has been also shown that GS is exhibited by the $p_{\text{T}}$ spectra at the LHC [@McLerran:2010ex]–[@Praszalowicz:2011rm] and that an onset of GS can be seen in heavy ion collisions at RHIC energies [@Praszalowicz:2011rm]. At low Bjorken $x<x_{\mathrm{max}}$ gluonic cloud in the proton is characterized by an intermediate energy scale $Q_{\text{s}}(x)$, called saturation scale [@sat1; @GolecBiernat:1998js]. $Q_{\text{s}}(x)$ is defined as the border line between dense and dilute gluonic systems (for review see *e.g.* Refs. [@Mueller:2001fv; @McLerran:2010ub]). In the present paper we study the consequences of the very existence of $Q_{\text{s}}(x)$; the details of saturation phenomenon are here not of primary importance.
Here we shall focus of four different pieces of data which exhibit both emergence and violation of geometrical scaling. In Sect. \[method\] we briefly describe the method used to assess the existence of GS. Secondly, in Sect. \[DIS\] we describe our recent analysis [@Praszalowicz:2012zh] of combined HERA data [@HERAdata] where it has been shown that GS in DIS works surprisingly well up to relatively large $x_{\text{max}}\sim0.1$ (see also [@Caola:2010cy]). Next, in Sect. \[ppLHC\], on the example of the CMS $p_{\rm T}$ spectra in central rapidity [@Khachatryan:2010xs], we show that GS is also present in hadronic collisions. For particles produced at non-zero rapidities, one (larger) Bjorken $x=x_{1}$ may be outside of the domain of GS, *i.e.* $x_{1}>x_{\text{max}}$, and violation of GS should appear. In Sect. \[ppNA61\] we present analysis of the pp data from NA61/SHINE experiment at CERN [@NA61] and show that GS is indeed violated once rapidity is increased. Finally in Sect. \[GSids\] we analyze identified particles spectra where the particle mass provides another energy scale which may lead to the violation of GS, or at least to some sort of its modification [@Praszalowicz:2013fsa]. We conclude in Sect. \[concl\].
Analyzing data with method of ratios {#method}
====================================
Geometrical scaling hypothesis means that some observable $\sigma$ depending in principle on two independent kinematical variables, like $x$ and $Q^2$, depends in fact only on a given combination of them, denoted in the following as $\tau$: $$\sigma(x,Q^{2})=F(\tau)/{Q_{0}^{2}}. \label{GSdef}$$ Here function $F$ in Eq. (\[GSdef\]) is a dimensionless universal function of scaling variable $\tau$: $$\tau=Q^{2}/Q_{\text{s}}^{2}(x).\label{taudef}$$ and $$Q_{\text{s}}^{2}(x)=Q_{0}^{2}\left( {x}/{x_{0}}\right) ^{-\lambda}
\label{Qsat}$$ is the saturation scale. Here $Q_{0}$ and $x_{0}$ are free parameters which, however, are not of importance in the present analysis, and exponent $\lambda$ is a dynamical quantity of the order of $\lambda\sim0.3$. Throughout this paper we shall test the hypothesis whether different pieces of data can be described by formula (\[GSdef\]) with [*constant*]{} $\lambda$, and what is the kinematical range where GS is working satisfactorily.
As a consequence of Eq. (\[GSdef\]) observables $\sigma(x_{i},Q^{2})$ for different $x_{i}$’s should fall on a universal curve, if evaluated in terms of scaling variable $\tau$. This means that ratios $$R_{x_{i},x_{\text{ref}}}(\lambda;\tau_{k})=\frac{\sigma(x_{i},\tau(x_{i},Q_{k}^{2};\lambda))}{\sigma(x_{\text{ref}},\tau(x_{\text{ref}},Q_{k,\text{ref}}^{2};\lambda))} \label{Rxdef}$$ should be equal to unity independently of $\tau$. Here for some $x_{\rm ref}$ we pick up all $x_i<x_{\rm ref}$ which have at least two overlapping points in $Q^2$.
For $\lambda\neq0$ points of the same $Q^{2}$ but different $x$’s correspond in general to different $\tau$’s. Therefore one has to interpolate $\sigma(x_{\text{ref}},\tau(x_{\text{ref}},Q^{2};\lambda))$ to $Q_{k,\text{ref}}^{2}$ such that $\tau(x_{\text{ref}},Q_{k,\text{ref}}^{2};\lambda)=\tau_{k}$. This procedure is described in detail in Refs. [@Praszalowicz:2012zh].
By adjusting $\lambda$ one can make $R_{x_{i},x_{\text{ref}}}(\lambda;\tau
_{k}) \rightarrow 1$ for all $\tau_{k}$ in a given interval. In order to find an optimal value $\lambda_{\rm min}$ which minimizes deviations of ratios (\[Rxdef\]) from unity we form the chi-square measure$$\chi_{x_{i},x_{\text{ref}}}^{2}(\lambda)=\frac{1}{N_{x_{i},x_{\text{ref}}}-1}{\displaystyle\sum\limits_{k\in x_{i}}}\frac{\left( R_{x_{i},x_{\text{ref}}}(\lambda;\tau_{k})-1\right) ^{2}}{\Delta R_{x_{i},x_{\text{ref}}}(\lambda;\tau_{k})^{2}} \label{chix1}$$ where the sum over $k$ extends over all points of given $x_{i}$ that have overlap with $x_{\text{ref}}$, and ${N_{x_{i},x_{\text{ref}}}}$ is a number of such points.
Geometrical scaling in DIS at HERA {#DIS}
==================================
In the case of DIS the relevant scaling observable is $\gamma^{\ast}p$ cross section and variable $x$ is simply Bjorken $x$. In Fig. \[xlamlog\] we present 3-d plot of $\lambda_{\min}({x,x_{\rm ref}})$ which has been found by minimizing (\[chix1\]).
![Three dimensional plot of $\lambda_{\mathrm{min}}(x,x_{\mathrm{ref}})$ obtained by minimization of Eq. (\[chix1\]).[]{data-label="xlamlog"}](lamX3D.pdf){width="8cm"}
Qualitatively, GS is given by the independence of $\lambda_{\text{min}}$ on Bjorken $x$ and by the requirement that the respective value of $\chi_{x,x_{\text{ref}}}^{2}(\lambda_{\text{min}})$ is small (for more detailed discussion see Refs. [@Praszalowicz:2012zh]). One can see from Fig. \[xlamlog\] that the stability corner of $\lambda_{\text{min}}$ extends up to $x_{\text{ref}}\lesssim0.1$, which is well above the original expectations. In Ref. [@Praszalowicz:2012zh] we have shown that: $$\lambda = 0.32 - 0.34 \,\,\,\,\, {\rm for} \,\,\,\,\, x \le 0.08.$$
Geometrical scaling of central rapidity $p_{\rm T}$ spectra at the LHC {#ppLHC}
======================================================================
![Ratios of CMS $p_{\mathrm{T}}$ spectra [Khachatryan:2010xs]{} at 7 TeV to 0.9 (blue circles) and 2.36 TeV (red triangles) plotted as functions of $p_{\mathrm{T}} $ (left) and scaling variable $\protect\sqrt{\protect\tau}$ (right) for $\protect\lambda=0.27$. []{data-label="ratios1"}](ratiopt-pp_v2.pdf "fig:") ![Ratios of CMS $p_{\mathrm{T}}$ spectra [Khachatryan:2010xs]{} at 7 TeV to 0.9 (blue circles) and 2.36 TeV (red triangles) plotted as functions of $p_{\mathrm{T}} $ (left) and scaling variable $\protect\sqrt{\protect\tau}$ (right) for $\protect\lambda=0.27$. []{data-label="ratios1"}](ratios27sqrtau_v2.pdf "fig:")
In hadronic collisions at c.m. energy $W=\sqrt{s}$ particles are produced in the scattering process of two patrons (mainly gluons) carrying Bjorken $x$’s $$x_{1,2}=e^{\pm y}\,p_{\text{T}}/W.\label{x12}$$ For central rapidities $x=x_1 \sim x_2$. In this case charged particles multiplicity spectra exhibit GS [@McLerran:2010ex] $$\left. \frac{dN}{dy d^{2}p_{\text{T}}}\right\vert _{y\simeq0}=\frac
{1}{Q_{0}^{2}}F(\tau)\label{GSinpp}$$ where $F$ is a universal dimensionless function of the scaling variable (\[taudef\]). Therefore the method of ratios can be applied to the multiplicity distributions at different energies ($W_i$ taking over the role of $x_i$ in Eq. (\[Rxdef\]))[^3]. For $W_{\rm ref}$ we take the highest LHC energy of 7 TeV. Hence one can form two ratios $R_{W_{\rm ref},W_i}$ with $W_1 =2.36$ and $W_2 = 0.9$ TeV. These ratios are plotted in Fig. \[ratios1\] for the CMS single non-diffractive spectra for $\lambda = 0$ and for $\lambda = 0.27$, which minimizes (\[chix1\]) in this case. We see that original ratios plotted in terms of $p_{\text{T}}$ range from 1.5 to 7, whereas plotted in terms of $\sqrt{\tau}$ they are well concentrated around unity. The optimal exponent $\lambda$ is, however, smaller than in the case of DIS. Why this so, remains to be understood.
Violation of geometrical scaling in forward rapidity region {#ppNA61}
===========================================================
For $y >0$ two Bjorken $x$’s can be quite different: $x_{1}>x_{2}$. Therefore by increasing $y$ one can eventually reach $x_{1}>x_{\mathrm{max}}$ and GS violation should be seen. For that purpose we shall use pp data from NA61/SHINE experiment at CERN [@NA61] at different rapidities $y=0.1-3.5$ and at five scattering energies $W_{1,\ldots,5}=17.28,\;12.36,\;8.77,\;7.75$, and $6.28$ GeV.
![Ratios $R_{1k}$ as functions of $\sqrt{\tau}$ for the lowest rapidity $y=0.1$: a) for $\lambda=0$ when $\sqrt{\tau}=p_{\mathrm{T}}$ and b) for $\lambda=0.27$ which corresponds to GS.[]{data-label="y01"}](eQCD3a.pdf "fig:"){width="6.0cm"} ![Ratios $R_{1k}$ as functions of $\sqrt{\tau}$ for the lowest rapidity $y=0.1$: a) for $\lambda=0$ when $\sqrt{\tau}=p_{\mathrm{T}}$ and b) for $\lambda=0.27$ which corresponds to GS.[]{data-label="y01"}](eQCD3b.pdf "fig:"){width="6.0cm"}
![Ratios $R_{1k}$ as functions of $\sqrt{\tau}$ for $\lambda=0.27$ and for different rapidities a) $y=0.7$ and b) $y=1.3$. With increase of rapidity, gradual closure of the GS window can be seen.[]{data-label="ys"}](eQCD4a.pdf "fig:"){width="6.0cm"} ![Ratios $R_{1k}$ as functions of $\sqrt{\tau}$ for $\lambda=0.27$ and for different rapidities a) $y=0.7$ and b) $y=1.3$. With increase of rapidity, gradual closure of the GS window can be seen.[]{data-label="ys"}](eQCD4b.pdf "fig:"){width="6.0cm"}
In Fig. \[y01\] we plot ratios $R_{1i}=R_{W_1,W_i}$ (\[Rxdef\]) for $\pi^-$ spectra in central rapidity for $\lambda=0$ and 0.27. For $y=0.1$ the GS region extends down to the smallest energy because $x_{\rm max}$ is as large as 0.08. However, the quality of GS is the worst for the lowest energy $W_5$. By increasing $y$ some points fall outside the GS window because $x_1 \ge x_{\rm max}$, and finally for $y\ge1.7$ no GS is present in NA61/SHINE data. This is illustrated nicely in Fig. \[ys\].
Geometrical scaling for identified particles {#GSids}
============================================
In Ref. [@Praszalowicz:2013fsa] we have proposed that in the case of identified particles another scaling variable should be used in which $p_{\mathrm{T}}$ is replaced by $\tilde{m}_{\mathrm{T}} =m_{\rm T} - m = \sqrt{m^2_{\rm T} +p^2_{\rm T}}-m $ ($\tilde
{m}_{\mathrm{T}}$ – scaling), *i.e.* $$\tau_{\tilde{m}_{\rm T}} =\frac{\tilde{m}_{\text{T}}^{2}}{Q_{0}^{2}}\left(
\frac{\tilde{m}_{\text{T}}}{x_0 W}\right) ^{\lambda}.\label{taumtdef}$$ This choice is purely phenomenological for the following reasons. Firstly, the gluon cloud is in principle not sensitive to the mass of the particle it finally is fragmenting to, so in principle one should take $p_{\mathrm{T}}$ as an argument of the saturation scale. In this case the proper scaling variable would be $$\tau_{\tilde{m}_{\rm T} p_{\rm T}} =\frac{\tilde{m}_{\text{T}}^{2}}{Q_{0}^{2}}\left(
\frac{p_{\text{T}}}{x_0 W}\right) ^{\lambda}.\label{taumtpdef}$$ However this choice ($\tilde{m}_{\mathrm{T}}$$p_{\mathrm{T}}$ – scaling) does not really differ numerically from the one given by Eq. (\[taumtdef\]).
To this end let us see how scaling properties of GS are affected by going from scaling variable $\tau_{p_{\rm T}}=\tau$ (\[taudef\]) to $\tau_{\tilde
{m}_{\mathrm{T}}}$ (\[taumtdef\]) and what would be the difference in scaling properties if we had chosen $p_{\mathrm{T}}$ as an argument in the saturation scale leading to scaling variable $\tau_{\tilde{m}_{\mathrm{T}}p_{\mathrm{T}}}$ (\[taumtpdef\]). This is illustrated in Fig. \[ratiosmandpT\] where we show analysis [@Praszalowicz:2013fsa] of recent ALICE data on identified particles [@ALICE]. In Fig. \[ratiosmandpT\].a-c full symbols refer to the $p_{\mathrm{T}}$ – scaling (\[taudef\]) and open symbols to $\tilde{m}_{\mathrm{T}}$ – scaling or $\tilde{m}_{\mathrm{T}}$$p_{\mathrm{T}}$ – scaling. One can see very small difference between open symbols indicating that scaling variables $\tau_{\tilde{m}_{\mathrm{T}}}$ (\[taumtdef\]) and $\tau_{\tilde
{m}_{\mathrm{T}}p_{\mathrm{T}}}$ (\[taumtpdef\]) exhibit GS of the same quality. On the contrary $p_{\mathrm{T}}$ – scaling in variable $\tau_{p_{\rm T}}$ (\[taudef\]) is visibly worse.
Finally in Fig. \[ratiosmandpT\].d, on the example of protons, we compare $\tilde{m}_{\mathrm{T}}$ – scaling (open symbols) and ${m}_{\mathrm{T}}$ – scaling (full symbols) in variable $$\tau_{m} =\frac{{m}_{\text{T}}^{2}}{Q_{0}^{2}}\left( \frac{m_{\text{T}}}{x_0 W}\right) ^{\lambda}.\label{taumdef}$$ for $\lambda=0.27$. One can see that no GS has been achieved in the latter case. Qualitatively the same behavior can be observed for other values of $\lambda$.
![Panels a) – c): comparison of geometrical scaling in three different variables: $\tau_{p_{\rm T}}$, $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{\tilde{m}_{\rm T} p_{\rm T}}$ for $\lambda=0.27$. Full symbols correspond to ratios $R_{W_{1}/W_{2}}$ plotted in terms of the scaling variable $\tau_{p_{\rm T}}$, open symbols to $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{\tilde{m}_{\rm T}p_{\rm T}}$, note negligible differences between the latter two forms of scaling variable. Panel a) corresponds to pions, b) to kaons and c) to protons. In panel d) we show comparison of geometrical scaling for protons in scaling variables $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{m_{\rm T}}$, no GS can be achieved in the latter case.[]{data-label="ratiosmandpT"}](pi_RmT_pT_027.pdf "fig:") ![Panels a) – c): comparison of geometrical scaling in three different variables: $\tau_{p_{\rm T}}$, $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{\tilde{m}_{\rm T} p_{\rm T}}$ for $\lambda=0.27$. Full symbols correspond to ratios $R_{W_{1}/W_{2}}$ plotted in terms of the scaling variable $\tau_{p_{\rm T}}$, open symbols to $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{\tilde{m}_{\rm T}p_{\rm T}}$, note negligible differences between the latter two forms of scaling variable. Panel a) corresponds to pions, b) to kaons and c) to protons. In panel d) we show comparison of geometrical scaling for protons in scaling variables $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{m_{\rm T}}$, no GS can be achieved in the latter case.[]{data-label="ratiosmandpT"}](Ka_RmT_pT_027.pdf "fig:")\
![Panels a) – c): comparison of geometrical scaling in three different variables: $\tau_{p_{\rm T}}$, $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{\tilde{m}_{\rm T} p_{\rm T}}$ for $\lambda=0.27$. Full symbols correspond to ratios $R_{W_{1}/W_{2}}$ plotted in terms of the scaling variable $\tau_{p_{\rm T}}$, open symbols to $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{\tilde{m}_{\rm T}p_{\rm T}}$, note negligible differences between the latter two forms of scaling variable. Panel a) corresponds to pions, b) to kaons and c) to protons. In panel d) we show comparison of geometrical scaling for protons in scaling variables $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{m_{\rm T}}$, no GS can be achieved in the latter case.[]{data-label="ratiosmandpT"}](pr_RmT_pT_027.pdf "fig:") ![Panels a) – c): comparison of geometrical scaling in three different variables: $\tau_{p_{\rm T}}$, $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{\tilde{m}_{\rm T} p_{\rm T}}$ for $\lambda=0.27$. Full symbols correspond to ratios $R_{W_{1}/W_{2}}$ plotted in terms of the scaling variable $\tau_{p_{\rm T}}$, open symbols to $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{\tilde{m}_{\rm T}p_{\rm T}}$, note negligible differences between the latter two forms of scaling variable. Panel a) corresponds to pions, b) to kaons and c) to protons. In panel d) we show comparison of geometrical scaling for protons in scaling variables $\tau_{\tilde{m}_{\rm T}}$ and $\tau_{m_{\rm T}}$, no GS can be achieved in the latter case.[]{data-label="ratiosmandpT"}](pr_RmT0_pT_027.pdf "fig:")
Conclusions {#concl}
===========
In Ref. [@Praszalowicz:2012zh] we have shown that GS in DIS works well up to rather large Bjorken $x$’s with exponent $\lambda = 0.32 - 0.34$. In pp collisions at the LHC energies in central rapidity GS is seen in the charged particle multiplicity spectra, however, $\lambda = 0.27$ in this case [@McLerran:2010ex]. By changing rapidity one can force one of the Bjorken $x$’s of colliding patrons to exceed $x_{\rm max}$ and GS violation is expected. Such behavior is indeed observed in the NA61/SHINE pp data [@Praszalowicz:2013uu]. Finally we have shown that for identified particles scaling variable $\tau$ of Eq. (\[taudef\]) should be replaced by $\tau_{\tilde{m}_{\rm T}}$ defined in Eq. (\[taumtdef\]) and the scaling exponent $\lambda \approx 0.3$ [@Praszalowicz:2013fsa].
Acknowledgemens {#acknowledgemens .unnumbered}
===============
Many thanks are due to the organizers of this successful series of conferences. This work was supported by the Polish NCN grant 2011/01/B/ST2/00492.
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[^1]: [email protected]
[^2]: Presented at the Low $x$ Workshop, May 30 - June 4 2013, Rehovot and Eilat, Israel
[^3]: For pp collisions we define ratios $R_{W_{\rm ref},W_i}$ as an inverse of (\[Rxdef\])
|
---
abstract: |
Here we prove that the minimal free resolution of a general space curve of large degree (e.g. a general space curve of degree $d$ and genus $g$ with $d \geq g+3$, except for finitely many pairs $(d,g)$) is the expected one. A similar result holds even for general curves with special hyperplane section and, roughly, $d \geq g/2$. The proof uses the so-called methode d’Horace.
[**[**Résumé**]{}**]{}
On montre ici que la résolution minimale de l’ideal d’une courbe générique de grand degree dans ${\mbox{\bf P}^{3}}$ est la résolution minimal attendue.
author:
- Edoardo Ballico
title: 'On the minimal free resolution of non-special curves in ${\mbox{\bf P}^{3}}$ '
---
\[subsection\][Theorem]{} \[subsection\][Remark]{} \[subsection\][Lemma]{} \[subsection\][Example]{} \[subsection\] \[subsection\][Definition]{}
Introduction {#0}
============
Many important properties of a closed subscheme $X$ of $\mbox{\bf P}^n$ are detected by the numerical data of its minimal free resolution, i.e. by its Betti numbers. Assume $n=3$ and that $X$ is a curve. We are mainly interested in the case $h^{1}(X,O_{X}(1)) = 0$. But we study also some curves, $X$, with $h^1(X,O_{X}(1)) \neq 0$ and $h^{1}(X,O_{X}(1)) = 0$ (see Theorem \[0.3\]). Set $\Pi:= {\mbox{\bf P}^{3}}$. $X$ is said to have maximal rank if for every integer $t$ the restriction map $r_{X,t}: H^{0}(\Pi,O_{\Pi}(t))
\rightarrow H^{0}(X,O_{X}(t))$ has maximal rank. It is known that for all integers $d$, $g$ with $d \geq g+3$ a general non-special smooth space curve of genus $g$ and degree $d$ has maximal rank (see [@BE1]). For such curves there are strong restrictions on the possible Betti numbers: roughly speaking, by Castelnuovo - Mumford’s lemma at each step of the minimal free resolution of $X$ at most two different Betti numbers may be non-zero. We will say that $X$ has the expected minimal free resolution if for every integer $t$ the line bundle $O_{\Pi}(t)$ is not a direct summand of two syzygy components of the minimal free resolution of X. This is equivalent to the fact that at most at one step of the minimal free resolution of $X$ two Betti numbers are non-zero (see e.g. the introduction of [@HS] or [@BG] in the case of points instead of curves, or here section \[1\]). By Koszul cohomology this condition is equivalent to the fact that for a “critical” integer $m$ the restriction maps $r_{X,\Omega (m+1)} : H^{0}(\Pi,\Omega (m+1)) \rightarrow
H^{0}(X,(\Omega (m+1)|X)$ and $r_{X,T \Pi (m-2)} : H^0(\Pi,T \Pi (m-2)) \rightarrow
H^{0}(X,(T \Pi (m-2)|X)$ have maximal rank, where $\Omega$ (resp. $T \Pi$) will denote the cotangent (resp. tangent) bundle of $\Pi$ (see section \[1\] or [@G] or the introductions of [@HS] or [@BG]); here $m$ is the only integer $ \geq 2$ with $h^{0}(\Pi,I_{X}(m-1)) = h^{1}(\Pi,I_{X}(m)) = 0$. For instance, the fact that the homogeneous ideal of the maximal rank non-special curve $X$ has the minimal possible number of generators in each degree is equivalent to the fact that the linear map $r_{X,\Omega (m+1)}$ has maximal rank (see e.g. [@B] or the introduction of [@I]). If $X$ has the expected minimal free resolution, then its Betti numbers depend only on $d$ and $g$ and may be given explicitely. In [@BE2] for all pairs $(d,g)$ of integers with $d \geq 3$ and $d-3 \leq g \leq 2d-9$ it was defined an irreducible component, $H(d,g)$, of the Hilbert scheme Hilb($\Pi$) of ${\mbox{\bf P}^{3}}$ with very nice properties (see section \[1\] for more details). Here we will prove the following results.
\[0.1\] There exists an integer $g_0$ such that for every integer $g \geq g_0$ and every integer $d \geq g+3$ a general non-special degree $d$ embedding in ${\mbox{\bf P}^{3}}$ of a general curve of genus $g$ has the expected minimal free resolution.
\[0.2\] For every integer $g \geq 0$ there is an integer $D(g)$ such that for every integer $d \geq D(g)$ a general degree $d$ embedding in ${\mbox{\bf P}^{3}}$ of a general curve of genus $g$ has the expected minimal free resolution.
\[0.3\] There exists a function $g: N \rightarrow R_{+}$ with $lim_{g \rightarrow + \infty}
\gamma (g) = 1/2$ such that for every $d \geq g \gamma (g)$ a general element of $H(d,g)$ has the expected minimal free resolution.
In the statement of Theorem \[0.1\] we may take $g_{0} = 899$ (see Remark \[5.1\]). In the statement of Theorem \[0.2\] we may take $D(g) = g+3$ if $g \geq 899$, $D(g) = 641927953$ if $0 < g \leq 898$ and $D(0) = 932$ (see Remark \[8.139\]). As in [@I], [@HS], [@B], [@BG] and in previous papers on the postulation of space curves ([@BE1] and [@BE2]) we will use the so - called Horace method introduced in [@H1]. To apply Horace method we need to control the cohomology of the restrictions $\Omega|Q$ and $T \Pi |Q$ of $\Omega$ and $T \Pi$ to a smooth quadric surface $Q$. This is done in section \[3\]. The main inductive assertions for Theorem \[0.1\] are stated and proved (modulo several initial cases) in section \[4\]. Section \[5\] contains the reduction of the proof of Theorem \[0.1\] (and, essentially, Theorems \[0.2\] and \[0.3\], too) to the proof of the main inductive assertions. Section \[6\] contains the proof of the huge number of initial cases of the inductive statements needed for \[0.1\]. Section \[7\] contains the proof of \[0.3\]. The last section contains the proof of \[0.2\]. As for \[0.1\], our proof \[0.2\] force us to check a huge number of initial cases. We work over an algebraically closed field [**K**]{}. To use freely the references [@BE1] and [@BE2] we assume char([**K**]{}) = 0. The author was partially supported by MURST and GNSAGA of CNR (Italy)
Preliminaries {#1}
=============
Set $\Pi:= {\mbox{\bf P}^{3}}$, $O:= O_{\Pi}$ and $\Omega := \Omega_{\Pi}^{1}$. For any sheaf $F$ on $\Pi$, set $h^{i}(F):= h^{i}(\Pi,F)$ and $H^{i}(F):= H^{i}(\Pi,F)$. By the Euler’s exact sequence of $T \Pi$ and its dual exact sequence we obtain $h^{0}(\Omega (t)) = (t+2)(t+1)(t-1)/2$ for $t \geq 2$, $h^{0}(\Omega (t)) = 0$ for $t \leq 1$, $h^{i}(\Omega (t)) = 0$ for $1 \leq
i \leq 2$, $t \in Z$ and $(i,t) \neq (1,0)$, $h^{1}(\Omega) = 1$, $h^{0}(T \Pi (t)) = (t+3)(t+2)(t+5)/2$ for $t \geq -1$, $h^{0}(T \Pi (t)) = 0$ for $t \leq -2$, $h^{i}(T \Omega (t)) = 0$ for $1 \leq i \leq 2$, $t \in Z$, $(i,t) \neq (2,-4)$.
\[1.1\] Let $Z \subset \Pi$ be a subscheme which is the union of a curve $Y$ and a 0-dimensional scheme. Fix an integer $m \geq 2$ and assume $h^{1}(Y,O_{Y}(m-1)) = 0$. By the Euler’s exact sequence of $T \Pi$ and its dual we obtain $h^{1}(Z,\Omega (m+1)|Z) =
h^{1}(Z,(T \Pi (m-2)|Z) = 0$. By Riemann - Roch we have:
$$h^{0}(Z,\Omega(m+1)|Z) = (3m-1)deg(Y) + 3\chi(O_{Z})
\label{e1}$$
$$h^{0}(Z,T \Pi (m-2)|Z) = (3m-2)deg(Y) + 3\chi(O_{Z})
\label{e2}$$
Note that by \[e1\] and \[e2\] the congruence classes modulo 3 of $h^{0}(Z,\Omega (m+1)|Z)$ and $h^{0}(Z,T\Pi(m-2)|Z)$ depend only on $deg(Y)$.
Fix integers $d$, $g$ with either $g \geq 0$ and $d \geq g+3$ or $d \geq 3$ and $d-3 \leq g \leq 2d-9$. There is an irreducible component $H(d,g)$ of the Hilbert scheme Hilb($\Pi$) of $\Pi$ with the following properties ([@BE2]). $H(d,g)$ is generically smooth, $dim(H(d,g)) = 4d$ and a general $D \in H(d,g)$ is a smooth connected curve with $deg(D) = d$, $p_{a}(D) = g$, $h^{1}(D,N_{D}) = 0$. If $d \geq g+3$ then a general $D \in H(d,g)$ is not special. If $d < g+2$, then $H(d,g)$ has the right number of moduli in the sense of [@Se]. Thus if $4d \geq 3g+12$ $H(d,g)$ contains curves with general moduli; furthermore, a general $D \in H(d,g)$ has $h^{1}(D,O_{D}(2)) = 0$. In the range of pairs $(d,g)$ in which we will use the schemes $H(d,g)$ a general $D \in H(d,g)$ has $h^{1}(D,O_{D}(-2)) = 0$ (see [@EH], [@W1], Th. 2.15, and [@W2], Th. 9.4 and 10.5, plus the definition of $H(d,g)$ given in [@BE2], def. \[1.3\]). If $d \geq g+3$, then a general $D \in H(d,g)$ has maximal rank ([@BE1]). If $4d \geq 3g+12$ a general $D \in H(d,g)$ has maximal rank ([@BE2], Th. 1). If $4d < 3g+12$ we know only a weaker statement, i.e. the existence of a function $u: N \rightarrow R$ with $lim_{g \rightarrow + \infty} u(g) = 1/2$ and such that if $d \geq g \cdot u(g)$ a general $D \in H(d,g)$ has maximal rank ([@BE2], Th. 2, and [@W2], Th. 7.1). It was proved in [@BE2] that the family of all schemes $H(d,g)$ for varying $d$ and $g$ has very strong stability properties with respect to the addition of “rational tails” in the sense explained by the following example which follows easily from a particular case of [@BE2], Lemma \[1.5\].
\[1.2\] Fix a general $A \in H(d,g)$ and integers $u$, $v$, $w$ with $0 \leq u \leq 2$, $0 \leq v \leq 3$ and $0 \leq w \leq 5$. Let $C$ be a line (resp. a smooth conic, resp. a rational normal curve) intersecting quasi-transversally $A$ and with $card(A \cap C) = u+1$ (resp. $v+1$, resp. $w+1$). Then $A \cup C \in H(d+1,g+u)$ (resp. $H(d+2,g+v)$, resp. $H(d+3,g+v)$). If $ \leq 1$, $v \leq 2$, $ \leq 3$ and $h^{1}(A,O_{A}(1)) = 0$, then a Mayer - Vietoris exact sequence gives $h_{1}(A \cup C,O_{A \cup C}(1)) = 0$.
For every pair of integers $(d,g)$ such that $H(d,g)$ is defined there is a unique integer $m(d,g)$ (called the critical value of $(d,g)$) with the following properties. If $(d,g) = (3,0)$ or $(4,1)$, set $m(d,g):= 1$; otherwise, let $m(d,g)$ be the maximal integer $k \geq 2$ such that $kd + 1 - g \leq (k+3)(k+2)(k+1)/6$. Fix $D \in H(d,g)$ with $h^{1}(D,O_{D}(2)) = 0$ (e.g. $D$ general in $H(d,g)$) and let $m:= m(d,g)$ be the critical value of the pair $(d,g)$. $D$ is said to have maximal rank if for all integers $t$ the restriction map $r_{D,t}: H^{0}(\Pi,O(t)) \rightarrow H^{0}(D,O_{D}(t))$ has maximal rank. $D$ has maximal rank if and only if $r_{D,m}$ is surjective and $r_{D,m-1}$ is injective, i.e. if and only if $h^{1}(I_{D}(m)) = h^{0}(I_{D}(m-1)) = 0$. Assume that $D$ has maximal rank. The first step of the minimal free resolution of $D$ is the expected one (i.e. the homogeneous ideal of $D$ has the minimal number of generators compatibly with its postulation) if and only if the homogeneous ideal of $D$ is generated by $h^{0}(I_{D}(m))$ forms of degree $m$ and by $max \{ 0, h^{0}(I_{D}(m+1)) - 4(h^{0}(I_{D}(m))) \}$ forms of degree $m+1$. By the dual of the Euler’s exact sequence for $T \Pi$ this is true if and only if the restriction map $r_{D,\Omega (m+1)} : H^{0}(\Pi,\Omega(m+1)) \rightarrow
H^{0}(D,\Omega (m+1)|D)$ has maximal rank. Thus the homogeneous ideal of the maximal rank curve $D$ has the expected number of generators if and only if either $h^{1}(I_{D} \otimes
\Omega (m+1)) = 0$ or $h_{0}(I_{D} \otimes \Omega (m+1)) = 0$. The last step of the minimal free resolution of the maximal rank curve $D$ is the expected one if and only if the restriction map $r_{D,T \Pi (m-2)} : H^{0}(\Pi,T \Pi (m-2)) \rightarrow
H^{0}(D,T \Pi (m-2)|D)$ has maximal rank. If the first and the last part of the minimal free resolution of $D$ are the expected ones, then by Euler characteristic reasons even the intermediate step of the minimal free resolution of $D$ must be the expected one and hence $D$ has the expected minimal free resolution. Note also that by the semicontinuity of cohomology groups in flat families of coherent sheaves all these conditions are satisfied in Zariski open subsets of $H(d,g)$ and the only problem is to show that each of these Zariski open subsets of $H(d,g)$ is not empty. Hence it is sufficient to find $D' \in H(d,g)$ and $D" \in H(d,g)$ with $h^{1}(D',(\Omega |D')(m+1)) = h^{1}(D",(T \Pi |D")(m-2)) = 0$ and with $r_{D',\Omega (m+1)}$ and $r_{D",T \Pi (m-2)}$ of maximal rank. Since $H(d,g)$ is irreducible, it is sufficient to prove the existence of some $X' \in H(d,g)$ with $r_{X',T \Pi (m-2)}$ injective (case $(3m-2)d + 3(1-g) \geq h^{0}(T \Pi (m-2))$) or the existence of some $D" \in H(d,g)$ with $r_{D",T \Pi (m-2)}$ surjective (case $(3m-2)d + 3(1-g) \leq h^{0}(T \Pi (m-2))$) and the existence some $D' \in H(d,g)$ with $r_{D',\Omega (m+1)}$ injective (case $(3m-1)d + 3(1-g) \geq h_{0}(\Omega (m+1))$) or the existence of some $D'\in H(d,g)$ with $r_{D',\Omega (m+1)}$ surjective (case $(3m-1)d + 3(1-g) \leq h^{0}(\Omega (m+1))$). To prove \[0.1\], \[0.2\] and \[0.3\] we will use the following elementary form of the so - called Horace lemma.
\[1.3\] Let $D$ be an effective Cartier divisor of the scheme $Z \subset \Pi$, $E$ a vector bundle on $Z$ and $X$ a closed subscheme of $Z$. Let $Y:= Res_{D}(X)$ be the residual scheme of $X$ with respect to $D$.
- [if $h^{1}(Z,E(m) \otimes O_{Z}(-D) \otimes I_{Y,Z}) = h^{1}(D,(E|D)(m) \otimes
I_{X \cap D,}D) = 0$, then $h^{1}(Z,E(m) \otimes I_{X,Z}) = 0$.]{}
- [ if $h^{0}(Z,E(m) \otimes O_{Z}(-D) \otimes I_{Y,Z}) = h^{0}(D,(E|D)(m) \otimes
I_{X \cap D,}D) = 0$, then $h^{0}(Z,E(m) \otimes I_{X,Z}) = 0$.]{}
\[1.4\] Here we discuss the unreduced curves which we will use in this paper. Fix $P \in \Pi$. We will call nilpotent $\chi (P)$ with support on $P$ the first infinitesimal neighborhood of $P$ in $\Pi$, i.e. the 0-dimensional subscheme of $\Pi$ with ideal sheaf $(I_{P})^{2}$. Let $A$, $B$ curves intersecting quasi-transversally at $P$; for every integer $x$ we have $\chi (O_{A \cup B \cup \chi (P)}(x)) = \chi (O_{A \cup B}(x)) + 1$. If ${P} = A \cap B$, then $A \cup B \cup \chi (P)$ is the flat limit in $\Pi$ of a family of curves $\{ A_{t} \cup B_{t} \}_{t \in T}$, $A_{o} = A$, $B_{o} = B$ with $A_{t}$ (resp. $B_{t}$) general translate of $A$ (resp. $B$) for $t \in (T\setminus{o})$; hence $A_{t}$ (resp. $B_{t}$) is projectively equivalent to $A$ (resp. $B$) and $A_{t} \cap B_{t} = \O$ for $t \in (T\setminus \{o\})$. When $card(A \cap B) > 1$ we will always met a situation in which a similar picture is true for the following reasons; $B$ will be a rational normal curve or a conic or a line, $card(A \cap B) \geq deg(B)+1$ and $B$ intersects quasi-transversally $A$; fix an integer $u$ with $0 \leq u < min\{card(A \cap B),deg(B)\}$ and let $X$ be the union of $A$, $B$ and the first infinitesimal neighborhoods of $u$ of the points of $A_{0} \cap B$; move $B$ in a family ${B_{t}}$ keeping the condition that each $B_{t}$ intersects quasi-transversally the fixed curve $A$ exactly at $u+1$ points; if $A \in H(d,g)$ we see in this way that $X \in H(d+deg(B),g+u)$.
\[1.5\] Let $A$, $B$ be space curves and $t$ an integer $\geq 1$. $B$ is said to be $t$-secant to $A$ if $card(A \cap B) = t$ and $B$ intersects $A$ quasi-transversally at every point of $A \cap B$.
Numerical invariants {#2}
====================
Let $m \geq 1$ be an integer, which should seen as the “critical value” for the pair $(d,g)$ we are interested in. As in [@H1] or [@BE1] we define integers $d"(m)$ and $r(m)$ which are related to the postulation problem for curves of genus 0. Define $d"(m)$ and $r(m)$ by the following relations: $$\label{e3}
m \cdot d"(m) + 1 + r(m) = (m+3)(m+2)(m+1)/6, 0 \leq r(m) \leq m-1$$ In [@BE1], p. 544, the integer $d"(m)$ (resp. $r(m)$) was denoted with $r(m,0)$ (resp. $q(m,0)$). Set $d"(0):= 1$ and $r(0):= 0$. For every integer $k \geq 0$ we have $d"(6k) = 6k^{2} + 6k + 1$, $r(6k) = 5k$, $d"(6k+1) = 6k^{2} + 8k + 3$, $r(6k+1) = 0$, $d"(6k+2) = 6k^{2} + 10k + 4$, $r(6k+2) = 3k+1$, $d"(6k+3) = 6k^{2} + 8k + 3$, $(6k+3) = 2k+1$, $d"(6k+4) = 6k^{2} + 14k + 8$, $r(6k+4) = 3k+2$, $d"(6k+5) = 6k^{2} + 16k + 11$, $r(6k+5) = 0$ ([@BE1], p. 544). Recall that for all integers $m \geq 0$ we have $h^{0}(\Omega (m+1)) = m(m+2)(m+3)/2$ and $h^{0}(T \Pi (m-2)) = m(m+1)(m+3)/2$. For all integers $m \geq 2$ define the integers $d(m)$, $a(m)$, $b(m)$, $d'(m)$, $a'(m)$, $b'(m)$ by the relations $$\label{e4}
(3m-1)d(m) + 3 + 3a(m) + b(m) = m(m+2)(m+3)/2$$ $$\label{e5}
0 \leq a(m) \leq m-1, 0 \leq b(m) \leq 2, (a(m),b(m)) \neq (m-1,2)$$ $$\label{e6}
(3m-2)d'(m) + 3 + 3a'(m) + b'(m) = m(m+1)(m+3)/2$$ $$\label{e7}
0 \leq a'(m) \leq m-1, 0 \leq b'(m) \leq 2, b'(m) = 0 \mbox{\quad \rm if \quad} a'(m) = m-1$$ Note that we have $lim_{m \rightarrow +\infty} |d(m) - m^{2}/6|/m^{2} = 0$, $lim_{m \rightarrow +\infty} |d'(m) - m^{2}/6|/m^{2} = 0$, $lim_{m \rightarrow +\infty} |d(m) - d(m-2) - (2/3)m|/m = 0$ and $lim_{m \rightarrow +\infty} |d'(m) - d'(m-2) - (2/3)m|/m = 0$. Since $3h^{0}(O(m)) \geq h^{0}(\Omega (m+1)) \geq h^{0}(T \Pi (m-2)) \geq 3h^{0}(O(m-1))$ for all integers $m \geq 0$, we have $d"(m-1) \leq d'(m) \leq d(m) \leq d"(m)$. Recall that $h^{0}(\Omega (m+1) \equiv 0 \quad mod(3)$ for $m \equiv 0,1 \quad mod(3)$, $h^{0}(\Omega (m+1)) \equiv 2 \quad mod(3)$ for $m \equiv 2 \quad mod(3)$, $h^{0}(T \Pi (m-2)) \equiv 0 \quad mod(3)$ for $m \equiv 0,2 \quad mod(3)$ and $h^{0}(T \Pi (m-2)) \equiv 1 \quad mod(3)$ for $m \equiv 1 \quad mod(3)$. Motivated by Remark \[1.1\] and Riemann - Roch formulas \[e1\] and \[e2\] we introduce the following definitions. Let $d(m)^{\ast}$ be the maximal integer $\leq d(m)$ such that $-d(m)^{\ast} \equiv h^{0}(\Omega (m+1)) \quad mod(3)$. Let $d'(m)^{\ast}$ be the maximal integer $\leq d'(m)$ such that $-2d'(m)^{\ast} \equiv h^{0}(T \Pi (m-2)) \quad mod(3)$. Note that $d(m) - 2 \leq d(m)^{\ast} \leq d(m)$ and $d'(m) - 2 \leq d'(m)^{\ast} \leq d'(m)$ for every $m$. Thus for every $m \geq 4$ we have $|(d(m) - d(m-2)) - (d(m)^{\ast} - d(m-2)^{\ast})| \leq 2$, $|(d'(m) - d'(m-2)) - (d'(m)^{\ast} - d'(m-2)^{\ast})| \leq 2$, $lim_{m \rightarrow +\infty}|d(m)^{\ast} - d(m-2)^{\ast} - (2/3)m|/m = 0$ and $lim_{m \rightarrow +\infty}|d'(m)^{\ast} - d'm-2)^{\ast} - (2/3)m|/m = 0$. Now we will define integers $\delta(m)$, $\delta'(m)$, $\delta"(m)$, $\alpha(m)$, $\beta(m)$, $\alpha'(m)$, $\beta'(m)$, $\delta"(m)$, $\rho(m)$, $\delta(m)^{\ast}$ and $\delta'(m)^{\ast}$ with the following modifications of the definitions given at the beginning of this section for the integers $\delta(m)$, $\delta'(m)$, $\delta"(m)$, $\alpha(m)$, $\beta(m)$, $\alpha'(m)$, $\beta'(m)$, $\delta"(m)$, $\rho(m)$, $\delta(m)^{\ast}$ and $\delta'(m)^{\ast}$. For $m \geq 2$ we use the following relations: $$\label{e8}
(m-1) \delta"(m) + 4 + \rho(m) = (m+3)(m+2)(m+1)/6, 0 \leq \rho(m) \leq m-2$$ $$(3m-4)\delta(m) + 12 + 3\alpha(m) + \beta(m) = m(m+2)(m+3)/2
\label{e9}$$ $$0 \leq \alpha(m) \leq m-2, 0 \leq \beta(m) \leq 2, (\alpha(m),\beta(m)) \neq (m-2,2)
\label{e10}$$ $$(3m-5)\delta'(m) + 12 + 3\alpha'(m) + \beta'(m) = m(m+1)(m+3)/2
\label{e11}$$ $$0 \leq \alpha'(m) \leq m-2, 0 \leq \beta'(m) \leq 2, \beta'(m) = 0 \mbox{\quad \rm if \quad} \alpha'(m) = m-2
\label{e12}$$ Note that we have $lim_{m \rightarrow +\infty} \: |\delta(m) - m^{2}/6|/m^{2} = 0$, $lim_{m \rightarrow +\infty} \: |\delta'(m) - m^{2}/6|/m^{2} = 0$, $lim_{m \rightarrow +\infty} \: |\delta(m) - \delta(m-2) - (2/3)m|/m = 0$ and $lim_{m \rightarrow +\infty} \: |\delta'(m) - \delta'(m-2) - (2/3)m|/m = 0$. Since $3h^{0}(O(m)) \geq h^{0}(\Omega (m+1)) \geq h^{0}(T \Pi (m-2)) \geq 3h^{0}(O(m-1))$ for all integers $m \geq 0$, we have $\delta"(m-1) \leq \delta'(m) \leq \delta(m) \leq \delta"(m)$. The integers $\delta"(m)$ and $\rho(m)$ where called $d(m)$ and $b(m)$ in [@BE1], §3. These integers are related to curves of genus $\gamma$ and degree $\gamma +3$ with critical value $m$ (see the Riemann - Roch formulas \[e1\] and \[e2\]). Let $\delta(m)^{\ast}$ be the maximal integer $\leq \delta(m)$ such that $-\delta(m)^{\ast} \equiv h^{0}(\Omega (m+1)) \quad mod(3)$. Let $\delta'(m)^{\ast}$ be the maximal integer $\leq \delta'(m)$ such that $-2\delta'(m)^{\ast} \equiv h^{0}(T \Pi (m-2)) \quad mod(3)$. Note that $\delta(m) - 2 \leq \delta(m)^{\ast} \leq \delta(m)$ and $\delta'(m) - 2 \leq \delta'(m)^{\ast} \leq \delta'(m)$ for every $m$. For every integer $m \geq 2$ define the integers $a(m)^{\ast}$,$a'(m)^{\ast}$, $\alpha(m)^{\ast}$, and $\alpha'(m)^{\ast}$ by the following relations: $$3a(m)^{\ast} = (3m-1)(d(m)-d(m)^{\ast}) + 3a(m) + b(m), 0 \leq b(m) \leq 2
\label{e13}$$ $$3a'(m)^{\ast} = (3m-2)(d'(m)-d'(m)^{\ast}) + 3a'(m) + b'(m), 0 \leq b'(m) \leq 2
\label{e14}$$ $$3\alpha(m)^{\ast} = (3m-4)(\delta(m)-\delta(m)^{\ast}) + 3\alpha(m) + \beta(m), 0 \leq \beta(m) \leq 2
\label{e15}$$ $$3\alpha'(m)^{\ast} = (3m-5)(\delta'(m)-\delta'(m)^{\ast}) + 3\alpha'(m) + \beta'(m), 0 \leq \beta'(m) \leq 2
\label{e16}$$
\[2.1\] Note that $a(m)^{\ast}$, $a'(m)^{\ast}$, $\alpha(m)^{\ast}$ and $\alpha'(m)^{\ast}$ are integers and not just rational numbers of the form $x / 3$ with $x \in Z$ by the definition of $d(m)^{\ast}$, $d'(m)^{\ast}$, $\delta(m)^{\ast}$ and $\delta'(m)^{\ast}$.
We introduced the integers $\delta(m)^{\ast}$, $\alpha(m)^{\ast}$, $\delta'(m)^{\ast}$, $\alpha'(m)^{\ast}$, $d(m)^{\ast}$, $a(m)^{\ast}$, $d'(m)^{\ast}$ and $a'(m)^{\ast}$ because $rank(\Omega) = rank(T\Pi)$ and hence by the Riemann - Roch formulas \[1\] and \[2\] using these integers we will be able to bypass several problems of congruences modulo 3 for the inductive proofs of Theorem \[0.1\], \[0.2\] and \[0.3\] and avoid the use of “ points of the geometric vector bundles $V(T\Pi)$ and $V(\Omega)$ ” or “ one-third of the point of ${\mbox{\bf P}^{3}}$ ” which is the key idea of [@H2] and a key point for the proofs in [@I]. The price we pay to avoid this notion is the huge number of initial cases handled in sections \[6\] and \[8\]. Anyway, even using this notion we were unable to avoid a lot of initial cases and we believe that the remark on the congruence classes modulo 3 and the corresponding introduction of the integers $\delta(m)^{\ast}$, $\alpha(m)^{\ast}$, $\delta'(m)^{\ast}$, $\alpha'(m)^{\ast}$, $d(m)^{\ast}$, $a(m)^{\ast}$, $d'(m)^{\ast}$ and $a'(m)^{\ast}$ is very useful for Horace - type proofs on the minimal free resolution of space curves.
\[2.2\] Since $0 \leq d(m) - d(m)^{\ast} \leq 2$, $0 \leq d'(m) - d'(m)^{\ast} \leq 2$, $0 \leq d(m) - d(m)^{\ast} \leq
2$ and $0 \leq d'(m) - d'(m)^{\ast} \leq 2$, we have $0 \leq a(m)^{\ast} \leq 3m-1$, $0 \leq a'(m)^{\ast} \leq
3m-2$, $0 \leq a(m)^{\ast} \leq 3m-4$ and $0 \leq\leq a(m)^{\ast} \leq 3m-5$. Fix an integer $g \geq 0$. For all integers $m \geq 2$ set $b(m,g):= b(m,0):= b(m)$, $b'(m,g):= b'(m)$ and define the integers $d(m,g)$, $a(m,g)$, $d'(m,g)$ and $a'(m,g)$ by the following relations: $$(3m-1)d(m,g) + 3(1-g) + 3a(m,g) + b(m) = m(m+2)(m+3)/2
\label{e17}$$ $$0 \leq a(m,g) \leq m-1, 0 \leq b(m,g) \leq 2, (a(m,g),b(m)) \neq (m-1,2)
\label{e18}$$ $$(3m-2)d'(m,g) + 3(1-g) + 3a'(m,g) + b'(m) = m(m+1)(m+3)/2
\label{e19}$$ $$0 \leq a'(m,g) \leq m-1, 0 \leq b'(m,g) \leq 2, b'(m) = 0 \mbox{\quad \rm if } a'(m,g) = 0
\label{e20}$$
\[2.3\] Note that $a(m,g) = a(m)+g$ and $d(m,g) = d(m)$ if $a(m)+g \leq 3m-2$. In the general case we have $d(m,g) \geq d(m,0)$ and $(d(m,g) - d(m))(3m-1) + a(m,g) = a(m) + g$.
\[2.4\] Note that $a'(m,g) = a'(m)+g$ and $d'(m,g) = d'(m)$ if $a'(m)+g \leq 3m-3$. In the general case we have $d'(m,g) \geq d'(m,0)$ and $(d'(m,g) - d'(m))(3m-2) + a'(m,g) = a'(m) + g$.
\[2.5\]
Here we give the explicit values of the integers $\delta(m)^{\ast}$, $\alpha(m)^{\ast}$, $\delta'(m)^{\ast}$ and $\alpha'(m)^{\ast}$ for $m \leq 51$. We have $\delta(2)^{\ast} = 4, \alpha(2)^{\ast} = 0, \delta'(2)^{\ast} = 3,
\alpha'(2)^{\ast} = 0, \delta(3)^{\ast} = 6, \alpha(3)^{\ast} = 1, \delta'(3)^{\ast} = 6, \alpha'(3)^{\ast} = 0,
\delta(4)^{\ast} = 9, \alpha(4)^{\ast} = 0, \delta'(4)^{\ast} = 7, \alpha'(4)^{\ast} = 3, \delta(5)^{\ast} = 10,
\alpha(5)^{\ast} = 6, \delta'(5)^{\ast} = 9, \alpha'(5)^{\ast} = 6, \delta(6)^{\ast} = 15, \alpha(6)^{\ast} = 2,
\delta'(6)^{\ast} = 13, \alpha'(6)^{\ast} = 12, \delta(7)^{\ast} = 15, \alpha(7)^{\ast} = 16, \delta'(7)^{\ast} = 15,
\alpha'(7)^{\ast} = 3, \delta(8)^{\ast} = 19, \alpha(8)^{\ast} = 12, \delta'(8)^{\ast} = 18, \alpha'(8)^{\ast} = 17,
\delta(9)^{\ast} = 24, \alpha(9)^{\ast} = 10, \delta'(9)^{\ast} = 24, \alpha'(9)^{\ast} = 0, \delta(10)^{\ast} = 28,
\alpha(10)^{\ast} = 20, \delta'(10)^{\ast} = 28, \alpha'(10)^{\ast} = 1, \delta(11)^{\ast} = 33, \alpha(11)^{\ast} = 4,
\delta'(11)^{\ast} = 30, \alpha'(11)^{\ast} = 24, \delta(12)^{\ast} = 39, \alpha(12)^{\ast} = 0, \delta'(12)^{\ast} = 36,
\alpha'(12)^{\ast} = 14, \delta(13)^{\ast} = 42, \alpha(13)^{\ast} = 26, \delta'(13)^{\ast} = 42, \alpha'(13)^{\ast} = 2,
\delta(14)^{\ast} = 49, \alpha(14)^{\ast} = 10, \delta'(14)^{\ast} = 45, \alpha'(14)^{\ast} = 36, \delta(15)^{\ast} = 54,
\alpha(15)^{\ast} = 23, \delta'(15)^{\ast} = 51, \alpha'(15)^{\ast} = 36, \delta(16)^{\ast} = 60, \alpha(16)^{\ast} = 28,
\delta'(16)^{\ast} = 58, \alpha'(16)^{\ast} = 26, \delta(17)^{\ast} = 67, \alpha(17)^{\ast} = 19, \delta'(17)^{\ast} = 66,
\alpha'(17)^{\ast} = 4, \delta(18)^{\ast} = 75, \alpha(18)^{\ast} = 6, \delta'(18)^{\ast} = 72, \alpha'(18)^{\ast} = 17,
\delta(19)^{\ast} = 81, \alpha(19)^{\ast} = 28, \delta'(19)^{\ast} = 79, \alpha'(19)^{\ast} = 20, \delta(20)^{\ast} = 88,
\alpha(20)^{\ast} = 40, \delta'(20)^{\ast} = 87, \alpha'(20)^{\ast} = 11, \delta(21)^{\ast} = 96, \alpha(21)^{\ast} = 44,
\delta'(21)^{\ast} = 93, \alpha'(21)^{\ast} = 46, \delta(22)^{\ast} = 105, \alpha(22)^{\ast} = 26, \delta'(22)^{\ast} = 103,
\alpha'(22)^{\ast} = 10, \delta(23)^{\ast} = 112, \alpha(23)^{\ast} = 61, \delta'(23)^{\ast} = 111, \alpha'(23)^{\ast} = 20,
\delta(24)^{\ast} = 123, \alpha(24)^{\ast} = 16, \delta'(24)^{\ast} = 120, \alpha'(24)^{\ast} = 16, \delta(25)^{\ast} = 132,
\alpha(25)^{\ast} = 22, \delta'(25)^{\ast} = 127, \alpha'(25)^{\ast} = 66, \delta(26)^{\ast} = 142, \alpha(26)^{\ast} = 12,
\delta'(26)^{\ast} = 138, \alpha'(26)^{\ast} = 31, \delta(27)^{\ast} = 150, \alpha(27)^{\ast} = 61, \delta'(27)^{\ast} = 147,
\alpha'(27)^{\ast} = 52, \delta(28)^{\ast} = 162, \alpha(28)^{\ast} = 16, \delta'(28)^{\ast} = 157,
\alpha'(28)^{\ast} = 57, \delta(29)^{\ast} = 177, \alpha(29)^{\ast} = 59, \delta'(29)^{\ast} = 168,
\alpha'(29)^{\ast} = 44, \delta(30)^{\ast} = 183, \alpha(30)^{\ast} = 30, \delta'(30)^{\ast} = 180,
\alpha'(30)^{\ast} = 11, \delta(31)^{\ast} = 195, \alpha(31)^{\ast} = 8, \delta'(31)^{\ast} = 190,
\alpha'(31)^{\ast} = 44, \delta(32)^{\ast} = 205, \alpha(32)^{\ast} = 56,
\delta'(32)^{\ast} = 201, \alpha'(32)^{\ast} = 59, \delta(33)^{\ast} = 216, \alpha(33)^{\ast} = 86,
\delta'(33)^{\ast} = 213, \alpha'(33)^{\ast} = 54, \delta(34)^{\ast} = 228, \alpha(34)^{\ast} = 96,
\delta'(34)^{\ast} = 226, \alpha'(34)^{\ast} = 27, \delta(35)^{\ast} = 241, \alpha(35)^{\ast} = 84,
\delta'(35)^{\ast} = 237, \alpha'(35)^{\ast} = 76, \delta(36)^{\ast} = 255, \alpha(36)^{\ast} = 48,
\delta'(36)^{\ast} = 252, \alpha'(36)^{\ast} = 2, \delta(37)^{\ast} = 267, \alpha(37)^{\ast} = 93,
\delta'(37)^{\ast} = 265, \alpha'(37)^{\ast} = 6, \delta(38)^{\ast} = 283, \alpha(38)^{\ast} = 6,
\delta'(38)^{\ast} = 276, \alpha'(38)^{\ast} = 95, \delta(39)^{\ast} = 297, \alpha(39)^{\ast} = 2,
\delta'(39)^{\ast} = 291, \alpha'(39)^{\ast} = 52, \delta(40)^{\ast} = 309, \alpha(40)^{\ast} = 88,
\delta'(40)^{\ast} = 304, \alpha'(40)^{\ast} = 100, \delta(41)^{\ast} = 325, \alpha(41)^{\ast} = 33,
\delta'(41)^{\ast} = 318, \alpha'(41)^{\ast} = 116, \delta(42)^{\ast} = 339, \alpha(42)^{\ast} = 70,
\delta'(42)^{\ast} = 333, \alpha'(42)^{\ast} = 110, \delta(43)^{\ast} = 354, \alpha(43)^{\ast} = 81,
\delta'(43)^{\ast} = 349, \alpha'(43)^{\ast} = 76, \delta(44)^{\ast} = 371, \alpha(44)^{\ast} = 64,
\delta'(44)^{\ast} = 366, \alpha'(44)^{\ast} = 12, \delta(45)^{\ast} = 387, \alpha(45)^{\ast} = 17,
\delta'(45)^{\ast} = 381, \alpha'(45)^{\ast} = 46, \delta(46)^{\ast} = 402, \alpha(46)^{\ast} = 72,
\delta'(46)^{\ast} = 397, \alpha'(46)^{\ast} = 52, \delta(47)^{\ast} = 418, \alpha(47)^{\ast} = 99,
\delta'(47)^{\ast} = 414, \alpha'(47)^{\ast} = 28, \delta(48)^{\ast} = 435, \alpha(48)^{\ast} = 96,
\delta'(48)^{\ast} = 429, \alpha'(48)^{\ast} = 111, \delta(49)^{\ast} = 453, \alpha(49)^{\ast} = 61,
\delta'(49)^{\ast} = 448, \alpha'(49)^{\ast} = 24, \delta(50)^{\ast} = 469, \alpha(50)^{\ast} = 142,
\delta'(50)^{\ast} = 465, \alpha'(50)^{\ast} = 46, \delta(51)^{\ast} = 486, \alpha(51)^{\ast} = 185,
\delta'(51)^{\ast} = 483, \alpha'(51)^{\ast} = 36$.
\[2.6\] For every integer $m \geq 52$ we have $\alpha(m)^{\ast} \leq 20(\delta(m+2)^{\ast} - \delta(m)^{\ast} - 23)$.
[Proof]{} We have $\delta(m+2)^{\ast} - \delta(m)^{\ast} \geq \delta(m+2) - \delta(m)^{\ast} - 2 \geq \delta(m+2) - \delta(m)
- 2$ and $\alpha(m)^{\ast} \leq 3m-4$. Hence it is sufficient to prove that $\delta(m+2) - \delta(m) \geq
25 + (3m-4)/20$. By eq. \[e9\] for $m+2$ and $m$ we obtain $\delta(m+2) \leq (m+5)(m+4)(m+2)/2(3m+2)$ and $(3m-4)(\delta(m+2) - \delta(m)) = (m+2)(3m+10) - 6\delta(m+2) - (3\alpha(m+2)+b(m+2)-3\alpha(m)-b(m)) \geq
(m+2)(3m+10) - 3(m+5)(m+4)(m+2)/(3m+2) - (3m+2)$. Hence it is sufficient to obtain the inequality $(25 + (3m-4)/20)(3m-4) \leq (m+2)(3m+10) - 3(m+5)(m+4)(m+2)/(3m+2) - (3m+3)$, i.e. the inequality $(3m+2)(9m^{2}-1500m-1992) \leq 20(3m+2)(3m^{2}+13m+17) - 60(m^{3}+11m^{2}+38m+20)$, i.e. the inequality $93m^{3} - 4722m^{2} + 8236m + 2864 \geq 0$, which is true for $m \geq 52$.
\[2.7\] For every integer $m \geq 52$ we have $\alpha'(m)^{\ast} \leq 20(\delta'(m+2)^{\ast} - \delta'(m)^{\ast} - 23)$.
For Theorem 0.3 we need the following lemma.
\[2.8\] For every integer $g \geq 0$ and every $m \geq max \{ 70,m_{o}(g)-2 \}$ we have $a(m,g)^{\ast} \leq 3m-1 \leq
20(d(m+2,g)^{\ast} - d(m,g)^{\ast} - 23)$.
[Proof]{} Since $a(m,g)^{\ast} \leq 3m-1$, it is sufficient to check the second inequality. Since $d(m+2,g)^{\ast} - d(m,g)^{\ast} \geq d(m+2,g) - d(m,g)^{\ast} - 2 \geq d(m+2,g) - d(m,g) - 2$, it is sufficient to check that $20(d(m+2,g) - d(m,g) - 25) \geq 3m-1$ for $m \geq max \{ 70,m_{o}(g)-2 \}$. By eq. \[e13\] for $m$ and $m+2$ we obtain $$\begin{array}{l}
(3m-1)(d(m+2,g) - d(m,g)) + 3a(m+2,g) - 3a(m,g) + \\
+ b(m+2,g) - b(m,g) + 6d(m+2,g) = (m+2)(3m+10)
\end{array}
\label{e21}$$ We have $a(m+2,g) \leq m+1$ and $b(m+2,g) \leq 2$. We have $d(m+2,g) \leq \delta(m+2)$ because $m+2 \geq m_{o}(g)$. Hence $d(m+2,g) \leq \delta(m+2) \leq (m+5)(m+4)(m+2)/2(3m+2)$. Thus by eq. \[e21\] to obtain the second inequality of \[2.8\] it is sufficient to check when $(3m-1)(3m+5)(3m+499) \leq
20(3m+5)(3m^{2}+3m+12) - 60(m^{3}+11m^{2}+38m+40)$, i.e. when $(3m+5)(51m^{2}-1440m-259) \geq 60(m^{3}+11m^{2}+38m+40)$. This is true for every $m \geq 70$.
In the same way we obtain the next lemma.
\[2.9\] For every $g \geq 0$ and every $m \geq max \{ 70,m_{o}(g)-2 \}$ we have $a'(m,g)^{\ast} \leq 3m-2 \leq
20(d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - 23)$.
\[2.10\] For every $g \geq 0$ and $m \geq max \{ 70,m_{o}(g)-2 \} $ we have $\alpha(m_{o}(g)-1)^{\ast} \leq
20(d(m_{o}(g)+1,g)^{\ast} - \delta(m_{o}(g)-2)^{\ast} - 23)$ and $\alpha(m_{o}(g)-1)^{\ast} \leq 20(d(m_{o}(g)+1,g)^{\ast}
- \delta(m_{o}(g)-2)^{\ast} - 23)$.
[Proof]{} By definition we have $\delta(x)^{\ast} = d(x,d(x)-3)^{\ast}$. If $g' \geq g"$ we have $d(x,g')^{\ast} \geq d(x,g")$ and hence we conclude by lemma \[2.8\].
\[2.11\] For every $g \geq 0$ and $m \geq max \{ 70,m_{o}(g)-2 \}$ we have $\alpha'(m_{o}(g)-1)^{\ast} \leq
20(d'(m_{o}(g)+1,g)^{\ast} - \delta'(m_{o}(g)-2)^{\ast} - 23)$ and $\alpha'(m_{o}(g)-1)^{\ast} \leq
20(d'(m_{o}(g)+1,g)^{\ast} - \delta'(m_{o}(g)-2)^{\ast} - 23)$.
\[2.12\]
Here we give the values of the integers $d(m)^{\ast}$, $a(m)^{\ast}$, $d'(m)^{\ast}$ and $a'(m)^{\ast}$ for $m \leq 70$. We have $d(2)^{\ast} = 1, a(2)^{\ast} = 4, d'(2)^{\ast} = 3, a'(2)^{\ast} = 0, d(3)^{\ast} = 3, a(3)^{\ast} = 6, d'(3)^{\ast} = 3,
a'(3)^{\ast} = 4, d(4)^{\ast} = 6, a(4)^{\ast} = 5, d'(4)^{\ast} = 4, a'(4)^{\ast} = 9, d(5)^{\ast} = 7, a(5)^{\ast} = 13,
d'(5)^{\ast} = 9, a'(5)^{\ast} = 0, d(6)^{\ast} = 12, a(6)^{\ast} = 3, d'(6)^{\ast} = 9, a'(6)^{\ast} = 14, d(7)^{\ast} = 15,
a(7)^{\ast} = 4, d'(7)^{\ast} = 13, a'(7)^{\ast} = 10, d(8)^{\ast} = 19, a(8)^{\ast} = 0, d'(8)^{\ast} = 15, a'(8)^{\ast} = 21,
d(9)^{\ast} = 21, a(9)^{\ast} = 15, d'(9)^{\ast} = 21, a'(9)^{\ast} = 4, d(10)^{\ast} = 24, a(10)^{\ast} = 27, d'(10)^{\ast} = 25,
a'(10)^{\ast} = 4, d(11)^{\ast} = 31, a(11)^{\ast} = 2, d'(11)^{\ast} = 27, a'(11)^{\ast} = 28, d(12)^{\ast} = 33,
a(12)^{\ast} = 34, d'(12)^{\ast} = 33, a'(12)^{\ast} = 15, d(13)^{\ast} = 39, a(13)^{\ast} = 25, d'(13)^{\ast} = 37,
a'(13)^{\ast} = 28, d(14)^{\ast} = 46, a(14)^{\ast} = 5, d'(14)^{\ast} = 42, a'(14)^{\ast} = 34, d(15)^{\ast} = 51,
a(15)^{\ast} = 16, d'(15)^{\ast} = 48, a'(15)^{\ast} = 31, d(16)^{\ast} = 57, a(16)^{\ast} = 18, d'(16)^{\ast} = 55,
a'(16)^{\ast} = 17, d(17)^{\ast} = 64, a(17)^{\ast} = 9, d'(17)^{\ast} = 60, a'(17)^{\ast} = 39, d(18)^{\ast} = 69,
a(18)^{\ast} = 40, d'(18)^{\ast} = 69, a'(18)^{\ast} = 0, d(19)^{\ast} = 78, a(19)^{\ast} = 6, d'(19)^{\ast} = 73,
a'(19)^{\ast} = 54, d(20)^{\ast} = 85, a(20)^{\ast} = 14, d'(20)^{\ast} = 81, a'(20)^{\ast} = 43, d(21)^{\ast} = 93,
a(21)^{\ast} = 9, d'(21)^{\ast} = 90, a'(21)^{\ast} = 17, d(22)^{\ast} = 99, a(22)^{\ast} = 54, d'(22)^{\ast} = 97,
a'(22)^{\ast} = 38, d(23)^{\ast} = 109, a(23)^{\ast} = 20, d'(23)^{\ast} = 105, a'(23)^{\ast} = 46, d(24)^{\ast} = 117,
a(24)^{\ast} = 38, d'(24)^{\ast} = 114, a'(24)^{\ast} = 39, d(25)^{\ast} = 126, a(25)^{\ast} = 41, d'(25)^{\ast} = 124,
a'(25)^{\ast} = 15, d(26)^{\ast} = 136, a(26)^{\ast} = 27, d'(26)^{\ast} = 132, a'(26)^{\ast} = 48, d(27)^{\ast} = 144,
a(27)^{\ast} = 74, d'(27)^{\ast} = 141, a'(27)^{\ast} = 66, d(28)^{\ast} = 156, a(28)^{\ast} = 23, d'(28)^{\ast} = 151,
a'(28)^{\ast} = 67, d(29)^{\ast} = 166, a(29)^{\ast} = 35, d'(29)^{\ast} = 162, a'(29)^{\ast} = 49, d(30)^{\ast} = 177,
a(30)^{\ast} = 28, d'(30)^{\ast} = 174, a'(30)^{\ast} = 10, d(31)^{\ast} = 189, a(31)^{\ast} = 0, d'(31)^{\ast} = 184,
a'(31)^{\ast} = 39, d(32)^{\ast} = 199, a(32)^{\ast} = 44, d'(32)^{\ast} = 195, a'(32)^{\ast} = 49, d(33)^{\ast} = 216,
a(33)^{\ast} = 83, d'(33)^{\ast} = 207, a'(33)^{\ast} = 38, d(34)^{\ast} = 222, a(34)^{\ast} = 73, d'(34)^{\ast} = 220,
a'(34)^{\ast} = 4, d(35)^{\ast} = 235, a(35)^{\ast} = 54, d'(35)^{\ast} = 231, a'(35)^{\ast} = 54, d(36)^{\ast} = 249,
a(36)^{\ast} = 10, d'(36)^{\ast} = 243, a'(36)^{\ast} = 71, d(37)^{\ast} = 261, a(37)^{\ast} = 49, d'(37)^{\ast} = 256,
a'(37)^{\ast} = 71, d(38)^{\ast} = 274, a(38)^{\ast} = 65, d'(38)^{\ast} = 270, a'(38)^{\ast} = 46, d(39)^{\ast} = 288,
a(39)^{\ast} = 56, d'(39)^{\ast} = 282, a'(39)^{\ast} = 109, d(40)^{\ast} = 303, a(40)^{\ast} = 20,
d'(40)^{\ast} = 298, a'(40)^{\ast} = 31, d(41)^{\ast} = 316, a(41)^{\ast} = 77, d'(41)^{\ast} = 312,
a'(41)^{\ast} = 43, d(42)^{\ast} = 330, a(42)^{\ast} = 109, d'(42)^{\ast} = 327, a'(42)^{\ast} = 28,
d(43)^{\ast} = 345, a(43)^{\ast} = 114, d'(43)^{\ast} = 340, a'(43)^{\ast} = 111, d(44)^{\ast} = 361,
a(44)^{\ast} = 90, d'(44)^{\ast} = 357, a'(44)^{\ast} = 39, d(45)^{\ast} = 378, a(45)^{\ast} = 35,
d'(45)^{\ast} = 372, a'(45)^{\ast} = 67, d(46)^{\ast} = 393, a(46)^{\ast} = 84, d'(46)^{\ast} = 388,
a'(46)^{\ast} = 66, d(47)^{\ast} = 409, a(47)^{\ast} = 104, d'(47)^{\ast} = 405, a'(47)^{\ast} = 34,
d(48)^{\ast} = 426, a(48)^{\ast} = 93, d'(48)^{\ast} = 420, a'(48)^{\ast} = 111, d(49)^{\ast} = 444,
a(49)^{\ast} = 49, d'(49)^{\ast} = 439, a'(49)^{\ast} = 14, d(50)^{\ast} = 460, a(50)^{\ast} = 119,
d'(50)^{\ast} = 456, a'(50)^{\ast} = 28, d(51)^{\ast} = 480, a(51)^{\ast} = 6, d'(51)^{\ast} = 474, a'(51)^{\ast} = 27,
d(52)^{\ast} = 498, a(52)^{\ast} = 27, d'(52)^{\ast} = 490, a'(52)^{\ast} = 109, d(53)^{\ast} = 514, a(53)^{\ast} = 135,
d'(53)^{\ast} = 510, a'(53)^{\ast} = 21, d(54)^{\ast} = 534, a(54)^{\ast} = 69, d'(54)^{\ast} = 528, a'(54)^{\ast} = 54,
d(55)^{\ast} = 552, a(55)^{\ast} = 128, d'(55)^{\ast} = 547, a'(55)^{\ast} = 52, d(56)^{\ast} = 571, a(56)^{\ast} = 152,
d'(56)^{\ast} = 567, a'(56)^{\ast} = 13, d(57)^{\ast} = 591, a(57)^{\ast} = 139, d'(57)^{\ast} = 585, a'(57)^{\ast} = 104,
d(58)^{\ast} = 613, a(58)^{\ast} = 87, d'(58)^{\ast} = 604, a'(58)^{\ast} = 160, d(59)^{\ast} = 631, a(59)^{\ast} = 170,
d'(59)^{\ast} = 627, a'(59)^{\ast} = 4, d(60)^{\ast} = 654, a(60)^{\ast} = 37, d'(60)^{\ast} = 645, a'(60)^{\ast} = 159,
d(61)^{\ast} = 675, a(61)^{\ast} = 41, d'(61)^{\ast} = 667, a'(61)^{\ast} = 98, d(62)^{\ast} = 697, a(62)^{\ast} = 4,
d'(62)^{\ast} = 687, a'(62)^{\ast} = 178, d(63)^{\ast} = 717, a(63)^{\ast} = 112, d'(63)^{\ast} = 711, a'(63)^{\ast} = 32,
d(64)^{\ast} = 738, a(64)^{\ast} = 181, d'(64)^{\ast} = 733, a'(64)^{\ast} = 29, d(65)^{\ast} = 763, a(65)^{\ast} = 15,
d'(65)^{\ast} = 753, a'(65)^{\ast} = 176, d(66)^{\ast} = 783, a(66)^{\ast} = 194, d'(66)^{\ast} = 777, a'(66)^{\ast} = 88,
d(67)^{\ast} = 807, a(67)^{\ast} = 134, d'(67)^{\ast} = 799, a'(67)^{\ast} = 152, d(68)^{\ast} = 832, a(68)^{\ast} = 27,
d'(68)^{\ast} = 822, a'(68)^{\ast} = 173, d(69)^{\ast} = 855, a(69)^{\ast} = 77, d'(69)^{\ast} = 846, a'(69)^{\ast} = 149,
d(70)^{\ast} = 879, a(70)^{\ast} = 82, d'(70)^{\ast} = 871, a'(70)^{\ast} = 78$.
For large $m$ we have also a stronger form of Lemmas \[2.5\] and \[2.6\] which will be used for the proof of theorem \[0.3\].
\[2.13\] For every integer $m \geq 100$ we have $\alpha(m)^{\ast} \leq 22(\delta(m+2)^{\ast} -
\delta(m)^{\ast} - 26)$.
On the quadric surface {#3}
======================
For every rational normal curve $C \subset \Pi, \Omega|C$ is the direct sum of 3 line bundles of degree -4 (see e.g. [@B], Lemma \[1.3\], or [@I], 3.2.2. Let $Q \subset \Pi$ be a smooth quadric surface. Consider the normal bundle sequence $$0 \rightarrow TQ \rightarrow T\Pi|Q \rightarrow O_{Q}(2,2) \rightarrow 0
\label{e22}$$ Since $\quad TQ \cong O_{Q}(2,0) \quad \oplus \quad O_{Q}(0,2)$, we obtain $\quad h^{1}(Q,(\Omega |Q)(x,y)) =
h^{2}(Q,(\Omega|Q)(x,y)) = h^{1}(Q,(T\Pi|Q)(u,v)) = h^{2}(Q,(T\Pi|Q)(u,v)) = 0$ for all integers $x, y, u, v$ with $x > 0, y > 0, u \geq - 1, v \geq - 1$. Hence by Riemann - Roch we have $h^{0}(Q,(\Omega|Q)(x,y)) = 7 + xy - 3x - 3y$ for all integers $x > 0, y > 0$ and $h^{0}(Q,(T\Pi|Q)(u,v)) = 15 + uv + 5u + 5v$ for all integers $u \geq -1, v \geq -1$. Note that all smooth curves $C \subset Q, D \subset Q$ with $C$ of type $(2,1)$ and $D$ of type $(1,2)$ are rational normal curves and hence $(\Omega|Q)(x,y)|C$ is the direct sum of 3 line bundles of degree $-4+2y+x$, $(\Omega|Q)(x,y)|D$ is the direct sum of 3 line bundles of degree $-4+2x+y$, $(T\Pi|Q)(x,y)|C$ is the direct sum of 3 line bundles of degree $4+2y+x$ and $(T\Pi|Q)(x,y)|D$ is the direct sum of 3 line bundles of degree $4+y+2x$. First, we will consider the cohomology of $(T\Pi|Q)(x,y)\otimes I_{S}$ and $(\Omega|Q)(x,y)\otimes I_{S}$ for a general $S \subset Q$. The following lemma is essentially a very particular case of [@I], Lemma 3.3.1, but the proof is different.
\[3.1\] Fix non-negative integers $x, y, u, w$ with $0 < y \leq x$ and $3u \leq
h^{0}(Q,(\Omega|Q)(x,y)) \leq 3w$. Let Let $M$ (resp. $N$) be the union of $u$ (resp. $w$) general points of $Q$. Then we have $\; h^{1}(Q\; ,\; (\Omega\; |\; Q)(x,y)\; \otimes \; I_{M}) \; =
h^{0}(Q,(\Omega|Q)(x,y)\otimes I_{N}) = 0$.
[Proof]{} Take a smooth curve $E$ of type $(2,1)$ on $Q$. Fix a subset of $E$ formed by $2y+x-3$ points and apply Horace Lemma \[1.3\] with respect to the Cartier divisor $E$ of $Q$ to reduce to a similar assertion for the integers $x':= x-2$, $y':= y-1$. Then we make the same construction using an irreducible curve of type $(1,2)$ of $E$. Then we continue. At each step we will use a different curve of $Q$; at the odd steps we use a component of type $(2,1)$ while at the even steps we use a component of type $(1,2)$. In this way we reduce to a problem about $H^{0}(Q,(\Omega|Q)(x",y"))$ with $y" = 1$ and $x" \geq 1$.
\[3.2\] Fix non-negative integers $x, y, u, w$ with $y \leq x$ and $3u \leq
\linebreak
h^{0}(Q,(T\Pi|Q)(x,y)) \leq 3w$. Let $M$ (resp. $N$) be the union of $u$ (resp. $w$) general points of $Q$. Then we have $h^{1}(Q \; ,\; (T\Pi|Q)(x,y) \; \otimes \; I_{M}) =
h^{0}(Q,(T\Pi|Q)(x,y)\otimes I_{N}) = 0$.
[Proof]{} Just copy the proof of Lemma \[3.1\]. For the last inductive step we use that $h^{1}(Q,(T\Pi|Q)(a,b)) = 0$ for all integers $a \geq -2$ and $b \geq -2$. Now we will consider the case of subsets $M$, $N$ of $Q$ containing many collinear points.
\[3.3\] Let $A, B, C$ be lines of $\Pi$ with $card(A \cap B) = card(B \cap C) = 1$ and $A\cap C = \O$. Set $T:= A\cup B\cup C$. Then $h^{0}(T,\Omega(1)|T) = 0$.
[Proof]{} Note that $T\in H(3,0)$ and that $h^{0}(A,\Omega(1)|A) = h^{0}(B,\Omega(1)|B) =
h^{0}(C,\Omega(1)|C) = 0$. Since $A\cup B$ and $B\cup C$ are plane conics, we see that $h^{0}(A\cup B,\Omega(1)|A\cup B) = h^{0}(B\cup C,\Omega(1)|B\cup C) = 1$. Hence the general section of $\Omega(1)|A$ and one of the sections of $\Omega(1)|B$ does not glue at the point $A\cup B$, but only one of their sections glues to give a section over $A\cup B$; the same is true for the pair $(B,C)$. Taking as $C$ a general line intersecting $B$, we obtain that the unique, up to a multiplicative constant, section of $\Omega(1)|A\cup B$ does not glue with any section of $\Omega(1)|C$ and hence $h^{0}(T,\Omega(1)|T) = 0$ for general $C$. Since any two configurations $(A,B,C)$ are projectively equivalent, we obtain the result for any such configuration.
In the same way we obtain the following result.
\[3.4\] Let $A, B, C$ be lines of $\Pi$ with $card(A\cap B) = card(B\cap C) = 1$ and $A\cap C = \O$. Set $T:= A\cup B\cup C$. Then $h^{0}(T,T\Pi(-1)|T) = 6$ and $h^{0}(T,T\Pi(-2)|T) = 0$.
\[3.5\] Fix integers $a, b, t$ with $a > 0$ and $0 < t \leq b$ and a smooth quadric $Q$. A net of type $(a,b;t)$ on $Q$ is given by $a$ distinct lines $D_{i}$, $1 \leq
i \leq a$, of type $(1,0)$ on $Q$, $b$ distinct lines $R_{j}$, $1 \leq j
\leq b$, of type $(0,1)$ on $Q$ and by the subset $S \subset Q$ with $card(S) = (a-1)b + t$ formed by the points $D_{i}\cap R_{j}$ for $1 \leq i < a$, $1 \leq j \leq b$, and the points $D_{a}\cap R_{j}$, $1 \leq j \leq t$. Abusing notations, we will often say that $S$ is a net of type $(a,b;t)$.
\[3.6\] Fix integers $x, y, a, b, t, u, w$ with $\; a > 0, \; 0 < t \leq b, \; x \geq
1 + 2[(a+1)/2], y \geq b+2[(a-1)/2]+1$ and $3((a-1)b+t) + 3u \leq h^{0}(Q,(\Omega|Q)(x,y)) \leq
3((a-1)b+t) + 3w$. Let $S$ be a net of type $(a,b;t)$ on a smooth quadric $Q$. Let $M$ (resp. $N$) be the union of $S$ and $u$ (resp.$w$) general points of $Q$. Then we have $h^{1}(Q,(\Omega|Q)(x,y)\otimes I_{M}) = h^{0}(Q,(\Omega|Q)(x,y)\otimes I_{N})
= 0$.
[Proof]{} Fix lines $D_{i}, 1 \leq i \leq a$, and $R_{j}, 1 \leq j \leq b$, associated to the net $S$. Set $A:= D_{1}, C:= D_{2}$ and take a general line $B$ of type $(0,1)$. Set $T:= A\cup B\cup C$. Take finite sets $A', C'$ and $B'$ with $A\cap S
\subseteq A' \subset A\setminus (B\cap A)$, $B\cap S \subseteq B' \subset
B\setminus (B\cap (A\cup C))$ and $C\cap S \subseteq C' \subset C\setminus (C\cap B)$ with $card(A') = card(C') = y-1$ and $card(B') = x-1$. Note that $A'\cup B'\cup C'$ is a Cartier divisor, $D$, on $T$. Apply Lemma \[3.3\] to the twist by $-D$ of $(\Omega|Q)(x,y)|T$. By Horace Lemma \[1.3\] applied to the Cartier divisor $T$ of $Q$ we reduce to a case with integers $x':=
x-2$, $y':= y-1$ and a net of type $(a-2,b;t)$. Then we continue. At the end we apply Lemma \[3.1\].
In the same way we obtain the following result.
\[3.7\] Fix integers $x, y, a, b, t, u, w$ with $\; a > 0, \; 0 < t \leq b, \;x \geq
-1+2[(a+1)/2], y \geq b+2[(a-1)/2]-1$ and $3((a-1)b+t) + 3u \leq h^{0}(Q,(T\Pi|Q)(x,y))
\leq 3((a-1)b+t) + 3w$. Let $S$ be a net of type $(a,b;t)$ on a smooth quadric $Q$. Let $M$ (resp. $N$) be the union of $S$ and $u$ (resp. $w$) general points of $Q$. Then we have $h^{1}(Q,(T\Pi|Q)(x,y)\otimes I_{M}) =
h^{0}(Q,(T\Pi|Q)(x,y)\otimes I_{N}) = 0$.
Lemmas \[3.6\] and \[3.7\] are sufficient for the inductive step from $m-2$ to $m$ when the integer $m$ is large. To obtain stronger lemmas we need, with the notations of the proof of Lemma \[3.5\], to take a different Cartier divisor $D$ on $T$. We take $y-2$ points on $A$, $y-2$ points on $C$, $x-3$ points on $B$, and the intersection of $T$ with the tangent planes to $Q$ at the two points of $Sing(T)$. In this way we may reduce to a case with a net of type $(a-2,b-1;t-1)$ if $t \geq 2$, of type $(a-2,b-2;b-3)$ if $t = 1$ and $b \geq 2$, while we have a complete victory if $b = 1$ or $b = 2$ and $t = 1$. With this trick we have the following lemmas.
\[3.8\] Fix integers $x, y, a, b, t, u, w$ with $a > 0, 0 < t \leq b, x \geq
1+2[(a+1)/2], y \geq b+1$ and $3((a-1)b+t) + 3u \leq h^{0}(Q,(\Omega|Q)(x,y)) \leq
3((a-1)b+t) + 3w$. Let $S$ be a net of type $(a,b;t)$ on a smooth quadric $Q$. Let $M$ (resp. $N$) be the union of $S$ and $u$ (resp. $w$) general points of $Q$. Then we have $h^{1}(Q,(\Omega|Q)(x,y)\otimes I_{M}) = h^{0}(Q,(\Omega|Q)(x,y)\otimes I_{N})
= 0$.
\[3.9\] Fix integers $x, y, a, b, t, u, w$ with $a > 0, 0 < t \leq b, x \geq
-1+2[(a+1)/2], y \geq b-1$ and $3((a-1)b+t) + 3u \leq h^{0}(Q,(T\Pi|Q)(x,y)) \leq
3((a-1)b+t) + 3w$. Let $S$ be a net of type $(a,b;t)$ on a smooth quadric $Q$. Let $M$ (resp. $N$) be the union of $S$ and $u$ (resp. $w$) general points of $Q$. Then we have $h^{1}(Q,(T\Pi|Q)(x,y)\otimes I_{M}) =
h^{0}(Q,(T\Pi|Q)(x,y)\otimes I_{N}) = 0$.
The inductive assumptions {#4}
=========================
We will try to copy [@BE1] with a few essential modifications. In this section will state and prove (modulo the numerical lemmas given in section \[2\] and the starting cases given in section \[6\]) two inductive hypotheses $HO(m)$ and $RO(m,g)$, $m \geq 2$, $g \geq 0$, related to $\Omega$ and two inductive hypothesis $HT(m)$ and $RT(m,g)$) related to $T\Pi$. In section \[5\] we will show how to use this work to prove Theorem \[0.1\]. In section \[7\] we will use both sections for the proof of Theorem \[0.3\]. The assertions $HO(m)$ and $HT(m)$ are related to curves of genus $\gamma$ and degree $\gamma+3$ for some $\gamma$ and with critical value $m$. The assertions $RO(m,g)$ and $RT(m,g)$ are related to curves of a fixed genus $g$ and degree $\geq g+3$ with critical value $m$. The definition of assertion $HO(m)$ (resp. $RO(m,g)$) will depend on the size of $\alpha(m)^{\ast}$ and $\delta(m+2)^{\ast} - \delta(m)^{\ast}$ (resp. $a(m,g)^{\ast}$ and $d(m+2,g)^{\ast} - d(m,g)^{\ast}$).
\[4.1\] We define the following assertion $HO(m), m \geq 0$, distinguishing two cases according to the value of $\alpha(m)^{\ast}$:
$HO(m), \; m \geq 2, \; 0 \leq \alpha(m)^{\ast} \leq \delta(m+2)^{\ast} - \delta(m)^{\ast} - 2$: There exists $(Y,Q,D,D',S)$ such that:
1. $Y\in H(\delta(m)^{\ast},\delta(m)^{\ast}-3)$;
2. $Q$ is a smooth quadric intersecting transversally $Y$; $D$ and $D'$ are lines of type $(1,0)$ on $Q$ which are 2-secant to $Y$; $S \subset D\setminus (D\cap Y), card(S) =
\alpha(m)^{\ast}$; for every $P\in S$ the line $R(P)$ of type $(0,1)$ on $Q$ containing $P$ intersects $Y$ but not $Y\cap D'$;
3. we have $h^{1}({\mbox{\bf P}^{3}},\Omega(m+1)\otimes I_{Y\cup S}) = 0$.
$HO(m), m \geq 2, w(\delta(m+2)^{\ast} - \delta(m)^{\ast} - w-2) \leq \alpha(m)^{\ast} \leq (w+1)(
\delta(m+2)^{\ast} - \delta(m)^{\ast} - w-4)$ for some integer $w$ with $1 \leq w \leq 18$: There exists $(Y,Q,D,D',R(j) (1\leq j\leq w+2),D(k) (1 \leq k \leq \delta(m+2)^{\ast} -
\delta(m)^{\ast} - w-4), S(i) (1\leq i\leq w+1))$ such that:
1. $Y\in H(\delta(m)^{\ast},\delta(m)^{\ast}-3)$;
2. $Q$ is a smooth quadric intersecting transversally $Y$; $D$ and $D'$ are lines of type $(1,0)$ on $Q$ which are 2-secant to $Y$; $R(j), 1 \leq j
\leq w+3$, are lines of type $(0,1)$ on $Q$ with $R(j)\cap Y = \O$ for every $j$; $D(k), 1 \leq k \leq
d(m+2)^{\ast} - d(m)^{\ast} - w-4$ are lines of type $(1,0)$ on $Q$ with $D(k)\cap Y \neq
\O$ if and only if $1 \leq k \leq \alpha(m)^{\ast} - w(\delta(m+2)^{\ast} - \delta(m)^{\ast} - w-4)$; we have $S(i) \subset R(i), 1 \leq
i \leq w+1, card(S(i)) = \delta(m+2)^{\ast} - \delta(m)^{\ast} - w-4$ for $1 \leq i \leq w, card(S(w+1)) =
\alpha(m)^{\ast} - w(\delta(m+2)^{\ast} - \delta(m)^{\ast} - w-4)$ and for every $P\in S(w+1)$ the line $D(P)$ of type $(1,0)$ containing $P$ intersects $Y$ but not $Y\cap (D\cup D')$;
3. set $S:= \cup _{1\leq i\leq w+1}S(i)$; we have $h^{1}({\mbox{\bf P}^{3}},\Omega(m+1)\otimes I_{Y\cup S}) = 0$.
In the definition of $HO(m)$ we impose that there is no coincidence or meeting relation except the prescribed ones; for instance we impose that $Y\cap S(i) = \O$ for every $i$ and, with the notations of condition (2) for $HO(m)$, the lines $D(P)$ or $R(P)$ are not 2-secant to $Y$, while the line $D(A), A\in S(1)$, does not intersect $Y$ unless $D(A) = D(P)$ for some $P\in S(w+1)$. Note that condition (3) in $HO(m)$ is equivalent to the bijectivity of the restriction map $r_{Y\cup S,\Omega(m+1)}$. Call $HO(m)"$ the existence of a pair $(Y,S)$ with $Y\in H(\delta(m)^{\ast},\delta(m)^{\ast}-3)$, $card(S) = \alpha(m)^{\ast}$ and $r_{Y\cup S,\Omega(m+1)}$ bijective.
\[4.2\] In a similar way we define the following assertion $HT(m), m \geq 2$, which is related to the last part of the minimal free resolution, i.e. the part controlled by $T\Pi$. Again, we distinguish two cases according to the value of $\alpha'(m)^{\ast}$:
$HT(m), \; m \geq 2, \; 0 \leq \alpha'(m)^{\ast} \leq \delta'(m+2)^{\ast} - \delta'(m)^{\ast} - 2$: There exists $(Y,Q,D,D',S)$ such that:
1. $Y\in H(\delta'(m)^{\ast},\delta'(m)^{\ast}-3)$;
2. $Q$ is a smooth quadric intersecting transversally $Y$; $D$ and $D'$ are lines of type $(1,0)$ on $Q$ which are 2-secant to $Y$; $S \subset D\setminus (D\cap Y)$, $card(S) =
\alpha'(m)^{\ast}$; for every $P\in S$ the line $R(P)$ of type $(0,1)$ on $Q$ containing $P$ intersects $Y$ but not $Y\cap D'$;
3. we have $h^{1}({\mbox{\bf P}^{3}},T\Pi(m-2)\otimes I_{Y\cup S}) = 0$.
$HT(m), m \geq 2, w(\delta'(m+2)^{\ast} - \delta'(m)^{\ast} - w-2) \leq \alpha'(m)^{\ast} \leq (w+1)( \delta'(m+2)^{\ast} -
\delta'(m)^{\ast} - w-4)$ for some integer $w$ with $1 \leq w \leq 18$: There exists $(Y,Q,D,D',R(j) (1\leq j\leq w+2),D(k) (1 \leq k \leq \delta'(m+2)^{\ast} - \delta'(m)^{\ast} - w-4), S(i)
(1\leq i\leq w+1))$ such that:
1. $Y\in H(\delta'(m)^{\ast},\delta'(m)^{\ast}-3)$;
2. $Q$ is a smooth quadric intersecting transversally $Y$; $D$ and $D'$ are lines of type $(1,0)$ on $Q$ which are 2-secant to $Y$; $R(j), 1 \leq j
\leq w+3$, are lines of type $(0,1)$ on $Q$ with $R(j)\cap Y = \O$ for every $j$; $D(k), 1 \leq k \leq
\delta'(m+2)^{\ast} - \delta'(m)^{\ast} - w-4$ are lines of type $(1,0)$ on $Q$ with $D(k)\cap Y \neq \O$ if and only if $1 \leq k \leq \alpha'(m)^{\ast} - w(\delta'(m+2)^{\ast} - \delta'(m)^{\ast} -
w-4)$; we have $S(i) \subset
R(i), 1 \leq i \leq w+1, card(S(i)) = \delta'(m+2)^{\ast} - \delta'(m)^{\ast} - w-4$ for $1 \leq i \leq w,
card(S(w+1)) = \alpha'(m)^{\ast} - w(\delta'(m+2)^{\ast} - \delta'(m)^{\ast} - w-4)$ and for every $P\in S(w+1)$ the line $D(P)$ of type $(1,0)$ containing $P$ intersects $Y$ but not $Y\cap (D\cup D')$;
3. set $S:= \cup _{1\leq i\leq w+1}S(i)$; we have $h^{1}({\mbox{\bf P}^{3}},T\Pi(m-2)\otimes I_{Y\cup S}) = 0$.
In the definition of $HT(m)$ we impose that there is no coincidence or meeting relation except the prescribed ones; for instance we impose that $Y\cap S(i) = \O$ for every $i$ and, with the notations of condition (2) for $HT(m)$, the lines $D(P)$ or $R(P)$ are not 2-secant to $Y$, while the line $D(A),
A\in S(1)$, does not intersect $Y$ unless $D(A) = D(P)$ for some $P\in S(w+1)$. Note that condition (3) in $HO(m)$ is equivalent to the bijectivity of the restriction map $r_{Y\cup S,T\Pi(m-2)}$. Call $HO(m)"$ the existence of a pair $(Y,S)$ with $Y\in H(\delta'(m)^{\ast},\delta'(m)^{\ast}-3), card(S) = a'(m)^{\ast} and r_{Y\cup S},T\Pi(m-2)$ bijective.
\[4.3\] For all integers $g \geq 0$ and $m \geq max \{ m_{0}(g)-2,70 \}$ we define the assertion $RO(m,g)$ in the following way.
$RO(m,g), g \geq 0, m \geq max \{ m_{0}(g)-2,70 \}, z(d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1) \leq
a(m,g)^{\ast} \leq (z+1)(d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1)$ for some integer $z$ with $0 \leq z
< d(m+2,g)^{\ast} - d(m,g)^{\ast} - z$: There exists $(Y,Q,D(i) (1 \leq i \leq z+1), R(j) (1 \leq
j \leq d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1), S(i) (1 \leq i \leq z+1))$ such that:
1. $Y\in H(d(m,g)^{\ast},g)$;
2. $Q$ is a smooth quadric intersecting transversally $Y$; $D(i), 1 \leq i \leq
z+1$, are lines of type $(1,0)$ on $Q$ which are 1-secant to $Y$; $R(j), 1 \leq j \leq
d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1$ are lines of type $(0,1)$ on $Q$ with $R(j)\cap Y \neq \O$ if and only if $j > z(d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1)$ (and in this case 1-secant to $Y$ and not intersecting any $D(i)\cap Y); S(i) \subset D(i)$ for every $i$; $card(S(i)) = d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1$ for $1 \leq i \leq z; card(S(z+1)) =
a(m,g)^{\ast} - z(d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1)$; we have $S(i) = {D(i)\cap R(j)}1 \leq j
\leq d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1$ for $1 \leq i \leq z$ and $S(z+1) =
\{ D(z+1)\cap R(j),
z(d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1) < j \leq
(z+1)(d(m+2,g)^{\ast} - d(m,g)^{\ast} - z - 1) \}$;
3. set $S:= \cup_{1\leq i\leq z+1}S(i)$; we have $h^{1}({\mbox{\bf P}^{3}},\Omega(m+1)\otimes I_{Y\cup S}) = 0$.
Note that condition (3) in the $RO(m,g)$ is equivalent to the bijectivity of the restriction map $r_{Y\cup S,\Omega(m+1)}$. Call $RO(m,g)"$ the existence of a pair $(Y,S)$ with $ Y\in H(d(m,g)^{\ast},g), card(S) = a(m,g)^{\ast}$ and $r_{Y\cup S,\Omega(m+1)}$ bijective.
\[4.4\] In the same way we define the corresponding assertion $RT(m,g)$ for the last part of the minimal free resolution, i.e. the part controlled by $T\Pi$. For all integers $m \geq max \{m_{0}(g)-2,70 \}, g \geq 0$ we define the assertion $RT(m,g)$ in the following way. $RT(m,g), m \geq max \{ m_{0}(g)-2,70 \}, g \geq 0, z(d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1) \leq
a'(m,g)^{\ast} \leq (z+1)(d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1)$ for some integer $z$ with $0
\leq z < d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z$: There exists $(Y,Q,D(i) (1 \leq i \leq z+1),
R(j) (1 \leq j \leq d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1), S(i) (1 \leq i \leq
z+1))$ such that:
1. $Y\in H(d'(m,g)^{\ast},g)$;
2. $Q$ is a smooth quadric intersecting transversally $Y$; $D(i), 1 \leq i \leq
z+1$, are lines of type $(1,0)$ on $Q$ which are 1-secant to $Y$; $R(j), 1 \leq j \leq
d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1$ are lines of type $(0,1)$ on $Q$ with $R(j)\cap Y \neq \O$ if and only if $j > z(d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1)$ (and in this case 1-secant to $Y$ and not intersecting any $D(i)\cap Y); S(i) \subset D(i)$ for every $i$; $card(S(i)) = d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1$ for $1 \leq i \leq z; card(S(z+1)) =
a'(m,g)^{\ast} - z(d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1)$; we have $S(i) = {D(i)\cap R(j)}1 \leq
j \leq d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1$ for $1 \leq i \leq z$ and $S(z+1) = \{ D(z+1)\cap R(j),
z(d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1) < j \leq (z+1)(d'(m+2,g)^{\ast} - d'(m,g)^{\ast} - z - 1) \}$;
3. set $S:= \cup _{1\leq i\leq z+1 } S(i)$; we have $h^{1}({\mbox{\bf P}^{3}},T\Pi(m-2)\otimes I_{Y\cup S}) = 0$.
Note that condition (3) in the $RT(m,g)$ is equivalent to the bijectivity of the restriction map $r_{Y\cup S,T\Pi(m-2)}$. Call $RT(m,g)"$ the existence of a pair $(Y,S)$ with $Y\in H(d'(m,g)^{\ast},g), card(S) = a'(m,g)^{\ast}$ and $r_{Y\cup S,T\Pi(m-2)}$ bijective.
\[4.5\] For every integer $m \geq 51 \quad HO(m-2)$ implies $HO(m)$.
[Proof]{} First, we will check $HO(m)"$. We distinguish two cases, according to the value of $\alpha(m-2)^{\ast}$.
- $0 \leq \alpha(m-2)^{\ast} \leq d(m)^{\ast} - d(m-2)^{\ast} - 2$. Take $(Y,Q,D,D',S)$ satisfying $HO(m-2)$. Let $X$ be the union of $Y, D, D', d(m)^{\ast} - d(m-2)^{\ast} - 2$ lines of type $(0,1)$ containing the lines $D(P)$ with $P\in S$, and the nilpotents $\chi (P), P\in S$. Note that $Res_{Q}(X) = Y\cup S$. Hence by Horace Lemma \[1.3\] and $HO(m-2)$ to check the bijectivity of $r_{X\cup S",\Omega(m+1)}$ for a suitable $S" \subset Q$ with $card(S") =
\alpha(m)^{\ast}$, it is sufficient to check that, calling $S'$ the subset of $Y\cap Q$ not on any line $D, D', D(k), 1 \leq k \leq d(m)^{\ast} - d(m-2)^{\ast} - w-4$, or $R(j), 1
\leq j \leq w+2$, we have $h^{0}(Q,(\Omega|Q)(m-1,m+3+d(m-2)^{\ast}-d(m)^{\ast})\otimes I_{S"\cup S'}) =
h^{1}(Q,(\Omega|Q)(m-1,m+3+d(m-2)^{\ast}-d(m)^{\ast})\otimes I_{S"\cup S'}) = 0$. This is true by Lemma \[3.5\] and the equality $h^{0}(Q,(\Omega|Q)(m-1,m+3+d(m-2)^{\ast}-d(m)^{\ast})) = 3(card(S')+\alpha(m)^{\ast})$ which follows subtracting eq. \[e11\] for $m$ and for $m-2$ and from the equalities $3\alpha(x)^{\ast} - 3\alpha(x) - b(x) = (3m-1)(d(x) - d(x)^{\ast}), x = m, m-2$ (eq. \[e15\]).
- there is an integer $w$ with $1 \leq w \leq 20$ and $w(d(m)^{\ast} - d(m-2)^{\ast} -
w-2) \leq \alpha(m-2)^{\ast} \leq (w+1)( d(m)^{\ast} - d(m-2)^{\ast} - w-4)$. Take $(Y,Q,D,D',R(j)
(1\leq j\leq w+2),D(k) (1 \leq k \leq d(m)^{\ast} - d(m-2)^{\ast} - w-4), S(i)
(1\leq i\leq w+1))$ satisfying $HO(m-2)$. Let $X$ be the union of $Y, D, D'$, the lines $R(j), 1 \leq j
\leq w+2$, the lines $D(k), 1 \leq k \leq d(m)^{\ast} - d(m-2)^{\ast} - w-4$, and the nilpotents $\chi (P)$ with $P\in S:= \cup _{1\leq i\leq w+1}S(i)$. We have $X\in H(d(m+2)^{\ast},d(m+2)^{\ast}-3)$ by \[1.4\]. Note that $Res_{Q}(X) = Y\cup S$. Hence by Horace Lemma \[1.3\] and $HO(m-2)$ to check the bijectivity of $r_{X\cup S",\Omega(m+1)}$ for a suitable $S" \subset Q$ with $card(S") = \alpha(m)^{\ast}$, it is sufficient to check that, calling $S'$ the subset of $Y\cap Q$ not on any line $D, D', D(k), 1 \leq k \leq d(m)^{\ast} - d(m-2)^{\ast} - w-4, or R(j), 1 \leq j
\leq w+2$, we have $h^{0}(Q,(\Omega|Q)(m-w-3,m+w+4+d(m-2)^{\ast}-d(m)^{\ast})\otimes I_{S"\cup S'}) =
h^{1}(Q,(\Omega|Q)(m-w-3,m+w+4+d(m-2)^{\ast}-d(m)^{\ast})\otimes I_{S"\cup S'}) = 0$. This is true by Lemma 3.5 and the equality $h^{0}(Q,(\Omega|Q)(m-w-3,m+w+4+d(m-2)^{\ast}-d(m)^{\ast})) =
3(card(S')+\alpha(m)^{\ast})$ which follows from eq. \[e11\] and \[e15\] for $m-2$ and $m$ as in case (i).
Note that we may deform $X$ to a reducible and reduced curve $X'\in
\linebreak
H(d(m)^{\ast},d(m)^{\ast}-3)$ containing $Y$ and transversal to $Q$. Now to prove $HO(m)$ we need to check that in both cases (i) and (ii) for $HO(m-2)$ and in both cases a priori possible for $HO(m)$ according to the value of $a(m)^{\ast}$ we may take a subset $S"$ (notations of cases (i) and (ii)) satisfying the condition (2) of $HO(m)$. We use Lemma \[3.5\] and that we may deform 2-secant lines very easily; much more than needed is contained in [@BE1], §8 (see in particular [@BE1], Lemma 8.2).
For every integer $g \geq 0$, let $m_{o}(g)$ be the critical value of the pair $(g+3,g)$.
\[4.6\] By Lemmas \[2.8\], \[2.9\], \[2.10\], \[2.11\] and \[2.13\] if $m
\geq max \{ m_{0}(g)-2,70 \}$ we may take $z \leq 20$ in the definitions of $RO(m,g)$ and $RT(m,g)$.
\[4.7\] For every integer $g \geq 0$ and every integer $m \geq max \{ m_{0}(g)-2,70 \}
RO(m-2,g)$ implies $RO(m,g)$.
[Proof]{} By Lemma \[2.8\] there is a minimal integer $z$ with $0 \leq z \leq 19$ such that $z(d(m,g)^{\ast} - d(m-2,g)^{\ast} - z - 1) \leq a(m-2,g)^{\ast} \leq (z+1)(d(m,g)^{\ast} - d(m-2,g)^{\ast} -
z - 1)$. Take $(Y,Q,D(i) (1 \leq i \leq z+1), R(j) (1 \leq j \leq d(m,g)^{\ast} - d(m-2,g)^{\ast} -
z - 1), S(i) (1 \leq i \leq z+1))$ satisfying $RO(m-2,g)$. Let $X$ be the union of $Y, D(i), 1 \leq i \leq z+1, R(j), 1 \leq j \leq d(m,g)^{\ast} - d(m-2,g)^{\ast} - z - 1$ and the nilpotents $c(P)$ with $P\in \cup 1\leq i\leq z+1S(i)$. A trivial modification of the proof of Lemma \[4.5\] concludes the proof.
\[4.8\] If $m_{o}(g) \geq 70, HO(m_{o}(g)-1)$ implies $RO(m_{o}(g)+1,g)$.
[Proof]{} As in the proofs of Lemmas \[4.5\] and \[4.7\] we will check $RO(m_{o}(g)+1,g)"$. By Lemma \[2.11\] we have $a(m-2)^{\ast} \leq 20(d(m,g)^{\ast} - d(m-2)^{\ast} -
23)$. Hence we may make the construction of Lemma \[4.5\] using $d(m,g)^{\ast} -
d(m-2)^{\ast}$ instead of $d(m,g)^{\ast} - d(m,g)^{\ast}$.
\[4.9\] If $m_{o}(g) \geq 70, HO(m_{o}(g)-2)$ implies $RO(m_{o}(g),g)$.
[Proof]{} Just copy the proof of Lemma \[4.8\].
In the same way we obtain the following results for $T\Pi$.
\[4.10\] For every integer $m \geq 51$ $HT(m-2)$ implies $HT(m)$.
\[4.11\] For every integer $g \geq 0$ and every integer $m \geq max \{ m_{0}(g)-2,70 \}
RT(m-2,g)$ implies $RT(m,g)$.
\[4.12\] If $m_{o}(g) \geq 70, HT(m_{o}(g)-1)$ implies $RT(m_{o}(g)+1,g)$.
\[4.13\] If $m_{o}(g) \geq 70, HT(m_{o}(g)-2)$ implies $RT(m_{o}(g),g)$.
The last part of the proof of \[0.1\] {#5}
=====================================
In this section we will conclude the proof of Theorem \[0.1\] modulo the initial cases with critical value $\leq 50$ which will be done in section \[6\]. We start with a pair $(d,g)$ with $d \geq g+3 \geq 3$ and $g \geq \delta(70)" - 3$. Hence the pair $(d,g)$ has critical value $m \geq
70$. We assume that $RO(m-2,g)$ is true and obtain the result for $\Omega$. In the same way assuming $RT(m-2,g)$ we will get the result for $T\Pi$. First assume $d = d(m,g)^{\ast}$. By Lemma \[4.7\] a general $Y\in H(d,g)$ has $r_{Y,\Omega(m+1)}$ surjective. Now assume $d < d(m,g)^{\ast}$. The proofs of Lemmas \[4.5\] and \[4.7\] gave a reducible and in general not reducible curve $Y \in H(d(m,g)^{\ast},g)$ with $r_{Y,\Omega(m+1)}$ surjective. By \[1.4\] the curve $Y$ obtained in this way has a flat deformation to a family of curves whose general element is a reducible but reduced curve $Y"\in H(d(m,g)^{\ast},g)$ such that there is a reducible $Y' \subset Y"$ with $Y'\in H(d,g)$. By semicontinuity we may assume $r_{Y",\Omega(m+1)}$ surjective. It is easy to see that this implies the surjectivity of $r_{Y',\Omega(m+1)}$. Alternatively, for large $m$ (e.g. if $20(d - d(m-2,g)^{\ast} - 22) \geq a(m-2,g)^{\ast}$ (case $m \geq
m_{o}(g)+2)$ or $20(d - \delta (m-2)^{\ast} - 22) \geq \alpha (m-2,g)^{\ast}$ (case $m =
m_{o}(g)$ or $m = m_{o}(g)+1)$ or if $(3m-4)(d(m,g) - d) - 3a(m,g) \geq a(m-
2,g)^{\ast})$ the proofs of Lemmas \[4.5\], \[4.7\] and \[4.8\] give the existence of a suitable curve $Z \subseteq Y"$ with $r_{Z,\Omega(m+1)}$ surjective. Now assume $d(m,g)^{\ast} < d \leq d(m,g)$. Now assume $d > d(m,g)$. Since $20(d - d(m-2,g)^{\ast} - 22) \geq a(m-2,g)^{\ast}$ (case $m \geq m_{o}(g)+2)$ or $20(d - \delta (m-2)^{\ast} - 22) \geq \alpha (m-2,g)^{\ast}$ (case $m = m_{o}(g)$ or $m
= m_{o}(g)+1)$ it is easy to modify the proofs of Lemmas \[4.5\], \[4.7\] and \[4.8\] to obtain a reducible and in general (i.e. if $a(m-2,g)^{\ast} \neq
0)$ unreduced curve $Y\in H(d,g)$ with $r_{Y,\Omega(m+1)}$ injective; in the smooth quadric $Q$ we add $d - d(m-2,g)^{\ast}$ (case $m \geq m_{o}(g)+2)$ or $4d
- \delta (m-2)^{\ast}$ (case $m = m_{o}(g)$ or $m = m_{o}(g)+1)$ suitable lines, no point, but add again $a(m-2,g)^{\ast}$ or $\alpha (m-2)^{\ast}$ suitable nilpotents. The proofs just sketched give that a general $X\in H(d,g)$ has $r_{X,T\Pi(m-
2)}$ with maximal rank. Hence a general $X \in H(d,g)$ has the expected minimal free resolution, concluding the proof of Theorem \[0.1\] (modulo to start the inductive machine).
\[5.1\] The proof shows that the statement of Theorem \[0.1\] is true for all pairs $(d,g)$ with critical value $m \geq 70$. Hence in Theorem \[0.1\] we may take $g_{0}:= \delta (70)" - 3 = 899$.
Starting the inductive machine {#6}
==============================
In this section we will construct the subschemes of $\Pi$ needed to start the inductive machine given in section \[4\] for Theorem \[0.1\]. We will do the initial cases for Theorem \[0.2\] at the end of the paper, i.e. in section \[8\]. Unless otherwise stated, $Q$ and $Q'$ will be smooth quadrics. If the integers $\alpha(m-2)^{\ast}, \alpha(m)^{\ast},
\alpha'(m-2)^{\ast}$ and $\alpha'(m)^{\ast}$ are “not bad” the following lemmas \[6.1\],..., \[6.6\] give the inductive step. In particular, these integers are “not bad” if $m$ is large by \[2.6\] and \[2.7\], but they are “not bad” also in a few cases with m very low.
\[6.1\] Fix an integer $m \geq 4$. Assume $\alpha(m-2)^{\ast} \leq \alpha(m)^{\ast}
\leq \alpha(m-2)^{\ast} + 2$ if $\alpha(m-2)^{\ast} \leq \delta(m)^{\ast} -
\delta(m-2)^{\ast} - 2$ and $\alpha(m-2)^{\ast} \leq \alpha(m)^{\ast} \leq
\alpha(m-2)^{\ast} + 2w + 2$ if in the definition of $HO(m-2)$ we introduced the integer $w > 0$. Assume $HO(m-2)$. Then, modulo the intersection with the quadric, i.e. modulo section \[3\], we will prove $HO(m)"$ in the following way. We fix data $(Y,Q,^{\ast\ast\ast},S'), card(S'):= \alpha(m-
2)^{\ast}$, satisfying $HO(m-2)$. To obtain $HO(m)$ we apply Horace Lemma \[1.3\] with respect to a general quadric $Q'$ intersecting transversally both $Y$ and the lines of $Q$ appearing in $(Y,Q,^{\ast\ast\ast})$. Hence each of these lines intersects $Q'$ in two points and we insert one or two or none points of intersection in such a way that the total number of added points is $\alpha(m)^{\ast} -\alpha(m-
2)^{\ast}$. Furthermore, we add in $Q'$ $\delta(m+2)^{\ast} -
\delta(m)^{\ast}$ suitable lines. The union $X\cup S$ of these lines and points with the configuration $Y\cup S'$ satisfies $HO(m)"$. In the same way we obtain the following result.
\[6.2\] Fix an integer $m \geq 4$. Assume $\alpha'(m-2)^{\ast} \leq
\alpha'(m)^{\ast} \leq \alpha'(m-2)^{\ast} + 2$ if $\alpha'(m-2)^{\ast} \leq
\delta'(m)^{\ast} - \delta'(m-2)^{\ast} - 2$ and $\alpha'(m-2)^{\ast} \leq
\alpha'(m)^{\ast} \leq \alpha'(m-2)^{\ast} + 2w + 2$ if in the definition of $HT(m-2)$ we introduced the integer $w > 0$. Assume $HT(m-2)$. Then $HT(m)"$ we can prove $HT(m)"$ if the intersection with a quadric gives no problem.To apply the inductive procedure we need $HO(m)$ and $HT(m)$, not just $HO(m)"$ or $HT(m)"$. Here we will see when we may modify the proofs of \[6.1\] and \[6.2\] to obtain $HO(m)$ and $HT(m)$.
\[6.3\] Fix an integer $m \geq 4$. Assume $\alpha(m-2)^{\ast} \leq \alpha(m)^{\ast}
\leq \alpha(m-2)^{\ast} + 2, \alpha(m-2)^{\ast} \leq \delta(m)^{\ast} -
\delta(m-2)^{\ast} - 2$ and $HO(m-2)$. Then, modulo the intersection with the quadric, we prove $HO(m)$ in the following way. To check that the configuration, $S$, of $\alpha(m)^{\ast}$ points constructed in $Q$ with $card(Q'\cap S) = \alpha(m)^{\ast} - \alpha(m-2)^{\ast}$ is allowable for $HO(m)$ it is sufficient to check that $\delta(m+2)^{\ast} -\delta(m)^{\ast}
\geq \delta(m)^{\ast} - \delta(m-2)^{\ast} + 2$. In the same way we obtain the corresponding result for $T\Pi$.
\[6.4\] Fix an integer $m \geq 4$. Assume $\alpha'(m-2)^{\ast} \leq
\alpha'(m)^{\ast} \leq \alpha'(m-2)^{\ast} + 2, \alpha'(m-2)^{\ast} \leq
\delta'(m)^{\ast} - \delta'(m-2)^{\ast} - 2$ and $HT(m-2)$. Then, modulo the intersection with the quadric, we can prove $HT(m)$.
\[6.5\] Fix an integer $m \geq 4$. Assume $\alpha(m)^{\ast} \leq \alpha(m-2)^{\ast}
\leq \alpha(m)^{\ast} + 2, \alpha(m-2)^{\ast} \leq \delta(m)^{\ast} -
\delta(m-2)^{\ast} - 2$ and $HO(m-2)$. Then, modulo the intersection with the quadric, we can prove $HO(m)$ in the following way. Take a solution $(Y,Q,D,D',S)$ of $HO(m-2)$ and a general quadric $Q'$ containing $\alpha(m-
2)^{\ast} - \alpha(m)^{\ast}$ of the points of $S$. In $Q'$ we may add a configuration, $T$, of lines and the nilpotents $\chi(P), P\in Q'\cap S$, so that the corresponding curve $X$ is in $H(\delta(m)^{\ast},\delta(m)^{\ast}-
3)$ and that for a suitable deformation $X"$ of $X$ $(X",Q,D,D',S\setminus(Q'\cap S))$ satisfies $HO(m)$; here we use that $\delta(m+2)^{\ast} - \delta(m)^{\ast} \geq \delta(m)^{\ast} - \delta(m-
2)^{\ast}$ which is true (see the proof of \[6.3\]). $\diamondsuit$
In the same way we obtain the corresponding result for $T\Pi$.
\[6.6\] Fix an integer $m \geq 4$. Assume $\alpha'(m)^{\ast} \leq \alpha'(m-
2)^{\ast} \leq \alpha'(m)^{\ast} + 2, \alpha'(m-2)^{\ast} \leq
\delta'(m)^{\ast} - \delta'(m-2)^{\ast} - 2$ and $HT(m-2)$. Then, modulo the intersection with the quadric, we can prove $HT(m)$.
Now we will use the constructions outlined in \[6.1\],....,\[6.6\] to check most of the initial inductive cases needed to start the proof of \[0.1\]. In a few cases we will needed more work, just for numerical resasons.
\[6.7\] $HO(2)$ is true because any degree 4 elliptic curve $X \subset \Pi$ is the complete intersection of 2 quadrics and hence its homogeneous ideal is generated by 2 quadrics.
We define the following assertions.
O{3}: There exist a smooth quadric $Q$ and a union $Z$ of 15 points of $\Pi$ with $card(Z\cap Q) = 10$ and $h^{0}(\Omega(5)\otimes I_{Z})
= h^{1}(\Omega(5)\otimes I_{Z}) = 0$.
O{4}: There exist a smooth quadric $Q$ and the union $Z \subset
\Pi$ of 28 points with $card(Z\cap Q) = 16$, $h^{0}(\Omega(5)\otimes I_{Z})
= h^{1}(\Omega(5)\otimes I_{Z}) = 0$ and such that $Z\cap Q$ is given by the following configuration. In $Q$ we take 3 general smooth curves $E, E',
E"$ of type $(2,1)$ and 3 general lines $F, F', F"$ of type $(0,1)$. Let $Z\cap Q$ be the union of $E'\cap E"$ and $(E'\cup E ")\cap (F\cup F'\cup
F")$.
\[6.8\] The proof given in [@B] for a general $Z \subset \Pi$ with $card(Z) =
15$ gives O{3}.
\[6.9\] The proof of the existence of a union $Z \subset \Pi$ of 28 points such that $h^{0}(\Omega(5)\otimes I_{Z}) = h^{1}(\Omega(5)\otimes I_{Z}) = 0$ given in [@H2] proves O{4}.
\[6.10\] $HO(6)$ is true.
[Proof]{} We start with a pair $(Q,Z)$ satisfying O{4}. In $Q$ we take 3 smooth curves $E, E', E"$ of type $(2,1)$ and 3 general lines $F, F', F"$ of type $(0,1)$ such that $Q\cap Z$ is the union of $E'\cap E"$ and $(E'\cup
E")\cap (F\cup F'\cup F")$. Set $S:= Z\setminus(Q\cap Z)$. Let $X$ be the union of $E, E', E", F, F', F"$ and the nilpotents $\chi(P)$ with $P\in A$. We have $X\in H(12,9)$ by \[1.4\]. Since $Res_{Q}(X\cup S) = Z$, by Horace Lemma \[1.3\] and O{4} we obtain that $(X,S)$ satisfies $HO(6)"$. Since $\alpha(6)^{\ast} = 2$ is very small, $HO(6)"$ implies $HO(6)$.
\[6.11\] $HO(5)$ is true.
[Proof]{} Take $(Q,Z)$ satisfying O{3} (Remark \[6.2\]). Take a general curve $U$ of type $(2,3)$ on $Q$ and a general curve $F$of type $(3,2)$ on $Q$. By the generality of $U$ and $F$ we may assume $Z\cap Q \subseteq U\cap F$. Set $Y:=
U\cup F$. Let $X$be the union of $Y$ and the nilpotents $\chi(P)$ with $P\in
Z\cap Q$. Set $S:= Z\setminus(Z\cap Q)$. Since $Res_{Q}(X\cup S) = Z$, by Horace Lemma \[1.3\] and O{3} we obtain $h^{0}(\Omega(6)\otimes I_{X\cup
S}) = h^{1}(\Omega(6)\otimes I_{X\cup S}) = 0$. Note that $Z\setminus(Z\cap
Q)$ is contained in a smooth quadric $Q'$. Furthermore, since $\delta(7)^{\ast} = 15 = \delta(6)^{\ast} + \alpha(6)^{\ast} + 3$, we may easily find $Z$ and Q’ with $Z$ satisfying O{3} and such that there are lines $D, D'$ on $Q$ with $(X,Q',D,D',Z\setminus (Z \cap Q))$ satisfying $HO(5)$; indeed it is sufficient to take $Q'$ containing one 2-secant line to $X$and intersecting transversally $X$.
\[6.12\] $HO(7)$ is true.
[Proof]{} Since $\alpha(5)^{\ast} = 6, \delta(5)^{\ast} = 10$ and $\delta(7)^{\ast} =
16$, the proof of Lemma \[4.5\] gives the existence of a pair $(X,S)$ satisfying $HO(7)"$. We have $\delta(9)^{\ast} = \delta(7)^{\ast} + 24$ and $\alpha(7)^{\ast} = 16$. Hence the same proof gives $HO(7)$ with respect to the integer $w = 2$ if we take a smooth quadric containing two lines of type $(1,0)$ and one line of type $(0,1)$ which are 2-secant to $X$.
\[6.13\] Since $\delta(11)^{\ast} = 33, \delta(9)^{\ast} = 24, \delta(7)^{\ast} = 15$ and $\alpha(9)^{\ast} = 10 \leq \delta(11)^{\ast} - \delta(9)^{\ast} - 2$, the general proof that $HO(m-2)$ implies $H(m)$ for large $m$ (Lemma \[4.5\]) proves $HO(9)$.
\[6.14\] $HO(8)$ is true: Take $(Y,Q)$ satisfying $HO(6)$. We obtain a solution of $HO(8)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$, both 2-secant to $Y$, and 2 lines of type $(0,1)$, both 1-secant to $Y$, plus 2 nilpotents at suitable singular points of $T$ and 12 points on the singular points of a configuration of lines of type $(6,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega Q)(7,7)$ we use Lemma \[3.6\] for the integers $x = y = 7, a = 2, b = 6$ and $t = 6$; note that $12 = (a-1)b+t$.
\[6.15\] $HO(10)$ is true: Take $(Y,Q)$ satisfying $HO(8)$. We obtain a solution of $HO(10)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 5 lines of type $(0,1)$ plus 12 nilpotents at suitable singular points of $T$ and 20 points on the singular points of a configuration of lines of type $(7,4)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 4 of the lines of type $(0,1)$ are 1-secant to $Y$.To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(6,5)$ we use Lemma \[3.6\] for the integers $x = 6, y = 7, a = 6, b = 4$ and $t = 2$; note that $20 = (a-
1)b+t$ and that $t$ is the number of 2-secant lines added in the next inductive step \[6.16\].
\[6.16\] $HO(12)$ is true: Take $(Y,Q)$ satisfying $HO(12)$. We obtain a solution of $HO(12)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 7 lines of type $(0,1)$ plus 20 nilpotents at suitable singular points of $T$ ; among the lines of $T$ 3 of the ones of type $(1,0)$ are 2-secant to $Y$ and all the other lines are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(9,6)$ we use Lemma \[3.1\].
\[6.17\] $HO(14)$ is true: Take $(Y,Q)$ satisfying $HO(12)$. We obtain a solution of $HO(14)$ adding in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 6 lines of type $(0,1)$ plus 20 nilpotents at suitable singular points of $T$ and 10 points on a configuration of lines of type $(8,3)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 5 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(10,9)$ we use Lemma \[3.6\] for the integers $x =
9, y = 10, a = 2, b = 7$ and $t = 3$.
\[6.18\] $HO(16)$ is true: Take $(Y,Q)$ satisfying $HO(14)$. We obtain a solution of $HO(16)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 8 lines of type $(0,1)$ plus 10 nilpotents at suitable singular points of $T$ and 28 points on the singular points of a configuration of lines of type $(11,4)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other line of type $(1,0)$ is 1-secant to $Y$ and 5 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(14,9)$ we use Lemma \[3.6\] for the integers $x = 9, y = 14, a = 3, b = 11$ and $t = 6$.
\[6.19\] $HO(18)$ is true: Take $(Y,Q)$ satisfying $HO(16)$. We obtain a solution of $HO(18)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 11 lines of type $(0,1)$ plus 28 nilpotents at suitable singular points of $Y\cup T$ and 6 points on the singular points of a configuration of lines of type $(15,3)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 8 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(15,8)$ we use Lemma \[3.6\].
\[6.20\] $HO(20)$ is true: Take $(Y,Q)$ satisfying $HO(18)$. We obtain a solution of $HO(20)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 11 lines of type $(0,1)$ and 40 points on the singular points of a configuration of lines of type $(12,5)$; among the lines of $T$, the 2 ones of type $(1,0)$ are 2-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(19,10)$ we use Lemma \[3.6\].
\[6.21\] $HO(22)$ is true: Take $(Y,Q)$ satisfying $HO(20)$. We obtain a solution of $HO(22)$ adding in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 12 lines of type $(0,1)$ plus 40 nilpotents at suitable singular points of $Y\cup T$ and 26 points on the singular points of a configuration of lines of type ($15,3)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 7 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(18,12)$ we use Lemma \[3.6\].
\[6.22\] $HO(24)$ is true: Take $(Y,Q)$ satisfying $HO(22)$. We obtain a solution of $HO(24)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 15 lines of type $(0,1)$ plus 26 nilpotents at suitable singular points of $T$ and 16 points on the singular points of a configuration of lines of type $(17,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 10 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(22,10)$ we use Lemma \[3.6\].
\[6.23\] $HO(26)$ is true: Take $(Y,Q)$ satisfying $HO(24)$. We obtain a solution of $HO(26)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 17 lines of type $(0,1)$ plus 16 nilpotents at suitable singular points of $T$ and 12 points on the singular points of a configuration of lines of type (18,2); among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 16 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Lemma \[3.6\] and Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(25,10)$.
\[6.24\] $HO(28)$ is true: Take $(Y,Q)$ satisfying $HO(26)$. We obtain a solution of $HO(28)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 18 lines of type $(0,1)$ plus 12 nilpotents at suitable singular points of $Y\cup T$ and 16 points on the singular points of a configuration of lines of type $(19,2)$; among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 12 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Lemma \[3.6\] and Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(27,9)$.
\[6.25\] $HO(30)$ is true: Take $(Y,Q)$ satisfying $HO(28)$. We obtain a solution of $HO(30)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 19 lines of type $(0,1)$ plus 16 nilpotents at suitable singular points of $T$ and 30 points on the singular points of a configuration of lines of type $(19,3)$; among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 16 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Lemma \[3.6\] and Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(29,12)$.
\[6.26\] $HO(32)$ is true: Take $(Y,Q)$ satisfying $HO(30)$. We obtain a solution of $HO(32)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 19 lines of type $(0,1)$ plus 30 nilpotents at suitable singular points of $T$ and 56 points on the singular points of a configuration of lines of type $(19,4)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other line of type $(1,0)$ is 1-secant to $Y$ and 12 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Lemma \[3.6\] and Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid
Q)(30,16)$.
\[6.27\] $HO(34)$ is true: Take $(Y,Q)$ satisfying $HO(32)$. We obtain a solution of $HO(34)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 19 lines of type $(0,1)$ plus 56 nilpotents at suitable singular points of $T$ and 96 points on the singular points of a configuration of lines of type $(21,6)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 16 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Lemma \[3.6\] and Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid
Q)(31,16)$.
\[6.28\] $HO(36)$ is true: Take $(Y,Q)$ satisfying $HO(34)$. We obtain a solution of $HO(36)$ adding in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 21 lines of type $(0,1)$ plus 96 nilpotents at suitable singular points of $T$ and 48 points on the singular points of a configuration of lines of type $(25,3)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 16 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Lemma \[3.6\] and Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid
Q)(31,16)$.
\[6.29\] $HO(38)$ is true: Take $(Y,Q)$ satisfying $HO(36)$. We obtain a solution of $HO(38)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 25 lines of type $(0,1)$ plus 48 nilpotents at suitable singular points of $T$ and 6 points on the singular points of a configuration of lines of type $(24,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other line of type $(1,0)$ is 1-secant to $Y$ and 24 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Lemma \[3.6\] and Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid
Q)(36,14)$.
\[6.30\] $HO(40)$ is true: Take $(Y,Q)$ satisfying $HO(38)$. We obtain a solution of $HO(40)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 24 lines of type $(0,1)$ plus 6 nilpotents at suitable singular points of $T$ and 88 points on the singular points of a configuration of lines of type $(25,5)$; among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 6 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Lemma \[3.6\] and Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(36,16)$.
\[6.31\] $HO(42)$ is true: Take $(Y,Q)$ satisfying $HO(40)$. We obtain a solution of $HO(42)$ adding in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 25 lines of type $(0,1)$ plus 88 nilpotents at suitable singular points of $T$ and 70 points on the singular points of a configuration of lines of type $(28,4)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 16 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(38,18)$.
\[6.32\] $HO(44)$ is true: Take $(Y,Q)$ satisfying $HO(42)$. We obtain a solution of $HO(44)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 28 lines of type $(0,1)$ plus 70 nilpotents at suitable singular points of $T$ and 64 points on the singular points of a configuration of lines of type $(27,4)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 16 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(41,17)$.
\[6.33\] $HO(46)$ is true: Take $(Y,Q)$ satisfying $HO(44)$. We obtain a solution of $HO(46)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 27 lines of type $(0,1)$ plus 64 nilpotents at suitable singular points of $T$ and 72 points on the singular points of a configuration of lines of type $(29,4)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 12 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(43,20)$.
\[6.34\] $HO(48)$ is true: Take $(Y,Q)$ satisfying $HO(46)$. We obtain a solution of $HO(48)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 29 lines of type $(0,1)$ plus 72 nilpotents at suitable singular points of $T$ and 96 points on the singular points of a configuration of lines of type $(29,5)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 16 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(45,20)$.
\[6.35\] $HO(50)$ is true: Take $(Y,Q)$ satisfying $HO(48)$. We obtain a solution of $HO(50)$ adding in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 29 lines of type $(0,1)$ plus 96 nilpotents at suitable singular points of $T$ and 142 suitable points; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 12 of the lines of type $(0,1)$ are 1-secant to $Y$. We apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(46,22)$.
\[6.36\] We define the following assertion $HO(7)'$. $HO(7)'$: There exists $(Y,Q',D,D',D",S,S',S")$ with $Y\in H(15,12)$, $Q'$ smooth quadric intersecting transversally $Y D, D'$ and $D"$ lines of type $(0,1)$ on $Q'$, each of them 2-secant to $Y$, $S \subset D, S' \subset D, S" \subset D"$, with $card(S) = card(S') = 6, card(S") = 4$ such that $h^{0}(\Omega(8)\otimes I_{Y\cup S\cup S'\cup S"}) = h^{1}(\Omega(8)\otimes
I_{Y\cup S\cup S'\cup S"}) = 0$, and there are 6 lines $R_{i}, 1 \leq i \leq
6$, of type $(1,0)$ on $Q'$ with $R_{i}$ 1-secant to $Y$ for every $i$, $R_{i}\cap S' \neq \O$ for every $i$, $R_{j}\cap S" \neq \O$ if and only if $1 \leq j \leq 4$.
\[6.37\] $HO(7)'$ is true: Just take a solution $(Y',Q',^{\ast \ast \ast})$ of $HO(5)$ and copy the proof of $HO(7)$ given in \[6.12\] using the quadric $Q'$.
\[6.38\] $HO(9)$ is true: Take $(Y,Q',D,^{\ast \ast},S,^{\ast \ast})$ satisfying $HO(7)'$. Take a smooth quadric $Q \neq Q'$ containing $D$, say as a line of type $(1,0)$. We obtain a solution of $HO(9)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$, one of them being $D$, and 7 lines of type $(0,1)$ plus 6 nilpotents $\chi(P), P\in S$; among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and the 6 lines of type $(0,1)$ intersecting S are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(8,3)$ we use Lemma \[3.1\]
\[6.39\] $HO(11)$ is true: Take $(Y,Q)$ satisfying $HO(9)$. We obtain a solution of $HO(11)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 5 lines of type $(0,1)$ plus 16 nilpotents at suitable singular points of $Y\cup T$ and 4 points on the singular points of a configuration of lines of type $(7,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 4 of the lines of type (0,1) are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(6,5)$ we use Lemma 3.6 for $a
= 1$ and $b = t = 4$.
\[6.40\] $HO(13)$ is true: Take $(Y,Q)$ satisfying $HO(11)$. We obtain a solution of $HO(13)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 7 lines of type $(0,1)$ plus 4 nilpotents at suitable singular points of $T$ and 26 points on the singular points of a configuration of lines of type $(11,3)$; among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 4 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(14,7)$ we use Lemma \[6.3\] for $a = 3, b = 11, t = 4$.
\[6.41\] We define the following assertion $HO(13)'$. $HO(13)'$: There exists $(Y,Q',D,D',S,S',Z)$ with $Y\in H(42,39)$, $Q'$ smooth quadric intersecting transversally $Y, D, D'$ are lines of type $(0,1)$ on $Q'$, each of them 2-secant to $Y$, $S \subset D, S' \subset D'$, with $card(S) = card(S') = 3,
card(S") = 4$, $Z \subset Q', card(Z) = 6$, such that $h^{0}(\Omega(14)\otimes I_{Y\cup S\cup S'\cup Z}) = h^{1}(\Omega(14)\otimes
I_{Y\cup S\cup S'\cup S"}) = 0$; furthermore there are 6 lines $R_{i}$, $1
\leq i \leq 6$, of type $(1,0)$ with $R_{i}$ 1-secant to $Y$ for every $i$, $R_{i}\cap Z \neq \O$ for every $i$.
\[6.42\] The proof of $HO(13)$ given in \[6.40\] shows that $HO(13)'$ is true.
\[6.43\] We define the following assertion $HO(15)'$. $HO(15)'$: There exists $(Y,Q',D',S',Z)$ with $Y\in H(42,39)$, $Q'$ smooth quadric intersecting transversally $Y, D'$ is a line of type $(0,1)$ on Q’ which 2-secant to $Y$, $S \subset D, S' \subset D'$, with $card(S) = card(S') = 3, card(S") = 4, Z
\subset Q', card(Z) = 6$, such that $h^{0}(\Omega(16)\otimes I_{Y\cup S'\cup
Z}) = h^{1}(\Omega(16)\otimes I_{Y\cup S\cup S'\cup S"}) = 0$; furthermore there are 6 lines $R_{i}$, $1 \leq i \leq 6$, of type $(1,0)$ with $R_{i}$ 1-secant to $Y$ for every $i$, $R_{i}\cap Z \neq \O$ for every $i$.
\[6.44\] $HO(15)$ and $HO(15)'$ are true: Take $(Y,Q',D,D',S,S',Z)$ satisfying $HO(13)'$. Take a smooth quadric $Q$ containing $D$, say as a line of type $(1,0)$, but not $D'$. We obtain a solution of $HO(15)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$, one of them being $D$, and 10 lines of type $(0,1)$ and the 3 nilpotents $\chi(P), P\in S$; among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 3 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(14,6)$ we use Lemma \[3.6\]. In the same way we obtain $HO(15)'$.
\[6.45\] $HO(17)$ is true: Take $(Y,Q',D',S',Z)$ satisfying $HO(15)'$. Take a smooth quadric $Q \neq Q'$ and containing $D'$, say as a line of type $(1,0)$, but not $D$. We obtain a solution of $HO(17)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$, $D'$ being one of them, and 11 lines of type $(0,1)$ plus the 4 nilpotents $\chi(P), P\in S'$; among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 4 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(16,7)$ we use Lemma \[3.6\].
\[6.46\] $HO(19)$ is true: Take $(Y,Q)$ satisfying $HO(17)$. We obtain a solution of $HO(19)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 11 lines of type $(0,1)$ plus 19 nilpotents at suitable singular points of $T$ and 28 points on the singular points of a configuration of lines of type $(11,4)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other line of type $(1,0)$ is 1-secant to $Y$ and 8 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(17,9)$ we use Lemma \[3.6\].
\[6.47\] $HO(21)$ is true: Take $(Y,Q)$ satisfying $HO(19)$. We obtain a solution of $HO(21)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 11 lines of type $(0,1)$ plus 28 nilpotents at suitable singular points of $Y\cup T$ and 44 points on the singular points of a configuration of lines of type $(10,6)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 8 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(18,11)$ we use Lemma \[3.6\].
\[6.48\] $HO(23)$ is true: Take $(Y,Q)$ satisfying $HO(21)$. We obtain a solution of $HO(23)$ adding in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 10 lines of type $(0,1)$ plus 44 nilpotents at suitable singular points of $T$ and 61 points on the singular points of a configuration of lines of type $(14,6)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 8 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(18,14)$ we use Lemma \[3.6\].
\[6.49\] $HO(25)$ is true: Take $(Y,Q)$ satisfying $HO(23)$. We obtain a solution of $HO(25)$ adding in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 14 lines of type $(0,1)$ plus 61 nilpotents at suitable singular points of $T$ and 22 points on the singular points of a configuration of lines of type $(15,3)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 9 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(20,12)$ we use Lemma \[3.6\].
\[6.50\] We define the following assertion $\quad HO \; (\; 27\; )' \quad$. There exists $(Y,Q',D,S,Z)$ with $Y\in H(150,147)$, $Q'$ smooth quadric, $D$ is a line of type $(0,1)$ which is 2-secant to $Y, S \subset D, card(S) = 2, Z
\subset Q', card(Z) = 59$ and $h^{0}(\Omega(14)\otimes I_{Y\cup S\cup Z}) =
h^{1}(\Omega(14)\otimes I_{Y\cup S\cup Z}) = 0$; furthermore, $(Y,Q',Z)$ may be prolonged (adding suitable 2 points of $Q'$) to a solution of $HO(27)$.
\[6.51\] $HO(27)$ and $HO(27)'$ are true: Take $(Y,Q)$ satisfying $HO(25)$. We obtain a solution of $HO(27)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 15 lines of type $(0,1)$ plus 22 nilpotents at suitable singular points of $T$ and 61 points on the singular points of a configuration of lines of type $(12,6)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other line of type $(1,0)$ is 1-secant to $Y$ and 6 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid
Q)(25,13)$ we use Lemma \[3.6\].
\[6.52\] $HO(29)$ is true: We start with a solution of $HO(27)'$ and copy the proof that $HO(15)'$ implies $HO(17)$.
\[6.53\] $HO(31)$ is true: Take $(Y,Q)$ satisfying $HO(29)$. We obtain a solution of $HO(31)$ adding in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 12 lines of type $(0,1)$ plus 59 nilpotents at suitable singular points of $T$ and 8 points on the singular points of a configuration of lines of type $(19,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$,the other lines of type $(1,0)$ are 1-secant to $Y$ and 4 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(26,20)$ we use Lemma \[3.6\].
\[6.54\] We define the following assertion $HO(33)'$. $HO(33)'$: There exists $(Y,Q',D,S,Z)$ with $Y\in H(216,213)$, $Q'$ smooth quadric, $D$ is a line of type $(0,1)$ which is 2-secant to $Y, S \subset D, card(S) = 2, Z \subset
Q', card(Z) = 84$ and $h^{0}(\Omega(34)\otimes I_{Y\cup S\cup Z}) =
h^{1}(\Omega(34)\otimes I_{Y\cup S\cup Z}) = 0$; furthermore, $(Y,Q',Z)$ may be prolonged (adding suitable 2 points of $Q'$) to a solution of $HO(27)$.
\[6.55\] $HO(33)$ and $HO(33)'$ are true: Take $(Y,Q)$ satisfying $HO(31)$. We obtain a solution of $HO(10)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 19 lines of type $(0,1)$ plus 8 nilpotents at suitable singular points of $T$ and 86 points on the singular points of a configuration of lines of type $(21,5)$; among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 8 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(32,15)$ we use Lemma \[3.6\].
\[6.56\] $HO(35)$ is true: Start with a solution of $HO(33)'$ and copy the proof of $HO(17)$.
\[6.57\] $HO(37)$ is true: Take $(Y,Q)$ satisfying $HO(35)$. We obtain a solution of $HO(37)$ adding in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 20 lines of type $(0,1)$ plus 84 nilpotents at suitable singular points of $T$ and 93 points on the singular points of a configuration of lines of type $(25,5)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 8 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(32,18)$ we use Lemma \[3.6\].
\[6.58\] $HO(39)$ is true: Take $(Y,Q)$ satisfying $HO(37)$. We obtain a solution of $HO(39)$ adding in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 25 lines of type $(0,1)$ plus 93 nilpotents at suitable singular points of $T$ and 2 points on the singular points of a configuration of lines of type $(26,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 15 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(35,15)$ we use Lemma \[3.6\].
\[6.59\] $HO(41)$ is true: Take $(Y,Q)$ satisfying $HO(39)$. We obtain a solution of $HO(41)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 26 lines of type $(0,1)$ plus 2 nilpotents at suitable singular points of $T$ and 33 points on the singular points of a configuration of lines of type $(26,3)$; among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 2 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid
Q)(40,16)$ we use Lemma \[3.6\].
\[6.60\] $HO(43)$ is true: Take $(Y,Q)$ satisfying $HO(41)$. We obtain a solution of $HO(43)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 26 lines of type $(0,1)$ plus 33 nilpotents at suitable singular points of $T$ and 81 points on the singular points of a configuration of lines of type $(29,4)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other line of type $(1,0)$ is 1-secant to $Y$ and 6 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(41,18)$ we use Lemma \[3.6\].
\[6.61\] $HO(45)$ is true: Take $(Y,Q)$ satisfying $HO(43)$. We obtain a solution of $HO(45)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 29 lines of type $(0,1)$ plus 81 nilpotents at suitable singular points of $T$ and 17 points on the singular points of a configuration of lines of type ($29,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other ones of type $(1,0)$ are 1-secant to $Y$ and 28 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(44,17)$ we use Lemma \[3.6\].
\[6.62\] $HO(47)$ is true: Take $(Y,Q)$ satisfying $HO(45)$. We obtain a solution of HO(47) adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 29 lines of type $(0,1)$ plus 17 nilpotents at suitable singular points of $T$ and 99 points on the singular points of a configuration of lines of type (30,5); among the lines of $T$ the ones of type $(1,0)$ are 2-secant to $Y$ and 17 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(46,19)$ we use Lemma \[3.6\].
\[6.63\] $HO(49)$ is true: Take $(Y,Q)$ satisfying $HO(47)$. We obtain a solution of $HO(49)$ adding in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 30 lines of type $(0,1)$ plus 99 nilpotents at suitable singular points of $T$ and 61 points on the singular points of a configuration of lines of type (29,4); among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 12 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(45,20)$ we use Lemma \[3.6\].
\[6.64\] $HO(51)$ is true: Take $(Y,Q)$ satisfying $HO(35)$. We obtain a solution of $HO(51)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 29 lines of type $(0,1)$ plus 61 nilpotents at suitable singular points of $T$ and 185 suitable points; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ are 1-secant to $Y$ and 1 of the lines of type $(0,1)$ is 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega\mid Q)(49,23)$ we use Lemma \[3.6\].
Now we prove the initial cases for $T\Pi$ needed for Theorem \[0.1\]. We define the following assertions.
T{2}: The general union $Z$of 5 points of $\Pi$ has $h^{0}(T\Pi\otimes
I_{Z}) = h^{1}(T\Pi\otimes I_{Z}) = 0$.
T{1}: For every line $D \subset \Pi$ we have $h^{0}(T\Pi(-1)\otimes I_{D})
= h^{1}(T\Pi(-1)\otimes I_{D}) = 0$.
T{3}: For a general $C\in H(3,0)$ and a general $Z \subset \Pi$ with $card(Z) = 3$ we have $h^{0}(T\Pi(1)\otimes I_{C\cup Z}) =
h^{1}(T\Pi(1)\otimes I_{C\cup Z}) = 0$.
\[6.65\] A proof of assertion T{2} was not done in [@BG], §3, only because assertion T{2} was previously known.
\[6.66\] $HT(4)$ is true.
[Proof]{} Take $Z \subset \Pi$ with $card(Z) = 5$ and satisfying T{2}. Take a smooth quadric $Q$ with $card(Z\cap Q) = 2$ and such that $Z\cap Q$ contains 2 points contained in a line $F$of $Q$ of type $(1,0)$. Set $S:=
Z\setminus(Z\cap Q)$. Let $E$ and $E'$ be general curves of type $(2,1)$ on $Q$ with $E'\cap F = Z\cap Q$. Set $Z\cap Q = {P, P'}$ and $X:= E\cup E'\cup
F\cup \chi(P)\cup \chi(P')$. By \[1.4\] we have X$\in H(7,4)$ and $Res_{Q}(X\cup S) = Z$. Hence in the usual way we obtain $HT(4)"$.Since $\alpha'(4)^{\ast} = 3 \leq 6 = \delta'(6)^{\ast} - \delta'(4)^{\ast}$ we obtain $HT(6)$, too.
\[6.67\] The Euler sequence gives easily that T{1} is true.
\[6.68\] Assertion T{3} is true.
[Proof]{} Fix a line $D$ and a smooth quadric $Q$ intersecting transversally $D$. Take a general curve $E$ of type $(1,1)$ on $Q$ containing $D\cap Q$ and set $X:=
D\cup E$. Take as $Z$ the union of 3 general points of $Q$. Then use Remark \[6.67\], \[1.4\] and Horace Lemma \[1.3\].
\[6.69\] Assertion $HT(5)$ is true.
[Proof]{} Take $(C,Z)$ satisfying T{3} and a smooth quadric $Q$ intersecting transversally $C$, with $card(Z\cap Q) = 2$ and such that $Z\cap Q$ is contained in a line $F$, say of type $(0,1)$. We take 2 smooth curves $E,
E'\subset Q$ of type $(2,1)$ with $card(E\cap C) = 4, E'\cap C = \O$ and ${P,P'}:= Z\cap Q \subset E'$. Take as $S$ the union of $Z\setminus{,P'}$ and a general point of $Q$. Set $X:= C\cup E\cup E'\cup
F\cup \chi(P)\cup \chi(P')$. In the usual way we obtain $HT(5)"$. Since $\alpha'(5)^{\ast} = 2 \leq 6 =
\delta'(7)^{\ast} - \delta(5)^{\ast} - 2$, the same proof gives $HT(5)$.
\[6.70\] $HT(7)$ is true.
[Proof]{} Since $a'(5)^{\ast} = 2, HT(5)$ implies $HT(7)$.
\[6.71\] $HT(6)$ is true.
[Proof]{} Take a solution $(Y,Z)$ of $HT(4)$ and a smooth quadric $Q$ intersecting transversally $Y$, containing 3 disjoint 2-secant lines of $Q$, say as lines of type $(1,0)$, each of them containg a point of $Z$; this is possible for general $Y$ because we just need to find a smooth quadric through 3 disjoint lines of $\Pi$. Let $E$ be a union of e of type $(2,3)$ on $Q$ with $E$ 6-secant to $Y$. Set $X:= Y\cup E$ and take as $S$ the union of $Z$and 4 general points of $Q$. To handle the cohomology of $(Y\cup S)\cap (Q\setminus
E)$, use Lemma 3.2 for $x = 2$ and $y = 1$.
\[6.72\] $HT(9)$ is true.
[Proof]{} Take $(X',S')$ satisfying $HT(7)"$ (Lemma \[6.70\] and a smooth quadric $Q$ containg $S'$. Take general curves $E, E'$ on $Q$ with $E$ of type $(3,2), E$ 6-secant to $X', E'$ containing $S'$ and $E'\cap X' = \O$. Take as $H\in H(24,21)$ the union of $X', E, E'$ and the points $\chi(P)$ with $P\in S'$.
\[6.73\] $HT(8)$ is true. Take a solution $(Y,Q,S')$ of $HT(6)$ (Lemma \[6.67\]) and a smooth quadric $Q'$ intersecting transversally $Y$ and containing two 2-secant lines to $Y$. We obtain a solution of HT(8) by adding in $Q'$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 2 lines of type $(0,1)$ plus 5 suitable points in the intersection with $Q'$ of the 3 lines of $Q$ containing the 12 points of $S'$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(3,3)$ we use Lemma \[3.7\] for $x = y
= 3, a = 3, b = 2, t = 1$.
\[6.74\] $HT(10)$ is true. Take a solution $(Y,Q)$ of $HT(8)$. We obtain a solution of $HT(10)$ by adding in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 5 lines of type $(0,1)$ plus 17 nilpotents at suitable singular points of $T$ and 1 point ; among the lines of $T$ two of the 2 ones of type $(1,0)$ are 2-secant to $Y$, while other lines are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(3,3)$ we use Lemma \[3.2\].
\[6.75\] We define the following modification, called $HT(12)'$, of the assertion $HT(12)$; $HT(12)'$: There exists $(Y,Q,T,S)$ with $Y\in H(36,33)$, $Q$ smooth quadric intersecting transversally $Y$, $T$ union of lines of type $(5,4)$ of $Q$ such that one of the lines of type $(0,1)$ of $T$ is 3-secant to $Y$, one of the lines of type $(0,1)$ is 2-secant to $Y$, the remaing lines of $T$ are 1-secant to $Y$, $S \subseteq Sing(T), card(S) =
\alpha'(12)^{\ast} = 14$ and $r_{Y\cup S,T\Pi(10)}$ is bijective. Here we will check $HT(12)'$. Take a solution $(Y,Q)$ of $HT(10)$. We obtain a solution of $HT(12)'$ by adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$, both 2-secant to $Y$, and 6 lines of type $(0,1)$, one of them 1-secant to $Y$, plus one nilpotent and 14 points on the singular points of a configuration of lines of type $(5,5)$ with the prescribed secant behaviour with respect to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(8,4)$ we use Lemma \[3.7\].
\[6.76\] We define the following modification, called $HT(14)'$, of the assertion $HT(14)$; $HT(14)'$: There exists $(Y,Q,A,S)$ with $Y\in H(45,42)$, $Q$ smooth quadric intersecting transversally $Y$, A union of lines of type $(7,6)$ of $Q$ such that one of the lines of type $(0,1)$ of $T$ is 3-secant to $Y$, one of the lines of type $(0,1)$ is 2-secant to $Y$, the remaing lines of $T$ are 1-secant to $Y$, $S \subseteq Sing(T), card(S) =
\alpha'(14)^{\ast} = 36$ and $r_{Y\cup S,T\Pi(12)}$ is bijective. Here we will check $HT(12)'$. Take a solution $(Y,Q,T,S)$ of $HT(12)'$. We obtain a solution of $HT(14)'$ by adding in $Q$ to $Y$ $T$ plus 14 nilpotent supported by $Sing(T)$ and 36 points on the the singular set of a configuration of lines of type $(7,6)$ with the prescribed secant behaviour with respect to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(8,7)$ we use Lemma \[3.9\] for $x = 8, y = 7, a = 7,
b = t = 6$.
\[6.77\] $HT(16)$ is true. Take a solution $(Y,Q,A,S)$ of $HT(14)'$. We obtain a solution of $HT(16)$ by adding in $Q$ to $Y$ A plus 36 nilpotents at suitable singular points of $A$ and 26 points on the singular points of a configuration of lines of type $(10,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(8,7)$ we use Lemma \[3.7\].
\[6.78\] $HT(18)$ is true. Take a solution $(Y,Q)$ of $HT(16)$. We obtain a solution of $HT(18)$ by adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 10 lines of type $(0,1)$ plus 26 nilpotents at suitable singular points of $T$ and 17 points on the singular points of a configuration of lines of type $(14,2)$; among the lines of $T$ two of the 2 ones of type $(1,0)$ are 2-secant to $Y$ and 10 of the other lines of type are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(14,6)$ we use Lemma \[3.7\].
\[6.79\] $HT(20)$ is true. Take a solution $(Y,Q)$ of $HT(18)$. We obtain a solution of $HT(20)$ by adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 14 lines of type $(0,1)$ plus 17 nilpotents at suitable singular points of $T$ and 11 points on the singular points of a configuration of lines of type $(14,2)$; among the lines of $T$ two of ones of type $(1,0)$ are 2-secant to $Y$ and 7 of the other ones are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(18,6)$ we use Lemma \[3.7\].
\[6.80\] $HT(22)$ is true. Take a solution $(Y,Q)$ of $HT(20)$. We obtain a solution of $HT(22)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 14 lines of type $(0,1)$ plus 11 nilpotents at suitable singular points of $T$ and 10 points on the singular points of a configuration of lines of type $(15,2)$; among the lines of $T$ the 2 ones of type $(1,0)$ are 2-secant to $Y$ and 11 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(18,6)$ we use Lemma \[3.7\].
\[6.81\] $HT(24)$ is true. Take a solution $(Y,Q)$ of $HT(22)$. We obtain a solution of $HT(24)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 15 lines of type $(0,1)$ plus 10 nilpotents at suitable singular points of $T$ and 16 points on the singular points of a configuration of lines of type $(16,2)$; among the lines of $T$ the 2 ones of type $(1,0)$ are 2-secant to $Y$ and 10 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(20,7)$ we use Lemma \[3.7\].
\[6.82\] $HT(26)$ is true. Take a solution $(Y,Q)$ of $HT(24)$. We obtain a solution of $HT(26)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 16 lines of type $(0,1)$ plus 16 nilpotents at suitable singular points of $Y\cup T$ and 31 points on the singular points of a configuration of lines of type $(16,3)$; among the lines of $T$ the 2 ones of type $(1,0)$ are 2-secant to $Y$ and the ones of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(22,8)$ we use Lemma \[3.7\].
\[6.83\] $HT(28)$ is true. Take a solution $(Y,Q)$ of $HT(26)$. We obtain a solution of $HT(28)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 16 lines of type $(0,1)$ plus 31 nilpotents at suitable singular points of $T$ and 57 points on the singular points of a configuration of lines of type $(18,5)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other one of type $(1,0)$ is 1-secant to $Y$ and 16 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(23,10)$ we use Lemma \[3.7\].
\[6.84\] $HT(30)$ is true. Take a solution $(Y,Q)$ of $HT(28)$. We obtain a solution of $HT(30)$ adding in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 18 lines of type $(0,1)$ plus 57 nilpotents at suitable singular points of $T$ and 11 points on the singular points of a configuration of lines of type $(19,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the remaining ones of type $(1,0)$ are 1-secant to $Y$ and 6 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(23,10)$ we use Lemma \[3.7\].
\[6.85\] $HT(32)$ is true. Take a solution $(Y,Q)$ of $HT(30)$. We obtain a solution of $HT(32)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 19 lines of type $(0,1)$ plus 11 nilpotents at suitable singular points of $T$ and 59 points on the singular points of a configuration of lines of type $(21,4)$; among the lines of $T$ the 2 ones of type $(1,0)$ are 2-secant to $Y$ and 11 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(28,11)$ we use Lemma \[3.7\].
\[6.86\] $HT(34)$ is true. Take a solution $(Y,Q)$ of $HT(32)$. We obtain a solution of $HT(34)$ adding in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 21 lines of type $(0,1)$ plus 59 nilpotents at suitable singular points of $T$ and 27 points on the singular points of a configuration of lines of type $(16,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other ones of type $(1,0)$ are 1-secant to $Y$ and 19 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(28,11)$ we use Lemma \[3.7\].
\[6.87\] $HT(36)$ is true. Take a solution $(Y,Q)$ of $HT(34)$. We obtain a solution of $HT(36)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 23 lines of type $(0,1)$ plus 27 nilpotents at suitable singular points of $T$ and 2 points on the singular points of a configuration of lines of type $(22,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other one of type $(1,0)$ is 1-secant to $Y$ and 5 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(31,11)$ we use Lemma \[3.7\].
\[6.88\] $HT(38)$ is true. Take a solution $(Y,Q)$ of $HT(36)$. We obtain a solution of $HT(38)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 22 lines of type $(0,1)$ plus 2 nilpotents at suitable singular points of $T$ and 95 points on the singular points of a configuration of lines of type $(22,6)$; among the lines of $T$ the 2 ones of type $(1,0)$ are 2-secant to $Y$ and 2 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(34,14)$ we use Lemma \[3.7\].
\[6.89\] $HT(40)$ is true. Take a solution $(Y,Q)$ of $HT(38)$. We obtain a solution of $HT(40)$ adding in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 22 lines of type $(0,1)$ plus 95 nilpotents at suitable singular points of $T$ and 100 points on the singular points of a configuration of lines of type $(16,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other ones of type $(1,0)$ are 1-secant to $Y$ and 11 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(32,16)$ we use Lemma \[3.7\].
\[6.90\] $HT(42)$ is true. Take a solution $(Y,Q)$ of $HT(40)$. We obtain a solution of $HT(42)$ adding in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 23 lines of type $(0,1)$ plus 100 nilpotents at suitable singular points of $T$ and 110 points on the singular points of a configuration of lines of type (27,6); among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$, the other ones of type $(1,0)$ are 1-secant to $Y$ and 12 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(34,17)$ we use Lemma \[3.7\].
\[6.91\] $HT(44)$ is true. Take a solution $(Y,Q)$ of $HT(42)$. We obtain a solution of $HT(44)$ adding in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 27 lines of type $(0,1)$ plus 110 nilpotents at suitable singular points of $T$ and 12 points on the singular points of a configuration of lines of type $(29,2)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$ and all other lines are disjoint from $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(36,15)$ we use Lemma \[3.7\].
\[6.92\] $HT(46)$ is true. Take a solution $(Y,Q)$ of $HT(44)$. We obtain a solution of $HT(46)$ adding in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ and 29 lines of type $(0,1)$ plus 12 nilpotents at suitable singular points of $T$ and 52 points on the singular points of a configuration of lines of type $(30,3)$; among the lines of $T$ the 2 ones of type $(1,0)$ are 2-secant to $Y$ and 12 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(42,15)$ we use Lemma \[3.7\].
\[6.93\] $HT(48)$ is true. Take a solution $(Y,Q)$ of $HT(46)$. We obtain a solution of $HT(48)$ adding in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 30 lines of type $(0,1)$ plus 52 nilpotents at suitable singular points of $T$ and 111 points on the singular points of a configuration of lines of type $(30,6)$; among the lines of $T$ 2 of the ones of type $(1,0)$ are 2-secant to $Y$ and 23 of the lines of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(43,16)$ we use Lemma \[3.7\].
\[6.94\] $HT(50)$ is true. Take the following deformation $(Y,Q)$ of a solution of $HT(48)$. We obtain a solution of $HT(50)$ adding in $Q$ to $Y$ the union, $T$, of 7 lines of type $(1,0)$ and 29 lines of type $(0,1)$ plus 111 nilpotents at suitable singular points of $T$ and 46 suitable points of $Q$; we assume that one of the line of type $(1,0)$ of $T$ is 2-secant to $Y$ and another one is 1-secant to $Y$, while the remaining lines of $T$ are disjoint from $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(41,19)$ we use Lemma \[3.7\].
\[6.95\] $HT(11)$ is true. Take $(Y,Q)$ satisfyng $HT(9)$ (Lemma \[6.74\]). To obtain a solution of $H(11)$ we add in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$, both 2-secant to $Y$, and 4 lines of type $(0,1)$ disjoint from $Y$ and 24 points on the singular points of a configuration of type $(6,6)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(7,3)$ we use Lemma \[3.7\].
\[6.96\] $HT(13)$ is true. Take $(Y,Q)$ satisfying $HT(11)$. To obtain a solution of $HT(13)$ we add in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 6 lines of type $(0,1)$ plus 24 nilpotents at points of $Sing(T)$ and 2 points on a line of type $(0,1)$; we assume that 2 of the lines of type $(1,0)$ of $T$ are 2-secant to $Y$, the other lines of type $(1,0)$ of $Y$ are 1-secant to $Y$ and 4 of the lines of type $(0,1)$ of $T$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(5,5)$ we use Lemma \[3.7\].
\[6.97\] $HT(15)$ is true. Take $(Y,Q)$ satisfying $HT(13)$. To obtain a solution of $HT(15)$ we add in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$, both 2-secant to $Y$ and 7 lines of type $(0,1)$, 2 of them 1-secant to $Y$ plus 2 nilpotents supported at suitable singular points of $T$ and 36 points at suitable singular points on a configuration of lines of type $(10,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(11,5)$ we use Lemma \[3.7\].
\[6.98\] $HT(17)$ is true. Take $(Y,Q)$ satisfying $HT(15)$. To obtain a solution of $HT(17)$ we add in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 10 lines of type $(0,1)$ plus 36 suitable nilpotents supported by $Sing(T)$ and 4 points on a line of type $(0,1)$; we assume that 2 of the lines of $T$ are 2-secant to $Y$ and that 12 of the other lines are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(10,5)$ we use Lemma \[3.7\].
\[6.99\] $HT(19)$ is true. Take $(Y,Q)$ satisfying $HT(17)$. To obtain a solution of $HT(19)$ we add in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ 2-secant to $Y$ and 11 lines of type $(0,1)$, 4 of them 1-secant to $Y$, plus 4 suitable nilpotents at points of $Sing(T)$ and 20 points on the singular set of a configuration of lines of type $(10,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(15,6)$ we use Lemma \[3.7\].
\[6.100\] $HT(21)$ is true. Take $(Y,Q)$ satisfying $HT(19)$. To obtain a solution of $HT(21)$ we add in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 10 lines of type $(0,1)$ plus 20 suitable nilpotents at the singular points of $T$ and 46 points on the singular set of a configuration of lines of type $(14,5)$; we assume that 2 of the lines of $T$ of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ of $T$ are 1-secant to $Y$ and 9 of the lines of $T$ of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(12,9)$ we use Lemma \[3.7\].
\[6.101\] $HT(23)$ is true. Take $(Y,Q)$ satisfying $HT(21)$. To obtain a solution of $HT(23)$ we add in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 14 lines of type $(0,1)$ plus 17 suitable nilpotents at the singular points of $T$ and 20 points on a configuration of lines of type $(13,3)$; we assume that 2 of the lines of $T$ of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ of $T$ are 1-secant to $Y$ and 7 of the lines of $T$ of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(16,7)$ we use Lemma \[3.7\].
\[6.102\] $HT(25)$ is true. Take $(Y,Q)$ satisfying $HT(23)$. To obtain a solution of $HT(25)$ we add in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 13 lines of type $(0,1)$ plus 20 suitable nilpotents at the singular points of $T$ and 66 points on a configuration of lines of type $(14,6)$; we assume that 2 of the lines of $T$ of type $(1,0)$ are 2-secant to $Y$, the other lines of type $(1,0)$ of $T$ is 1-secant to $Y$ and 5 of the lines of $T$ of type $(0,1)$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(20,10)$ we use Lemma \[3.7\].
\[6.103\] $HT(27)$ is true. Take $(Y,Q)$ satisfying $HT(25)$. To obtain a solution of $HT(27)$ we add in $Q$ to $Y$ the union, $T$, of 6 lines of type $(1,0)$ and 14 lines of type $(0,1)$ plus 66 suitable nilpotents at singular points of $T$ and 52 points on a configuration of lines of type (16,5); we assume that two lines of $T$ of type $(1,0)$ are 2-secant to $Y$ and that the other lines of $T$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(19,11)$ we use Lemma \[3.7\].
\[6.104\] $HT(29)$ is true. Take $(Y,Q)$ satisfying $HT(27)$. To obtain a solution of $H(29)$ we add in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 16 lines of type $(0,1)$ plus 52 nilpotents at singular points of $T$ and 44 points on the singular set of a configuration of lines of type $(5,18)$; a small difference with respect to the general case considered in Lemma \[2.6\]: here we take a configuration of type $(4,19)$ and not $(4,17)$ because we need it for the proof of $HT(33)$; we assume that two of the lines of $T$ of type $(1,0)$ are 2-secant to $Y$ and that 10 of the other lines of $Y$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(22,11)$ we use Lemma \[3.7\].
\[6.105\] $HT(31)$ is true. Take $(Y,S), card(S) = 44$, satisfying $HT(29)$. Let $Q$ be a general quadric containing two disjoint 2-secant lines, $R$ and $R'$, to $Y$. To obtain a solution of $HT(31)$ we add in $Q$ to $Y\cup S$ the union, $T$, of $R, R'$ and 20 general lines of type $(0,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(27,9)$ we use Lemma \[3.7\].
\[6.106\] $HT(33)$ is true. Take $(Y,Q,S)$ satisfying $HT(31)$ with $S \subset Q,
card(S) = 44$, in a singular set of a suitable configuration, $T$, of lines of type $(5,18)$; we assume that 2 of the lines of $T$ are 2-secant to $Y$ and that 10 of the other lines of $T$ are 1-secant to $Y$. To obtain a solution of $HT(33)$ we add to $Y$ the union of $T$, the nilpotents $\chi(P), P\in S$, and 54 of the singular points of a configuration of lines of type $(20,4)$ on $Q$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(26,13)$ we use Lemma \[3.7\].
\[6.107\] $HT(35)$ is true. Take $(Y,Q)$ satisfying $HT(33)$. To obtain a solution of $HT(35)$ we add in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 20 lines of type $(0,1)$ plus 54 suitable nilpotents at singular points of $T$ and 76 points on a configuration of lines of type $(23,5)$; we assume that two lines of $T$ of type $(1,0)$ are 2-secant to $Y$ and that 18 of the other lines of $T$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(19,11)$ we use Lemma \[3.7\].
\[6.108\] $HT(37)$ is true. Take $(Y,Q)$ satisfying $HT(35)$. To obtain a solution of $HT(37)$ we add in $Q$ to $Y$ the union, $T$, of 5 lines of type $(1,0)$ and 23 lines of type $(0,1)$ plus 76 suitable nilpotents at singular points of $T$ and 6 points on a line of type $(0,1)$; we assume that two lines of $T$ of type $(1,0)$ are 2-secant to $Y$ and that 13 of the other lines of $T$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadic $Q$ for $(T\Pi\mid Q)(30,12)$ we use Lemma \[3.7\].
\[6.109\] $HT(39)$ is true. Take $(Y,Q,D,S), card(S) = 6, S \subset D, D$ line of type $(1,0)$ 2-secant to $Y$, satisfying $HT(37)$. To obtain a solution of $HT(39)$ we add in $Q$ to $Y$ the union, $T$, of 2 lines of type $(1,0)$ (one of them being $D$) and 24 lines of type $(0,1)$ plus the 6 nilpotents $\chi(P), P\in S$, and 52 points on a configuration of lines of type $(23,4)$; we assume that the two lines of $T$ of type $(1,0)$ are 2-secant to $Y$ and that the 6 other lines of $T$ intersecting S are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid
Q)(35,13)$ we use Lemma \[3.7\].
\[6.110\] $HT(41)$ is true. Take $(Y,Q)$ satisfying $HT(39)$. To obtain a solution of $HT(41)$ we add in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 23 lines of type $(0,1)$ plus 52 suitable nilpotents at singular points of $T$ and 116 points on a configuration of lines of type $(24,7)$; we assume that two lines of $T$ of type $(1,0)$ are 2-secant to $Y$ and that 10 of the other lines of $T$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(35,16)$ we use Lemma \[3.7\].
\[6.111\] $HT(43)$ is true. Take $(Y,Q)$ satisfying $HT(41)$. To obtain a solution of $HT(43)$ we add in $Q$ to $Y$ the union, $T$, of 7 lines of type $(1,0)$ and 23 lines of type $(0,1)$ plus 116 suitable nilpotents at singular points of $T$ and 76 points on a configuration of lines of type $(28,4)$; we assume that two lines of $T$ of type $(1,0)$ are 2-secant to $Y$ and that 6 of the other lines of $T$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(34,18)$ we use Lemma \[3.7\].
\[6.112\] $HT(45)$ is true. Take $(Y,Q)$ satisfying $HT(43)$. To obtain a solution of $HT(45)$ we add in $Q$ to $Y$ the union, $T$, of 4 lines of type $(1,0)$ and 28 lines of type $(0,1)$ plus 76 suitable nilpotents at singular points of $T$ and 46 points on a configuration of lines of type $(30,3)$; we assume that two lines of $T$ of type $(1,0)$ are 2-secant to $Y$ and that 24 of the other lines of $T$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(39,15)$ we use Lemma \[3.7\].
\[6.113\] $HT(47)$ is true. Take $(Y,Q)$ satisfying $HT(45)$. To obtain a solution of $HT(47)$ we add in $Q$ to $Y$ the union, $T$, of 3 lines of type $(1,0)$ and 30 lines of type $(0,1)$ plus 46 suitable nilpotents at singular points of $T$ and 28 points on a line of type $(1,0)$; we assume that two lines of $T$ of type $(1,0)$ are 2-secant to $Y$ and that 18 of the other lines of $T$ are 1-secant to $Y$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(42,15)$ we use Lemma \[3.7\].
\[6.114\] $HT(49)$ is true. Take $(Y,Q,D,S)$ satisfying $HT(33)$ with $S \subset D,
card(S) = 22, D$ 2-secant line of type $(1,0)$ to $Q$ and $Q$ containing another 2-secant line $R$ of type $(1,0)$. To obtain a solution of $HT(49)$ we add in $Q$ to $Y$ the union of $D, R$, another 1-secant line of type $(1,0)$, 31 lines of type $(0,1)$, 22 of them containing a point of $S$ and 1-secant to $Y$, another one 1-secant to $Y$, the nilpotents $\chi(P), P\in
S$4, and 24 points on a line of type $(1,0)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi\mid Q)(19,11)$ we use Lemma \[3.7\].
\[6.115\] The proof of $HT(49)$ shows that $HT(51)$ is true.
Proof of Theorem \[0.3\] {#7}
========================
Here we show how to modify the proof of Theorem \[0.1\] to obtain a proof of Theorem \[0.3\]. We start with an integer $m(0)$, say $m(0) = 100$, such that for every integer $m \geq m(0)$ properties $HO(m)$ and $HT(m)$ are true and we may apply Lemma \[2.13\]. We will do the construction and introduce new notations just for $Omega$; the interested reader can do the same for $T\Pi$ in the same way (e.g. calling $u'(s,x), g'(s,x)$ and $v'(s,x)$ the corresponding integers). We fix an integer $s \geq m(0)$. For all integers $x \geq s$ with $x \equiv s
mod(2)$ we define the non-negative integers $u(s,x), g(s,x)$ and $v(s,x)$ in the following way. Set $u(s,s):= \delta(s)^{\ast}, g(s,s):= u(s,s) - 3$ and $v(s,s) =
\alpha(s,s)^{\ast}$. Assume defined the integers $u(s,x), g(s,x)$ and $v(s,x)$ satisfying the relations $$g(s,x) = u(s,s) - 3 + 2(u(s,x) - u(s,s) - 10(x-s))
\label{e23}$$ $$(3x-1)u(s,x) + 3(1-g(s,x)) + v(s,x) = (x+3)(x+2)x/2 = h^{0}(\Omega(x+1))
\label{e24}$$ $$u(s,x) \geq u(s,s)
\label{e25}$$ $$0 \leq v(s,x) \leq 3x-1
\label{e26}$$ By \[e23\] we have $2u(s,x) - 9 \geq g(s,x)$ and hence it is defined the component $H(u(s,x),g(s,x))$ of Hilb($\Pi$). Furthermore, the pair $(u(s,x),g(s,x))$ has critical value $x$. Define the integers $u(s,x+2), g(s,x+2)$ and $v(s,x+2)$ in the following way. Let $z$ be the maximal integer such that $(3x+5)z +
3(1-2(x-u(s,x)-20)) \leq h^{0}(\Omega(x+3))$. Let $u(s,x+2)$ be the only integer with $z-2 \leq u(s,x+2) \leq z$ such that $h^{0}(\Omega(x+3)) - (3x+5)u(s,x+2) \equiv h^{0}(\Omega(x+3))
\quad mod(3)$. Set $g(s,x+2):= u(s,s) - 3 + 2(u(s,x+2) - u(s,s) - 10(x+2-s)) =
g(s,x) + 2(u(s,x+2) - u(s,x) - 20)$. There is a unique integer $v(x,s+2)$ such that $$(3x+5)u(s,x+2) + 3(1-g(s,x+2)) + 3v(s,x+2) = h^{0}(\Omega(x+3))
\label{e27}$$ We need to check that $0 \leq v(s,x+2) \leq 3x+5$; this is obvious by the definition of $z$ if $v(s,x+2) = z$ and true if $v(s,x+2) = z-1$ or $z-2$ by the definition of $g(s,x+2)$. By Lemma \[2.13\] we see by induction on $x$ that $u(s,x+2) \geq u(s,x) + 20$ and that $u(s,t) \geq d(t)^{\ast}$ for every $t$. This implies $g(s,x+2) \geq g(s,x)$. Take any smooth curve $Y \in H(u(s,x),g(s,x))$ we may obtain a reducible element of $H(u(s,x+2),g(s,x+2))$. We may obtain a reducible element of $H(u(s,x+2),g(s,x+2))$ adding to $Y$ the union of 20 lines which are 1-secant to $Y$ and $u(s,x+2) - u(s,x) - 20$ lines that are 3-secant to $Y$. We fix a triple $(d,g,m)$ such that $d < g+3, g \leq 2d-9$ (i.e. $H(d,g)$ is defined), $m \geq m(0)+2$, and $(d,g)$ has critical value $m$. We will assume that a general $C\in H(d,g)$ has maximal rank; as discussed in section \[1\] we may make this assumption (for some function $\gamma(g)$ in the statement of \[0.3\]) by [@BE2] Th. \[0.2\]. Since H(d,g) is an irreducible component, it is sufficient to prove that for a general $X\in H(d,g)$ and a general $Y\in H(d,g)$ the restriction maps $r_{X,Omega(m+1)}$ and $r_{Y,T\Pi(m-2)}$ have maximal rank. To obtain curves with special hyperplane section we added in [@BE2] to “good curves” 3-secant lines. The reader should have no problem to modify the assertions $HO(m)$ and $HT(m)$ so that (after taking cares of the nilpotents $\chi(P)$ introduced in \[1.4\]) in the inductive step from $HO(m-2)$ to $HO(m)$ (or from $HT(m-2$) to $HT(m)$) we add two, three or four 2-secant lines (all of them of type $(1,0)$), while the other lines are 3-secant; this is exactly what is done in [@BE2], §6. Then to arrive exactly to a pair $(d,g)$ we repeat section \[5\]. It is obvious that in this way asymptotically we cover exactly the range claimed in the statement of \[0.3\] (as was the case in [@BE2] for the postulation, i.e. for the vector bundle $O_{\Pi}$).
Proof of Theorem 0.2 {#8}
====================
Theorem \[0.1\] implies Theorem \[0.2\] for large $g$, but to obtain it for low $g$ we will need to modify the inductive machine. Fix an integer $g \geq 0$. Essentially we will do first the genus 0 case for large degree until we arrive at rational curves with expected minimal free resolution and with critical value bigger than, say, $69g+27$. Then we apply one inductive step, say a proof that $R(m-2,0)$ implies $R(m,g)$ for large $m$, to increase the genus to $g$ and then consider again the general machine (i.e. $HO(m-2,g)$ implies $HO(m,g)$ and $HT(m-2,g)$ implies $HT(m,g)$ for $m \geq 70$ proved in section \[2\]) (see the last part of this section for more details). The final part is given by section \[5\] without any modification. To carry over this program we have to deal with smooth rational curves. For these curves we may use the explicit values of $d(m,0), a(m,0), d'(m,0)$ and $a'(m,0)$ for $m \leq
70$ given in \[2.12\]. To start the inductive machine for genus 0 we consider the following assertions. We fix a triple $(d,g,m)$ such that $d \geq g+3, m \geq m(0)+2, m
\geq 69g+27$ and $(d,g)$ has critical value $m$. By [@BE1] a general $C\in H(d,g)$ has maximal rank. Since by its very definition $H(d,g)$ is irreducible, it is sufficient to prove that for a general $X\in H(d,g)$ and a general $Y\in H(d,g)$ the restriction maps $r_{X,\Omega(m+1)}$ and $r_{Y,T\Pi(m-2)}$ have maximal rank. Just to fix the notations, we assume that to check that the homogeneous ideal of a general $X\in H(d,g)$ has the right number of generators, we need to check the surjectivity of $r_{X,\Omega(m+1)}$ and the injectivity of $r_{X,\Omega(m)}$. Since $m \geq 67g+27$ we have $d(m-1,0) \leq d(m-1,g) \leq d(m-1,0)+1$ and $d(m,0) \leq d(m,g) \leq d(m,0) + 1$. Hence $d(m-1,0)^{\ast} \leq d(m-1,g)^{\ast} \leq d(m-1,0)^{\ast} + 3$ and $d(m,0)^{\ast} \leq d(m,g)^{\ast} \leq
d(m,0)^{\ast} + 3$. Hence we have $20(d(m-1,g)^{\ast} - d(m-3,0)^{\ast} -27) \geq 4m$ and $20(d(m,g)^{\ast} - d(m-2,0) - 27) \geq 4m$. Since $(d,g)$ has critical value $m$ we have $d(m-1,g) < d \leq d(m,g)$. Hence as in section \[5\] we obtain just increasing the genus by $g$ that a suitable $Y\in H(d(m,g),g)$ has $r_{Y,\Omega(m+1)}$ surjective; furthermore, we obtain such $Y$ of the form $Y = X\cup A$ with $X\in H(d,g)$ and $A \quad d(m,g)-d$ lines 1-secant to $X$. Hence $r_{X,\Omega(m+1)}$ is surjective. Similarly we find $(Z,S)$ with $Z\in H(d(m-1,g)^{\ast},g), card(S) = a(m-1,g)^{\ast}$, with $r_{Z\cup S,\Omega(m)}$ bijective and such that $S$ is contained in $d(m-1,g) - d(m-1,g)^{\ast} +
1$ lines 1-secant to $Z$. Since $d > d(m-1,g)$, we have $d-d(m-1,g)^{\ast} \geq d(m-1,g)
- d(m-1,g)^{\ast} + 1$, there is a reducible element $\Omega\in H(d,g)$, $\Omega$ union of $Z$ and $d(m-1,g) - d(m-1,g)^{\ast} + 1$ lines containing $S$ with $r_{\Omega,\Omega(m)}$ injective. Hence we conclude, modulo the initial cases for $g = 0$ needed to start the induction. Here we will do these initial cases and conclude the paper. Here are the initial cases for $RO(m,0)$ and $RT(m,0), m \leq 56$, needed for the proof of Theorem \[0.2\].
\[8.1\] $RO(2,0)$ is true. Take a line $Y$ and a general $S \subset \Pi$ with $card(S) =
4$. Take a smooth quadric $Q$ containing $Y$ and S. We may assume that $S$ is general in $Q$. Hence we may apply Lemma \[3.1\] to $(\Omega|Q)(2,3)$.
\[8.2\] $RO(4,0)$ is true. Take a line $Y$ and a general $S \subset \Pi$ with $card(S) =
4$. By \[8.1\] $(Y,S)$ satisfies $RO(2,0)$. There is a smooth quadric $Q$ and distinct lines $D, D'$ of type $(1,0)$ on $Q, R, R', R"$ of type $(0,1)$ on $Q$ such that $D\cap Y \neq \O, D'\cap Y \neq \O, S = {D\cap R, D'\cap R, D\cap R",
D'\cap R'}$. Take as solution of $RO(4,0)$ the union of $Y, D, D', R, R', R"$, the nilpotents $\chi(P), P\in S$, and a suitable subset S’ of $Q$ with $card(S') = 5$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(3,2)$ it is sufficient to use Lemma \[3.1\] .
\[8.3\] $RO(6,0)$ is true. Take $(Y,Q)$ satisfying $RO(4,0)$. To obtain a solution of $RO(6,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 5 lines of type $(0,1)$ (10 all the lines of $T$ being 1-secant to $Y$), 5 nilpotents supported by $Sing(T)$ and 3 points on the singular set of a configuration of lines of type $(4,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(6,2)$ we use Lemma \[3.1\] .
\[8.4\] $RO(8,0)$ is true. Take $(Y,Q)$ satisfying $RO(6,0)$. To obtain a solution of $RO(8,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$ 1-secant to $Y$, 6 lines of type $(0,1)$ (3 of them being 1-secant to $Y$) and 3 nilpotents supported by $Sing(T)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(8,3)$ we use Lemma \[3.1\] .
\[8.5\] $RO(10,0)$ is true. Take $Y\in H(19,0)$ satisfying $RO(8,0)$ and a general quadric $Q$. To obtain a solution of $RO(10,0)$ we add in $Q$ to $Y$ a line of type $(1,0)$ 1-secant to Y, 4 lines of type $(0,1)$ and a subset $S$ with $card(S) = 27$ of the singular points of a configuration of lines of type $(7,6)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(10,7) $ we use Lemma \[3.8\] for the integers $y= 10, x = 7, a = 4, b = 8, t = 3$. We will call $RO(10,0)'$ the special configuration obtained in this way.
\[8.6\] $RO(12,0)$ is true. Take $(X,Q,S)$ satisfying $R(10,0)'$ and a general quadric $Q"$. We add in $Q"$ a line of type $(1,0)$ 1-secant to $X$, 8 general lines of type $(0,1)$ and a suitable set $S'$ of the intersection with $Q"$ of the configuration of lines of $Q$ supporting $S$ and with $card(S') = 7$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q"$ for $(\Omega|Q")(12,5)$ we use Lemma \[3.6\] for the integers $y = 12, x = 5, a = 3, b = 3, t = 2$. We will call $RO(12,0)'$ the special configuration obtained in this way.
\[8.7\] $RO(14,0)$ is true. Take $(Z,Q,S\cup S')$ the configuration satisfying $RO(12,0)'$ just obtained. To obtain a solution of $RO(14,0)$ we add in $Q$ the configuration, $T$, of type $(6,7)$, the nilpotents $\chi(P), P\in S\cup S'$ and 5 collinear points. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(9,8)$ we use Lemmas \[3.6\] or \[3.8\] for the integers $a =
1, b = t = 5$.
\[8.8\] $RO(16,0)$ is true. Take $(Y,Q,S)$ satisfying $RO(14,0)$. To obtain a solution of $RO(16,0)$ add in $Q$ the line of type $(1,0)$ containing $S$ and intersecting $Y$, 10 lines of type $(0,1)$ (among them the ones intersecting $S$ and exactly these ones 1-secant to $Y$), the nilpotents $\chi(P), P\in S$, and 18 of the singular points of a configuration of lines of type $(2,10)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(16,7)$ we use Lemma \[3.8\] for the integers $y = 16, x = 7, a = 2, b = 10, t = 8$.
\[8.9\] $RO(18,0)$ is true. Take $(Y,Q)$ satisfying $RO(14,0)$. To obtain a solution of $RO(18,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 10 lines of type $(0,1)$ (10 of the lines of $T$ being 1-secant to $Y$), 18 nilpotents supported by $Sing(T)$ and 40 points on the singular set of a configuration of lines of type $(12,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(17,9)$ we use Lemma \[3.8\].
\[8.10\] $RO(20,0)$ is true. Take $(Y,Q)$ satisfying $RO(18,0)$. To obtain a solution of $RO(20,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 12 lines of type $(0,1)$ (8 of the lines of T being 1-secant to Y), 40 nilpotents supported by $Sing(T)$ and 14 points on the singular set of a configuration of lines of type $(12,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(17,9)$ we use Lemma \[3.8\].
\[8.11\] $RO(22,0)$ is true. Take $(Y,Q)$ satisfying $RO(20,0)$. To obtain a solution of $RO(22,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 12 lines of type $(0,1)$ (4 of the lines of $T$ being 1-secant to $Y$), 14 nilpotents supported by $Sing(T)$ and 54 points on the singular set of a configuration of lines of type $(14,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(21,11)$ we use Lemma \[3.8\].
\[8.12\] $RO(24,0)$ is true. Take $(Y,Q)$ satisfying $RO(22,0)$. To obtain a solution of $RO(24,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 14 lines of type $(0,1)$ (4 of the lines of $T$ being 1-secant to $Y$), 54 nilpotents supported by Sing(T) and 38 points on the singular set of a configuration of lines of type (16,3). To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(21,11)$ we use Lemma \[3.8\].
\[8.13\] $RO(26,0)$ is true. Take $(Y,Q)$ satisfying $RO(24,0)$. To obtain a solution of $RO(26,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 16 lines of type $(0,1)$ (9 of the lines of $T$ being 1-secant to $Y$), 38 nilpotents supported by $Sing(T)$ and 27 points on the singular set of a configuration of lines of type $(18,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(24,11)$ we use Lemma \[3.8\].
\[8.14\] $RO(28,0)$ is true. Take $(Y,Q)$ satisfying $RO(26,0)$. To obtain a solution of $RO(28,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 18 lines of type $(0,1)$ (12 of the lines of $T$ being 1-secant to $Y$), 27 nilpotents supported by $Sing(T)$ and 23 points on the singular set of a configuration of lines of type $(19,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(27,11)$ we use Lemma \[3.8\].
\[8.15\] $RO(30,0)$ is true. Take $(Y,Q)$ satisfying $RO(28,0)$. To obtain a solution of $RO(30,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 19 lines of type $(0,1)$ (11 of the lines of $T$ being 1-secant to $Y$), 23 nilpotents supported by $Sing(T)$ and 28 points on the singular set of a configuration of lines of type $(20,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(29,12)$ we use Lemma \[3.8\].
\[8.16\] $RO(32,0)$ is true. Take $(Y,Q)$ satisfying $RO(30,0)$. To obtain a solution of $RO(32,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 20 lines of type $(0,1)$ (10 of the lines of $T$ being 1-secant to $Y$), 28 nilpotents supported by $Sing(T)$ and 44 points on the singular set of a configuration of lines of type $(20,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(31,13)$ we use Lemma \[3.8\].
\[8.17\] $RO(34,0)$ is true. Take $(Y,Q)$ satisfying $RO(32,0)$. To obtain a solution of $RO(34,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 20 lines of type $(0,1)$ (7 of the lines of $T$ being 1-secant to $Y$), 44 nilpotents supported by $Sing(T)$ and 73 points on the singular set of a configuration of lines of type $(23,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(32,15)$ we use Lemma \[3.8\].
\[8.18\] $RO(36,0)$ is true. Take $(Y,Q)$ satisfying $RO(34,0)$. To obtain a solution of $RO(36,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 23 lines of type $(0,1)$ (8 of the lines of $T$ being 1-secant to $Y$), 73 nilpotents supported by $Sing(T)$ and 10 points on the singular set of a configuration of lines of type $(24,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(33,14)$ we use Lemma \[3.8\].
\[8.19\] $RO(38,0)$ is true. Take $(Y,Q)$ satisfying $RO(36,0)$. To obtain a solution of $RO(38,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 24 lines of type $(0,1)$ (11 of the lines of $T$ being 1-secant to $Y$), 10 nilpotents supported by $Sing(T)$ and 65 points on the singular set of a configuration of lines of type $(25,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(38,15)$ we use Lemma \[3.8\].
\[8.20\] $RO(40,0)$ is true. Take $(Y,Q)$ satisfying $RO(38,0)$. To obtain a solution of $RO(40,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 26 lines of type $(0,1)$ (14 of the lines of $T$ being 1-secant to $Y$), 65 nilpotents supported by $Sing(T)$ and 20 points on the singular set of a configuration of lines of type $(26,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(38,15)$ we use Lemma \[3.8\].
\[8.21\] RO(42,0) is true. Take $(Y,Q)$ satisfying $RO(40,0)$. To obtain a solution of $RO(42,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 26 lines of type $(0,1)$ (21 of the lines of $T$ being 1-secant to $Y$), 10 nilpotents supported by $Sing(T)$ and 109 points on the singular set of a configuration of lines of type $(26,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(42,17)$ we use Lemma \[3.8\].
\[8.22\] $RO(44,0)$ is true. Take $(Y,Q)$ satisfying $RO(42,0)$. To obtain a solution of $RO(44,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 26 lines of type $(0,1)$ (10 of the lines of $T$ being 1-secant to $Y$), 109 nilpotents supported by $Sing(T)$ and 90 points on the singular set of a configuration of lines of type $(27,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(40,19)$ we use Lemma \[3.8\].
\[8.23\] $RO(46,0)$ is true. Take $(Y,Q)$ satisfying $RO(44,0)$. To obtain a solution of $RO(46,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 27 lines of type $(0,1)$ (12 of the lines of $T$ being 1-secant to $Y$), 90 nilpotents supported by $Sing(T)$ and 84 points on the singular set of a configuration of lines of type $(27,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(42,20)$ we use Lemma \[3.8\].
\[8.245\] $RO(48,0)$ is true. Take $(Y,Q)$ satisfying $RO(46,0)$. To obtain a solution of $RO(48,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 27 lines of type $(0,1)$ (6 of the lines of $T$ being 1-secant to $Y$), 84 nilpotents supported by $Sing(T)$ and 93 points on the singular set of a configuration of lines of type $(30,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(44,22)$ we use Lemma \[3.8\].
\[8.25\] $RO(50,0)$ is true. Take $(Y,Q)$ satisfying $RO(48,0)$. To obtain a solution of $RO(50,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 30 lines of type $(0,1)$ (7 of the lines of $T$ being 1-secant to $Y$), 93 nilpotents supported by $Sing(T)$ and 119 points on the singular set of a configuration of lines of type $(34,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(47,21)$ we use Lemma \[3.8\].
\[8.26\] $RO(52,0)$ is true. Take $(Y,Q)$ satisfying $RO(50,0)$. To obtain a solution of $RO(52,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 34 lines of type $(0,1)$ (21 of the lines of $T$ being 1-secant to $Y$), 23 nilpotents supported by $Sing(T)$ and 27 points on the singular set of a configuration of lines of type $(35,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(49,19)$ we use Lemma \[3.8\].
\[8.27\] $RO(54,0)$ is true. Take $(Y,Q)$ satisfying $RO(52,0)$. To obtain a solution of $RO(54,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 35 lines of type $(0,1)$ (28 of the lines of $T$ being 1-secant to $Y$), 27 nilpotents supported by $Sing(T)$ and 69 points on the singular set of a configuration of lines of type $(35,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(54,20)$ we use Lemma \[3.8\].
\[8.28\] $RO(56,0)$ is true. Take $(Y,Q)$ satisfying $RO(54,0)$. To obtain a solution of $RO(56,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 35 lines of type $(0,1)$ (36 of the lines of $T$ being 1-secant to $Y$), 23 nilpotents supported by $Sing(T)$ and 152 points on the singular set of a configuration of type $(37,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(55,22)$ we use Lemma \[3.8\].
\[8.29\] $RO(58,0)$ is true. Take $(Y,Q)$ satisfying $RO(56,0)$. To obtain a solution of $O(58,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 37 lines of type $(0,1)$ (9 of the lines of $T$ being 1-secant to $Y$), 152 nilpotents supported by $Sing(T)$ and 87 points on the singular set of a configuration of lines of type $(38,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(54,22)$ we use Lemma \[3.8\].
\[8.30\] $RO(60,0)$ is true. Take $(Y,Q)$ satisfying $RO(58,0)$. To obtain a solution of $RO(60,0$ add in $Q$ the union, $T$, of 3 line of type $(1,0)$, 38 lines of type $(0,1)$ (14 of the lines of $T$ being 1-secant to $Y$), 87 nilpotents supported by $Sing(T)$ and 37 points on the singular set of a configuration of lines of type $(42,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(60,19)$ we use Lemma \[3.8\].
\[8.31\] $RO(62,0)$ is true. Take $(Y,Q)$ satisfying $RO(60,0)$. To obtain a solution of $RO(62,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 42 lines of type $(0,1)$ (38 of the lines of $T$ being 1-secant to $Y$), 37 nilpotents supported by $Sing(T)$ and 4 points on the singular set of a configuration of lines of type $(40,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(62,21)$ we use Lemma \[3.8\].
\[8.32\] $RO(64,0)$ is true. Take $(Y,Q)$ satisfying $RO(62,0)$. To obtain a solution of $RO(64,0)$ add in $Q$ the union, $T$, of 1 line of type $(1,0)$, 40 lines of type $(0,1)$ (5 of the lines of $T$ being 1-secant to $Y$), 4 nilpotents supported by $Sing(T)$ and 181 points on the singular set of a configuration of lines of type $(40,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(64,25)$ we use Lemma \[3.8\].
\[8.33\] $RO(66,0)$ is true. Take $(Y,Q)$ satisfying $RO(64,0)$. To obtain a solution of $RO(66,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 40 lines of type $(0,1)$ (26 of the lines of $T$ being 1-secant to $Y$), 27 nilpotents supported by $Sing(T)$ and 194 points on the singular set of a configuration of lines of type $(44,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(62,27)$ we use Lemma \[3.8\].
\[8.34\] $RO(68,0)$ is true. Take $(Y,Q)$ satisfying $RO(66,0)$. To obtain a solution of $RO(68,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 44 lines of type $(0,1)$ (23 of the lines of $T$ being 1-secant to $Y$), 194 nilpotents supported by $Sing(T)$ and 27 points on the singular set of a configuration of lines of type $(46,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(64,25)$ we use Lemma \[3.8\].
\[8.35\] $RO(70,0)$ is true. Take $(Y,Q)$ satisfying $RO(68,0)$. To obtain a solution of $RO(70,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 46 lines of type $(0,1)$ (28 of the lines of $T$ being 1-secant to $Y$), 27 nilpotents supported by $Sing(T)$ and 82 suitable points. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(70,25)$ we use Lemma \[3.8\].
\[8.36\] $RO(3,0)$ is true. Take a point $P$ and a smooth quadric $Q'$ with $P \not\in Q'$. Abusing notations, we may say that $\{P\}$ satisfies $RO(1,0)$ because $\{P\}$ is the complete intersection of 3 planes and hence its minimal free resolution is as it should be. To obtain a solution of $RO(3,0)$ add in $Q'$ to $\{P\}$ the union, $T$, of 1 line of type $(1,0)$, 2 lines of type $(0,1)$ and 5 suitable points; and note that any 6 “suitable” points are contained in a smooth quadric. To apply Horace Lemma \[1.3\] in the smooth quadric $Q'$ for $(\Omega|Q')(3,2)$ we use Lemma \[3.1\].
\[8.37\] $RO(5,0)$ is true. Take $(Y,Q,S)$ satisfying $RO(3,0)$ and a smooth quadric $Q'$ intersecting transversally $Y$ and with $S\cap Q' = \O$. To obtain a solution of $RO(5,0)$ add in $Q'$ the union, $T$, of 1 line of type $(1,0)$ 1-secant to $Y$, 3 general lines of type $(0,1)$ and 7 points on the intersection with $Q'$ of the 4 lines of $Q$ containing $S$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(5,3)$ we use Lemma \[3.8\] for the integers $y = 5, x = 3, a = 2, b = 4$, $t = 3$.
\[8.38\] $RO(7,0)$ is true. Take $(Y,Q)$ satisfying $RO(5,0)$. To obtain a solution of $RO(7,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 4 lines of type $(0,1)$ (5 of the lines of $T$ being 1-secant to $Y$), 13 nilpotents supported by $Sing(T)$ and 4 points on the singular set of a configuration of lines of type $(3,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(4,4)$ we use Lemma \[3.8\].
\[8.39\] $RO(9,0)$ is true. Take $(Y,Q)$ satisfying $RO(7,0)$. To obtain a solution of $RO(9,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$ (both of them 1-secant to $Y$), 4 lines of type $(0,1)$ disjoint from $Y$, 4 non-collinear nilpotents supported by $Sing(T)$ and 15 points on the singular set of a configuration of lines of type $(8,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(9,5)$ we use Lemma \[3.8\] with $a =
2, b = 8$ and $t = 7$.
\[8.40\] $RO(11,0)$ is true. Take $(Y,Q)$ satisfying $RO(9,0)$. To obtain a solution of $RO(11,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 8 lines of type $(0,1)$ (9 of the lines of $T$ being 1-secant to $Y$), 15 nilpotents supported by $Sing(T)$ and 2 points on the singular set of a configuration of lines of type $(7,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(10,4)$ we use Lemma \[3.8\].
\[8.41\] $RO(13,0)$ is true. Take $(Y,Q)$ satisfying $RO(11,0)$. To obtain a solution of $RO(13,0)$ add in $Q$ the union, $T$, of 1 line of type $(1,0)$, 7 lines of type $(0,1)$ (3 of the lines of $T$ being 1-secant to $Y$), 2 nilpotents supported by $Sing(T)$ and 25 points on the singular set of a configuration of lines of type $(9,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(13,7)$ we use Lemma \[3.8\].
\[8.42\] $RO(15,0)$ is true. Take $(Y,Q)$ satisfying $RO(13,0)$. To obtain a solution of $RO(15,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 9 lines of type $(0,1)$ (10 of the lines of $T$ being 1-secant to $Y$), 25 nilpotents supported by $Sing(T)$ and 16 points on the singular set of a configuration of lines of type $(11,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(13,7)$ we use Lemma \[3.8\].
\[8.43\] $RO(17,0$ is true. Take $(Y,Q)$ satisfying $RO(15,0)$. To obtain a solution of $RO(17,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 11 lines of type $(0,1)$ (7 of the lines of $T$ being 1-secant to $Y$), 16 nilpotents supported by $Sing(T)$ and 9 points on the singular set of a configuration of lines of type $(13,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(16,7)$ we use Lemma \[3.8\].
\[8.44\] $RO(19,0)$ is true. Take $(Y,Q)$ satisfying $RO(17,0$. To obtain a solution of $RO(19,0)$ add in $Q$ the union, $T$, of 1 line of type $(1,0)$, 13 lines of type $(0,1)$ (10 of the lines of $T$ being 1-secant to $Y$), 9 nilpotents supported by $Sing(T)$ and 6 points on the singular set of a configuration of lines of type $(14,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(19,7)$ we use Lemma \[3.8\].
\[8.45\] $RO(21,0)$ is true. Take $(Y,Q)$ satisfying $RO(19,0)$. To obtain a solution of $RO(21,0)$ add in $Q$ the union, $T$, of 1 line of type $(1,0)$, 14 lines of type $(0,1)$ (7 of the lines of $T$ being 1-secant to $Y$), 6 nilpotents supported by $Sing(T)$ and 9 points on the singular set of a configuration of lines of type $(15,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(21,8)$ we use Lemma \[3.8\].
\[8.46\] $RO(23,0)$ is true. Take $(Y,Q)$ satisfying $RO(21,0)$. To obtain a solution of $RO(23,$ add in $Q$ the union, $T$, of 1 line of type $(1,0)$, 15 lines of type $(0,1)$ (10 of the lines of $T$ being 1-secant to $Y$), 9 nilpotents supported by $Sing(T)$ and 20 points on the singular set of a configuration of lines of type $(15,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(23,9)$ we use Lemma \[3.8\].
\[8.47\] $RO(25,0)$ is true. Take $(Y,Q)$ satisfying $RO(23,0)$. To obtain a solution of $RO(25,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 15 lines of type $(0,1)$ (6 of the lines of $T$ being 1-secant to $Y$), 20 nilpotents supported by $Sing(T)$ and 41 points on the singular set of a configuration of lines of type $(15,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(24,9)$ we use Lemma \[3.8\].
\[8.48\] $RO(27,0)$ is true. Take $(Y,Q)$ satisfying $RO(25,0)$. To obtain a solution of $RO(27,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 15 lines of type $(0,1)$ (14 of the lines of $T$ being 1-secant to $Y$), 41 nilpotents supported by $Sing(T)$ and 74 points on the singular set of a configuration of lines of type $(17,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(31,13)$ we use Lemma \[3.8\].
\[8.49\] $RO(29,0)$ is true. Take $(Y,Q)$ satisfying $RO(27,0)$. To obtain a solution of $RO(29,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 17 lines of type $(0,1)$ (5 of the lines of $T$ being 1-secant to $Y$), 74 nilpotents supported by $Sing(T)$ and 25 points on the singular set of a configuration of lines of type $(21,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(25,13)$ we use Lemma \[3.8\].
\[8.50\] $RO(31,0)$ is true. Take $(Y,Q)$ satisfying $RO(29,0$. To obtain a solution of $RO(31,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 21 lines of type $(0,1)$ (6 of the lines of $T$ being 1-secant to $Y$) and 23 nilpotents supported by $Sing(T)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(30,11)$ we use Lemma \[3.8\].
\[8.51\] $RO(33,0)$ is true. Take $(Y,Q)$ satisfying $RO(31,0)$. To obtain a solution of $RO(33,0)$ add in $Q$ the union, $T$, of 1 line of type $(1,0)$, 26 lines of type $(0,1)$ (one of the lines of $T$ being 1-secant to $Y$) and 83 points on the singular set of a configuration of lines of type $(12,7)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(33,8)$ we use Lemma \[3.8\].
\[8.52\] $RO(35,0)$ is true. Take $(Y,Q)$ satisfying $RO(33,0)$. To obtain a solution of $RO(35,0)$ add in $Q$ the union, $T$, of 7 lines of type $(1,0)$, 12 lines of type $(0,1)$ (18 of the lines of $T$ being 1-secant to $Y$), 83 nilpotents supported by $Sing(T)$ and 54 points on the singular set of a configuration of lines of type $(23,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(29,24)$ we use Lemma \[3.8\].
\[8.53\] $RO(37,0)$ is true. Take $(Y,Q)$ satisfying $RO(35,0)$. To obtain a solution of $RO(37,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 23 lines of type $(0,1)$ (11 of the lines of $T$ being 1-secant to $Y$), 54 nilpotents supported by $Sing(T)$ and 49 points on the singular set of a configuration of lines of type $(25,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(35,15)$ we use Lemma \[3.8\].
\[8.54\] $RO(39,0)$ is true. Take $(Y,Q)$ satisfying $RO(37,0)$. To obtain a solution of $RO(39,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 25 lines of type $(0,1)$ (26 of the lines of $T$ being 1-secant to $Y$), 49 nilpotents supported by $Sing(T)$ and 56 points on the singular set of a configuration of lines of type $(25,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(38,15)$ we use Lemma \[3.8\].
\[8.55\] $RO(41,0)$ is true. Take $(Y,Q)$ satisfying $RO(39,0)$. To obtain a solution of $RO(41,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 25 lines of type $(0,1)$ (9 of the lines of $T$ being 1-secant to $Y$), 56 nilpotents supported by $Sing(T)$ and 77 points on the singular set of a configuration of lines of type $(26,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(39,17)$ we use Lemma \[3.8\].
\[8.56\] $RO(43,0)$ is true. Take $(Y,Q)$ satisfying $RO(41,0)$. To obtain a solution of $RO(43,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 26 lines of type $(0,1$ (9 of the lines of $T$ being 1-secant to $Y$), 77 nilpotents supported by $Sing(T)$ and 114 points on the singular set of a configuration of lines of type $(29,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(41,18)$ we use Lemma \[3.8\].
\[8.57\] $RO(45,0)$ is true. Take $(Y,Q)$ satisfying $RO(43,0)$. To obtain a solution of $RO(45,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 29 lines of type $(0,1)$ (28 of the lines of $T$ being 1-secant to $Y$), 114 nilpotents supported by $Sing(T)$ and 35 points on the singular set of a configuration of lines of type $(29,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(42,17)$ we use Lemma \[3.8\].
\[8.58\] $RO(47,0)$ is true. Take $(Y,Q)$ satisfying $RO(45,0)$. To obtain a solution of $RO(47,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 29 lines of type $(0,1)$ (10 of the lines of $T$ being 1-secant to $Y$), 35 nilpotents supported by $Sing(T)$ and 104 points on the singular set of a configuration of lines of type $(31,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(46,19)$ we use Lemma \[3.8\].
\[8.59\] $RO(49,0)$ is true. Take $(Y,Q)$ satisfying $RO(47,0)$. To obtain a solution of $RO(49,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 31 lines of type $(0,1)$ (15 of the lines of $T$ being 1-secant to $Y$), 104 nilpotents supported by $Sing(T)$ and 49 points on the singular set of a configuration of lines of type $(34,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(46,19)$ we use Lemma \[3.8\].
\[8.60\] $RO(51,0)$ is true. Take $(Y,Q)$ satisfying RO(49,0). To obtain a solution of $RO(51,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 34 lines of type $(0,1)$ (17 of the lines of $T$ being 1-secant to $Y$), 49 nilpotents supported by $Sing(T)$ and 6 points on the singular set of a configuration of lines of type $(33,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(50,18)$ we use Lemma \[3.8\].
\[8.61\] $RO(53,0)$ is true. Take $(Y,Q)$ satisfying $RO(51,0)$. To obtain a solution of $RO(53,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 33 lines of type $(0,1)$ (7 of the lines of $T$ being 1-secant to $Y$), 6 nilpotents supported by $Sing(T)$ and 135 points on the singular set of a configuration of lines of type $(34,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(53,21)$ we use Lemma \[3.8\].
\[8.62\] $RO(55,0)$ is true. Take $(Y,Q)$ satisfying $RO(53,0)$. To obtain a solution of $RO(55,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 34 lines of type $(0,1)$ (37 of the lines of $T$ being 1-secant to $Y$), 135 nilpotents supported by $Sing(T)$ and 128 points on the singular set of a configuration of lines of type $(32,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(52,22)$ we use Lemma \[3.8\].
\[8.63\] $RO(57,0)$ is true. Take $(Y,Q)$ satisfying $RO(55,0)$. To obtain a solution of $RO(57,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 32 lines of type $(0,1)$ (7 of the lines of $T$ being 1-secant to $Y$), 128 nilpotents supported by $Sing(T)$ and 139 points on the singular set of a configuration of lines of type $(32,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(53,26)$ we use Lemma \[3.8\].
\[8.64\] $RO(59,0)$ is true. Take $(Y,Q)$ satisfying $RO(57,0)$. To obtain a solution of $RO(59,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 35 lines of type $(0,1)$ (4 of the lines of $T$ being 1-secant to $Y$), 139 nilpotents supported by $Sing(T)$ and 170 points on the singular set of a configuration of lines of type $(39,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(55,25)$ we use Lemma \[3.8\].
\[8.65\] $RO(61,0)$ is true. Take $(Y,Q)$ satisfying $RO(59,0)$. To obtain a solution of $RO(61,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 39 lines of type $(0,1)$ (19 of the lines of $T$ being 1-secant to $Y$), 170 nilpotents supported by $Sing(T)$ and 41 points on the singular set of a configuration of lines of type $(41,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(57,23)$ we use Lemma \[3.8\].
\[8.66\] $RO(63,0)$ is true. Take $(Y,Q)$ satisfying $RO(61,0)$. To obtain a solution of $RO(53,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 41 lines of type $(0,1)$ (all the lines of $T$ being 1-secant to $Y$), 41 nilpotents supported by $Sing(T)$ and 112 points on the singular set of a configuration of lines of type $(43,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(63,23)$ we use Lemma \[3.8\].
\[8.67\] $RO(65,0)$ is true. Take $(Y,Q)$ satisfying $RO(63,0)$. To obtain a solution of $RO(65,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 43 lines of type $(0,1)$ (29 of the lines of $T$ being 1-secant to $Y$), 112 nilpotents supported by $Sing(T)$ and 15 points on the singular set of a configuration of lines of type $(47,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(63,23)$ we use Lemma \[3.8\].
\[8.68\] $RO(67,0)$ is true. Take $(Y,Q)$ satisfying $RO(65,0)$. To obtain a solution of $RO(67,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 47 lines of type $(0,1)$ (16 of the lines of $T$ being 1-secant to $Y$), 15 nilpotents supported by $Sing(T)$ and 134 points on the singular set of a configuration of lines of type $(44,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(53,21)$ we use Lemma \[3.8\].
\[8.69\] $RO(69,0)$ is true. Take $(Y,Q)$ satisfying $RO(67,0)$. To obtain a solution of $RO(69,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 44 lines of type $(0,1)$ (6 of the lines of $T$ being 1-secant to $Y$), 134 nilpotents supported by $Sing(T)$ and 77 suitable points. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(\Omega|Q)(52,22)$ we use Lemma \[3.8\].
\[8.70\] $RT(3,0)"$ is true. Take a line $Y$. Abusing notations, we may say that $Y$ satisfies $RT(1,0)$ because it is a complete intersection of two planes. Take a smooth quadric $Q$ intersecting transversally $Y$. To obtain a solution of $RT(3,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$ 1-secant to $Y$, one general line of type $(0,1)$ and 4 suitable points. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(0,0)$ we use Lemma \[3.2\].
\[8.71\] $RT(5,0)$ is true. Take $(Y,S)$ satisfying $RT(3,0)"$. Take a smooth quadric $Q$ such that the points on $S$ are the singular points of a configuration, $T'$, of lines of type $(2,2)$. To obtain a solution of $RT(5,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$ (one of them 1-secant to $Y$), 3 general lines of type $(0,1)$ and the 4 nilpotents $\chi(P)$; we assume $T' \subset T$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(0,0)$ we use Lemma \[3.2\].
\[8.72\] $RT(7,0)$ is true. Take $(Y,Q)$ satisfying $RT(5,0)$. To obtain a solution of $RT(7,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$ 1-secant to $Y$, 3 general lines of type $(0,1)$ and 10 points on the singular set of a configuration of lines of type $(6,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(4,2)$ we use Lemma \[3.9\] with $y = 4, x = 2, b = t = 5, a = 2$.
\[8.73\] $RT(9,0)$ is true. Take $(Y,Q)$ satisfying $RT(5,0)$. To obtain a solution of $RT(9,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 6 lines of type $(0,1)$ (6 of the lines of $T$ being 1-secant to $Y$), 10 nilpotents supported by $Sing(T)$ and 4 points on the singular set of a configuration of lines of type $(5,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(5,1)$ we use Lemma \[3.9\] with $y = 5, x = 1, b = t
= 4, a = 1$.
\[8.74\] In order to prove $RT(13,0)$ we define the assertion $RT(11,0)'$ as $RT(11,0)$ with the only modification that the corresponding 28 points are in the singular set of a configuration of lines of type $(6,5)$ on $Q$. Here we will check that $RT(11,0)'$ is true. Take $(Y,Q)$ satisfying RT(9,0). To obtain a solution of $RT(11,0)'$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 5 lines of type $(0,1)$ (5 of the lines of $T$ being 1- secant to $Y$), 4 nilpotents supported by $Sing(T)$ and 28 points on the singular set of a configuration of lines of type $(6,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(8,4)$ we use Lemma \[3.9\] for $y = 8, x = 4, a = 4, b = t = 7$.
\[8.75\] $RT(13)$ is true. Take $(Y,Q)$ satisfying $RT(11,0)'$ and a general quadric $Q'$. To obtain a solution of $RT(13,0)$ add in $Q'$ the union, $T$, of 5 lines of type $(1,0)$ and 6 lines of type $(0,1)$ (one of the lines of $T$ being 1-secant to $Y$), 28 nilpotents supported by $Sing(T)$ and 28 suitable points on a configuration of lines of type $(6,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(6,5)$ we use Lemma \[3.9\] for $y = 6, x = 5, a = 5, b = 6, t = 4$.
\[8.76\] $RT(15,0)$ is true. Take $(Y,Q)$ satisfying $RT(13,0)$. To obtain a solution of $RT(15,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 6 lines of type $(0,1)$ (9 of the lines of $T$ being 1-secant to $Y$), 28 nilpotents supported by $Sing(T)$ and 31 points on the singular set of a configuration of lines of type $(7,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(8,7)$ we use Lemma \[3.9\] for $y = 8,
x = 7, b = 7, a = 5, t = 3$.
\[8.77\] $RT(17,0)$ is true. Take $(Y,Q)$ satisfying $RT(15,0)$. To obtain a solution of $RT(17,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 7 lines of type $(0,1)$ (8 of the lines of $T$ being 1-secant to $Y$), 31 nilpotents supported by $Sing(T)$ and 39 points on the singular set of a configuration of lines of type $(8,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(10,8)$ we use Lemma \[3.9\].
\[8.78\] $RT(19,0)$ is true. Take $(Y,Q)$ satisfying $RT(17,0)$. To obtain a solution of $RT(19,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 8 lines of type $(0,1)$ (12 of the lines of $T$ being 1-secant to $Y$), 39 nilpotents supported by $Sing(T)$ and 54 points on the singular set of a configuration of lines of type $(12,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(12,9)$ we use Lemma \[3.9\].
\[8.79\] $RT(21,0)$ is true. Take $(Y,Q)$ satisfying $RT(19,0)$. To obtain a solution of $RT(21,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 12 lines of type $(0,1)$ (11 of the lines of $T$ being 1-secant to $Y$), 54 nilpotents supported by $Sing(T)$ and 17 points on the singular set of a configuration of lines of type $(13,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(14,7)$ we use Lemma \[3.9\].
\[8.80\] $RT(23,0)$ is true. Take $(Y,Q)$ satisfying $RT(21,0)$. To obtain a solution of $RT(23,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 13 lines of type $(0,1)$ (6 of the lines of $T$ being 1-secant to $Y$), 17 nilpotents supported by $Sing(T)$ and 46 points on the singular set of a configuration of lines of type $(16,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(19,8)$ we use Lemma \[3.9\].
\[8.81\] $RT(25,0)$ is true. Take $(Y,Q)$ satisfying $RT(23,0)$. To obtain a solution of $RT(25,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 16 lines of type $(0,1)$ (17 of the lines of $T$ being 1-secant to $Y$), 46 nilpotents supported by $Sing(T)$ and 15 points on the singular set of a configuration of lines of type $(16,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(20,7)$ we use Lemma \[3.9\].
\[8.82\] $RT(27,0)$ is true. Take $(Y,Q)$ satisfying $RT(25,0)$. To obtain a solution of $RT(27,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 16 lines of type $(0,1)$ (16 of the lines of $T$ being 1-secant to $Y$), 15 nilpotents supported by $Sing(T)$ and 66 points on the singular set of a configuration of lines of type $(17,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(24,9)$ we use Lemma \[3.9\].
\[8.83\] $RT(29,0)$ is true. Take $(Y,Q)$ satisfying $RT(27,0)$. To obtain a solution of $RT(29,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 17 lines of type $(0,1)$ (19 of the lines of $T$ being 1-secant to $Y$), 66 nilpotents supported by $Sing(T)$ and 49 points on the singular set of a configuration of lines of type $(19,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(23,10)$ we use Lemma \[3.9\].
\[8.84\] $RT(31,0)$ is true. Take $(Y,Q)$ satisfying $RT(29,0)$. To obtain a solution of $RT(31,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 19 lines of type $(0,1)$ (14 of the lines of $T$ being 1-secant to $Y$), 49 nilpotents supported by $Sing(T)$ and 39 points on the singular set of a configuration of lines of type $(21,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(26,10)$ we use Lemma \[3.9\].
\[8.85\] $RT(33,0)$ is true. Take $(Y,Q)$ satisfying $RT(31,0)$. To obtain a solution of $RT(33,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 21 lines of type $(0,1)$ (20 of the lines of $T$ being 1-secant to $Y$), 39 nilpotents supported by $Sing(T)$ and 38 points on the singular set of a configuration of lines of type $(22,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(29,10)$ we use Lemma \[3.9\].
\[8.86\] $RT(35,0)$ is true. Take $(Y,Q)$ satisfying $RT(33,0)$. To obtain a solution of $RT(35,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 22 lines of type $(0,1)$ (18 of the lines of $T$ being 1-secant to $Y$), 38 nilpotents supported by $Sing(T)$ and 54 points on the singular set of a configuration of lines of type $(22,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(31,11)$ we use Lemma \[3.9\].
\[8.87\] $RT(37,0)$ is true. Take $(Y,Q)$ satisfying $RT(35,0)$. To obtain a solution of $RT(37,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 22 lines of type $(0,1)$ (13 of the lines of $T$ being 1-secant to $Y$), 54 nilpotents supported by $Sing(T)$ and 71 points on the singular set of a configuration of lines of type $(22,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(32,13)$ we use Lemma \[3.9\].
\[8.88\] $RT(39,0)$ is true. Take $(Y,Q)$ satisfying $RT(37,0)$. To obtain a solution of $RT(39,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 22 lines of type $(0,1)$ (11 of the lines of $T$ being 1-secant to $Y$), 71 nilpotents supported by $Sing(T)$ and 109 points on the singular set of a configuration of lines of type $(25,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(33,15)$ we use Lemma \[3.9\].
\[8.89\] $RT(41,0)$ is true. Take $(Y,Q)$ satisfying $RT(39,0)$. To obtain a solution of $RT(41,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 25 lines of type $(0,1)$ (14 of the lines of $T$ being 1-secant to $Y$), 109 nilpotents supported by $Sing(T)$ and 43 points on the singular set of a configuration of lines of type $(26,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(34,14)$ we use Lemma \[3.9\].
\[8.90\] $RT(43,0)$ is true. Take $(Y,Q)$ satisfying $RT(41,0)$. To obtain a solution of $RT(43,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 26 lines of type $(0,1)$ (19 of the lines of $T$ being 1-secant to $Y$), 43 nilpotents supported by $Sing(T)$ and 111 points on the singular set of a configuration of lines of type $(28,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(39,15)$ we use Lemma \[3.9\].
\[8.91\] $RT(45,0)$ is true. Take $(Y,Q)$ satisfying $RT(43,0)$. To obtain a solution of $RT(45,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 28 lines of type $(0,1)$ (one of the lines of $T$ being 1-secant to $Y$), 111 nilpotents supported by $Sing(T)$ and 67 points on the singular set of a configuration of lines of type $(30,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(39,15)$ we use Lemma \[3.9\].
\[8.92\] $RT(47,0)$ is true. Take $(Y,Q)$ satisfying $RT(45,0)$. To obtain a solution of $RT(47,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 30 lines of type $(0,1)$ (10 of the lines of $T$ being 1-secant to $Y$), 67 nilpotents supported by $Sing(T)$ and 34 points on the singular set of a configuration of lines of type $(31,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(42,15)$ we use Lemma \[3.9\].
\[8.93\] $RT(49,0)$ is true. Take $(Y,Q)$ satisfying $RT(47,0)$. To obtain a solution of $RT(49,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 31 lines of type $(0,1)$ (5 of the lines of $T$ being 1-secant to $Y$), 34 nilpotents supported by $Sing(T)$ and 14 points on the singular set of a configuration of lines of type $(34,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(45,16)$ we use Lemma \[3.9\].
\[8.94\] $RT(51,0)$ is true. Take $(Y,Q)$ satisfying $RT(49,0)$. To obtain a solution of $RT(51,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 34 lines of type $(0,1)$ (15 of the lines of $T$ being 1-secant to $Y$), 14 nilpotents supported by $Sing(T)$ and 27 points on the singular set of a configuration of lines of type $(35,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(48,15)$ we use Lemma \[3.9\].
\[8.95\] $RT(53,0)$ is true. Take $(Y,Q)$ satisfying $RT(51,0)$. To obtain a solution of RT(53,0) add in $Q$ the union, $T$, of one line of type $(1,0)$, 35 lines of type $(0,1)$ (28 of the lines of $T$ being 1-secant to $Y$), 27 nilpotents supported by $Sing(T)$ and 21 points on the singular set of a configuration of lines of type $(36,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(50,16)$ we use Lemma \[3.9\].
\[8.96\] $RT(55,0)$ is true. Take $(Y,Q)$ satisfying $RT(53,0)$. To obtain a solution of $RT(55,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 36 lines of type $(0,1)$ (22 of the lines of $T$ being 1-secant to $Y$), 21 nilpotents supported by $Sing(T)$ and 52 points on the singular set of a configuration of lines of type $(36,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(52,17)$ we use Lemma \[3.9\].
\[8.97\] $RT(57,0)$ is true. Take $(Y,Q)$ satisfying $RT(55,0)$. To obtain a solution of $RT(57,0)$ add in $Q$ the union, $T$, of two lines of type $(1,0)$, 36 lines of type $(0,1)$ (18 of the lines of $T$ being 1-secant to $Y$), 52 nilpotents supported by $Sing(T)$ and 104 points on the singular set of a configuration of lines of type $(39,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(53,19)$ we use Lemma \[3.9\].
\[8.98\] $RT(59,0)$ is true. Take $(Y,Q)$ satisfying $RT(57,0)$. To obtain a solution of $RT(59,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 39 lines of type $(0,1)$ (29 of the lines of $T$ being 1-secant to $Y$), 104 nilpotents supported by $Sing(T)$ and 4 points on the singular set of a configuration of lines of type $(39,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(54,18)$ we use Lemma \[3.9\].
\[8.99\] $RT(61,0)$ is true. Take $(Y,Q)$ satisfying $RT(59,0)$. To obtain a solution of $RT(61,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 39 lines of type $(0,1)$ (5 of the lines of $T$ being 1-secant to $Y$), 4 nilpotents supported by $Sing(T)$ and 98 points on the singular set of a configuration of lines of type $(41,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(58,20)$ we use Lemma \[3.9\].
\[8.100\] $RT(63,0)$ is true. Take $(Y,Q)$ satisfying $RT(61,0)$. To obtain a solution of $RT(63,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 41 lines of type $(0,1)$ (29 of the lines of $T$ being 1-secant to $Y$), 98 nilpotents supported by $Sing(T)$ and 32 points on the singular set of a configuration of lines of type $(41,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(58,20)$ we use Lemma \[3.9\].
\[8.101\] $RT(65,0)$ is true. Take $(Y,Q)$ satisfying $RT(63,0)$. To obtain a solution of $RT(65,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 41 lines of type $(0,1)$ (33 of the lines of $T$ being 1-secant to $Y$), 32 nilpotents supported by $Sing(T)$ and 176 points on the singular set of a configuration of lines of type $(41,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(62,21)$ we use Lemma \[3.9\].
\[8.102\] $RT(67,0)$ is true. Take $(Y,Q)$ satisfying $RT(65,0)$. To obtain a solution of $RT(67,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 41 lines of type $(0,1)$ (17 of the lines of $T$ being 1-secant to $Y$), 176 nilpotents supported by $Sing(T)$ and 152 points on the singular set of a configuration of lines of type $(43,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(60,24)$ we use Lemma \[3.9\].
\[8.103\] $RT(69,0)$ is true. Take $(Y,Q)$ satisfying $RT(67,0)$. To obtain a solution of $RT(69,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 43 lines of type $(0,1)$ (27 of the lines of $T$ being 1-secant to $Y$), 152 nilpotents supported by $Sing(T)$ and 78 suitable points. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(63,24)$ we use Lemma \[3.9\].
\[8.104\] $RT(2,0)$ is true because a rational normal curve has the expected minimal free resolution.
\[8.105\] In order to prove $RT(8,0)$ we introduce the following variation, called $RT(4,0)'$, of the assertion $RT(4,0)$. $RT(4,0)'$: There exists $(Y,Q,S,A)$ with $Y\in H(4,0)$, $Q$ smooth quadric intersecting transversally $Y$, $S
\subset Q, card(S) = 9$, and $S$ is contained in the singular set of a configuration $A$ of lines of type $(6,4)$, such that 7 of the lines of $A$ are 1-secant to $Y$, and with $r_{Y\cup S,T\Pi(2)}$ bijective. Here we will check $RT(4,0)'$. Take $Y\in H(3,0)$ (hence satisfing $RT(2,0)$) and add in a general quadric $Q$ a line 1-secant to $Y$ and 9 suitable points. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(2,1)$ we use Lemma \[3.2\].
\[8.106\] To prove $RT(8,0)$ we introduce the following variation, called $RT \; ( \, 6 \, , \, 0 \, )'$, of the assertion $ \; RT \; ( \, 6 \, , \, 0 \, )$. $ \; RT \; ( \, 6 \, , \, 0 \, )'$: There exists $(Z,Q,Q',S,S',A)$ such that $Z\in H(9,0)$, $Q$ and $Q'$ are smooth quadrics intersecting transversally $Z, Q \neq Q', S \subset Q, card(S) = 9, S' \subset Q\cap Q',
\linebreak
card(S') = 5$, $A$ is a configuration of lines of type $(6,4)$, 7 of them 1-secant to $Z, S\cap S' \subset Sing(A)$ and with $r_{Z\cup S\cup S',T\Pi(4)}$ bijective. Here we will check $RT(6,0)'$. Take $(Y,Q,S,A)$ satisfying $RT(4,0)'$ and a quadric $Q'$ containg 7 suitable points of $Sing(A)\setminus S$; we may assume that no 3 of these 7 points are collinear and hence that $Q'$ does not contain any line of $A$. Add in $Q'$ one line of type $(1,0)$ 1-secant to $Y$, 5 lines of type $(0,1)$ and 7 points of $Sing(A)$.
\[8.107\] $RT(8,0)$ is true. Take $(Z,Q,Q',S,S',A)$ satisfying $RT(8,0)'$ and add in $Q$ to $Z$ $A$, the nilpotents $\chi(P), P\in S\cup S'$, and 4 points on a line of type $(0,1)$.
\[8.108\] $RT(10,0)$ is true. Take $(Y,Q)$ satisfying $RT(8,0)'$. To obtain a solution of $RT(10,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 4 lines of type $(0,1)$ (3 of the lines of $T$ being 1-secant to $Y$), 8 nilpotents supported by $Sing(T)$ and 4 points on the singular set of a configuration of lines of type $(7,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(7,4)$ we use Lemma \[3.9\].
\[8.109\] $RT(12,0)$ is true. Take $(Y,Q)$ satisfying $RT(10,0)$. To obtain a solution of $RT(12,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 7 lines of type $(0,1)$ (5 of the lines of $T$ being 1-secant to $Y$), 4 nilpotents supported by $Sing(T)$ and 15 points on the singular set of a configuration of lines of type $(6,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(9,3)$ we use Lemma \[3.9\].
\[8.110\] $RT(14,0)$ is true. Take $(Y,Q)$ satisfying $RT(51,0)$. To obtain a solution of $RT(19,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 6 lines of type $(0,1)$ (6 of the lines of $T$ being 1-secant to $Y$), 15 nilpotents supported by $Sing(T)$ and 34 points on the singular set of a configuration of lines of type $(9,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(9,6)$ we use Lemma \[3.9\] for $y = 7,
x = 4, a = 1, b = t = 4$.
\[8.111\] $RT(16,0)$ is true. Take $(Y,Q)$ satisfying $RT(14,0)$. To obtain a solution of $RT(16,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 9 lines of type $(0,1)$ (11 of the lines of $T$ being 1-secant to $Y$), 17 nilpotents supported by $Sing(T)$ and 21 points on the singular set of a configuration of lines of type $(12,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(10,5)$ we use Lemma \[3.9\] for $y =
9, x = 4, a = 3, b = 6, t = 3$.
\[8.112\] To prove $RT(20,0)$ and $RT(22,0)$ we define the following modification, call it $RT(18,0)'$, of the assertion $RT(18,0)$; for the assertion $RT(18,0)'$ we impose that the set $S$ with $card(S) = 40$ is contained in the singular set of a configuration of type $(12,4)$. Here we check $RT(18,0)'$. Take $(Y,Q)$ satisfying $RT(16,0)$. To obtain a solution of $RT(18,0)'$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 12 lines of type $(0,1)$ (7 of the lines of $T$ being 1-secant to $Y$), 17 nilpotents supported by $Sing(T)$ and 40 points on the singular set of a configuration of lines of type $(12,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(16,6)$ we use Lemma \[3.9\].
\[8.113\] $RT(20,0)$ is true. Take $(Y,Q,S)$ satisfying $RT(18,0)'$ and a general quadric $Q'$. To obtain a solution of $RT(20,0)$ add in $Q'$ to $Y$ the union, $T$, of one line of type $(1,0)$, 11 lines of type $(0,1)$ (one of the lines of $T$ being 1-secant to $Y$) and 3 points contained in the intersection with $Q'$ of the lines supporting $S$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q'$ for $(T\Pi|Q')(17,7)$ we use Lemma \[3.9\].
\[8.114\] $RT(22,0)$ is true. Take $(Y,Q)$ satisfying $RT(20,0)$. To obtain a solution of $RT(22,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 12 lines of type $(0,1)$ (11 of the lines of $T$ being 1-secant to $Y$), 43 nilpotents supported by $Sing(T)$ and 38 points on the singular set of a configuration of lines of type $(14,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(16,8)$ we use Lemma \[3.9\].
\[8.115\] $RT(24,0)$ is true. Take $(Y,Q)$ satisfying $RT(22,0)$. To obtain a solution of $RT(24,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 14 lines of type $(0,1)$ (13 of the lines of $T$ being 1-secant to $Y$), 38 nilpotents supported by $Sing(T)$ and 39 points on the singular set of a configuration of lines of type $(15,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(19,8)$ we use Lemma \[3.9\].
\[8.116\] $RT(26,0)$ is true. Take $(Y,Q)$ satisfying $RT(24,0)$. To obtain a solution of $RT(26,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 15 lines of type $(0,1)$ (12 of the lines of $T$ being 1-secant to $Y$), 39 nilpotents supported by $Sing(T)$ and 48 points on the singular set of a configuration of lines of type $(15,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(21,9)$ we use Lemma \[3.9\].
\[8.117\] $RT(28,0)$ is true. Take $(Y,Q)$ satisfying $RT(26,0)$. To obtain a solution of $RT(28,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 15 lines of type $(0,1)$ (7 of the lines of $T$ being 1-secant to $Y$), 48 nilpotents supported by $Sing(T)$ and 67 points on the singular set of a configuration of lines of type $(19,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(22,11)$ we use Lemma \[3.9\].
\[8.118\] $RT(30,0)$ is true. Take $(Y,Q)$ satisfying $RT(28,0)$. To obtain a solution of $RT(30,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 19 lines of type $(0,1)$ (14 of the lines of $T$ being 1-secant to $Y$), 67 nilpotents supported by $Sing(T)$ and 10 points on the singular set of a configuration of lines of type $(20,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(24,9)$ we use Lemma \[3.9\].
\[8.119\] $RT(32,0)$ is true. Take $(Y,Q)$ satisfying $RT(30,0)$. To obtain a solution of RT(32,0) add in $Q$ the union, $T$, of one line of type $(1,0)$, 20 lines of type $(0,1)$ (11 of the lines of $T$ being 1-secant to $Y$), 10 nilpotents supported by $Sing(T)$ and 49 points on the singular set of a configuration of lines of type $(20,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(29,10)$ we use Lemma \[3.9\].
\[8.120\] $RT(34,0)$ is true. Take $(Y,Q)$ satisfying $RT(32,0)$. To obtain a solution of $RT(34,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 20 lines of type $(0,1)$ (2 of the lines of $T$ being 1-secant to $Y$), 49 nilpotents supported by $Sing(T)$ and 4 points on the singular set of a configuration of lines of type $(22,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(29,12)$ we use Lemma \[3.9\].
\[8.121\] $RT(36,0)$ is true. Take $(Y,Q)$ satisfying $RT(34,0)$. To obtain a solution of $RT(36,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 22 lines of type $(0,1)$ (5 of the lines of $T$ being 1-secant to $Y$), 4 nilpotents supported by $Sing(T)$ and 71 points on the singular set of a configuration of lines of type $(24,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(33,12)$ we use Lemma \[3.9\].
\[8.122\] $RT(38,0)$ is true. Take $(Y,Q)$ satisfying $RT(36,0)$. To obtain a solution of $RT(38,0)$ add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 24 lines of type $(0,1)$ (one of the lines of $T$ being 1-secant to $Y$), 71 nilpotents supported by $Sing(T)$ and 46 points on the singular set of a configuration of lines of type $(26,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(33,12)$ we use Lemma \[3.9\].
\[8.123\] $RT(40,0)$ is true. Take $(Y,Q)$ satisfying $RT(38,0)$. To obtain a solution of $RT(40,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 26 lines of type $(0,1)$ (22 of the lines of $T$ being 1-secant to $Y$), 46 nilpotents supported by $Sing(T)$ and 31 points on the singular set of a configuration of lines of type $(27,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(36,12)$ we use Lemma \[3.9\].
\[8.124\] $RT(42,0)$ is true. Take $(Y,Q)$ satisfying $RT(40,0)$. To obtain a solution of $RT(42,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 27 lines of type $(0,1)$ (6 of the lines of $T$ being 1-secant to $Y$), 31 nilpotents supported by $Sing(T)$ and 28 points on the singular set of a configuration of lines of type $(29,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(38,13)$ we use Lemma \[3.9\].
\[8.125\] $RT(44,0)$ is true. Take $(Y,Q)$ satisfying $RT(42,0)$. To obtain a solution of $RT(44,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 29 lines of type $(0,1)$ (29 of the lines of $T$ being 1-secant to $Y$), 28 nilpotents supported by $Sing(T)$ and 39 points on the singular set of a configuration of lines of type $(29,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(43,13)$ we use Lemma \[3.9\].
\[8.126\] $RT(46,0)$ is true. Take $(Y,Q)$ satisfying $RT(44,0)$. To obtain a solution of $RT(46,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 29 lines of type $(0,1)$ (12 of the lines of $T$ being 1-secant to $Y$), 39 nilpotents supported by $Sing(T)$ and 46 points on the singular set of a configuration of lines of type $(30,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(42,15)$ we use Lemma \[3.9\].
\[8.127\] $RT(48,0)$ is true. Take $(Y,Q)$ satisfying $RT(46,0)$. To obtain a solution of $RT(48,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 30 lines of type $(0,1)$ (18 of the lines of $T$ being 1-secant to $Y$), 46 nilpotents supported by $Sing(T)$ and 111 points on the singular set of a configuration of lines of type $(32,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(44,16)$ we use Lemma \[3.9\].
\[8.128\] $RT(50,0)$ is true. Take $(Y,Q)$ satisfying $RT(48,0)$. To obtain a solution of $RT(50,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 32 lines of type $(0,1)$ (19 of the lines of $T$ being 1-secant to $Y$), 111 nilpotents supported by $Sing(T)$ and 28 points on the singular set of a configuration of lines of type $(33,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(44,16)$ we use Lemma \[3.9\].
\[8.129\] $RT(52,0)$ is true. Take $(Y,Q)$ satisfying $RT(50,0)$. To obtain a solution of $RT(52,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 33 lines of type $(0,1)$ (29 of the lines of $T$ being 1-secant to $Y$), 28 nilpotents supported by $Sing(T)$ and 109 points on the singular set of a configuration of lines of type $(34,4)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(49,17)$ we use Lemma \[3.9\].
\[8.130\] $RT(54,0)$ is true. Take $(Y,Q)$ satisfying $RT(52,0)$. To obtain a solution of $RT(54,0)$ add in $Q$ the union, $T$, of 4 lines of type $(1,0)$, 34 lines of type $(0,1)$ (11 of the lines of $T$ being 1-secant to $Y$), 109 nilpotents supported by $Sing(T)$ and 54 points on the singular set of a configuration of lines of type $(37,2)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(48,18)$ we use Lemma \[3.9\].
\[8.131\] $RT(56,0)$ is true. Take $(Y,Q)$ satisfying $RT(54,0)$. To obtain a solution of $RT(56,0)$ add in $Q$ the union, $T$, of 2 lines of type $(1,0)$, 37 lines of type $(0,1)$ (19 of the lines of $T$ being 1-secant to $Y$), 54 nilpotents supported by $Sing(T)$ and 13 points on a line of type $(0,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(52,17)$ we use Lemma \[3.9\].
\[8.132\] $RT(58,0)$ is true. Take $(Y,Q)$ satisfying $RT(56,0)$. To obtain a solution of $RT(58,0)$ add in $Q$ the union, $T$, of one line of type $(1,0)$, 36 lines of type $(0,1)$ (14 of the lines of $T$ being 1-secant to $Y$), 13 nilpotents supported by $Sing(T)$ and 160 points on the singular set of a configuration of lines of type $(36,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(55,20)$ we use Lemma \[3.9\].
\[8.133\] $RT(60,0)$ is true. Take $(Y,Q)$ satisfying $RT(58,0)$. To obtain a solution of $RT(60,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 36 lines of type $(0,1)$ (21 of the lines of $T$ being 1-secant to $Y$), 160 nilpotents supported by $Sing(T)$ and 159 points on the singular set of a configuration of lines of type $(37,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(53,22)$ we use Lemma \[3.9\].
\[8.134\] $RT(62,0)$ is true. Take $(Y,Q)$ satisfying $RT(60,0)$. To obtain a solution of $RT(62,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 37 lines of type $(0,1)$ (16 of the lines of $T$ being 1-secant to $Y$), 159 nilpotents supported by $Sing(T)$ and 178 points on the singular set of a configuration of lines of type $(39,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(55,23)$ we use Lemma \[3.9\].
\[8.135\] $RT(64,0)$ is true. Take $(Y,Q)$ satisfying $RT(62,0)$. To obtain a solution of $RT(64,0)$ add in $Q$ the union, $T$, of 5 line of type $(1,0)$, 39 lines of type $(0,1)$ (29 of the lines of $T$ being 1-secant to $Y$), 178 nilpotents supported by $Sing(T)$ and 29 points on the singular set of a configuration of lines of type $(43,1)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(57,23)$ we use Lemma \[3.9\].
\[8.136\] $RT(66,0)$ is true. Take $(Y,Q)$ satisfying $RT(64,0)$. To obtain a solution of RT(66,0) add in $Q$ the union, $T$, of one line of type $(1,0)$, 43 lines of type $(0,1)$ (30 of the lines of $T$ being 1-secant to $Y$), 29 nilpotents supported by $Sing(T)$ and 88 points on the singular set of a configuration of lines of type $(42,3)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(63,21)$ we use Lemma \[3.9\].
\[8.137\] RT(68,0) is true. Take $(Y,Q)$ satisfying RT(66,0). To obtain a solution of RT(68,0) add in $Q$ the union, $T$, of 3 lines of type $(1,0)$, 42 lines of type $(0,1)$ (7 of the lines of $T$ being 1-secant to $Y$), 88 nilpotents supported by $Sing(T)$ and 173 points on the singular set of a configuration of lines of type $(40,5)$. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(63,22)$ we use Lemma \[3.9\].
\[8.138\] $RT(70,0)$ is true. Take $(Y,Q)$ satisfying $RT(68,0)$. To obtain a solution of $RT(70,0)$ add in $Q$ the union, $T$, of 5 lines of type $(1,0)$, 40 lines of type $(0,1)$ (18 of the lines of $T$ being 1-secant to $Y$), 28 nilpotents supported by $Sing(T)$ and 78 suitable points. To apply Horace Lemma \[1.3\] in the smooth quadric $Q$ for $(T\Pi|Q)(63,28)$ we use Lemma \[3.7\].
\[8.139\] By Remark \[5.1\] in the statement of \[0.2\] we may take $D(g) = g+3$ if $g \geq 899$. By the first part of this section we may take $D(g) = d"(899) =
641927953$ for every integer $g \leq 898$ and $D(0) = 932$.
[BE2]{}
Ballico, E: Generators for the homogeneous ideal of s general points in ${\mbox{\bf P}^{3}}$. J. Algebra [**106**]{}, 46-52(1987) Ballico, E., Ellia, Ph.: The maximal rank conjecture for non special curves in ${\mbox{\bf P}^{3}}$. Invent. Math. 79, 541-555 (1985) Ballico, E., Ellia, Ph.: Beyond the maximal rank conjecture for curves in ${\mbox{\bf P}^{3}}$. in: Space Curves, Proc. Rocca di Papa 1985. Lect. Notes in Math. 1266. Springer-Verlag, 1987 Ballico, E., Geramita, A. V.: The minimal free resolution of the ideal of s general points of ${\mbox{\bf P}^{3}}$. Canadian Math. Soc. Conference Proceedings. Vol. 6. 1986, 1-10 Ellingsrud, G., Hirschowitz, A.: Sur le fibré normal des courbes gauches. C. R. Acad. Sci. Paris [**299**]{}, 245-248 (1984) Green, M.: Koszul cohomology and the geometry of projective varieties. J. Diff. Geom. [**19**]{}, 125-171 (1984) Hirschowitz, A.: Sur la postulation génerique des courbes rationnelles. Acta Math. [**146**]{}, 209-230 (1981) Hirschowitz, A.: Letter to R. Hartshorne, August 12, 1983 Hirschowitz, A., Simpson, C.: La résolution minimal de l’idéal d’un arrangement général d’un grand nombre de points dans $\mbox{\bf P}^{n}$. Invent. Math. [**126**]{}, 467-503 (1996) Id$\grave{a}$, M.: On the homogeneous ideal of the generic union of lines in ${\mbox{\bf P}^{3}}$. J. reine angew. Math. [**403**]{}, 67-153 (1990) Perrin, D.: Courbes passant par m points généraux de ${\mbox{\bf P}^{3}}$. Bull. Soc. Math. France, Mem. [**28/29**]{} (1987) Sernesi, E.: On the existence of certain families of curves. Invent. Math. 75, 25-57 (1984) Walter, C.: On the cohomology of normal bundles of curves in ${\mbox{\bf P}^{3}}$, II. preprint Walter, C.: Normal bundles and smoothing of algebraic space curves. preprint
Edoardo Ballico
Dept. of Mathematics, Universit$\grave{a}$ di Trento
38050 Povo (TN), Italy
[*fax*]{}: italy + 461881624 [*e-mail*]{}: [email protected]
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abstract: 'The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities (the largest distance) in each row and each column. In this paper, we give a characterization of the star graph, among the trees, in terms of invertibility of the associated eccentricity matrix. The largest eigenvalue of $\varepsilon(G)$ is called the $\varepsilon$-spectral radius, and the eccentricity energy (or the $\varepsilon$-energy) of $G$ is the sum of the absolute values of the eigenvalues of $\varepsilon(G)$. We establish some bounds for the $\varepsilon$-spectral radius and characterize the extreme graphs. Two graphs are said to be $\varepsilon$-equienergetic if they have the same $\varepsilon$-energy. For any $n \geq 5$, we construct a pair of $\varepsilon$-equienergetic graphs on $n$ vertices, which are not $\varepsilon$-cospectral.'
author:
- 'Iswar Mahato[^1]'
- 'R. Gurusamy[^2]'
- 'M. Rajesh Kannan[^3]'
- 'S. Arockiaraj[^4]'
bibliography:
- 'reference.bib'
title: On the spectral radius and the energy of eccentricity matrix of a graph
---
=0.25in
[**AMS Subject Classification(2010):**]{} 05C12, 05C50.
**Keywords.** Adjacency matrix, Distance matrix, Eccentricity matrix, Eigenvalue, Energy, Spectral radius.
Introduction {#sec1}
============
All graphs considered in this paper are finite and simple graphs, that is graphs without loops, multiple edges or directed edges. Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)=\{v_1,v_2,\hdots,v_n\}$ and edge set $E(G)=\{e_1, \dots , e_m \}$. The *adjacency matrix* of a graph $G$, denoted by $A(G) = (a_{uv})_{n\times n}$, is the $0-1$ matrix whose rows and columns are indexed by the vertices of $G$, and is defined by $a_{uv}=1$ if and only if the vertices $u$ and $v$ are adjacent, and $a_{uv}=0$ otherwise. For two vertices $u,v \in V(G)$, let $P(u,v)$ denote the path joining the vertices $u$ and $v$. The *distance* between the vertices $u,v\in V(G)$, denoted by $d_{G}(u,v)$, is the minimum length of the paths between $u$ and $v$. Let $D(G)=(d_{uv})_{n\times n}$ be the distance matrix of $G$, where $d_{uv}=d_{G}(u,v)$. The *eccentricity* $e(u)$ of the vertex $u$ is defined as $e(u)=max \{d(u,v): v \in V(G)\}$. A vertex $v$ is said to be an eccentric vertex of the vertex $u$ if $d_{G}(u,v)=e(u)$. The diameter $diam(G)$, and the radius $rad(G)$ of a graph $G$, is the maximum and the minimum eccentricity of all vertices of $G$, respectively. A vertex $u\in V(G)$ is said to be *diametrical vertex* of $G$ if $e(u)=diam(G)$. If each vertex of $G$ has a unique diametrical vertex, then $G$ is called the *diametrical graph* which is studied and referred to as even graphs in [@gobel1986even].
The *eccentricity matrix $\varepsilon(G)=(\epsilon_{uv})$* of a graph $G$, which is introduced in [@ran1; @ecc-main] and further studied in [@ours1; @ran1; @ecc-main], is defined as $$\epsilon_{uv}=
\begin{cases}
\text{$d_G(u,v)$} & \quad\text{if $d_G(u,v)=min\{e(u),e(v)\}$,}\\
\text{0} & \quad\text{otherwise.}
\end{cases}$$ In [@ran1; @ran2], the eccentricity matrix is known as $D_{\max}$-matrix. The eigenvalues of the eccentricity matrix of a graph $G$ is called the $\varepsilon$-eigenvalues of $G$. Since $\varepsilon(G)$ is symmetric, all of its eigenvalues are real. Let $\xi_1>\xi_2>\hdots >\xi_k$ be all the distinct $\varepsilon$-eigenvalues of $G$, then the $\varepsilon$-spectrum of $G$ can be written as $$spec_{\varepsilon}(G)=
\left\{ {\begin{array}{cccc}
\xi_1 & \xi_2 &\hdots & \xi_k\\
m_1& m_2& \hdots &m_k\\
\end{array} } \right\},$$ where $m_i$ be the algebraic multiplicity of $\xi_i$ for $i=1,2,\hdots,k$. The largest eigenvalue of $\varepsilon(G)$ is called the $\varepsilon$-spectral radius and is denoted by $\rho(\varepsilon(G))$.
It is well-known that *graph energy* is a vital chemical index in chemical graph theory. The energy ( or $A$-energy ) of a graph is introduced in [@gutman-energy], which is defined as $$E_{A}(G)=\sum_{i=1}^n |\lambda_i|,$$ where $\lambda_i$, $i=1,2,\hdots,n$ are the eigenvalues of the adjacency matrix of $G$. In a similar way, the eccentricity energy (or $\varepsilon$-energy ) of a graph $G$ is defined [@wang2019graph] as $$E_{\varepsilon}(G)=\sum_{i=1}^n |\xi_i|,$$ where $\xi_1,\xi_2,\hdots,\xi_n$ are the $\varepsilon$-eigenvalues of $G$. Two graphs are said to be *$\varepsilon$-cospectral* if they have the same $\varepsilon$-spectrum, and two graphs are said to be *$\varepsilon$-equienergetic* if they have the same $\varepsilon$-energy. We are, of course, interested in studying about $\varepsilon$-equienergetic graphs which are not $\varepsilon$-cospectral.
The *Wiener index* of a graph is an important and well studied topological index in mathematical chemistry. It is defined as $$W(G)=\frac{1}{2}\sum_{{u,v}\in V(G)}d_{G}(u,v).$$ Similarly, we define the *eccentric Wiener index* (or $\varepsilon$-Wiener index) of a connected graph $G$ as follows $$W_{\varepsilon}(G)=\frac{1}{2}\sum_{{u,v}\in V(G)}\epsilon_{uv}.$$ As usual let $K_{1,n-1}$, $P_n$ and $K_n$ denote the star, the path and the complete graph on $n$ vertices, respectively. For other undefined notations and terminology from graph theory, we refer to [@bon-mur-book]. We shall use the following results for the proof of our main results.
\[lem:comp\] Let $A=(a_{ij})$ and $B=(b_{ij})$ be two $n \times n$ matrices such that $b_{ij}\geq a_{ij}$ for all $i,j$. Then $\rho(B)\geq \rho(A)$.
[@lin2013distance Lemma 2.1] \[lem:star\] The graph $K_{1,n-1}$ is the unique graph, which have maximum distance spectral radius among all graphs with diameter 2.
[@hor-john-mat](Interlacing Theorem) Let $A$ be a symmetric matrix of order $n$ and let $B$ be its principal submatrix of order $m<n$. Suppose $\lambda_1(A) \leq \lambda_2(A) \hdots \leq \lambda_n(A)$are the eigenvalues of $A$ and $\beta_1(B) \leq \beta_2(B) \hdots \leq \beta_m(B)$ are the eigenvalues of $B$. Then, $\lambda_i(A) \leq \beta_i(B) \leq \lambda_{i+n-m}(A)$ for $i = 1,\hdots , m$, and if $ m = n-1$, then $\lambda_1(A)\leq \beta_1(B) \leq \lambda_2(A) \leq \beta_2(B) \leq \hdots \leq \beta_{n-1}(B) \leq \lambda_n(A)$.
[@hor-john-mat] (Equitable partitions) Let $A$ be a real symmetric matrix whose rows and columns are indexed by $X=\{1,2,\hdots,n\}$. Let $\pi=\{X_1,X_2,\hdots,X_m\}$ be a partition of $X$. The characteristic matrix $C$ is the $n\times m$ matrix whose $j$-th column is the characteristic vector of $X_j$ $(j=1,2,\hdots,m)$. Let $A$ be partitioned according to $\pi$ as $$A=\left[ {\begin{array}{cccc}
A_{11} & A_{12} &\hdots & A_{1m}\\
A_{21} & A_{22} &\hdots & A_{2m}\\
\vdots &\hdots & \ddots & \vdots\\
A_{m1} & A_{m2}& \hdots &A_{mm}\\
\end{array} } \right],$$ where $A_{ij}$ denotes the submatrix (block) of $A$ formed by rows in $X_i$ and the columns in $X_j$. If $q_{ij}$ denote the average row sum of $A_{ij}$, then the matrix $Q=(q_{i,j})$ is called the quotient matrix of $A$. If the row sum of each block $A_{ij}$ is a constant, then the partition $\pi$ is called equitable partition.
[@cvetkovic2009introduction]\[quo-spec\] Let $Q$ be a quotient matrix of any square matrix $A$ corresponding to an equitable partition. Then the spectrum of $A$ contains the spectrum of $Q$.
This article is organized as follows: In section $2$, we show that the eccentricity matrix of a tree, other than $P_4$, is invertible if and only if it is the star. In section $3$, we obtain bounds for $\varepsilon$-spectral radius of graphs and characterize the extreme graphs. In section $4$, we construct a pair of non-cospectral $\varepsilon$-equienergetic graphs.
A characterization of star graph
================================
In this section, we prove that among all trees, other than $P_4$, star is the only graph for which the eccentricity matrix is always invertible.
Let $T$ be a tree, other than $P_4$, then the eccentricity matrix of $T$ is invertible if and only if $T$ is the star.
Let $T$ be the star on $n$ vertices. As the distance matrix and the eccentricity matrix of the star are same, so $\det(\varepsilon(T)=(-1)^{n-1}(n-1)2^{n-2}$. Thus $\varepsilon(T)$ is invertible.
To prove the converse, first, let us consider the trees of order up to $4$. For $n = 2, 3$, the proof is trivial. For $n=4$, $P_4$ and $K_{1,3}$ are the only trees of order $4$, and the eccentricity matrix of both the trees are invertible.
Let $T$ be a tree on $n \geq 5$ vertices other than the star. We will show that $det(\varepsilon(T))=0$. Let $P(v_1,v_m)=v_1v_2\hdots v_{m-1}v_m$ be a diametrical path of length $m-1$ in $T$. Now consider the following two cases:\
**Case(I)**: Let either $v_2$ or $v_{m-1}$ be adjacent to at least one pendant vertex other than the vertices $v_1$ and $v_m$. Without loss of generality, assume that $v_{m-1}$ is adjacent to $p$ pendant vertices, say, $u_1,u_2,\hdots,u_p$. Then the rows corresponding to the vertices $u_1,u_2,\hdots,u_p$ and $v_m$ are the same in $\varepsilon(T)$. Thus $det(\varepsilon(T))=0$.\
**Case(II)**: Let both the vertices $v_2$ and $v_{m-1}$ are not adjacent to any of the pendant vertices in $G$ other than $v_1$ and $v_m$, respectively. Since $T$ is a tree other than the star, so $diam(T)\geq 3$. If $T$ is a tree on $n\geq 5$ vertices and $diam(T)=3$, then one of the vertices $v_2$ or $v_{m-1}$ must be adjacent to at least two pendent vertices, and the proof follows from case(I). Let $diam(T)\geq 4$. Let us show that at least two rows of $\varepsilon(T)$ are linearly dependent. Now we consider the following two subcases:\
Subcase(I): Let $diam(T)=4$, and let $P(v_1,v_5)=v_1v_2v_3 v_4v_5$ be a diametrical path in $T$. Let $u_1,u_2,\hdots,u_p$ be the vertices, other than $v_1$ and $v_5$, such that each $u_i$ has exactly one common neighbour, say $w_i$, with $v_3$. It is easy to see that, the vertices $u_1,u_2,\hdots,u_p$ are pendant. The rows corresponding to the vertices $w_1,w_2,\hdots,w_p,v_2,v_4$ and the row corresponding to the vertex $v_3$, in $\varepsilon(T)$, are linearly dependent .\
Subcase(II): Let $diam(T)\geq 5$, and let $P(v_1,v_m)=v_1v_2v_3\hdots v_{m-1}v_m$ be a diametrical path in $T$. Then the rows corresponding to the vertices $v_2$ and $v_3$ are linearly dependent in $\varepsilon(T)$.
Thus $\det(\varepsilon(T)) =0$ in all the above cases. Therefore, if the eccentricity matrix of $T$ is invertible, then $T$ is the star.
Bounds for $\varepsilon$-spectral radius of graphs
==================================================
In this section, we establish bounds for the $\varepsilon$-spectral radius of graphs, and characterize the extreme graphs. In the next theorem, we derive a characterization for the star, among all connected graphs with diameter $2$, in terms of the $\varepsilon$-spectral radius.
Among all connected graphs on $n$ vertices with diameter $2$, the star $K_{1,n-1}$ is the unique graph, which has maximum $\varepsilon$-spectral radius.
Let $G$ be a connected graph on $n$ vertices such that $diam(G)=2$. From the definition, it follows that the eccentricity matrix $\varepsilon(G)$ of $G$ is entrywise dominated by the distance matrix $D(G)$. So by Lemma \[lem:comp\], $\rho(\varepsilon(G))\leq \rho(D(G))$. For $K_{1,n-1}$, the star on $n$ vertices, the eccentricity matrix and the distance matrix are the same, and hence $\rho(D(K_{1,n-1}))=\rho(\varepsilon(K_{1,n-1}))$. By Lemma \[lem:star\], $\rho(D(G))\leq \rho(D(K_{1,n-1}))=(n-2)+\sqrt{n^2-3n+3}$, and the equality holds if and only if $G$ is the star. Therefore, $\rho(\varepsilon(G))\leq \rho(D(G))\leq \rho(D(K_{1,n-1}))=\rho(\varepsilon(K_{1,n-1}))$, and the equality holds only for the star.
Next we establish an lower bound for the $\varepsilon$-spectral radius of a graph with given diameter, and characterize the extreme graph.
\[prop:diam\] If $G$ is a connected graph with diameter $d\geq 2$, then $\rho(\varepsilon(G))\geq d$, and the equality holds if and only if $G$ is the diametrical graph with diameter $d$.
Let $G$ be a connected graph with diameter $d\geq 2$. Then there exists a $2 \times 2$ principal submatrix $ \left [ {\begin{array}{cc}
0 & d\\
d & 0\\
\end{array} } \right ]$ whose eigenvalues are $ d,-d$. Thus, by interlacing theorem, we have $\rho(\varepsilon(G))\geq d$.
Let $G$ be a diametrical graph with diameter $d$. Then for each vertex $v$ of $G$, the eccentricity $e(v)=diam(G)=d$, and the eccentricity attains for a unique vertex. So the eccentricity matrix of $G$ can be written as $ \left [ {\begin{array}{cc}
0 & dI_k\\
dI_k & 0\\
\end{array} } \right ]$, whose $\varepsilon$-spectrum is $ \left \{ {\begin{array}{cc}
d & -d\\
k & k\\
\end{array} } \right \}$. Thus $\rho(\varepsilon(G))=d$.
Conversely, let $\rho(\varepsilon(G))=d$. Suppose $G$ is not the diametrical graph. Then we have the following cases:\
**Case(I):** Let $G$ be a graph such that $rad(G)=diam(G)=d$. Then $$B = \left[ {\begin{array}{ccc}
0 & d & d\\
d & 0 & 0\\
d & 0 & 0 \\
\end{array} } \right]$$ is a principal submatrix of $\varepsilon(G)$, and $\rho(B) = d\sqrt{2}$. Therefore, by interlacing theorem, we have $\rho(\varepsilon(G)) \geq d \sqrt{2}>d$, which is not possible.\
**Case(II):** Let $G$ be a graph such that $rad(G) \neq diam(G)=d$. Then there exists a vertex $v_k$ with eccentricity $e(v_k)=k<d$. Let $v_1$ be a vertex of $G$ with $e(v_1)=d$. Since $G$ is a connected graph, there is a path $P(v_1,v_k)$ between the vertices $v_1$ and $v_k$. It is easy to see that the eccentricity of any vertex which is adjacent to $v_1$ is either $d$ or $d-1$. Hence, in the path $P(v_1,v_k)$ there always exists a pair of adjacent vertices $u$ and $v$ such that $e(u)=d$ and $e(v)=d-1$. Let $w$ be the eccentric vertex of $u$, that is, $d(u,w)=d$. Then $d(v,w)=d-1$. Since $e(v)=d-1$ and $w$ is an eccentric vertex of $v$, the $vw$-th entry of $\varepsilon(G)$ is $d-1$. Therefore, $$C = \left[ {\begin{array}{ccc}
0 & 0 & d\\
0 & 0 & d-1\\
d & d-1 & 0 \\
\end{array} } \right]$$ is a principal submatrix of $\varepsilon(G)$, corresponding to the vertices $u,v$ and $w$. Now, since $\rho(C)$ equals to $\sqrt{(d-1)^2+d^2}$, by interlacing theorem, we have $\rho(\varepsilon(G)) \geq \sqrt{(d-1)^2+d^2}>d$, which is a contradiction.
Therefore, $G$ is a diametrical graph. This completes the proof.
Among the connected bipartite graphs on $2n$ $(n\geq 3)$ vertices, the graph $W_{n,n}$ has the minimum $\varepsilon$-spectral radius, where $W_{n,n}$ is the graph obtained by deleting $n$ independent edges from the complete bipartite graph $K_{n,n}$.
Since $W_{n,n}$ is obtained by deleting $n$ independent edges from $K_{n,n}$, each vertex of $W_{n,n}$ has a unique diametrical vertex with eccentricity 3. Therefore, $W_{n,n}$ is a diametrical graph with diameter 3. So, by Theorem \[prop:diam\], we have $\rho(\varepsilon(W_{n,n}))=3$.
Among the bipartite graphs on $2n$ vertices, $K_{1,2n-1}$ and $K_{n,n}$ are the only graphs of diameter 2 and $\rho(\varepsilon(K_{1,2n-1}))=2(n-1)+\sqrt{4n^2-6n+3} \geq 3$, $\rho(\varepsilon(K_{n,n}))=2(n-1)\geq 3$. Therefore, the proof follows from Theorem \[prop:diam\].
The $\varepsilon$-degree of a vertex $v_i\in V(G)$ is defined as $\varepsilon(i)=\sum_{j=1}^n\epsilon_{ij}$. A graph $G$ is said to be $\varepsilon$-regular if $\varepsilon(i)=k$ for all $i$ [@wang2019graph]. Now let us establish a lower bound for the $\varepsilon$-spectral radius of a graph in terms of eccentric Wiener index.
\[bou-wie\] Let $G$ be a connected graph on $n$ vertices with eccentric Wiener index $W_{\varepsilon}$. Then $\rho(\varepsilon(G))\geq \frac{2W_{\varepsilon}}{n}$ and the equality holds if and only if $G$ is $ \varepsilon$-regular graph.
Let $x=\frac{1}{\sqrt{n}}[1,1,\hdots,1]^T$ be the unit positive vector of order $n$. By applying Rayleigh Principle to the eccentricity matrix $\varepsilon(G)$ of the graph $G$, we get $$\begin{aligned}
\rho(\varepsilon(G)) \geq \frac{x^T\varepsilon(G)x}{x^Tx} &=&\frac{1}{\sqrt{n}}[1,1,\hdots,1] \frac{1}{\sqrt{n}}[\varepsilon(1),\varepsilon(2),\hdots,\varepsilon(n)]^T\\
&=& \frac{1}{n}\sum_{i=1}^n \varepsilon(i)\\
&=& \frac{2W_{\varepsilon}}{n}.\end{aligned}$$ Now, if $G$ is $\varepsilon$-regular, then each row sum of $\varepsilon(G)$ is a constant, say $k$ and hence $\rho(\varepsilon(G))=k$. Therefore, $\rho(\varepsilon(G))=k=\frac{nk}{n}= \frac{2W_{\varepsilon}}{n}$, and hence the equality holds.
Conversely if equality holds, then $x$ is an eigenvector corresponding to $\rho(\varepsilon(G))$ and hence $\varepsilon(G)x=\rho(\varepsilon(G))x$. Therefore, $\varepsilon(i)=\rho(\varepsilon(G))$ for all $i$. Thus $G$ is $\varepsilon$-regular. This completes the proof.
Let $G$ be a connected graph on $n$ vertices and $m$ edges with diameter $2$. If $G$ has $k$ vertices of degree $n-1$, then $$\label{eq1}
\rho(\varepsilon(G)) \geq \frac{2(n^2-n-2m)+k(2n-k-1)}{n}.$$
Let $G$ be a connected graph of diameter 2 with vertex set $\{v_1,v_2,\hdots,v_k,v_{k+1},\hdots,v_n\}$, where $v_1,v_2,\hdots,v_k$ are the vertices of degree $n-1$. Therefore, $e(v_i)=1$ for $i=1,2,\hdots,k$ and $e(v_i)=2$ for $i=k+1,\hdots,n$. Then $$\begin{aligned}
2W_{\varepsilon}(G)=\sum_{i=1}^n \varepsilon(i) &=& k(n-1)+ \sum_{i=k+1}^n \Big( k+2\big((n-k)-(d_i-k)-1\big)\Big)\\
&=&k(n-1)+ \sum_{i=k+1}^n \big( k+2(n-d_i-1)\big)\\
&=& 2(n^2-n-2m)+k(2n-k-1).\end{aligned}$$ Thus the proof follows from Theorem \[bou-wie\].
In the next result, we obtain a lower bound for the $\varepsilon$-spectral radius in terms of $\varepsilon$-Wiener index, and $\varepsilon$-degree sequence.
\[wie-deg\] Let $G$ be a connected graph of order $n$ with $\varepsilon$-Wiener index $W_{\varepsilon}$ and $\varepsilon$-degree sequence $\{\varepsilon(1),\varepsilon(2),\hdots,\varepsilon(n)\}$. Then $$\rho(\varepsilon(G))\geq \max_{i}\Big\{\frac{1}{n-1}\Big(\big(W_{\varepsilon}-\varepsilon(i)\big)+\sqrt{\big(W_{\varepsilon}-\varepsilon(i)\big)^2+(n-1){\varepsilon}^2(i)}\Big)\Big\}.$$
Let $v_i$ be a vertex of the graph $G$ and $\varepsilon(i)$ be its $\varepsilon$-degree. Let us partition the eccentricity matrix of $G$ with respect to the row corresponding to the vertex $v_i$. Then the quotient matrix corresponding to this partition is $$A=\left[ {\begin{array}{cc}
0 & \varepsilon(i)\\
\frac{\varepsilon(i)}{n-1} & \frac{2\big(W_{\varepsilon}-\varepsilon(i)\big)}{n-1} \\
\end{array} } \right].$$ The eigenvalues of $A$ are $$\mu_1=\frac{1}{n-1}\Big\{\big(W_{\varepsilon}-\varepsilon(i)\big)+\sqrt{\big(W_{\varepsilon}-\varepsilon(i)\big)^2+(n-1){\varepsilon}^2(i)}\Big\}$$ and $$\mu_2=\frac{1}{n-1}\Big\{\big(W_{\varepsilon}-\varepsilon(i)\big)-\sqrt{\big(W_{\varepsilon}-\varepsilon(i)\big)^2+(n-1){\varepsilon}^2(i)}\Big\}.$$ From Lemma \[quo-spec\], we have $$\rho(\varepsilon(G))\geq \mu_1=\frac{1}{n-1}\Big\{\big(W_{\varepsilon}-\varepsilon(i)\big)+\sqrt{\big(W_{\varepsilon}-\varepsilon(i)\big)^2+(n-1){\varepsilon}^2(i)}\Big\}.$$ Since this is true for all $i$, the proof is done.
The counterparts of the bounds provided in Theorem \[bou-wie\] and Theorem \[wie-deg\] for the distance matrix case is known in the literature [@indu-laa].
Construction of $\varepsilon$-equienergetic graphs
==================================================
The problem of constructing non-cospectral equienergetic graphs is an interesting problem in spectral graph theory. Motivated by this, in this section we discuss the construction of $\varepsilon$-equienergetic graphs.
\[bi-ener\] Let $K_{p,q}$ be a complete bipartite graph on $n=p+q$ vertices. If $p, q \geq 2$, then the $\varepsilon$-energy of $K_{p,q}$ is $4(p+q-2)$.
The eccentricity matrix of $K_{p,q}$ can be written as $$\varepsilon(K_{p,q})=\left[ {\begin{array}{cc}
2(J_p-I_p) & 0\\
0 & 2(J_q-I_q) \\
\end{array} } \right]$$ Therefore, $$\label{bi-spec} spec_{\varepsilon}(K_{p,q})=\left\{ {\begin{array}{ccc}
2(p-1) & 2(q-1)& -2\\
1 & 1& p+q-2 \\
\end{array} } \right\},$$ and hence $E_{\varepsilon}(K_{p,q})=4(p+q-2)$.
There are only two connected graphs of order $3$, and only six connected graphs of order $4$. By an elementary calculation, we can say that there does not exist any $\varepsilon$-equienergetic graphs of order 3 and 4. Let us consider the graphs $G_1$ and $G_2$ of order $5$ as shown in Figure $1$.
{height="4.5" width="10.5"}
Therefore, $$\begin{aligned}
spec_{\varepsilon}(G_1)=\left\{ {\begin{array}{ccc}
2(1\pm \sqrt{2}) & -4 & 0\\
1 & 1 & 2 \\
\end{array} } \right\} \qquad \mbox{and} \qquad
spec_{\varepsilon}(G_1)=\left\{ {\begin{array}{ccc}
\pm 2\sqrt{2}) & \pm 2& 0\\
1 & 1 & 1 \\
\end{array} } \right\}. \end{aligned}$$ Hence, $E_{\varepsilon}(G_1)=4+4\sqrt{2}=E_{\varepsilon}(G_2)$. So, the graphs $G_1$ and $G_2$ are non-$\varepsilon$-cospectral $\varepsilon$-equienergetic graphs of order $5$. In the next theorem we show that, for $n\geq 6$, there exists a pair of non-$\varepsilon$-cospectral graphs which are $\varepsilon$-equienergetic.
For $n\geq 6$ and $p,q \geq 2$, the graphs $K_{p,n-p}$ and $K_{q,n-q}$ are $\varepsilon$-equienergetic, but not $\varepsilon$-cospectral .
Proof follows from Lemma \[bi-ener\].
Also, we have more general result. If $K_{n_1,n_2,\hdots,n_k}$ is a complete $k$-partite graphs on $n$ vertices, where $n=\sum_{i=1}^nn_i$ with $n_i\geq 2$, then by Theorem $4.6$ in [@ours1], we have $$spec_{\epsilon}(K_{n_1,\hdots,n_k})=
\left\{ {\begin{array}{ccccc}
-2 & 2(n_1-1) & 2(n_2-1)&\hdots & 2(n_k-1)\\
n-k &1 &1&\hdots &1\\
\end{array} } \right\}.$$ Thus, $E_{\varepsilon}(K_{n_1,n_2,\hdots,n_k})=4(n-k)$, which is independent of $n_1,n_2,\hdots,n_k$. Therefore, every complete $k$-partite graphs are $\varepsilon$-equienergetic but not $\varepsilon$-cospectral if they have at least $2$ vertices in each partition.
**Acknowledgement:** Iswar Mahato and M. Rajesh Kannan would like to thank the Department of Science and Technology, India, for financial support through the Early Career Research Award (ECR/2017/000643).
[^1]: Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India. Email: [email protected]
[^2]: Department of Mathematics, Mepco Schlenk Engineering College, Sivakasi 626005, Tamil Nadu, India. Email: [email protected]
[^3]: Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India. Email: [email protected], [email protected]
[^4]: Department of Mathematics, Government Arts and Science College, Sivakasi 626124, Tamil Nadu, India. Email: [email protected]
|
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abstract: 'We report the discovery of a kiloparsec-scale dual active galactic nucleus (AGN) in J0038+4128. From the [*Hubble Space Telescope*]{} ([*HST*]{}) Wide Field Planetary Camera 2 (WFPC2) images, we find two optical nuclei with a projection separation of 4.7 kpc (344). The southern component (J0038+4128S) is spectroscopically observed with the [*HST*]{} Goddard High Resolution Spectrograph in the UV range and is found to be a Seyfert 1 galaxy with a broad Ly$\alpha$ emission line. The northern component (J0038+4128N) is spectroscopically observed during the Large Sky Area Multi-Object Fibre Spectroscopic Telescope (also named the Guoshoujing Telescope) pilot survey in the optical range. The observed line ratios as well as the consistency of redshift of the nucleus emission lines and the host galaxy’s absorption lines indicate that J0038+4128N is a Seyfert 2 galaxy with narrow lines only. These results thus confirm that J0038+4128 is a Seyfert 1-Seyfert 2 AGN pair. The [*HST*]{} WFPC2 $F$336$W$/[*U*]{}-band image of J0038+4128 also reveals for the first time for a dual AGN system two pairs of bi-symmetric arms, as are expected from the numerical simulations of such system. Being one of a few confirmed kiloparsec-scale dual AGNs exhibiting a clear morphological structure of the host galaxies, J0038+4128 provides an unique opportunity to study the co-evolution of the host galaxies and their central supermassive black holes undergoing a merging process.'
title: '[*HST*]{} and LAMOST discover a dual active galactic nucleus in J0038+4128'
---
galaxies: active — galaxies: individual: J0038+4128 — galaxies: interactions — galaxies: nuclei — galaxies: Seyfert.
Introduction {#sec:optaxrat_int}
============
In the hierarchical $\Lambda$ cold dark matter ($\Lambda {\rm CDM}$) cosmology, galaxies built up via mergers (Toomre & Toomre 1972). Binary supermassive black holes (SMBHs) are natural outcomes of galaxy mergers (Begelman et al. 1980; Milosavljević & Merritt 2001; Yu 2002), since almost all massive galaxies are believed to host a central SMBH. In the gas-rich case, the strong tidal interactions caused by galaxy mergers can trigger the active galactic nucleus (AGN) by sending a large amount of gas to the central regions (Hernquist 1989; Kauffmann & Haehnelt 2000; Hopkins et al. 2008). A dual AGN could emerge if the two merging SMBHs are both simultaneously accreting gas in a gas-rich major merger. Finding dual AGNs, especially those exhibiting two black holes on a kiloparsec-scale (Liu et al. 2013), will provide important clues to understand the relation between the AGN activity and the galaxy evolution and AGN physics (Colpi & Dotti 2011; Yu et al. 2011).
Over the past few years, hundreds of dual AGNs of separations greater than 10 kpc have been discovered (e.g. Myers et al. 2007, 2008; Green et al. 2010; Piconcelli et al. 2010). However, such systems represent only the earliest stage of the binary SMBH evolution. No more than dozens of close dual AGNs (separations between 1 and 10 kpc) are found both from either systematic searches (e.g. Liu et al. 2010, 2013; Rosario et al. 2011) or by serendipitous discoveries (e.g. Junkkarinen et al. 2001; Komossa et al. 2003; Ballo et al. 2004; Bianchi et al. 2008; Fu et al. 2011; Koss et al. 2011; McGurk et al. 2011). However, most of them only show two prominent nuclei either in the radio, X-ray or the optical band, the properties of the host galaxies are poorly known. The latter are extremely important for probing the relation between the galaxy evolution and nucleus activity (Shields et al. 2012).
Here, we report the discovery of a new dual AGN in J0038+4128 ([*z*]{} = 0.0725) with a spatial separation of 4.7 kpc (344). A $F$336$W$/[*U*]{}-band image of J0038+4128 obtained with the Wide Field Planetary Camera 2 (WFPC2) on board the [*Hubble Space Telescope*]{} ([*HST*]{}) reveals for the first time for a dual AGN system two pairs of bi-symmetric arms. In Section 2.1, the [*HST*]{} photometric and spectroscopic observation and data reduction of J0038+4128 are introduced. We describe the Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST; also known as the Guoshoujing Telescope) spectroscopic observation and data reduction in Section 2.2. The results are discussed and concluded in Section 3, followed by a brief summary in Section 4. We adopt a $\Lambda$CDM cosmology with $\Omega_{\rm m}$=0.3, $\Omega_{\Lambda}$=0.7, and $H_0$= 70 km s$^{-1} $Mpc$^{-1}$ throughout. All quoted wavelengths are in vacuum units.
OBSERVATIONS AND REDUCTIONS
===========================
[*HST*]{} Images and Spectra
----------------------------
The images presented here were obtained with the WFPC2 on board the [*HST*]{}[^1] as parts of the General Observer (GO) programme GO-6749 (PI: Laura Danly) on 1996 August 27. WFPC2 contains four Loral CCD detectors of 800$\times$800 pixels. The field of view (FOV) of the Planetary Camera is about 34$\times$34 arcsec$^{2}$ (0046 pixel$^{-1}$) , whereas that of each of the three Wide Field (WF) arrays is 150$\times$150 arcsec$^{2}$ (01 pixel$^{-1}$). Fig.1 shows the WFPC2 combined images of J0043+4128 (located in one of the three WFs) of two deep exposures (600s) in the $F$336$W$/[*U*]{}-band and two deep exposures (400s) and one shallow exposure (30s) in the $F$555$W$/[*V*]{}-band. The combined images were produced at the Canadian Astronomical Data Center (CADC) with MultiDrizzle with improved astrometry and geometric distortions correction[^2]. J0038+4128 is well resolved into two components, a main (S) and a companion (N), separated by 344 in both bands.
The UV spectrum of J0038+4128S was obtained with the [*HST*]{} Goddard High Resolution Spectrograph (GHRS) on 1997 February 4, also under project GO-6749. Six exposures were taken with the G140L grating through the Large Science Aperture (20$\times$20) for a total integration time of 18,822 s. The resolving power is about 1000 at 1200 [Å]{}, which corresponds to $\sim$300 km s$^{-1}$. The spectra were processed using the standard CALHRS v1.3.14 reduction pipeline with the latest reference files. We compute the wavelength zero-points of the spectra with the nearest SPYBAL[^3] calibration spectrum observations in time using the STSDAS IRAF task [*waveoff*]{} and find offsets ranging from 0.6956 to 0.9948 [Å]{} for individual exposures. After correcting for the wavelength zero-points for each exposure we present the final combined spectrum in Fig.2. The spectrum was rebinned to a common wavelength grid with a constant step size of 0.57 [Å]{}.
LAMOST spectra
--------------
--------------- ------------- -------------- ---------------- ------------- ---------- ----------- ------------------- -----------------
Area RA Dec. Facility Date Seeing Exp. time Spectral coverage Resolving power
(c.f. Fig. 1) (arcsec) (second) (Å)
Yellow circle 00 38 33.05 +41 28 53.48 LAMOST 2011 Oct 05 4.0 1800 3700–9000 1800
Red circle 00 38 33.36 +41 28 51.95 LAMOST 2011 Oct 24 3.4 3600 3700–9000 1800
Red circle 00 38 33.36 +41 28 51.95 LAMOST 2011 Oct 28 2.8 2100 3700–9000 1800
Yellow box 00 38 33.12 +41 28 50.30 [*HST*]{}/GHRS 1997 Feb 04 — 18,822 1275–1561 1000
--------------- ------------- -------------- ---------------- ------------- ---------- ----------- ------------------- -----------------
The optical spectrum of J0038+4128N were observed during the LAMOST pilot survey on 2011 October 5. LAMOST is a 4 metre quasi-meridian reflecting Schmidt telescope equipped with 4000 fibres, each of an angular diameter of 33 projected on the sky, in an FOV of 5$^{\circ}$ in diameter (Cui et al. 2012). The spectra has a resolution of [*R*]{} $\sim$ 1800 and covers wavelengths from 3700 to 9000 [Å]{}. In addition to J0038+4128N, LAMOST also observed a region a few arcsec east of the two nuclei marked as J0038+4128H shown in Fig.1 on 2011 October 24 and 28. The weather of the three nights was clear but of relatively poor seeing ($\sim$3). Long-term monitoring of the LAMOST fibre positioning accuracy (Yuan et al., in preparation) shows that it varies from fibre to fibre. However, the observations of J0038+4128N and J0038+4128H were obtained with fibres of the highest precision (better than 05). The data were processed using the LAMOST standard pipeline, with flux calibration better than $\sim$ 10% (Liu et al. 2013).
We summarize the details of all spectral observations of J0038+4128 in Table 1, including the three positions of the observed regions, central positions of aperture, telescopes used, observational dates, seeing, total exposure times, spectral coverage and resolving power. In addition, the photometric properties of J0038+4128 from archival observations are summarized in Table 2.
![Left: [*HST*]{}/WFPC2 $F$336$W$/[*U*]{}-band pseudo colour image of J0038+4128 on a logarithmic scale (smoothed using the Gaussian function available in SAOImage DS9 6.2). Right: [*HST*]{}/WFPC2 $F$555$W$/[*V*]{}-band pseudo colour image of J0038+4128 on a logarithmic scale. The two nuclei are clearly resolved in both bands into two components, the S and the N components. North is up and east is to the left. Spatial scale is also shown in each panel. Apertures used to obtain the spectra presented in this work are marked, including one 2$\times$ 2rectangular aperture for the [*HST*]{}/GHRS spectroscopy, three circular apertures of 33 diameter for the LAMOST spectroscopy. See Table 1 for detail.](fig1a.eps "fig:"){width="3.75cm" height="5.5cm"} ![Left: [*HST*]{}/WFPC2 $F$336$W$/[*U*]{}-band pseudo colour image of J0038+4128 on a logarithmic scale (smoothed using the Gaussian function available in SAOImage DS9 6.2). Right: [*HST*]{}/WFPC2 $F$555$W$/[*V*]{}-band pseudo colour image of J0038+4128 on a logarithmic scale. The two nuclei are clearly resolved in both bands into two components, the S and the N components. North is up and east is to the left. Spatial scale is also shown in each panel. Apertures used to obtain the spectra presented in this work are marked, including one 2$\times$ 2rectangular aperture for the [*HST*]{}/GHRS spectroscopy, three circular apertures of 33 diameter for the LAMOST spectroscopy. See Table 1 for detail.](fig1b.eps "fig:"){width="3.75cm" height="5.5cm"}
{width="15cm" height="7.5cm"} -0.25truein
Band $\lambda$ ($\mu$m) J0038+418S J0038+418N Facility
---------- -------------------- ------------------ ------------------ ----------
FUV 0.16 [*GALEX*]{}
NUV 0.23 [*GALEX*]{}
$u^{a}$ 0.36 16.889$\pm$0.015 24.635$\pm$0.975 SDSS
$g^{a}$ 0.47 15.604$\pm$0.003 19.985$\pm$0.020 SDSS
$r^{a}$ 0.62 14.831$\pm$0.003 18.752$\pm$0.011 SDSS
$i ^{a}$ 0.75 14.261$\pm$0.003 18.386$\pm$0.012 SDSS
$z^{a}$ 0.89 13.943$\pm$0.005 18.004$\pm$0.023 SDSS
$J^{b}$ 1.25 2MASS
$H^{b}$ 1.65 2MASS
$K^{b}$ 2.17 2MASS
$W$1 3.40 [*WISE*]{}
$W$2 4.60 [*WISE*]{}
$W$3 12.00 [*WISE*]{}
$W$4 22.00 [*WISE*]{}
: Photometric data for J0038+4128
Note: The two nuclei just can be separated in SDSS images, we list magnitudes from SDSS for both components and total magnitudes from other bands for the whole system.\
$^{a}$ SDSS model magnitude.\
$^{b}$ Magnitudes are for isophotal fiducial elliptical-aperture.
RESULTS AND DISCUSSIONS
=======================
Confirmation of a dual AGN
--------------------------
Fig.1 (c.f. also §3.4) shows clear evidence of interaction between the two components and the chance of the system being a coincidental superposition of two physically unrelated objects or the possibility of being a gravitationally lensed system can both be squarely ruled out. However, images alone are insufficient to distinguish starbursts from AGNs of types 1 or 2. With the spectra obtained from the [*HST*]{} and LAMOST, we can constrain the nature of the two components of J0038+4128 by examining the widths of observed emission lines as well as the locations of the measured line flux ratios on the line-ratio diagnostic diagrams (Baldwin et al. 1981; Veilleux & Octerbrock 1987; Kewley et al. 2001, 2006; Kauffmann et al. 2003). Fig.2 shows the [*HST*]{} UV spectrum of J0038+4128S, which clearly exhibits broad emission features identified as H [i]{} Ly$\alpha$ and the Si [iv]{} $\lambda\lambda$1394,1403 lines. Since Ly$\alpha$ and the N [v]{} doublet ($\lambda\lambda$1239,1243) are blended, we fit the observed profile by the sum of two pairs of two Gaussian, one broad and one narrow, with one pair for Ly$\alpha$ and the other for the N [v]{} doublet. When fitting, data points corresponding to most prominent absorption features are masked and excluded from the fitting. The fitting yields a full width at half maximum (FWHM) for the broad component of Ly$\alpha$ of 4700 $\pm$ 70 km s$^{-1}$ , indicating J0038+4128S harbors a Seyfert 1 type nucleus. Furthermore, a close examination of the UV spectra reveals absorption lines that are seen among $\sim$ 50% Seyfert 1 galaxies (Crenshaw et al. 1999; Dunn et al. 2007).
J0038+4128N is clearly not a type 1 AGN since the optical spectrum obtained with the LAMOST reveals only narrow emission lines. Its nature, whether it is a Seyfert 2, a starburst, or a composite of both can be established by examining the locations of the diagnostic line ratios \[O [iii]{}\] $\lambda$5008/H$\beta$ versus \[N [ii]{}\] $\lambda$6585/H$\alpha$, \[S [ii]{}\] $\lambda\lambda$6718,6733/H$\alpha$ and \[O [i]{}\] $\lambda$6302/H$\alpha$ on the line-ratio diagnostic diagrams. We measure line fluxes by fitting Gaussians to profiles of detected emission lines, plus a first or second order polynomial for the continuum[^4]. From the measured Balmer decrement[^5], we estimate a colour excess $E(B-V) = 0.71$ for J0038+4128N, a value consistent with the expectation for Seyfert 2 galaxy. The extinction corrected (for both the host and the foreground Galactic extinction) line ratios of J0038+4128N (Fig.3) show clearly that it is a Seyfert-type galaxy. We use the diagnostic doublet line ratio of \[S [ii]{}\] $\lambda$6718/$\lambda$6733, assuming a typical electron temperature of $10^{4}$ K, to estimate the electron density $n_{\rm{e}}$ of the narrow-line region of J0038+4128N. The result log $n_{\rm e}/{\rm cm}^{-3}$ $\sim$ 2.5 is consistent with typical values found for narrow-line regions of Seyfert galaxies. However, it is still ambiguous whether the gas of the narrow-line region in J0038+4128N is ionized by nuclear emission from J0038+4128S or by that from J0038+4128N itself. The host galaxy spectrum of J0038+4128N is seen clearly in Fig.2, with the Fe [i]{} $\lambda$5268 and the Fe [i]{} $\lambda$5617 lines well detected. The redshift derived from the Fe [i]{} $\lambda$5617 line of the host galaxy is 0.07351$\pm$0.00011, consistent with the value derived from the narrow emission lines of J0038+4128N (see §3.2). The signal-to-noise ratio of the spectrum of the host galaxy taken at J0038+4128H with the LAMOST is unfortunately too low to see the underlying absorption features, let alone measurement of the Fe [i]{} lines. Nevertheless, the consistency in redshift of emission lines from the ionized gas and absorption lines from the host galaxy strongly supports that the gas around the nuclear region of J0038+4128N is ionized locally. In other words, there are two ionized sources, J0038+4128S and J0038+4128N.
On the other hand, we used Yunnan Faint Object Spectrograph and Camera on Yunnan 2.4 m telescope (pixel size 0283) to obtain the long-slit medium resolution ($R \sim 2200$, $\lambda = 4970-9830{\rm \AA}$) spectra of J0038+4128 on 2013 November 10. With clear sky condition and $\sim$ 10 seeing, the J0038+418S and J0038+4128N are spatially resolved as shown in Fig.4, which confirms that J0038+4128S is a Seyfert 1 galaxy revealed by the [*HST*]{} UV spectrum and J0038+4128N is a Seyfert 2 galaxy revealed by the LAMOST optical spectrum.
In addition, the diagnostic line ratios of J0038+4128H are presented in Fig.3 and the results also suggest that J0038+4128H is consistent with AGN ionization. The classical size of a narrow-line region of Seyfert 1galaxy is about 1-2 kpc (e.g. Bennert et al. 2006) and corresponding to 13-27 in J0038+4128. Considering the small 28 distance between the centre of the aperture towards J0038+4128H and J0038+4128S core and the high seeing, the spectra taken at J0038+4128H may mainly come from the narrow-line region of J0038+4128S. That is why the measured line ratios at J0038+4128H are consistent with AGN ionization. But it is a little strange that the J0038+4128H and J0038+4128N share the similar ionization conditions as shown in Fig.3. Future optical integral-field or long-slit spectroscopy of high spatial resolution are needed to reveal the ionization conditions for the whole system.
In summary, the existing data, both imaging and spectroscopy, strongly suggest that J0038+4128 is a Seyfert 1-Seyfert 2 AGN pair.
![ Diagnostic diagrams and measured line ratios from the LAMOST spectra of J0038+4128N (black dot) and J0038+4128H (red dot for 20111024 and blue dot for 20111028). In panel a, the red curve represents the theoretical starburst limit of Kewley et al. (2001) and the blue dotted curve represents the empirical separation between H [ii]{} regions and AGNs (Kauffmann et al. 2003). H [ii]{} regions fall lie below the blue dotted curve, AGNs dominate the region above the red curve, whereas objects of composite H [ii]{}-AGN type (Comp) lie between the two curves. In panels b and c, all the curves are taken from Kewley et al. (2006). H [ii]{}-regions fall in the area below the red curve, LINERs lie above the red curve but below the blue dotted curve, and Seyferts lie above both the red and blue dotted curves.](fig3.eps){width="8.5cm" height="3.1cm"}
![Segment of the two-dimensional long-slit spectrum of J0038+4128 that exhibits spatially resolved broad H${\alpha}$ emission line in the location of J0038+4128S and narrow emission lines (H${\alpha}$+\[N [ii]{}\]) in the location of J0038+4128N. The position of the long-slit is set to cross the centres of J0038+4128N and J0038+4128S.](fig4.eps){width="8.4cm" height="4.0cm"}
Radio and X-ray properties of J0038+4128
----------------------------------------
J0038+4128 is detected (but unresolved) at 1.4GHz by the NRAO VLA Sky Survey (NVSS; Condon et al. 1998) and at 325 MHz in a radio survey of M31 (Gelfand et al. 2004), with an integrated flux of 5.2 $\pm$ 0.4 mJy and 8.3 $\pm$ 0.72 mJy, respectively. The radio spectral index $\alpha$ ($F_{\nu} \propto \nu^{-\alpha}$) is about 0.32, and indicates that J0038+4128 is a compact radio source. The rest frame radio luminosity at 5 GHz is estimated at 2.12$\times10^{{39}}$ ergs$^{{-1}}$, assuming the above power-law spectral index. The rest-frame $B$-band luminosity is about 5.17$\times10^{{43}}$ erg s$^{{-1}}$ estimated from the SDSS $u$ and $g$ magnitudes by the equation of Vanden Berk et al. (2001)[^6]
The radio and optical luminosities thus place J0038+4128 as a typical Seyfert galaxy (Sikora et al. 2007). J0038+4128 has also been detected in the [*XMM-Newton*]{} slew survey (Saxton et al. 2008). The survey yields two X-ray sources close to J0038+4128. Of the two sources, the one with the larger position offset has parameter [*Ver\_Pusp*]{} set to true which means that its position is poorly constrained. Thus, we have adopted measurements of the X-ray source closest to J0038+4128S (with an offset of only 21 from J0038+4128S). The source has an X-ray flux of (1.3 $\pm$ 0.31 $)\times10^{-11}$ ergcm$^{-2}$s$^{-1}$ at band 0 (0.2–12 keV) and (4.6 $\pm$ 1.1) $\times10^{-12}$ ergcm$^{-2}$s$^{-1}$ band 5 (0.2–2 keV), and is not detected at band 4 (2–12 keV). Given a Galactic absorption value of $N_{\rm H}$ = 4.4$\times$10$^{20}$cm$^{-2}$, estimated from $E(B - V) = 0.069$, using the relation of Predehl & Schmitt (1995), we find an X-ray luminosity [^7] $L_{0.5-10\,\rm keV}\sim$1.5 $\times10^{44}$ ergs$^{-1}$ for a photon index $\Gamma = 1.8$. The above X-ray luminosity and the rest-frame $B$-band absolute magnitude $M_{B} = -20.5$ also show that J0038+4128 is a Seyfert-type galaxy (Brusa et al. 2007).
Relative line-of-sight velocity of the dual AGN
-----------------------------------------------
For J0038+4128N, we measured the redshift by fitting Gaussian profiles to emission lines detected in the LAMOST red arm spectra only (5800–9000[Å]{}), given the higher accuracy of wavelength calibration of the red arm spectra compared to those of the blue arm (3700–6000[Å]{}). By fitting the \[N [ii]{}\] $\lambda\lambda$6550,6585, H$\alpha$ and the \[S [ii]{}\] $\lambda\lambda$6718,6733, we derive an average redshift of $z_{\rm N} = 0.07328\,\pm\,0.00014$ for J0038+4128N. For J0038+4128S, only the broad Ly$\alpha$ can be fitted reliably (the N [v]{} and Si [iv]{} lines are too weak to obtain reliable redshifts). The measured central wavelengths of the broad and narrow components are not the same, which is normal since in most AGNs the Ly$\alpha$ emission line shows an asymmetric profile. The broad component yields a redshift of $0.07330\,\pm\, 0.00030$. However, from the UV spectrum plotted in Fig.2, we find that the narrow component is closer to the overall centroid of the whole Ly$\alpha$ profile, thus adopt the redshift $z_{\rm S} = 0.07177\,\pm\, 0.00015$ of narrow component of Ly$\alpha$ as the system value of J0038+4128S. The value is also consistent with the estimate of Barbieri & Romano (1976). From $z_{\rm N}$ and $z_{\rm S}$, we find that the dual AGN nuclei have a relative line-of-sight velocity of 453 $\pm$ 87 km s$^{-1}$.
Host galaxy morphology
----------------------
With the high spatial resolution images provided by the [*HST*]{}, the morphology and structure of J0038+4128 can be studied in detail. In the $F$336$W$/[*U*]{}-band image, which reveals the star-forming activities, two pairs of bi-symmetric spiral arms are detected in a binary AGN system for the first time. The results are consistent with the numerical simulations of Di Matteo et al. (2005). They have simulated the merging of two discs galaxies of the size of the Milky Way and found the tidal interactions can distort the discs into a pair of bi-symmetric spirals as the two galaxies begin to coalesce. In addition to the bi-symmetric spiral arms, there are several compact knots scattered around the two nuclei. Knots triggered by interaction are common features in merging galaxies (Villar-Martín et al. 2011). The ongoing process of star formation in those knots can be confirmed by further spectroscopy. In the $F$555$W$/[*V*]{}-band image, we see a stream-like substructure along the north-western edge. This may indicate that the two galaxies may have encountered more than once.
Variability in the optical and infrared bands
---------------------------------------------
J0038+4128 is found to be a fast variable in the optical , as shown by Barbieri & Romano (1976). They detect irregular variations of large amplitudes ($\sim$ 1.5 mag) on short time-scales (few days) in $B$-band based on hundreds of measurements accumulated with the 67/92 cm Schmidt Telescope of Asiago and the 182 cm Telescope of Cima Ekar from 1965 September 5 to 1975 January 13. The strong, fast optical variability indicates the presence of a fast flare component (tens of days) in the light curve of J0038+4128.
Finally, J0038+4128 is also detected by the [*Wide-Field Infrared Survey Explorer*]{} ([*WISE*]{}; Wright et al. 2010) in all bands. [*WISE*]{} maps the entire sky at 3.4, 4.6, 12, and 22 $\mu$m (bands $W1$, $W2$, $W3$ and $W4$, respectively) at an angular resolution of 61, 64, 65 and 120, respectively. In the [*WISE*]{} All-Sky Data Release Source Catalog, a variability flag (of integer values 0–9), [*var\_flg*]{}, is assigned to every detected object in each band, indicating the probability of possible flux variations. The greater the value of [*var\_flg*]{}, the higher the possibility of variability (see Hoffman et al. 2012 for detail). For J0038+4128, the flag has the highest value of 9 in both $W1$ and $W2$ bands and the light curves in the two bands show variability of amplitude of about 0.15 mag over a time-scale of half year.
SUMMARY
=======
J0038+4128 is resolved to contain two compact nuclei in both $F$336$W$/[*U*]{}- and $F$555$W$/[*V*]{}-band images secured with the [*HST*]{} WFPC2. The [*HST*]{} GHRS UV spectrum of the southern nucleus shows broad Ly$\alpha$ emission (FWHM $\sim$ 4700kms$^{-1}$), which indicates that it is a Seyfert 1 galaxy. The LAMOST optical spectrum of the northern nucleus shows narrow emission lines only. Line diagnostics as well as the consistency in redshift between emission lines from the ionized gas and absorption lines from the host galaxy suggest it is a Seyfert 2. Therefore, the [*HST*]{} and LAMOST data confirm that J0038+4128 is a Seyfert 1-Seyfert 2 AGN pair with a projected spatial separation of 4.7 kpc (344) and a line-of-sight relative velocity of 453 km s$^{-1}$.
The $F$336$W$/[*U*]{}-band image of J0038+4128, also reveals for the first time for a dual AGN system two pairs of bi-symmetric spiral arms. The [*HST*]{} images also reveal a number of compact star-forming knots as well as some tidal stream features.
Future optical integral-field spectroscopy of high spatial resolution would be extremely useful for further study of J0038+4128, in order to reveal the physical conditions and chemical properties of both the nuclear ionized gas and gas of the host galaxy for the whole entire system. Such a study will provide us much needed information of the co-evolution of the host galaxy and the central black holes.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by National Key Basic Research Program of China 2014CB845700. We thank Professor Fukun Liu and Dr. Andreas Schulze for providing valuable comments and suggestions of this paper.
This work has made use of data products from the Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST), Sloan Digital Sky Survey (SDSS), [*Galaxy Evolution Explorer*]{} ([*GALEX*]{}), Two Micron All Sky Survey (2MASS), [*Wide-field Infrared Survey Explorer*]{} ([*WISE*]{}), NRAO VLA Sky Survey (NVSS), [*XMM-Newton*]{} and Yunnan 2.4 m telescope.
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[^1]: Based on observations made with the NASA/ESA [*Hubble Space Telescope*]{}, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre (CADC/NRC/CSA).
[^2]: http://hla.stsci.edu/
[^3]: SPYBAL stands for SPectrum Y BALance, and it is performed to ensure proper alignment of the spectrum on the science diodes.
[^4]: For the H$\alpha$ and \[N [ii]{}\] doublet, we fit the continuum by a second order polynomial due to the $B$-band absorption to the blue of these three emission lines and possibly the weak contamination from J0038+4128S caused by the high seeing.
[^5]: We use an intrinsic, dust-free Balmer decrement of ${\rm H}\alpha/{\rm H}\beta$ ratio of 3.1 for Case B recombination at an electron temperature $T_{\rm e} = 10^4$K and a density $n_{\rm{e}}$ $\sim$ $10^{2-4} $cm$^{-3}$ for Seyfert galaxies (Osterbrock 1989).
[^6]: We modelled the SDSS $u$ and $g$-band images of J0038+4128 using the galaxy structure fitting code GAFIT v3.0 (Peng et al. 2002) by two Gaussian profiles for the dual AGN and a Sérsic component for the extended host galaxy.
[^7]: webPIMMS:http://heasarc.nas.gov/Tools/w3pimms.html
|
---
abstract: |
In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: Let $\ell$ be a prime, $q$ a prime power and consider the ensemble $\mathcal{H}_{g,\ell}$ of $\ell$-cyclic covers of ${\mathbb{P}}^1_{{\mathbb{F}}_q}$ of genus $g$.
We assume that $q\not\equiv 0,1\mod \ell$. If $2g+2\ell-2\not\equiv0\mod (\ell-1){\rm ord}_\ell(q)$, then $\mathcal{H}_{g,\ell}$ is empty. Otherwise, the number of rational points on a random curve in $\mathcal{H}_{g,\ell}$ distributes as $\sum_{i=1}^{q+1} X_i$ as $g\to \infty$, where $X_1,\ldots, X_{q+1}$ are i.i.d. random variables taking the values $0$ and $\ell$ with probabilities $\frac{\ell-1}{\ell}$ and $\frac{1}{\ell}$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$-th-root of unity, the presence of which was crucial in previous studies.
address:
- 'School of Mathematical Science, Tel Aviv University, Ramat Aviv, Tel Aviv, 6997801, Israel '
- 'School of Mathematical Science, Tel Aviv University, Ramat Aviv, Tel Aviv, 6997801, Israel '
author:
- 'Lior Bary-Soroker'
- Patrick Meisner
bibliography:
- 'SecondDraft.bib'
title: On the distribution of the rational points on cyclic covers in the absence of roots of unity
---
Introduction {#intro}
============
For a given smooth projective curve $C$ defined over a finite field $\mathbb{F}_q$ of genus $g=g(C)$, the Hasse-Weil bound says that $$|\# C(\mathbb{F}_q) - (q+1)| \leq 2g \sqrt{q}.$$ Here $C(\mathbb{F}_q)$ denotes the set of ${\mathbb{F}}_q$-rational points on $C$ and $q+1 = \# \mathbb{P}^1(\mathbb{F}_q)$. The problem we are after in this paper is the distribution of $\#C(\mathbb{F}_q)$ in families of covers of $\mathbb{P}^1_{{\mathbb{F}}_q}$ of genus $g$ in the limit $g\to \infty$.
Kurlberg and Rudnick [@KR09] study the ensemble of hyperelliptic curves $C$ of genus $g\to \infty$ under the assumption that $q$ is odd. They show that $\#C(\mathbb{F}_q) $ distributes as $\sum_{i=1}^{q+1} Y_i$, with $Y_1,\ldots, Y_{q+1}$ i.i.d. (independent and identically distributed) random variables taking the values $0,1,2$ with probabilities $\frac{q}{2(q + 1)}$, $\frac{1}{q + 1}$, $\frac{q}{2(q+1)}$, respectively. (In *loc.cit.*, a different normalization is used, the above formulation is similar to the one appearing in [@BDFL10a Theorem 1.1] in terms of the trace of Frobenius.)
The result of Kurlberg and Rudnick has been a subject to generalizations and extensions in many directions, see e.g. [@BJ; @BDFL10b; @CWZ]. One direction of generalization is to consider abelian covers of $\mathbb{P}^1$: Artin-Schreier covers [@BDFLS12; @BDFL16], biquadratic covers [@LMM] in odd characteristic, cyclic [@BDFKLOW; @BDFL10a; @BDFL11; @Meisner17; @Xiong10] and abelian [@Meisner_arxiv] covers of exponent dividing $q-1$. The latter assumption is used in a crucial way in those papers: it implies that $\mathbb{F}_q$ contains a primitive root of unity of order equal to the exponent of the covers, which allows the application of Kummer theory.
To be more precise, for a prime power $q$, a non-negative integer $g$, and a prime $\ell$ consider the moduli space $$\label{moduli_space}
{\mathcal{H}}_{g,\ell} = \{\varphi \colon C \to \mathbb{P}^1_{{\mathbb{F}}_q} : {\textnormal{Gal}}({\mathbb{F}}_q(C)/{\mathbb{F}}_q(\mathbb{P}^1_{{\mathbb{F}}_q})) \cong {\mathbb{Z}}/\ell{\mathbb{Z}},\quad g(C)=g \},$$ of $\ell$-cyclic covers of ${\mathbb{P}}^1_{{\mathbb{F}}_q}$. Here ${\mathbb{F}}_q(C)$ denotes the function field of $C$ and ${\mathbb{F}}_q({\mathbb{P}}^1_{{\mathbb{F}}_q}) = {\mathbb{F}}_q(X)$ is the field of rational functions. The state-of-the-art result [@BDFKLOW Theorem 1.3] for primes $\ell$ not dividing $q$ says that if $\ell\mid q-1$ and if $C$ is a random curve chosen uniformly from ${\mathcal{H}}_{g,\ell}$, then $$\label{eq:BDFKLOW}
{\textnormal{Prob}}\left(\#C({\mathbb{F}}_q)=N\right) = {\textnormal{Prob}}\left(\sum_{i=1}^{q+1} Y_i = N\right) + O\left(\frac{1}{g}\right),$$ as $g\to \infty$, where $Y_1,\ldots, Y_{q+1}$ are i.i.d. random variables taking the values $0,1$ and $\ell$ with probabilities $\frac{(\ell-1)q}{\ell(q+\ell-1)}$, $\frac{\ell-1}{q+\ell-1}$, and $\frac{q}{\ell(q+\ell-1)}$, respectively.
The primes $\ell$ to which may be applied are bounded by $q-1$; in particular, for ${\mathbb{F}}_2$ the result is empty, for ${\mathbb{F}}_3$ and ${\mathbb{F}}_5$ one must have $\ell = 2$, and the main term of coincides with that of [@KR09]. Of course, for larger fields (such as ${\mathbb{F}}_7$) is more general.
Our main result treats $q\not\equiv 0,1\mod \ell$. Not every genus may be obtained in this case: If $n_q$ is the multiplicative order of $q$ modulo $\ell$, then the genus $g$ of an $\ell$-cyclic cover must satisfy the congruence $$\label{genus_congruence}
2g+2\ell -2 \equiv 0 \mod (\ell-1)n_q,$$ see Proposition \[genformprop1\].
\[thm:main\] Let $q$ be a prime power and $\ell$ a prime such that $q\not\equiv 0,1\mod{\ell}$. Let $C$ be a random curve chosen uniformly from ${\mathcal{H}}_{g,\ell}$ (see ). Then, as $g\to\infty$ satisfying the congruence relation , $$\label{eq:Main}
{\textnormal{Prob}}\left( \#C({\mathbb{F}}_q)=N\right) = {\textnormal{Prob}}\left(\sum_{i=1}^{q+1} X_i = N \right) +
O\left(q^{-\frac{(1-\epsilon)g}{\ell-1}}\right).$$ Here the $X_1,\ldots, X_{q+1}$ are i.i.d. random variables taking the values $0$ and $\ell$ with probabilities $\frac{\ell-1}{\ell}$ and $\frac{1}{\ell}$, respectively.
In each of the $q+1$ random variables $Y_i$ models the number of rational points on $C$ lying above the $i$-th rational point $x_i$ of ${\mathbb{P}}^1({\mathbb{F}}_q)$ (under some arbitrary order). The number of points is $0$ if $x_i$ is inert in $C$ (i.e., the fiber is irreducible as an ${\mathbb{F}}_q$-scheme), $1$ if ramified, and $\ell$ if totally split. The probabilities are derived from the probabilities for the point to have each of these splitting types.
In , when $q\not\equiv 0,1\mod\ell$, the points of ${\mathbb{P}}^1({\mathbb{F}}_q)$ behave similarly to except for the fact that they cannot be ramified, see Remark \[genformrem\]. So each of these $q$ points contributes either $0$ or $\ell$. The probabilities are derived from the heuristic that the Frobenius element is a random element of ${\mathbb{Z}}/\ell{\mathbb{Z}}$, and the point splits if and only if the Frobenius is trivial.
The error term in decays exponentially in $g$ as the error term in [@KR09], while in the error term decays linearly in $g$. In the latter results, the big error term comes from points of ${\mathbb{P}}^1({\mathbb{F}}_q)$ that ramify in $C$, while in our setting the points of ${\mathbb{P}}^1({\mathbb{F}}_q)$ are always unramified in $C$.
We conclude the introduction with a few words on the methods and an outline of the proof. When $\ell \mid q-1$, the field ${\mathbb{F}}_q$ contains a primitive $\ell$-th root of unity $\zeta_\ell$. Hence the $\ell$-cyclic covers of ${\mathbb{P}}^1$ over ${\mathbb{F}}_q$ are birationally equivalent to a plane curve of the form $$\label{eq:SE}
Y^\ell = F(X), \qquad \mbox{$F$ is $\ell$-th-power free}.$$ This leads to a parameterization of the moduli space ${\mathcal{H}}_{g,\ell}$. Then one derives an analytic formula for the number of rational points lying above a fixed point of ${\mathbb{P}}^1({\mathbb{F}}_q)$ in terms of Dirichlet characters. The next step is to apply generating function techniques to derive good asymptotic formulas for each of the terms.
When $q\not\equiv0,1\mod\ell$, Equation generates a non-Galois and in particular non-abelian cover. We use the ray class group and explicit Galois descent to obtain a parametrization of the $\ell$-cyclic Galois covers (Section \[classify\]) together with a genus formula (Section \[genformsec\]). We control the number of rational points lying over the points of ${\mathbb{P}}^1({\mathbb{F}}_q)$ (Proposition \[numptsprop\]). From this point, we follow a similar path as described above: We derive an analytic formula (Proposition \[numptscor\]) and then use generating function techniques (Section \[SetCount\]) to prove Theorem \[thm:main\] (Section \[sec:proof\]).
Acknowledgments {#acknowledgments .unnumbered}
---------------
The authors thank Dan Haran for a useful conversation on wreath product actions on fields, Chantal David for crucial comments, and Will Sawin for finding a critical mistake in a previous version.
L.B.S. was partially supported by a grant of the Israel Science Foundation and by the Simons CRM Scholars program. P.M. has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n$^{\text{o}}$ 320755.
Classifying the Curves {#classify}
======================
In this section we use explicit class field theory to classify the $\ell$-cyclic covers $C\to {\mathbb{P}}^1_{{\mathbb{F}}_q}$ which in terms of function fields corresponds to ${\mathbb{Z}}/\ell {\mathbb{Z}}$-extensions of $K={\mathbb{F}}_q(X)$.
Class Field Theory Preliminaries {#CFTprel}
--------------------------------
We begin by recalling the basic objects and results of class field theory which we use. For more extensive background on this topic we refer the reader to [@CF] or to [@HM], for a more specific treatment in the case of function fields.
For now, we allow $K$ to be an arbitrary global field and $L$ a tamely ramified abelian extension of $K$. Denote $\mathcal{D}(K)$, $\mathcal{D}(L)$ as the groups of divisors of $K$ and $L$, respectively. Define the conorm map on the set of prime divisors in $\mathcal{D}(K)$ by $$i_{L/K}(P) = \sum_{\mathfrak{P}|P}e(\mathfrak{P}/P)\mathfrak{P},$$ where the sum is over all primes of $L$ dividing $P$. We then extend $i_{L/K}$ linearly to all of $\mathcal{D}(K)$.
For any effective divisor $\mathfrak{m}$ of $K$, denote $$\mathcal{D}_{\mathfrak{m}}(L) = \{D\in \mathcal{D}(L) : {\textnormal{supp}}(D)\cap {\textnormal{supp}}(i_{L/K}(\mathfrak{m})) = \emptyset\}.$$
If $L=K$ then $i_{L/K}$ is the identity map and $\mathcal{D}_{\mathfrak{m}}(K)$ is the set of divisors of $\mathcal{D}(K)$ that are coprime to $\mathfrak{m}$.
Let $P$ be a prime of $K$ that is unramified in $L$. Then there is a unique automorphism $\sigma_P \in {\textnormal{Gal}}(L/K)$, called the **Frobenius automorphism**, such that $$\sigma_P(x) \equiv x^{N(P)} \mod{\mathfrak{P}}$$ for all $x\in \mathcal{O}_{\mathfrak{P}}$ and all places $\mathfrak{P}$ of $L$ lying above $P$, where $N(P)$ is the norm of $P$ and $\mathcal{O}_{\mathfrak{P}}$ is the local ring at $\mathfrak{P}$. Let $\mathfrak{m}$ be an effective divisor of $K$ such that $L/K$ is unramified outside of $\mathfrak{m}$ and define the **Artin map** as
------------- --------------------------------- ----------- ------------------------------------------------
$A_{L/K}$ : $\mathcal{D}_{\mathfrak{m}}(K)$ $\to$ ${\textnormal{Gal}}(L/K)$
$D$ $\mapsto$ $\prod_{P}\sigma_P^{{\textnormal{ord}}_P(D)},$
------------- --------------------------------- ----------- ------------------------------------------------
where the product is over all primes of $K$ dividing $D$.
Define the **ray** of $L$ modulo $\mathfrak{m}$ as $$\mathcal{P}_{\mathfrak{m}}(L) = \{(f) : f\in L^{*}, f\equiv 1 \mod{\mathfrak{P}^{{\textnormal{ord}}_{\mathfrak{P}}(i_{L/K}(\mathfrak{m}))}} \mbox{ for all places $\mathfrak{P}$ of $L$} \}$$ where we use the notation $(f)$ for the principle divisor of $f$. Notice that $\mathcal{P}_{\mathfrak{m}}(L)$ is a subgroup of $\mathcal{D}_{\mathfrak{m}}(L)$.
We say that $\mathfrak{m}$ is a **modulus** for $L$ if $\mathcal{P}_{\mathfrak{m}}(L)\subset \ker(A_{L/K})$. In this case, we can view $A_{L/K}$ as taking elements from the ray class group: $\mathcal{C}\ell_{\mathfrak{m}}(L) := \mathcal{D}_{\mathfrak{m}}(L)/\mathcal{P}_{\mathfrak{m}}(L)$. The following are the class field theory results which we need, borrowed from [@HM].
1. $\mathfrak{m}$ is a modulus of $L$ if and only if $L/K$ is unramified outside of ${\textnormal{supp}}(\mathfrak{m})$.
2. If $L/K$ is unramified outside of ${\textnormal{supp}}(\mathfrak{m})$ then $A_{L/K} : \mathcal{C}\ell_{\mathfrak{m}}(K)\to {\textnormal{Gal}}(L/K)$ is surjective.
3. Let $L_1,L_2/K$ be two finite abelian extensions with modulus $\mathfrak{m}$. Then $L_1=L_2$ if and only if $\ker(A_{L_1/K}) = \ker(A_{L_2/K})$.
4. For every effective divisor $\mathfrak{m}$ and finite index subgroup $H$ of $\mathcal{C}\ell_{\mathfrak{m}}(K)$ there exists a finite abelian extension $L$ of modulus $\mathfrak{m}$ such that $\ker(A_{L/K})=H$. Moreover, $n\mathcal{C}\ell_{\mathfrak{m}}(K) \subset H$ where $n$ is the exponent of ${\textnormal{Gal}}(L/K)$.
Abstract Parametrization of Geometric $\ell$-cyclic Extensions {#ltor}
--------------------------------------------------------------
From now on we restrict to the case $K={\mathbb{F}}_q(X)$, $n=\ell$ a prime not dividing $q$, and we want to parametrize the set $$\mathcal{H}_{\mathfrak{m}} = \{ L/K \mbox{ $\ell$-cyclic Galois extension, unramified outside $\frak{m}$ and $L\cap \overline{\mathbb{F}}_q=\mathbb{F}_q$}\}.$$ Let $n_q$ be the multiplicative order of $q$ modulo $\ell$. So, $n_q = [K(\mu_\ell):K]=[{\mathbb{F}}_q(\mu_{\ell}):{\mathbb{F}}_q]$ where $\mu_\ell$ is the group of $\ell$-th-roots of unity.
First we compute the ray class group in this setting.
\[ltorlem\] Let $\mathfrak{m}$ be an effective divisor of $K$, then the ray class group $\mathcal{C}\ell_{\mathfrak{m}}(K)$ is finite and $$\mathcal{C}\ell_{\mathfrak{m}}(K)/\ell \mathcal{C}\ell_{\mathfrak{m}}(K) \cong {\mathbb{Z}}/\ell{\mathbb{Z}}\times \prod_{\substack{P\in {\textnormal{supp}}(\mathfrak{m}) \\ n_q|\deg(P)}} {\mathbb{Z}}/\ell{\mathbb{Z}}.$$
We may write $\mathcal{D}_{\mathfrak{m}} = \mathcal{D}^0_{\mathfrak{m}}(K) \times {\mathbb{Z}}$ with $\mathcal{D}^0_{\mathfrak{m}}(K)$ the subgroup of divisors of degree $0$. Since $\mathcal{P}_{m}(K)\subseteq \mathcal{D}^0_{\mathfrak{m}}(K)$, we have $$\mathcal{C}\ell_{\mathfrak{m}}(K)/\ell \mathcal{C}\ell_{\mathfrak{m}}(K) \cong \mathcal{C}\ell^0_{\mathfrak{m}}(K)/\ell \mathcal{C}\ell^0_{\mathfrak{m}}(K) \times {\mathbb{Z}}/\ell{\mathbb{Z}}$$ with $\mathcal{C}\ell^0_{\mathfrak{m}}(K)=\mathcal{D}^0_{\mathfrak{m}}/\mathcal{P}_{\mathfrak{m}}(K)$. So it suffices to prove that $$\mathcal{C}\ell^0_{\mathfrak{m}}(K)/\ell \mathcal{C}\ell^0_{\mathfrak{m}}(K) \cong \prod_{\substack{P\in {\textnormal{supp}}(\mathfrak{m}) \\ n_q|\deg(P)}} {\mathbb{Z}}/\ell{\mathbb{Z}}.$$
First, suppose $\mathfrak{m} = P$ for some prime $P$ of $K$. Since ${\mathbb{F}}_q[X]$ is a principle ideal domain, we get that $$\mathcal{D}^0_{\mathfrak{m}}(K) = \{(f) : f \in K^*, {\textnormal{ord}}_P(f)=0\} = \{(f) : f \in K^*, f \not\equiv 0,\infty \mod{P}\}.$$ Hence, $$\begin{aligned}
\begin{split}
\mathcal{C}\ell^0_{\mathfrak{m}}(K) = & \{(f) : f \in K^*, f \not\equiv 0 ,\infty \mod{P}\}/\{(f) : f \in K^*, f \equiv 1 \mod{P}\} \\
\cong & \left({\mathbb{F}}_q[X]/P\right)^* = {\mathbb{F}}^*_{q^{\deg(P)}}
\end{split}\end{aligned}$$
Since ${\mathbb{F}}^*_{q^{\deg(P)}}$ is cyclic, and its order is divisible by $\ell$ if and only if $n_q\mid \deg(P)$, we have $$\begin{aligned}
\label{ltoreq}
\mathcal{C}\ell^0_{\mathfrak{m}}(K)/\ell \mathcal{C}\ell^0_{\mathfrak{m}}(K) \cong \begin{cases}{\mathbb{Z}}/\ell{\mathbb{Z}}& n_q|\deg(P) \\ 0 & \mbox{otherwise.} \end{cases}\end{aligned}$$
Likewise, if $\mathfrak{m} = eP$ for some positive integer $e$, then $$\mathcal{C}\ell_{\mathfrak{m}}(K) \cong \left({\mathbb{F}}_q[X]/P^e\right)^* \cong \left({\mathbb{F}}_q[X]/P\right)^*\times H$$ for some $p$-group $H$ (where $p=\textnormal{char}(K)\neq \ell$). Hence holds.
Finally, assume $\mathfrak{m} = \sum_{P}e_P P$ with $e_P\geq 0$ and only finitely many nonzero. Then by the Chinese Remainder Theorem, $$\mathcal{D}^0_{\mathfrak{m}}(K) \cong \prod_{P\in {\textnormal{supp}}(\mathfrak{m})}\mathcal{D}^0_{e_PP}(K)
\quad \mbox{and}\quad
\mathcal{P}_{\mathfrak{m}}(K) \cong \prod_{P\in {\textnormal{supp}}(\mathfrak{m})}\mathcal{P}_{e_PP}(K)$$ and hence, by $$\mathcal{C}\ell^0_{\mathfrak{m}}(K)/\ell \mathcal{C}\ell^0_{\mathfrak{m}}(K) = \prod_{\substack{P\in {\textnormal{supp}}(\mathfrak{m}) \\ n_q|\deg(P)}} {\mathbb{Z}}/\ell{\mathbb{Z}},$$ as needed.
\[def:n-divisible\] We say a polynomial $F$ is **$n$-divisible** if
1. $n\mid \deg(P)$ for all $P\mid F$ and
2. $F$ is monic.
A trivial observation, which plays a crucial point in the establishment of the analytic formula below, is that if $f$ is $n$-divisible for $n>1$, then $f(x) \neq 0$ for all $x\in {\mathbb{F}}_q$.
\[ltorcor\] For every effective divisor $\mathfrak{m}$ of $K$, let $\Omega_1$ be the set of pairs $(F,b)$ with $F\in {\mathbb{F}}_q[X]$ an $n_q$-divisible polynomial with prime factors in ${\rm supp}(\mathfrak{m})$ which is not an $\ell$-th-power and $b\in {\mathbb{F}}_{q^{n_q}}^*$ both $F,b$ taken up to $\ell$-th-powers in the respective groups and let $\Omega_2$ be the set of $\ell$-cyclic extensions $L/K$ of modulus $\mathfrak{m}$ such that $L\cap \bar{{\mathbb{F}}}_q={\mathbb{F}}_q$. Then $$|\Omega_1| = (\ell-1)|\Omega_2|.$$
By Proposition \[CFTprop\], the $\ell$-cyclic extensions $L/K$ of modulus $\mathfrak{m}$ are in bijection with subgroups of index $\ell$ of $\mathcal{C}\ell_{\mathfrak{m}}(K)/\ell \mathcal{C}\ell_{\mathfrak{m}}(K)$. By Lemma \[ltorlem\], the latter are in $1$-to-$(\ell-1)$ correspondence with non-zero elements of $${\mathbb{Z}}/\ell{\mathbb{Z}}\times \prod_{\substack{P\in {\textnormal{supp}}(\mathfrak{m}) \\ n_q|\deg(P)}} {\mathbb{Z}}/\ell{\mathbb{Z}}.$$ We identify the first copy of ${\mathbb{Z}}/\ell{\mathbb{Z}}$ with ${\mathbb{F}}_{q^{n_q}}^* /({\mathbb{F}}_{q^{n_q}}^* )^\ell$.
Now, a nonzero element $(b,(a_P))$ corresponds to the pair $(b,\prod_{P} P^{a_P})$ bijectively. To conclude the proof, we need to note that there is exactly one $\ell$-cyclic extension $L/K$ with $L\cap \bar{{\mathbb{F}}}_q \neq {\mathbb{F}}_q$ (namely, $L={\mathbb{F}}_{q^\ell}$) and there are exactly $\ell-1$ pairs with $F$ an $\ell$-th-power, namely $(1,b)$, $b\in {\mathbb{F}}_{q^{n_q}}^*/({\mathbb{F}}_{q^{n_q}}^*)^\ell$.
Explicit Correspondence {#basefield}
-----------------------
The correspondence described in Corollary \[ltorcor\] is given non-explicitly. In this section, we construct such a correspondence, using explicit Galois descent. We denote the to-be-constructed extension corresponding to $(F,b)$ by $L_{F,b}$. The case $n_q=1$ is treated by Kummer theory:
\[kummerprop\] Assume $n_q=1$. Then $L_{F,b}=K(\sqrt[\ell]{bF})$.
This is the main result of Kummer theory, cf. [@DF Proposition §14.37].
From now on assume $n_q>1$. As mentioned above, the extension $K(\sqrt[\ell]{F})/K$ is not Galois and in particular not abelian. To deal with this, we extend the base field in order to apply Kummer theory and then descend the extension (using wreath products) to describe the model over the original field.
Let $K' := {\mathbb{F}}_{q^{n_q}}(X)$. Then, by the definition of $n_q$, we have $\mu_{\ell} \subseteq K'$. Here we can apply Kummer theory, but $F$ gives rise to many $\ell$-cyclic extensions which, in order to descend back to $K$, we need to describe explicitly together with the Galois action.
\[basefieldrem2\] One word on the terminology we use, since both $K$ and $K'$ are rational function fields in $X$, we shall use the term ‘prime polynomial’ in $K$ (resp. $K'$) usually denoted by $P$ (resp. $\mathfrak{P}$) to indicate both a monic irreducible polynomial in $X$ with coefficients in ${\mathbb{F}}_q$ (resp. ${\mathbb{F}}_{q^{n_q}}$) and the corresponding prime of the field (which is not $X=\infty$).
We denote by $\phi$ the $q$-th-power Frobenius. We extend its action from ${\mathbb{F}}_{q^{n_q}}$ to $K'$ by letting it act trivially on $X$. So, $$\label{def:Frob}
\phi(a_nX^n+a_{n-1} X^{n-1} + \cdots + a_0 ) = a_n^qX^n+a_{n-1}^q X^{n-1} + \cdots + a_0^q$$ and ${\textnormal{Gal}}(K'/K) = \left<\phi\right>$.
Every $x\in {\mathbb{F}}_q$ is fixed by $\phi$, i.e., $x^q=x$. Thus, $$\label{eq:phiqx}
\phi(f)(x) = (f(x))^q, \qquad x\in {\mathbb{F}}_q, f\in {\mathbb{F}}_{q^{n_q}}[X].$$
We begin with a lemma about factorization.
\[basefieldlem\]
Let $P$ be a prime polynomial in $K$ and let $m=\gcd(n_q,\deg(P))$. Then there exist distinct prime polynomials $\mathfrak{P}_1,\dots,\mathfrak{P}_m$ in $K'$ such that $$P = \mathfrak{P}_1 \cdots \mathfrak{P}_m$$ and $\phi(\mathfrak{P}_i) = \mathfrak{P}_{i+1}$, where $\phi$ is as defined in and with the convention that $\mathfrak{P}_{m+1}=\mathfrak{P}_1$.
Since $K'/K$ is a cyclic unramified extension, each prime can be decomposed as $$P = \mathfrak{P}_1 \cdots \mathfrak{P}_g,$$ with the $\mathfrak{P}_i$ pairwise distinct. Since the Galois group ${\textnormal{Gal}}(K'/K) = \left<\phi\right>$ acts transitively on the primes above $P$, by relabeling the primes we may assume that $\mathfrak{P}_{i+1} = \phi(\mathfrak{P}_i)$. Finally, $${\mathbb{F}}_{q^{n_q}}[X]/P = ({\mathbb{F}}_{q}[X]/P) \otimes {\mathbb{F}}_{q^{n_q}} = {\mathbb{F}}_{q^{\deg(P)}}\otimes {\mathbb{F}}_{q^{n_q}} = \prod_{i=1}^m {\mathbb{F}}_{q^{n_q\deg(P)/m}},$$ whence $g=m$.
We extend the above factorization of primes to $n_q$-divisible polynomials.
\[basefieldcor\]
If $F$ is an $n_q$-divisible polynomial in $K$, then there exist coprime polynomials in $K'$, $F_1,\dots,F_{n_q}$ such that $$\begin{aligned}
\label{stablefact}
\begin{split}
F&=F_1\cdots F_{n_q} \\
F_{i+1} &= \phi(F_i)
\end{split}\end{aligned}$$ where $\phi$ is defined in and $F_{n_q+1}:=F_1$. We call a **$\mu_{\ell}$-stable factorization** and the $F_i$ the corresponding **$\mu_{\ell}$-stable factors**.
Setting $F_i = \prod_{P|F} \mathfrak{P}^{{\textnormal{ord}}_P(F)}_i$ with $\mathfrak{P}_i$ as in Lemmas \[basefieldlem\] suffices.
From now on we fix our pair $(F,b)$ and explicitly construct a ${\mathbb{Z}}/\ell{\mathbb{Z}}$ extension $L = L_{F,b}$. Since $F$ is taken up to $\ell$-th powers, we may assume that $F$ is an $\ell$-th-power free polynomial in $K$ that is $n_q$-divisible. We also fix notation: Let $F_1,\ldots, F_{n_q}$ be the $\mu_{\ell}$-stable factorization factors of $F$ as in , let $b_i=b^{q^{i-1}}$, and $$u_i = \sqrt[\ell]{b_iF_i} ,$$ for $i=1,\ldots, n_q$. We put $$E' = K(u_1,\ldots, u_{n_q}).$$
Recall that the wreath product $A\wr B$ of two finite groups $A$ and $B$ is defined as the semi-direct product $A^B\rtimes B$ with $B$ acting on $A^B$ by permuting the indices. To be more precise, if for example $B ={\mathbb{Z}}/n_q{\mathbb{Z}}$ and $x=1\in B$ is a generator, then $$(-x) (v_{1}, \ldots, v_{n_q}) x = (v_{2}, \ldots, v_{n_q}, v_1).$$
\[basefieldlem2\]
Let $\zeta_\ell\in \mu_{\ell}$ be a primitive $\ell$-th-root of unity. Then
1. ${\textnormal{Gal}}(E'/K')=\left({\mathbb{Z}}/\ell{\mathbb{Z}}\right)^{n_q}$ with action given by $$\label{actionofv}
(h_1, \ldots, h_{n_q}).u_i = \zeta_{\ell}^{q^{i-1}h_i} u_i, \qquad i=1,\ldots, n_q.$$
2. The action of $\phi$ on $K'$ as defined in may be lifted to an action on $E'$ by setting $$\label{actionofphi}
\phi(u_i) = u_{i+1},$$ with our usual convention $u_{n_q+1} := u_1$.
3. The extension $E'/K$ is Galois with ${\textnormal{Gal}}(E'/K) \cong {\mathbb{Z}}/\ell{\mathbb{Z}}\wr {\mathbb{Z}}/n_q{\mathbb{Z}}$.
4. The fixed field $E$ of $\left<\phi\right>$ in $E'$ is $$E = K(u_1+\cdots + u_{n_q}).$$
For $(1)$, it is clear that $K'(u_1,\ldots, u_{n_q})\subseteq E'$ since $K'(u_j) = L_{e_j} \subseteq E'$. The other inclusion is obvious since $L_{\mathbf{v}}\subseteq K'(u_1,\ldots, u_{n_q})$ for all $\mathbf{v}$. This proves that $E'=K'(u_1,\ldots, u_{n_q})$. Now, since the $F_i$-s are pairwise co-prime, we have that ${\textnormal{Gal}}(E'/K')=\left({\mathbb{Z}}/\ell{\mathbb{Z}}\right)^{n_q}$. We may choose coordinates for $({\mathbb{Z}}/\ell {\mathbb{Z}})^{n_q}$ to have the stated action.
$(2)$ and $(3)$ are well known consequences of $(1)$, but we give a full account because we need the precise actions.
For $(2)$, lift $\phi$ in any way to a map $\psi$ from $E'$ to its algebraic closure. Since $$\psi(u_i)^{\ell} = \psi(u_i^\ell) = \psi(b_iF_i) = \phi(b_iF_i) = b_{i+1}F_{i+1},$$ we have that $\psi(u_i) = \zeta_{\ell}^{v_{i+1}} u_{i+1}$, for some tuple $(v_1,\ldots, v_{n_q})\in ({\mathbb{Z}}/\ell {\mathbb{Z}})^{n_q}$ (as usual $v_{n_q+1}=v_1$). From this we conclude that $\psi\colon E'\to E'$ and in particular $E'/K$ is a Galois extension. To get the desired lift, we change $\psi$ by an element of ${\textnormal{Gal}}(E'/K')$: $$(-v_{1},-v_2q^{-1},\ldots, -v_{n_q}q^{1-n_q})\psi,$$ and this lift maps each $u_i$ to $u_{i+1}$. By abuse of notation, we denote the latter lift also by $\phi$.
For $(3)$, by $(1)$ and $(2)$, we have that $$\begin{split}
(\phi^{-1} (v_1,\ldots, v_{n_q}) \phi) (u_i) &= \phi^{-1} (v_1,\ldots, v_{n_q}) . u_{i+1}=\phi^{-1}\zeta_\ell^{q^{i}v_{i+1}} = \zeta_{\ell}^{q^{i-1}v_{i+1}} u_i\\
&= (v_2,\ldots, v_{n_q}, v_1).u_i.
\end{split}$$ This shows that the Galois group ${\textnormal{Gal}}(E'/K)$ is the wreath product.
For $(4)$, put $u= \sum_{i=1}^n u_i$. Since $K(u)\subseteq E$, by the fundamental theorem of Galois theory to show equality, it suffices to show that $K(u)$ is not fixed by any element $\sigma$ in ${\mathbb{Z}}/\ell{\mathbb{Z}}\wr {\mathbb{Z}}/n_q {\mathbb{Z}}$ which is not in $\left<\phi\right>$, which has the form $\sigma = ((h_1,\ldots, h_{n_q}), \phi^k)$ with $(h_1,\ldots, h_{n_q})$ a nonzero vector in $({\mathbb{Z}}/\ell{\mathbb{Z}})^{n_q}$. Assume by contradiction that $\sigma(u)\in K(u)$. Now, by and since $u$ is fixed by $\phi$, $$\sigma (u) = \sum_{i}\zeta_\ell^{q^{i-1}h_i} u_i.$$ Since $\sigma(u)$ is also fixed by $\phi$, all the powers of $\zeta_\ell$ must be equal; i.e., $\sigma(u) = \zeta_{\ell}^a u$ with $a \equiv q^{i-1}h_i\mod\ell$ for all $i$. But then $\zeta_\ell^a \in K(u)$ and thus fixed by $\phi$, which implies that $a\equiv 0 \mod \ell$. So all $h_i \equiv0\mod\ell$, contradiction.
Now that we have computed the Galois group, to study $\ell$-cyclic subextensions we need to find $\ell$-cyclic quotients of the wreath product.
\[subclassifylem1\] The wreath product $G={\mathbb{Z}}/\ell{\mathbb{Z}}\wr{\mathbb{Z}}/n_q{\mathbb{Z}}$ has a unique $\ell$-cyclic quotient. More precisely, for $H\lhd G$ we have $G/H\cong {\mathbb{Z}}/\ell {\mathbb{Z}}$ if and only if $H = \tilde{H}\rtimes{\mathbb{Z}}/n_q{\mathbb{Z}}$, where $$\tilde{H} = \{ h = (h_1,\dots,h_{n_q}) : h_1+\dots+h_{n_q} \equiv 0 \mod{\ell}\}\leq \left({\mathbb{Z}}/\ell{\mathbb{Z}}\right)^{n_q}.$$
Let $G'$ be the commutator of $G$. Direct computation shows that $$G' \leq \{(h,0) : h_1+\dots+h_{n_q} = 0\}$$ and since the quotient by the right hand side is abelian we have an equality. Here $h=(h_1,\dots,h_{n_q})\in\left({\mathbb{Z}}/\ell{\mathbb{Z}}\right)^{n_q}$. Hence we have the isomorphism $$\begin{split}
G/G' &\to {\mathbb{Z}}/\ell{\mathbb{Z}}\times{\mathbb{Z}}/n_q{\mathbb{Z}}\cong {\mathbb{Z}}/\ell n_q{\mathbb{Z}}\\
(h,\sigma)&\mapsto \Big(\sum_j h_j, \sigma\Big)
\end{split}$$ which implies that $G/G'$ has a unique $\ell$-cyclic quotient as a cyclic group, and so does $G$. Moreover this quotient is given by $(h,\sigma)\mapsto \sum_j h_j$, and so the kernel is $\tilde{H}\rtimes {\mathbb{Z}}/n_q{\mathbb{Z}}$.
We enumerate all the $\ell$-cyclic subextensions of $E'/K'$ by nonzero vectors $\mathbf{v}=(v_1,\dots,v_{n_q})$ of elements in $\mathbb{Z}/\ell \mathbb{Z}$. We write $$\label{eq:DefFv}
F_{\mathbf{v}} := \prod_{i=1}^{n_q} (b_iF_i)^{v_i'},$$ where $0\leq v_i'\leq \ell-1$ is a minimal non-negative representative of $v_i$, $i=1,\ldots, n_q$ and we let $$\label{eq:DefLv}
L_{\mathbf{v}} = K'(\sqrt[\ell]{F_{\mathbf{v}}})$$ be the corresponding Kummer extension.
\[subclassifylem2\] Let $\tilde{H}\leq H \leq {\mathbb{Z}}/\ell{\mathbb{Z}}\wr{\mathbb{Z}}/n_q{\mathbb{Z}}\cong {\textnormal{Gal}}(E'/K)$ be as in Lemma \[subclassifylem1\] and let $L=E'^H$ and $\tilde{L}=E'^{\tilde{H}}$. Then, $$\begin{aligned}
\label{Lv0}\tilde{L} &= L_{\mathbf{v_0}}, \\
\label{ExtensionofScalars}\tilde{L} &= L {\mathbb{F}}_{q^{n_q}},\quad \mbox{and}\\
\label{explicitequation}L &= {\mathbb{F}}_q(X)[Y]/\Bigg(\prod_{j=0}^{\ell-1} \Big( Y - \sum_{k=0}^{n_q-1} \zeta_{\ell}^{jq^k} \sqrt[\ell]{F_{\mathbf{v}_k}}\Big)\Bigg),\end{aligned}$$ where $\mathbf{v}_k = (q^{k},q^{k-1},\dots,q^{k+1-n_q})$.
By , , and since $q^{j-1}v_j\equiv 1\mod \ell$, we have $$h\left(\sqrt[\ell]{F_{\mathbf{v}_0}}\right) = \prod_{j=1}^{n_q} \zeta_\ell^{q^{j-1}h_jv_j} \sqrt[\ell]{F_{\mathbf{v}_0}} = \zeta_\ell ^{\sum_{j=1}^{n_q}h_j } \sqrt[\ell]{F_{\mathbf{v}_0}} = \sqrt[\ell]{F_{\mathbf{v}_0}},$$ for all $h=(h_1,\ldots, h_{n_q})\in \tilde{H}$ Therefore, $L_{\mathbf{v}_0} \subseteq \tilde{L}$. On the other hand, $[L_{\mathbf{v}_0}:K']=\ell = [\tilde{L}:K']$, hence $\tilde{L}=L_{\mathbf{v}_0}$.
Since $\tilde{H} = H\cap ({\mathbb{Z}}/\ell{\mathbb{Z}})^{n_q}$ and $({\mathbb{Z}}/\ell{\mathbb{Z}})^{n_q} \cong {\textnormal{Gal}}(E'/K')$, the fundamental theorem of Galois theory implies that $$\tilde{L} = LK' = L{\mathbb{F}}_{q^{n_q}}.$$
Finally, let $u\in L$ be the trace of $\sqrt[\ell]{F_{\mathbf{v}_0}}\in \tilde{L}$ in the extension $\tilde{L}/L$. Since ${\textnormal{Gal}}(\tilde{L}/L)$ is generated by the restriction of $\phi$, by , we have $$u = \sum_{k=0}^{n_q-1} \sqrt[\ell]{F_{\mathbf{v}_k}}.$$ The conjugates of $u$ over $K$ are $(j,0,\ldots, 0) u$, $j\in {\mathbb{Z}}/\ell{\mathbb{Z}}$ which by satisfy $$(j,0,\ldots,0)u = \sum_{k=0}^{n_q-1} \zeta_{\ell}^{jq^{k}} \sqrt[\ell]{F_{\mathbf{v}_k}}.$$ Thus $u$ generates the $\ell$-cyclic extension $L/K$ and has $$\prod_{j=0}^{\ell-1} \left( Y - \sum_{k=0}^{n_q-1} \zeta_{\ell}^{jq^{k}} \sqrt[\ell]{F_{\mathbf{v}_k}}\right)$$ as its minimal polynomial.
We summarize the construction in the following diagram.
\(K) [$K$]{}; (L) \[above of=K\] [$L$]{}; (E) \[above of=L\] [$E$]{}; (K’) \[right of=K\] [$K'$]{}; (L’) \[above of=K’\] [$L_{\mathbf{v}}$]{}; (E’) \[above of=L’\] [$E'$]{}; (K) to node (L); (L) to node (L’); (L) to node (E); (K) to node (K’); (K’) to node (L’); (L’) to node (E’); (E) to node (E’);
Now we are ready to introduce an explicit description of the correspondence of Corollary \[ltorcor\].
\[subclassifyprop\] The correspondence $$\begin{split}
\Omega_1 &\to \Omega_2\\
(F,b)&\mapsto L=L_{F,b}
\end{split}$$ induces an $(\ell-1)$-to-$1$ map between the sets $\Omega_1,\Omega_2$ defined in Corollary \[ltorcor\].
Let $\frak m$ be an effective divisor and let $F$ and $G$ be two $n_q$-divisible polynomials that are supported on $\frak m$. For each prime polynomial $P\in {\mathbb{F}}_q[X]$ of degree divisible by $n_q$, let $P=\frak P_1\cdots \frak P_{n_q}$ be a $\mu_{\ell}$-stable factorization. Let $F = F_1\cdots F_{n_q}$ and $G=G_1\cdots G_{n_q}$ be the corresponding $\mu_\ell$-stable factorizations of $F$ and $G$. So, $\mathfrak{P}_1$ appears only in $F_1$ if $P\mid F$ and only in $G_1$ if $P\mid G$.
Assume that $L_{F,b} = L_{G,c}$. By , , and there exists $0\leq r\leq \ell-1$ such that $F_{\mathbf{v}_0}$ and $G_{\mathbf{v}_0}^r$ are equal up to $\ell$-th-powers in $K'={\mathbb{F}}_{q^{n_q}}(X)$. Comparing leading coefficients gives that $b^{n_q} = c^{n_q r}$ and so $b = c^r$ up to $\ell$-th-powers.
Let $P$ be a prime dividing $F$ (resp. $G$) with multiplicity $\alpha$ (resp. $\beta$). Then $\frak P_1$ divides $F_1$ (resp. $G_1$) with the same multiplicity. So $\alpha\equiv r\beta \mod \ell$. This implies that $F=G^r$ up to $\ell$-th-powers. So we got that up to respective $\ell$-th powers, $(F, b) = (G^r , c^r )$. This implies the correspondence between the pairs and extensions is $(\ell-1)$-to-$1$ and by Corollary \[ltorcor\] we also obtain all the extensions.
\[rmk:modelforcovers\] Let us translate the above construction in terms of covers and equations. Let $C_{F,b}$ be the smooth projective model of $L_{F,b}$. By , $C_{F,b}$ is birationally equivalent to the affine plane curve $$\Bigg\{\prod_{j=0}^{\ell-1} \left( Y - \sum_{k=0}^{n_q-1} \zeta_{\ell}^{jq^k} \sqrt[\ell]{F_{\mathbf{v}_k}}\right) = 0\Bigg\}.$$ By and , $C_{F,b}$ is geometrically birationally equivalent to $$\label{rmk:modelforcovers}
\{ Y^\ell = F_{\mathbf{v}_0}\}.$$
If $n_q=1$, then $F_{\mathbf{v}_0}=bF$ and the characteristic polynomial becomes $$\prod_{j=0}^{\ell-1} \left(Y- \zeta_{\ell}^i \sqrt[\ell]{bF}\right) = Y^{\ell}-bF$$ which recovers the statement in Kummer theory.
Genus Formula {#genformsec}
=============
From now on we fix an $n_q$-divisible polynomial $F$ which is $\ell$-th-power free and $b\in {\mathbb{F}}_{q^{n_q}}^*$. We let $L=L_{F,b}$ be as in and $C=C_{F,b}$ the smooth projective model of $L$. We also assume that $$n_q\neq1.$$ Our goal is to give a formula for the genus $g=g(C)$ of $C$.
The genus of projective curves is preserved under separable base change, so the genus $g$ equals the genus of the smooth projective curve which is birationally equivalent to the affine plane curve . By the Riemann-Hurwitz formula the genus of the curve in may be computed in terms of a factorization of the $\ell$-th-power free polynomial $F_{\mathbf{v}_0}$ as explained below.
Since $F$ is $\ell$-th-power free, there exist monic, square-free polynomials $f_1,\dots,f_{\ell-1}$ which are pairwise coprime such that $$F = f_1f_2^2\cdots f_{\ell-1}^{\ell-1}.$$ Since $F$ is $n_q$-divisible, all the $f_i$ are $n_q$-divisible as well. Hence, by Corollary \[basefieldcor\], for each $i$, we have a $\mu_\ell$-stable factorization of $f_i$: $$\label{eq:f_i}
f_i = f_{i,1}\cdots f_{i,n_q}$$ We now define $$\begin{aligned}
\label{stablefact2}
F_j := f_{1,j}f_{2,j}^2\cdots f_{\ell-1,j}^{\ell-1},\end{aligned}$$ which gives us a $\mu_\ell$-stable factorization of $F$, namely $$F=F_1\cdots F_{n_q},$$ and $F_{i} = \phi^{i-1}(F_1)$, with $\phi$ the $q$-th-power Frobenuis. Taking $v_i$ to be the minimal non-negative integer with $v_i \equiv q^{1-i} \mod \ell$, we get by and that $$\begin{aligned}
\label{F_v}
F_{\mathbf{v}_0} = b' \prod_{j=1}^{n_q} F_j^{v_j} = b' \prod_{i=1}^{\ell} \prod_{j=1}^{n_q} f_{i,j}^{iv_j},\end{aligned}$$ for some $b'\in {\mathbb{F}}_{q^{n_q}}^*$. We apply the Riemann-Hurwitz formula which in this setting gives that $$\label{eq:genus1}
2g+2\ell-2 = (\ell-1) \sum_{i=1}^{\ell}\sum_{j=1}^{n_q} \deg(f_{i,j}) + \begin{cases} 0 & \mbox{if } \deg(F_{\mathbf{v}_0}) \equiv 0 \mod{\ell} \\ \ell-1 & \mbox{otherwise} \end{cases}.$$ (cf. [@Meisner17 Equation (1.2)] with our $\ell$ replacing $r$ in *loc.cit.* and using the fact that $\ell$ is coprime to all of the $v_i$).
Since $\deg(f_{i,j})=\deg(f_{i,1})$ for all $j$, $$\deg(F_{\mathbf{v}_0}) \equiv \sum_{i=1}^{\ell-1} \sum_{j=1}^{n_q} iq^{1-j}\deg(f_{i,j}) \equiv \sum_{i=1}^{\ell-1}i\deg(f_{i,1})\sum_{j=1}^{n_q} q^{1-j} \mod{\ell}.$$ Since we assume $n_q\not=1$ and since $q^{n_q}\equiv 1\mod\ell$, we have $$\sum_{j=1}^{n_q} q^{1-j} \equiv q\cdot \frac{1-q^{-n_q}}{1-q^{-1}} \equiv 0 \mod{\ell}.$$ Furthermore, by , $\sum_{j=1}^{n_q} \deg(f_{i,j}) = \deg(f_i)$. Plugging the above computations into yields the genus formula for $C$: $$\begin{aligned}
\label{genform}
2g+2\ell-2 = (\ell-1) \sum_{i=1}^{\ell} \deg(f_i).\end{aligned}$$
\[genformrem\] The Riemann-Hurwitz formula and imply that the only primes that ramify in ${\mathbb{F}}_q(C)$ are those that divide $F$. Consequently, no linear primes ramify in ${\mathbb{F}}_q(C)$ if $n_q\not=1$. Hence, all the points of ${\mathbb{P}}^1({\mathbb{F}}_q)$ are unramified in $C$.
We use to parametrize the moduli space $\mathcal{H}_{g,\ell}$ as defined in . Define the following set $$\label{eq:FqD}
{\mathcal{F}}_{n_q}(D) = \left\{(f_1,\dots,f_{\ell-1})\in {\mathbb{F}}_q[X]\ :\ \parbox{15em}{$f_1,\ldots, f_{\ell-1}$ are square free, pairwise coprime, $n_q$-divisible, and
$\deg(f_1\cdots f_{\ell-1})=D$}\right\}.$$ Each tuple $(f_1,\dots,f_{\ell-1})\in{\mathcal{F}}_{n_q}(D)$ corresponds to the $\ell$-th-power free $n_q$-divisible polynomial $F= f_1f_2^2\cdots f_{\ell-1}^{\ell-1}$. By , the genus of a corresponding curve $C=C_{F,b}$ satisfies: $$\label{eq:gtoD}
D = \frac{2g+2\ell-2}{\ell-1}.$$ Thus, by Corollary \[ltorcor\], the correspondence $(F,b)\mapsto C_{F,b}$ defines an $(\ell-1)$-to-$1$ correspondence between the sets $${\mathcal{F}}_{n_q}(D) \times {\mathbb{F}}_{q^{n_q}}^*/({\mathbb{F}}_{q^{n_q}}^*)^{\ell} \to {\mathcal{H}}_{g,\ell}.$$ Since $n_q$-divisible polynomials have degree divisible by $n_q$, we have that ${\mathcal{F}}_{n_q}(D) = \emptyset$ if and only if $D\not \equiv 0 \mod n_q$. Thus we immediately get
\[genformprop1\] For $n_q>1$, ${\mathcal{H}}_{g,\ell}=\emptyset$ if $2g+2\ell-2 \not\equiv 0 \mod{(\ell-1)n_q}$.
When $D\equiv 0\mod n_q$ or equivalently $2g+2\ell-2\equiv 0 \mod(\ell-1)n_q$, it suffices to count the number of points on curves when we parameterize by ${\mathcal{F}}_{n_q}(D)$ instead of by ${\mathcal{H}}_{g,\ell}$. More rigorously,
\[H\_gtoF\_n\] Let $g, D$ be related by and assume that $n_q>1$ and that $2g+2\ell-2\equiv 0 \mod(\ell-1)n_q$. Choose a random $(f_1,\ldots, f_{\ell-1})$ uniformly in ${\mathcal{F}}_{n_q}(D)$ and a random $b\in {\mathbb{F}}_{q^{n_q}}^*$ and put $F$ and $C_{F,b}$ as above. Choose a random $C$ uniformly in ${\mathcal{H}}_{g,\ell}$. Then, for every $N\geq 0$ $${\textnormal{Prob}}(\#C_{F,b}({\mathbb{F}}_q) = N) = {\textnormal{Prob}}(\#C({\mathbb{F}}_q) = N).$$
Number of Points Formula {#numpts}
========================
The goal of this section is to give an analytic formula for the number of rational points on cyclic covers. Let $\pi \colon C\to {\mathbb{P}}^1_{{\mathbb{F}}_q}$ be an $\ell$-cyclic cover of smooth projective ${\mathbb{F}}_q$-curves and let $$C' = C\times_{{\mathbb{F}}_q} {\mathbb{F}}_{q^{n_q}}$$ be the scalar extension of $C$ to ${\mathbb{F}}_{q^{n_q}}$. We start by a simple general observation connecting the number of ${\mathbb{F}}_q$-rational points of $C$ and ${\mathbb{F}}_{q^{n_q}}$-rational points of $C'$ lying above ${\mathbb{P}}^1({\mathbb{F}}_q)$.
It is convenient to introduce the following piece of notation: For $x\in {\mathbb{P}}^1({\mathbb{F}}_q)$, let $$\label{eq:Nx}
N_x = \# \{ y\in C({\mathbb{F}}_q) : \pi(y) = x\} \quad \mbox{and} \quad N_x' = \# \{ y\in C'({\mathbb{F}}_{q^{n_q}}) : \pi(y) = x\}$$ be the number of ${\mathbb{F}}_q$-rational and ${\mathbb{F}}_{q^{n_q}}$-rational points on $C$ and $C'$ lying above $x$, respectively.
\[numptsprop\] Let $x\in {\mathbb{P}}^1({\mathbb{F}}_q)$. Then $N_x=N_x'$.
Let $x\in \mathbb{P}^1({\mathbb{F}}_q)$, let $O=O_{{\mathbb{P}}^1_{{\mathbb{F}}_q},x}$ be the local ring at $x$ (which is the localization of ${\mathbb{F}}_q[X]$ at $X-x$ if $x$ is finite or of ${\mathbb{F}}_q[X^{-1}]$ at $X^{-1}$ if $x=\infty$) and $\frak p$ the corresponding maximal ideal (which is $(X-x)$ if $x$ is finite or $(X^{-1})$ if $x=\infty$). Since $L={\mathbb{F}}_q(C)$ is Galois with cyclic group of prime order $\ell$, there are $3$ possible factorizations of $\frak p = \frak{P}_1^{e}\cdots \frak{P}_g^e$; namely,
1. $g=\ell$, $e=f=1$, (where $f=\deg \frak P_i$) in which case $N_x=\ell$
2. $e=\ell$, $f=g=1$, in which case $N_x=1$,
3. $f=\ell$, $e=g=1$, in which case $N_x=0$.
Now we base change to ${\mathbb{F}}_{q^{n_q}}$, and denote it as before by adding a tag. Since $n_q= [{\mathbb{F}}_{q^{n_q}}:{\mathbb{F}}_q]$ is co-prime to $\ell$ (as we add $\ell$-th root of unity), we get that $f=f'$. Since the extension ${\mathbb{F}}_{q^{n_q}}/{\mathbb{F}}_q$ is unramified, we get that $e=e'$. Since $\ell=[L:K]=[L':K']$, we conclude that $g=g'$. As $N_x'$ is determined by $e',f',g'$ in the same manner as $N_x$ is determined by $e,f,g$, we conclude that $N_x= N'_x$.
Next we compute the number of rational points in terms of the parametrization of Section \[genformsec\]. Recall that $b\in {\mathbb{F}}_{q^{n_q}}^*$, $$\label{eq:DefF}
F=f_1f_2^2\cdots f_{\ell-1}^{\ell-1}$$ for $(f_1,\ldots, f_{\ell-1})\in {\mathcal{F}}_{n_q}(D)$ and that $C=C_{F,b}$ in the notation of . Also recall that $C'$ is birationally equivalent to the affine plane curve $Y^\ell = F_{\mathbf{v}_0}$, see , where $F_{\mathbf{v}_0}$ is given in .
Let $\chi_\ell$ be a primitive multiplicative character of ${\mathbb{F}}_{q^{n_q}}$ of order $\ell$ (there exists such, as $n_q$ by definition is the minimal positive integer with $q^{n_q}\equiv 1\mod \ell$). For $x\in {\mathbb{F}}_q$ and for a polynomial $G$, we define $$\label{eq:chi_x}
\chi_x(G) = \chi_\ell(G(x)),$$ which is a multiplicative Dirichlet character modulo $X-x$ of order $\ell$. We extend this definition to $x=\infty\in \mathbb{P}^1({\mathbb{F}}_q)$ by setting $$\chi_{\infty}(G) = \chi_{\ell}(g_n),$$ with $g_n$ being the leading coefficient of $G$.
\[numptscor\] We have $$\label{analytic_formula}
\#C({\mathbb{F}}_q) = \sum_{w=0}^{\ell-1}\chi_\ell^w(b) + \sum_{x\in{\mathbb{F}}_q} \sum_{w=0}^{\ell-1} \chi_{x}^w\left(F_{\mathbf{v}_0}\right).$$
In [@Meisner17] it is established that for $x\in \mathbb{P}^1({\mathbb{F}}_q)$ $$N_x' = \sum_{w=0}^{\ell-1} \chi_x^{w}(F_{\mathbf{v}_0})$$ (To see it follows from *loc. cit.*, plug $\ell$, $F_{\mathbf{v}_0}$ for $r$, $F$ in the notation of *loc. cit.* For finite $x$, we note that $F_{\mathbf{v}_0}(x) \neq 0$ since $f_i(x)\neq 0$ as they are $n_q$-divisible polynomial. Thus the formula is given in the first paragraph of the proof of Lemma 2.1 in *loc. cit.* For $x=\infty$, see page 536.)
Then, as $N_x=N_x'$ (Proposition \[numptsprop\]) we get that $$\#C({\mathbb{F}}_q) = \sum_{x\in \mathbb{P}^1(\mathbb{F}_q)} N_x = \sum_{x\in \mathbb{P}^1(\mathbb{F}_q)} N_x' = \sum_{x\in\mathbb{P}^1({\mathbb{F}}_q)} \sum_{w=0}^{\ell-1} \chi_{x}^w\left(F_{\mathbf{v}_0}\right).$$ The leading coefficient of $F_{\mathbf{v_0}}$ is $b^{n_q}$ and so $\sum_{w=0}^{\ell-1}\chi_{\ell}^w(b) = \sum_{w=0}^{\ell-1}\chi_{\ell}^w(b^{n_q})$.
Set Count {#SetCount}
=========
In light of the analytic formula , the study of the distribution of the number of points on $\ell$-cyclic covers parameterized by ${\mathcal{F}}_{n_q}(D)\times {\mathbb{F}}_{q^{n_q}}^*$ may be reduced to the computation of the size of sets of the form $$\label{eq:FknqD}
{\mathcal{F}}^k_{n_q}(D)={\mathcal{F}}^k_{n_q,\epsilon}(D) = \{(f_1,\dots,f_{\ell-1})\in{\mathcal{F}}_{n_q}(D) : \chi_{\ell}(F_{\mathbf{v}_0}(x_i)) = \epsilon_i, i=1,\dots,k\},$$ where $x_1,\ldots, x_k \in{\mathbb{F}}_q$ are pairwise distinct elements, $\epsilon=(\epsilon_1,\ldots, \epsilon_{k})\in\mu_{\ell}^k$ is fixed, and, as usual, $F$ and $F_{\mathbf{v}_0}$, $\mathbf{v}_0 = (1,q^{-1}, \ldots, q^{1-n_q})$ are as defined in and , respectively. The computation is done by analyzing an appropriate generating function ${\mathcal{G}}_k(u)$.
While the definition of ${\mathcal{F}}^k_{n_q}(D)$ depends on the choice of $b$ and of $(\epsilon_1,\dots,\epsilon_k)$, its asymptotic size — which is computed below in — is independent of this choice. Hence we omit the $\epsilon_i$’s from the notation.
Generating Series
-----------------
Let $\zeta_{n_q} \in \mathbb{C}$ be a primitive $n_q$-th root of unity and define the following auxiliary functions: $$\begin{split}
\mathcal{I}_{\infty}(F) &:= \mu^2(F)\prod_{P|F}\frac{1}{n_q}\left(\sum_{i=0}^{n_q-1} \zeta_{n_q}^{i\deg(P)} \right)
\\
\mathcal{I}_{x_i}(F) &:= \frac{1}{\ell}\left(\sum_{w=0}^{\ell-1} \epsilon_{i}^{-w}\chi_{x_i}^w(F)\right),
\end{split}$$ with the convention that $\chi_{x}^0$ is the trivial character modulo $X-x$. Using these functions we define a function in $\ell-1$ variables (which are always assumed to be monic polynomials) $${\mathcal{I}}(f_1,\ldots, f_{\ell-1}) = {\mathcal{I}}_{\infty}(f_1\cdots f_{\ell-1}) \prod_{i=1}^k{\mathcal{I}}_{x_i}(F_{\mathbf{v}_0}) ,$$ with $F$ and $F_{\mathbf{v}_0}$ defined as in and .
For a tuple of monic polynomials $(f_1,\ldots,f_{\ell - 1})$ we have $${\mathcal{I}}(f_1,\ldots, f_{\ell-1}) = \begin{cases}
1, & \mbox{if }(f_1,\ldots, f_{\ell-1})\in {\mathcal{F}}_{n_q}^k(\deg(f_1\cdots f_{\ell-1}))\\
0, & \mbox{otherwise.}
\end{cases}$$ In particular, $$\begin{aligned}
\label{generatfun}
{\mathcal{G}}_k(u) := \sum_{f_1,\dots,f_{\ell-1}}{\mathcal{I}}(f_1,\ldots, f_{\ell-1})u^{\deg(f_1\cdots f_{\ell-1})} = \sum_{D=0}^{\infty} |{\mathcal{F}}^k_{n_q}(D)|u^D,\end{aligned}$$ where the first sum is over monic polynomials.
By the orthogonality relations, for a tuple of monic polynomials $(f_1,\ldots, f_{\ell-1})$, we have that $\mathcal{I}_{\infty}(f_1\cdots f_{\ell-1}) = 1$ if $(f_1,\dots,f_{\ell-1})\in {\mathcal{F}}_{n_q}(D)$, and $=0$ otherwise. Similarly, if $(f_1,\ldots, f_{\ell-1})\in {\mathcal{F}}_{n_q}(D)$, then ${\mathcal{I}}_{x_i} (F_{\mathbf{v}_0}) =1$ if $\chi_{x_i}(F_{\mathbf{v}_0})=\epsilon_i$ and $=0$ otherwise. This completes the proof of the lemma.
Expanding the product $\prod_{i} {\mathcal{I}}_{x_i}(F)$ gives $$\prod_{i=1}^k \mathcal{I}_{x_i}(F) = \frac{1}{\ell^k} \prod_{i=1}^k \sum_{w=0}^{\ell-1} \epsilon_{i}^{-w}\chi_{x_i}^w(F)
= \frac{1}{\ell^k}\sum_{\mathbf{w}} \left(\prod_{i=1}^k \epsilon_i^{-w_i}\right) \chi_{\mathbf{w}}(F),$$ where the sum is over all vectors $\mathbf{w}=(w_1,\dots,w_k)$ such that $0\leq w_i \leq \ell-1$ and $$\label{eq:chiw}
\chi_{\mathbf{w}}(F) = \prod_{i=1}^k \chi_{x_i}^{w_i}(F).$$ This is a Dirichlet character of modulo $\prod (X-x_i)$ and is non-trivial if $\mathbf{w} \neq \vec{0}$. Thus we get a decomposition of ${\mathcal{G}}_k(u)$, $$\begin{aligned}
\label{eq:factorization_generating_function}
{\mathcal{G}}_k(u) = \frac{1}{\ell^k}\sum_{\mathbf{w}} \left(\prod_{i=1}^k \epsilon_i^{-w_i}\right) {\mathcal{G}}_{\mathbf{w}}(u),\end{aligned}$$ where $$\label{Gw}
{\mathcal{G}}_{\mathbf{w}}(u) = \sum_{f_1,\dots, f_{\ell-1}} {\mathcal{I}}_{\infty}(f_1\cdots f_{\ell-1}) \chi_{\mathbf{w}}(F_{\mathbf{v}_0}) u^{\deg(f_1...f_{\ell-1})}.$$
Euler Product
-------------
Recall that a nonzero multivariate function $\psi$ in is called *firmly multiplicative* if $$\psi(a_1b_1,\ldots, a_rb_r)=\psi(a_1,\ldots, a_r)\psi(b_1,\ldots, b_r)$$ whenever $\gcd(a_i,b_i)=1$, for all $i=1,\ldots, r$. These functions are determined by their values on $(1,\ldots, 1, p^\alpha,1\ldots, 1)$ and behave as multiplicative functions in one variable. We assume that the reader is familiar with the standard properties of firmly multiplicative functions, or at least that the reader may complete the details of how those are derived from the one variable case; if this is not the case, one may consult the survey paper [@Toth].
As usual, let $K={\mathbb{F}}_q(X) = {\mathbb{F}}_q({\mathbb{P}}^1_{{\mathbb{F}}_q})$ and $K'={\mathbb{F}}_{q^{n_q}}(X)$. For each prime $P$ of $K$ we fix a prime $\mathfrak{P}$ of $K'$ lying above $P$. Then we have the Euler decomposition $$\begin{aligned}
\label{eulprod}
{\mathcal{G}}_{\mathbf{w}}(u) = \prod_{\substack{P \\ n_q|\deg(P)}} \left(1+\sum_{j=1}^{\ell-1}\chi^{j}_{\mathbf{w}}(\mathfrak{P})u^{n_q \deg(\frak P)}\right),\end{aligned}$$ with ${\mathcal{G}}_{\mathbf{w}}$ and $\chi_{\mathbf{w}}$ as defined in and .
\[rem:sumchi\] If $\frak P$ and $\frak P'$ are two primes over $P$, then $\phi^k(\frak P) = \frak P'$ for some $k$, and so by , the inner sum in equals to the same sum with $\frak P'$ replacing $\frak P$.
The function $$\Psi (f_1,\ldots, f_{\ell-1}) = {\mathcal{I}}_{\infty}(f_1\cdots f_{\ell-1}) \chi_{\mathbf{w}}(F_{\mathbf{v}_0})$$ is firmly multiplicative, hence to compute the Euler decomposition of ${\mathcal{G}}_{\mathbf{w}}(u)$, it suffices to evaluate $\Psi$ on $$e_{i}(P^{\nu})=(1,\ldots, 1,P^\nu,1\ldots, 1),$$ with $P^\nu$ appearing in the $i$-th place and $P$ is a prime polynomial in $K$.
Since ${\mathcal{I}}_{\infty}(P^\nu)=1$ only when $\nu=1$ and $n_q\mid \deg(P)$ (and $=0$ otherwise), we may restrict to this case.
By Corollary \[basefieldcor\], $P$ has a $\mu_{\ell}$-stable factorization $$P = \mathfrak{P}_1 \cdots \mathfrak{P}_{n_q},$$ in which we may assume without loss of generality that $\mathfrak{P}_1=\mathfrak{P}$. We note that as $P$ is prime in $K$, the $\mathfrak{P}_i$ are prime in $K'$. Then $F = P^{i}$, $F_{j} = \frak P_j^i$, and $$F_{\mathbf{v}_0}=\prod_{j=1}^{n_q} \frak P_j^{iv_j},$$ with $v_{j}\equiv q^{1-j} \mod \ell$. By , $\frak P_j^{iv_j}(x_r)$ equals to $\frak P(x_i)^{i}$ up to $\ell$-th-powers. As $\chi_{\mathbf{w}}$ is defined modulo $\prod (X-x_i)$ and is trivial on $\ell$-th-powers, we conclude that $$\Psi(e_i(P)) = \chi_{\mathbf{w}} (F_{\mathbf{v}_0}) = \chi_{\mathbf{w}}^{in_q}(\frak P).$$ Thus we get the following Euler decomposition: $${\mathcal{G}}_{\mathbf{w}}(u) =
\prod_{\substack{P \\ n_q|\deg(P)}} \left(1+\sum_{i=1}^{\ell-1}\chi^{in_q}_{\mathbf{w}}(\mathfrak{P})u^{\deg(P)}\right).$$ By Lemma \[basefieldlem\], $\deg (P) = n_q \deg (\mathfrak P)$. Thus, since $(n_q,\ell)=1$, we have that $\sum_{i=1}^{\ell-1}\chi^{in_q}_{\mathbf{w}}=\sum_{j=1}^{\ell-1}\chi^{j}_{\mathbf{w}}$, and the proof is done.
Analytic Continuation {#analcont}
---------------------
For a non-trivial character $\chi$ of ${\mathbb{F}}_{q^{n_q}}[X] \subseteq K'$ we define $$\label{eq:Lfn}
L_{K'}(u,\chi) = \prod_{\mathfrak{P}} \left( 1-\chi(\mathfrak{P})u^{\deg(\mathfrak{P})} \right)^{-1},$$ where the product is over all prime polynomials in $K'$. Then, $L_{K'}(u,\chi)$ is a polynomial (due to the orthogonality relations) and its zeros lie on the circle $|u|=q^{-n_q/2}$ (due to the Riemann Hypothesis for curves). We shall use repeatedly that a product $$\label{eq:radiusconvegence}
\prod_{P}(1+O(u^{c\deg P}))$$ over prime polynomials absolutely converges in the disc $|u|<q^{-1/c}$.
\[analcontprop\] For any $\mathbf{w}$, there exists a non-vanishing function $H_{\mathbf{w}}(u)$ which is analytic in the open disc $|u|<q^{-1/2}$ such that
1. if $\mathbf{w}\neq \vec{0}$, then $${\mathcal{G}}_{\mathbf{w}}(u) = \prod_{j=1}^{\ell-1}\left(\frac{L_{K'}(u^{n_q},\chi^j_{\mathbf{w}})}{L_{K'}(u^{2n_q},\chi^{2j}_{\mathbf{w}})}\right)^{\frac{1}{n_q}} H_{\mathbf{w}}(u).$$
2. if $\mathbf{w} = \vec{0}$, then $${\mathcal{G}}_{\vec{0}}(u) = \prod_{j=0}^{n_q-1}\left(1-\zeta_{n_q}^jqu \right)^{-\frac{\ell-1}{n_q}} H_{\vec{0}}(u),$$ and $H_{\vec{0}}(\zeta_{n_q} u)= H_{\vec{0}}(u)$.
In particular,
1. ${\mathcal{G}}_{\mathbf{w}}(u)$ has a meromorphic continuation to $|u|<q^{-1/2}$ which is analytic if $\mathbf{w}\neq \vec0$ and has poles of order $\frac{\ell-1}{n_q}$ at $u=(\zeta_{n_q}^jq)^{-1}$, $j=0,\ldots, n_q-1$ if $\mathbf{w}=\vec{0}$.
By the Riemann Hypothesis, the poles and zeros of $\frac{L_{K'}(u^{n_q},\chi^j_{\mathbf{w}})}{L_{K'}(u^{2n_q},\chi^{2j}_{\mathbf{w}})}$ lies on the circles $|u|=q^{-1/4}$ and $|u|=q^{-1/2}$, respectively, hence it has an analytic $n_q$-th root in the open disc $|u|< q^{-1/2}$ and so (3) follows from (1) and (2).
Now we prove (1). Since each irreducible $P$ in ${\mathbb{F}}_q[X]$ with $n_q \mid \deg (P)$ has $n_q$ prime divisors in ${\mathbb{F}}_{q^{n_q}}[X]$ (Lemma \[basefieldlem\]) we get by applied to all the primes dividing $P$ that $${\mathcal{G}}_{\mathbf{w}}(u)^{n_q} = \prod_{\substack{P \\ n_q\mid \deg(P)}} \prod_{\frak P\mid P} \left(1+\sum_{j=1}^{\ell-1}\chi^{j}_{\mathbf{w}}(\mathfrak{P})u^{n_q\deg(\mathfrak{P})}\right).$$ We take $n_q$-th root and break this product into two parts: Denote $$F_1(u) = \prod_{{\substack{P \\ n_q\nmid\deg(P)}}}\prod_{\mathfrak{P}|P} \left(1+\sum_{j=1}^{\ell-1}\chi^{j}_{\mathbf{w}}(\mathfrak{P})u^{n_q\deg(\mathfrak{P})}\right)^{-\frac{1}{n_q}}$$ and $$F_2(u) = \prod_{\mathfrak{P}} \left(1+\sum_{j=1}^{\ell-1}\chi^{j}_{\mathbf{w}}(\mathfrak{P})u^{n_q\deg(\mathfrak{P})}\right)^{\frac{1}{n_q}}$$ so $$\begin{aligned}
\label{factorization}
{\mathcal{G}}_{\mathbf{w}}(u) = F_1(u)F_2(u).\end{aligned}$$ We study each of the factors starting with $F_1$. By Lemma \[basefieldlem\] and Remark \[rem:sumchi\] for $|u|<q^{-1/2}$, $$\begin{aligned}
F_1(u) &= \prod_{\substack{P \\ n_q\nmid\deg(P)}} \left(1+\sum_{j=1}^{\ell-1}\chi^{j}_{\mathbf{w}}(\mathfrak{P})u^{\frac{n_q}{(\deg(P),n_q)}\deg(P)}\right)^{-\frac{(\deg(P),n_q)}{n_q}}.\end{aligned}$$ Since for $n_q\nmid \deg P$ we have $$\left(1 + \chi^{j}_{\mathbf{w}}(\mathfrak{P})u^{\frac{n_q}{(\deg(P),n_q)}\deg(P)}\right)^{-\frac{(\deg(P),n_q}{n_q}}=1+O(q^{2\deg P}),$$ by , we get that $F_1(u)$ is analytic in this open disc $|u|<q^{-1/2}$.
Next we study $F_2$. Expanding the product in its definition we get $$\label{eqF2}
F_2(u) = \prod_{j=1}^{\ell-1} \prod_{\mathfrak{P}} \left(1 + \chi^{j}_{\mathbf{w}}(\mathfrak{P})u^{n_q\deg(\mathfrak{P})}\right)^{\frac{1}{n_q}} \prod_{\mathfrak{P}} \left(1 + O(u^{2 n_q \deg (\mathfrak P)}) \right)$$ By , the right product absolutely converges in $|u|<q^{-1/2}$ and by , $$\prod_{\mathfrak{P}} \left(1 + \chi^{j}_{\mathbf{w}}(\mathfrak{P})u^{n_q\deg(\mathfrak{P})}\right) = \frac{L_{K'}(u^{n_q},\chi^j_{\mathbf{w}})}{L_{K'}(u^{2n_q},\chi^{2j}_{\mathbf{w}})},$$ which together with the analyticity of $F_1$, with , and with finishes the proof of (1).
For (2), we use , to get that $${\mathcal{G}}_{\vec{0}}(u) = \prod_{\substack{P \\ n_q|\deg(P)}} \left(1+(\ell-1)u^{\deg(P)}\right).$$ By the orthogonality relations $1_{\{n\equiv 0\mod n_q\}} = \frac{1}{n_q} \sum_{j=0}^{n_q-1} \zeta_{n_q}^{jn}$ we have $${\mathcal{G}}_{\vec{0}}(u) = \prod_{P} \Big(1 + \frac{\ell-1}{n_q} \sum_{j=0}^{n_q-1}\zeta_{n_q}^{j\deg(P)} u^{\deg(P)} \Big).$$ Then we define $H_{\vec{0}} (u)$ by the equation $$\begin{aligned}
{\mathcal{G}}_{\vec0}(u) &= \prod_{j=0}^{n_q-1}\left( \prod_{P} \left(1 - (\zeta_{n_q}^ju)^{\deg(P)}\right)\right)^{-\frac{\ell-1}{n_q}} H_{\vec{0}}(u)\\
& = \prod_{j=0}^{n_q-1}\left(\frac{1}{1-q\zeta_{n_q}^ju} \right)^{\frac{\ell-1}{n_q}} H_{\vec{0}}(u).\end{aligned}$$ As $H_{\vec{0}}(u) = \prod_{P} (1+O(u^{2\deg(P)}))$ holds true, implies that $H_{\vec{0}}(u)$ is analytic in $|u|<q^{-1/2}$. Further, $H_{\vec{0}}(\zeta_{n_q}u) = H_{\vec{0}}(u)$ since both ${\mathcal{G}}_{\mathbf{w}}$ and $\prod_{j=0}^{n_q-1}\left(\frac{1}{1-q\zeta_{n_q}^ju} \right)^{\frac{\ell-1}{n_q}}$ satisfy this relation.
Computing $|{\mathcal{F}}^k_{n_q}(D)|$
--------------------------------------
We determine the size of the set ${\mathcal{F}}^k_{n_q}(D)$ using the residue theorem applied to the contour $C_{\epsilon} = \{u : |u|=q^{-1/2-\epsilon}\}$ and to the function ${\mathcal{G}}_k(u)/u^{D+1}$. By Proposition \[analcontprop\], by , and by we have $$\label{eq:contourint}
\begin{split}
\frac{1}{2\pi i} \oint_{C_{\epsilon}}\frac{{\mathcal{G}}_k(u)}{u^{D+1}} du & = {\textnormal{Res}}_{u=0} \left(\frac{{\mathcal{G}}_k(u)}{u^{D+1}}\right) + \sum_{j=0}^{n_q-1} {\textnormal{Res}}_{u=(\zeta_{n_q}^jq)^{-1}} \left(\frac{{\mathcal{G}}_k(u)}{u^{D+1}}\right)\\
& = |{\mathcal{F}}^k_{n_q}(D)| +\frac{1}{\ell^k} \sum_{j=0}^{n_q-1}{\textnormal{Res}}_{u=(\zeta_{n_q}^jq)^{-1}} \left(\frac{{\mathcal{G}}_{\vec{0}}(u)}{u^{D+1}}\right).
\end{split}$$
We compute each of the terms separately. Put $N=\frac{\ell-1}{n_q}-1$ and $u_j = (\zeta_{n_q}^jq)^{-1}$. Since ${\mathcal{G}}_{\vec0}$ has a pole at $u=u_j$ of order $N+1$ and using Proposition \[analcontprop\], we have
$$\begin{aligned}
\label{zerosatnonzeropoints}
\begin{split}
N!\cdot{\textnormal{Res}}_{u=u_j} \left(\frac{{\mathcal{G}}_{\vec{0}}(u)}{u^{D+1}}\right)
& = \frac{d^N}{du^N} \left[ (u-u_j)^{N+1}\frac{{\mathcal{G}}_{\vec{0}}(u)}{u^{D+1}} \right] \bigg|_{u=u_j}\\
& = \frac{d^{N}}{du^{N}}\Bigg[ \frac{u_j^{N+1}}{u^{D+1}} \prod_{\substack{i=0 \\ i\not=j}}^{n_q-1}u_i^{N+1}\left(u_i-u\right)^{-(N+1)} H_{\vec{0}}(u)\Bigg]\Bigg|_{u=u_j}\\
&= \sum_{m=0}^{N} \binom{N}{m} \frac{d^m}{du^m} \frac{u_j^{N+1}}{u^{D+1}}\Bigg|_{u=u_j} \frac{d^{N-m}}{du^{N-m}} \Bigg[\prod_{\substack{i=0 \\ i\not=j}}^{n_q-1}\left(\frac{1}{1-u/u_i}\right)^{N+1} H_{\vec{0}}(u)\Bigg]\Bigg|_{u=u_j} \\
= & -\frac{1}{n_q}P(D) u_j^{-D}= -\frac{1}{n_q} P(D)q^D\zeta_{n_q}^{jD},
\end{split}\end{aligned}$$
where $P(D)$ is a polynomial of degree $N$ which is independent of $j$.
Since ${\mathcal{G}}_k(u)$ is continuous on $C_{\epsilon}$, it is $O(1)$ there, and we have $$\left| \frac{1}{2\pi i} \oint_{C_{\epsilon}}\frac{{\mathcal{G}}_k(u)}{u^{D+1}} du \right| = O\left(q^{(1/2+\epsilon)D}\right).$$ Plug this bound and in to get $$\begin{aligned}
\label{Fsize1}
|{\mathcal{F}}^k_{n_q}(D)| = \begin{cases} \frac{1}{\ell^k}P(D)q^D + O\left(q^{(1/2+\epsilon)D}\right), & D\equiv 0 \mod{n_q} \\ 0, & \mbox{otherwise}.\end{cases}\end{aligned}$$
We emphasize that the main term is independent on the actual choice of the values $\epsilon_1,\ldots, \epsilon_k$ of the characters. The error term may depend on this choice.
Proof of Theorem \[thm:main\] {#sec:proof}
=============================
Fix $D\equiv 0 \mod n_q$. Let $N\geq 0$ and $\epsilon_0,\epsilon_1,\ldots, \epsilon_{q}\in \mu_{\ell}$ such that $$\sum_{i=0}^{q} \sum_{w=0}^{\ell-1} \epsilon_i^w = N.$$
Conditioning on $b$, gives that the probability $\pi$ that $\chi_{x_i}(F_{\mathbf{v}_0}) = \epsilon_i$ for $i=1,\ldots, q$ and $\chi_{\ell}(b)=\epsilon_0$ is $$\pi = \frac{1}{\ell^{q+1}} + O(q^{D(-1/2+\epsilon)}).$$ Thus, implies that $${\textnormal{Prob}}(\#C_{F,b}({\mathbb{F}}_q) = N) = \sum_{\substack{\epsilon_0,\ldots, \epsilon_q\in \mu_{\ell}\\ \sum_i\sum_w \epsilon_i^w = N}} \frac{1}{\ell^{q+1}} + O(q^{D(-1/2+\epsilon)}).$$ Putting $\epsilon_i' = \sum_{w=0}^{\ell-1} \epsilon_i^w$, we have that $\epsilon'_i\in \{0,\ell\}$. Moreover, $\epsilon'_i=0$ if and only if $\epsilon_i\neq 1$. This means that if $\epsilon'_i=0$ there are $\ell-1$ different choices for $\epsilon_i$, whereas if $\epsilon'_{i}=\ell$, then there is one choice: $\epsilon_i=1$. We plug this in the previous equation and get $$\begin{split}
{\textnormal{Prob}}(\#C_{F,b}({\mathbb{F}}_q) = N)
&= \sum_{\substack{\epsilon_0',\ldots, \epsilon_q'\in \{0,\ell\}\\ \sum_i\epsilon_i' = N}} \frac{(\ell-1)^{\#\{i:\epsilon'_i=0\}}}{\ell^q} + O(q^{D(-1/2+\epsilon)})\\
&=\sum_{\substack{\epsilon_0',\ldots, \epsilon_q'\in \{0,\ell\}\\ \sum_i\epsilon_i' = N}} \left(\frac{\ell-1}{\ell}\right)^{\#\{i:\epsilon'_i=0\}}\left(\frac{1}{\ell}\right)^{\#\{i:\epsilon'_i=\ell\}}
+ O(q^{D(-1/2+\epsilon)})\\
&= {\textnormal{Prob}}(\sum_{i=0}^q X_i = N),
\end{split}$$ with $X_i$ as in the formulation of the theorem. Since by Proposition \[H\_gtoF\_n\] $${\textnormal{Prob}}(\#C_{F,b}({\mathbb{F}}_q) = N) ={\textnormal{Prob}}(\#C({\mathbb{F}}_q)=N)$$ with $C$ random curve in $\mathcal{H}_{g,\ell}$, this finishes the proof.
|
---
abstract: 'We study Andreev reflection in a normal conductor-molecule-superconductor junction using a first principles approach. In particular, we focus on a family of molecules consisting of a molecular backbone and a weakly coupled side group. We show that the presence of the side group can lead to a Fano resonance in the Andreev reflection. We use a simple theoretical model to explain the results of the numerical calculations and to make predictions about the possible sub-gap resonance structures in the Andreev reflection coefficient.'
author:
- 'A. Kormányos, I. Grace, and C. J. Lambert'
title: Andreev reflection through Fano resonances in molecular wires
---
Fano resonances[@ref:fano] are a universal interference phenomenon which can affect coherent electrical transport through nanostructures in many different systems. Examples of Fano lineshape in mesoscopic systems include scanning tunnelling microscope measurements on a single magnetic atom absorbed on a gold surface[@ref:madhavan; @ref:schneider], single-electron transistors fabricated into a gated two-dimensional electron gas[@ref:gores], quantum dots embedded into an Aharonov-Bohm ring[@ref:kobayashi-2002; @ref:kobayashi-2003], multiwall carbon nanotubes[@ref:kim; @ref:yi; @ref:chandrasekhar] and recently single-wall carbon nanotubes[@ref:babic] and double-wall nanotubes[@ref:iain]. Fano resonances (FRs) also appear in the the conductance of quasi one-dimensional quantum wires with donor impurities[@ref:tekman] and in the case of quantum wires with a side coupled quantum dot[@ref:franco]. In molecular electronics, due to the realistic treatment of the metal electrodes, FRs have been found in the transmission of dithiol benzene[@ref:grigoriev]. More generally, theoretical calculations predict that Fano-lineshape should appear in the transmission through molecular wires with attached side groups[@ref:thodoris] or as a consequence of quantum interference between surface states of the measuring electrodes and the molecular orbitals[@ref:shi].
If one of the measuring probes is superconducting, the conductance for energies $E$ smaller than the superconducting pair potential $\Delta$ depends on the Andreev reflection probability $R_A(E)$. The Andreev reflection in various mesoscopic systems has been studied for a long time (see e.g Refs. and references therein) but the interest has recently renewed when Andreev reflection through carbon nanotubes was measured experimentally[@ref:morpurgo; @ref:graeber]. These experiments have sparked numerous theoretical studies both in the absence of the electron-electron interaction[@ref:wei; @ref:pan] and in the presence of the interaction[@ref:vishveshwara; @ref:titov; @ref:cuevas; @ref:schwab; @ref:clerk; @ref:sun; @ref:avishai; @ref:splett; @ref:doma; @ref:domanski]. In many of these studies it was assumed that it was sufficient to consider resonant transport through a single energy level and as a consequence, the Andreev reflection as a function of energy exhibited Breit-Wigner type resonances. A notable exception is Ref. where transport through an Aharonov-Bohm ring with an interacting quantum dot situated in one of its arms was considered and a Fano-type asymmetric resonance was found in the conductance. Very recently, Tanaka *et al.*[@ref:tanaka] studied Andreev transport through side-coupled interacting quantum dots focusing on the interplay of Andreev scattering and Kondo effect.
It was demonstrated in Ref. that Fano resonances are a generic feature of molecular wires with attached side groups. It was also shown that for a certain type of molecular wires a FR can appear in the normal conductance $G_N(E)$ very close to the Fermi energy $E_F$. In a normal metal-molecule-superconductor (N-Mol-S) junction therefore these FRs would also affect the sub-gap transport. The aim of this paper is to study Andreev reflection through molecular wires when the normal conductance exhibits FRs close to the Fermi energy. Performing *ab initio* simulations of molecular wires in N-Mol-S junctions we show how FRs influence the sub-gap transport. We elucidate the results of the numerical calculations using a simple analytic model. We also predict that for finite energies the differential conductance can reach the unitary limit if there is a strong asymmetry in the coupling to the leads.
![Possible experimental setup. The molecule is contacted by gold electrodes, one of which is superconducting due to the proximity effect. \[fig:exp-setup\]](exper-setup-2.eps)
A possible experimental setup to measure Andreev reflection in N-Mol-S junctions is shown in Fig. \[fig:exp-setup\]. The molecule is contacted with thin gold electrodes on both sides. On top of one of the electrodes a second layer of e.g. aluminium or niobium is deposited, which at low enough temperature becomes superconducting. Due to the proximity effect this top layer induces superconductivity in the gold electrode beneath (in our calculations we assume that the induced superconductivity is $s$ type). We note that this setup was successfully used in Ref. to study Andreev-reflecion in normal conductor - carbon nanotube - superconductor (N-Cn-S) junctions.
To study FRs in a N-Mol-S system, we choose the smallest molecule of a recently synthesized family of molecular wires[@ref:wang1; @ref:wang2; @ref:wang3]. Since these molecules have terminal thiol groups they can easily bind to gold surfaces making them ideal for experiments on single-molecule transport properties. The central part of the molecule consists of a single fluorenone unit, which could be chemically modified, e.g. by replacing the oxygen with bipyridine rings, as shown schematically in Fig. \[fig:exp-setup\]. The differential conductance of the system was calculated using a combination of the DFT code SIESTA[@ref:siesta] and a Green’s function scattering approach explained in Refs. . Initially the isolated molecule is relaxed to find the optimum geometry, then the molecule is extended to include surface layers of the gold leads. In this way, charge transfer at the gold-molecule interface is included self-consistently. The number $N_g$ of gold layers is increased until computed transport properties between the (normal conducting) gold leads no longer changed with increasing $N_g$. Typically, this extended molecule contained $N_g=3$ to $4$ gold layers on each side, and the layers consisted of 9 atoms on the (111) plane. The leads, which were assumed to be periodic in the transport direction, also consisted of gold layers containing 9 atoms on the (111) plain. Using a double–$\zeta$ basis plus polarization orbitals, Troullier–Martins pseudopotentials[@ref:pseudopot] and the Ceperley–Alder LDA method to describe the exchange correlation [@ref:zunger], effective tight-binding Hamiltonians $H_{M}$, $H_{L}$ of the extended molecule and of the leads, respectively, were obtained. To investigate the generic physics of this system, we employ the simplest possible approximation for the order parameter, namely that it changes in a step-function-like manner at the superconducting lead - extended molecule interface. Therefore the superconducting lead was modelled by introducing couplings of constant magnitude $\Delta$ between the electron and hole degrees of freedom in $H_L$, while no such coupling was present in the extended molecule and in the normal lead. We focus on sub-gap transport and therefore compute the Andreev-reflection probability $R_A(E)$, because at zero temperature for $E<\Delta$ the differential conductance is given by $G_{NS}(E)=\frac{4 e^2}{h} R_A(E)$.
The calculations of Ref. have shown that by changing by rotational conformation of the bipyridine unit it is possible to change the position of the Fano resonance with respect to $E_F$. The definition of the angle of rotation $\theta$ of the bipyridine group is the following: $\theta=0^{\circ}$ when the rings of the sidegroup are parallel to the molecule axis and it is $90^{\circ}$ when they lie perpendicular.
We consider the molecule whose rotational conformation is $\theta=71.4^{\circ}$. Assuming first that both leads are normal conducting (N-Mol-N junction), close to the Fermi energy there is a FR in the differential conductance $G_N(E)=\frac{2 e^2}{h} T_N(E)$ (where $T_N(E)$ is the normal transmission) as it can be seen in Fig. \[fig:trans2\]. Since for conventional supercondcutors the typical superconducting gap values are $0.1-1.5$meV, we first consider the transport for energies $E<\Delta=1.35$ meV. The Fano peak in $G_N(E)$ is at $\delta E\approx 3.7$meV above $E_F$, therefore $\delta E$ is bigger than $\Delta$ and the influence of this resonance on the sub-gap transport can be understood by considering the zero bias conductance $G_{NS}(0)$. Indeed, as Fig. \[fig:trans1\] shows, the Andreev reflection is almost constant apart from the region $E\approx \Delta$ where a sharp peak can be observed which is due to the singularity in the density of states of the superconductor at this energy.
![Normal differential conductance $G_N(E)$ (in units of $2 e^2/h$, dashed line) and sub-gap conductance $G_{NS}(E)$ (in units of $4 e^2/h$, solid line) in logarithmic scale as a function of energy. We used $\Delta=41$ meV. \[fig:trans2\]](transport2.eps)
One can see that off-resonance $G_{NS}(0)$ is smaller than $G_N(0)$.
More generally however, if there is a narrow $\Gamma \lesssim \Delta$ resonance at some $|\delta E| < \Delta$ above or below $E_F$, the energy dependence of the Andreev reflection becomes important. (In case e.g. of carbon nanotubes, which can be gated, this scenario should be easily attainable, as in Ref where the width of the Fano peak was $\approx 0.2$meV.) To illustrate this case, we performed computations using the same molecule but much bigger $\Delta$.
![$T_N(E)$ (dashed line) and $R_{A}(E)$ (solid line) as a function of energy. We used $\Delta=1.35$ meV. The sharp peak in $R_A(E)$ is at $E\approx \Delta$ (see main text). \[fig:trans1\]](transport1.eps)
The results of the computations are shown in Fig. \[fig:trans2\]. As one can see, a Fano resonance now appears both $G_{N}$ in $G_{NS}$. However, a closer inspection reveals that the width of the Fano peak in $G_{NS}$ at $\approx 3.7$ meV is roughly half of the width of the corresponding peak in $G_N$.
To explain the results of the numerical calculations we consider a simple model, introduced in Ref. , which was shown to capture the essential features of the transport between normal conducting leads. Close to a resonance, it is sufficient to consider a single backbone state $|f_1\rangle$ with resonant energy $\tilde{{\varepsilon}}_1$ and a state $|f_2\rangle$ of energy $\tilde{{\varepsilon}}_2$ which is associated with a side group of the molecule ($\tilde{{\varepsilon}}_1$ and $\tilde{{\varepsilon}}_2$ are measured relative to the $E_F$). The weak coupling between the backbone of the molecule and the side group is described by a matrix $H_{12}$. We denote by $t_c=\langle f_2|H_{12}|f_1\rangle$ the coupling between the two states, whereas the coupling of the backbone state to the normal (superconductor) lead is described by matrices $W_N$ ($W_S$). A brief derivation of the Andreev reflection probability $R_A(E)$ for this system is given in Appendix \[sec:r\_a-deriv\], here we only summarize the main results.
The linear conductance is given by $$G_{NS}(0)=\frac{4e^2}{h}
\frac{4 \Gamma_L^2\Gamma_R^2 \tilde{{\varepsilon}}_2^4}
{[(\tilde{{\varepsilon}}_{+}\tilde{{\varepsilon}}_{-})^2+(\Gamma_L^2+\Gamma_R^2)
\tilde{{\varepsilon}}_2^2]^2}.
\label{eq:gns-zero}$$ Here $\Gamma_L$, ($\Gamma_R$) is the normal state tunnelling rate to the left (right) lead at $E_F$ and $\tilde{{\varepsilon}}_{\pm}=\bar{{\varepsilon}}\pm \sqrt{\delta{\varepsilon}^2+t^2}$ where $\bar{{\varepsilon}}=(\tilde{{\varepsilon}}_1+\tilde{{\varepsilon}}_2)/2$, $\delta{\varepsilon}=(\tilde{{\varepsilon}}_1-\tilde{{\varepsilon}}_2)/2$. The maximal conductance is attained at $\Gamma_L=\Gamma_R$, $\tilde{{\varepsilon}}_{\pm}=0$ when it is twice as large as the normal conductance. Note that the conductance maximum is not attained when $\tilde{{\varepsilon}}_1$ is aligned with $E_F$ as one might expect. The hybridization between $\tilde{{\varepsilon}}_1$ and $\tilde{{\varepsilon}}_2$ due to the coupling $t_c$ leads to a different resonance condition for this system. Off-resonance, i.e. when $\tilde{{\varepsilon}}_{\pm}\neq 0$ , $G_{NS}$ falls off more rapidly as a function of $\tilde{{\varepsilon}}_{\pm}$ than $G_{N}$ \[see Eq. (1) in Ref. \]. Therefore the $G_{NS}(0)$ is usually smaller than $G_{N}(0)$. Moreover, $G_{NS}$ is zero if $\tilde{{\varepsilon}}_2=0$, i.e. when the energy of the side coupled state equals $E_F$.
![$G_{N}$ (in units of $2e^2/h$, dashed) and $G_{NS}$ (in units of $4e^2/h$, solid) in logarithmic scale as a function of energy. We used $\sqrt{\Gamma_N^e \Gamma_N^h}/|\sigma_S^{eh}|=4$ in the case of a), b), c) and $\sqrt{\Gamma_N^e \Gamma_N^h}/|\sigma_S^{eh}|=0.25$, in the case of d). \[fig:theorycurves\]](theorycurves.eps)
For finite energies $E<\Delta$ the most important features of the differential conductance of our model are the following. If the coupling to the normal lead is stronger than to the superconducting one, i.e. when $\sqrt{\Gamma_N^e \Gamma_N^h}\gtrsim |\sigma_S^{eh}|$ where $\Gamma_N^e$ ($\Gamma_N^h$) are tunnelling rates for electrons (holes) from the normal lead and $\sigma_S^{eh}$ is an off-diagonal element of the self-energy matrix $\Sigma_S$ (see Appendix \[sec:r\_a-deriv\] for the precise definitions, as well as for the definitions of ${{\varepsilon}}_+^e$, ${{\varepsilon}}_-^e$, to be introduced below), in good approximation $$G_{NS}(E)=\frac{4e^2}{h} A(E)\, T_e(E)
\label{eq:gns-finitE}$$ where the amplitude $A(E)$ is a slowly varying function of the energy and $$T_e(E)=
\frac{\Gamma_N^e \Gamma_N^h (E-\tilde{{\varepsilon}}_2)^2}
{[(E-{{\varepsilon}}_+^e)(E-{{\varepsilon}}_-^e)]^2+
(\Gamma_N^e)^2 (E-\tilde{{\varepsilon}}_2)^2}.
\label{eq:T_F}$$ Assuming a weak coupling between $|f_1\rangle$ and $|f_2\rangle$ i.e. that $
t_c \ll
\delta{\varepsilon}=|{\varepsilon}_+^e -{\varepsilon}_-^e|
$ (which also means that ${{\varepsilon}_-^e} \approx \tilde{{\varepsilon}}_2$), for energies close to ${{\varepsilon}_-^e}$ the probability amplitude $T_e(E)$ can be further approximated by $$T_e(E)\approx \mathcal{A}
\frac{(\epsilon + q)^2}{\alpha^2 \epsilon^2 +1}
\label{eq:Tfano}$$ where $\mathcal{A}=\Gamma_N^e\Gamma_N^h/({\varepsilon}_-^e-\tilde{{\varepsilon}}_2)^2$, $\epsilon=(E-{\varepsilon}_-^e)/\Gamma_N^e$, $\alpha^2=({\varepsilon}_-^e-{\varepsilon}_+^e)^2/({\varepsilon}_-^e-\tilde{{\varepsilon}}_2)^2$ and $q=({\varepsilon}_-^e-\tilde{{\varepsilon}}_2)/ \Gamma_N^e$. Therefore, if $0<{\varepsilon}_-^e,\tilde{{\varepsilon}}_2<\Delta$ a FR will appear in the subgap transport \[see Fig.\[fig:theorycurves\](a)\]. For strong coupling such that $
\Gamma_N^e \gg |{\varepsilon}_-^e-\tilde{{\varepsilon}}_2|
$ the Fano lineshape would become a symmetric dip. If however ${\varepsilon}_+^e<\Delta < {\varepsilon}_-^e,\tilde{{\varepsilon}}_2$ is satisfied, a Breit-Wigner resonance (BWR) of width ${\Gamma_N^e}$ occurs \[shown in Fig. \[fig:theorycurves\](b)\], while for $0<{\varepsilon}_+^e,{\varepsilon}_-^e,\tilde{{\varepsilon}}_2 <\Delta$ the Andreev reflection exhibits both a FR and a BWR \[Fig. \[fig:theorycurves\](c)\]. Note, that $T_e(E)$ is very similar to the transmission amplitude $T_N(E)$ calculated in Ref. for normal conducting leads. Since for $\sqrt{\Gamma_N^e \Gamma_N^h}\gtrsim |\sigma_S^{eh}|$ the resonance energies ${\varepsilon}_-^e$, ${\varepsilon}_+^e$ are usually very close to the resonance energies appearing in the expression of $T_N(E)$, one finds that the resonance structures of the normal conductance will also appear in the sub-gap transport if the relevant resonance energies are smaller than the superconducting pair potential. This explains the occurrence of a Fano resonance in $G_{NS}(E)$ in Fig. \[fig:trans2\]. However, since $A(E)$ in Eq. (\[eq:gns-finitE\]) is usually much smaller than unity, $G_{NS}(E)$ itself can also be smaller than $G_N(E)$. The widths of the resonances in the Andreev-reflection coefficient can be significantly smaller than in the normal transmission. This happens because coupling to the superconductor does not lead to the broadening of the resonant levels. Therefore if $\sqrt{\Gamma_N^e \Gamma_N^h}\gg |\sigma_S^{eh}|$ the peaks in the normal and in the Andreev transport have roughly the same width while for $\sqrt{\Gamma_N^e \Gamma_N^h}\gtrsim |\sigma_S^{eh}|$ the width of the peaks in the Andreev reflection is *half* of the width of the corresponding peaks in the normal transmission. This can also be observed in Fig. \[fig:trans2\]. We note that for $E\approx\Delta$ where $\sigma_S^{eh}$ changes rapidly with energy the formula shown in Eq. (\[eq:gns-finitE\]) is not applicable because in the derivation of Eq. (\[eq:gns-finitE\]) we have assumed that the self energy $\sigma_S^{eh}$ is a slowly varying function of the energy.
Finally, we briefly discuss the predictions of our model for the case when the coupling to the superconductor is stronger than to the normal lead, i.e. when $|\sigma_S^{eh}| \gtrsim \sqrt{\Gamma_N^e \Gamma_N^h}$. The conductance can no longer be approximated by Eq. (\[eq:gns-finitE\]) because $\sigma_S^{eh}$ introduces hybridization between electron and hole levels. We find that in the most general case the conductance exhibits both a FR and a BWR, if the corresponding resonance energies are smaller than the superconducting gap. These peaks, as mentioned before, can be much narrower than the ones in the normal transmission because the superconductor does not broaden them. Moreover, we find that for $|\sigma_S^{eh}| \gg \sqrt{\Gamma_N^e \Gamma_N^h}$ the conductance can even reach the unitarity limit. This could not happen in the opposite, $|\sigma_S^{eh}| \ll \sqrt{\Gamma_N^e \Gamma_N^h}$ case because a resonance in $T_e(E)$ is not accompanied by a resonance in $A(E)$ and therefore the conductance is always smaller than $4e^2/h$. We illustrate this in Fig. \[fig:theorycurves\](d) where $G_{NS}$ is shown along with $G_N$. One can see that $G_N<2e^2/h$ because the couplings to the leads are asymmetric and there is a broad resonance at $E/\Delta\approx 0.15$ along with an almost symmetric, narrow dip at $E/\Delta\approx 0.32$. In contrast, $G_{NS}$ has a narrow FR and also a BWR, the latter peak reaching the unitarity limit.
In summary, we have studied the Andreev reflection through a class of molecules which exhibit Fano resonances in the normal conductance. Our numerical calculations based on *ab initio* methods indicate that Fano resonances may also appear in the sub-gap transport. A simple theoretical model that we used to understand the results of the numerical calculations predicts that a) if the coupling to the normal lead is weaker than the coupling to the superconducting one, the resonance structure of the normal conductance can manifest itself in the Andreev reflection coefficient if the resonance energies are smaller than the superconducting gap and b) if the coupling to the superconductor is strong, the resonances in the normal conductance and in the Andreev reflection can be very different, both in position and in width.
Acknowledgment
==============
This work is supported partly by European Commission Contract No. MRTN-CT-2003-504574 and by EPSRC.
{#sec:r_a-deriv}
There are numerous equivalent approaches to calculate transport coefficients through phase coherent normal-superconductor hybrid systems[@ref:methods]. Here we employ the Green’s function technique presented in Ref. in which the Hilbert space is divided into a sub-space $A$ containing the external leads and a sub-space $B$ containing the molecule.
Assuming for a moment that the molecule is isolated, for energies close to a resonance it can be described by quantum states $|f_1\rangle$, $|f_2\rangle$ with resonant energies ${\varepsilon}_1$, ${\varepsilon}_2$. These states are coupled together by a hamiltonian $H_{12}$ with matrix element $t_c = \langle f_1\vert H_{12}\vert f_2\rangle$. The effect of coupling of the molecule to the normal conducting (superconducting) lead via a coupling matrix $W_N$ ($W_S$) is represented by the energy dependent self-energy matrices $\mbox{\boldmath $\Sigma$}_N=\mbox{\boldmath $\sigma$}_N-i\mbox{\boldmath $\Gamma$}_N$ ($\mbox{\boldmath $\Sigma$}_S=\mbox{\boldmath $\sigma$}_S-i\mbox{\boldmath $\Gamma$}_S$) where $\mbox{\boldmath $\sigma$}_N$, $\mbox{\boldmath $\Gamma$}_N$ ($\mbox{\boldmath $\sigma$}_S$, $\mbox{\boldmath $\Gamma$}_S$) are hermitian. We assume that the coupling matrices are diagonal in the quasiparticle $e,h$ space: $$W_{N,S}=
\left( \begin{array}{cc}
W_{N,S}^e & 0 \\
0 & W_{N,S}^h \\
\end{array} \right),\\$$ where $W_{N\,(S)}^e=-(W_{N\,(S)}^h)^*$. Since the Green’s function of the (isolated) normal lead is also diagonal in the quasiparticle space, so will be $\mbox{\boldmath $\Sigma$}_N=
\textnormal{Diag}(\mbox{\boldmath $\sigma$}_{N}^{e}
-i\mbox{\boldmath $\Gamma$}_N^{e},
\mbox{\boldmath $\sigma$}_{N}^{h}-i\mbox{\boldmath $\Gamma$}_N^{h})
$, too. The self energy coming from the coupling to the superconductor has both diagonal and off-diagonal parts, but for $E\le\Delta$ it reads $$\mbox{\boldmath $\Sigma$}_S =\left( \begin{array}{cc}
\mbox{\boldmath $\sigma$}_S^{e} & \mbox{\boldmath $\sigma$}_S^{eh}\\
\mbox{\boldmath $\sigma$}_S^{he}& \mbox{\boldmath $\sigma$}_S^{h} \\
\end{array} \right)\\$$ i.e. the superconducting lead does not broaden the levels. Moreover, since $|f_2\rangle$ is only coupled with $|f_1\rangle$ but not with any of the leads, the self-energy matrix elements of the matrices $\mbox{\boldmath $\Sigma$}_N$, $\mbox{\boldmath $\Sigma$}_S$ will only affect the resonance energy ${\varepsilon}_1$ of the backbone state but not the energy $\tilde{{\varepsilon}}_2={\varepsilon}_2-E_F$ of the side coupled state. We now introduce the following notations: $\tilde{{\varepsilon}}_1^{e,h}={\varepsilon}_1-E_F-({\sigma}_N^{e,h}+{\sigma}_S^{e,h})$ \[where $\tilde{\sigma}_{N,S}^{e,h}$ are the (only) nonzero element of the matrices $\mbox{\boldmath $\sigma$}_{N,S}^{e,h}$\], $\bar{{\varepsilon}}^{\,e,h}=(\tilde{{\varepsilon}}_1^{\,e,h}+\tilde{{\varepsilon}}_2)/2$, $\delta{{\varepsilon}}^{\,e,h}=(\tilde{{\varepsilon}}_1^{\,e,h}-\tilde{{\varepsilon}}_2)/2$, ${\varepsilon}_{\pm}^{e,h}=\bar{{\varepsilon}}^{\,e,h}\pm\sqrt{(\delta{\varepsilon}^{\,e,h})^2+t_c^2}$. Denoting by $\mathbf{G}_{BB}(E)$ the retarded Green’s function of the molecule and using the formula[@ref:colin] $$R_A=\mbox{Tr}[\mbox{\boldmath $\Gamma$}_N^{e}\mathbf{G}_{BB}(E)\mbox{\boldmath $\Gamma$}_N^{h}\mathbf{G}^{\dagger}_{BB}(E)]$$ to calculate the probability of the Andreev reflection, we find after straightforward calculations that $$R_A=
\frac{4 \Gamma_N^e\Gamma_N^h(E-\tilde{{\varepsilon}}_2)^2(E+\tilde{{\varepsilon}}_2)^2 (\sigma_S^{eh})^2}{|D|^2}.$$ Here the denominator is
$$D=[(E-{\varepsilon}_{+}^{e})(E-{\varepsilon}_{-}^{e})+ i (E-\tilde{{\varepsilon}}_2)\Gamma_N^e]
[(E+{\varepsilon}_{+}^{h})(E+{\varepsilon}_{-}^{h})+ i (E+\tilde{{\varepsilon}}_2)\Gamma_N^h]
-(\sigma_S^{eh})^2(E-\tilde{{\varepsilon}}_2)(E+\tilde{{\varepsilon}}_2).$$
and $\sigma_S^{eh}$, $\Gamma_N^e$, $\Gamma_N^h$ are the only non-zero elements of the matrices $\mbox{\boldmath $\sigma$}_S^{eh}=\mbox{\boldmath $\sigma$}_S^{he}$, $\mbox{\boldmath $\Gamma$}_N^{e}$, $\mbox{\boldmath $\Gamma$}_N^{h}$.
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abstract: 'Multilayer infrastructure is often interdependent, with nodes in one layer depending on nearby nodes in another layer to function. The links in each layer are often of limited length, due to the construction cost of longer links. Here, we model such systems as a multiplex network composed of two or more layers, each with links of characteristic geographic length, embedded in 2-dimensional space. This is equivalent to a system of interdependent spatially embedded networks in two dimensions in which the connectivity links are constrained in length but varied while the length of the dependency links is always zero. We find two distinct percolation transition behaviors depending on the characteristic length, $\zeta$, of the links. When $\zeta$ is longer than a certain critical value, $\zeta_c$, abrupt, first-order transitions take place, while for $\zeta<\zeta_c$ the transition is continuous. We show that, though in single-layer networks increasing $\zeta$ decreases the percolation threshold $p_c$, in multiplex networks it has the opposite effect: increasing $p_c$ to a maximum at $\zeta=\zeta_c$. By providing a more realistic topological model for spatially embedded interdependent and multiplex networks and highlighting its similarities to lattice-based models, we provide a new direction for more detailed future studies.'
author:
- 'Michael M. Danziger'
- 'Louis M. Shekhtman'
- Yehiel Berezin
- Shlomo Havlin
bibliography:
- 'explength.bib'
title: Two distinct transitions in spatially embedded multiplex networks
---
Introduction
============
Several models have been proposed for spatially embedded networks [@doar1993bad; @wei1993comparison; @zegura1997quantitative; @watts-nature1998; @penrose2003random; @kleinberg2000small; @kosmidis-epl2008; @li-epl2011; @barthelemy-physicsreports2011; @mcandrew2015robustness]. In lattice-based models, links are only formed to nearest or next nearest neighbors. In random geometric models, links are formed to all neighbors within some distance [@rozenfeld-prl2002; @bradonjic2007giant]. In models of power grid topology, links are formed with the $m$ nearest neighbors, statically [@hines2010topological] or as a generative model [@deka-sitis2013]. Some models utilize a cost function [@manna-jphysicsa2003; @gastner2006spatial; @emmerich2014structural; @ren-naturecomm2014] or a characteristic distance distribution [@wang-proceedings2008; @grassberger2013sir] to determine link lengths. The model which we study here has spatiality expressed via characteristic link lengths. We utilize exponentially distributed link lengths, similar to the Waxman model [@waxman1988routing].
![**Examples of real-world networks with links of characteristic length.** We examine the distribution of the geographic lengths of the edges in both the European power grid [@zhou-ieee2005] (1851 edges) and the inter-station local railway lines in Japan [@japanrail] (20745 edges). These networks have links of characteristic length and longer links are exponentially unlikely, as indicated by the linear drop on the semi-logarithmic plot. To compare the two datasets, we rescale the lengths so that they are measured in units of their own minimum length, which we determine as the peak of the distribution (mode length). The normalization value ($l=1$) corresponds to 3.7 km (power) and 1.0 km (rail). The characteristic length as determined by the mean is 4.8 km (power) and 1.2 km (rail). As the slope of the exponential fit, it is 3.3 km (power) and 2.0 km (rail). The Japan local railway data is formed from the complete railway network from [@japanrail] with bullet train lines and internal station tracks removed.[]{data-label="fig:realworld"}](power-japan-local.pdf){width="0.93\linewidth"}
Interdependent networks have been studied mainly on random topologies where analytic calculations are possible [@buldyrev-nature2010; @gao-naturephysics2012; @peixoto-prl2012; @radicchi-naturephysics2013; @kivela-jcomnets2014; @boccaletti-physicsreports2014]. However, since many real-world complex systems are embedded in space, it is important to understand the properties of interdependent networks with topology reflecting the dimensionality of the space [@gastner2006spatial; @li-naturephysics2011]. This is particularly important when dealing with critical infrastructure which is heavily influenced by spatial constraints [@rinaldi-ieee2001; @barthelemy-physicsreports2011; @hokstad-book2012; @helbing-nature2013]. An important first step in that direction is the model presented by Li et al. [@wei-prl2012] which models the networks as lattices and includes dependency links which are of limited geographic length (described by the parameter $r$). This model was shown to include a number of surprising properties including three regimes of phase transitions depending on the value of $r$. For $r<r_c$, the phase transition is second-order and appears to be in the same universality class as 2-dimensional lattices [@danziger-newjphysics2015]. The percolation threshold increases as $r$ increases and reaches a maximum value at $r_c$, where the transition switches to first-order [@wei-prl2012; @danziger-jcomnets2014]. For $r_c<r<\infty$ the transition is first-order and characterized by a spreading process, with $p_c$ decreasing monotonically. For $r=\infty$, the transition is an abrupt simultaneous first and second order transition [@danziger-jcomnets2014; @dong-pre2014]. The model was studied under partial dependency with $r=\infty$ and it was shown to have first-order transitions for any fraction of dependency [@bashan-naturephysics2013]. When there is partial dependency and finite dependency link lengths, $r_c$ is shown to increase as the fraction of dependent nodes decreases, diverging at $q=0$ [@danziger-sitis2013; @danziger-jcomnets2014]. This model was also studied for general networks formed of interdependent lattices [@shekhtman-pre2014] for interdependent resistor networks with process-based dependency [@danziger-newjphysics2015] and in the presence of healing [@stippinger-physa2014].
However, in many real-world systems, the length of the dependency links may not be longer than the length of the connectivity links. For instance, in the example of the power grid and communications network, it is unlikely that a communications station will skip the nearer power stations and depend on a power station that is farther away. Also, real-world networks including the power grid do not have uniform link lengths like a lattice, but rather have links of a characteristic length, consistent with an exponential distribution (see example in Fig. \[fig:realworld\]) [@li-naturephysics2011]. To address these two issues, here we model interdependent networks where every node has a bi-directional dependency link with the nearest geographic node in the other network. We treat each pair of nodes as single nodes in a multiplex network, where the links in each layer are different but of the same characteristic length $\zeta$. We thus interpret each node as a geographic entity (e.g. a city or neighborhood) which is linked via two types of links (e.g. electricity and communications) to other nodes. Each node requires both of its constituents to function and each constituent requires connectivity within its layer.
Our main focus in this paper is to examine the role of the characteristic length ($\zeta$) of the links on the robustness of the multiplex. We find that, though increasing $\zeta$ *decreases* $p_c$ in single networks–making them more robust–it has the opposite effect on multiplex networks. Increasing $\zeta$ *increases* $p_c$ for multiplex networks until a critical length $\zeta_c$ where $p_c$ is maximal. At $\zeta_c$, the percolation transition changes to first-order and the multiplex network is susceptible to spreading cascading failures similar to the ones observed in interdependent lattices [@wei-prl2012; @danziger-sitis2013; @danziger-jcomnets2014; @berezin-scireports2015].
By demonstrating that comparable critical behavior emerges in more realistic topologies, we show that the critical behavior demonstrated in previous lattice-based models is not limited to the specific implementation or the lattice topology but is rather a generic property of interdependent spatially embedded networks. Furthermore, by providing a topological model that more closely matches real-world systems, we provide a more realistic topological framework for future studies of interdependent critical infrastructure.
Model
=====
To model connectivity links of characteristic length in each layer, we construct the network as follows. We begin by assigning each node an $(x,y)$ coordinate with integers $x,y\in [0,1,\ldots,L)$. To construct the links in each layer, we select a source node at random with coordinates $(x_0,y_0)$ and draw a length $l$ according to $P(l) \sim e^{-l/\zeta}$. We choose the permitted link length $(dx,dy)$ which is closest to fulfilling $l = \sqrt{dx^2 + dy^2}$, select one of the eight length-preserving permutations $(dx\leftrightarrow-dx, dy\leftrightarrow-dy, dx\leftrightarrow dy)$ uniformly at random and make a link to node $(x_1,y_1)$ with $x_1 = x_0 + dx$ , $y_1 = y_0 + dy$. This process is executed independently in each layer and is continued until the desired number of links ($N{\langle k \rangle}/2$) is obtained. For simplicity, we use the same characteristic length $\zeta$ and average degree ${\langle k \rangle}$ in each layer. However, because they are constructed independently, the links in each layer are different (as demonstrated in Fig. \[fig:layers\] and Fig. \[fig:overlap\]) and this disorder enables the critical behavior which we describe below.
We then perform site percolation by removing a fraction $1-p$ of the nodes from the system and finding the mutual giant component [@buldyrev-nature2010]. When a node is removed, it causes damage to nodes in both layers due to the connectivity links in each layer, which are also removed. However, since the connectivity links are not the same in both layers, there will be nodes that are connected to the giant component in one layer but are disconnected in the other layer. Since the node functionality requires connectivity in *both* layers, such nodes will fail, causing further damage in the system. This leads to the cascading failures as demonstrated in Figs. \[fig:cascade-diagram\] and Fig. \[fig:hole\], which are similar to those described in [@buldyrev-nature2010; @parshani-epl2010; @wei-prl2012; @hu-pre2013; @danziger-jcomnets2014; @dong-pre2014; @berezin-scireports2015].
Results
=======
Since we are not aware of a discussion of the percolation properties of this topology for single networks, we briefly describe those properties here and in the Appendix. In the limit of $\zeta \rightarrow 0$, the only permitted links will be to nearest neighbors (because links of length $<1$ are not accessible) and a square lattice is recovered. As such, in the case of ${\langle k \rangle}= 4$, we recover the standard 2-dimensional percolation behavior with $p_c \approx 0.5927$ [@bunde1991fractals; @ziff-prl1992] (Fig. \[subfig:single\_perc\]). When slightly longer (i.e., next nearest neighbor) links are allowed, the system becomes slightly less robust as discussed in the Appendix. As $\zeta$ increases further, the robustness increases and in the limit $\zeta \rightarrow \infty$, all lengths are equally likely to be drawn and the topology reverts to purely random ([Erdős-Rényi ]{} topology) with $p_c = 1 / {\langle k \rangle}= 0.25$ (see Fig. \[subfig:single\_perc\].) Thus, similar to rewiring probability in the original small world model [@watts-nature1998], we have a single parameter $\zeta$ which allows us to smoothly transition from lattice to random topology. For all values of $\zeta$, a single network undergoes a second-order transition (Fig. \[subfig:njumpk3\] and \[subfig:njumpk4\]).
In multiplex networks, where connectivity to the giant component in both layers is required, cascading failures emerge [@buldyrev-nature2010; @baxter-prl2012; @peixoto-prl2012; @danziger-ndes2014]. For $\zeta\approx 0$, large cascading failures do not emerge. This is because the multi-layer structure is mostly redundant and the difference between connectivity in one layer or both layers is negligible. However, once $\zeta$ becomes long enough (above $\zeta_c$), intensive cascading failures emerge and the system undergoes an abrupt, first-order transition, similar to the transition in interdependent lattices [@wei-prl2012], as shown in Fig. \[fig:pc\_Njump\] and \[fig:hole\]. As $\zeta$ becomes even longer, $p_c$ decreases and slowly approaches its asymptotic value of $2.4554/{\langle k \rangle}$ as known from interdependent [Erdős-Rényi ]{} networks [@buldyrev-nature2010], (Fig. \[subfig:pck3\] and \[subfig:pck4\]).
In interdependent lattices the mechanism of the first-order transition for dependency links of large finite length is a propagating spinodal interface [@wei-prl2012; @danziger-sitis2013; @danziger-jcomnets2014; @berezin-scireports2015]. On a microscopic level, the first-order transition takes place by the emergence of a hole due to random fluctuations of characteristic length $\xi$ (the percolation correlation length) which then propagates through the system. This propagation is enabled by the cascade dynamics (cf. Fig. \[fig:cascade-diagram\]) and dependency links which relay the damage caused by the hole into a concentrated area around the hole’s edge. It would seem that in order for this phenomenon to take place, long dependency links are needed and longer connectivity link lengths would be insufficient to sustain damage propagation. The reason that the dependency links have a stronger influence on robustness is that in order for a node to fail due to connectivity, all of its links to the giant component need to fail whereas with dependency, an otherwise well-connected node will fail if the single node that it depends on fails.
We find that, in fact, this is not the case. Connectivity link lengths which are above a critical length, $\zeta_c$, but much smaller than the total system size are sufficient to cause the cascading failures observed in lattice models, even when the dependency links have zero length. Indeed, the first order transition lacks scaling behavior just above $p_c$ (Fig. \[subfig:multi\_perc\]) and is characterized by a slow spreading process (Fig. \[fig:hole\]), just like interdependent lattices described in Refs. [@wei-prl2012; @danziger-sitis2013; @danziger-jcomnets2014]. However, the maximum $p_c$ in this system is lower than the comparable lattice system: $\approx 0.64$ (${\langle k \rangle}= 4$ here) vs. $\approx 0.74$ (interdependent lattices). This is because, in order for the connectivity links to effectively relay the damage from the hole, the system must be closer to criticality. Indeed, only when the average degree of each node is 1 do the connectivity and dependency links have the same effect upon it.
Surprisingly, the highly localized topology which emerges from spatial embedding makes the system more susceptible to cascading failures. This is due to the fact that, because the damage from an emergent (or induced [@berezin-scireports2015]) hole is relayed by the cascade dynamics to the neighborhood of its interface, the nodes near the edge of the hole are far more likely to become disconnected. This is related to work on information diffusion in social networks, where it was shown that high modularity makes viral cascades more likely to occur due to the increased likelihood of multiple exposure to the information [@centola-science2010; @weng-scireports2013; @nematzadeh-prl2014].
\
In single layer networks, $p_c$ decreases monotonically as $\zeta$ increases from $\zeta \approx 0.5$ to the limit of $\zeta = \infty$. In contrast, in multiplex networks, $p_c$ increases until $\zeta_c$ and then decreases monotonically thereafter. The peak in $p_c(\zeta)$ is due to the fact that the size of the critical hole that is needed to trigger the transition scales linearly with $\zeta$ [@berezin-scireports2015] and the size of emergent holes above the percolation threshold, $\xi(p)$, decreases with $p$ [@bunde1991fractals]. This would indicate that the smaller $\zeta$ is, the smaller the critical hole needs to be and that $p_c$ would increase monotonically as $\zeta$ decreases. However, when $\zeta < \zeta_c$ there is not enough space between the emergent hole and the extent of the damage propagation ($\zeta$) for the network to disintegrate and the small emergent holes remain in place [@wei-prl2012; @danziger-sitis2013; @danziger-jcomnets2014; @berezin-scireports2015].
For interdependent lattices with dependency links of finite length, the single network case is recovered when the dependency links have length zero ($r=0$) [@wei-prl2012]. In spatially embedded multiplex networks the same limit is recovered as $\zeta\rightarrow 0$ due to overlapping links. The fraction of common connectivity links between two interdependent networks or between two layers in a multiplex network is called intersimilarity [@parshani-epl2010; @hu-pre2013] or overlap [@cellai-pre2013; @li-newjphysics2013]. The cascading failures and abrupt transitions which characterize interdependent networks decrease as overlap increases. In the limit of total overlap, they disappear altogether because the system is composed of two exact copies of the same network and the cascade shown in Fig. \[fig:cascade-diagram\] does not take place at all.
In multiplex networks with links of characteristic length, the extent of the overlap can be estimating by considering the probability that, given the same source node, two links lead to the same target node. In the continuum limit this is proportionate to the probability that the links have the same length and the same direction. The system is isotropic by construction so the directional condition is simply $1/2\pi r$, the size of a ring of radius $r$. We obtain $$P(overlap) \sim \int_1^\infty \frac{P^2(r)}{2\pi r} dr \sim \frac{1}{\zeta^2} \int_1^\infty \frac{e^{-2r/\zeta}}{2\pi r} dr \sim \frac{1}{\zeta^2}$$ We find that the scaling in our system is scale-free, with an exponent of $\approx -1.8$ (Fig. \[fig:overlap\]). We hypothesize that the deviation from the continuum calculation is due to the fact that with the discretization of space that we introduce, links that would otherwise have been distinct are unified and the critical exponent is reduced from $-2$ to $\approx -1.8$.
Unlike studies of random multilayer networks with overlap [@parshani-epl2010; @hu-pre2013; @cellai-pre2013; @li-newjphysics2013], decreased overlap alone is not enough to enable the critical behavior observed here. It is only the combination of the disorder (as indicated by decreased overlap) with the spatially embedded links that enables the distinctive first-order transition which we observe here.
![**The fraction of overlapping nodes.** The fraction of overlapping nodes is determined as the number of common links across both layers divided by the total number of links in each layer. When $\zeta \approx 0$, the overlap is maximal and the networks are identical (for ${\langle k \rangle}= 4$, as in this figure). As $\zeta$ increases, the fraction decreases with an exponent of $\approx -1.8$, see text for discussion. []{data-label="fig:overlap"}](Overlaps_multipleL){width="0.9\linewidth"}
Discussion
==========
Previous models of spatially embedded interdependent networks [@wei-prl2012; @bashan-naturephysics2013; @danziger-sitis2013; @danziger-jcomnets2014; @shekhtman-pre2014; @berezin-scireports2015] have used two-dimensional lattices with dependency links connecting nodes from one network to the other. The dependency links were also affected by the spatial embedding via the restriction that they have length of up to $r$, a system parameter. This model led to many important results, but left several important issues unaddressed. First, the topology of real-world spatially embedded networks is not lattice-like or even strictly planar and it was not clear that results derived on lattices would accurately describe real-world topologies. Second, the assumption that dependency links are longer than connectivity links does not correspond with what we would expect from critical infrastructure: It is not reasonable to expect a communications station to get power from a distant power plant and not the one nearest to it. Here we address these problems by modeling spatially embedded interdependent networks as multiplex networks where the dependency relationship is to the nearest node in the other layer and the connectivity links are of finite characteristic length but not uniform or regular.
We find that the most important features of the lattice-based models are reproduced by our new model: second-order and first-order transitions depending on the link length, a first-order transition characterized by the emergence and spreading of a hole and substantially higher vulnerability compared to single-layer networks. Furthermore, $p_c$ has a maximal value at the $\zeta$ value ($\zeta_c$) that separates the two types of transitions. This validates previous work based on lattices and also shows a new way forward for the modelling of critical infrastructure and other spatially embedded multilayer networks.
We acknowledge the MULTIPLEX (No. 317532) EU project, the Deutsche Forschungsgemeinschaft (DFG), the Israel Science Foundation, ONR and DTRA for financial support. We also thank Sergey V. Buldyrev for helpful discussions and comments on the model.
Percolation threshold in single networks and in multiplex and single networks for $\zeta<3$
===========================================================================================
In single networks, for $\zeta>1$, $p_c$ decreases monotonically with increasing $\zeta$. This is evident in Fig. \[fig:pc\_zeta\_single\], where $p_c$ is shown to rapidly approach $1/{\langle k \rangle}$, the [Erdős-Rényi ]{} value, as $\zeta$ is increased [@newman-book2010; @cohen-book2010]. Thus, we find that the effect of increasing $\zeta$ on single networks is to make them *more* robust, the opposite of its effect in multiplex networks.
The behavior of single and multiplex spatially embedded networks is different for $0<\zeta<3$. Since, this is not the main focus of this study and since it does not affect the critical phenomena, we have avoided discussing it in the main text. When $\zeta = 0$, only the shortest possible links will be drawn (those with $l = 1$). Thus, the topology of the network layer will be a pure lattice for ${\langle k \rangle}= 4$ and a diluted lattice for ${\langle k \rangle}< 4$. However, for $0<\zeta<3$, some of the nearest neighbor links are exchanged for next nearest neighbor (diagonal) links which are less robust than nearest neighbor only links. This is a well-known result from bond percolation on lattices with complex neighborhoods, where it was found that $\langle k_c \rangle$ for a next-nearest neighbor lattice ($z=8$) is higher than $\langle k_c \rangle$ for a nearest neighbor lattice ($z=4$) [@feng-pre2008]. For instance, for $\zeta \approx 0.6$ the network layer is a mixture of approximately 85% nearest neighbor links and 15% next nearest neighbor links. As $\zeta$ increases, the links are less redundant and tend to strengthen the network, as seen in Fig. \[fig:pc\_zeta\_single\] and Fig. \[fig:shortzeta\] for single networks.
The robustness is more severely impacted in the multiplex case. This is due to the effects of clustering and disorder. Combining nearest and next-nearest neighbor links leads to high clustering (number of triangles) which, though having only a minor impact on the robustness of single networks, substantially weakens interdependent networks [@huang-epl2013; @shao-pre2014]. Furthermore, the increased disorder decreases the fraction of overlapping links substantially ($\approx 50\%$ as in Fig. \[fig:overlap\]). These effects combine in the multiplex case to cause a substantial increase in $p_c$ for $0.3< \zeta <1$. At $\zeta \approx 1$, the links are no longer redundant, clustering decreases, and the system becomes more stable again. However, it is important to note that even though $p_c$ is increased in this interval, the transition is still second-order. The first-order transition only takes place when the links are substantially longer ($\zeta > \zeta_c$), and the spreading process described above can take place.
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abstract: 'We present a new method for supermassive black hole (SMBH) mass measurements in Type 1 active galactic nuclei (AGN) using polarization angle across broad lines. This method gives measured masses which are in a good agreement with reverberation estimates. Additionally, we explore the possibilities and limits of this method using the STOKES radiative transfer code taking a dominant Keplerian motion in the broad line region (BLR). We found that this method can be used for the direct SMBH mass estimation in the cases when in addition to the Kepler motion, radial inflows or vertical outflows are present in the BLR. Some advantages of the method are discussed.'
---
Introduction
============
Supermassive Black Holes (SMBHs, $10^6-10^9\ M_\odot$) are supposed to reside in the bulges of spiral and elliptical galaxies, and they are in action to shape their cosmic environment, i.e. it seems that there is a connection between the central SMBH and the host galaxy structure. The close connection between the formation and evolution of galaxies and of their central SMBHs involves a variety of physical phenomena of great relevance in modern astrophysics (see e.g. Heckman & Best 2014). The parameters which define a SMBH are mass, spin and electricity, where mass of SMBH is the most important and probably is in the correlation with the galaxy bulge mass. Therefore, one of the most important issue in astrophysics today is to measure masses of SMBHs.
One of the most powerful objects in the Universe are active galactic nuclei (AGNs) which emit huge amount of energy that is created around SMBHs. The emission gas is very close to the central SMBH, and therefore the gas kinematics is directly influenced by the mass of the central black hole. Especially if the gas emits the broad lines from the so called broad line region (BLR). The width of lines emitted from the BLR corresponds to the rotational velocity that can be used for estimation of the black hole masses in AGNs.
In principle there are several methods for the SMBH mass measurements (see Peterson 2014), out of which the reverberation method is the one often used in the case of AGNs. Using this method one can estimate the SMBH mass as:
$$M_{\rm BH}=f{R_{\rm BLR} \sigma_V^2\over G},$$ where $R_{\rm BLR}$ is the photometric dimension of the BLR obtained from the reverberation mapping, and $\sigma_V$ is the corresponding orbital velocity that can be estimated from the broad line widths. $G$ is the gravitational constant and the virial factor $f$ depends on the BLR inclination and geometry which are unknown and, consequently there is a problem to accurately determine the virial factor (Peterson 2014).
On the other side, the polarization in the broad lines can be very useful for investigation of the nature of the BLR (Smith et al. 2002, 2004, 2005; Afanasiev et al 2014; Afanasiev & Popović 2015, Afanasiev et al. 2018 etc.). The polarization in the broad lines, especially the polarization angle ($\varphi$) that shows horizontal S shape across the broad line profile, which indicates a dominant Keplerian-like motion in the BLR, and dominant equatorial scattering as a polarization mechanism in the broad lines (see Afanasiev & Popović 2015).
In this contribution we shortly describe the method and give some overview of the numerical tests of the validity (limitations) of the method which have been performed by STOKES code (see Savić et al. 2018)
![The observed (left) and modeled (right) polarization angle changes across the H$\alpha$ line profile (panels up) for NGC 4151 and $\log({V\over c})$ vs. $\tan(\Delta\varphi)$ (panels down) relationship (in more detail see Savić et al. 2018).[]{data-label="fig1"}](NGC4151_2-2.eps){width="12cm"}
Polarization method for SMBH measurements
=========================================
The method is described in more details in [@ap15]. We used the fact that in the case of equatorial scattering, the polarization angle ($\varphi_i$) is connected with the distance ($R_i$) of an emitting cloud (see Fig. 1 in Afanasiev & Popović 2015) as $$\tan\varphi_i\sim R_i,$$ that in the case of a Keplerian like motion can be connected with the velocity ($V_i=(\lambda_i-\lambda_0)/\lambda_0$, where $\lambda_0$ is the transition wavelength) of the emitting cloud (i.e. $\lambda_i$ is the wavelength emitted by the cloud) as $$V_i=\sqrt{GM_{\rm BH}\over{R_i}},$$ where $M_{\rm BH}$ is the mass of SMBH.
It is not difficult to obtain that $R_i= R_{sc}\cdot\tan\varphi_i,$ where $R_{\rm sc}$ is the the distance between the SMBH and the scattering region (supposed to be in the inner part of the torus, see Fig. 1 in Afanasiev & Popović 2015). Considering the above relations one can obtain the relationship between the velocity ($V_i$) and $\tan\varphi$ as:
$$\log({V_i\over c})=a-0.5\cdot \log(\tan(\Delta\varphi_i)),$$ where $c$ is the speed of light, and constant $a$ depends on the SMBH mass ($M_{\rm BH}$) as
$$a=0.5\log\bigl({{GM_{\rm BH} \cos^2(\theta)}\over{c^2R_{\rm sc}}}\bigr),$$ that can be used for the SMBH mass measurements.
We show in [@ap15] that the method gives SMBH mass estimates which can be compared with the reverberation mapping results. The method seems to be very perspective, since, as it can be seen in Fig. 1 (left panel), very often we can observe a horizontal S-shaped polarization angle in the case of Type 1 AGNs (AGNs with the broad emission lines, see Afanasiev et al. 2018).
Modeling and method validity and limitations
============================================
Following the method described by [@ap15], we theoretically model the broad line polarization due to equatorial scattering. We assume that the unpolarized lines are emitted from the disk-like BLR with dominant Keplerian motion and scattered by free electrons at the inner part of the dusty torus. We used 3D Monte Carlo radiative transfer code <span style="font-variant:small-caps;">STOKES</span> (see, e.g. Goosmann & Gaskell 2007, Marin et al. 2015, Marin 2018, Rojas Lobos et al. 2018) The size of the BLR was obtained from the reverberation mapping with the outer edge set due to dust sublimation. Model parameters of the BLR and the scattering region (SR) are described in more details in [@sa18], and here will not be repeated. The models are showing that the mass estimates with the polarization method are in the frame of 10% error-bars in the case that the SR distance which is twice the size of the outer limit of the BLR. The largest number of photons is scattered only once from the inner edge of the SR. The number of multiple scattering events is negligible and the assumption of a single scattering approximation is valid. In Fig. 1 (right panel) we show the modeled polarization angle across the broad H$\alpha$ profile for NGC 4151, and as one can see from Fig 1, the modeled profile is very similar to the observed one. Also the relationship between velocity and $\tan\varphi$ across the line profile can be reproduced (Fig 1. bottom panels). This indicates that the single element equatorial approximation is valid.
Additionally, we considered the possible inflows/outflows in the BLR (see Savić et al. 2018) and we have shown that the method for SMBH mass measurements can be used when the contribution of the inflows/outflow is much smaller in comparison with Keplerian motion. Therefore, this method could potentially be used for highly ionized lines such as C III\] and C IV that are observed in the optical domain for high-redshifted quasars.
For high inclinations, the polar scattering becomes dominant and we have Type-2 objects for which the method is no longer valid. For face-on AGNs, the optical polarization is usually much lower than 1% (Smith et al. 2002, Marin 2014) and the amount of interstellar polarization can dominate the equatorial scattering induced polarization in the innermost part of the AGNs. The variability of Type-1 AGNs must also be taken into account. The flux variations of the continuum and broad lines can be up to a factor of 10 or greater between minimum and maximum activity state (Shapovalova et al. 2008, 2010). When observed in the minimum activity (up to Type-2), the absence or the weak flux in the BLRs could not be detected and the method cannot be used.
Discussion and conclusions
==========================
In this paper we shortly describe the polarization method for the SMBH mass measurement using the shape of polarization angle across the broad line profiles (Afanasiev et al. 2014, Afanasiev & Popović 2015, Savić et al. 2018). We found that our method gives very good results in comparison with the reverberation one.
As conclusions we can list the advantages and disadvantages of the method. First, let us mention the advantages, which are: i) the method does not need assumption of virialization (as in the case of the reverberation one), since the horizontal S-like shape of polarization angle across the broad line indicates presence of the Keplerian-like motion (see Fig. 1); ii) the method is not time consuming, i.e. from one epoch observation of broad line, one can obtain the SMBH mass; iii) the method can be applied to all broad lines from the UV to the optical, i.e. one can measure the SMBH mass of different redshifted AGNs in the same (consistent) way.
On the other hand, there are some disadvantages of the method which are: i) the polarization in the broad line of AGNs is not so big, therefore the effect can be hidden by the other (additional) effects of polarization or depolarization; ii) in the case of very strong outflow and inflows, the effect cannot be clearly seen, and could not be used for the SMBH mass determination and iii) there can be a problem with the estimation of distance of the equatorial scattering region ($R_{\rm sc}$).
Finally, let us conclude that the method is very useful and it represents a new method for the SMBH mass determination that can be used also in the case of high redshifted AGNs.
This work was supported by the Russian Foundation for Basic Research ([project N15-02-02101]{}) and the Ministry of Education, Science and Technological Development (Republic of Serbia) through the project Astrophysical Spectroscopy of Extragalactic Objects (176001).
Afanasiev, V. L., Popović, L. Č. 2015, ApJ, 800L, 35 Afanasiev, V. L., Popović, L. Č., Shapovalova, A. I. 2018, MNRAS sent Afanasiev, V. L., Popović, L. Č., Shapovalova, A. I., Borisov, N. V., Ilić, D. 2014, MNRAS, 440, 519 , R. W. & [Gaskell]{}, C. M. 2007, A&A, 465, 129 Heckman T. M. and Best P. N. 2014 ARAA 52 589 , F. 2014, MNRAS, 441, 551 , F. 2018, ArXiv e-prints \[\[arXiv\][1805.09098]{}\] , F., [Goosmann]{}, R. W., & [Gaskell]{}, C. M. 2015, A&A, 577, A66 2014, *SSRev*, 183, 253 , P. A., [Goosmann]{}, R. W., [Marin]{}, F., & [Savi[ć]{}]{}, D. 2018, A&A, 611, A39 Savić, Dj., Goosmann, R.; Popović, L. Č., Marin, F., Afanasiev, V. L. 2018, A&A, accepted (arXiv:1801.06097) , A. I., [Popovi[ć]{}]{}, L. [Č]{}., [Burenkov]{}, A. N., [et al.]{} 2010, A&A, 509, A106 , A. I., [Popovi[ć]{}]{}, L. [Č]{}., [Collin]{}, S., [et al.]{} 2008, A&A, 486, 99 Smith, J. E., Young, S., Robinson, A., et al. 2002, MNRAS, 335, 773 Smith, J. E., Robinson, A., Alexander, D. M., et al. 2004, MNRAS, 350, 140 Smith, J. E., Robinson, A., Young, S., et al. 2005, MNRAS, 359, 846
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author:
- 'Stukopin V.'
title: 'Quantum Double of Yangian of Lie Superalgebra $A(m,n)$ and computation of Universal $R$-matrix'
---
Introduction
============
Last time along with Yangians of simple (and reductive) Lie algebras it is became studied the Yangians of Lie superalgebras of classical type (see [@N], [@N1], [@St]).
Itself notion of Yangian of simple Lie algebra was introduced by V.Drinfel’d as a quantization Lie bialgebra of polynomial currents (with values in this simple Lie algebra) and with coalgebra structure defined rational Yang $r$-matrix. But dual the Yangian object (for linear algebra Lie $gl(n)$)) was became studied earlier in framework Quantum Invers Scaterring Method (QISM). The V. Drinfel’d shows that this object is isomorphic the Yangian. In many papers is ised namely this assignment of Yangian in terms of generators which are matrix elements of irredicible representations of Yangian in sense of Drinfel’d. As noted above this two approaches essentually isomorphic and its employment is dictated of solving problems. In the article [@St] it was defined Yangian of the Lie superalgebra $A(m,n)$ type in the framework Drinfel’d approach and it was formulated Poincare-Birkgoff-Witt theorem (PBW-theorem) and theorem on existence pseudotriangular structure that is theorem on existence of universal $R$-matrix. This article is natural continuation of the article [@St] and its final result is explicite formula for universal $R$-matrix. As a corollary we also receive such formula and for the partial case of Yangian $Y(sl_2)$ Lie algebra $sl_2$. I remined that universal $R$-matrix of the Yangian $Y(\mathfrak{g})$ of the simple Lie algebra $\mathfrak{g}$ was introduced by V.G. Drinfel’d (see [@Dr], [@Dr2]) as a formal power series $\it{R}(\lambda)= 1 + \sum_{k=0}^{\infty}\it{R}_k\lambda^{-k-1}$ with coefficients $\it{R}_k \in Y(\mathfrak{g})^{\otimes2}$, which conjugates the comultiplication $\Delta$ and opposite comultiplication $\Delta' = \tau \circ \Delta, \tau (x \otimes y =
y \otimes x)$ ($\tau (x \otimes y = (-1)^{deg(x)deg(y)}y \otimes x)$ for Lie superalgebras. Explicite defintions will be done later in the text of this article.) More exactly, $\it{R}(\lambda)$ conjugate the images $\Delta$ and $\Delta'$ under action operator $id \otimes T_{\lambda}$, where $T_{\lambda}$ is a quantum counterpart of shift operator, аnd $id$ be an identical operator. The $\it{R}(\lambda)$ behave as it is image of some hypotetical R-matrix $R$ under action $id \otimes T_{\lambda}$ conjugates $\Delta$ and $\Delta'$. Drinfel’d called such formal power series $\it{R}(\lambda)$ the pseudotriangular structure and proved it existence for $Y(\mathfrak{g})$, when $\mathfrak{g}$ be a simple Lie algebra. But explicite formula for $\it{R}(\lambda)$ it hasn’t received up to now. If we shall see on classical counterparts of notions $\it{R}(\lambda)$ and $R$, namely, on classical $r$-matrices $\it{r}(\lambda)$ and $r$, then $r$ be an element of a topological tensor square of a classical double and $\it{r}(\lambda)=(id \otimes T_{\lambda})r$, where $T_{\lambda}f(u)=f(u+\lambda)$ be a shift operatorа. Then we can naturally to expect that and in the quantum case $\it{R}(\lambda)$ will be an image of universal R-matrix $R$ of quantum double of Yangian under the action of some shift operator. When V.Drinfel’d defined pseudotriangular structure he didn’t know good description of Yangian double in terms of generators and defining relations and universal R-matrix of Yangian double. But in the middle of 90-th S.Khoroshkin and V.Tolstoy receeived the description of Yangian double in the terms of generators and defining relations and they computed the multiplicative formula of universal R-matrix of Yangian Double (see [@Kh-T]).
In this article we describe quantum double $DY(\mathfrak{g})$ of Yangian of Lie superalgebra $\mathfrak{g}=A(m,n)$ in the terms of generators and defining relations. We also calculate the universal R-matrix of Yangian Double follow the plan suggested in article [@Kh-T]. The main result of this article is a such formula for universal R-matrix for $DY(A(m,n))$. This formula is represented in the factorable form as an product of three factors each of them is an infinite ordered product. It should be mentioned that computation of universal R-matrix of Yangian Double based on the same ideas as a computation of universal R-matrix of quantized universal enveloping algebra of affine Lie algebra (see [@L-S-S], [@T-Kh]). In the [@L-S-S] it was quantum Weyl group for computation the multiplicative formula of universal R-matrix. In the case of the Yangian Double we havn’t full counterpart quantum Weyl group. But partial analogies we use completely. Namely, operator $t^{\infty}$ we can consider as an counterpart of longest element of affine Weyl group. We also can interpret the twist $F$ which can use for construction of universal R-matrix in the terms of counterpert of elements of affine Weyl group. After them as a formula of universal R-matrix of Yangian double is received we calculate the universal R-matrix of Yangian applying to receiving formula the operator $id \otimes T_{\lambda}$. Further calculation bases on the description action this operators $id \otimes T_{\lambda}$ on the generators of dual Hopf superalgebra to Yangian in quantum double.
Quantum Yangian Double of Lie Superalgebra $A(m,n)$.
====================================================
I recall that the Yangian $Y(\mathfrak{g})$ of basic Lie superalgebra $\mathfrak{g}$ (see [@F-S], [@K]) is a deformation of universal enveloping superalgebra $U(\mathfrak{g}[t])$ of bisuperalgebra Lie $\mathfrak{g}[t]$ of polynomial currents. The structure of the Lie bisuperalgebra is defined by cocycle $\delta: \mathfrak{g} \rightarrow \mathfrak{g} \bigwedge \mathfrak{g}$
$$\delta: a(u) \rightarrow [a(u) \otimes 1 + 1 \otimes a(v),
r(u,v)], \label{coc1}$$
where $$r(u,v)= \frac{\mathfrak{t}}{u-v},$$ and $\mathfrak{t}$ be a Casimir operator, defined nondegenerate scalar product $(\cdot, \cdot)$ on basic Lie superalgebra $\mathfrak{g}$ (which exist on very basic Lie superalgebra (see [@F-S], [@K])). Other words, let $\{e_i\}, \{e^i\}$ be dual bases in $\mathfrak{g}$ relatively this scalar product. Then $\mathfrak{t}= \sum_i e_i \otimes e^i$. Further, let $\mathfrak{g}=A(m,n)$. The Lie superalgebra $\mathfrak{g}$ as an each basic Lie superalgebra is defined by itself Cartan matrix $A=(a_{i,j})_{i,j=1}^{m+n+1}$.
Nonzero elements of Cartan matrix are follows: $a_{i,i}=2, a_{i,i+1}=a_{i+1,i}=-1, i < m+1; a_{i-1,i}=a_{i,i-1}=1, a_{i,i}=-2,
m+1<i, i \in I=\{1, \cdots , m+n+1\}.$ Then Lie superalgebra $\mathfrak{g}$ is generated by the generators $h_i, x^{\pm}_i, i \in I,$ where generators $x^{\pm}_{m+1}$ are odd and other generators are even. These generators satisfy the following defining relations:
$$\begin{aligned}
&[h_i,h_j]=0
&[h_i,x_j^{\pm}]=\pm a_{ij}x_j^{\pm}, \quad\\
&[x_i^+, x_j^-]=\delta_{ij} h_i, \quad\\
&[[x^{\pm}_{m+1},x^{\pm}_{m+2}], [x^{\pm}_{m+2}, x^{\pm}_{m+1}]]=0\\
&[x_i^{\pm},[x_i^{\pm}, x_j^{\pm}]]=0 \quad\end{aligned}$$
As usual, $[\cdot, \cdot]$ denotes supercommutator: $[a,b]= ab - (-1)^{p(a)p(b)}ba$. Let $\Pi = \{\alpha_1, \cdots, \alpha_{m+1}, \cdots, \alpha_{m+n+1}\}$ be a set of simple roots, $\Delta (\Delta_+)$ be a set of all roots (positive roots). Let also $\{x_{\alpha}, x_{-\alpha}\}, \alpha \in \Delta_+$ be a Cartan-Weyl base, normalized by condition $(x_{\alpha}, x_{-\alpha}\})=1$. We shall use notation $(\alpha_i, \alpha_j) = a_{ij}$.
Below, we also use notation $\mathfrak{g}:= A(m,n)$.
[**Definition 1.1.**]{} (see. [@St]) Yangian $Y(\mathfrak{g})$ of Lie superalgebra $\mathfrak{g}$ be a Hopf superalgebra over $\cal{C}$, generated as an associative superalgeba by generators $h_{i,k}:= h_{\alpha_i, k}, x^{\pm}_{i,k}:=x^{\pm}_{\alpha_i,k}, i \in I, k \in Z_+$, which satisfy the following defining relations: $$\begin{aligned}
&[h_{i,k},h_{j,l}]=0, \quad \label{2.1}\\
&\delta_{i,j} h_{i,k+l}=[x_{i,k}^+,x_{j,l}^{-} ], \label{2.2}\\
&[h_{i,k+1},x_{j,l}^{\pm}]=[h_{i,k},x_{j,l+1}^{\pm}]+
(b_{ij}/2)(h_{i,k}x_{j,l}^{\pm}+x_{j,l}^{\pm} h_{i,k}), \label{2.3} \\
&[h_{i,0},x_{j,l}^{\pm}] = \pm b_{ij}x_{j,l}^{\pm}, \label{2.4}\\
&[x_{i,k+1}^{\pm},x_{j,l}^{\pm}]=[x_{i,k}^{\pm},x_{j,l+1}^{\pm}]+
(b_{ij}/2)(x_{i,k}^{\pm}x_{j,l}^{\pm}+x_{j,l}^{\pm}x_{i,k}^{\pm}), \label{2.5} \quad \\
& \sum_{\sigma}[x_{i,k_{\sigma (1)}}^{\pm}, \cdots
[x_{i,k_{\sigma (r)}}^{\pm},x_{j,l}^{\pm}]...]=0,
i\neq j, r=n_{ij}=2 \label{2.6}\\
&[[x^{\pm}_{m,k},x^{\pm}_{m+1,k}],[x^{\pm}_{m+2,k},
x^{\pm}_{m+1,k}]]=0 \label{2.7}\end{aligned}$$
The sum taken over all permutations $\sigma$ of set $\{1,...,r\}.$ Parity function take the following values on generators: $p(x_{j,k}^{\pm})=0,$ for $k\in Z_+, j \in I\setminus \tau$ $p(h_{i,k})=0,$ for $i \in I, k\in Z_+,$ $p(x_{i,k}^{\pm})=1, k \in Z_+, i \in \tau.$
Comultiplication on generators $h_{i,k}, x^{\pm}_{i,k}, i \in I, k =0,1$ is defined by following formulas: $$\begin{aligned}
&\Delta(x) = x \otimes 1 + 1 \otimes x, x \in \mathfrak{g} \label{2.8} \\
&\Delta(h_{i,1})= h_{i,1} \otimes 1 + 1 \otimes h_{i,1} + [h_{i,0} \otimes 1, \mathfrak{t}_0] + h_{i,0}\otimes h_{i,0} = \nonumber \\
&h_{i,1} \otimes 1 + 1 \otimes h_{i,1}+ h_{i,0}\otimes h_{i,0} - \sum _{\alpha \in \Delta_+} (-1)^{deg(x_{\alpha})} (\alpha_i, \alpha) x_{-\alpha} \otimes x_{\alpha}; \label{2.9}\\
&\Delta(x^-_{i,1})= x^-_{i,1} \otimes 1 + 1 \otimes x^-_{i,1} + [1 \otimes x^-_{i,0}, \mathfrak{t}_0] = \nonumber \\
&x^-_{i,1} \otimes 1 + 1 \otimes x^-_{i,1} + \sum _{\alpha \in \Delta_+} (-1)^{deg(x_{\alpha})} [x_{-\alpha_i}, x_{-\alpha}] \otimes x_{\alpha};\label{2.10}\\
& \Delta(x^+_{i,1})= x^+_{i,1} \otimes 1 + 1 \otimes x^+_{i,1} + [x^+_{i,0} \otimes 1, \mathfrak{t}_0] = \nonumber \\
&x^+_{i,1} \otimes 1 + 1 \otimes x^+_{i,1} - \sum _{\alpha \in \Delta_+} (-1)^{deg(x_{\alpha})} x_{-\alpha} \otimes [x_{\alpha_i},x_{\alpha}];\label{2.11} \end{aligned}$$
Let’s note that universal enveloping superalgebra $U(\mathfrak{g})$ naturally embedded in $Y(\mathfrak{g})$.
Let’s introduce the quantum double $DY(\mathfrak{g})$ of Yangian $Y(\mathfrak{g})$. I recall the definition of quantum double (see [@Dr1]). Let $A$ be a Hopf superalgebra. Let’s denote by $A^0$ the dual Hopf superalgebra $A^*$ with opposite comultiplication. Then quantum double $DA$ of Hopf superalgebra $A$ be a such quasitriangular Hopf superalgebra $(DA, R)$, that $DA$ contains $A, A^0$ as a Hopf subsuperalgebras; $R$ be an image of canonical element of $A \otimes A^0,$ corresponding the identical operator under embedding in $DA \otimes DA$; linear map $A \otimes A^0 \rightarrow DA, a \otimes b \rightarrow ab$ be a bijection.
Let’s note if Hopf superalgebra $A$ be a quantization of bisuperalgebra Lie $\mathfrak{g}$, then quantum double $DA$ of $A$ be a quantization of a classical double $\mathfrak{g} \oplus \mathfrak{g}^*$ of a Lie bisuperalgebra $\mathfrak{g}$. Moreover the cobracket in a classical double is defined by formula: $\delta = \delta_{\mathfrak{g}} \oplus (-\delta_{\mathfrak{g}^*}).$
Let $C(\mathfrak{g})$ (see [@Dr3], [@Kh-T]) be an associative superalgebra generated by generators $h_{i,k}, x^{\pm}_{i,k}, i \in I, k \in Z$, which satisfy above mentioned defining relations (\[2.1\])-(\[2.7\]). If to define the degrees of generators of $C(\mathfrak{g}$ by formula: $deg(h_{i,k})=deg(x^{\pm}_{i,k})=k$, then we receive the following filtration on $C(\mathfrak{g})$: $$\cdots C_{-n} \subset \cdots \subset C_{-1} \subset C_0 \subset \cdots \subset C_m \subset \cdots C(\mathfrak{g}),$$ where $C_k= \{ x \in C(\mathfrak{g}): deg(x) \leq k\}$.
Let $\bar{C}(\mathfrak{g})$ be a formal completion of $C(\mathfrak{g})$ relatively this filtration. The generators $x^{\pm}_{i,k}, h_{i,k}, i \in I, k \in Z_+$ generate Hopf subsuperalgebra $Y^+(\mathfrak{g})$ в $\bar{C}(\mathfrak{g})$, isomorphic to $Y(\mathfrak{g})$. Let $Y^-(\mathfrak{g})$ be a closed subsuperalgebra in $\bar{C}(\mathfrak{g})$, generated by generators $x^{\pm}_{i,k}, h_{i,k}, i \in I, k < 0.$
[**Theorem 1.1.**]{} Hopf Superalgebra $Y^0(\mathfrak{g})$ isomorphic to $Y^-(\mathfrak{g})$.
This theorm will be follows from the results which will be formulated below. From theorem 1.1 it follows that Hopf superalgebra $Y^-(\mathfrak{g})$ be a quantization of bisuperalgebra Lie $t^{-1} \mathfrak{g}[[t^{-1}]]$ (with cocycle (\[coc1\])).
For description $DY(\mathfrak{g})$ it is convenient to introduce the generating functions (“fields”) $e^+_i(u):= \sum_{k \geq 0} x^+_{i,k}u^{-k-1}, e-_i(u):= -\sum_{k < 0} x^+_{i,k}u^{-k-1},
f^+_i(u):= \sum_{k \geq 0} x^-_{i,k}u^{-k-1}, h^+_i(u):= 1 + \sum_{k \geq 0} h_{i,k}u^{-k-1},
f^-_i(u):= -\sum_{k < 0} x^-_{i,k}u^{-k-1}, h_i(u):= 1 - \sum_{k < 0} h_{i,k}u^{-k-1}.$
[**Proposition 1.1.**]{} Defining relations (\[2.1\])-(\[2.7\]) in superalgebra $\bar{C}(\mathfrak{g})$ are equivalent the following relations for generating functions $$\begin{aligned}
&[h^{\pm}_i(u), h^{\pm}_j(v)]=0, [h^+_i(u), h^-_j(u)]=0,\\
&[e^{\pm}_i(u), f^{\pm}_j(v)]=-\delta_{i,j}\frac{h^{\pm}_i(u)- h^{\pm}_i(v)]}{u - v},\\
&[e^{\pm}_i(u), f^{\mp}_j(v)]=-\delta_{i,j}\frac{h^{\mp}_i(u)- h^{\pm}_i(v)]}{u - v},\\
&[h^{\pm}_i(u), e^{\pm}_j(v)]=-\frac{(\alpha_i, \alpha_j)}{2} \frac{\{h^{\pm}_i(u), (e^{\pm}_j(u)- e^{\pm}_j(v))\}}{u - v}, \quad\\
&[h^{\pm}_i(u), e^{\mp}_j(v)]=-\frac{(\alpha_i, \alpha_j)}{2} \frac{\{h^{\pm}_i(u), (e^{\pm}_j(u)- e^{\mp}_j(v))\}}{u - v},\\
&[h^{\pm}_i(u), f^{\pm}_j(v)]=\frac{(\alpha_i, \alpha_j)}{2} \frac{\{h^{\pm}_i(u), (f^{\pm}_j(u)- f^{\pm}_j(v))\}}{u - v},\quad\\
&[h^{\pm}_i(u), f^{\mp}_j(v)]=\frac{(\alpha_i, \alpha_j)}{2} \frac{\{h^{\pm}_i(u), (e^{\pm}_j(u)- e^{\mp}_j(v))\}}{u - v},\quad\\
&[e^{\pm}_i(u), e^{\pm}_j(v)] + [e^{\pm}_j(u), e^{\pm}_i(v)] =-\frac{(\alpha_i, \alpha_j)}{2} \frac{\{(e^{\pm}_i(u)- e^{\pm}_i(v)), (e^{\pm}_j(u)- e^{\pm}_j(v))\}}{u - v}, \qquad \\
&[e^+_i(u), e^-_j(v)] + [e^+_j(u), e^-_i(v)] =-\frac{(\alpha_i, \alpha_j)}{2} \frac{\{(e^+_i(u)- e^-_i(v)), (e^+_j(u)- e^-_j(v))\}}{u - v}, \qquad\\
&[f^{\pm}_i(u), f^{\pm}_j(v)] + [f^{\pm}_j(u), f^{\pm}_i(v)] =-\frac{(\alpha_i, \alpha_j)}{2} \frac{\{(f^{\pm}_i(u)- f^{\pm}_i(v)), (f^{\pm}_j(u)- f^{\pm}_j(v))\}}{u - v},\quad\\
&[f^+_i(u), f^-_j(v)] + [f^+_j(u), f^-_i(v)] =-\frac{(\alpha_i, \alpha_j)}{2} \frac{\{(f^+_i(u)- f^-_i(v)), (f^+_j(u)- f^-_j(v))\}}{u - v},\qquad\\
&[e_i^{\epsilon_1}(u_1), [e_i^{\epsilon_2}(u_2), e_j^{\epsilon_3}(u_3)]] + [e_i^{\epsilon_2}(u_2), [e_i^{\epsilon_1}(u_1), e_j^{\epsilon_3}(u_3)]] =0 ,\qquad \\
&[f_i^{\epsilon_1}(u_1), [f_i^{\epsilon_2}(u_2), f_j^{\epsilon_3}(u_3)]] + [f_i^{\epsilon_2}(u_2), [f_i^{\epsilon_1}(u_1), f_j^{\epsilon_3}(u_3)]] =0 ,\qquad \\
&[[e_m^{\epsilon_1}(u_1), e_{m+1}^{\epsilon_2}(u_2)], [e_{m+2}^{\epsilon_3}(u_3), e_{m+1}^{\epsilon_4}(u_4)]]=0,\quad \\
&[[f_m^{\epsilon_1}(u_1), f_{m+1}^{\epsilon_2}(u_2)], [f_{m+2}^{\epsilon_3}(u_3), f_{m+1}^{\epsilon_4}(u_4)]]=0.\quad\end{aligned}$$
Triangular decomposition and pairing formulas.
==============================================
Let $Y'_+, Y'_0, Y'_-$ be subsuperalgebras (without unit) in $Y(\mathfrak{g})$, generating of elements $x^+_{ik}, h_{ik}, x^-_{ik}, (i \in I, k \in Z_+),$ correspondingly. Let $Y_+, Y_0, Y_-$ be subsuperalgebras $Y'_+, Y'_0, Y'_-$ with unit element.
[**Proposition 2.1.**]{} Multiplication in $Y(\mathfrak{g})$ induces isomorphism of vector superspaces
$$Y_+ \otimes Y_0 \otimes Y_- \rightarrow Y(\mathfrak{g})$$
This proposition is partial case of theorem 3 from [@St]. Let’s extend this proposition on $DY(\mathfrak{g})$.
For these we need some simple properties of comultiplication on $Y(\mathfrak{g})$, which is proved by induction using formulas (\[2.8\])-(\[2.11\]) and relations (\[2.1\])-(\[2.7\]), and using also that fact that comultiplication be homomorphism of associative superalgebra, i.e. $\Delta(a \cdot b)= \Delta(a) \cdot \Delta(b)$.
[**Proposition 2.2.**]{} 1) $\Delta(x) = x \otimes 1 (mod Y \otimes Y'_+)$, for all $x \in Y'_+;$\
2) $\Delta(y) = 1 \otimes y (mod Y'_- \otimes Y)$, for all $y \in Y'_-.$\
[**Corollary.**]{} 1) $\Delta(Y_+) \subset Y\otimes Y_+;$\
2) $\Delta(Y_-) \subset Y_-\otimes Y.$
So, we have that $Y_+ (Y_-)$ be a right (left) coideal in $Y = Y(\mathfrak{g})$.
Let’s also $BY'_{\pm}$ be a subsuperalgebra (without unit) in $Y(\mathfrak{g})$, generated by $x^{\pm}_{ik}, h_{jr}, (i, j \in I, k,r \in Z_+).$
[**Proposition 2.3.**]{} 1) $\Delta(e) = e \otimes 1 (mod Y \otimes BY'_+)$, for all $e \in BY'_+;$\
2) $\Delta(f) = 1 \otimes f (mod BY'_- \otimes Y)$, for all $f \in BY'_-.$\
3) $\Delta(h) = h \otimes 1 (mod Y \otimes BY'_+)= 1 \otimes h (mod BY'_- \otimes Y)$, for all $f \in Y'_0.$\
Properties 1), 2) is proved also as analogous properties in proposition 2.2, property 3) follows from 1), 2).
Let $<\cdot, \cdot> : Y(\mathfrak{g}) \otimes Y^0(\mathfrak{g}) \rightarrow C$ be a canonical bilinear pairing $Y(\mathfrak{g})$ and its dual Hopf superalgebra $Y^*(\mathfrak{g})$ with opposite comultiplication. (We denote by $Y^0(\mathfrak{g})$ the $Y^*(\mathfrak{g})$ with opposite comultiplication.) From definition imply the next properties of this pairing.
$<xy, x'y'>= <\Delta(xy), x' \otimes y'> =
(-1)^{p(x)p(y)}<y \otimes x, \Delta(x'y')>, \\
<x \otimes y><x^{\prime}\otimes y^{\prime}> =
(-1)^{p(x)p(y)}<x, x^{\prime}><y, y^{\prime}>,$\
для $\forall x, y \in Y(\mathfrak{g}), \forall x^{\prime}, y^{\prime} \in
Y^0(\mathfrak{g})$
Let $A, B$ are subsuperalgebras of $Y(\mathfrak{g})$. Let’s also $(AB)_{\bot} := \{x' \in Y^0(\mathfrak{g}): <ab, x'>=0,
\forall a \in A, b \in B\}$. It is easy to check that $(Y \cdot BY'_-)_{\bot}, (BY'_+ \cdot Y)_{\bot}, (Y \cdot Y'_-)_{\bot},
(Y'_+ \cdot Y)_{\bot}$ are subsuperalgebras of $Y^0(\mathfrak{g})$. Let\
$Y^*_+ := (Y \cdot BY'_-)_{\bot}, BY^*_+ := (Y \cdot Y'_-)_{\bot},
Y^*_- := (BY'_+ \cdot Y)_{\bot}, \\
(BY)^*_- := (Y'_+ \cdot Y)_{\bot}, Y^*_0 := BY^*_+ \bigcap BY^*_-.$
[**Proposition 2.4.**]{} 1) For all $x \in Y_+, h \in Y_0, y \in Y_-, x' \in Y^*_+, h' \in Y^*_0, y' \in Y^*_-$ canonical pairing is factorized as $$<xhy, x'h'y'> = (-1)^{deg(x')deg(y)}<x, x'><h, h'><y, y'>.$$ 2) Multiplication in $Y^0(\mathfrak{g})$ induces isomorphism of vector spaces:\
$Y^*_+ \otimes Y^*_0 \otimes Y^*_- \rightarrow Y^0(\mathfrak{g}).$\
3) PBW-theorem is fulfilled for $Y^0(\mathfrak{g})$.
[**Proof.**]{} Let’ prove 1).\
$<xhy, x'h'y'>= <\Delta(xh)\cdot \Delta(y), x'h' \otimes y'>= <\Delta(x) \Delta(h) \Delta(y), x'h' \otimes y'> =\\
<(x \otimes 1 + \sum a_n \otimes x_n)(h \otimes 1 + \sum \tilde{a}_s \otimes \tilde{x}_s) (1 \otimes y + \sum y_m \otimes a'_m), x'h' \otimes y'>= \\
<xh \otimes y, x'h' \otimes y'> + < \sum c_r \otimes d_r, x'h' \otimes y'> = \\
(-1)^{deg(x')deg(y)},xh, x'h'> <y, y'>$.
Let’s noticed that $<\sum c_r \otimes d_r, x'h' \otimes y'> = \sum_r (-1)^{deg(x')deg(d_r)} <c_r, x'h'><d_r, y'>=0.$ As $<d_r, y'>=0$, $d_r \in Y'_+Y, y' \in (BY'_+Y)_{\bot}$ and we have $<xh, x'h'>=
< (x \otimes 1 + \sum a_n \otimes x_n)(1\otimes h + \sum y_m \otimes b_m), x' \otimes h'> =
<x \otimes h, x' \otimes h'> + 0 = <x, x'><h, h'>$ therefore we received proposition of 1).
Let’s note that 2) follows from 3). Let’ prove 3). Let’s choose PBW base $Y(\mathfrak{g})$. Every vector of this base can be represented in the followin form: $xhy$, where $x \in Y_+, h \in H, y \in Y_-.$ Then biorthogonal vector in view of 1), can be represented in the form: $x'h'y'$, where $ x' \in Y^*_+, h \in H^*, y \in Y^*_-.$ These vectors also form base in $Y^0(\mathfrak{g})$. This fact proves 3). Let’s study this pairing in detail. First, let’s describe the PBW base for $Y(\mathfrak{g})$ in detail (see alternative description in [@St]). Let’s as above $\Delta, \Delta_+$ denote the set of roots, set of positive roots of Lie superalgebra $A(m,n)$. Let’s consider also the set $\hat{\Delta}^{re}$ of real roots of affine (nontwisted) Lie superalgebra $A(m,n)^{(1)}$ (see [@F-S]). For generators of $DY(\mathfrak{g})$ $x^{\pm}_{i,k}$ we shall use the next notation:\
$x_{\alpha_i + k\delta} := x^{+}_{i,k}, \\
x_{-\alpha_i + k\delta} := x^{-}_{i,k}, i \in I, k \in Z, \alpha_i \in \Delta_+.$
In this case $\pm\alpha_i + k\delta \in \hat{\Delta}^{re}.$ Let $\Xi \subset \hat{\Delta}^{re}$. Total linear order $\precneqq$ на $ \Xi$ is called convex (normal), if for all roots $\alpha, \beta, \gamma \in \Xi$ such that $\gamma= \alpha + \beta$ we have :\
$\alpha \precneqq \gamma \precneqq \beta$ or $\beta \precneqq \gamma \precneqq \alpha$.\
Let’s introduce subsets $\Xi_+, \Xi_-$ of set $\hat{\Delta}^{re}$:\
$\Xi_{\pm} := \{\pm \gamma +k\delta: \gamma \in \hat{\Delta}^{re}_+ \}$.\
Let’s introduce on $\Xi_+, \Xi_-$ convex orderings $\precneqq_+, \precneqq_-$, saisfying the following conditions: $$\gamma + k\delta \precneqq_+ \gamma + l\delta \qquad \mbox{and}\quad -\gamma +
l\delta \precneqq_- -\gamma + k\delta,
\qquad \mbox{if}\quad k \le l, \quad \mbox{for} \quad \forall \gamma \in \Delta_+ \label{no}$$
Let’s define root vectors $x_{\pm \beta}, \beta \in \Xi_+ \cup \Xi_-$ by induction in the following way. Let vectors $x_{\beta_1}, x_{\beta_2}$ are already being constructed. If root $x_{\beta_3}$ satisfy conditions: $x_{\beta_1} \precneqq x_{\beta_3} \precneqq x_{\beta_2}$ and in the segment $(x_{\beta_1}, x_{\beta_2})$ we havn’t root vecors (which was already being constructed), the let’s define root vectors $x_{\pm\beta_3}$ by formulas:\
$x_{\beta_3} = [x_{\beta_1}, x_{\beta_2}], x_{-\beta_3} = [x_{-\beta_2}, x_{-\beta_1}].$
Let’s note that convex (normal) ordering connects with natural ordering of elements of affine Weyl group in the case of Lie algebras.
We need the following description of $Y(\mathfrak{g})$, which is an analog of description of quantized universal enveloping superalgebra of affine Lie superalgebra. First, let’s fix the following convex ordering on $\mathfrak{g}=A(m,n)$:\
$(\epsilon_1 - \epsilon_2, \epsilon_1 - \epsilon_3, \cdots, \epsilon_1 - \epsilon_{m+n+2}),
(\epsilon_2 - \epsilon_3, \epsilon_2 - \epsilon_4, \cdots, \epsilon_2 - \epsilon_{m+n+2}), \cdots
(\epsilon_{m+n+1} - \epsilon_{m+n+2}).$\
Here $\epsilon_i - \epsilon_j = \alpha_i + \cdots + \alpha_{j-1}$.
Let’s add affine root $\alpha_0$ to the set of simple roots. I remind that $\alpha_0 = \delta - \theta, \theta := \alpha_1 + \cdots + \alpha_{m+n+1} = \epsilon_1 - \epsilon_{m+n+1}$ be a highest root, $\delta$ be a minimal imaginary root. Let’s consider the next convex ordering on the set $\hat{\Delta}^{re}$ affine real roots:\
$(\alpha_1, \alpha_1 + \delta, \alpha_1 + 2\delta, \cdots, \alpha_1 + n\delta,
\cdots ),
(\cdots \alpha_1 + \alpha_2 + n \delta,
\cdots, \alpha_1 + \alpha_2 + \delta, \alpha_1 + \alpha_2),
(\cdots \alpha_1 + \alpha_2+ \alpha_3 + n \delta,
\cdots, \alpha_1 + \alpha_2 + \alpha_3 + \delta, \alpha_1 + \alpha_2 +\alpha_3),
\cdots , (\cdots \alpha_{n+m+1} + n \delta, \cdots, \alpha_{n+m+1} + \delta,
\alpha_{n+m+1})$.
Let’s calculate pairing for root vectors. Let $h^*_{i,k}, e^*_{i,k}, f^*_{i,k}$ are generators of $Y^*=Y_-$. Let $e_{i,k} := x^+_{i,k}, f_{i,k} := x^-_{i,k}$.
[**Proposition 2.5.**]{} The following two conditions are equivalent.\
1) $<e_{i,k}, e^*_{j, -l-1}> = -\delta_{i,j} \delta_{k,l};$\
$<f_{i,k}, f^*_{j, -l-1}> = -\delta_{i,j} \delta_{k,l};$\
$ <h_{i,k}, h^*_{j, -l-1}> = -(-\frac{a_{ij}}{2})^{k-l}\frac{a_{ij}k!}{l!(k-l)!} \qquad{for} k \geq l \geq 0.$\
2) $$\begin{aligned}
&[h^*_{i,-k}, h^*_{j,-l}] = 0, \quad \\
&\delta_{i,j} h^*_{i,-k-l} = [e_{i,-k}, f_{j,-l}], \label{equation61}\\
&[h^*_{i,-k-1}, e^*_{j,-l}]=[h^*_{i,-k}, e^*_{j,-l-1}]+
(b_{ij}/2)(h^*_{i,-k}e_{j,-l} + e^*_{j,-l} h^*_{i,-k}), \\
&[h^*_{i,-k-1}, f^*_{j,-l}] = [h^*_{i,-k}, f^*_{j,-l-1}]-
(b_{ij}/2)(h^*_{i,-k}f_{j,-l}+f^*_{j,-l}h^*_{i,-k}), \\
&[h^*_{i,0},e^*_{j,l}] = b_{ij}e^*_{j,l}, \label{equation63}\\
&[h^*_{i,0}, f^*_{j,l}] = -b_{ij}f^*_{j,l}\end{aligned}$$ $$\begin{aligned}
&[e^*_{i,-k+1}, e^*_{j,-l}]=[e^*_{i,-k},e^*_{j,-l+1}]+
(b_{ij}/2)\{e^*_{i,-k}, e^*_{j,-l}\}, \quad \\
&[f^*_{i,-k+1},f^*_{j,-l}]=[f^*_{i,-k},f^*_{j,-l+1}]-
(b_{ij}/2)\{f^*_{i,-k}, f^*_{j,-l}\}, \quad \\
& \sum_{\sigma}[e^*_{i,-k_{\sigma (1)}}, \cdots
[e^*_{i,-k_{\sigma (r)}},e^*_{j,-l}]...]=0,
i\neq j, r=n_{ij}=2\\
& \sum_{\sigma}[f^*_{i,-k_{\sigma (1)}}, \cdots
[f^*_{i,-k_{\sigma (r)}},f^*_{j,-l}]...]=0,
i\neq j, r=n_{ij}=2\\
&[[e^*_{m,-k_1},e^*_{m+1,-k_2}],[e^*_{m+2,-k_3}, e^*_{m+1,-k_4}]]=0,\\
&[[f^*_{m,-k_1},f^*_{m+1,-k_2}],[f^*_{m+2,-k_3}, f^*_{m+1,-k_4}]]=0.\end{aligned}$$
[**Proof.**]{} The proof of this proposition is inconveniently and we having marked basic points of proof, omiting technical details. We shall lead the proof by induction on values of indexes $k,l$. First of all it is easy to prove next formulas.\
$\Delta(e_{i,k}) = e_{i,k} \otimes 1 + 1 \otimes e_{i,k} + \sum_{r=0}^{k-1} h_{i,r} \otimes e_{i,k-r} (mod Y Y_- \otimes Y'_+);$\
$\Delta(f_{i,k}) = f_{i,k} \otimes 1 + 1 \otimes f_{i,k} + \sum_{r=0}^{k-1} h_{i,r} \otimes e_{i,k-r} (mod Y'_- \otimes Y'_+Y); \\
\Delta(h_{i,k}) = h_{i,k} \otimes 1 + 1 \otimes h_{i,k} + \sum_{r=0}^{k-1} h_{i,r} \otimes h_{i,k-r} (mod Y Y'_- \otimes Y'_+Y);$\
From this formulas it is follows the next equalities:\
$\Delta(e_{i,k}e_{j,l}) = e_{i,k}e_{j,l} \otimes 1 + 1 \otimes e_{i,k}e_{j,l} + e_{i,k} \otimes e_{j,l} +
(-1)^{deg(e_{i,k})deg(e_{j,l})} e_{j,l} \otimes e_{i,k} (mod YY'_- \otimes Y'_+); \\
\Delta(f_{i,k}f_{j,l}) = f_{i,k}f_{j,l} \otimes 1 + 1 \otimes f_{i,k}f_{j,l} + f_{i,k} \otimes f_{j,l} +
(-1)^{deg(f_{i,k})deg(f_{j,l})} f_{j,l} \otimes f_{i,k} (mod Y'_- \otimes Y'_+Y); \\
\Delta(h_{i,k}h_{j,l}) = h_{i,k}h_{j,l} \otimes 1 + 1 \otimes h_{i,k}h_{j,l} + h_{i,k} \otimes h_{j,l} +
h_{j,l} \otimes h_{i,k} (mod YY'_- \otimes Y'_+Y).$\
Using these formulas and definition of quantum double we can prove by induction the invariance of this pairing on generators of Yangian Double.
$$<[a,b],c> = <a,[b,c]> \label{p1}$$
We omit the proof of this fact realizing that proof of such simple fundamental fact it must be short and idea’s. We have only proof bases on induction using above written formulas and next definition of Hopf pairing. $$\begin{aligned}
&<ab,cd> = <\Delta(ab),c\otimes d> = (-1)^{deg(a)deg(b)}<b\otimes a, \Delta(cd)>
\label{s1}\\
& <a, 1>=\epsilon(a)>, <1, b>= \epsilon(b) \qquad \label{s2}\end{aligned}$$
Now we can to show how the condition 1) follows from condition 2). Let’s show, for example, how by induction it is derived pairing formula on Cartan generators of Yangian Double from commutative relations using formula \[p1\]. For $m=n=0$, proving formulas coincide with its quasiclassical limits for which they are evidently correct. Let these formulas correct for $m \ge k, n<l+1$. Let’s show that they correct for $n=l+1$.\
$<h_{i,k}, h_{j,l}> = -<e_{i,0},[f_{i,k}, h_{j,l}]> = <e_{i,0},
[h_{j,0},f_{i,k-l-1}] + \\
\frac{1}{2}a_{ij}\sum_{s=0}^l\{h_{j,s-l-1}, f_{k-s-1}\}> =
-\frac{1}{2}a_{ij}(<e_{i,0}, \sum_{s=0}^l ([h_{j,s-l-1}, f_{k-s-1}]
+ \\
2f_{i,k-s-1}h_{j,s-l-1})>) =
-\frac{1}{2}a_{ij}(<e_{i,0}, \sum_{s=0}^l [h_{j,s-l-1},
f_{k-s-1}]>\\
+ 2<e_{i,0}, \sum_{s=0}^l [f_{i,k-s-1}, h_{j,s-l-1}]>)$.\
Second summand equal zero in view of inductive assumption. Let’s transform first summand. Let’s decrease degree of right-hand side in pairing formula using defining relations in Yangian Double.
$<h_{i,k}, h_{j,l}> =a_{ij}<e_{i,0}, \sum_{s=0}^l(l+s-1) [h_{j,-s}, f_{k-l+s-2}]a_{ij}/2> =
-(\frac{1}{2}a_{ij})^2<e_{i,0}, \sum_{s=0}^l(l+s-1) [h_{j,-s}, f_{k-l+s-2}]> = \cdots \\
=-(\frac{1}{2}a_{ij})^{k-l}(<e_{i,0},[h_{j,0}, f_{i,-1}]>C^{k-l-1}_{k-l-1} + C^{k-l-1}_{k-l} +
\cdots \\
C^{k-l-1}_{k-l+l-1})=
-(\frac{1}{2}a_{ij})^{k-l}C^{k-l}_{k}a_{ij}.$\
First pairing formula is proved. The second formula is proved simpler by analogous arguments. The proving of sufficiency rather inconvin inconveniently and we omit it here. Note, only, that actually it is also leaded by induction and based on formulas (\[s1\]), (\[s2\]).
[**Theorem 2.1.**]{} 1) Subsuperalgebras $Y^*_+, H^*, Y^*_-$ of superalgebra $Y_-$ are generated by fields $e^-_i(u), h^-_i(u), f^-_i(u);$\
2) Pairing of generators of subsuperalgebras $Y_+, Y_-$ of superalgebra $DY(\mathfrak{g})$ is assigned the next relations for $|v|< 1 <|u|$:\
$$\begin{aligned}
&<e^+_i(u), f^-_j(v)> = <f^+_i(u), e^-_j(v)> = \frac{\delta_{i,j}}{u-v}; \\
&<h^+_i(u), h^-_j(v)> = \frac{u-v + \frac{1}{2}(\alpha_i, \alpha_j)}{u-v - \frac{1}{2}(\alpha_i, \alpha_j)}\label{2.2}\end{aligned}$$
[**Proof.**]{} Theorem imply from proposition 2.5.
Computation of Universal $R$- matrix of Yangian Double $DY(\mathfrak{g})$.
==========================================================================
First, I remind the definition of universal $R$- matrix for quasitriangular Hopf topological superalgebra, which is natural generalization of notion of universal $R$-matrix for quasitriangular Hopf algebra (see [@Dr1]).
Universal $R$-matrix for quasitriangular topological Hopf superalgebra $A$ is called such invertible element $R$ from some extension of completion of tensor square $A \hat{\otimes} A$ and satisfying next conditions :\
$$\Delta^{op}(x) = R \Delta(x) R^{-1}, \qquad \forall x \in A;$$ $$(\Delta \otimes id) R = R^{13}R^{23}, (id \otimes \Delta) R = R^{13} R^{12},$$ где $\Delta^{op}= \sigma \circ \Delta, \sigma(x \otimes y) = (-1)^{p(x)p(y)} y \otimes x.$
If $A$ be a quantum double of Hopf superalgebra $A^+$, i.e. $A \cong A^+ \otimes A^-$, $A^-:=A^0$ be a dual to $A$ Hopf superalgebra with opposite comultiplication, then $A$ be a quasitriangular Hopf superalgebra and universal $R$- matrix in $A$ assume next canonical presentation: $$R = \sum e_i \otimes e^i,$$ where $\{e_i\}, \{e^i\}$ are dual bases in $A^+, A^-$, respectively.
Let $Y^{\pm}_+, Y^{\pm}_0, Y^{\pm}_-$ are subsuperalgebras in $DY(\mathfrak{g})$, generated by fields $e^{\pm}_i(u), h^{\pm}_i(u), \\
f^{\pm}_i(u), i \in I,$ respectively.
[**Proposition 3.1.**]{} 1) Universal $R$-matrix of Yangian Double can be presented in the next factorizable form: $$R = R_+ R_0 R_-,$$ where $R_+ \in Y^+_+ \otimes Y^-_-, R_0 \in Y^+_0 \otimes Y^-_0, R_- \in Y^+_- \otimes Y^-_+.$\
2)Pairing on the base elements can be computed according to the next formulas: $$\begin{aligned}
&<e^{n_0}_{\beta_0}e^{n_1}_{\beta_1} \cdots e^{n_k}_{\beta_k}, e^{m_0}_{-\beta_0- \delta}e^{m_1}_{-\beta_1- \delta}
\cdots e^{m_k}_{-\beta_k- \delta}> = \quad \nonumber\\
&(-1)^{n_0 + \cdots + n_k} \delta_{n_0, m_0} \cdots
\delta_{n_0, m_0}\cdot n_0!n_1! \cdots n_k! \cdot \alpha(\gamma_0)^{n_0} \cdots
\alpha(\gamma_k)^{n_k}(-1)^{\theta(\beta_0)+ \cdots + \theta(\beta_k)};\qquad\\
&< e^{n_k}_{-\beta_k} \cdots e^{n_1}_{-\beta_1} e^{n_0}_{\beta_0}, e^{m_k}_{-\beta_k- \delta}
\cdots e^{m_1}_{-\beta_1- \delta} e^{m_0}_{-\beta_0- \delta}> = \nonumber\\
&(-1)^{n_0 + \cdots + n_k} \delta_{n_0, m_0} \cdots
\delta_{n_0, m_0}\cdot n_0!n_1! \cdots n_k! \cdot \alpha(\gamma_0)^{n_0} \cdots
\alpha(\gamma_k)^{n_k} (-1)^{\theta(\beta_0)+ \cdots + \theta(\beta_k)}; \qquad\end{aligned}$$
Here $\beta_k= \beta'_k + n'_k\delta$, and coefficients $\alpha(\beta)$ can be calculated from condition $[e_{\beta}, e_{-\beta}]=\alpha(\beta)h_{\beta'}$.
From proposition 3.1 follows
[**Lemma 3.1.**]{} The elements $R_+, R_-$ in decomposition of universal $R$-matrix of $DY(\mathfrak{g})$ can be presented in the following form $$\begin{aligned}
R_+ = & \overrightarrow{\prod}_{\beta \in \Xi_+} \exp(-(-1)^{\theta(\beta)}a(\beta) e_{\beta} \otimes e_{-\beta}), \label{3.3}\\
R_- = & \overleftarrow{\prod}_{\beta \in \Xi_-}\exp(-(-1)^{\theta(\beta)}a(\beta) e_{\beta} \otimes e_{-\beta}), \label{3.4}\end{aligned}$$ where product taken according to normal orderings $\precneqq_+, \precneqq_-$, satisfying conditions \[no\].
Normalizing constants $a(\beta)$ can be found from the following condition: $$\begin{aligned}
&[e_{\beta}, e_{-\beta}] = (a(\beta))^{-1} h_{\gamma} \quad \mbox{if} \quad \beta = \gamma + n\delta \in \Xi_+, \gamma \in \Delta_+(\mathfrak{g}),\quad\\
&[e_{\beta}, e_{-\beta}] = (a(\beta))^{-1} h_{\gamma} \quad \mbox{if} \quad \beta = \gamma + n\delta \in \Xi_+, \gamma \in \Delta_+(\mathfrak{g}),\quad\\\end{aligned}$$
and $\theta(\beta)= deg(e_{\beta})=deg(e_{-\beta})$ denotes parity of element $e_{\pm \beta}$.
For description of term $R_0$ we need some auxiliary notions. First of all, let’s introduce “logarithmic” generators $\phi^{\pm}_i(u), i=1, \cdots,r$ by formulas $$\phi^{+}_i(u):= \sum_{k=0}^{\infty}\phi_{i,k}u^{-k-1} = \ln h^+_i(u);
\phi^{-}_i(u):= \sum_{k=0}^{\infty}\phi_{i,-k-1}u^k = \ln h^-_i(u)$$
Let’s introduce vector-functions
${\phi}^{\pm}(u)=
\begin{pmatrix}
\phi^{\pm}_1(u)\\ \phi^{\pm}_2(u)\\ \cdots \\
\phi^{\pm}_r(u)
\end{pmatrix}
h^{\pm}(u)=
\begin{pmatrix}
h^{\pm}_1(u)\\ h^{\pm}_2(u)\\ \cdots \\
h^{\pm}_r(u)
\end{pmatrix}$
From theorem 2.1 implyes pairing formula in the terms of generating vector-functions $$<((h^+(u))^T, h^-(v))>=
(\frac{u-v + \frac{1}{2}(\alpha_i, \alpha_j)}{u-v -\frac{1}{2}(\alpha_i, \alpha_j)})_{i,j=1}^r$$ Therefore, for generating functions $\phi^+_i(u), \phi^-_j(u)$ pairing formula has the following form $$<\phi^+_i (u), \phi^-_j (v)> = ln(\frac{u-v + \frac{1}{2}(\alpha_i,\alpha_j)}
{u-v - \frac{1}{2}(\alpha_i,\alpha_j)}) \label{3.5}$$
These formulas we can rewritten in the matrix form as $$<(\phi^+(u))^T, \phi^-(v)> =
(\ln(\frac{u-v + \frac{1}{2}(\alpha_i, \alpha_j)}
{u-v - \frac{1}{2}(\alpha_i, \alpha_j)}))_{i,j=1}^r$$
$$<(\phi^+(u))^T, \phi^-(v)> = (\ln(\frac{u-v + \frac{1}{2}(\alpha_i - \alpha_j)}
{u-v - \frac{1}{2}(\alpha_i - \alpha_j)}))_{i,j=1}^r \label{3.6}$$
Further calculation we shall conduct on the scheme suggested in [@Kh-T]. Using this way we can attach to this calculations some gemetrical sense.
Along with Yangian Double $DY(\mathfrak{g})$ let’s consider Hopf superalgebra $\widehat{DY}(\mathfrak{g})$, isomorphic to as associative superalgebra to $DY(\mathfrak{g})$, but with another comultiplication defined next formulas: $$\begin{aligned}
&\tilde{\Delta}(h^{\pm}_i(u))=h^{\pm}_i(u) \otimes h^{\pm}_i(u) \quad \label{3.k} \\
&\tilde{\Delta}(e_i(u))=e_i(u) \otimes 1 + h^{-}_i(u) \otimes e_i(u) \quad \\
&\tilde{\Delta}(f_i(u)) =1 \otimes f_i(u) + f_i(u)\otimes h^{+}_i(u) \quad\end{aligned}$$
Here\
$e_i(u):= e^+_i(u) - e^-_i(u) = \sum _{k \in Z} e_{i,k} u^{-k-1},\\
f_i(u):= f^+_i(u) - f^-_i(u) = \sum _{k \in Z} f_{i,k} u^{-k-1}$
Such comultiplication it was introduced by V.Drinfel’d ([@Dr3]) in the case of Yangians (and Yangian Doubles) of simple Lie algebras and it convenient by that the pairing formulas relatively this comultiplication has a simple form. It is possible to check (see [@Kh-T]), that comultiplications $\Delta$ и $\tilde{\Delta}$ conjugated by limit operator $\hat{t}^{\infty}:= lim_{n \rightarrow \infty} \hat{t}^n$, $\hat{t}(e_{i,k})=e_{k+1}, \hat{t}(f_{i,k})=f_{k-1}, \hat{t}(h_{i,k})=h_{k}$. In other words, $$\tilde{\Delta}(x) = lim_{n \rightarrow \infty}(\hat{t}^n \otimes \hat{t}^n)\Delta(\hat{t}^{-n}(x)),$$ for $\forall x \in DY(\mathfrak{g})$. (The convergence it is implied in suitable topology of $DY(\mathfrak{g})\otimes DY(\mathfrak{g})$.)
Let $\widehat{DY}^+(\mathfrak{g})$ ($\widehat{DY}^-(\mathfrak{g})$) be a Hopf subsuperalgebra of Hopf superalgebra $\widehat{DY}(\mathfrak{g})$, generated by elements $e_{i,k}, k \in Z, h_{i,m}, m \in Z_+$ ($f_{i,k}, k \in Z, h_{i,m}, m <0$). Then $\widehat{DY}^-(\mathfrak{g})$ isomorphic to dual Hopf superalgebra $(\widehat{DY}^-(\mathfrak{g}))^*$. From comultiplication formula (\[3.k\]) imply that elements $\phi^{\pm}_{i,k}$ are primitive elements in $\widehat{DY}(\mathfrak{g})$.
Let $\Phi^+ = <\phi^{+}_{i,k}: i \in I= \{1, \cdots, r\}, k \in Z_+>,
\Phi^- = <\phi^{-}_{i,-k-1}: i \in I= \{1, \cdots, r\}, k \in Z_+>$ are linear superspaces (generated by indicating in brackets sets of vectors).
Let also $\{\tilde{\phi}_{i,m}\}, \{\tilde{\phi}^{i,m}\}$ are dual bases relatively form (\[3.5\]) bases in superspaces $\Phi^+, \Phi^-$, respectively.
We have the following
[*Proposition 3.3.*]{} The element $R_0$ from proposition 3.1 has the following form $$R_0 = exp (\sum_{i,m}(-1)^{deg(\tilde{\phi}_{i,m})}\tilde{\phi}_{i,m}\otimes \tilde{\phi}^{i,m})$$
[**Proof**]{}. Let $B_+ = C[\Phi^+], B_- = C[\Phi^-]$ are commutative function algebras on $\Phi^+, \Phi^-$, respectively and $\{\tilde{\phi}_{i,m}\}, \{\tilde{\phi}^{i,m}\}$ are above mentioned dual bases.
Let’s fix some total linear ordering of basic and below we’ll use notation $\{\tilde{\phi}_a \}, \{ \tilde{\phi}^a \}, a \in N.$ Let’s prove by induction next formula $$<\tilde{\phi}_{i_1}^{n_1} \cdots \tilde{\phi}_{i_k}^{n_k}, (\tilde{\phi}^{i_1})^{m_1} \cdots (\tilde{\phi}^{i_k})^{m_k}> =
\delta_{n_1,m_1} \cdots \delta_{n_k,m_k} n_1! \cdots n_k!$$
It is easy to verify the base of induction for $k=1, n_1=1$ $<\tilde{\phi}_{i_1}, \tilde{\phi}^{i_1}>=1, <\tilde{\phi}_{i_1}, 1> = 0$. Further, let $<\tilde{\phi}_{i_1}^{n}, (\tilde{\phi}^{i_1})^n> = n!$. Let’s show that $<\tilde{\phi}_{i_1}^{n+1}, (\tilde{\phi}^{i_1})^{n+1}> = (n+1)!$. In fact,\
$<\tilde{\phi}_i^{n+1}, (\tilde{\phi}^i)^{n+1}> =
<\Delta(\tilde{\phi}_i)\Delta((\tilde{\phi}_i)^n), \tilde{\phi}^i \otimes (\tilde{\phi}_i)^n>= \\
<(\tilde{\phi}_i\otimes 1 + 1\otimes \tilde{\phi}_i)(\sum_{k=0}^{n} C^k_n (\tilde{\phi}_i)^k(\tilde{\phi}_i)^{n-k}), \tilde{\phi}^i \otimes (\tilde{\phi}_i)^n > = \\
(n+1)<\tilde{\phi}_i,\tilde{\phi}^i>
<(\tilde{\phi}_i)^n,(\tilde{\phi}^i)^n > = (n+1)!$. Using proved formula by induction on $k$ it is proved statement of theorem. Theorem is proved.
Let now $(f(u))' =\frac{d}{du}(f(u))$. Let’s differentiate equality (\[3.5\]) on parameter $u$. We derive\
$\frac{d}{du}<\phi^+_i(u), \phi^-_j(u)>= <(\phi^+_i(u))', \phi^-_j(u)> =
\frac{1}{u-v + \frac{1}{2}(\alpha_i, \alpha_j)} - \frac{1}{u-v - \frac{1}{2}(\alpha_i, \alpha_j)}$. Let $\tilde{\phi}^+_i(u)= \sum_{k=0}^{\infty}\tilde{\phi}_{i,k}u^{-k-1}$, $\tilde{\phi}^-_i(u)= \sum_{k=0}^{\infty}\tilde{\phi}_{i,-k-1}u^k$. Then in terms of generaing functions the pairing $<\tilde{\phi}_{i,k}, \tilde{\phi}_{j,l}>=\delta_{ij}\delta_{kl}$ can be rewritten in the next form: $<\tilde{\phi}^+_i(u), \tilde{\phi}^-_j(v)> =
\sum_{k,l}<\tilde{\phi}_{i,k}, \tilde{\phi}^{j,l}>u^{-k-1}v^l= \\
(v<1<u)=\delta_{ij}\sum_{k=1}^{\infty}u^{-1}(\frac{v}{u})^k=
\frac{\delta_{ij}}{u-v}$.
Thus we receive that $$<\tilde{\phi}^+_i(u), \tilde{\phi}^-_j(v)> =
\frac{\delta_{ij}}{u-v} \label{3.p}$$
Let’ introduce a generating vector-functions
$\tilde{\phi}^{\pm}(u)=
\begin{pmatrix}
\tilde{\phi}^{\pm}_1(u)\\ \tilde{\phi}^{\pm}_2(u)\\ \cdots \\
\tilde{\phi}^{\pm}_r(u)
\end{pmatrix}$
Then pairing (\[3.p\]) we can rewrite in the next matrix equality: $$<(\tilde{\phi}^+(u))^T, \tilde{\phi}^-(u)> = \frac{E_r}{u-v},$$ where $E_r$ be a unit $r\times r$-matrix.
Let $T: f(v) \rightarrow f(v-1)$ be a shift operator. Clearly that $$\begin{aligned}
<(\phi^-_i(v))', \phi^+_j(v)>=
\frac{1}{u-v + \frac{1}{2}(\alpha_i, \alpha_j)} - \frac{1}{u-v - \frac{1}{2}(\alpha_i, \alpha_j)}= \\
(id\otimes (T^{b_{ij}}-T^{-b_{ij}}))\frac{\delta_{ij}}{u-v}=
<\tilde{\phi}^-_i(v), (T^{b_{ij}}-T^{-b_{ij}})\tilde{\phi}_j^+(u)>.\end{aligned}$$
Here $b_{ij}=\frac{1}{2}a_{ij}=\frac{1}{2}(\alpha_i,\alpha_j)$. Let $A = (a_{ij})_{i,j=1}^r$ be a symmetric Cartan matrix of Lie superalgebra $A(m,n)$. $\mathfrak{g}$, i. e. $a_{ij} = (\alpha_i, \alpha_j)$. Let also $A(q)=(a_{ij}(q))_{i,j=1}^r$ be a q-analog of Cartan matrix, where $a_{ij}(q)=[a_{ij}]_q = [(\alpha_i, \alpha_j)]_q=
\frac{q^{a_{ij}}-q^{-a_{ij}}}{q-q^{-1}}.$
Let also $D(q)$ be an inverse matrix to $A(q)$ and $A^T$ denote a transposition of matrix $A$. Then we can rewrite previous equality in the next matrix form $$\begin{aligned}
<(\phi^+(v))^T, \phi^-(u)> =
<(\tilde{\phi}^+(u))^T, A(T^{-\frac{1}{2})}(T-T^{-1})\tilde{\phi}^-(v)>\end{aligned}$$ Therefore $$\begin{aligned}
&<(\tilde{\phi}^-(v))^T, \tilde{\phi}^+(u)>& = \nonumber \\
&<((T-T^{-1})^{-1}D(T^{-\frac{1}{2}})(\phi^-(v))^T, (\phi^+(u))'>&\end{aligned}$$
Thus we have the next equality $$\begin{aligned}
\frac{E_r}{u-v} =
<((T-T^{-1})^{-1} D(T^{-\frac{1}{2}})(\phi^-(v)), ((\phi^+(u))')^T>\end{aligned}$$
So we have diagonalize pairing. Let’s present matrix $D(q)$ in the form $D(q)= \frac{1}{[l(\mathfrak{g})]}C(q)$, where $C(q)$ be a matrix with matrix coefficirnts being polynomials of $q$ and $q^{-1}$ with positive integer coefficients (i.e. $c_{ij} \in Z[q,q^{-1}]$). Let also $l(\mathfrak{g})=\check h(\hat{\mathfrak{g}})$ be a dual Coxeter number. In these notations the previous formula can be written in the next form
$$\begin{aligned}
&\frac{E_r}{u-v}= &\nonumber \\
&<((T^{l(\mathfrak{g})}-T^{-l(\mathfrak{g})})^{-1}C(T^{-\frac{1}{2}})(\phi^-(v)),
((\phi^+(u))')^T>&\end{aligned}$$
From this equality imply formula for the term $R_0$ in the factorizable formula for the universal $R$-matrix.
[**Theorem 3.1.**]{} $$\begin{aligned}
&R_0 = & \nonumber \\
&\prod_{n \ge 0} \exp\sum_{i,j=1}^r \sum_{k \ge 0} ((\phi^+_i(u))')_k \otimes
c_{ji}(T^{-\frac{1}{2}})(\phi^-_j(v+(n+\frac{1}{2})l(\mathfrak{g})))_{-k-1}& \qquad\end{aligned}$$
Computation of the Universal $R$- matrix of the Yangian $Y(\mathfrak{g})$.
==========================================================================
First of all let’s consider the classical analogs of the argumrnts which will be leaded below. Classical $r-$matrix $r(u,v)$ of the classical double\
$(\mathfrak{g}((u^{-1}))), u^{-1}\mathfrak{g}[[u^{-1}]], \mathfrak{g}[u])$ of the current algebra $\mathfrak{g}[u]$ has next form: $r(u,v)= \sum_{i,k} e_{i,k} \otimes e^{i,k}$, where $\{e_{i,k} = e_i u^k\}, \{e^{i,k} = e^i u^{-k-1}\}$ are the dual bases in the $\mathfrak{g}[u]), u^{-1}\mathfrak{g}[[u^{-1}]]$, respectively, with relate to pairing $$<f,g>= res(f(u), g(u)),$$ where $(\cdot, \cdot)$ be an invariant bilinear form on $\mathfrak{g}$ and $\{e_i\}, \{e^i\}$ are dual bases in $\mathfrak{g}$ relative to this form.
It is easy to see that\
$r = \sum_i \sum_{k=0}^{\infty}e_i\cdot u^k \otimes e^i\cdot
v^{-k-1} = \sum_{k+0}^{\infty} \sum_i e_i \otimes e^i \cdot
v^{-1}(\frac{u}{v})^k = (u<1<v) = \sum_i e_i \otimes e^i
\frac{v^{-1}}{1 - u/v} = \frac{\mathfrak{t}}{v-u}, $\
where $\mathfrak{t} = \sum_i e_i \otimes e^i$ be an Casimir operator of universal enveloping superalgebra $U(\mathfrak{g})$ of Lie superalgebra $\mathfrak{g}=
A(m,n)$. Thus we have that $$r = \frac{\mathfrak{t}}{v-u}$$ Note that this classical $r$-matrix don’t belong to $\mathfrak{g}[t]^{\otimes2}$. Let’s introduce shift operator $T_{\lambda}: f(u) \rightarrow f(u+\lambda)$. Let’s act by operator $id \otimes T_{\lambda}$ on $r$. We derive\
$(id \otimes T_{\lambda})r(u,v) = \frac{\mathfrak{t}}{\lambda -
(u-v)} = \frac{\mathfrak{t}}{\lambda(1 - \lambda^{-1}(u-v))} = \\
\sum_{k=0}^{\infty} \mathfrak{t}(u-v)^k \lambda^{-k-1} =
\sum_{k=0}^{\infty}r_k \lambda^{-k-1}, $\
where $r_k \in \mathfrak{g}[t]^{\otimes2}$.
We’ll derive this arguments another equivalent way in order that to do the analogy with quantum case more evident.\
$(id \otimes T_{\lambda})r(u,v)=
\sum_i \sum_{k=0}^{\infty}e_i\cdot u^k \otimes e^i\cdot
(v+\lambda)^{-k-1} = \\
\sum_i \sum_{k=0}^{\infty}e_i\cdot u^k \otimes e^i\cdot
\frac{1}{(\lambda(1 - (-v/\lambda)))^{k+1}} =
\\ \sum_i \sum_{k=0}^{\infty}\sum_{m=0}^{\infty}e_i\cdot u^k \otimes e^i\cdot
(-1)^m C^k_{m+k}v^m\lambda^{-m-k-1}= (n=m+k) = \\
\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\sum_i (-1)^{n-k}C^k_n e_iu^k\otimes e^i
v^{n-k}\lambda^{-n-1}=
\sum_{k=0}^{\infty}\sum_i e_i \otimes e^i(u-v)^n \lambda^{-n-1}$
Let’s try to repeat this argument in the quantum case keeping in mind that Yangian be a quantization of the bisuperalgebra of polynomial current $\mathfrak{g}[t]$, Yangian Double be a quantization of classical double $\mathfrak{g}((t))$ and universal $R$-matrix of Yangian Double be a quantum analog of classical $r$-matrix $r$, and above considered $r$-matrix $(id \otimes T_{\lambda})r(u,v)$ is that classical analog of the universal $R$- matrix of Yangian wgich we are going to compute.
Let’s define homomorphism $T_{\lambda}$ in the quantum case $$T_{\lambda}: Y(\mathfrak{g}) \rightarrow Y(\mathfrak{g})$$ by formulas: $T_{\lambda}(x)=x$ for $x \in \mathfrak{g}$, $T_{\lambda}(a_{i,1}) = a_{i,1} + \lambda a_{i,0}$ for $a \in \{e,f,h\}$.
[**Proposition 4.1.**]{} The action of $T_{\lambda}$ on the generators $a_{i,n}, a \in \{e,f,h\}, n \in Z,$ of Yangian Double $DY(\mathfrak{g})$ is defined by next formulas: $$\begin{aligned}
T_{\lambda}a_{i,n} = \sum_{k=0}^n C^k_n a_{i,k}\lambda^{n-k}, n \in Z_+ \label{4.6} \\
T_{\lambda}a_{i,-n} = \sum_{k=0}^{\infty}(-1)^k C^{n-1}_{k+n-1} a_{i,k}\lambda^{-n-k-1}, n \in N \label{4.7}\end{aligned}$$
[*Proof.*]{} First of all note that the same formulas define action of $T_{\lambda}$ in classical case also. Let $\tilde{h}_{i,1} = h_{i,1} - \frac{1}{2} h_{i,0}^2$. Then it is fulfilled next relations in the Yangian Double $DY(\mathfrak{g})$. $$\begin{aligned}
&[\tilde{h}_{i,1}, e_{j,n}]=a_{ij}e_{i,n+1}, [\tilde{h}_{i,1}, f_{j,n}]= -a_{ij}f_{i,n+1}, \label{4.8}\\
&[e_{i,k}, f_{j,m}]=\delta_{ij}h_{j, k+m}. \label{4.9} \quad\end{aligned}$$ Sufficiently to check this formulas for generators $e_{j,n}, f_{j,n}$. Because the relations (\[4.7\]), (\[4.8\]) for generators $h_{j,n}$ imply for these relations for generators $e_{j,n}, f_{j,n}$ and formula \[4.9\]. Let’s prove the relations (\[4.7\]), (\[4.8\]) for generators $e_{j,n}$. For generators $f_{j,n}$ the arguments are the same. Let’s prove the formula (\[4.7\]). For $n=0$ formula (\[4.7\]) is true on definition. Let this formula is true for all $k \le n$. Let’s prove that this formula is true for $k= n+1.$ Let $j$ such that $a_{j,i} \ne 0$. Let’s note that $T_{\lambda}$ be a homomorphism. Then\
$T_{\lambda}(e_{i,n+1})=T_{\lambda}(a_{ji}^{-1}[\tilde{h}_{j,1}, e_{i,n}])=
a_{ji}^{-1}[T_{\lambda}\tilde{h}_{j,1}, T_{\lambda}e_{j,n+1}]=\\
a_{ji}^{-1}[\tilde{h}_{j,1} + \frac{1}{2}h_{j,0}, \sum_{k=0}^n C^k_n e_{i,k}\lambda^{n-k}]= \\
a_{ji}^{-1}(\sum_{k=0}^n C^k_n[\tilde{h}_{j,1}, e_{i,k}]\lambda^{n-k} +\sum_{k=0}^n C^k_n[h_{j,0}, e_{i,k}]\lambda^{n+1-k})=\\
\sum_{k=0}^{n+1} C^k_{n+1}e_{i,k}\lambda^{n+1-k}.$\
Formula (\[4.7\]) is proved. Let’s prove the formula (\[4.8\]). Let $n=1$. Then $a_{ji}^{-1}[\tilde{h}_{j,1}, e_{i,-1}]= e_{i,0}$. Let’s act on the left-hand and right-hand sides by the operator $T_{\lambda}$. First let’s act on the left-hand side. We have\
$a_{ji}^{-1}[T_{\lambda}(\tilde{h}_{j,1}), T_{\lambda}(e_{i,-1})]=
a_{ji}^{-1}[\tilde{h}_{j,1} + \frac{1}{2}h_{j,0}, \sum_{k=0}^{\infty}(-1)^k C^0_k e_{i,k}\lambda^{-k-2}]=\\
a_{ji}^{-1}[\tilde{h}_{j,1} + \frac{1}{2}h_{j,0}, \sum_{k=0}^{\infty} e_{i,k}\lambda^{-k-2}]=\\
\sum_{k=0}^{\infty}(-1)^k e_{i,k+1}\lambda^{-k-2} + \sum_{k=0}^{\infty}(-1)^k e_{i,k}\lambda^{-k-1} =\\
e_{i,0} + \sum_{k=0}^{\infty}((-1)^k +(-1)^{k+1})e_{i,k+1}\lambda^{-k-2}= e_{i,0}$\
As the right-hand side don’t change under the action $T_{\lambda}$ on definition then we have that left-side and right-side hands are coincide. Formula for $n=1$ is verifyed. Let formula is proved for all $k \le n$. Let’s prove it for $k=n+1$. Let’s act as above by operator $T_{\lambda}$ on left-hand and right-hand sides of formula $a_{ji}^{-1}[\tilde{h}_{j,1}, e_{i,-n-1}]=e_{i,-n}$. Let’s act on the left-hand side. $a_{ji}^{-1}[T_{\lambda}\tilde{h}_{j,1}, T_{\lambda}e_{i,-n-1}] =
a_{ji}^{-1}[\tilde{h}_{j,1} + \frac{1}{2}h_{j,0}, \sum_{k=0}^{\infty}(-1)^k C^{n}_{k+n} e_{i,k}\lambda^{-n-k-2}] =
\sum_{k=0}^{\infty}(-1)^k C^{n}_{k+n} e_{i,k+1}\lambda^{-n-k-2} + \sum_{k=0}^{\infty}(-1)^k C^{n}_{k+n} e_{i,k}\lambda^{-n-k-1}=\\
e_{i,0} + \sum_{k=0}^{\infty}(-1)^{k+1} (-C^{n}_{k+n}+C^{n}_{k+n+1})e_{i,k}\lambda^{-n-k-2}= \\
\sum_{k=0}^{\infty}(-1)^k C^{n-1}_{k+n-1} e_{i,k}\lambda^{-n-k-1}= T_{\lambda}e_{i,-n}$\
We have that left-hand side is equal to right-hand side, and therefore the formula (\[4.8\]) for all natural numbers $n$ is proved by induction. Proposition is proved.\
[*Remark* ]{} Note that series defining the value of operator $T_{\lambda}$ on generators $a_{i,-n}$ converges for enough large values of $\lambda$.
Now we can to calculate the universal $R$-matrix ${\it R}(\lambda)$ of Yangian $Y(\mathfrak{g})$ by formula: $${\it R}(\lambda) = (id \otimes T_{-\lambda})R,$$ where $R$ be an universal $R$-matrix of double $DY(\mathfrak{g})$. As $R= R_+ R_0 R_-$, then acting by operator $id \otimes T_{\lambda}$ on $R$ and using the fact that $T_{\lambda}$ be a homomorphism and therefore $id \otimes T_{\lambda}$ be a homomorphism also, we have, $${\it R}(\lambda)= {\it R}_+(\lambda) {\it R}_0(\lambda) {\it R}_-(\lambda),$$ where ${\it R}_+(\lambda)= (id \otimes T_{-\lambda})R_+, {\it R}_0(\lambda)= (id \otimes T_{-\lambda})R_0,
{\it R}_-(\lambda)= (id \otimes T_{-\lambda})R_-.$
Note that $${\it R}(\lambda)= 1 + \sum_{k=0}^{\infty}{\it R}_k \lambda^{-k-1},$$ where $1$ be an unit element in $Y(\mathfrak{g})^{\otimes 2}$, ${\it R}_k \in Y(\mathfrak{g}) \otimes Y(\mathfrak{g})$. Such form is not enough suitable as coefficients ${\it R}_k$ have heavy visible form. Becouse final result let’s present in other more visible form. Let’s act by operator $id \otimes T_{-\lambda}$ on right-hand side of \[3.3\]. We have $${\it R}_+(\lambda) = \overrightarrow{\prod}_{\beta \in \Xi_+} \exp(-(-1)^{\theta(\beta)} a(\beta) e_{\beta} \otimes T_{-\lambda} e_{-\beta}), \label{4.12}$$
Let’s calculate separately element $T_{-\lambda} e_{-\beta}$. As $\beta = \beta' + n \delta$, then in view \[4.7\] we have $${\it R}_-(\lambda) = \overleftarrow{\prod}_{\beta \in \Xi_-} \exp(-(-1)^{\theta(\beta)} a(\beta) (\sum_{k=0}^{\infty}(-1)^k C^{n-1}_{k+n-1}(e_{\beta}\otimes e_{-\beta +(n+m)\delta})\lambda^{-n-k-1}), \label{4.13}\\$$
Similarly it is calculated element ${\it R_-}(\lambda)$. Summarizing stated above we get
[**Proposition 4.2.**]{} Terms ${\it R_+}(\lambda),
{\it R_-}(\lambda)$ of universal $R$-matrix of Yangian have the next form $$\begin{aligned}
&{\it R}_+(\lambda) = \overrightarrow{\prod}_{\beta \in \Xi_+} \exp(-(-1)^{\theta(\beta)} a(\beta) (\sum_{k=0}^{\infty}(-1)^k C^{n-1}_{k+n-1}(e_{\beta}\otimes e_{-\beta +(n+m)\delta})\lambda^{-n-k-1}),\qquad \label{4.14}\\
&{\it R}_-(\lambda) = \overleftarrow{\prod}_{\beta \in \Xi_-} \exp(-(-1)^{\theta(\beta)} a(\beta) (\sum_{k=0}^{\infty}(-1)^k C^{n-1}_{k+n-1}(e_{\beta}\otimes e_{-\beta +(n+m)\delta})\lambda^{-n-k-1}),\qquad \label{4.15}\end{aligned}$$
[*Example 4.1.*]{} Let’s consider example of calculation of terms ${\it R_+}(\lambda),
{\it R_-}(\lambda)$ in the case of the simple Lie algebra $\mathfrak{sl}_2$. Then\
${\it R}_+(\lambda)= \overrightarrow{\prod}_{n \ge 0}
\exp(-e_n \otimes T_{\lambda}(f_{-n-1} = \\
\overrightarrow{\prod}_{n \ge 0} \exp((-1)^n \sum_{m=0}^{\infty}
C^n_{m+n} e_n \otimes f_m \lambda^{-n-m-1}) = \\
\exp(\sum_{n=0}^{\infty} \sum_{m=0}^{\infty}C^n_{m+n}(-1)^n e_n \otimes f_m
\lambda^{-n-m-1})= \exp(\sum_{k=0}^{\infty} (\sum_{m=0}^{k}C^n_{k}(-1)^n e_n \otimes
f_{k-n})\lambda^{-k-1})= \\
\overrightarrow{\prod}_{n \ge 0} \exp((\sum_{m=0}^{k}C^n_{k}(-1)^n e_n \otimes
f_{k-n})\lambda^{-k-1})$\
Similarly it can be calculated the term ${\it R_-}(\lambda)$. Thus we get the next formulas $$\begin{aligned}
&{\it R}_+(\lambda)= \overrightarrow{\prod}_{n \ge 0} \exp((\sum_{m=0}^{k}C^n_{k}(-1)^n e_n \otimes
f_{k-n})\lambda^{-k-1}), \quad\\
&{\it R}_-(\lambda)= \overleftarrow{\prod}_{n \ge 0} \exp((\sum_{m=0}^{k}C^n_{k}(-1)^n f_n \otimes
e_{k-n})\lambda^{-k-1}), \quad\end{aligned}$$
Let $ord(\beta):= n$, if $\beta = \beta' + n \delta,
\beta' \in \Delta_+(\mathfrak{g})$. Then the proposition 4.2 we can rewrite in the next form $$\begin{aligned}
&{\it R}_+(\lambda) =
\overrightarrow{\prod}_{\beta \in \Xi_+} \exp(-(-1)^{\theta(\beta)}
a(\beta) (\sum_{k=0}^{\infty} (-1)^k \quad \nonumber\\
&C^{ord(\beta)-1}_{k + ord(\beta)-1}(e_{\beta} \otimes
e_{-\beta +(ord(\beta)+k)\delta})\lambda^{-ord(\beta)-k-1})),\quad \label{4.16}\\
&{\it R}_-(\lambda) = \overleftarrow{\prod}_{\beta \in \Xi_-} \exp(-(-1)^{\theta(\beta)} a(\beta)
(\sum_{k=0}^{\infty} \quad \nonumber\\
&(-1)^k C^{ord(\beta)-1}_{k + ord(\beta)-1}(e_{\beta}\otimes e_{-\beta +(ord(\beta)+k)\delta})\lambda^{-ord(\beta)-k-1}), \label{4.17}\end{aligned}$$
Let’s calculate the term ${\it R}_0(\lambda)$. For this it is required next
[**Proposition 4.3.**]{} Shift operator acts on generating function of Cartan generators in the following way $$T_{\lambda}(h^-_i(u)) = h^+_i(u + \lambda)$$
[*Proof.*]{} $T_{\lambda}(h^-_i(u)) = T_{\lambda}(1- \sum_{k=0}^{\infty}h_{i, -k-1}u^k)
= 1 - \sum _{k=0}^{\infty}T_{\lambda}(h_{i, -k-1})u^k =
1 - \sum_{k=0}^{\infty}\sum_{m=0}^{\infty}
C^k_{m+k}h_{i,m})\lambda^{-m-k-1}u^k = 1 + \sum_{m=0}^{\infty}
h_{i,m}(\lambda^+ u)^{-m-1} = h^+_i(u+ \lambda).$ Proposition is proved.\
[**Corollary 4.2.**]{} $T_{\lambda}(\varphi^-_i(u)) = \varphi^+_i(u+ \lambda)$.
[*Proof.*]{} $T_{\lambda}(\varphi^-_i(u))=
T_{\lambda}(\ln(h^-_i(u)))= \ln(T_{\lambda}(h^-_i(u)))= \ln(h^+_i(u+
\lambda))= \varphi^+_i(u+ \lambda)$.
Now we can calculate the term ${\it R}_0(\lambda)$.
[**Proposition 4.4.**]{} Term ${\it R}_0(\lambda)$ has next form $${\it R}_0(\lambda) =
\prod_{n \ge 0} \exp(\sum_{i,j \in I} \sum_{k \ge 0} ((\phi^+_i(u))')_k \otimes
c_{ji}(T^{-\frac{1}{2}})(\phi^+_j(v+(n+\frac{1}{2})l(\mathfrak{g})+\lambda))_{k})
\label{4.18}$$
[*Example 4.2.*]{} In the case of the simple Lie algebra $\mathfrak{sl}_2$ formula \[4.18\] admit the next simple form $${\it R}_0(\lambda) =
\prod_{n \ge 0} \exp(-\sum_{k \ge 0}((\phi^+(u))')_k \otimes
(\phi^+(v-2n-1+\lambda))_k$$
We can present results of this section in the form of the next theorem.
[**Theorem 4.4.**]{} Universal R-matrix of Yangian $Y(A(m,n)$ has the next form $${\it R}(\lambda)= {\it R}_+(\lambda) {\it R}_0(\lambda) {\it R}_-(\lambda),$$ where terms ${\it R}_+(\lambda), {\it R}_0(\lambda), {\it R}_-(\lambda)$ is described by, respectively, formulas (\[4.16\]), (\[4.17\]), (\[4.18\]).
Author are thankful to S.M. Khoroshkin for useful discussions of this work.
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Math.Dep., Don State Technical University, Gagarin sq.,1, Rostov-na-Donu, 344010, Russia.
|
---
author:
- 'F. Bouchy, F. Pont, N.C. Santos,C. Melo, M. Mayor, D. Queloz'
- 'S. Udry[^1]'
date: 'Received ; accepted '
title: |
Two new ”very hot Jupiters”\
among the OGLE transiting candidates
---
Introduction
============
Since 1995, more than 120 planetary candidates have been discovered, mainly by radial velocity surveys. Although they provide a lot of information concerning the orbital parameters and the host star properties, such surveys do not yield the real mass of the planet (only $m\sin i$) and do not give any information about its size. The discovery of HD209458 by Doppler measurements (Mazeh et al. [@mazeh]) and photometric transit (Charbonneau et al. [@charbonneau]; Henry et al. [@henry]) led to the first complete characterization of an extra-solar planet, illustrating the real complementarity of the two methods. The OGLE survey (Optical Gravitational Lensing Experiment) announced the detection of 137 short-period multi-transiting objects (Udalski et al., [@udalski1], [@udalski2], [@udalski3], [@udalski4]). Recently Konacki et al. ([@konacki1], [@konacki2]), thanks to a Doppler follow-up, announced the characterization of the first extra-solar planet with the unexpected extremely short period of 1.2 day, much below the lower end of the period distribution of planets detected by Doppler surveys (Udry et al. [@udry]). We present in this letter two new cases of extra-solar planets with very short orbital periods: OGLE-TR-113 and OGLE-TR-132.
Observations and reductions
===========================
The FLAMES facilities on the VLT (available since march 2003, Pasquini et al. 2002) is a very efficient way to conduct the Doppler follow-up of OGLE candidates. FLAMES is a multi-fiber link which feeds into the spectrograph UVES up to 7 targets on a field-of-view of 25 arcmin diameter in addition to the simultaneous thorium calibration. The fiber link produces a stable illumination at the entrance of the spectrograph and permits the use of simultaneous ThAr calibration in order to track instrumental drift. Forty-five minutes on a $17^{th}$ magnitude star yield a signal-to-noise ratio (SNR) of about 8, corresponding to a photon noise uncertainty of about 30 on a non-rotating K dwarf star. We have obtained 3.2 nights in visitor mode in March 2004 on this instrument (program 72.C-0191) in order to observe all OGLE candidates of the Carina field suspected to have a planetary companion. We present here the results and the analysis of three of these candidates, OGLE-TR-113, OGLE-TR-131 and OGLE-TR-132.
The spectra obtained from the FLAMES+UVES spectrograph were extracted using the standard ESO-pipeline with bias, flat-field and background correction. Wavelength calibration was performed with ThAr spectra. The radial velocities were obtained by cross-correlation with a G2 digital mask. The instrumental drift was computed by cross-correlation of the simultaneous ThAr spectrum with a Thorium mask. Radial velocity errors were computed as a function of the SNR of the spectrum and the width ($FWHM$) of the Cross-Correlation Function (CCF) through the following relation based on photon noise simulations: $\sigma_{RV}=\,0.025\,FWHM / \,SNR$.
However, our measurement are clearly not photon noise limited and we added quadratically an error of 35 in order to take into the account systematic errors probably due to wavelength calibration errors, fiber-to-fiber contamination, and residual cosmic rays. This value is based on our experience with FLAMES+UVES and are confirmed by the velocity residuals for OGLE-TR-131.
Results
=======
Radial velocities
-----------------
Our radial velocity measurements are listed in Table \[tablevr\]. Figure \[doppler\] shows the radial velocity data phased with the period and transit epoch from Udalski et al. ([@udalski3], [@udalski4]). If the radial velocity variations are caused by the transiting objects, then phase $\phi=0$ must correspond to the passage of the curve at center-of-mass velocity with decreasing velocity, which provides a further constraint. We fitted the data with a sinusoid (assuming a circular orbit) and determined the velocity semi-amplitude $K$ and the center-of-mass velocity $V_0$. The orbital parameters are reported in Table \[tablespectro\].
-------------------- ---------- ------- ------ ------ ---------------
BJD RV depth FWHM SNR $\sigma_{RV}$
[\[$-$2453000\]]{} \[\] \[%\] \[\] \[\]
OGLE-113
78.60419 $-$7.862 40.23 9.4 11.0 0.041
79.64235 $-$8.268 40.07 9.5 11.7 0.040
80.65957 $-$7.931 39.73 9.6 10.2 0.042
81.59492 $-$7.858 38.94 9.5 9.3 0.043
82.71279 $-$8.077 40.19 9.5 11.5 0.041
83.66468 $-$8.098 39.60 9.4 10.8 0.041
84.65149 $-$7.574 40.91 9.4 12.0 0.040
85.60720 $-$8.027 40.32 9.4 11.5 0.041
OGLE-131
78.57421 18.902 20.40 11.0 3.2 0.093
79.69280 18.944 36.18 10.5 7.5 0.049
80.69588 18.910 33.25 10.8 6.0 0.057
81.72914 19.001 32.90 10.2 5.8 0.056
82.64055 19.066 31.21 10.6 4.9 0.064
83.70039 18.954 33.55 10.1 6.2 0.054
84.61585 19.016 27.87 10.3 4.5 0.067
85.64254 18.879 28.34 10.6 4.6 0.067
OGLE-132
81.72913 39.724 30.90 10.0 9.0 0.045
82.64054 39.700 29.41 10.3 7.6 0.049
83.70038 39.564 31.09 10.0 9.6 0.044
84.61585 39.822 30.56 10.3 8.2 0.047
85.64254 39.493 30.66 10.1 8.7 0.045
-------------------- ---------- ------- ------ ------ ---------------
: Radial velocity measurements (in the barycentric frame) and CCF parameters for OGLE-TR-113, 131 and 132.[]{data-label="tablevr"}
The existence of an orbital signal for OGLE-TR-113 is clear. For OGLE-TR-132, the reduced $\chi^2$ of a constant velocity curve without orbital motion is 33.1 ($P(\chi^2)\sim 10^{-6}$). Even if unrecognized systematics caused our error bars to be underestimated by a factor 1.5, the reduced $\chi^2$ would still be 14.7 ($P(\chi^2)=0.5$%). As a foolproof check of the detection confidence, we also applied a bootstrap procedure to the data. Bootstrapping gives an estimation of the significance of a signal without any assumption on the size of the uncertainties (see e. g. Press et al. 1992). This yields a positive value of K in 97% of the cases. Therefore, even with the assumption of large unrecognized systematics, the detection of orbital motion for OGLE-TR-132 on the correct period and phase is robust.
------------- ------------ ------------- -------------------- -------------------- ------ ---------------- --------------------- ---------------
Name P$_{OGLE}$ T0$_{OGLE}$ K V$_0$ O-C $T_{eff}$ $\log g$ \[Fe/H\]
\[days\] $-$2452000 \[\] \[\] \[\]
OGLE-TR-113 1.4325 324.36394 0.287$\pm$0.042 $-$7.944$\pm$0.027 66 4752$\pm$130 K 4.50$\pm$0.53 0.14$\pm$0.14
OGLE-TR-131 1.8699 324.94513 $-$0.026$\pm$0.049 18.964$\pm$0.023 56 5244$\pm$136 K 3.30$\pm$0.73 0.11$\pm$0.20
OGLE-TR-132 1.68965 324.70067 0.141$\pm$0.042 39.676$\pm$0.032 53 6411$\pm$179 K 4.86$\pm$0.14$^{a}$ 0.43$\pm$0.18
------------- ------------ ------------- -------------------- -------------------- ------ ---------------- --------------------- ---------------
Spectral line bisectors and blend scenarios
-------------------------------------------
It is known that in certain circumstances, the combination of a single star with a background unresolved eclipsing binary can mimic both a planet transit signal and velocity variations (Santos et al. [@santos2]). If the velocity variation are caused by a background binary system, however, the CCF bisector is expected to vary. In order to examine the possibility that the radial velocity variation is due to a blend scenario, we computed the CCF bisectors as described by Santos et al. ([@santos2]). Figure \[bisspan\] indicates that there is no correlation of the line asymmetries with phase. Furthermore the CCF was computed with different masks without significant change in the radial velocity value (as discussed by Santos et al. 2002, most blend scenarios produce mask-dependent velocities). Moreover, a background binary of such short period would be expected to be synchronized, and thus to show a cross-correlation function very broadened by rotation. Simulations show that a broad background CCF contaminating a narrow foreground CCF is very inefficient in causing an apparent velocity variation. In order to provoke variations of the observed amplitude, any broadened background CCF would have to be large enough to be clearly visible in the total CCF, which is not the case. Therefore, the scenario “foreground single star plus background short-period eclipsing binary” can be eliminated with confidence. While other more intricate scenarios could conceivably be possible, we were not able to contrive any that could explain both the photometric and the velocity signals while remaining credible.
Spectral classification
-----------------------
On the summed spectra, the intensity and equivalent width of some spectral lines were analyzed to give temperature, gravity and metallicity estimates for the primaries in the manner described in Santos et al. ([@santos3]). The results are given in Table \[tablespectro\].
light curve analysis and physical parameters
--------------------------------------------
The shape of the transit light curve depends in a non-linear way on the latitude of the transit, the radius ratio $r/R$, the sum of masses $m+M$ and the primary radius $R$, where $R$, $M$, $r$, $m$ are the radius and mass values for the eclipsed and eclipsing bodies respectively. Synthetic transit curves computed with the procedure of Mandel & Agol ([@mandel]) were fitted to the photometry data by least-squares. A quadratic limb darkening with $u1=u2=0.3$ was assumed (based on the values for HD209458 used by Brown et al. [@brown]). Some possible sources of systematic uncertainties are not taken into account in this letter: variations in the limb darkening coefficients, possible contamination by background stars, uncertainties in the stellar evolution predictions, uncertainties in the orbital period. They will be included in the paper presenting the complete spectroscopic follow-up (Pont et al., in preparation).
The constraints on M and R were combined using a Bayesian procedure similar to that described in Pont & Eyer ([@pont]): the posterior probability distribution of $M$ and $R$ was computed as the product of the combined likelihood from the light curve fit and the spectroscopic determinations of $T_{eff}$, $\log g$ and \[Fe/H\], and a prior probability distribution obtained from the Padua stellar evolution models (Girardi et al. 2002). The value of $r$ was then derived from $r/R$ and $R$, and $m$ from $M$ and the semi-amplitude of the radial velocity orbit, assuming circular Keplerian motion with $\sin i=1$ (The shortness of the period ensures that the orbits are circularized, and the presence of a transit indicates that $\sin i$ is very near to unity). Table \[tablezoo\] summarize the resulting physical parameters of OGLE-TR-113 and OGLE-TR132 and their planetary companions.
Discussion and Conclusion
=========================
[c c c c c]{} Name& M & R & m & r\
& \[$M_\odot$\] & \[$R_\odot$\]& \[$M_{Jup}$\]& \[$R_{Jup}$\]\
\
OGLE-113 & 0.77$\pm$0.06 & 0.765$\pm$0.025 & 1.35$\pm$0.22 & 1.08$^{+0.07}_{-0.05}$\
OGLE-132 & 1.34$\pm$0.10 & 1.41$^{+0.49}_{-0.10}$ & 1.01$\pm$0.31 & 1.15$^{+0.80}_{-0.13}$\
The parameters of OGLE-TR-113b are very accurately defined, first because the transit shape is clearly delineated by the photometric data, second because there is only a narrow range of possible parameters allowed by stellar evolution models for a cool K dwarf. As a result, $r$ could be computed with a very small formal uncertainty.
No radial velocity variations were detected in OGLE-TR-131. Moreover, the spectroscopy indicates that it is most probably a sub-giant ($\log g=3.30\pm 0.73$), which renders the existence of a close companion unlikely. OGLE-TR-131 is included here to show that our estimates of the radial velocity uncertainties are coherent with the residuals in the absence of detectable orbital motion.
The photometric transit signal of OGLE-TR-132 is near the detectability limit, but the existence of a radial velocity variation at the precise period and phase of the transit gives confidence in the reality of the transiting companion. In contrast with OGLE-TR-113, the parameters are not very well constrained. The transit shape is too poorly defined to constrain the transit latitude, so that there is a strong degeneracy between the impact parameter $b$ and the primary radius $R$. Furthermore the values of temperature and gravity measured from the spectra are compatible with a wide variety of young to evolved F dwarfs, with radii ranging from 1.3 to 1.9 $R_\odot$. As a result, the upper uncertainty interval on $r$ is wide, as indicated by the dotted error line in Fig. \[massradius\].
The semi-major axis of OGLE-TR-113b and OGLE-TR-132b are $a=0.0228 \pm 0.0006$ AU and $a=0.0306 \pm 0.0008$ AU respectively.
The detection of OGLE-TR-113b and OGLE-TR-132b show that the case of OGLE-TR-56 is not isolated and that “very hot Jupiters” (i.e. Jovian exoplanets with periods much smaller than 3 days) are not extremely uncommon. Therefore, the accumulation of hot Jupiters near periods of 3 days (Udry et al. [@udry]) does not reflect an absolute limit for the existence of planets. The parent stars of OGLE-TR-56, 113 and 132 are very different, ranging from F to K dwarfs, indicating that very hot Jupiters are possible around different type of stars.
Fig. \[massradius\] gives the mass-radius relation for the four known transiting exoplanets. It is noteworthy that the three OGLE objects do not seem to have such an inflated radius as HD209458b, despite their much shorter periods.
The two new detections bring to 3 the number of “very hot Jupiters” detected in the OGLE transit survey, among 155’000 light curves examined for transits. Given the geometric probability of transit ($\sim $17% for $P=1.5$ days), this results implies a total number of 21 $\pm$ 10 very hot Jupiters among the targets. The detection completeness of the OGLE survey toward transiting very hot Jupiter should be computed with detailed simulations, but it may be quite high because for such low values of period the phase coverage is very good. Assuming a detection probability of 50%, then the total number is $42 \pm 20$, therefore one in every 2500 to 7000 targets. Even if there is a proportion of giants stars among the OGLE candidate that would have been weeded out of the radial velocity surveys, the absence of “very hot Jupiters” among the $\sim$3000 field dwarfs surveyed in radial velocity in the solar neighborhood is therefore not incompatible with our result. This estimate also indicates that “very hot Jupiters” are not out of reach of future radial velocity surveys. It is noteworthy though that for such low periods, photometric transit surveys are a more efficient detection method than radial velocity monitoring.
We are grateful to C. Moutou for very useful comments and J. Smoker for support at Paranal. The data presented herein were obtained as part of an ESO visitor mode run (program 72.C-0191). F.P. gratefully acknowledges the support of CNRS through the fellowship program of CNRS. Support from Fundação para a Ciência e Tecnologia (Portugal) to N.C.S. in the form of a scholarship is gratefully acknowledged.
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[^1]: Based on observations collected with the FLAMES+UVES spectrograph at the VLT/UT2 Kueyen telescope (Paranal Observatory, ESO, Chile)
|
=0.0in =0.15in =6ex =2ex
plus 0.2pt minus 0.2pt plus 0.2pt minus 0.2pt
Michael Martin Nieto[^1]\
[*Theoretical Division, Los Alamos National Laboratory\
University of California\
Los Alamos, New Mexico 87545, U.S.A.\
*]{} and\
[*Abteilung für Quantenphysik\
Universität Ulm\
D-89069 Ulm, GERMANY\
*]{}
D. Rodney Truax[^2]\
[*Department of Chemistry\
University of Calgary\
Calgary, Alberta T2N 1N4, Canada\
*]{}
[ABSTRACT]{}
=.33in
> As an aid to understanding the [*displacement operator*]{} definition of squeezed states for arbitrary systems, we investigate the properties of systems where there is a Holstein-Primakoff or Bogoliubov transformation. In these cases the [*ladder-operator or minimum-uncertainty*]{} definitions of squeezed states are equivalent to an extent displacement-operator definition. We exemplify this in a setting where there are operators satisfying $[A, A^{\dagger}] = 1$, but the $A$’s are not necessarily the Fock space $a$’s; the multiboson system. It has been previously observed that the ground state of a system often can be shown to to be a coherent state. We demonstrate why this must be so. We close with a discussion of an alternative, effective definition of displacement-operator squeezed states.
=.33in
Introduction
============
As has now been known and studied for some time, there are three equivalent, widely-used definitions of the coherent states of the harmonic oscillator [@sch]-[@2]. These are (1) the minimum-uncertainty, (2) annihilation- (or, more generally, ladder-) operator, and (3) displacement-operator methods. These methods have been extended to the squeezed states of the harmonic oscillator. Further, with one exception, general coherent and squeezed states have been obtained for general systems by these three methods. That exception is a general definition of squeezed states by the displacement-operator method.
With an aim towards understanding a general method, we can study systems where such a definition works. Specifically, after reviewing the coherent and squeezed states for the harmonic oscillator and more general systems, we focus on why displacement-operator squeezed states often can not be obtained by a naive generalization of the harmonic-oscillator case: this is when there is, in general, no Bogoliubov transformation.
This problem does not exist in certain systems. In particular, we here study the multiboson formalism of Brandt and Greenberg [@brandt], where the multi-boson operators obey canonical commutation relations, and hence one can proceed with calculations in the standard way. Elsewhere [@II], we will study time-dependent systems which have isomorphic symmetry algebras.
We also explain the property of these various definitions of squeezed and coherent states which is that the ground state is a member of the set of coherent states. In closing, we discuss an alternative, effective method for defining displacement-operator squeezed states.
The Coherent and Squeezed States of the Harmonic Oscillator
===========================================================
Coherent states
---------------
Given the canonical commutation relations $$[a,~a^{\dagger}] = 1~, ~~~~~~ [a,~ a] = 0~, \label{coma}$$ where we adopt the realization $$a=\frac{1}{\sqrt{2}}(x+ip),~~~a^{\dag} = \frac{1}{\sqrt{2}}(x-ip).\label{adef}$$ The definitions of displacement-operator and ladder-operator coherent states are well known. They are $$D(\alpha)|0\rangle = |\alpha\rangle \label{docs}$$ and $$a|\alpha\rangle = \alpha|\alpha\rangle~, \label{locs}$$ where $$D(\alpha) = \exp[\alpha a^{\dagger} - \bar{\alpha} a]
=\exp\left[-\frac{1}{2}|\alpha|^2\right]
\exp[\alpha a^{\dagger}] \exp[-\bar{\alpha} a] \label{D}$$ and $$|\alpha\rangle =
\exp\left[-\frac{1}{2}|\alpha|^2\right] \sum_{n}
\frac{\alpha ^n}{\sqrt{n!}}
|n\rangle ~.$$ The last equality in Eq. (\[D\]) comes from using a Baker-Campbell-Hausdorff relation. Observe that the definition (\[locs\]) follows from the definition (\[docs\]) by $$[a,~D(\alpha)] = \alpha D(\alpha)~.$$
The coherent-state wave functions are ($m\omega/\hbar \rightarrow 1$) $$\psi_{cs}(x) = \pi^{-1/4}\exp\left[-\frac{(x-x_0)^2}{2}+ip_0x\right]~,
\label{psi}$$ $$x_0 = \langle x\rangle~,
~~~~ p_0 =\langle p\rangle~,$$ $$Re(\alpha)=x_0/2^{1/2}~,
\hspace{.5in} Im(\alpha)=p_0/2^{1/2}~.$$ That is, the states are Gaussians with the width being that of the ground state.
Squeezed states
---------------
Squeezed states [@ss1]-[@ss5] can be defined by the displacement-operator method as the product of a unitary displacement operator and a unitary squeeze operator acting on the ground state: $$D(\alpha)S(z)|0\rangle \equiv |\alpha,z\rangle~,
~~~~~z = z_1+iz_2 = re^{i\theta}~.$$ $\theta$ is a phase which defines the starting time, $t_0 =
(\theta/2\omega)$. $S(z)$ is given by $$\begin{aligned}
S(z) & = & \exp\left[{\frac{1}{2}}za^{\dagger}a^{\dagger} -
{\frac{1}{2}}\bar{z}aa\right]
\label{a} \\
& = & \exp\left[{\frac{1}{2}}e^{i\theta}(\tanh r)
a^{\dagger}a^{\dagger}\right]
\left({\frac{1}{\cosh r}}\right)^{({\small{\frac{1}{2}}+a^{\dagger}a})}
\exp\left[-{\frac{1}{2}}e^{-i\theta}(\tanh r)aa\right] \label{b} \\
& = & \exp\left[{\frac{1}{2}}e^{i\theta}(\tanh r)
a^{\dagger}a^{\dagger}\right]
(\cosh r)^{-1/2}\sum_{n=0}^{\infty}
\frac{({\rm sech} r -1)^n}{n!}(a^{\dagger})^n(a)^n \nonumber \\
&~& ~~~~~~~~\times~
\exp\left[-{\frac{1}{2}}e^{-i\theta}(\tanh r)aa\right]~, \label{c}\end{aligned}$$ where Eqs. (\[b\]) and (\[c\]) are obtained from BCH relations. Observe that $$D(\alpha)S(z) = S(z)D(\gamma)~, ~~~~~~
\gamma = \alpha\cosh r - \bar{\alpha} e^{i\theta}\sinh r~.$$ Therefore, the ordering of $D$ and $S$ is only a convention.
The squeezed-state wave functions are given by a more complicated form of Eq. (\[psi\]). Specifically, they are [@bchst] $$\begin{aligned}
\psi_{ss} &=& D(\alpha)S(z)\psi_0 \nonumber \\
&=& \frac{1}{\pi^{1/4}}\frac{\exp[-ix_0p_0/2]}{[{\cal S}(1 +i2\kappa)]^{1/2}}
\exp\left[-(x-x_0)^2 \left(\frac{1}{2{\cal S}^2(1+i2\kappa)}-i\kappa\right)
+ip_0x\right]~,\end{aligned}$$ where $${\cal{S}} = \cosh r + \frac{z_1}{r} \sinh r
= e^r \cos^2\frac{\phi}{2} +e^{-r}\sin^2\frac{\phi}{2}~ \label{squeeze}$$ and $$\kappa = \frac{z_2 \sinh r}{2rs}~.$$ These wave functions are Gaussians which, in general, do not have the width of the ground state; i.e., they are squeezed by the squeeze parameters ${\cal S},\kappa$. The most commonly studied example is when $z$ is real and positive, giving $$\psi_{ss}(x) = [\pi s^2]^{-1/4}
\exp\left[-\\frac{(x-x_0)^2}{2s^2}+ip_0x\right]~, ~~~~s=e^r~.
\label{psiss}$$
The elements involved in $S$ actually are an SU(1,1) group defined by $$K_+ = \frac{1}{2}a^{\dagger}a^{\dagger}~, \hspace{0.5in}
K_- = \frac{1}{2}aa~, \hspace{0.5in}
K_0 = \frac{1}{2}\left(N + \frac{1}{2}\right)~, \label{j}$$ where $N=a^{\dagger}a$. The operators $K_0, K_{\pm}$ satisfy the commutation relations $$[K_{0},~K_{\pm}] = {\pm}K_{\pm}~,~~~~[K_{+},~K_{-}] = -2K_{0}~.
\label{comj}$$ Therefore, $S$ can be given by $$\begin{aligned}
S(z) & = & \exp[zK_+ - \bar{z}K_-] \label{aa} \\
& = & \exp[e^{i\theta}(\tanh r)K_+]
\left({\frac{1}{\cosh r}}\right)^{2K_0}
\exp[-e^{-i\theta}(\tanh r)K_-]~. \label{bb}\end{aligned}$$ The commutation relations (\[coma\]) and (\[comj\]) close with $$[K_{+}, ~a^{\dagger}]=0~,~~~~~~ [K_-,~a^{\dagger}] = a~, ~~~~~~
[K_{+}, ~a] = -~a^{\dagger}~,$$ $$[K_{-},~a]=0~, ~~~~~~
[K_{0}, ~a^{\dagger}]=
\mbox{$1\over2$}~a^{\dagger}~,
~~~~~~[K_{0},a]=-\mbox{$1\over2$}~a.$$
The ladder-operator definition of the squeezed states is $$[\mu a - \nu a^{\dagger}]|\alpha,z\rangle = \beta|\alpha,z\rangle~.$$ Again this follows from the displacement-operator definition because $$\begin{aligned}
b \equiv S(z)^{-1}aS(z) &=&
(\cosh r)a +e^{i\theta}(\sinh r) a^{\dagger}~, \nonumber \\
b^{\dagger} \equiv S(z)^{-1}aS(z) &=&
(\cosh r)a^{\dagger} +e^{-i\theta}(\sinh r) a~. \label{btrans}\end{aligned}$$ where $$[b,~b^{\dagger}]=1~, ~~~~~~b\equiv\mu a +\nu a^{\dagger}~, ~~~~~~
|\mu|^2 - |\nu|^2 = 1~.$$ Eq. (\[btrans\]) is a Holstein-Primakoff [@hp] or Bogoliubov [@bog] transformation. When such a transformation exits, such as for the harmonic oscillator and for some other cases [@bt1]-[@bt3], there is no problem defining displacement-operator squeezed states. However, such a transformation does not always exist, and that is at the crux of the problem of finding a general definition for displacement-operator squeezed states.
Lastly, we note the time-dependent uncertainties in $x$ and $p$. They are [@uncert] $$[\Delta x(t)]^2_{(\alpha,z)}
= \mbox{$1\over2$}\left[s^2\cos^2\omega t
~+~\frac{1}{s^2}\sin^2\omega t\right]~,$$ $$[\Delta p(t)]^2_{(\alpha,z)}
= \mbox{$1\over2$}\left[\frac{1}{s^2}\cos^2\omega t~
+~s^2\sin^2\omega t\right]~,$$ $$[\Delta x(t)]^2_{(\alpha,z)}[\Delta p(t)]^2_{(\alpha,z)}
=\frac{1}{4}\left[ 1~+~\frac{1}{4}\left(s^2-\frac{1}{s^2}\right)^2
\sin^2[\omega t]\right]~.$$
Generalized Coherent and Squeezed States
========================================
As discussed in Ref. [@nt], generalizations of the displacement-operator and ladder-operator coherent states have been widely discussed and studied [@jb; @gcs1; @gcs2; @gcs3]. Also, a generalization of the minimum-uncertainty coherent states was found [@n1; @n2], and this method turned out to also yield the generalized squeezed states as a byproduct.
Recently, we gave a generalized ladder operator method to define squeezed states for arbitrary systems [@nt], and there we pointed out the problem which is at the crux of the present study. In general there is no Bogoliubov transformation and hence no connection between the ladder-operator and displacement-operator methods for defining squeezed states.
This can be exemplified by considering the ordinary squeeze operator acting on the ground state, with no displacement operator: $$S(z)|0\rangle = |z\rangle ~.$$ In this form, $S(z)$ is the SU(1,1) displacement operator, and hence the states $|z\rangle$ are the SU(1,1) coherent states. Note that these coherent states have only even occupation numbers in the number basis. (Indeed, recall that one of the early names for the squeezed states was “two-photon coherent states" [@ss1].)
But if $S$ is the displacement operator for SU(1,1), what is the SU(1,1) squeeze operator? A first guess would be to square the elements of $S$, i.e., to square $aa$ and $a^{\dagger}a^{\dagger}$ to yield operators exponentiated to the fourth power. But this leads to operators that are not well-defined [@us1; @us2]; that is, the operators $$U_j = exp[\hat{z}_j(a^{\dagger})^j - \hat{\bar{z}}_j(a)^j]~,
~~~~j = 3,4,5,\dots .$$ So, there is no naive higher-order squeezing. Another way to state this is that there exist no simple operators which obey $$\hat{S}(y)^{-1} aa \hat{S}(y) = \mu aa + \nu a^{\dagger}a^{\dagger}~.$$ That is, there is no Bogoliubov transformation for the SU(1,1) elements. Hence, there is no obvious way to define the SU(1,1) squeezed states by the displacement-operator method.
Multiboson Operators
====================
In a program to circumvent the problems with naive multiboson squeezing, a productive collaboration [@d1]-[@d6] proposed using the generalized Bose operators of Brandt and Greenberg [@brandt]. These latter two observed that if one defines the operators $$A_j = \sum_{k=0}^{\infty} \alpha_{jk} (a^{\dagger})^ka^{k+j}~,
~~~~j \ge 2~,$$ $$\alpha_{jk} = \sum_{l=0}^k\frac{(-1)^{k-l}}{(k-l)!}
\left[\frac{1+[[l/j]]}{l!(l+j)!}\right]^{1/2} e^{i\rho_l}~,$$ where we denote the greatest-integer function by $[[y]]$, and the ${\rho_l}$ are arbitrary phases. Then, we have $$[A_j^{\dagger},~A_j] = 1~. \label{Aj}$$ That is, these functions satisfy the canonical commutation relations even though they are not the ordinary boson operators. They also satisfy $$[N,A_j] = [a^{\dagger}a,A_j]= -jA_j~,$$ and $$\begin{aligned}
A_j|jn + k\rangle &=& \sqrt{n}|j(n-1) + k\rangle~, \\
A_j^{\dagger}|jn+k\rangle &=& \sqrt{(n+1)}|j(n+1) + k\rangle~,
~~~~~~0\leq k < j~.\end{aligned}$$ Note that for a given $j$ we have $j$ different sets of states. Each of them starts at a different lowest state $|k\rangle$, where $0 \leq k <
j$; i.e., $|0\rangle$, $|1\rangle$, $|2\rangle$, …$|j-1\rangle$.
If one acts on eigenstates of $N$, then from the normal-ordering theorems of Wilcox [@wilcox], a very useful form of $A_j$ can be given [@r] $$A_j^{\dagger} =
\left[[[\tilde{N}/j]]\frac{(\tilde{N}-j)!}{\tilde{N}!}\right]^{1/2}
(a^{\dagger})^j~,$$ where $\tilde{N}$ is the eigenvalue of the operator $N$ in the number operator basis.
The collaboration of Refs. [@d1]-[@d6] concentrated on investigating the properties of the states defined by $$D(\alpha)V(z) |0\rangle =
D(\alpha)\exp[z A_j^{\dagger} - \bar{z} A_j] |0\rangle =
|\alpha, z_j\rangle~. \label{them}$$ In other words, they took an ordinary coherent state and then squeezed this state by the j-photon operators of $A_j$ and $A_j^{\dagger}$. (Also, they studied [@d5] the properties of states obtained from a generalized set of Weyl-Heisenberg operators, $A_j^{\eta}$.)
Coherent and Squeezed States for the Multiboson Systems
=======================================================
Coherent states
---------------
Now, from our point of view, of finding general and consistent methods of obtaining coherent and squeezed states, another path suggests itself. Since the $A_j$’s obey the canonical commutation relations of Eq. (\[Aj\]), which are identical in form to Eq. (\[coma\]), this means one can use [*these*]{} operators in displacement operators. That is, we consider the operator $V$ of equation (\[them\]) not to be a multiboson squeeze of a coherent state, but rather a multiboson displacement operator: $$D_j(\alpha) = \exp[\alpha A_j^{\dagger} - \bar{\alpha} A_j]
=\exp\left[-\frac{1}{2}|\alpha|^2\right]
\exp[\alpha A_j^{\dagger}] \exp[-\bar{\alpha} A_j] ~.$$ Therefore, the multi-boson coherent states are $$|\alpha (j,k)\rangle = D_j(\alpha)|k\rangle
= \exp\left[-\frac{1}{2}|\alpha|^2\right]
\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}} |jn+k\rangle~.$$ Again observe that for a given $j$ we have $j$ different sets of (coherent) states. Each of them again starts at a different lowest state $|k\rangle$, where $0 \leq k <
j$; i.e., $|0\rangle$, $|1\rangle$, $|2\rangle$, …$|j-1\rangle$. That is why we label the states by the couple $(j,k)$. \[The states $|\alpha(j,0)\rangle$ were studied in Ref. [@d4].\]
These coherent states are, of course, consistent with the ladder-operator definition, $$A_j |\alpha (j,k)\rangle = \alpha |\alpha (j,k)\rangle ~.$$ By using the number-state basis of the wave functions, $$\psi_n = \left(\frac{a_o}{\pi^{1/2} 2^n n!}\right)^{1/2}
\exp\left[-\frac{1}{2}a_0^2x^2\right]
H_n(a_0x)~,$$ where $a_0^2= (m\omega/\hbar)$ will now be set to $1$ and the $H$ are the Hermite polynomials, one can write the normalized coherent state wave functions as $$\psi_{cs}(j,k)(x) = \pi^{-1/4}
\exp\left[-\frac{1}{2}\left(|\alpha|^2 + x^2\right)\right]
I_{(j,k)}(\alpha,x)~,$$ where $I$ is the sum $$I_{(j,k)}(\alpha,x)= \sum_{n=0}^{\infty}
\frac{\alpha^n H_{jn+k}(x)}{[n!(jn+k)!2^{jn+k}]^{1/2}} ~.$$ Note that for $(j,k)= (1,0)$, we obtain the usual generating function [@genf] for the ordinary coherent states result, $$I_{(1,0)}(x) = \exp[\sqrt{2}\alpha x -\alpha^2/2]~.$$
The “natural quantum operators" for this system are [@n1; @n2] (in dimensionless units) $$X_j \equiv \frac{1}{\sqrt{2}}[A_j + A_j^{\dagger}]~, ~~~~~~~
P_j \equiv \frac{1}{i\sqrt{2}}[A_j - A_j^{\dagger}]~,$$ But then, the Heisenberg-Weyl algebra tells us immediately that these are the operators directly connected to the minimum-uncertainty method. Therefore, we have that [@nt] $$(\Delta X_j)^2_{(j,k)}= 1/2~,~~~~(\Delta P_j)^2_{(j,k)} = 1/2~.$$
We can also obtain information for the uncertainties of the physical position and momentum, $x$ and $p$. We immediately observe that $$\langle x \rangle_{(j,k)} = \langle p \rangle_{(j,k)} = 0~, ~~~~j>1~.$$ (For j=1 we have the ordinary harmonic oscillator). For $j>2$, we have, then, that $$\begin{aligned}
\langle x^2 \rangle_{(j,k)}
&=&
(\Delta x)^2_{(j,k)}
=\langle p^2 \rangle_{(j,k)}= (\Delta p)^2_{(j,k)} \nonumber \\
&=&
\exp[-|\alpha|^2] \sum_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}
[jn+k+\mbox{$1\over2$}]~
\nonumber \\
&=&
\mbox{$1\over2$} + k +j|\alpha|^2~, ~~~~~~j>2~.\end{aligned}$$
The case $j=2$ is slightly more complicated because the operators $x^2$ and $p^2$ connect different number states in the expectation values. In particular, $$\begin{aligned}
\langle x^2 \rangle_{(2,k)}&=&(\Delta x)^2_{(2,k)}
=\mbox{$1\over2$} + k +2|\alpha|^2 + C_{(2,k)} \\
\langle p^2 \rangle_{(2,k)}&=&(\Delta p)^2_{(2,k)}
=\mbox{$1\over2$} + k +2|\alpha|^2 - C_{(2,k)} ~,\end{aligned}$$ where $$C_{(2,k)}=\mbox{$1\over2$}[\langle a^2 \rangle_{(2,k)}
+\langle (a^{\dagger})^2 \rangle_{(2,k)}]~,$$ which evaluates to $$C_{(2,k)}=(\alpha + \bar{\alpha})\exp[-|\alpha|^2]
\sum_{n=0}^{\infty}\frac{|\alpha|^{2n}}{n!}
\left[\frac{(n+1+k/2)(n+1/2+k/2)}{n+1}\right]^{1/2}~.$$
Squeezed states
---------------
Because the $A_j$’s define a Heisenberg-Weyl algebra, one can therefore define an SU(1,1) squeeze algebra in the normal way:
$$K_{j+} = \frac{1}{2}A_j^{\dagger}A_j^{\dagger}~, \hspace{0.5in}
K_{j-} = \frac{1}{2}A_jA_j~, \hspace{0.5in}
K_{j0} = \frac{1}{2}\left(A_j^{\dagger}A_j + \frac{1}{2}\right)~.
\label{jj}$$
Then all these $A_j$’s and $K_j$’s again have the same commutation relations as before, and so all the results of the ordinary harmonic oscillator coherent and squeezed states goes through in the same manner, only with the $a$’s being changed into the $A_j$’s. The squeeze operators are therefore $$\begin{aligned}
S_j(z) & = & \exp[zK_{j+} - \bar{z}K_{j-}] \label{jaa} \\
& = & \exp[e^{i\theta}(\tanh r)K_{j+}]
\left({\frac{1}{\cosh r}}\right)^{2K_{j0}}
\exp[-e^{-i\theta}(\tanh r)K_{j-}]~, \label{jbb}\end{aligned}$$ where $$z = r e^{i\theta}~,$$ meaning the squeezed states are $$D_j(\alpha)S_j(z)|k\rangle = |\alpha, z(j,k)\rangle~.$$
Furthermore, all the mathematics of the ordinary squeezed states follows automatically, just changing notation. For example, there is a Bogoliubov transformation: $$\begin{aligned}
B_j \equiv S_j(z)^{-1}A_jS_j(z) &=&
(\cosh r)A_j +e^{i\theta}(\sinh r) A_j^{\dagger}~,
\label{btransj} \\
B_j^{\dagger} \equiv S_j(z)^{-1}A_jS_j(z) &=&
(\cosh r)A_j^{\dagger} +e^{-i\theta}(\sinh r) A_j~.
\label{bdtransj}\end{aligned}$$ where $$[B_j,~B_j^{\dagger}]=1~, ~~~~~~B_j\equiv\mu A_j +\nu A_j^{\dagger}~,
~~~~~~ |\mu|^2 - |\nu|^2 = 1~.$$ This means, of course, that there is an equivalent ladder-operator definition of these squeezed states: $$[\mu A_j - \nu A_j^{\dagger}]|\alpha,z(j,k)\rangle
= \beta|\alpha,z(j,k)\rangle~.$$
Again, from the the Heisenberg-Weyl algebra, it follows that $$(\Delta X_j)^2_{ss}(\Delta P_j)^2_{ss} = 1/4~.$$ Of course, being squeezed states the above equality holds at $t=0$ and oscillates, and the uncertainty in each quadrature also oscillates.
The Ground State as a Coherent State
====================================
In finding the coherent and squeezed states for general systems, it has been noted that the ground state (or extremal state) is always a member of the set of coherent states [@nt; @n1]. This is also true in the multi-boson case and we want to show that why it is true in general. Before continuing, however, note that this makes intuitive sense. The ground state is the closest quantum state to zero motion, which corresponds to a classical particle at rest. Therefore, the most-classical like states should include this state.
Starting from a minimum-uncertainty Hamiltonian system, the classical Hamiltonian is transformed to classical variables that vary as the sin and the cosine of the classical $\omega t$. In these variables the Hamiltonian can be written as $$H_{cl} = X^2/2 + P^2/2~.$$ This is harmonic-oscillator like. Indeed, for the rest of this discussion keep the harmonic oscillator in mind for intuitive aid.
When the classical variables are changed to quantum operators, it is found that $$X = \frac{{\cal A} + {\cal A^{\dagger}}}{\sqrt2}~, ~~~~~
P = \frac{{\cal A} - {\cal A^{\dagger}}}{i\sqrt2}~,$$ where the ${\cal A}$’s are the lowering operators of the system. In general, these operators may be $n$-dependent or have to be made Hermitian with respect to the adjoint, but the statement holds.
Therefore, the states which minimize the uncertainty relation between $X$ and $P$, $$[X,P] = iG~, \label{com}$$ are those (squeezed) states which satisfy the eigenvalue equation $$\left[X +\frac{i\Delta X}{\Delta P}P\right]\psi_{mus} =
\left[\langle X\rangle +
\frac{i\Delta X}{\Delta P}\langle P\rangle \right]\psi_{mus} ~.
\label{mus}$$ \[When dealing with symmetry, non-Hamiltonian systems, the starting point for the study is here, simply considering the implications of the commutation relation (\[com\]).\]
These states are, in general, squeezed states. This can be seen by writing $X$ and $P$ in terms of ${\cal A}$ and ${\cal A}^{\dagger}$. Then the left hand side of Eq. (\[mus\]) is proportional to a linear combination of ${\cal A}$ and ${\cal A}^{\dagger}$, just like after any Bogoliubov transformation. To change to a coherent state, the relative uncertainties must be equal, i.e., $(\Delta X)/(\Delta P)= 1$. but then the left hand side of Eq. (\[mus\]) is proportional simply to ${\cal A}$. Then taking the case corresponding to the smallest classical motion, $\langle X\rangle=\langle P\rangle = 0$, one is left with the equation $${\cal A}\psi_{mus} = 0~.$$ But the state that is annihilated by the lowering operator is the ground state.
An Alternative, Effective Definition for Displacement-Operator Squeezed States
==============================================================================
We close with a comment on how an alternative method can be used to define “displacement operator” squeezed states. This method can be used for the systems under discussion: systems with minimum-uncertainty, ladder-operator, and displacement operator coherent states, but only minimum-uncertainty or ladder-operator squeezed states. An example, where it has been used, suffices to explain the procedure.
The even and odd coherent states [@eo; @eocs] are defined in terms of the double destruction operator: $$aa|\alpha\rangle_{\pm} = \alpha^2|\alpha\rangle_{\pm}~.$$ They also can be defined in terms of an unusual displacement operator, $$|\alpha\rangle_{\pm}=
D_{\pm}(\alpha)|0\rangle
=\left[2(1\pm \exp[-2|\alpha|^2]\right]^{-1/2}
\left[D(\alpha) \pm D(-\alpha)\right] |0\rangle~,$$ where $D$ is the ordinary coherent state displacement operator.
The even and odd squeezed states are generalized to those states satisfying the eigenvalue equation $$\left[\left(\frac{1+q}{2}\right) aa
+\left(\frac{1-q}{2}\right) a^{\dagger} a^{\dagger} \right]\psi_{ss}
=\alpha^2 \psi_{ss} . \label{Kss}$$ The solutions are [@nt] $$\psi_{Ess}=N_E\exp{\left[-\frac{x^2}{2}(q+\sqrt{q^2-1})\right]}
\Phi\left(\left[\frac{1}{4}+\frac{\alpha^2}{2\sqrt{q^2-1}}\right],~
\frac{1}{2};~x^2\sqrt{q^2-1}\right)~,$$ $$\psi_{Oss}=N_O ~x\exp{\left[\frac{-x^2}{2}(q+\sqrt{q^2-1})\right]}
\Phi\left(\left[\frac{3}{4}+\frac{\alpha^2}{2\sqrt{q^2-1}}\right],~
\frac{3}{2};~x^2\sqrt{q^2-1}\right)~,$$ where $\Phi(a,b;c)$ is the confluent hypergeometric function $ \sum_{n=0}^{\infty} \frac{(a)_n c^n}{(b)_n~n!}$. In the limit $q\rightarrow 1$, these become the even and odd coherent states.
But there are no displacement operator squeezed states because there does not exist a unitary (Bogoliubov) transformation that can rotate $aa$ into a linear combination of $aa$ and $a^{\dagger}a^{\dagger}$. Therefore, an alternative idea is to simply use the ordinary squeeze operator, $S$, with the given displacement operator, and call these the displacement-operator squeezed states [@kss]. Here that would be $$D_{\pm}(\alpha)S(z)|0\rangle = |\alpha,z\rangle_{\pm}~.$$
In Ref. [@eoss] these minimum-uncertainty and “displacement-operator" were compared and found to be similar in nature. Since the “displacement-operator" states are more amenable to analytic calculations, they were then used for exploration of the time-dependence of the even and odd states of a trapped ion.
This, then, is a viable alternative to mathematically rigorous siplacment-operator squeezed states.
Acknowledgements {#acknowledgements .unnumbered}
================
We wish to thank G. M. D’Ariano and Wolfgang Schleich for helpful conversations on these topics. MMN acknowledges the support of the United States Department of Energy and the Alexander von Humboldt Foundation. DRT acknowledges a grant from the Natural Sciences and Engineering Research Council of Canada.
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[^1]: Email: [email protected]
[^2]: Email: [email protected]
|
---
abstract: 'In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers of finite mutation type. In particular, we show that a mutation finite cluster quiver has a maximal green sequence unless it arises from a once-punctured closed marked surface, or one of the two quivers in the mutation class of $\mathbb{X}_7$. We develop a procedure to explicitly find maximal green sequences for cluster quivers associated to arbitrary triangulations of closed marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with boundary has a maximal green sequence. We also compute explicit maximal green sequences for exceptional quivers of finite mutation type.'
address: 'Department of Mathematics, University of Nebraska - Lincoln, Lincoln, NE, USA'
author:
- 'Matthew R. Mills'
title: Maximal green sequences for quivers of finite mutation type
---
[^1]
Introduction {#sec:in}
============
Cluster algebras were introduced by Fomin and Zelevinsky in [@cluster1]. Cluster algebras have become an important tool in the study of many areas of mathematics and mathematical physics. They play a role in the study of Teichmüller theory, canonical bases, total positivity, Poisson-Lie groups, Calabi-Yau algebras, noncommutative Donaldson-Thomas invariants, scattering amplitudes, and representations of finite dimensional algebras.
One very important property of a quiver associated to a cluster algebra is whether or not it has a maximal green sequence. Quiver mutation is a transformation of a quiver, determined by a choice of a vertex of the quiver, into a new quiver. A maximal green sequence is a certain sequence of quiver mutations given by a sequence of vertices of the quiver. The idea of maximal green sequences of cluster mutations was introduced by Keller in [@keller]. He explored quantum dilogarithm identities by utilizing these sequences in the explicit computation of noncommutative Donaldson-Thomas invariants of quivers which were introduced by Kontsevich and Soibelman in [@kontsevich]. If a quiver with potential has a maximal green sequence, then its associated Jacobi algebra is finite dimensional [@brustle; @keller2]. In [@amiot] an explicit construction of a cluster category from a quiver with potential whose Jacobian algebra is finite dimensional is given.
The first result in this paper focuses on the existence of maximal green sequences for quivers that are associated to triangulations of surfaces.
\[thm:main\] Suppose that a marked surface $\Sigma$ is not once-punctured and closed, that is, $\Sigma$ is
1. of genus at least one with at least two punctures;
2. of genus zero with at least four punctures;
3. or of arbitrary genus with at least one boundary component.
Then for any triangulation of $\Sigma$ there exists a maximal green sequence for the associated quiver.
The proof of Theorem \[thm:main\] for $(A)$ and $(B)$ is an explicit construction of maximal green sequences for these surfaces. The construction is given in Section \[sec:proof\]. The proof for $(C)$ follows from the previous cases together with a theorem of Muller that we recall here as Theorem \[thm:subquiver\]. The author had originally provided a construction for the case of higher genus surfaces and discussed surfaces with boundary in an extended abstract [@mills]. The construction given in this paper is a refinement of the one given there that also applies to punctured spheres. This construction of maximal green sequences requires many choices so we in fact get many different maximal green sequences for these quivers. It follows from the work of Keller [@keller] that our work here provides many quantum dilogorithm identities.
It is straightforward from the definition of quiver mutation (formally stated in Definition \[def:mutation\]) that mutation imposes an equivalence relation on the set of all quivers. For a quiver $Q$ we let $\operatorname{Mut}(Q)$ denote the equivalence class of $Q$ under this relation.
Muller showed that in general the existence of a maximal green sequence is not mutation invariant [@muller]. It is already known that every quiver of type $\mathbb{A}, \mathbb{D},$ and $\mathbb{E}$ has a maximal green sequence [@brustle]. It was shown by Ladkani that quivers associated to once-punctured closed surfaces of genus at least one do not admit maximal green sequences [@ladkani]. The existence of maximal green sequences for specific triangulations of various marked surfaces has been shown in many papers [@alim; @bucher; @bm]. In [@garver] Garver and Musiker give a combinatorial approach to construct maximal green sequences for type $\mathbb{A}$ quivers, which are exactly the quivers associated to triangulations of unpunctured disks. Cormier *et al*., give an explicit construction of minimal length maximal green sequences for this case in [@cormier].
In [@brustle2] Brüstle, Hermes, Igusa, and Todorov use semi-invariants to prove two conjectures about maximal green sequences. One particularly usefull result from this paper is the Rotation Lemma ([@brustle2 Theorem 3]). In part, it shows that if a maximal green sequence for a quiver $Q$ first mutates vertex $k$, then the quiver obtained from mutating $Q$ at $k$ also has a maximal green sequence. Repeated application of this result then shows that any intermediate quiver in a maximal green sequence has a maximal green sequence. The Rotation Lemma gives the existence of a maximal green sequence for many quivers in a mutation class, but does not prove Theorem \[thm:main\].
A quiver $Q$ is said to be of finite mutation type if $\operatorname{Mut}(Q)$ is finite. It is known that all finite mutation type quivers arise from triangulations of surfaces except for the rank 2 case and 11 exceptional cases [@felikson]. These 11 cases are $\mathbb{E}_6,\mathbb{E}_7,\mathbb{E}_8,\widetilde{\mathbb{E}_6},\widetilde{\mathbb{E}_7},\widetilde{\mathbb{E}_8},\mathbb{E}_6^{(1,1)},\mathbb{E}_7^{(1,1)},\mathbb{E}_8^{(1,1)}, \mathbb{X}_6,$ and $\mathbb{X}_7.$ Among these exceptional cases it has been shown that there exists a quiver with a maximal green sequence for all but $\mathbb{X}_7$ [@alim]. It was shown in [@seven] that neither of the two quivers in the mutation class of $\mathbb{X}_7$ have a maximal green sequence. All rank 2 cluster algebras have a maximal green sequence given by first mutating at the source vertex and then mutating at the other vertex.
We use the cluster algebra package in the computer program Sage to produce an explicit maximal green sequence for every quiver in the outstanding exceptional cases to obtain our second result.
\[thm:main2f\] If $Q$ is a quiver in the mutation class of $\widetilde{\mathbb{E}_6},\widetilde{\mathbb{E}_7},\widetilde{\mathbb{E}_8},\mathbb{E}_6^{(1,1)},\mathbb{E}_7^{(1,1)},$ $\mathbb{E}_8^{(1,1)},$ or $\mathbb X_6$, then $Q$ has a maximal green sequence.
It is already known that every quiver in the mutation class of $\mathbb{E}_6,\mathbb{E}_7,$ and $\mathbb{E}_8$ have maximal green sequences by [@brustle]. By combining previous results with Theorem \[thm:main\] and Theorem \[thm:main2f\] we have a complete classification of which quivers of finite mutation have a maximal green sequence. Furthermore, we have shown that the existence of a maximal green sequence is mutation-invariant for quivers of finite mutation type.
\[con:muteq\] Let $Q$ be a quiver of finite mutation type, then a maximal green sequence exists for every quiver in $\operatorname{Mut}(Q),$ or there is no maximal green sequence for any quiver in $\operatorname{Mut}(Q)$. In particular, $Q$ has a maximal green sequence unless it arises from a triangulation of a once-punctured closed surface, or is one of the two quivers in the mutation class of $\mathbb{X}_7$.
The existence of a maximal green sequence for a quiver also seems to be related to whether the cluster algebra $\mathcal{A}$ it generates is equal to its upper cluster algebra $\mathcal{U}$. Gross, Hacking, Keel and Kontsevich showed that if $\mathcal{A}= \mathcal{U}$ and a maximal green sequence exists, then the Fock-Goncharov canonical basis conjecture holds [@gross]. It is still unknown as to whether or not $\mathcal{A}=\mathcal{U}$ for closed higher genus surfaces with at least two punctures and punctured closed spheres. For all other quivers of finite mutation type it is known that $\mathcal{A}=\mathcal{U}$ if and only if there exists a quiver with a maximal green sequence. See [@lee] and references therein for more information on when $\mathcal{A}=\mathcal{U}.$
In Section \[sec:quivers\] we give background on quivers and maximal green sequences. In Section \[sec:surfaces\] we give background on marked surfaces and their triangulations. In Section \[sec:cycleindep\] we discuss two mutation sequences that are used in Section \[sec:proof\], where we give the construction for maximal green sequences for closed surfaces. In Section \[sec:boundary\] we prove the existence of maximal green sequences for surfaces with boundary. We then discuss the maximal green sequences for exceptional cases in Section \[sec:exceptional\].
Quivers and maximal green sequences {#sec:quivers}
===================================
We recall the definitions from [@keller2], but use the conventions given in [@brustle].
A **(cluster) quiver** is a directed graph with no loops or 2-cycles. An **ice quiver** is a pair $(Q, F)$ where $Q$ is a quiver and $F$ is a subset of the vertices of $Q$ called **frozen vertices;** such that there are no edges between frozen vertices. If a vertex of $Q$ is not frozen it is called **mutable**. For convenience, we assume that the mutable vertices are labelled $\{1,\ldots, n\}$, and frozen vertices are labeld by $\{n+1, \ldots, n+m\}$.
\[def:mutation\] Let $(Q,F)$ be an ice quiver, and $k$ a mutable vertex of $Q$. The **mutation** of $(Q,F)$ at vertex $k$ is denoted by $\mu_k$, and is a transformation $(Q,F)$ to a new ice quiver $(\mu_k(Q),F)$ that has the same vertices, but making the following adjustment to the edges:
1. For every 2-path $i \rightarrow k \rightarrow j$, add a new arrow $i \rightarrow j$.
2. Reverse the direction of all arrows incident to $k$.
3. Delete any 2-cycles created during the first two steps, and any arrows between frozen vertices.
Mutation at a vertex is an involution, and an equivalence relation. We define $\operatorname{Mut}(Q)$ to be the equivalence class of all quivers that can be obtained from $Q$ by a sequence of mutations.
Let $Q_0$ be the set of vertices of $Q$. The **framed quiver** associated with a quiver $Q$ is the ice quiver $(\hat{Q},Q_0')$ such that:
$$Q_0'=\{i'\text{ }|\text{ }i\in Q_0\}, \hspace{.6cm} \hat{Q}_0 = Q_0 \sqcup Q_0'$$ $$\hat{Q}_1 = Q_1 \sqcup \{i \to i'\text{ }|\text{ }i \in Q_0\}$$
Since the frozen vertices of the framed quiver are so natural we will simplify the notation and just write $\hat{Q}$. Now we must discuss what is meant by red and green vertices.
\[def:green1\] Let $R \in \operatorname{Mut}(\hat{Q})$.\
A mutable vertex $i \in R_0$ is called **green** if $$\{j'\in Q_0'\text{ }| \text{ } \exists \text{ } j' \rightarrow i \in R_1 \}=\emptyset.$$ It is called **red** if $$\{j'\in Q_0'\text{ }| \text{ } \exists \text{ } j' \leftarrow i \in R_1 \}=\emptyset.$$
It is not clear from the definition that every mutable vertex in $R_0$ is either red or green. In the case of quivers this result is due to [@derksen] and then it was also shown in a more general setting in [@gross].
[@derksen; @gross]\[thm:signcoh\] Let $R \in \operatorname{Mut}(\hat{Q})$. Then every mutable vertex in $R_0$ is either red or green.
A **green sequence** for $Q$ is a sequence $\textbf{i}=(i_1, \dots, i_l) \subset Q_0$ such that $i_1$ is green in $\hat{Q}$ and for any $2\leq k \leq l$, the vertex $i_k$ is green in $\mu_{i_{k-1}}\circ \cdots \circ \mu_{i_1}(\hat{Q})$. A green sequence **i** is called maximal if every mutable vertex in $\mu_{i_{l}}\circ \cdots \circ \mu_{i_1}(\hat{Q})$ is red.
Marked surfaces and their triangulations {#sec:surfaces}
========================================
\(a) at (0,0); (b) at (6,0); (c) at (6,6); (d) at (0,6); (g) at (2,2); (f) at (4,2); (e) at (3,4); (a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt);
\(a) node\[left\][[$P$]{}]{}; (d) node\[left\][[$P$]{}]{}; (b) node\[right\][[$P$]{}]{}; (c) node\[right\][[$P$]{}]{};
\(a) to node \[midway, below\] (b); (b) to node \[midway, right\] (c); (c) to node \[midway, above\] (d); (d) to node \[midway, left\] (a);
\(d) to node \[midway,fill=white\][3]{} (e); (e) to node \[midway,fill=white\][4]{} (c); (e) to\[bend left\] node \[midway,fill=white\][5]{} (b); (f) to node \[midway,fill=white\][6]{} (b); (f) to node \[midway,fill=white\][7]{} (a); (g) to node \[midway,fill=white\][8]{} (a); (g) to node \[midway,fill=white\][9]{} (d); (g) to node \[midway,fill=white\][10]{} (e); (e) to node \[midway,fill=white\][11]{} (f); (g) to node \[midway,fill=white\][12]{} (f);
0;<.3pt,0pt>:<0pt,-.2pt>:: (0,0) \*+[1]{} =“0”, (500,0) \*+[2]{} =“1”, (100,175) \*+[3]{} =“2”, (100,75) \*+[4]{} =“3”, (400,100) \*+[5]{} =“4”, (400,225) \*+[6]{} =“5”, (400,400) \*+[7]{} =“6”, (100,400) \*+[8]{} =“7”, (100,300) \*+[9]{} =“8”, (200,250) \*+[10]{} =“9”, (300,200) \*+[11]{} =“10”, (300,300) \*+[12]{} =“11”, “3”, [“0”]{}, “0”, [“4”]{}, “7”, [“0”]{}, “0”, [“8”]{}, “2”, [“1”]{}, “1”, [“3”]{}, “5”, [“1”]{}, “1”, [“6”]{}, “3”, [“2”]{}, “8”, [“2”]{}, “2”, [“9”]{}, “4”, [“3”]{}, “4”, [“5”]{}, “10”, [“4”]{}, “6”, [“5”]{}, “5”, [“10”]{}, “6”, [“7”]{}, “11”, [“6”]{}, “8”, [“7”]{}, “7”, [“11”]{}, “9”, [“8”]{}, “9”, [“10”]{}, “11”, [“9”]{}, “10”, [“11”]{},
To begin the section we recall the definition of a marked surface given in [@fomin]. Let $S$ be an orientable 2-dimensional Riemann surface with or without boundary. We designate a finite number of points, $M$, in the closure of $S$ as marked points. We require at least one marked point on each boundary component. We call marked points in the interior of $S$ **punctures.** Together the pair $\Sigma=(S,M)$ is called a **marked surface.** For technical reasons we exclude the cases when $\Sigma$ is one of the following:
- a sphere with less than four punctures;
- an unpunctured or once punctured monogon;
- an unpunctured digon; or
- an unpunctured triangle.
Note that the construction allows for spheres with four or more punctures.
Up to homeomorphism a marked surface is determined by four things. The first is the genus $g$ of the surface. The second is the number of boundary components $b$ of $S$. The third is the number of punctures $p$ in $M$, and the fourth is the set $m=\{m_i\}_{i=1}^{b}$ where $m_i\in \mathbb{Z}_{>0}$ denotes the number of marked points on the $i$th boundary component of $S.$ We say a marked surface is **closed** if it has no boundary.
An **arc** $\gamma$ in $(S,M)$ is a curve in $S$ such that:
- The endpoints of $\gamma$ are in $M$.
- $\gamma$ does not intersect itself, except that its endpoints may coincide.
- $\gamma$ is disjoint from $M$ and the boundary of $S$, except at its endpoints.
- $\gamma$ is not isotopic to the boundary, or the identity.
An arc is called a **loop** if its two endpoints coincide. Each arc is considered up to isotopy. Two arcs are called compatible if there exists two arcs in their respective isotopy classes that do not intersect in the interior of $S$.
A **taggd arc** is constructed by taking an arc that does not cut out a once-punctured monogon and marking or “tagging” its ends as either **plain** or **notched** so that:
- an endpoint lying on the boundary of $S$ is tagged plain; and
- both ends of a loop must be tagged in the same way.
We use a $\bowtie$ to denote the tagging of an arc in figures. Two tagged arcs are considered compatible if:
- Their underlying untagged arcs are the same, and their tagging agrees on exactly one endpoint.
- Their underlying untagged arcs are distinct and compatible, and any shared endpoints have the same tagging.
A maximal collection of pairwise compatible tagged arcs is called a **(tagged) triangulation** of $(S,M)$.
(0,0) to\[bend left\] node\[above\][$\ell$]{} (9,0) to\[bend left\] node \[below\][$i$]{} (0,0); (0,0) to\[bend left\] node \[below\][$k$]{} node\[near end,sloped,rotate=90\][$\bowtie$]{} (5,0) to\[bend left\] node \[above\][$j$]{} (0,0);
(0,0) circle (1.5pt); (5,0) circle (1.5pt); (9,0) circle (1.5pt); (5,0) node\[right\] [$P$]{};
We call a puncture $P$ a **radial puncture in a tagged triangulation** if and only if $P$ is the unique puncture in the interior of a digon and there exists two arcs in the interior of this digon that differ only by their tagging at $P$. See Figure \[fig:radial\].
$$\begin{xy} 0;<1pt,0pt>:<0pt,-1pt>::
(0,75) *+{i} ="0",
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"0", {\ar"1"},
"0", {\ar"2"},
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\[def:quivfromtri\] Let $T$ be a triangulation of a marked surface. The **quiver associated to $T$**, which we will denote as $Q_T$, is the quiver obtained from the following construction. For each arc $\alpha$ in a triangulation $T$ add a vertex $v_\alpha$ to $Q_T$. If $\alpha_i$ and $\alpha_j$ are two edges of a triangle in $T$ with $\alpha_j$ following $\alpha_i$ in a clockwise order, then add an edge to $Q_T$ from $v_{\alpha_i} \rightarrow v_{\alpha_j}$. If $\alpha_k$ and $\alpha_j$ have the same underlying untagged arc as in Figure \[fig:radial\] we refer you to Figure \[fig:arc\] for the construction in this situation. Note that the quiver is the same whether $\alpha_j$ or $\alpha_k$ is tagged. More generally distinct triangulations may yield the same quiver.
We now define the analog of quiver mutation for triangulations of a marked surface.
A **flip** is a transformation of a triangulation that removes an arc $\gamma$ and replaces it with a (unique) different arc $\gamma'$ that, together with the remaining arcs, forms a new triangulation $T'$. In this case we define $\mu_\gamma(T)=T'$. This makes sense by the following lemma.
[@fomin Lemma 9.7]\[lem:mut=flip\] Let $T$ and $T'$ be two triangulations related by a flip of an arc $\gamma$. Suppose $\gamma$ corresponds to vertex $k$ of $Q_T$, then $Q_{T'} = \mu_k(Q_T)$.
In Sections \[sec:cycleindep\] and \[sec:proof\] we will exclusively refer to a flip of an arc in a triangulation as a mutation.
Thurston’s theory of laminations and shear coordinates provide a way to introduce frozen vertices in the geometric setting.
[@ft Definition 12.1] A **lamination** on a marked surface $(S,M)$ is a finite collection of non-self-intersecting and pairwise non-intersecting curves in $S$ up to isotopy. Each curve must be one of the following:
- a closed curve;
- a curve connecting two unmarked points on the boundary of $S$;
- a curve starting at an unmarked point on the boundary, and at its other end spiraling into a puncture (either clockwise or counter clockwise);
- a curve whose ends both spiral into punctures (not necessarily distinct).
We forbid any curves that bound an unpunctured or once-punctured disk, curves with two endpoints on the boundary which are isotopic to a piece of boundary containing zero or one marked points, and a curve with two ends spiraling into the same puncture in the same direction without enclosing anything else.
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(3,0) to (4,0); (4,0) to (4,1); (4,1) to (3,1); (3,1) to (3,0);
(3,0) to node\[near end,below\][$\alpha$]{}(4,1); (.5,1.25) node\[above\][$\ell$]{} to\[out=260, in= 80,looseness=1\] (.5,-0.25); (3.5,1.25) node\[above\][$\ell$]{} to\[out=260, in= 80,looseness=1\] (3.5,-.25);
[@ft Definition 12.2,13.1]\[def:lamination\] Let $L$ be a lamination, and let $T$ be a triangulation without any arcs that are notched. Note that this requires that there are no radial punctures in $T$. For each $\alpha \in T$, the corresponding **shear coordinate** of $L$ with respect to $T$ denoted $b_\alpha(T,L),$ is defined as a sum of contributions from all intersections of curves in $L$ with $\alpha$. An intersection contributes a +1 (resp., -1) to $b_\alpha(T,L)$ if the corresponding segment of a curve in $L$ cuts through the quadrilateral surrounding $\alpha$ cutting through the edges in the shape of an “$S$” (resp., in the shape of a $``Z$”), as in Figure \[fig:shearcoord\]. See Figure \[fig:shearcoord\]. Note that at most one of these two types of intersections can occur and $b_\alpha(T,L)$ is always finite.
For a triangulation $T$ that contains notched arcs the shear coordinates are uniquely defined by the following rules:
1. Suppose that tagged triangulations $T_1$ and $T_2$ coincide except that at a particular puncture $P$, the tags of the arcs in $T_1$ are all different from the tags of their counterparts in $T_2$. Suppose that laminations $L_1$ and $L_2$ coincide except that each curve in $L_1$ that spirals into $P$ has been replaced in $L_2$ by a curve that spirals in the opposite direction. Then $b_{\alpha_1}(T_1,L_1)=b_{\alpha_2}(T_2,L_2)$ for each tagged arc $\alpha_1 \in T_1$ and its counterpart $\alpha_2 \in T_2$.
2. By performing tag-changing transformations $L_1 \rightarrow L_2$ with $L_1$ and $L_2$ as above, we can convert any triangulation into a triangulation $T$ that does not contain any notches except possibly at radial punctures. If $\alpha \in T$ is not notched at any puncture, then we define $b_\alpha(T,L)$ as above for the underlying plain arc.
If $\alpha \in T$ is the arc incident to a radial puncture with different taggings at its endpoints, then we apply the tag-changing transformation in rule (1) to the radial puncture and then use rule (2) to compute $b_\alpha(T,L)$.
Note that the quiver constructed from a triangulation in Definition \[def:quivfromtri\] does not include any frozen vertices. We can use laminations and shear coordinates to extend Theorem \[lem:mut=flip\] to show that the flips in triangulations of surfaces together with laminations agree with mutations of ice quivers.
A **multi-lamination** is a finite family of laminations. Let $T$ be a triangulation for a marked surface and $\mathcal{L}$ a multi-lamination. Let $Q_T$ be the quiver constructed in Definition \[def:quivfromtri\]. Suppose that the arcs in $T=\{\alpha_i\}_{i=1}^n$ are indexed by their corresponding vertex of $Q_T$ and that $\mathcal{L}=\{L_j\}_{j=n+1}^m$. We define an ice quiver $(\widetilde{Q_T},F_\mathcal{L})$ where $$F_\mathcal{L}=\{j | L_j \in \mathcal{L}\}, (\widetilde{Q_T})_0=(Q_T)_0 \sqcup F_\mathcal{L},$$ and the edges of $\widetilde{Q_T}$ are the edges of $Q_T$ together with $b_{\alpha_i}(T,L_j)$ edges $i \rightarrow j$ for all $i=1,\dots,n$ and $j=n+1,\dots,m$. Note that a negative value for $b_{\alpha_i}(T,L_j)$ corresponds to adding $|b_{\alpha_i}(T,L_j)|$ edgess $j \rightarrow i$.
[@ft Theorem 13.5]\[lem:flip=mut2\] Let $T$ and $T'$ be two triangulations related by a flip of an arc $\gamma$. Suppose $\gamma$ corresponds to vertex $k$ of $\widetilde{Q_T}$, then $\widetilde{Q_{T'}} = \mu_k(\widetilde{Q_T})$.
\[def:elelam\] Let $\alpha$ be a tagged arc of a marked surface. Denote by $L_\alpha$ a lamination consisting of a single curve defined as follows. The curve $L_\alpha$ runs along $\alpha$ within a small neighborhood of it. If $\alpha$ has an endpoint $a$ on a component $C$ of the boundary of $S$, then $L_\alpha$ begins at a point $a' \in C$ located near $a $ in the clockwise direction, and proceeds along $\alpha$. If $\alpha$ has an endpoint at a puncture, then $L_\alpha$ spirals into $a$: clockwise if $\alpha$ is tagged plain at $a$, and counterclockwise if it is notched. We call $L_\alpha$ the **elementary lamination associated to $\alpha$**.
Note that for any arc $\alpha$ in a triangulation $T,$ $L_\alpha$ is the unique lamination such that $$b_\gamma(T,L_\alpha) = \begin{cases} -1 & \text{if } \gamma = \alpha, \\ 0 &\text{if } \gamma \neq \alpha.\end{cases}$$ If we fix the multi-lamination $\mathcal{L}= \{L_\alpha | \alpha \in T\}$ then the ice quiver $\widetilde{Q_T}$ is identical to the framed quiver $\widehat{Q_T}.$
The elementary laminations defined in Definition \[def:elelam\] are not the same elementary laminations given in [@ft]. There $L_\alpha$ is the unique lamination that contributes a +1 to $b_\alpha(T,L)$ and 0 for all other arcs.
We now give the geometric characterization for what it means for an arc to be green or red.
\[def:green2\] Let $T$ be a triangulation of a marked surface. Fix the multi-lamination $\mathcal{L}=\{L_\alpha\}_{\alpha\in T}$ where $L_\alpha$ is the elementary lamination associated to $\alpha$. Let $T'$ be a triangulation obtained from $T$ by some finite sequence of flips. Then $\alpha' \in T'$ is said to be **green** if $$\{L \in L^\circ | b_{\alpha'}(T',L) > 0\} = \emptyset.$$ It is called **red** if $$\{L \in L^\circ | b_{\alpha'}(T',L) < 0\} = \emptyset.$$
It follows from Lemma \[lem:flip=mut2\] that an arc $\alpha' \in T'$ is green (resp., red) in the sense of Definition \[def:green2\] if and only if its corresponding vertex in $(\widetilde{Q_{T'}},F_\mathcal{L})$ is green (resp., red) in the sense of Definition \[def:green1\].
We also provide one more lemma about green sequences that we will use in the sequel.
\[lem:vertexdone\] Let $Q$ be a quiver and $i$ a vertex in $Q$. If there exists a frozen vertex $j'$ such that there is exactly one arrow incident to $j'$ and it points at $i$, then $i$ is red and will never be mutated at in any green sequence for $Q$.
Since $j'$ is a frozen vertex pointing at $i$, we have that $i$ is not green and by Theorem \[thm:signcoh\] it must be red and therefore cannot be mutated at in a green sequence. Mutating at any vertex other than $i$ in the quiver will not affect the edge $j' \rightarrow i$, so this edge will persist through any mutation sequence and $i$ will always be red.
Translating this lemma into the language of laminations, it says that if there exists an arc $\alpha$ and a lamination $L$ such that $$b'_{\gamma}(T,L)=\begin{cases} 1 & \gamma=\alpha, \\ 0 & \gamma \neq \alpha.\end{cases}$$ then $\alpha$ will be red in any green mutation sequence for $T$.
Cycle Lemma and Independent mutation sequence {#sec:cycleindep}
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In any triangulation with at least three distinct arcs (none of which are loops) incident to a puncture, these arcs form an oriented cycle in the corresponding quiver. The mutation sequences given in this paper make use of the following maximal green sequence for oriented cycles.
Let $i$ and $j$ be vertices of a quiver $Q$. Suppose $\mu$ is some mutation sequence for $Q$. Let $\overline{\mu(Q)}$ denote the quiver $\mu(Q)$ with the relabelling of the vertices that fixes the label on vertex $k$ if $k \neq i,j;$ but relabels $i$ as $j$ and vice versa. We say that $\mu$ **interchanges $i$ and $j$** if $Q = \overline{\mu(Q)}$.
[[@bucher Lemma 4.2]]{} \[cycle\] Let $C$ be a quiver that is an oriented $n$-cycle with vertices labeled $1,\ldots,n$, and edges $i \rightarrow (i-1)$ for $ 2 \leq i \leq n$ and $1 \rightarrow n$. Define the mutation sequence $${\mu_{\text{cycle}}}:=(n,{n-1}\ldots ,2,1, 3 ,4, \ldots, ,{n-1},n).$$ Then ${\mu_{\text{cycle}}}$ is a maximal green sequence for $C$ that interchanges 1 and 2.
It is helpful in the next section to understand how the Cycle Lemma affects triangulations.
\[lem:cyclesurface\] Let $T$ be a triangulation of a marked surface. Suppose that a puncture $P$ is incident to at least three distinct arcs, none of which are loops, so these arcs correspond to an oriented cycle in $Q_T.$ Let ${\mu_{\text{cycle}}}^P$ be the mutation sequence from Lemma \[cycle\] for this oriented cycle. Then ${\mu_{\text{cycle}}}^P(T)$ coincides with $T$ (up to relabelling of arcs) except all of the taggings of the arcs at $P$ differ.
Let $P$ and $T$ be as above. Without loss of generality we assume that all of the arcs incident to $P$ are tagged plain. Suppose ${\mu_{\text{cycle}}}^P = (\alpha_n, \dots, \alpha_2,\alpha_1, \alpha_3, \dots, \alpha_n).$ Let $\lambda = (\alpha_n , \dots , \alpha_3)$ be the first part of ${\mu_{\text{cycle}}}^P$. Note that in $\mu_{\alpha_2}\lambda(T)$ we have that $P$ is a radial puncture incident to $\alpha_1$ and $\alpha_2$, with $\alpha_2$ notched at $P$. Now when we mutate $\alpha_1$ it will again be incident to $P$ and must be tagged at $P$ by the compatibility rules for tagged arcs. Furthermore, the triangulation $\mu_{\alpha_1}\mu_{\alpha_2}\lambda(T)$ is identical to $\lambda(T)$ except that $\alpha_1$ and $\alpha_2$ are now notched at $P$. It follows then that the rest of ${\mu_{\text{cycle}}}^P$ mutates all the other arcs back into place, but by the compatibility rules for tagged arcs they must all be notched at $P$.
In the degenerate case when there are exactly two arcs incident to a puncture $P$, the corresponding vertices in the quiver are not connected by an edge. If $P$ is not a radial puncture then we define ${\mu_{\text{cycle}}}^P:=(1,2)$ and note that the result of Lemma \[lem:cyclesurface\] still applies in this case.
Independent mutation sequence
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Brústle and Qiu showed that it is necessary that a maximal green sequence for a triangulation of a surface with punctures must change the tagging at every puncture [@brustleqiu]. One might hope that we could apply ${\mu_{\text{cycle}}}$ to every puncture of a closed marked surface to obtain a maximal green sequence, but that is not the case.
The two issues with this approach are when there are two distinct punctures $P$ and $R$ with an arc between them and loops at punctures. The mutation sequence ${\mu_{\text{cycle}}}^R{\mu_{\text{cycle}}}^P$ would not be a green sequence as the arc between $P$ and $R$ will be red in ${\mu_{\text{cycle}}}^P(T)$. These problems motivate the following mutation sequence. This sequence mutates us to a triangulation where every puncture in a designated proper subset of punctures will not be the base point of any loops, and will not share an arc with any other puncture in the subset. Then we may proceed to apply the Cycle Lemma to each oriented cycle around each puncture in this subset.
\[def:independent\] Let $P$ and $R$ be two not necessarily distinct punctures of a closed surface with at least two punctures. We say that $P$ is **independent** of $R$ in $T$ if there is no arc in $T$ with one endpoint at $P$ and the other at $R$. A set $\mathcal{P}$ of marked points is called **independent** in $T$ if for any two not necessarily distinct points $P,R \in \mathcal{P}$ we have $P$ is independent of $R$.
\[def:indpath\] Let $\alpha$ be an arc. Let $\iota(\alpha)$ denote the underlying untagged arc of $\alpha$ if $\alpha$ is not incident to a radial puncture, or incident to a radial puncture with both endpoints tagged the same way. If $\alpha$ is incident to a radial puncture and its endpoints have different taggings let $\iota(\alpha)$ be a loop enclosing the radial puncture based at the other endpoint of $\alpha$.
Let $T$ be a triangulation of a closed marked surface and $\mathcal{P}$ a proper subset of its punctures. Define $$E_T^\mathcal{P}:=\{\alpha \in T | \iota(\alpha) \text{ has two endpoints in } \mathcal{P}\}.$$ An **independence path** for $\alpha \in E_T^\mathcal{P}$ is a path from some point $x_\alpha \in \alpha$ to some puncture not in $\mathcal{P}$, such that the path is disjoint from punctures of $\mathcal{P}$ and disjoint from any arcs not contained in $E_T^\mathcal{P}$.
\[ex:indpath\] Consider the triangulation given in Figure \[fig:indpath\]. Take $\mathcal{P}=\{P_i\}_{i=1}^8$. The set $$E_T^\mathcal{P}=\{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,\alpha,\beta\}.$$ An independence path for the arc labelled $\alpha$ is shown in the figure in green. Note that the arc $\beta \in E_T^\mathcal{P}$ since $\iota(\beta)$ is a loop based at $P_1$ enclosing the puncture $S$. We draw $\iota(\beta)$ on the triangulation in blue.
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(p1) at (3,0); (p2) at (0,3); (p3) at (-3,0); (p4) at (0,-3); (p5) at (1,1); (p6) at (-1,1); (p7) at (-1,-1); (p8) at (1,-1); (p9) at (5,0);
(p1) node\[above\][$P_1$]{} circle (1.5pt); (p2) node\[above\][$P_2$]{} circle (1.5pt); (p3) node\[left\][$P_3$]{} circle (1.5pt); (p4) node\[below\][$P_4$]{} circle (1.5pt); (p5) node\[above right\][$P_5$]{} circle (1.5pt); (p6) node\[above left\][$P_6$]{} circle (1.5pt); (p7) node\[below left\][$P_7$]{} circle (1.5pt); (p8) node\[below right\][$P_8$]{} circle (1.5pt); (p9) circle (1.5pt);
\(N) to node\[fill=white\][1]{} (E) to node\[fill=white\][2]{} (S) to node\[fill=white\][1]{} (W) to node\[fill=white\][2]{} (N); (p1) to node\[above right\][3]{} (p2) to node\[above left\][4]{} (p3) to node\[below left\][5]{} (p4) to node\[below right\][6]{} (p1); (p5) to node\[fill=white\][7]{} (p6) to node\[fill=white\][8]{} (p7) to node\[fill=white\][9]{} (p8) to node\[fill=white\][10]{} (p5); (p1) to node\[fill=white\][11]{} (p5) to node\[fill=white\][12]{} (p2) to node\[fill=white\][13]{} (p6) to node\[fill=white\][14]{} (p3) to node\[fill=white\][15]{} (p7) to node\[fill=white\][16]{} (p4) to node\[fill=white\][17]{} (p8) to node\[fill=white\][18]{} (p1); (p5) to node\[near start,fill=white\][$\alpha$]{}(p7); (N) to node\[fill=white\][19]{} (p2) to node\[fill=white\][20]{} (W) to node\[fill=white\][21]{} (p3) to node\[fill=white\][22]{} (S) to node\[fill=white\][23]{} (p4) to node\[fill=white\][24]{} (E) to node\[fill=white\][25]{} (p1) to node\[fill=white\][26]{} (N); (p1)\[bend left\] to node\[midway,fill=white\] [$\beta$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p9) node\[right\][$S$]{} to\[bend left\] node\[fill=white\][27]{} (p1); (p1) to\[out=-40,in=180\] (5.3,-.8) to\[out=0,in=240\] node\[fill=white\][28]{} (N); (p1) to\[out=40,in=90\] (5.5,0) to\[out=-90,in=-35\] (p1); (5,1) node [$\iota(\beta)$]{}; (0,0) to\[out=90,in=-45\] (W); (0,0) node\[below right\][$x_\alpha$]{}circle (2pt);
\[lem:indpathexists\] For any triangulation $T$, proper subset of punctures $\mathcal{P}$, and any arc $\alpha \in E_T^\mathcal{P}$ there exists an independence path for $\alpha$.
Let $T$, $\mathcal{P}$, and $\alpha$ be as above. Note two simple facts about independence paths. First, any arc in $E_T^\mathcal{P}$ that is an edge of a triangle which contains a vertex not in $\mathcal{P}$ has an independence path. Second, if one arc in a triangle has an independence path then every other arc in the triangle that is in $E_T^\mathcal{P}$ has an independence path.
Now suppose that some arc $\alpha \in E_T^\mathcal{P}$ does not have an independence path. Then any arc that shares a triangle with $\alpha$ also does not have an independence path. Continuing inductively, no arc in $E_T^\mathcal{P}$ is an edge of a triangle which has vertex not in $\mathcal{P}$. But since our marked surface is connected this is only possible if $T = E_T^\mathcal{P}$ or rather every puncture is in $\mathcal{P}$. A contradiction to our stipulation that $\mathcal{P}$ is a proper subset of punctures.
We define $$\sigma_T^{\mathcal{P}}(\alpha)= \inf \{ \text{number of arcs of } E_T^\mathcal{P} \text{ that are crossed by a separation path for } \alpha \}.$$ Note that $\sigma_T^{\mathcal{P}}$ is well-defined by Lemma \[lem:indpathexists\] and $\sigma_T^{\mathcal{P}}(\alpha) \geq 0$ where equality holds if and only if $\alpha$ is in a triangle with a vertex not in $\mathcal{P}$.
\[lem:sep\] Index the arcs of $E_T^\mathcal{P}=\{\alpha_i\}_{i=1}^m$ so that $i < j $ if and only if $\sigma_T^{\mathcal{P}}(\alpha_i) \leq \sigma_T^{\mathcal{P}} (\alpha_j)$. Then we define the mutation sequence $${\mu_{\text{Ind}}}^{\mathcal{P}} := (\alpha_1, \dots, \alpha_n).$$
Continuing with Example \[ex:indpath\] the triangulation given in Figure \[fig:indpath\] we have $\sigma_T^\mathcal{P}(\gamma)=0$ if $\gamma \in \{3,4,5,6,\beta\}$. The function $\sigma_T^\mathcal{P}(\gamma)=1$ for $\gamma \in \{11,12,13,14,15,16,17,18\}$, $\sigma_T^\mathcal{P}(\gamma)=2$ for $\gamma \in \{7,8,9,10\}$ and finally $\sigma_T^\mathcal{P}(\alpha)=3.$ Therefore one possible mutation sequence for ${\mu_{\text{Ind}}}^\mathcal{P}$ would be $${\mu_{\text{Ind}}}^\mathcal{P}=(3,4,5,6,\beta,11,12,13,14,15,16,17,18,7,8,9,10,\alpha).$$ We can rearrange the order of the arcs in the mutation sequence that take the same value on $\sigma_T^\mathcal{P}$ to obtain other mutation sequences.
\[lem:ind\] The set of punctures $\mathcal{P}$ is an independent set in ${\mu_{\text{Ind}}}^\mathcal{P}(T)$.
We show the claim by inducting on $\max\{\sigma_T^{\mathcal{P}}(\alpha) | \alpha \in E_T^\mathcal{P}\}$. Suppose $\sigma_T^{\mathcal{P}}$ takes the value 0 for all arcs in $E_T^\mathcal{P}$. Let $\alpha \in E_T^\mathcal{P}$. Then $\sigma_T^{\mathcal{P}}(\alpha) = 0$ if and only if $\alpha$ is in a triangle with a vertex not in $\mathcal{P}$, so let $R \not \in \mathcal{P}$ be a puncture that is a vertex of a triangle with $\alpha$ as an edge. Since $\alpha$ has both endpoints in $\mathcal{P}$ and is contained in a triangle with $R$ the arc obtained from flipping $\alpha$ will have an endpoint at $R$ so it will not be in $E_{\mu_\alpha(T)}^\mathcal{P}$. So $E_{{\mu_{\text{Ind}}}^\mathcal{P}(T)}^\mathcal{P}$ is empty and therefore $\mathcal{P}$ must be an independent set, so the claim is true when $\max\{\sigma_T^{\mathcal{P}}(\alpha) | \alpha \in E_T^\mathcal{P}\}=0$.
Assume that the claim is true for $k < \max\{\sigma_T^{\mathcal{P}}(\alpha) | \alpha \in E_T^\mathcal{P}\}$. Let $\lambda=(\alpha_1,\dots,\alpha_j)$ be the initial part of ${\mu_{\text{Ind}}}^\mathcal{P}$ that runs over all arcs in $E_T^\mathcal{P}$ where $\sigma_T^{\mathcal{P}}$ takes value 0. Let $\chi$ be the remaining part of ${\mu_{\text{Ind}}}^\mathcal{P}$ so that $\mu_\chi\mu_\lambda={\mu_{\text{Ind}}}^\mathcal{P}.$ Observe that $\lambda$ cannot be empty since $\mathcal{P}$ is a proper subset of punctures of the surface. It follows easily that for any arc $\alpha \in E_{{\mu_\lambda}(T)}^\mathcal{P}$ we have $\sigma_{\mu_\lambda(T)}^\mathcal{P}(\alpha)=\sigma_{T}^\mathcal{P}(\alpha)-1$ since any independence path for such an arc must have crossed one of the $\alpha_\ell$ in $T$, but will no longer cross the mutated arc $\alpha'_\ell$ in $\mu_\lambda(T)$. Furthermore the arcs of $E^\mathcal{P}_{\mu_\lambda(T)}$ are indexed by increasing value of $\sigma_{\mu_\lambda(T)}^\mathcal{P}$ so $\mu_\chi$ is a sequence that could be constructed in Construction \[lem:sep\] so by our inductive hypothesis $\mathcal{P}$ is in an independent set of punctures in $\mu_\chi(\mu_\lambda(T)).$ But $\mu_\chi\mu_\lambda = {\mu_{\text{Ind}}}^\mathcal{P}$ so we have proven the claim.
\[lem:greenind\] If every arc of $E_{T}^\mathcal{P}$ is green, then ${\mu_{\text{Ind}}}$ is a green sequence.
If $\mu_k$ is a green mutation then the only vertex that goes from green to red is $k$, and every arc that is mutated, is mutated exactly once in the sequence. Therefore ${\mu_{\text{Ind}}}$ is a green sequence.
\(a) at (-1,1); (b) at (1,1); (p2) at (0,1); (p1) at (1,0); (p) at (-1,0); (r) at (0,-1);
\(p) to (a) to (p2) to (b) to (p1) to node\[below\][$\beta$]{} (p2) to node\[left\][$\gamma$]{} (p) to (r) to (p1) to node\[below\][$\alpha$]{} (p); (-.5,1.5) node\[above\] [$L$]{} to\[out=-90,in=180\] (1.1,.5);
(p1) node\[right\][$P'$]{} circle (1.5pt); (p2) node\[above\][$P''$]{} circle (1.5pt); (p) node\[left\][$P$]{} circle (1.5pt); (r) node\[below\][$R$]{} circle (1.5pt); (a) circle (1.5pt); (b) circle (1.5pt);
\(a) at (-1,1); (b) at (1,1); (p2) at (0,1); (p1) at (1,0); (p) at (-1,0); (r) at (0,-1);
(p1) node\[right\][$P'$]{} circle (1.5pt); (p2) node\[above\][$P''$]{} circle (1.5pt); (p) node\[left\][$P$]{} circle (1.5pt); (r) node\[below\][$R$]{} circle (1.5pt); (a) circle (1.5pt); (b) circle (1.5pt);
\(p) to (a) to (p2) to (b) to (p1)to (r) to (p); (p) to node\[left\][$\gamma'$]{}(p2); (r) to node\[left\][$\alpha'$]{} (p2); (r) to node\[right\][$\beta'$]{}(b); (-.5,1.5) node\[above\] [$L$]{} to\[out=-90,in=180\] (1.1,.5);
\(a) at (-1,1); (b) at (1,1); (p2) at (0,1); (p1) at (1,0); (p) at (-1,0); (r) at (0,-1);
\(p) to (a) to (p2) to (b) to (p1) to (r) to (p); (r) to node\[left\][$\gamma'$]{}(a); (r) to node\[left\][$\alpha'$]{}(p2); (r) to node\[right\][$\beta'$]{}(b); (-.5,1.5) node\[above\] [$L$]{} to\[out=-90,in=180\] (1.1,.5);
(p1) node\[right\][$P'$]{} circle (1.5pt); (p2) node\[above\][$P''$]{} circle (1.5pt); (p) node\[left\][$P$]{} circle (1.5pt); (r) node\[below\][$R$]{} circle (1.5pt); (a) circle (1.5pt); (b) circle (1.5pt);
\[lem:cycleok\] Assume there are no radial punctures in $\mathcal{P}$. Suppose ${\mu_{\text{Ind}}}^\mathcal{P} = (\gamma_1,\dots,\gamma_n)$, and each $\gamma_i$ is green in a triangulation $T$. Note that $\gamma_i \neq \gamma_j$ for all $i \neq j$. Let $\alpha = \gamma_j$ for some $j = 1, \dots, n$. Let $\mu=(\gamma_1,\dots,\gamma_{j-1}).$ Let $\alpha'$ be the unique new arc obtained from mutating $\alpha$ in $\mu(T)$. By Lemma \[lem:ind\] $\alpha'$ can have at most one endpoint in $\mathcal{P}$.
1. If $\alpha'$ has exactly one endpoint in $\mathcal{P}$, then $\alpha'$ is green in ${\mu_{\text{Ind}}}^\mathcal{P}(T)$.
2. If $\alpha'$ is not incident to any puncture in $\mathcal{P}$, then $\alpha'$ is red in ${\mu_{\text{Ind}}}^\mathcal{P}(T)$.
Since the arc $\alpha$ was mutated in ${\mu_{\text{Ind}}}^{\mathcal{P}}$ it is in $E^\mathcal{P}_{T}$ and hence has both endpoints in $\mathcal{P}$. Let $P, P' \in \mathcal{P}$ be the endpoints of $\alpha.$ Note that it is possible for $P=P'$.
Suppose that $\alpha'$ has one endpoint $P'' \in \mathcal{P}$ and its other endpoint $R \not \in\mathcal{P}$. Then there exists a quadrilateral $PP''P'R$ in $\mu(T)$ with diagonals $\alpha$ and $\alpha'.$ Let $\beta$ be the arc counter-clockwise from $\alpha$ in the quadrilateral with both endpoints in $\mathcal{P}$. Then $\beta \in E_T^\mathcal{P}$ and $\beta = \gamma_k$ for some $k > j$. See the left triangulation in Figure \[fig:cycleok\]. We will show that when $\beta$ is mutated during ${\mu_{\text{Ind}}}^{\mathcal{P}}$ the arc $\alpha'$ will turn green. Suppose that $L$ is some lamination with $b_\beta(T,L)=-1$ and let $\gamma$ is the third arc of the triangle with edges $\alpha$ and $\beta.$ Then by our assumption that $\gamma$ is green $L$ must curve upwards in Figure \[fig:cycleok\]. It is then easy to see that $b_{\alpha'}(\mu_\beta \dots \mu_\alpha\mu(T),L)=-1$ so $\alpha'$ is green in $ \mu_\beta \dots \mu_\alpha\mu(T)$. Note that we must check both the case when $\gamma$ is mutated before $\beta$ in ${\mu_{\text{Ind}}}^\mathcal{P}$ and the case when $\beta$ is mutated first. We provide a picture of the triangulation $\mu_\beta \dots \mu_\alpha\mu(T)$ in both cases in Figure \[fig:cycleok\]. Since $\alpha$ is not mutated again in ${\mu_{\text{Ind}}}^\mathcal{P}$ it will still be green in ${\mu_{\text{Ind}}}^\mathcal{P}(T)$.
Suppose that $\alpha'$ is not incident to any puncture in $\mathcal{P}$. Let $\mu(T)$. Let $R,R' \not\in \mathcal{P}$ be the endpoints of $\alpha'$. Then there exists a quadrilateral $PRP'R'$ in $\mu(T)$ with diagonals $\alpha$ and $\alpha'.$ But notice that each arc composing the boundary of this quadrilateral is not in $E_{T'}^\mathcal{P}$ so none of them will be mutated in the remaining mutations of ${\mu_{\text{Ind}}}.$ Therefore if $L$ is some lamination such that $b_\alpha(T,L)=-1$ then we have $b_{\alpha'}({\mu_{\text{Ind}}}^{\mathcal{P}}(T),L)=1$ so $\alpha'$ is red in $\mu_\alpha\mu(T)$. Now as none of the arcs in the quadrilateral containing $\alpha'$ are mutated after $\alpha$ in ${\mu_{\text{Ind}}}^\mathcal{P}$ the arc $\alpha$ will remain red in ${\mu_{\text{Ind}}}^\mathcal{P}(T)$.
Construction of maximal green sequences for closed surfaces {#sec:proof}
===========================================================
Let $\Sigma$ be a closed marked surface of genus zero with at least four punctures, or a closed marked surface of genus at least one with at at least two punctures. It was shown by Ladkani in [@ladkani] that any once-punctured closed surface has no maximal green sequence.
Let $T$ be a triangulation of $\Sigma$. We assume for simplicity that all arcs are tagged plain at all punctures except for radial punctures. We may make this assumption because the corresponding quivers of two triangulations that differ only by the tagging at a puncture are isomorphic. For all $\alpha\in T$ let $\alpha^\circ$ denote the elementary lamination associated to $\alpha$. Fix the multi-lamination $\mathcal{L}=\{\alpha^\circ\}_{\alpha \in T}$.
$$\begin{xy} 0;<.75pt,0pt>:<0pt,-.65pt>::
(0,0) *+{1} ="1",
(25,200) *+{2} ="2",
(400,300) *+{3} ="3",
(400,225) *+{4} ="4",
(350,250) *+{5} ="5",
(300,300) *+{6} ="6",
(300,200) *+{7} ="7",
(400,150) *+{8} ="8",
(400,400) *+{9} ="9",
(275,400) *+{10} ="10",
(275,350) *+{11} ="11",
(175,400) *+{12} ="12",
(0,400) *+{13} ="13",
(125,350) *+{14} ="14",
(50,325) *+{15} ="15",
(225,325) *+{16} ="16",
(250,275) *+{17} ="17",
(125,300) *+{18} ="18",
(215,235) *+{19} ="19",
(150,250) *+{20} ="20",
(250,200) *+{21} ="21",
(300,100) *+{22} ="22",
(400,0) *+{23} ="23",
(225,50) *+{24} ="24",
(225,25) *+{25} ="25",
(200,200) *+{26} ="26",
(135,205) *+{27} ="27",
(75,250) *+{28} ="28",
(95,185) *+{29} ="29",
(225,150) *+{30} ="30",
(50,150) *+{31} ="31",
(75,33) *+{32} ="32",
(50,100) *+{33} ="33",
(150,125) *+{34} ="34",
(100,125) *+{35} ="35",
(91,86) *+{36} ="36",
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"33", {\ar"36"},
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"34", {\ar"36"},
"36", {\ar"35"},
\end{xy}$$
\(a) at (1.00000000000000,0.000000000000000); (b) at (0.707106781186548,0.707106781186548); (c) at (0.00000000000000,1.00000000000000); (d) at (-1,1); (e) at (-1.6,0); (f) at (-1,-1); (g) at (0,-1.00000000000000); (h) at (0.707106781186548,-0.707106781186548); (i) at (0.0,0); (q5) at (-.1,0.5); (q2) at (-1,0); (q3) at (-.4,-.15); (q4) at (-.5,-.3); (r1) at (-1.1,-.2); (r2) at (-.9,-.2); (r3) at (-1,-.4); (r4) at (-.5,-.9);
(mb1) at (.2357,.098-1); (mb2) at (2\*.2357,2\*.098-1); (mt1) at (.2357,-.098+1); (mt2) at (2\*.2357,-2\*.098+1); (mt3) at (.855,0.707106781186548/2); (mb3) at (.855,-0.707106781186548/2);
\(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (2.5pt); (q5) circle (2.5pt); (q4) circle (2.5pt); (q3) circle (2.5pt); (q2) circle (2.5pt); (r1) circle (2.5pt); (r2) circle (2.5pt); (r3) circle (2.5pt); (r4) circle (2.5pt);
\(c) to node \[above\] [2]{} (d) to node \[left\] [1]{} (e) to node \[left\] [2]{} (f) to node \[below\] [1]{} (g);
\(a) to (b) to (c); (g) to (h) to (a);
(mb1) to node \[midway,fill=white\][4]{} (i) to node \[midway,fill=white\][3]{} (mt1); (mb2) to node \[midway,fill=white\][5]{} (i) to node \[midway,fill=white\][5]{} (mt2); (mb3) to node \[midway,fill=white\][3]{} (i) to node \[midway,fill=white\][4]{} (mt3);
(mt4) at (.9,.23) ; (mb4) at (.9,-.23); (mt4) to node \[left\][5]{} (mb4);
\(i) to\[bend right\] node \[midway,fill=white\][14]{} (e); (g) to node \[midway,fill=white\][8]{} (i); (i) to node \[midway,fill=white\][9]{} (c); (i) to\[out=135, in = 225, min distance=25mm\] node \[left,fill=white\][12]{} (c); (i) to node \[midway,fill=white\][10]{} (q5); (q5) to node \[midway,fill=white\][11]{} (c); (e) to node \[midway,fill=white\][13]{} (c); (f) to\[out=125,in=180\] node \[midway,fill=white\][28]{} (q2); (f) to\[out=45, in = 0\] node \[midway,fill=white\][27]{} (q2); (i) to node \[midway,fill=white\][6]{} (h); (i) to node \[midway,fill=white\][7]{} (b); (q2) to node \[midway,fill=white\][15]{} (e); (q2) to node \[midway,fill=white\][16]{} (i); (q3) to node \[midway,fill=white\][17]{} (i); (q3) to node \[midway,fill=white\][19]{} (q4); (q3) to\[out=155,in=0\] node \[near start,fill=white\][18]{} (q2); (q4) to\[out=135,in=0\] node \[near start,fill=white\][20]{} (q2); (q4) to\[out=235,in=45\] node \[midway,fill=white\][26]{} (f); (q4) to node \[midway,fill=white\][21]{} (i); (i) to\[out=250,in=45\] node \[midway,fill=white\][22]{}(f); (g) to\[out=90,in=45\] node \[midway,fill=white\][23]{} (f); (q2) to node \[midway,fill=white\][35]{} (r1); (q2) tonode \[midway,fill=white\][34]{} (r2); (q2) to\[out=-15,in = 0,min distance=12mm\] node \[near end,fill=white\][30]{} (r3); (q2) to\[out = 195, in = 180,min distance=12mm\] node \[near end,fill=white\][31]{} (r3); (r2) to node \[midway,fill=white\][36]{} (r1); (r1) to node \[midway,fill=white\][33]{} (r3); (r2) to node \[midway,fill=white,inner sep=0pt,outer sep=1pt\][32]{} (r3);
(q2) to\[out=0,in = 0,min distance=14mm\] (-1,-.5); (q2) to\[out = 180, in = 180,min distance=14mm\] (-1,-.5); (-1,-.5) node \[fill=white\][29]{}; (g) to node \[midway,fill=white\][25]{} (r4); (g) to\[bend right\] node \[near end,sloped,rotate=90\][$\bowtie$]{} node \[midway,fill=white\][24]{} (r4);
\(a) node\[right\][[$X$]{}]{}; (d) node\[left\][[$X$]{}]{}; (b) node\[right\][[$X$]{}]{}; (c) node\[right\][[$X$]{}]{}; (e) node\[left\][[$X$]{}]{}; (f) node\[left\][[$X$]{}]{}; (g) node\[right\][[$X$]{}]{}; (h) node\[right\][[$X$]{}]{};
(.2,0) node [$P_1$]{}; (q2) node\[above\] [$P_2$]{}; (q3) node\[above\] [$P_3$]{}; (q4) node\[below right\] [$P_4$]{}; (q5) node\[above left\] [$P_5$]{}; (r4) node\[left\][$S_1$]{};
Choose any puncture that is not in the interior of a monogon or a radial puncture and label it $X$. Let $\mathcal{S}$ denote the set of all radial punctures of $\Sigma$ in $T$. Let $\mathcal{M}$ be the set of all punctures of $\Sigma$ that are not in $\mathcal{S}$ and are not $X$. We now define a partition on the set $\mathcal{M}$.
Let $\mathcal{M}_0$ be the set of all punctures of $\mathcal{M}$ that are not in the interior of any monogon. Define $\mathcal{M}_{i+1}$ to be the punctures of $\mathcal{M}$ that
1. are in the interior of a monogon based at a puncture in $\mathcal{M}_i$;
2. but not in the interior of any monogon based at a puncture in $\mathcal{M} \setminus \bigcup_{j=0}^{i}\mathcal{M}_j.$
Consider the triangulation $T^*$ given in Figure \[fig:main\_example\_triangulation\]. We choose $X$ to be the puncture in the exterior of the diagram. Then $$\mathcal{S}=\{S_1\}, \mathcal{M}_0=\{P_1, P_2,P_3,P_4,P_5\}, \text{ and } \mathcal{M}_1=\{R_1,R_2,R_3\}.$$
Our maximal green sequence will initially focus on changing the taggings at the punctures of $\mathcal{M}$ and then change the tagging at $X$. The taggings of punctures of $\mathcal{S}$ will be changed during this process.
Independence of punctures.
--------------------------
Note that it is possible for the set $\mathcal{M}_0$ to be empty. If this is the case then we may skip ahead to Subsection \[sec:muI\].
In this step we take $\mathcal{P}= \mathcal{M}_0$ in Construction \[lem:sep\] and apply ${\mu_{\text{Ind}}}^{\mathcal{M}_0}$ to $T$. No arcs have been mutated yet so ${\mu_{\text{Ind}}}^{\mathcal{M}_0}$ is a green sequence for $T$ by Lemma \[lem:greenind\].
\[ex:main\_sep\] In the triangulation $T^*$ in Figure \[fig:main\_example\_triangulation\] we have $\mathcal{M}_0= \{P_1, P_2,P_3,P_4,P_5\}$ and it follows that $$E_T^{\mathcal{M}_0} = \{ 3,4,5,10,16,20,21,29,17,18,19\}.$$ Note that $\sigma_T^{\mathcal{M}_0}(\alpha)=0$ for $ \alpha \in \{3,4,5,10,16,20,21,29\},$ and $\sigma_T^{\mathcal{M}_0}(\alpha)=1$ for $\alpha \in \{17,18,19\},$ so $${\mu_{\text{Ind}}}^{\mathcal{M}_0}=(3, 4, 5, 10, 16, 17, 18, 19, 20, 21, 29).$$ The order that we mutate the arcs $3,4,5,10,16,20,21,$ and $29$ does not matter, but we will order them by their labels. We adopt this convention in the sequel when we have freedom to do so. See Figure \[fig:main\_example\_triangulation\_3\] for a picture of the triangulation ${\mu_{\text{Ind}}}^{\mathcal{M}_0}(T^*).$
Mutating cycles around punctures.
---------------------------------
Label the punctures $\mathcal{M}_0=\{P_i\}_{i=1}^n$ so that in ${\mu_{\text{Ind}}}^{\mathcal{M}_0}(T)$ we have $P_i$ is not a radial puncture for $i=1, \dots, t$, and $P_i$ is a radial puncture for $i=t+1, \dots, n$.
Now in ${\mu_{\text{Ind}}}^{\mathcal{M}_0}(T)$ we apply ${\mu_{\text{cycle}}}^{P_i}$ from Lemma \[lem:cyclesurface\] to the arcs incident to $P_i$ for $i=1,\dots, t$. By Lemma \[lem:ind\] we have that there are no loops based at each $P_i$ and all of these mutation sequences ${\mu_{\text{cycle}}}^{P_i}$ will be disjoint. We define the sequence $${\mu_{\text{cycle}}}^{\mathcal{M}_0}:={\mu_{\text{cycle}}}^{P_t} \dots {\mu_{\text{cycle}}}^{P_1}.$$ The mutation sequence ${\mu_{\text{cycle}}}^{\mathcal{M}_0}$ is a green sequence for ${\mu_{\text{Ind}}}^{\mathcal{M}_0}(T)$ by Lemma \[lem:cycleok\].
\[ex:main\_3\] In ${\mu_{\text{Ind}}}^{\mathcal{M}_0}(T^*)$ we have that $P_4$ and $P_5$ are radial punctures so we will not mutate any arcs adjacent to them in ${\mu_{\text{cycle}}}^{\mathcal{M}_0}.$ For the other three punctures of ${\mathcal{M}_0}$ we have $${\mu_{\text{cycle}}}^{P_1}= (3, 6, 7, 9, 12, 14, 22, 8, 14, 12, 9, 7, 6, 3),$$ which interchanges arcs 22 and 8; $${\mu_{\text{cycle}}}^{P_2}= (15, 28, 31, 35, 34, 30, 27, 34, 35, 31, 28, 15),$$ which interchanges arcs 30 and 27; and finally $${\mu_{\text{cycle}}}^{P_3}= (16, 20, 21, 16),$$ which interchanges arcs 20 and 21. See Figure \[fig:main\_example\_triangulation\_3\].
Mutating back to our initial triangulation. {#subsec:indstar}
-------------------------------------------
We now apply a slightly modified version of $({\mu_{\text{Ind}}}^{\mathcal{M}_0})^{-1}$ to mutate back to our original triangulation. This modification accounts for the punctures $P_i \in {\mathcal{M}_0}$ for $i = t+1, \dots, n$, which we did not apply ${\mu_{\text{cycle}}}$ in the previous step and arcs that were interchanged during ${\mu_{\text{cycle}}}^{\mathcal{M}_0}.$
For each $P_i \in \mathcal{M}_0$ with $i = t+1, \dots, n$, that is each puncture that is a radial puncture in ${\mu_{\text{cycle}}}^{\mathcal{M}_0}{\mu_{\text{Ind}}}^{\mathcal{M}_0}(T)$, let $\alpha_{P_i}$ be the arc mutated during ${\mu_{\text{Ind}}}^{\mathcal{M}_0}$ and $\beta_{P_i}$ the other arc incident to ${P_i}$. We define a sequence ${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}$ that is the same as ${\mu_{\text{Ind}}}^{-1}$ except;
1. We replace $\alpha_{P_i}$ with $\beta_{P_i}$ in ${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}$ for $i = t+1, \dots, n$;
2. If $\alpha$ appears in ${\mu_{\text{Ind}}}^{\mathcal{M}_0}$ and was interchanged with $\beta$ during ${\mu_{\text{cycle}}}^{\mathcal{M}_0}$ then we replace $\alpha$ with $\beta$ in ${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}$.
Note that for both $\alpha_{P_i}$ and $\beta_{P_i}$ their other endpoint opposite of $P_i$ is $X$, so they were not mutated during ${\mu_{\text{cycle}}}^{\mathcal{M}_0}$.
\[ex:main\_4\] In our running example we have ${\mu_{\text{Ind}}}^{\mathcal{M}_0}=(3, 4, 5, 10, 16, 20, 21, 29, 17, 18, 19).$ To construct ${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}$ we look at $( {\mu_{\text{Ind}}}^{\mathcal{M}_0})^{-1}$ then (1) tells us to replace 10 with 11 and 19 with 26; (2) tells us to replace 20 with 21 and vice versa. That is $${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}=(26, 18, 17, 29, 20, 21, 16, 11, 5, 4, 3).$$ See Figure \[fig:main\_example\_triangulation\_4\].
\[lem:reverse\] ${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}$ is a green sequence for $T'={\mu_{\text{cycle}}}^{\mathcal{M}_0}{\mu_{\text{Ind}}}^{\mathcal{M}_0}(T)$.
Suppose ${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}=(\alpha_1, \dots, \alpha_n)$. Define $\mu_j=\mu_{\alpha_j}\dots \mu_{\alpha_1}$. Let $\alpha_i^T$ denote the arc in our initial triangulation $T$ that corresponds to the same vertex of the associated quiver as $\alpha_i$. Consider the elementary lamination $(\alpha_{j'}^{T})^\circ$, where $j'=j$ if $\alpha_j$ was not interchanged with another arc during ${\mu_{\text{cycle}}}^{\mathcal{M}_0}, $ and $j'=i$ if $\alpha_j$ was interchanged with $\alpha_i$ during ${\mu_{\text{cycle}}}^{\mathcal{M}_0}.$ Then we have $$b_{\alpha_j}(\mu_{j-1}(T'),(\alpha_{j'}^{T})^{\circ})
=-1.$$ Therefore at every step of ${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}$ we mutate at a green vertex.
It is worthwhile to note that ${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}{\mu_{\text{Ind}}}^{\mathcal{M}_0}{\mu_{\text{Ind}}}^{\mathcal{M}_0}(T)$ coincides with our original triangulation except that the taggings at all of the punctures of $\mathcal{M}_0$ are different. This observation gives rise to the fact that any arcs whose endpoints are exclusively punctures of ${\mathcal{M}_0}$ in ${\mu_{\text{Ind}}}^{\mathcal{M}_0,*}{\mu_{\text{cycle}}}^{\mathcal{M}_0}{\mu_{\text{Ind}}}^{\mathcal{M}_0}(T)$ will no longer have to be mutated in our mutation sequence. This also applies for any arcs whose endpoints are a radial puncture in $\mathcal{S}$ and a puncture in $\mathcal{M}_0$.
\[lem:done1\] Any arc with endpoint exclusively in $\mathcal{M}_0$ in $T'={\mu_{\text{Ind}}}^{\mathcal{M}_0,*}{\mu_{\text{cycle}}}^{\mathcal{M}_0}{\mu_{\text{Ind}}}^{\mathcal{M}_0}(T)$ is done being mutated.
Let $\alpha$ be an arc whose endpoints are both in $\mathcal{M}_0$ and let $\alpha^T$ denote the arc in $T$ that corresponds to the same vertex of the associated quiver as $\alpha$. Then $$b_{\gamma}(T',(\alpha^{T})^{\circ})=\begin{cases} 1 & \gamma=\alpha, \\ 0 & \gamma \neq \alpha.\end{cases}$$ Therefore $\alpha$ is done being mutated by Lemma \[lem:vertexdone\].
\[lem:done2\] If $\alpha$ is an arc with one endpoint in $\mathcal{M}_0$ and the other at a puncture of $\mathcal{S}$ then $\alpha$ is done being mutated.
Suppose the end points of $\alpha$ are $P \in \mathcal{M}_0$ and $S \in \mathcal{S}$. Let $\beta$ be the unique other arc with endpoints $P$ and $S$. Then $$b_{\alpha}(T',\beta^{\circ})= 1.$$ Therefore $\alpha$ is done being mutated by Lemma \[lem:vertexdone\].
We now iteratively apply a similar mutation sequence to each set of punctures $\mathcal{M}_i$ for $i \geq 1.$
Punctures in the interior of monogons. {#sec:muI}
--------------------------------------
For $i >0$;
- Let ${\mu_{\text{Ind}}}^{\mathcal{M}_i}$ be a mutation sequence from Construction \[lem:sep\] with $\mathcal{P}=\mathcal{M}_i$.
- Take ${\mu_{\text{cycle}}}^{\mathcal{M}_i}$ to be the composition of mutation sequences ${\mu_{\text{cycle}}}^{P}$ for each cycle around a puncture $P \in \mathcal{M}_i,$ that is not a radial puncture in $ {\mu_{\text{Ind}}}^{\mathcal{M}_i}{\mu_{\text{Ind}}}^{\mathcal{M}_{i-1},*}{\mu_{\text{cycle}}}^{\mathcal{M}_{i-1}}{\mu_{\text{Ind}}}^{\mathcal{M}_{i-1}} \cdots {\mu_{\text{Ind}}}^{\mathcal{M}_0,*}{\mu_{\text{cycle}}}^{\mathcal{M}_0}{\mu_{\text{Ind}}}^{\mathcal{M}_0}(T).$
- Construct ${\mu_{\text{Ind}}}^{\mathcal{M}_i,*}$ as we did in Subsection \[subsec:indstar\].
Suppose the highest index for nonempty $\mathcal{M}_i$ is $k$. Define $$\mu^\mathcal{M}:={\mu_{\text{Ind}}}^{\mathcal{M}_k,*}{\mu_{\text{cycle}}}^{\mathcal{M}_k}{\mu_{\text{Ind}}}^{\mathcal{M}_k} \cdots {\mu_{\text{Ind}}}^{\mathcal{M}_0,*}{\mu_{\text{cycle}}}^{\mathcal{M}_0}{\mu_{\text{Ind}}}^{\mathcal{M}_0}.$$
The proof that each mutation sequence ${\mu_{\text{Ind}}}^{\mathcal{M}_i,*}{\mu_{\text{cycle}}}^{\mathcal{M}_i}{\mu_{\text{Ind}}}^{\mathcal{M}_i}$ is a green sequence follows from an identical argument as in the sections above.
\[ex:main\_5\] In our running example of $T^*$ we have: $${\mu_{\text{Ind}}}^{\mathcal{M}_1} = (32,33,36).$$ After ${\mu_{\text{Ind}}}^{\mathcal{M}_1}$ we have two degenerate cycles consisting of 27 and 31, and 32 and 35 so $${\mu_{\text{cycle}}}^{\mathcal{M}_1} = (27, 31,32, 35).$$ Now in $({\mu_{\text{Ind}}}^{\mathcal{M}_1})^{-1}$ we replace 36 by 34 as they are incident to a radial puncture, and as 35 was interchanged with 32 we replace 32 with 35 so that $${\mu_{\text{Ind}}}^{\mathcal{M}_1,*} = (24,33,35).$$ See Figures \[fig:main\_example\_triangulation\_4\] and \[fig:main\_example\_triangulation\_5\].
As mentioned previously, Ladkani showed that once-punctured closed surfaces do not admit quivers with maximal green sequences. In this case the procedure we describe here reduces to mutating all the loops based at the puncture so they are no longer loops, applying Lemma \[cycle\], and then mutating the loops back into place. However, our approach fails since every arc in the triangulation is a loop and there is no way to mutate them into arcs that are not loops.
Changing the tagging at X
-------------------------
Let $T'=\mu^\mathcal{M}(T).$ Take $\mathcal{P}= \{X\}$ in Construction \[lem:sep\] to construct a mutation sequence ${\mu_{\text{Ind}}}^X$. Note that this mutation sequence only mutates loops at $X$ and none of these arcs have been mutated yet so they are all green. Therefore ${\mu_{\text{Ind}}}^X$ is a green sequence for $T'$. We apply a mutation sequence ${\mu_{\text{cycle}}}^X$ to the arcs incident to $X$ in ${\mu_{\text{Ind}}}^X(T')$. This changes the tagging at puncture $X.$ The fact that ${\mu_{\text{cycle}}}^X$ is a green sequence for ${\mu_{\text{Ind}}}^X(T')$ follows from Lemma \[lem:cycleok\].
We also define a mutation sequence ${\mu_{\text{Ind}}}^{X,*}$ in a similar way as we did above. However since $X$ is the only puncture in our independent set we do not create any new radial punctures during ${\mu_{\text{Ind}}}^X$. Therefore when constructing ${\mu_{\text{Ind}}}^{X,*}$ we only need to apply the modification from above which replaces arcs that were interchanged during ${\mu_{\text{cycle}}}^X$. The fact that ${\mu_{\text{Ind}}}^{P,*}$ is green sequence for ${\mu_{\text{cycle}}}^P{\mu_{\text{Ind}}}^P(T')$ follows the same proof as in Lemma \[lem:reverse\].
\[ex:main\_6\] We construct ${\mu_{\text{Ind}}}^X$, ${\mu_{\text{cycle}}}^X$ and ${\mu_{\text{Ind}}}^{X,*}$ for $\mu^\mathcal{M}(T^*)$ similar to as we did above. We first make $X$ independent of itself by applying the mutation sequence $${\mu_{\text{Ind}}}^X = (2,13, 23, 24, 1).$$ Note that $24$ is mutated here because $\iota(24)$ is a loop based at $X$. Then we apply the Cycle Lemma to the arcs incident to $X$ to obtain the sequence $${\mu_{\text{cycle}}}^X = (6, 7, 22, 25, 23, 15, 14, 2, 8, 19, 30, 28, 12, 10, 9, 12, 28, 30, 19, 8, 2, 14, 15, 23, 25, 22, 7, 6).$$ There is no modification to make to $({\mu_{\text{Ind}}}^{X})^{-1}$ to obtain ${\mu_{\text{Ind}}}^{X,*}$ because arcs 10 and 11 are not mutated during ${\mu_{\text{Ind}}}^{X}$ so $${\mu_{\text{Ind}}}^{X,*} = (1, 24, 23, 13, 2).$$ See Figures \[fig:main\_example\_triangulation\_5\] and \[fig:main\_example\_triangulation\_6\].
Maximal Green Sequence
----------------------
Putting together the previous lemmas we have the following theorem.
\[thm:mgsclosed\] For any triangulation $T$ of a closed marked surface with at least two punctures the sequence $${\mu_{\text{Ind}}}^{X,*}{\mu_{\text{cycle}}}^X{\mu_{\text{Ind}}}^X\mu^\mathcal{M}$$ is a maximal green sequence for $Q_T$.
From our work above we have shown that this mutation sequence is a green sequence. It remains to be shown that it is in fact maximal. This follows from the same kind of arguments made at the end of Subsection \[subsec:indstar\] in the proofs of Lemmas \[lem:done1\] and \[lem:done2\].
We have that ${\mu_{\text{Ind}}}^{X,*}{\mu_{\text{cycle}}}^X{\mu_{\text{Ind}}}^X\mu^\mathcal{M}(T)$ coincides with our original triangulation $T$ except that the tagging of arcs differs at every puncture of $\Sigma$. Therefore if the underlying untagged arc of $\alpha \in {\mu_{\text{Ind}}}^{X,*}{\mu_{\text{cycle}}}^X{\mu_{\text{Ind}}}^X\mu^\mathcal{M}(T)$ coincides with the underlying untagged arc of $\beta \in T$ then $$b_\alpha({\mu_{\text{Ind}}}^{X,*}{\mu_{\text{cycle}}}^X{\mu_{\text{Ind}}}^X\mu^\mathcal{M}(T),\beta^\circ) = 1.$$ Therefore every arc in ${\mu_{\text{Ind}}}^{X,*}{\mu_{\text{cycle}}}^X{\mu_{\text{Ind}}}^X\mu^\mathcal{M}(T)$ is red.
Existence of maximal green sequences for surfaces with nonempty boundary {#sec:boundary}
========================================================================
\(a) at (0,0); (b) at (6,0); (c) at (6,6); (d) at (0,6); (g) at (2,2); (f) at (4,2); (e) at (3,4); (a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt);
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\(d) to node \[midway,fill=white\][3]{} (e); (e) to node \[midway,fill=white\][4]{} (c); (e) to\[bend left\] node \[midway,fill=white\][5]{} (b); (f) to node \[midway,fill=white\][6]{} (b); (f) to node \[midway,fill=white\][7]{} (a); (g) to node \[midway,fill=white\][8]{} (a); (g) to node \[midway,fill=white\][9]{} (d); (g) to (e); (e) to (f); (g) to (f);
\(a) node\[left\][[$P$]{}]{}; (d) node\[left\][[$P$]{}]{}; (b) node\[right\][[$P$]{}]{}; (c) node\[right\][[$P$]{}]{};
\(g) – (e) – (f) – (g);
0;<.3pt,0pt>:<0pt,-.25pt>:: (0,0) \*+[1]{} =“0”, (250,225) \*+[2]{} =“1”, (100,175) \*+[3]{} =“2”, (100,75) \*+[4]{} =“3”, (400,75) \*+[5]{} =“4”, (400,225) \*+[6]{} =“5”, (400,400) \*+[7]{} =“6”, (100,400) \*+[8]{} =“7”, (100,300) \*+[9]{} =“8”, “3”, [“0”]{}, “0”, [“4”]{}, “7”, [“0”]{}, “0”, [“8”]{}, “2”, [“1”]{}, “1”, [“3”]{}, “5”, [“1”]{}, “1”, [“6”]{}, “3”, [“2”]{}, “8”, [“2”]{}, “4”, [“3”]{}, “4”, [“5”]{}, “6”, [“5”]{}, “6”, [“7”]{}, “8”, [“7”]{},
We first recall a theorem from Muller and the definition of an induced subquiver.
Given a subset $V$ of vertices of a quiver $Q,$ the **induced subquiver**, is the quiver with vertex set $V$ and edges consisting of the edges between pairs of vertices in $V$ that are in $Q$.
[[@muller Lemma 1.4.1]]{}\[thm:subquiver\] If a quiver admits a maximal green sequence, then any induced subquiver admits a maximal green sequence.
By the previous theorem it suffices to show that the quiver obtained from any triangulation of a surface with boundary is an induced subquiver of a quiver from a closed surface.
Suppose $\Sigma$ is a marked surface with boundary. We construct a marked surface without boundary $\overline{\Sigma}$ by gluing disks to the boundary components of $\Sigma$. To be more precise, for each boundary component $b_i$ of $\Sigma$ with $m_i$ marked points we glue a disk $D_i$ with $m_i$ marked points on its boundary and a single puncture if $m_i=1$ or $2$.
If $T$ is a triangulation of $\Sigma$ we can extend it to a triangulation $\overline{T}$ of $\overline{\Sigma}$. Let $B$ be a set of arcs of $\overline{\Sigma}$ that are isotopic to sections of the boundary component $\Sigma$. Then consider the collection of arcs $T \cup B$. This is a partial triangulation of the surface $\overline \Sigma$. We add more arcs to this collection to obtain a triangulation $\overline T$ of $\overline{\Sigma}$. Furthermore, by deleting vertices corresponding to the arcs of $\overline T \setminus T$ from $Q_{\overline{T}}$ we see that resulting quiver is $Q_T.$ That is $Q_T$ is an induced subquiver of $Q_{\overline{T}}$. To summarize we have the following lemma.
\[lem:subquiver\] Let $T$ be any triangulation of $\Sigma$. Let $\overline{\Sigma}$ be as above. Then the quiver $Q_T$ is an induced subquiver of the quiver $Q_{\overline{T}}$ corresponding to the triangulation $\overline{T}$ of $\overline{\Sigma}$.
\[thm:bdy\] The quiver $Q_T$ has a maximal green sequence.
By Theorem \[thm:mgsclosed\] $Q_{\overline{T}}$ has a maximal green sequence and by Lemma \[lem:subquiver\] $Q_T$ is an induced subquiver of $Q_{\overline{T}}$. Therefore by Theorem \[thm:subquiver\] the quiver $Q_T$ has a maximal green sequence.
Exceptional quivers of finite mutation type {#sec:exceptional}
===========================================
Recall, a quiver $Q$ is said to be of finite mutation type if $\operatorname{Mut}(Q)$ is finite. Felikson, Shapiro, and Tumarkin showed that every quiver of finite mutation type has rank 2, arises from a triangulation of marked surface, or is in one of 11 exceptional mutation classes [@felikson]. As we mentioned in the introduction every rank 2 quiver has a simple maximal green sequence. The 11 exceptional mutation classes are represented by the quivers $\mathbb{E}_6,\mathbb{E}_7,\mathbb{E}_8,\widetilde{\mathbb{E}_6},\widetilde{\mathbb{E}_7},\widetilde{\mathbb{E}_8},\mathbb{E}_6^{(1,1)},\mathbb{E}_7^{(1,1)},\mathbb{E}_8^{(1,1)}, \mathbb{X}_6,$ and $\mathbb{X}_7.$
For four of these cases it is known whether or not these quivers have a maximal green sequence.
[[@brustle Theorem 4.1]]{}\[thm:finite\] Every quiver mutation equivalent to $\mathbb{E}_6,\mathbb{E}_7,$ and $\mathbb{E}_8$ has a maximal green sequence.
[[@seven]]{}\[thm:x7\] Neither quiver in the mutation class of $\mathbb{X}_7$ has a maximal green sequence.
We used the cluster algebra package in Sage developed by Gregg Musiker and Christian Stump to compute an explicit maximal green sequence for every quiver in the remaining 7 exceptional mutation classes. The maximal green sequences can be found on the authors webpage [@webpage].
\[thm:main2\] If $Q$ is a quiver that is mutation equivalent to $\widetilde{\mathbb{E}_6},\widetilde{\mathbb{E}_7},\widetilde{\mathbb{E}_8},\mathbb{E}_6^{(1,1)},\mathbb{E}_7^{(1,1)},\mathbb{E}_8^{(1,1)}$ or $\mathbb{X}_6,$ then $Q$ has a maximal green sequence.
Combining Theorems \[thm:mgsclosed\], \[thm:bdy\], \[thm:finite\], \[thm:x7\], and \[thm:main2\] and the fact about rank 2 quivers we have a complete classification of all quivers of finite mutation type which have a maximal green sequence.
\[thm:main3\] If $Q$ is a quiver of finite mutation type, then $Q$ has a maximal green sequence unless it arises from a triangulation of a once-punctured closed surface, or is one of the two quivers in the mutation class of $\mathbb{X}_7$.
Acknowledgements
================
The author would like to thank Kyungyong Lee for many helpful discussions, Pierre-Guy Plamondon for translating the abstract of the original work [@mills] to French, and Khrystyna Serhiyenko and Greg Muller for helpful suggestions to improve the manuscript.
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Canakci, I., Lee, K., Schiffler, R., *On cluster algebras from unpunctured surfaces with one marked point,* Proc. Amer. Math. Soc. Ser. B 2 (2015), 35-49.
Cormier, E., Dillery, P., Resh, J., Serhiyenko, K., Whelan, J., *Minimal Length Maximal green Sequences and Triangulations of Polygons*, arXiv:1508.02954.
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Fock, V., Goncharov, A., *Moduli Spaces of Local Systems and Higher Teichmuller Theory*. Publ. Math. Inst. Hautes Etudes Sci. **103** (2006), 1-211.
Fock, V. and Goncharov, A., *Dual Teichmüller and lamination spaces.* Handbook of Teichmüller theory. Vol. I, 647-684, IRMA Lect. Math. Theor. Phys., 11, Eur. Math. Soc., Zürich, 2007.
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\(c) to node \[above\] [2]{} (d) to node \[left\] [1]{} (e) to node \[left\] [2]{} (f) to node \[below\] [1]{} (g);
\(a) to (b); (b) to node \[above\] [5]{} (c); (g) to (h); (h) to node \[right\] [5]{} (a);
\(i) to node \[midway,fill=white\][3]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(mb2); (mt3) to node \[midway,fill=white\][3]{} (h); (b) to node \[midway,fill=white\][4]{} (h); (mt4) at (.9,.23) ; (mb4) at (.9,-.23); (i) to\[bend right\] node \[midway,fill=white\][14]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(e); (g) to node \[midway,fill=white\][22]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(i); (i) to\[bend right\] node \[midway,fill=white\][9]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(c); (i) to\[out=135, in = 225, min distance=25mm\] node \[left,fill=white\][12]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(c); (c) to\[bend right\] node \[midway,fill=white\][10]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q5); (q5) to node \[midway,fill=white\][11]{} (c); (e) to node \[midway,fill=white\][13]{} (c); (f) to\[out=125,in=180\] node \[midway,fill=white\][28]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (f) to\[out=45, in = 0\] node \[midway,fill=white\][30]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (i) to node \[midway,fill=white\][6]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (h); (i) to node \[midway,fill=white\][7]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(b); (q2) to node \[midway,fill=white\][15]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (e); (q3) to\[out=90, in = 5\] node \[midway,fill=white\][16]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (e); (e) to\[out=15,in=145\] (-.25,0) node \[fill=white\][17]{} to\[out=0,in=25\] (f) ; (q4) to\[out=-90, in= 45\] node \[midway,fill=white,outer sep=0pt\][19]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f); (e) to\[out=5,in=165\] (-.6,0) node \[fill=white\][18]{} to\[out=235,in=45\] (f); (q3) to\[out=200, in = 45\] node \[near start,fill=white\][21]{} node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} (f); (q4) to\[out=235,in=45\] node \[midway,fill=white\][26]{} (f); (q3) to\[out=300, in = 35\] node \[midway,fill=white\][20]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f); (i) to\[out=250,in=25\] node \[midway,fill=white\][8]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(f); (g) to\[out=90,in=25\] node \[midway,fill=white\][23]{} (f); (q2) to node \[midway,fill=white\][35]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (r1); (q2) tonode \[midway,fill=white\][34]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (r2); (q2) to\[out=-15,in = 0,min distance=12mm\] node \[near end,fill=white\][27]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(r3); (q2) to\[out = 195, in = 180,min distance=12mm\] node \[near end,fill=white\][31]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (r3); (r2) to node \[midway,fill=white\][36]{} (r1); (r1) to node \[midway,fill=white\][33]{} (r3); (r2) to node \[midway,fill=white\][32]{} (r3);
(r3) to node \[midway,fill=white\][29]{} (f) ; (g) to node \[midway,fill=white\][25]{} (r4); (g) to\[bend right\] node \[near end,sloped,rotate=90\][$\bowtie$]{} node \[midway,fill=white\][24]{} (r4);
\(a) node\[right\][[$X$]{}]{}; (d) node\[left\][[$X$]{}]{}; (b) node\[right\][[$X$]{}]{}; (c) node\[right\][[$X$]{}]{}; (e) node\[left\][[$X$]{}]{}; (f) node\[left\][[$X$]{}]{}; (g) node\[right\][[$X$]{}]{}; (h) node\[right\][[$X$]{}]{};
(.2,0) node [$P_1$]{}; (q2) node\[above\] [$P_2$]{}; (q3) node\[above\] [$P_3$]{}; (q4) node\[below right\] [$P_4$]{}; (q5) node\[above left\] [$P_5$]{}; (r4) node\[left\][$S_1$]{};
\(a) at (1.00000000000000,0.000000000000000); (b) at (0.707106781186548,0.707106781186548); (c) at (0.00000000000000,1.00000000000000); (d) at (-1,1); (e) at (-1.6,0); (f) at (-1,-1); (g) at (0,-1.00000000000000); (h) at (0.707106781186548,-0.707106781186548); (i) at (0.0,0); (q5) at (-.1,0.5); (q2) at (-1,0); (q3) at (-.4,-.15); (q4) at (-.5,-.3); (r1) at (-1.1,-.2); (r2) at (-.9,-.2); (r3) at (-1,-.4); (r4) at (-.5,-.9);
(mb1) at (.2357,.098-1); (mb2) at (2\*.2357,2\*.098-1); (mt1) at (.2357,-.098+1); (mt2) at (2\*.2357,-2\*.098+1); (mt3) at (.855,0.707106781186548/2); (mb3) at (.855,-0.707106781186548/2);
\(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (2.5pt); (q5) circle (2.5pt); (q4) circle (2.5pt); (q3) circle (2.5pt); (q2) circle (2.5pt); (r1) circle (2.5pt); (r2) circle (2.5pt); (r3) circle (2.5pt); (r4) circle (2.5pt);
\(c) to node \[above\] [2]{} (d) to node \[left\] [1]{} (e) to node \[left\] [2]{} (f) to node \[below\] [1]{} (g);
\(a) to (b) to (c); (g) to (h) to (a);
(mb1) to node \[midway,fill=white\][4]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][3]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt1); (mb2) to node \[midway,fill=white\] [5]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][5]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt2); (mb3) to node \[midway,fill=white\][3]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][4]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt3);
(mt4) at (.9,.23) ; (mb4) at (.9,-.23); (mt4) to node \[left\][5]{} (mb4);
\(i) to\[bend right\] node \[midway,fill=white\][14]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (e); (g) to node \[midway,fill=white\][22]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (i) to node \[midway,fill=white\][9]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (i) to\[out=135, in = 225, min distance=25mm\] node \[left,fill=white\][12]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (i) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q5); (q5) to node \[midway,fill=white\][10]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (e) to node \[midway,fill=white\][13]{} (c); (f) to\[out=125,in=180\] node \[midway,fill=white\][28]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (f) to\[out=45, in = 0\] node \[midway,fill=white\][30]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (i) to node \[midway,fill=white\][6]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (h); (i) to node \[midway,fill=white\][7]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (b); (q2) to node \[midway,fill=white\][15]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (e); (q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q4); (q3) to\[out=155,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=135,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=235,in=45\] node \[midway,fill=white\][19]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f); (q4) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (i) to\[out=250,in=45\] node \[midway,fill=white\][8]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f); (g) to\[out=90,in=45\] node \[midway,fill=white\][23]{} (f); (q2) to node \[midway,fill=white\][35]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (r1); (q2) to node \[midway,fill=white\][34]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (r2); (q2) to\[out=-15,in = 0,min distance=12mm\] node \[near end,fill=white\][27]{} (r3); (q2) to\[out = 195, in = 180,min distance=12mm\] node \[near end,fill=white\][31]{} (r3); (r2) to node \[midway,fill=white\][36]{} (r1); (r1) to node \[midway,fill=white\][33]{} (r3); (r2) to node \[midway,fill=white,inner sep=0pt,outer sep=1pt\][32]{} (r3);
(q2) to\[out=0,in = 0,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (q2) to\[out = 180, in = 180,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (-1,-.5) node \[fill=white\][29]{}; (g) to node \[midway,fill=white\][25]{} (r4); (g) to\[bend right\] node \[near end,sloped,rotate=90\][$\bowtie$]{} node \[midway,fill=white\][24]{} (r4);
\(a) node\[right\][[$X$]{}]{}; (d) node\[left\][[$X$]{}]{}; (b) node\[right\][[$X$]{}]{}; (c) node\[right\][[$X$]{}]{}; (e) node\[left\][[$X$]{}]{}; (f) node\[left\][[$X$]{}]{}; (g) node\[right\][[$X$]{}]{}; (h) node\[right\][[$X$]{}]{};
(.2,0) node [$P_1$]{}; (q2) node\[above\] [$P_2$]{}; (q3) node\[above\] [$P_3$]{}; (q4) node\[below right\] [$P_4$]{}; (q5) node\[above left\] [$P_5$]{}; (r4) node\[left\][$S_1$]{};
(p2) at (0,2); (l) at (0,-2); (r1) at (0,-1); (r2) at (.5,0.5); (r3) at (-.5,0.5);
(r1) to node \[fill=white\][32]{} (r2); (r2) to node \[fill=white\][36]{} (r3); (r3) to node \[fill=white\][33]{} (r1); (r2) to node \[fill=white\][34]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (r3) to node \[fill=white\][35]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (r1) to\[out=0,in=-10,min distance=14mm\] node \[fill=white\][27]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (r1) to\[out=180,in=190,min distance=14mm\] node \[fill=white\][31]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2);
(p2) to\[out=0,in=0,min distance=24mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (l) to\[out=180,in=180,min distance=24mm\] node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2);
\(l) node \[fill=white\][29]{};
(p2) node\[above\][$P_2$]{} circle (1.5pt); (r1) circle (1.5pt); (r2) circle (1.5pt); (r3) circle (1.5pt);
(p2) at (0,2); (l) at (0,-2); (r1) at (0,-1.5); (r2) at (0,.5); (r3) at (0,-.5); (l2) at (0,-1);
(p2) to\[out=-10,in=0\] node\[near end,fill=white\][32]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (r3); (r2) to\[bend right\] node \[fill=white\][36]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (p2) to\[out=180,in=180\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (l2) node \[fill=white\][33]{} to\[out=0,in=0\] node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (r2) to\[bend left\] node \[fill=white\][34]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (r3) to\[out=180,in=190\] node \[near start, fill=white\][35]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (r1) to\[out=0,in=-10,min distance=14mm\] node \[near start,fill=white\][27]{} (p2); (r1) to\[out=180,in=190,min distance=14mm\] node \[near start, fill=white\][31]{} (p2);
(p2) to\[out=0,in=0,min distance=24mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (l) to\[out=180,in=180,min distance=24mm\] node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (l) node \[fill=white\][29]{};
(p2) node\[above\][$P_2$]{} circle (1.5pt); (r1) circle (1.5pt); (r2) circle (1.5pt); (r3) circle (1.5pt);
(p2) at (0,2); (l) at (0,-2); (r1) at (0,-1.5); (r2) at (0,.5); (r3) at (0,-.5); (l2) at (0,-1);
(p2) to\[out=-10,in=0\] node\[near end,fill=white\][35]{} node\[pos=.9,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (r3); (r2) to\[bend right\] node \[fill=white\][36]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (p2) to\[out=180,in=180\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (l2) node \[fill=white\][33]{} to\[out=0,in=0\] node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (r2) to\[bend left\] node \[fill=white\][34]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2); (r3) to\[out=180,in=190\] node \[near start, fill=white\][32]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} (p2); (r1) to\[out=0,in=-10,min distance=14mm\] node \[near start,fill=white\][31]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} (p2); (r1) to\[out=180,in=190,min distance=14mm\] node \[near start, fill=white\][27]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} (p2);
(p2) to\[out=0,in=0,min distance=24mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (l) to\[out=180,in=180,min distance=24mm\] node\[near end,sloped, rotate=90\][$\bowtie$]{} (p2);
\(l) node \[fill=white\][29]{};
(p2) node\[above\][$P_2$]{} circle (1.5pt); (r1) circle (1.5pt); (r2) circle (1.5pt); (r3) circle (1.5pt);
\(a) at (1.00000000000000,0.000000000000000); (b) at (0.707106781186548,0.707106781186548); (c) at (0.00000000000000,1.00000000000000); (d) at (-1,1); (e) at (-1.6,0); (f) at (-1,-1); (g) at (0,-1.00000000000000); (h) at (0.707106781186548,-0.707106781186548); (i) at (0.0,0); (q5) at (-.1,0.5); (q2) at (-1,0); (q3) at (-.4,-.15); (q4) at (-.5,-.3); (r1) at (-1.1,-.2); (r2) at (-.9,-.2); (r3) at (-1,-.4); (r4) at (-.5,-.9);
(mb1) at (.2357,.098-1); (mb2) at (2\*.2357,2\*.098-1); (mt1) at (.2357,-.098+1); (mt2) at (2\*.2357,-2\*.098+1); (mt3) at (.855,0.707106781186548/2); (mb3) at (.855,-0.707106781186548/2);
\(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (2.5pt); (q5) circle (2.5pt); (q4) circle (2.5pt); (q3) circle (2.5pt); (q2) circle (2.5pt); (r1) circle (2.5pt); (r2) circle (2.5pt); (r3) circle (2.5pt); (r4) circle (2.5pt);
\(c) to node \[above\] [2]{} (d) to node \[left\] [1]{} (e) to node \[left\] [2]{} (f) to node \[below\] [1]{} (g);
\(a) to (b) to (c); (g) to (h) to (a);
(mb1) to node \[midway,fill=white\][4]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][3]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt1); (mb2) to node \[midway,fill=white\] [5]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][5]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt2); (mb3) to node \[midway,fill=white\][3]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][4]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt3);
(mt4) at (.9,.23) ; (mb4) at (.9,-.23); (mt4) to node \[left\][5]{} (mb4);
\(i) to\[bend right\] node \[midway,fill=white\][14]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (e); (g) to node \[midway,fill=white\][22]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (i) to node \[midway,fill=white\][9]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (i) to\[out=135, in = 225, min distance=25mm\] node \[left,fill=white\][12]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (i) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q5); (q5) to node \[midway,fill=white\][10]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (e) to node \[midway,fill=white\][13]{} (c); (f) to\[out=125,in=180\] node \[midway,fill=white\][28]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (f) to\[out=45, in = 0\] node \[midway,fill=white\][30]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (i) to node \[midway,fill=white\][6]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (h); (i) to node \[midway,fill=white\][7]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (b); (q2) to node \[midway,fill=white\][15]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (e); (q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q4); (q3) to\[out=155,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=135,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=235,in=45\] node \[midway,fill=white\][19]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f); (q4) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (i) to\[out=250,in=45\] node \[midway,fill=white\][8]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f); (g) to\[out=90,in=45\] node \[midway,fill=white\][23]{} (f);
(q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r1); (q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r2); (q2) to\[out=-15,in = 0,min distance=12mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r3); (q2) to\[out = 195, in = 180,min distance=12mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r3); (r2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r1); (r1) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r3); (r2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r3);
(q2) to\[out=0,in = 0,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (q2) to\[out = 180, in = 180,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (-1,-.5) node \[fill=white\][29]{}; (g) to node \[midway,fill=white\][25]{} (r4); (g) to\[bend right\] node \[near end,sloped,rotate=90\][$\bowtie$]{} node \[midway,fill=white\][24]{} (r4);
\(a) node\[right\][[$X$]{}]{}; (d) node\[left\][[$X$]{}]{}; (b) node\[right\][[$X$]{}]{}; (c) node\[right\][[$X$]{}]{}; (e) node\[left\][[$X$]{}]{}; (f) node\[left\][[$X$]{}]{}; (g) node\[right\][[$X$]{}]{}; (h) node\[right\][[$X$]{}]{};
(.2,0) node [$P_1$]{}; (q2) node\[above\] [$P_2$]{}; (q3) node\[above\] [$P_3$]{}; (q4) node\[below right\] [$P_4$]{}; (q5) node\[above left\] [$P_5$]{}; (r4) node\[left\][$S_1$]{};
\(a) at (1.00000000000000,0.000000000000000); (b) at (0.707106781186548,0.707106781186548); (c) at (0.00000000000000,1.00000000000000); (d) at (-1,1); (e) at (-1.6,0); (f) at (-1,-1); (g) at (0,-1.00000000000000); (h) at (0.707106781186548,-0.707106781186548); (i) at (0.0,0); (q5) at (-.1,0.5); (q2) at (-1,0); (q3) at (-.4,-.15); (q4) at (-.5,-.3); (r1) at (-1.1,-.2); (r2) at (-.9,-.2); (r3) at (-1,-.4); (r4) at (-.13,-.5);
(mb1) at (.2357,.098-1); (mb2) at (2\*.2357,2\*.098-1); (mt1) at (.2357,-.098+1); (mt2) at (2\*.2357,-2\*.098+1); (mt3) at (.855,0.707106781186548/2); (mb3) at (.855,-0.707106781186548/2);
\(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (2.5pt); (q5) circle (2.5pt); (q4) circle (2.5pt); (q3) circle (2.5pt); (q2) circle (2.5pt); (r1) circle (2.5pt); (r2) circle (2.5pt); (r3) circle (2.5pt); (r4) circle (2.5pt);
\(c) to (d); (d) to (e); (e) to (f); (f) to (g);
\(a) to (b) to (c); (g) to (h) to (a);
(mb1) to node \[midway,fill=white\][4]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][3]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt1); (mb2) to node \[midway,fill=white\] [5]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][5]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt2); (mb3) to node \[midway,fill=white\][3]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][4]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt3);
(mt4) at (.9,.23) ; (mb4) at (.9,-.23); (mt4) to node \[left\][5]{} (mb4);
\(i) to\[bend right\] node \[midway,fill=white\][14]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (e); (g) to node \[midway,fill=white\][22]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (i) to node \[midway,fill=white\][9]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (i) to\[out=135, in = 225, min distance=25mm\] node \[above left,fill=white\][12]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (i) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q5); (q5) to node \[midway,fill=white\][10]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (f) to\[out=125,in=180\] node \[midway,fill=white\][28]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (f) to\[out=45, in = 0\] node \[midway,fill=white\][30]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (i) to node \[midway,fill=white\][6]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (h); (i) to node \[midway,fill=white\][7]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (b); (q2) to node \[midway,fill=white\][15]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (e); (q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q4); (q3) to\[out=155,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=135,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=235,in=45\] node \[midway,fill=white\][19]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f); (q4) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (i) to\[out=210,in=30\] node \[midway,fill=white\][8]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f); (g) to\[out=170,in=200\] node \[midway,fill=white\][23]{} (i);
(q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r1); (q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r2); (q2) to\[out=-15,in = 0,min distance=12mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r3); (q2) to\[out = 195, in = 180,min distance=12mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r3); (r2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r1); (r1) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r3); (r2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r3);
(q2) to\[out=0,in = 0,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (q2) to\[out = 180, in = 180,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (-1,-.5) node \[fill=white\][29]{}; (g) to node \[midway,fill=white\][25]{} (r4); (i) to node \[near start,sloped,rotate=90\][$\bowtie$]{} node \[midway,fill=white\][24]{} (r4);
\(a) node\[right\][[$X$]{}]{}; (d) node\[left\][[$X$]{}]{}; (b) node\[right\][[$X$]{}]{}; (c) node\[right\][[$X$]{}]{}; (e) node\[left\][[$X$]{}]{}; (f) node\[left\][[$X$]{}]{}; (g) node\[right\][[$X$]{}]{}; (h) node\[right\][[$X$]{}]{};
(.2,0) node [$P_1$]{}; (q2) node\[above\] [$P_2$]{}; (q3) node\[above\] [$P_3$]{}; (q4) node\[below right\] [$P_4$]{}; (q5) node\[above left\] [$P_5$]{}; (r4) node\[left\][$S_1$]{};
(t2) at (-.5,1); (b2) at (-1.3,-.5); (e) to node\[fill=white\][2]{} (t2); (b2) to\[out=90,in=180\] node\[near start, fill=white\][2]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2);
\(i) to\[out=150,in=-90\] node\[pos=.85, fill=white\][13]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (-.35,1); (-1.2,-.6666667) to\[out=95, in =180,min distance = 16 mm\] (q2);
\(i) to\[out=210,in=80\] node \[near end,fill=white\][1]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(-0.5,-1); (-1.3,.5) to node \[fill=white\][1]{} (-.75,1); (-1.4,-.333) to\[out=90,in=180\] node \[near start, fill=white\][1]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2);
\(a) at (1.00000000000000,0.000000000000000); (b) at (0.707106781186548,0.707106781186548); (c) at (0.00000000000000,1.00000000000000); (d) at (-1,1); (e) at (-1.6,0); (f) at (-1,-1); (g) at (0,-1.00000000000000); (h) at (0.707106781186548,-0.707106781186548); (i) at (0.0,0); (q5) at (-.1,0.5); (q2) at (-1,0); (q3) at (-.4,-.15); (q4) at (-.5,-.3); (r1) at (-1.1,-.2); (r2) at (-.9,-.2); (r3) at (-1,-.4); (r4) at (-.13,-.5);
(mb1) at (.2357,.098-1); (mb2) at (2\*.2357,2\*.098-1); (mt1) at (.2357,-.098+1); (mt2) at (2\*.2357,-2\*.098+1); (mt3) at (.855,0.707106781186548/2); (mb3) at (.855,-0.707106781186548/2);
\(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (2.5pt); (q5) circle (2.5pt); (q4) circle (2.5pt); (q3) circle (2.5pt); (q2) circle (2.5pt); (r1) circle (2.5pt); (r2) circle (2.5pt); (r3) circle (2.5pt); (r4) circle (2.5pt);
\(c) to (d); (d) to (e); (e) to (f); (f) to (g);
\(a) to (b) to (c); (g) to (h) to (a);
(mb1) to node \[midway,fill=white\][4]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][3]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt1); (mb2) to node \[midway,fill=white\] [5]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][5]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt2); (mb3) to node \[midway,fill=white\][3]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][4]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt3);
(mt4) at (.9,.23) ; (mb4) at (.9,-.23); (mt4) to node \[left\][5]{} (mb4);
\(i) to\[bend right\] node \[midway,fill=white\][14]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(e); (g) to node \[midway,fill=white\][22]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (i); (i) to node \[midway,fill=white\][10]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (i) to\[out=135, in = 225, min distance=25mm\] node \[above left,fill=white\][12]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (c); (i) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q5); (q5) to node \[midway,fill=white\][9]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(c); (f) to\[out=125,in=180\] node \[midway,fill=white\][28]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (f) to\[out=45, in = 0\] node \[midway,fill=white\][30]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (q2); (i) to node \[midway,fill=white\][6]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(h); (i) to node \[midway,fill=white\][7]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (b); (q2) to node \[midway,fill=white\][15]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (e); (q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q4); (q3) to\[out=155,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=135,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=235,in=45\] node \[midway,fill=white\][19]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (f); (q4) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (i) to\[out=210,in=30\] node \[midway,fill=white\][8]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (f); (g) to\[out=170,in=200\] node \[midway,fill=white\][23]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i);
(q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r1); (q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r2); (q2) to\[out=-15,in = 0,min distance=12mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r3); (q2) to\[out = 195, in = 180,min distance=12mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r3); (r2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r1); (r1) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r3); (r2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r3);
(q2) to\[out=0,in = 0,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (q2) to\[out = 180, in = 180,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (-1,-.5) node \[fill=white\][29]{}; (g) to node \[midway,fill=white\][25]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (r4); (i) to node \[near start,sloped,rotate=90\][$\bowtie$]{} node \[midway,fill=white\][24]{} (r4);
\(a) node\[right\][[$X$]{}]{}; (d) node\[left\][[$X$]{}]{}; (b) node\[right\][[$X$]{}]{}; (c) node\[right\][[$X$]{}]{}; (e) node\[left\][[$X$]{}]{}; (f) node\[left\][[$X$]{}]{}; (g) node\[right\][[$X$]{}]{}; (h) node\[right\][[$X$]{}]{};
(.2,0) node [$P_1$]{}; (q2) node\[above\] [$P_2$]{}; (q3) node\[above\] [$P_3$]{}; (q4) node\[below right\] [$P_4$]{}; (q5) node\[above left\] [$P_5$]{}; (r4) node\[left\][$S_1$]{};
(t2) at (-.5,1); (b2) at (-1.3,-.5); (e) to node\[fill=white\][2]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (t2); (b2) to\[out=90,in=180\] node\[near start, fill=white\][2]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2);
\(i) to\[out=150,in=-90\] node\[pos=.85, fill=white\][13]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (-.35,1); (-1.2,-.6666667) to\[out=95, in =180,min distance = 16 mm\] (q2);
\(i) to\[out=210,in=80\] node \[near end,fill=white\][1]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(-0.5,-1); (-1.3,.5) to node \[fill=white\][1]{} (-.75,1); (-1.4,-.333) to\[out=90,in=180\] node \[near start, fill=white\][1]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2);
\(a) at (1.00000000000000,0.000000000000000); (b) at (0.707106781186548,0.707106781186548); (c) at (0.00000000000000,1.00000000000000); (d) at (-1,1); (e) at (-1.6,0); (f) at (-1,-1); (g) at (0,-1.00000000000000); (h) at (0.707106781186548,-0.707106781186548); (i) at (0.0,0); (q5) at (-.1,0.5); (q2) at (-1,0); (q3) at (-.4,-.15); (q4) at (-.5,-.3); (r1) at (-1.1,-.2); (r2) at (-.9,-.2); (r3) at (-1,-.4); (r4) at (-.5,-.9);
(mb1) at (.2357,.098-1); (mb2) at (2\*.2357,2\*.098-1); (mt1) at (.2357,-.098+1); (mt2) at (2\*.2357,-2\*.098+1); (mt3) at (.855,0.707106781186548/2); (mb3) at (.855,-0.707106781186548/2);
\(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (2.5pt); (q5) circle (2.5pt); (q4) circle (2.5pt); (q3) circle (2.5pt); (q2) circle (2.5pt); (r1) circle (2.5pt); (r2) circle (2.5pt); (r3) circle (2.5pt); (r4) circle (2.5pt);
\(c) to node \[above\] [2]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (d) to node \[left\] [1]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(e) to node \[left\] [2]{}node\[near end,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f) to node \[below\] [1]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(g);
\(a) to (b) to (c); (g) to (h) to (a);
(mb1) to node \[midway,fill=white\][4]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][3]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt1); (mb2) to node \[midway,fill=white\] [5]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][5]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt2); (mb3) to node \[midway,fill=white\][3]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i) to node \[midway,fill=white\][4]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (mt3);
(mt4) at (.9,.23) ; (mb4) at (.9,-.23); (mt4) to node \[left\][5]{} (mb4);
\(i) to\[bend right\] node \[midway,fill=white\][14]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (e); (g) to node \[midway,fill=white\][22]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (i) to node \[midway,fill=white\][10]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(c); (i) to\[out=135, in = 225, min distance=25mm\] node \[left,fill=white\][12]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}node\[near end,sloped, rotate=90\][$\bowtie$]{} (c); (i) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q5); (q5) to node \[midway,fill=white\][9]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (c); (e) to node \[midway,fill=white\][13]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (c); (f) to\[out=125,in=180\] node \[midway,fill=white\][28]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(q2); (f) to\[out=45, in = 0\] node \[midway,fill=white\][30]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(q2); (i) to node \[midway,fill=white\][6]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(h); (i) to node \[midway,fill=white\][7]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (b); (q2) to node \[midway,fill=white\][15]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (e); (q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (q3) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q4); (q3) to\[out=155,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=135,in=0\] node\[pos=.1,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (q2); (q4) to\[out=235,in=45\] node \[midway,fill=white\][19]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} (f); (q4) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (i); (i) to\[out=250,in=45\] node \[midway,fill=white\][8]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (f); (g) to\[out=90,in=45\] node \[midway,fill=white\][23]{} node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (f);
(q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r1); (q2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r2); (q2) to\[out=-15,in = 0,min distance=12mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r3); (q2) to\[out = 195, in = 180,min distance=12mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r3); (r2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r1); (r1) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{}(r3); (r2) to node\[near start,sloped, rotate=90\][$\bowtie$]{} node\[near end,sloped, rotate=90\][$\bowtie$]{} (r3);
(q2) to\[out=0,in = 0,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (q2) to\[out = 180, in = 180,min distance=14mm\] node\[near start,sloped, rotate=90\][$\bowtie$]{} (-1,-.5); (-1,-.5) node \[fill=white\][29]{}; (g) to node \[midway,fill=white\][25]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(r4); (g) to\[bend right\] node \[near end,sloped,rotate=90\][$\bowtie$]{} node \[midway,fill=white\][24]{} node\[near start,sloped, rotate=90\][$\bowtie$]{}(r4);
\(a) node\[right\][[$X$]{}]{}; (d) node\[left\][[$X$]{}]{}; (b) node\[right\][[$X$]{}]{}; (c) node\[right\][[$X$]{}]{}; (e) node\[left\][[$X$]{}]{}; (f) node\[left\][[$X$]{}]{}; (g) node\[right\][[$X$]{}]{}; (h) node\[right\][[$X$]{}]{};
(.2,0) node [$P_1$]{}; (q2) node\[above\] [$P_2$]{}; (q3) node\[above\] [$P_3$]{}; (q4) node\[below right\] [$P_4$]{}; (q5) node\[above left\] [$P_5$]{}; (r4) node\[left\][$S_1$]{};
[^1]: Supported by University of Nebraska - Lincoln, and by NSA grant H98230-14-1-0323.
|
---
abstract: 'In an adaptive population which models financial markets and distributed control, we consider how the dynamics depends on the diversity of the agents’ initial preferences of strategies. When the diversity decreases, more agents tend to adapt their strategies together. This change in the environment results in dynamical transitions from vanishing to non-vanishing step sizes. When the diversity decreases further, we find a cascade of dynamical transitions for the different signal dimensions, supported by good agreement between simulations and theory. Besides, the signal of the largest step size at the steady state is likely to be the initial signal.'
author:
- 'H. M. Yang, Y. S. Ting and K. Y. Michael Wong'
date: 31 August 2006
title: Cascades of Dynamical Transitions in an Adaptive Population
---
Introduction
============
Many natural and artificial systems consist of a population of agents with coupled dynamics. Through their mutual adaptation, they are able to exhibit interesting collective behavior. Although the individuals are competing to maximize their own payoffs, the system is able to self-organize itself to globally efficient states. Examples can be found in economic markets and communication networks [@Anderson1988; @Challet1997; @Wei1995; @Schweitzer2002].
An important factor affecting the behavior of an adaptive population is the dependence of the payoffs on the environment experienced by the individual agents. The payoffs facilitate the agents to assess the preferences of their decisions, hence inducing them to take certain actions when they experience similar dynamical environment in the future. Thus, the payoff function is crucial to the mechanism of adaptation.
As a prototype of an adaptive population, the Minority Game (MG) considers the dynamics of the buyers and sellers in a model of the financial market, in which the minority group is the winning one [@Challet1997]. A good indicator of the mutual adaptation of the agents is the reduction of the variance of the buyer population to values below those of random fluctuations [@Challet1997]. Furthermore, this variance has a universal dependence on the complexity of the strategies adopted by the agents, dropping to a minimum when the complexity is reduced to a universal critical value, and rapidly rising thereafter [@Savit1999; @Manuca2000]. Theoretical studies using the replica method [@Challet2000; @Marsili2000] and the generating functional [@Heimel2001; @Coolen2005] confirmed these general trends.
The agents in the original version of MG uses a *step* payoff function [@Challet1997; @Savit1999; @Manuca2000], meaning that the payoffs received by the winning group are the same, irrespective of the *winning margin* (the difference between the majority and minority group). Latter versions of MG uses a *linear* payoff function [@Challet2000; @Marsili2000; @Heimel2001; @Coolen2005], in which the payoffs increase with the winning margin. Other payoff functions yield the same macroscopic behavior in their dependence of the population variance on the complexity of strategies [@Li2000; @Lee2003]. Thus, the behavior of the population is universal as long as the payoff function favors the minority group. A recent extension of the MG considers payoff functions which reward the minority agents only when they win by a large margin, but punish them when the winning margin is small [@de; @Martino2004]. The extended model displays a smooth crossover from a minority game to a majority game when the payoff function is tuned.
However, when one considers details beyond the population variance, one can find that the agents self-organize in different ways induced by different payoff functions. For a payoff function that favors a large winning margin, the distribution of the buyer population is doubled-peaked [@Challet1997]. This shows that the dynamics of the population self-organizes to favor large winning margins of either the buyers or sellers, since the agents have adapted themselves to maximize their payoffs.
In this paper, we compare the behavior of MGs using step and linear payoffs. Previously, we found that the population variance scales as a power law of the *diversity* for a step payoff [@Wong2004; @Wong2005]. Diversity refers to the variance of the initial biases of the strategy payoffs of the agents. In a population with diverse preferences of strategies, the adaptation rate is slow, resulting in small fluctuations of the buyer or seller population. As we shall see, when the payoff function becomes linear, the scaling relation between the variance and the diversity for the step payoff is replaced by a continuous dynamical transition from a vanishing variance at high diversity to a finite variance at low diversity. The dynamical transition is due to the payoffs being enhanced by large winning margins at low diversity. Furthermore, for systems with multi-dimensional signals feeding the strategies, the dynamical transition in each dimension do not take place at the same transition point. Rather, there is a cascade of dynamical transitions for the different signal dimensions. This rich behavior demonstrates the flexibility of an adaptive population for self-organizing to states in which agents maximize their payoffs, and is hence important in the modeling of economics and distributed control.
The Minority Game
=================
The Minority Game model consists a population of *N* agents competing for limited resources, *N* being odd [@Challet1997]. Each agent makes a decision 1 or 0 at each time step, and the minority group wins. For economic markets, the decisions 1 and 0 correspond to buying and selling respectively, so that the buyers can win by belonging to the minority group, which pushes the price down, and vice versa. For typical control tasks such as the distribution of shared resources, the decisions 1 and 0 may represent two alternative resources, so that less agents utilizing a resource implies more abundance. The decisions of each agent are responses to the environment of the game, described by *signal* $\mu^{*}(t)$ at time *t*, where $\mu^{*}(t)=1,...,D$. These responses are prescribed by *strategies*, which are binary functions mapping the *D* signals to decisions 1 or 0. In this paper, we consider *endogenous* signals, which are the *history* of the winning bits in the most recent *m* steps. Thus, the strategies have an input dimension of $D=2^{m}$, and the parameter $\alpha\equiv\textit{D}/N$ is referred to as the *complexity*. Before the game starts, each agent randomly picks *s* strategies. Out of her *s* strategies, each agent makes decisions according to the most successful one at each step. The success of a strategy is measured by its cumulative payoff, as explained below.
Let $\xi_{i}^{\mu}(t)=\pm1$ when the decisions of strategy *a* are 1 or 0 respectively, responding to signal *$\mu$*. Let $a^{*}(i,t)$ be the strategy adopted by agent *i* at time *t*. Then $A(t)\equiv\sum_{i}\xi_{a^{*}(i,t)}^{\mu^{*}(t)}/N$ is the excess demand of the game at time *t*. The payoff received by strategy $a$ is then $-\xi_{a}^{\mu^{*}(t)}\varphi(\sqrt{N}A(t))$, where $\varphi$ is the payoff function. For step and linear payoffs, $\varphi(\chi)=\mathrm{sgn}\chi$ and $\chi$ respectively. (Here, we have implicitly assumed that an agent does not consider the impact of adopting a strategy, although the excess demand is only dependent on the adopted ones.) Let $\Omega_{a}(t)$ be the cumulative payoff of strategy *a* at time *t*. Then its updating dynamics is described by $$\begin{aligned}
\Omega_{a}(t+1)=\Omega_{a}(t)-\xi_{a}^{\mu^{*}(t)}\varphi(\sqrt{N}A(t)).\end{aligned}$$ Diversity of initial preferences of strategies is introduced by adding random biases $\omega_{ia}$ to the cumulative payoffs of strategy *a* (*a*$=2,...,s$) of agent *i* with respect to her first one. The biases are drawn from a Gaussian or binomial distribution with mean 0 and variance *R*. The ratio $\rho\equiv\textit{R}/N$ is referred to as the *diversity*.
To monitor the mutual adaptive behavior of the population, we measure the variance $\sigma^{2}/N$ of the population making decision 1, defined by $$\begin{aligned}
\frac{\sigma^{2}}{N}\equiv\frac{N}{4}\langle[A^{\mu^{*}(t)}(t)-\langle\mathrm{A}^{\mu^{*}(t)}(t)\rangle]^{2}\rangle\end{aligned}$$ where the average is taken over time when the system reaches the steady state, and over the random distribution of strategies and biases.
Dynamical Transitions
=====================
As shown in Fig. 1, the dependence of the variance $\sigma^{2}/N$ on the complexity $\alpha$ for linear payoffs is very similar to that for step payoffs [@Wong2004; @Wong2005]. For $\alpha$ above a universal critical value $\alpha_c(\approx 0.3)$, the variance drops when $\alpha$ is reduced. The effects of introducing the diversity is also similar to that for step payoffs, namely, the variance remains unaffected when $\alpha>\alpha_{c}$, but decreases significantly with the diversity when $\alpha<\alpha_{c}$.
However, there are differences when one goes beyond this general trend. As shown in Fig. 2, the variance curves at different values of $\alpha$ cross at at $\rho=\rho_c\approx0.16$, indicating the existence of a continuous phase transition at $\rho_c$ from a phase of vanishing variance at large $\rho$ to a phase of finite variance at small $\rho$.
This behavior is very different from that for step payoffs, where the variance scales as $\rho^{-1}$ and there are no dynamical transitions (Fig. 2 inset). The picture is confirmed by analyzing the dynamics of the game for small $m$. The dynamics can be conveniently described by introducing the $D$-dimensional vector $A^{\mu}(t)\equiv\sum_{i}\xi_{a^{*}(i,t)}^{\mu}/N$. While only one of the $D$ signals corresponds to the historical signal $\mu^{*}(t)$ of the game, the augmentation to $D$ components is necessary to describe the attractor structure of the game dynamics. Fig. 3 illustrates the attractor structure in this phase space for the visualizable case of $m=1$. The dynamics proceeds in the direction which tends to reduce the magnitude of the components of $A^{\mu}(t)$ [@Challet2000]. However, the components of $A^{\mu}(t)$ overshoot, resulting in periodic attractors of period $2D$. For $m=1$, the attractor is described by the sequence $\mu^{*}(t)=0,1,1,0$, and takes the L-shape as shown in Fig. 3 [@Wong2005]. Note that the displacements in the two directions may not have the same amplitude.
Following steps similar to those in [@Wong2005], we find that for $m$ not too large, and for convergence within time steps much less than $\sqrt{R}$, $$\begin{aligned}
A^{\mu}(t+1)=A^{\mu}(t)-\sqrt{\frac{2}{\mathrm{\pi}R}}\varphi(\sqrt{N}A^{\mu}(t))\delta_{\mu\mu^{*}(t)}.
\label{step}\end{aligned}$$ For step payoffs, Eq. (\[step\]) converges to an attractor confined in a $D$-dimensional hypercube of size $\sqrt{2/{\mathrm{\pi}R}}$, irrespective of the value of $R$. On the other hand, for linear payoffs, $A^{\mu}(t+1)$ becomes a linear function of $A^{\mu}(t)$ with a slope of $1-\sqrt{2/\pi\rho}$. Hence, for $\rho>\rho_{c}=1/{2\pi}\sim0.16$, the step sizes $\mid$A$^{\mu}(t+1)-A^{\mu}(t)\mid$ converge to zero, whereas for $\rho<\rho_{c}$, steps of vanishing sizes become unstable, resulting in a continuous dynamical transition at $\rho_{c}$.
The Phase of Finite Variance
============================
However, when $\rho<\rho_{c}$, the step sizes for each of the $D$ signals may not be equal. To see this, we monitor the variance for each of the $D$ signals and rank them. The *r*th maximum variance is then given by $$\begin{aligned}
S_{r}=\mathrm{large}_{\mu}\left(\frac{N}{4}[\langle(\mathrm{A}^\mathrm{\mu})^2\rangle|_{\mu=\mu^{*}(t)}-(\langle\mathrm{A}^\mathrm{\mu}\rangle|_{\mu=\mu^{*}(t)})^2],r\right)\end{aligned}$$ where $\mathrm{large}_{\mu}(f(\mu),r)$ is the *r*th largest function $f(\mu)$ for $\mu=1,...,D$.
As shown in Figs. 4-5, the step sizes for each of the $D$ signals do not bifurcate simultaneously at $\rho=\rho_{c}$, Rather, only their first maximum bifurcates from zero when $\rho$ falls below $\rho_{c}$, while the step sizes for the remaining $D$-1 signals remain small. When the diversity further decreases to around 0.05, the second maximum becomes unstable as well, and a further bifurcation takes place. For $m\geq2$, there are further bifurcations of the third or higher order maxima, resulting in a *cascade* of dynamical transitions when the diversity decreases.
This cascade of transitions is confirmed by analysis. For $m=1$, we can generalize Eq. (\[step\]) to convergence times of the order $\sqrt{R}$. Assuming without loss of generality that $\mathrm{A}^{1}$ bifurcates while $\mathrm{A}^{0}$ remains small, the variance of the buyer population, as derived in [@Ting2004], is $$\begin{aligned}
\frac{\sigma^{2}}{N}=\frac{N}{32}(\Delta\mathrm{A}^{1})^{2},
\quad\Delta\mathrm{A}^{1}=\mathrm{erf}\left(\frac{\Delta\mathrm{A}^{1}}{\sqrt{8\rho}}\right),\end{aligned}$$ where $\Delta\mathrm{A}^{1}$ is the step size responding to signal 1. As Fig. 4 inset shows, the analytical and simulation results well agree down to $\rho \sim 0.05$. However, when the diversity decreases further, this simple analysis implies that the variance will saturate to a constant $N/32$, whereas simulation results are clearly higher.
This discrepancy is due to a further bifurcation of the minimum step size. This can be analyzed by considering the effect of a perturbation $\delta\mathrm{A}^{0}(t)$ in the direction of $\mathrm{A}^{0}$. After a period of 4 steps, the accumulated perturbation becomes $$\begin{aligned}
\delta\mathrm{A}^{0}(t+4)=\left[1-\frac{1}{\sqrt{2\pi\rho}}(1+e^{-\frac{(\Delta\mathrm{A}^{1})^{2}}{8\rho}})\right]^{2}\delta\mathrm{A}^{0}(t)\label{stability}.\end{aligned}$$ At $\rho=0.0459$, where $\Delta$A$^{1} = 0.9775$, the coefficient on the right hand side of Eq. (\[stability\]) reaches the value 1, and $\delta\mathrm{A}^{0}(t)$ diverges on further reduction of $\rho$. Numerical iterations of the analytical equations for $\mathrm{A}^{\mu}(t)$, averaged over samples of different initial conditions, yield the theoretical curves in Fig. 4 and inset, agreeing very well with simulation results. Similarly, the agreement between analytical and simulation results are satisfactory for $m=2$.
Since the attractors have asymmetric responses to different signals, we also study their dependence on the initial states. Letting the system start from a certain state (say, state 1 for $m=1$) for a given sample, Fig. 6 shows that the initial state is more likely to have the largest step size in the attractor for $m=1$. Simulations show that higher values of $m$ share the same trend.
Conclusion
==========
We have studied the behavior of an adaptive population using a payoff function that increases linearly with the winning margin. We found a continuous dynamical transition when the adaptation rate of the population is tuned by varying their diversity of preferences. This is in contrast with the case of payoff functions independent of the winning margin, in which no phase transitions are found. Furthermore, we found a cascade of dynamical transitions in the responses to different signals. This shows that an adaptive population has the ability to self-organize to globally efficient states and display a rich behavior, although the individual agents make selfish decisions. Hence, despite the simplicity of the population models, they are able to capture the essential features of economic markets and distributed control.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank S. W. Lim, and C. H. Yeung for discussions. This work is supported by the Research Grant Council of Hong Kong (DAG05/06.SC36).
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abstract: 'We answer negatively an open problem of Illusie on the DR-decomposability of the log de Rham complex of the special fiber of a semi-stable reduction over the Witt ring. We also show that $E_1$ degeneration of the Hodge to log de Rham spectral sequence does not imply DR-decomposability of semi-stable varieties.'
address:
- 'School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China'
- 'School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, China'
author:
- Mao Sheng
- Junchao Shentu
title: 'Some examples of DR-indecomposable special fibers of semi-stable reductions over Witt rings'
---
\[section\] \[Unity\][Definition]{} \[Unity\][Notation]{} \[Unity\][Claim]{}
\[Unity\][Theorem]{} \[Unity\][Proposition]{} \[Unity\][Corollary]{} \[Unity\][Lemma]{} \[Unity\][Conjecture]{} \[Unity\][Problem]{} \[Unity\][Question]{} \[Unity\][Remark]{} \[Unity\][Example]{}
[^1]
Introduction
============
The work of Deligne-Illusie [@Del_Ill1987] is fundamental in Hodge theory since it gives a new method to establish the $E_1$-degeneration property of the Hodge to de Rham spectral sequence. Let $k$ be a perfect field of positive characteristic and $X_0$ an algebraic variety over $k$. We have the following commutative diagram of Frobenius $$\xymatrix{
X_0 \ar[r]^{F=F_{X_0/k}}\ar[rd] & X_0'\ar[r]^{\pi} \ar[d] &X_0 \ar[d]\\
& {\rm Spec}\ k \ar[r]^{\sigma}&{\rm Spec}\ k,\\
}$$ The variety $X_0$ is said to be *DR-decomposable* if the complex $\tau_{<p}F_{\ast}\Omega^{\bullet}_{X_0}$ is quasi-isomorphic to $\bigoplus_{i=0}^{\dim X}\Omega^i_{X'_0/k}[-i]$, where $\Omega^{\bullet}_{X_0}$ is the de Rham complex of $X_0/k$. The main result of Deligne-Illusie asserts that for *smooth* varieties, $X_0$ is $W_2=W_2(k)$-liftable if and only if it is DR-decomposable. On the other hand, if $X_0$ is proper over $k$ and $\dim X_0<p$, the DR-decomposability of $X_0$ implies the $E_1$-degeneration of the Hodge to de Rham spectral sequence (for $\dim X_0=p$, the $E_1$-degeneration also holds by the Grothendieck duality). Properness on $X_0$ is required because of finite dimensionality of Hodge cohomologies. However, it is not clear whether one can remove the assumption on the dimension of $X_0$: this is exactly one of the two open problems posed by Illusie [@Illusie2002]. It is neither clear whether $E_1$-degeneration would imply the DR-decomposability.\
This note grew out from our study on the other open problem posed by Illusie in loc. cit, that is about the generalization of Deligne-Illusie’s main result to semi-stable varieties over $k$. Note that semi-stable varieties appear naturally in algebraic geometry as very typical singular varieties. The problem is stated as follows: let $k$ be as above and $W=W(k)$ the ring of Witt vectors. For a semi-stable reduction $X$ over $W$, we set $X_0=X\times_{W}k$, the special fiber of $X$, and $F: X_0\to X'_0$ the relative Frobenius. Consider the complex of ${\mathscr{O}}_{X_0}$-modules: $$\Omega^{\log\bullet}_{X_0}=\Omega^{\bullet}_{X}(\textrm{log}X_0)|_{X_0}.$$
\[prob\_Illusie\] Is the complex $\tau_{<p}F_{\ast}\Omega^{\log\bullet}_{X_0}$ decomposable in $D(X'_0)$?
Our answer to this problem is NO. Indeed, we constructed explicit examples of semi-stable reductions over $W$ negating the problem, whose dimension can be arbitrary large (in the curve case the answer is affirmative for cohomological reason) and the characteristic of $k$ can be arbitrary. See §3 for the construction. We also examined the $E_1$-degeneration property of these examples. It turns out that all examples we constructed whose dimensions are less than or equal to the characteristic of the residue field have the $E_1$-degeneration property. This is a direct consequence of Theorem \[thm\_degeneration\_E1\] and Deligne-Illusie’s decomposation theorem. Therefore, the $E_1$-degeneration property is NOT equivalent to the DR-decomposability in the semi-stable (non-smooth) case. We are not aware of similar results in the smooth case.
DR-decomposability and log deformation
======================================
We use the log geometry as developed in the work [@KKato1988] to study Problem \[prob\_Illusie\], and the construction of our examples is mainly based on a simple criterion of the DR-decomposability in terms of the existence of a log smooth deformation over the log scheme $(W_2(k),1\mapsto 0)$ (Theorem \[thm\_decom\_lifting\]).\
Let $X$ be a semi-stable reduction over $W$. Let $M_{X_0}$ (resp. $M_{{\textrm{Spec}}(k)}$) be the log structure on $X$ (resp. ${\textrm{Spec}}(W)$) attached to the reduced normal crossing divisor $X_0$ (resp. ${\textrm{Spec}}(k)$) (Example (1.5) [@KKato1988]). Then the extended morphism of log schemes $f: (X,M_{X_0})\to ({\textrm{Spec}}(W),M_{{\textrm{Spec}}(k)})$ is smooth. Let $(X_0,M_0)\to {\bf k}:=(k, 1\mapsto 0)$ be the base change of $f$ via the inclusion ${\textrm{Spec}}(k)\to {\textrm{Spec}}(W)$. When the context is clear, we denote the log scheme $(X_0,M_0)$ simply by $X_0$ (in some other occasion, we use $\underline{X}$ to denote the underlying scheme of a log scheme $X$). It is known that the morphism $X_0\to {\bf k}$ is smooth, and the de Rham complex $\Omega^{\bullet}_{X_0/\bf{k}}$ of the log variety $X_0/\bf{k}$ is naturally isomorphic to the complex $\Omega^{\log\bullet}_{X_0}$ considered in §1 (1.7 [@KKato1988]). Moreover, it is known that the log structure $M_0$ of $X_0$ is of semi-stable type:
([@Ollson2003])\[defn\_semistable\_type\] A log variety $X$ over $\bf k$ is called semi-stable type if étale locally over each closed point $x\in \underline{X}$ it is strict smooth over $$({\textrm{Spec}}(k(x)[x_1,\cdots,x_r]/(x_1\cdots x_r)),\bigoplus_{i=1}^r \mathbb{N}e_i,e_i\mapsto x_i),$$ where the log structure is induced by the homomorphism of monoids $\bigoplus_{i=1}^r \mathbb{N}e_i\rightarrow{\mathscr{O}}_{\underline{X}}$ defined by $e_i\mapsto x_i$.
Let $F$ be the absolute Frobenius of the log scheme $\mathbf{k}$ which is given by the commutative diagram $$\xymatrix{
k\ar[r]^{F_{k}} & k\\
\mathbb{N} \ar[u]^0 \ar[r]^{\times p} & \mathbb{N}\ar[u]^0.\\
}$$ It is easy to verify that $F$ is liftable to the log scheme ${\bf W}_2:=(W_2, 1\mapsto 0)$ (but not to the log scheme $(W_2, 1\mapsto p)$!), and an obvious lifting $G$ over ${\bf W}_2$ is given by the following commutative diagram: $$\xymatrix{
W_2\ar[r]^{F_{W_2}} & W_2\\
\mathbb{N} \ar[u]^0 \ar[r]^{\times p} & \mathbb{N}\ar[u]^0,\\
}$$ where $F_{W_2}$ is the Frobenius automorphism of $W_2$. A special case of the Kato’s decomposition theorem is the following
\[Kato decomposition\] Let $X/\bf k$ be a log variety of semi-stable type and $X'$ the base change of $X$ via the absolute Frobenius of $\bf k$. Let $F_{X/\bf k}: X\to X'$ be the relative Frobenius. Then the complex $\tau_{<p}F_{X/{\bf k}\ast}\Omega^{\bullet}_{X/\bf k}$ is decomposable if and only if $X'$ is liftable to ${\bf W}_2$.
Remark that Kato’s decomposition theorem works for a log variety of Cartier type which is more general than semi-stable type (Definition 4.8 [@KKato1988]). In the following, we show further that $X'$ is liftable to ${\bf W}_2$ if and only if $X$ itself is liftable to ${\bf W}_2$, and hence we obtain the following criterion for DR-decomposability:
\[thm\_decom\_lifting\] Notation and assumption as Theorem \[Kato decomposition\]. Then the complex $\tau_{<p}F_{X/{\bf k}\ast}\Omega^{\bullet}_{X/\bf k}$ is decomposable if and only if $X$ is liftable to ${\bf W}_2$.
Via the base change by $G$, one obtains a $\mathbf{W}_2$-lifting of $X'$ from that of $X$. Since $G$ is not an isomorphism of log schemes, our argument is to show the converse nevertheless is still true. Let $\omega_{X}\in H^2(X,T_{X/\mathbf{k}})$ (resp. $\omega_{X'}\in H^2(X',T_{X'/\mathbf{k}})$) be the obstruction class of the lifting of $X$ (resp. $X'$) to $\mathbf{W}_2$. Recall that $\omega_{X}$ is constructed as follows: Let $\{U_i\}$ be an affine cover of $X$. Choosing for each $U_i$ a log smooth lifting ${\mathscr{U}}_i$ on $\mathbf{W}_2$, we then have that on each overlap $U_{ij}=U_i\cap U_j$ there exists an isomorphism $\alpha_{ij}:{\mathscr{U}}_j|{U_{ij}}\rightarrow {\mathscr{U}}_i|{U_{ij}}$. Then $\omega_{X}$ is represented by $\{(U_i\cap U_j\cap U_k, \alpha_{ij}\alpha_{jk}\alpha_{ki})\}$. Because of the existence of $G$, $\{(G^{-1}(U_i\cap U_j\cap U_k), G^\ast\alpha_{ij}\alpha_{jk}\alpha_{ki})\}$ represents $\omega_{X'}$. Thus we have that $\sigma^\ast(\omega_{X})=\omega_{X'}$ through the canonical map $$\xymatrix{
H^2(X',T_{X'/\mathbf{k}}) \ar@{}[r]|= & H^2(X',\sigma^\ast T_{X/\mathbf{k}})\\
& H^2(X,T_{X/\mathbf{k}})\ar[u]^{\sigma^\ast}
}.$$ The above equality uses the fact that $\sigma^*\Omega_{X/\mathbf{k}}=\Omega_{X'/\mathbf k}$ and that both sheaves are locally free (see 1.7 and Proposition 3.10 [@KKato1988]). However, since $\sigma$ is an isomorphism of schemes, the map $\sigma^*$ in the vertical line is bijective. It follows immediately that $\omega_{X}=0$ under the assumption that $X'$ is $\mathbf{W}_2$-liftable and hence $X$ itself is $\mathbf{W}_2$-liftable.
The following corollary ensures it is valid to assume $k$ is algebraically closed in the study of Problem \[prob\_Illusie\].
\[cor\_change\_field\] Let $f:X\rightarrow \mathbf{k}$ be a smooth morphism of semistable type and $k'$ be a perfect field containing $k$. Denote by $\mathbf{k'}$ the field $k'$ with the induced log structure from $\mathbf{k}$ and by $X_{\mathbf{k'}}$ the log base change. Then $\tau_{<p}F_{X/{\bf k}\ast}\Omega^{\bullet}_{X/\mathbf{k}}$ is decomposable if and only if $\tau_{<p}F_{X_{\mathbf{k'}}/{\bf k'}\ast}\Omega^{\bullet}_{X_{\mathbf{k'}}/{\mathbf{k'}}}$ is decomposable.
By Theorem \[thm\_decom\_lifting\], it is enough to show that a $(W_2(k'), \mathbb{N}\mapsto 0)$-lifting of $X_{\mathbf{k'}}$ induces a $(W_2(k), \mathbb{N}\mapsto 0)$-lifting of $X$. By the flat base change, one has the isomorphism $H^2(X,T_{X/\mathbf{k}})\otimes_kk'=H^2(X_{\mathbf{k'}}, T_{X/\mathbf{k'}})$ and hence the injection $\alpha: H^2(X,T_{X/\mathbf{k}})\to H^2(X_{\mathbf{k'}}, T_{X/\mathbf{k'}})$. Then, by the same arguments in Theorem \[thm\_decom\_lifting\], the obstruction class $ob_k$ to lifting $X$ to $\mathbf{W_2(k)}$ is mapped to to the obstruction class $ob_{k'}$ of lifting $X_{\mathbf{k'}}$ to $(W_2(k'),\mathbb{N}\mapsto 0)$ via the map $\alpha$. By the condition that $\alpha(ob_k)=ob_{k'}=0$, it follows that $ob_k=0$.
Noticing that the obstruction whether $X_0$ can be lifted to $(W_2,1\mapsto 0)$ lies in $H^2(X_0,T^{\log}_{X_0})$ ([@KKato1988], Proposition 3.14), where $T^{\log}_{X_0}$ is the log tangent sheaf of $X_0$ over the canonical log point $(k,1\mapsto 0)$, we have:
\[cor\_sufficient\_condition\] If the special fiber $X_0$ satisfies that $H^2(X_0,T^{\log}_{X_0})=0$, then $\tau_{<p}F_{X_0\ast}\Omega^{\log\bullet}_{X_0}$ is decomposable. In particular, the answer to Problem \[prob\_Illusie\] is affirmative if
1. $X_0$ is affine, or
2. $X_0$ is a curve, or
3. $X_0$ is a combinatorial $K3$ surface which appears in the semi-stable degenerations of a $K3$ surface [@Nakkajima2000].
After presenting our results, Weizhe Zheng provided us a more conceptual proof of Theorem \[thm\_decom\_lifting\]: Denote by $\textrm{Lift}(X)$ (resp.$\textrm{Lift}(X')$) the groupoid of liftings of $X$ (rsep. $X'$) over $\mathbf{W}_2$. Let $G:\mathbf{W}_2\rightarrow\mathbf{W}_2$ be a lifting of the log Frobenius morphism $F:\mathbf{k}\rightarrow\mathbf{k}$. Given a lifting $X^{(1)}\in \textrm{Lift}(X)$, the pullback of $X^{(1)}$ along $G$ gives an object in $\textrm{Lift}(X')$. With the obvious assignments on morphisms, one can get a functor $$A: \textrm{Lift}(X)\rightarrow\textrm{Lift}(X').$$ Conversely, let $X'^{(1)}\in \textrm{Lift}(X')$ be a lifting of $X'$. Denote by $i:X'\hookrightarrow X'^{(1)}$ the canonical strict closed immersion and by $\sigma:X'\rightarrow X$ the base change of $F:\mathbf{k}\rightarrow\mathbf{k}$. Recall that $\underline{\sigma}:\underline{X'}\rightarrow\underline{X}$ is an isomorphism and ${\mathscr{M}}_{X'}\simeq{\mathscr{M}}_X\oplus_{{\mathscr{K}}_k}{\mathscr{M}}_k$. One can construct the pushout $X'^{(1)}\amalg_{X'}X$ of the diagram $$\xymatrix{
X' \ar[r]^{\sigma} \ar[d]^i & X\\
X'^{(1)} &
}$$ as follows:
- The underlying scheme $\underline{X'^{(1)}\amalg_{X'}X}$ is defined to be $\underline{X'^{(1)}}$,
- the log structure of $X'^{(1)}\amalg_X'X$ is defined to be ${\mathscr{M}}_{X'^{(1)}}\times_{{\mathscr{M}}_{X'}}{\mathscr{M}}_X$.
With the obvious assignments on morphisms, the pushout process along $\sigma:X'\rightarrow X$ gives a functor $$B: \textrm{Lift}(X')\rightarrow\textrm{Lift}(X).$$ It is straightforward to check the following proposition.
The functor $A$ gives an equivalence of groupoids, and the functor $B$ is its quasi-inverse.
Examples
========
In this section, $k$ is an algebraically closed field of characteristic $p>0$. We proceed to construct examples of semi-stable reductions over $W$ whose special fibers do not admit log deformation to ${\bf W}_2$, which negate Problem \[prob\_Illusie\] because of Theorem \[thm\_decom\_lifting\].
More preparations
-----------------
The first lemma is another characterization of semi-stable reductions over $W=W(k)$.
\[dejong lemma\] Let $K_0$ be the fractional field of $W$. Then an $W$-scheme $X$ is a semi-stable reduction over $W$ if and only if the following two properties hold:
1. the generic fiber $X_{K_0}=X\times_W{K_0}$ is smooth over $K_0$,
2. the special fiber $X_{k}=X\times_Wk$ is a normal crossing variety over $k$.
See [@deJong1996], 2.16.
The second lemma is rather standard.
\[lem\_w2lifting\] Let $X/{\bf k}$ be a log variety of semi-stable type. Assume the irreducible components $\{X_i, i\in I\}$ of the underlining variety $\underline{X}$ to be smooth. Let ${\mathscr{X}}$ be a smooth deformation $X$ over ${\bf W}_2$. Then the underlying scheme of ${\mathscr{X}}$ is written into the schematic union of closed subschemes $\underline{{\mathscr{X}}}=\bigcup_{i\in I} {\mathscr{X}}_i$s with the property that, for each nonempty $J\subseteq I$, the schematic intersection $\bigcap_{j\in J}{\mathscr{X}}_j$ is a $W_2$-lifting of $\bigcap_{j\in J}X_j$.
Set $${\mathscr{I}}_i=I_i+pI_i,$$ where $I_i$ is the ideal sheaf of $X_i$ in $X$. Then, ${\mathscr{I}}_i$ is an ideal sheaf of ${\mathscr{O}}_{\underline{{\mathscr{X}}}}$. We claim that the closed subschemes ${\mathscr{X}}_i$s defined by ${\mathscr{I}}_i$s have the property in the lemma. To show this it suffices to prove the following properties:
1. ${\mathscr{O}}_{\underline{{\mathscr{X}}}}/{\mathscr{I}}_i$ is flat over $W_2$,
2. $\bigcap {\mathscr{I}}_i=0$, and
3. for each nonempty $J\subseteq I$, ${\mathscr{O}}_{\underline{{\mathscr{X}}}}/\cup_{j\in J}{\mathscr{I}}_j$ is flat over $W_2$.
Since $\widehat{{\mathscr{O}}_{\underline{{\mathscr{X}}},x}}$ is faithfully flat over ${\mathscr{O}}_{\underline{{\mathscr{X}}},x}$ for each point $x\in \underline{{\mathscr{X}}}$, it suffices to verify the above claim after tensoring with $\widehat{{\mathscr{O}}_{\underline{{\mathscr{X}}},x}}$ for every $x\in\underline{{\mathscr{X}}}$. By ([@KKato1988] Theorem 3.5, Proposition 3.14), there is an étale morphism $U\rightarrow \underline{{\mathscr{X}}}$ such that we have $$\xymatrix{
U\ar[r]^-f \ar[dr]_{\pi'|_{U}} &\textrm{Spec}(W_2[x_1,\cdots,x_n]/(x_1\cdots x_r)) \ar[d]\\
& {\textrm{Spec}}(W_2)
},$$ where $f$ is an étale morphism. As a consequence, there is an isomorphism $$\alpha:\widehat{{\mathscr{O}}_{\underline{{\mathscr{X}}},x}}\cong W_2[[x_1,\cdots,x_n]]/(x_1\cdots x_r)$$ such that each ${\mathscr{I}}_i\widehat{{\mathscr{O}}_{\underline{{\mathscr{X}}},x}}$ (whenever it is nonempty) is generated by $\alpha^{-1}(\Pi_{j\in J_i}x_j)$ for some nonempty set $J_i\subseteq\{1,\cdots,r\}$. Moreover, $\{1,\cdots,r\}$ is the disjoint union of $J_i$s. Then the claim follows from direct calculations.
By the above two lemmas, we can conclude the following
\[simple lemma\] Let $Z$ be a smooth scheme over $W$. Let $Y_0$ be a smooth closed subvariety of $Z_0=Z\times_Wk$. Set $X=Bl_{Y_0}Z$, the blowup of $Z$ along the closed subscheme $Y_0$. Then $X$ is a semi-stable reduction over $W$, whose special fiber $X_0$ is a simple normal crossing divisor consisting of two smooth components $Bl_{Y_0}Z_0$ and $\mathbb{P}(N_{Y_0/Z})$ (the projective normal bundle of $Y_0$ in $Z$) which intersect transversally along $\mathbb{P}(N_{Y_0/Z_0})$ (the projective normal bundle of $Y_0$ in $Z_0$). Furthermore, if the normal crossing variety $X_0$ over ${\bf k}$ admits a smooth deformation over ${\bf W}_2$, then both pairs $(Bl_{Y_0}Z_0,\mathbb{P}(N_{Y_0/Z_0}))$ and $(\mathbb{P}(N_{Y_0/Z}),\mathbb{P}(N_{Y_0/Z_0}))$ are $W_2(k)$-liftable.
The first statement follows from Lemma \[dejong lemma\] (the remaining fact is fairly standard and therefore omitted, see [@Fulton1998]). The second statement follows from Lemma \[lem\_w2lifting\].
\[cynklemma\] Let $\pi:Y\rightarrow X$ be a morphism of schemes over $k$ and let $S={\textrm{Spec}}(A)$, where $A$ is artinian with residue field $k$. Assume that ${\mathscr{O}}_X =\pi_\ast{\mathscr{O}}_Y$ and $R^1\pi_\ast({\mathscr{O}}_Y)=0$. Then for every lifting ${\mathscr{Y}}\rightarrow S$ of $Y$ there exists a preferred lifting ${\mathscr{X}}\rightarrow S$ making a commutative diagram $$\xymatrix{
Y \ar@{^{(}->}[r]\ar[d] & {\mathscr{Y}}\ar[d] \\
X \ar@{^{(}->}[r] & {\mathscr{X}}}$$
\[prop\_conterexample2\] Notation as in Proposition \[simple lemma\]. If $Y_0$ is not $W_2(k)$-liftable, then the special fiber $X_0$ of $X$ (regarded as a log variety over ${\bf k}$) does not admit any smooth deformation over ${\bf W}_2$.
Use Propositions \[simple lemma\] and \[cynklemma\] which assert that the $W_2$-liftability of $\mathbb{P}(N_{Y_0/Z})$ implies that of $Y_0$.
Example 1
---------
Corollary \[prop\_conterexample2\] provides direct examples: take a smooth projective variety $Y_0$ over $k$ which is non $W_2$-liftable, and take a closed embedding $Y_0\hookrightarrow Z_0$ over $k$ into a smooth projective variety such that the codimension $\textrm{Cod}_{Z_0}Y_0\geq 2$ and $Z_0$ admits a smooth lifting $Z$ over $W$ (for example take $Z_0$ to be a projective space of high dimension). Set $X=\textrm{Bl}_{Y_0}Z$, the blowup of $Z$ along the closed subscheme $Y_0$. Then $X$ is a semi-stable reduction over $W$ whose special fiber $X_0/{\bf k}$ does not admit ${\bf W}_2$-deformation.
Example 2
---------
Notice that Mukai [@Mukai2013] has obtained a nice generalization to higher dimension of Raynaud’s classical example [@Raynaud1978] of non $W_2$-liftable smooth projective surface over $k$. His construction, together with an idea of Liedtke-Satriano (Theorem 1.1 (a) [@LM2014]), allows us to make concrete examples of all relative dimensions $\geq 2$.\
Take a closed embedding $Y_0\hookrightarrow Z_0$ over $k$ such that both $Y_0$ and $Z_0$ are smooth, $Z_0$ admits a smooth lifting $Z$ over $W$, and such that the pair $(\textrm{Bl}_{Y_0}Z_0,E)$ is not $W_2$-liftable, where $E$ is the exceptional divisor. Set $X=\textrm{Bl}_{Y_0}Z$. Then $X$ is a semi-stable reduction over $W$ whose special fiber $X_0$ does not admit any **Type II** deformation.
Such examples exist by the recent work of Liedtke-Satriano [@LM2014]. See Theorem 1.1 (a) [@LM2014] (more specifically Theorem 2.3 (a) and Theorem 2.4 loc. cit.). This approach (with a little modification) provides examples of DR-indecomposable special fibers of semi-stable reductions over the Witt rings $W(k)$ of relative dimension $d\geq 2$ with any algebraically closed field $k$ of arbitrary positive characteristic.
Let us recall first the following
\[defn\_Tango\] A smooth curve $C$ over $k$ of genus $\geq2$ is called a Tango-Raynaud curve if there exists a rational function $f$ on $C$ such that $df\neq0$ and that $(df)=pD$ for some ample divisor $D$.
A typical example of Tango-Raynaud curve is the plane curve defined by the affine polynomial $$G(x^p)-x=y^{pe-1},$$ where $G$ is a polynomial of degree $e\geq 1$ in the variable $x$. The following lemma is well known.
\[lem\_tango\_curve\] Let $C$ be a Tango-Raynaud curve, then there exists a rank two vector bundle $E$ on $C$ together with a smooth curve $D$ in the projectification $\mathbb{P}_C(E)$ of $E/C$, such that the composite $D\rightarrow \mathbb{P}_C(E)\rightarrow C$ is the relative Frobenius $F_0: D\rightarrow D^{(p)}=C$.
\[examples of all dimensions\] Notation as in Lemma \[lem\_tango\_curve\]. Let ${\mathscr{C}}$ be a $W$-lifting of $C$ and ${\mathscr{E}}$ a lifting of $E$ over ${\mathscr{C}}$. For $d\geq 2$, set $Z_d=\mathbb{P}_{{\mathscr{C}}}({\mathscr{E}}\oplus {\mathscr{O}}_{{\mathscr{C}}}^{d-2})$ and $X_d=Bl_{D}Z_d$. Then $X_d$ is a semi-stable reduction over $W$ of relative dimension $d$, whose special fiber, regarded as a log variety over $\bf k$, is non ${\bf W}_2$-liftable and therefore DR-indecomposable.
We prove the statement for $d=2$ only (the proof for $d\geq 3$ is the same). Denote $$C_0=C, \quad Y_0=D, \quad Z_0=\mathbb{P}_C(E), \quad Z=Z_2.$$ Assume the contrary that the special fiber $X_0$ of $Bl_{Y_0}Z$, regarded as a log variety over ${\bf k}$, admit a smooth deformation over ${\bf W}_2$. It follows from Proposition \[lem\_w2lifting\] that the pair $(Z_0, Y_0)$ consisting of the component $Z_0=Bl_{Y_0}Z_0$ of $X_0$ together with the divisor $Y_0=\mathbb{P}(N_{Y_0/Z})\cap Z_0\subset X_0$ lift to a pair $(Z_1,Y_1)$ over $W_2$ (The scheme $Z_1$ is not necessarily the mod $p^2$-reduction of $Z$). On the other hand, Proposition \[cynklemma\] implies that the projection $Z_0\to C_0$ is the reduction of a certain $W_2$-morphism $Z_1\to C_1$. Therefore, the composite $F_0: Y_0\hookrightarrow Z_0\to C_0$ lifts to the composite $F_1: Y_1\hookrightarrow Z_1\to C_1$ over $W_2$. But this leads to a contradiction: the nonzero morphism $dF_1: F_1^*\Omega_{C_1}\to \Omega_{Y_1}$ is divisible by $p$ and it induces a nonzero morphism over $k$ $$\frac{dF_1}{p}:F_0^*\Omega_{C_0}\to \Omega_{Y_0},$$ which is impossible because of the degree. Therefore, $X_0/{\bf k}$ is indeed non ${\bf W}_2$-liftable as claimed.
An $E_1$-degeneration result
============================
This section is devoted to prove the following
\[thm\_degeneration\_E1\] Let $k$ be an algebraically closed field and $R$ a DVR with the residue field $k$. Let $Z/R$ be a smooth proper $R$-scheme and $X/R$ be a blow-up of $X$ along a closed regular center $Y_0$ supported in $Z_0=Z\times_Rk$. If the Hodge to de Rham spectral sequence $$E_1^{pq}=H^q(Z_0,\Omega^p_{Z_0})\Rightarrow H^{p+q}(\Omega_{Z_0}^{\bullet})$$ degenerates at $E_1$ (e.g. when $\textrm{char}(k)=0$ or $\dim Z_0\leq \textrm{char}(k)$ and $R$ is of mixed characteristic), then the Hodge to log de Rham spectral sequence $$E_1^{pq}=H^q(X_0,\wedge^p\Omega_{X_0}^{\log})\Rightarrow H^{p+q}(\Omega_{X_0}^{\log\bullet})$$ degenerates at $E_1$.
Recall from Proposition \[simple lemma\] that $X_0$ is a simple normal crossing divisor consisting of two smooth components $X_1=Bl_{Y_0}Z_0$ and $X_2=\mathbb{P}(N_{Y_0/Z})$ which intersect transversally along $D=\mathbb{P}(N_{Y_0/Z_0})$. The blowdown morphism of the log pairs $(Z,Z_0)\rightarrow(X,X_0)$ restricts on the special fiber to a log morphism $\pi:X_0\rightarrow (Z_0,1\mapsto 0)$ between log varieties over $({\textrm{Spec}}(k),1\mapsto0)$. This induces a canonical morphism $$\pi^{\ast i}:\Omega^i_{Z_0}\rightarrow R\pi_\ast\bigwedge^i\Omega_{X_0}^{\textrm{log}}.$$ Our main technical step in proving Theorem \[thm\_degeneration\_E1\] is the following
\[prop\_KEY\] Let $Z/R$ be a smooth proper $R$-scheme and $X/R$ be a blow-up of $X$ along a closed regular center $Y_0$ supported in $Z_0$. Denote by $\pi:X_0\rightarrow Z_0$ the restriction morphism. Then for each $i$ the canonical morphism (defined in the proof) $$\Omega_{Z_0}^i \rightarrow R\pi_\ast\wedge^i\Omega_{X_0}^{\textrm{log}}$$ is an isomorphism in $D^b(Z_0)$.
From Proposition \[prop\_KEY\], we may derive the main result of the section.
We actually prove that the two spectral sequences $$\begin{aligned}
\label{align_derham_1}
E_1^{pq}=H^q(Z_0,\Omega^p_{Z_0})\Rightarrow H^{p+q}(\Omega_{Z_0}^{\bullet})\end{aligned}$$ and $$\begin{aligned}
\label{align_derham_2}
E_1^{pq}=H^q(X_0,\wedge^p\Omega_{X_0}^{\log})\Rightarrow H^{p+q}(\Omega_{X_0}^{\log\bullet})\end{aligned}$$ are isomorphic. First recall that (\[align\_derham\_1\]) is induced by the hypercohomology of the complex $\Omega^{\bullet}_{Z_0}$ with respect to the truncated filtration $$F^i=\tau^{\textrm{st}}_{\geq i}\Omega^{\bullet}_{Z_0},$$ where $\tau^{\textrm{st}}$ is the stupid truncation. (\[align\_derham\_2\]) is induced by the hypercohomology of the complex $\Omega_{X_0}^{\textrm{log}\bullet}$ with respect to the truncated filtration $$F^i=\tau^{\textrm{st}}_{\geq i}\Omega_{X_0}^{\textrm{log}\bullet}.$$ By Proposition \[prop\_KEY\], there are natural quasi-isomorphisms $$R\pi_{\ast}\Omega_{X_0}^{\log\bullet}\simeq\pi_{\ast}\Omega_{X_0}^{\log\bullet}\simeq\Omega^{\bullet}_{Z_0},$$ and the isomorphisms respect the filtration $$F^i=R\pi_{\ast}\tau^{\textrm{st}}_{\geq i}\Omega_{X_0}^{\textrm{log}\bullet}\simeq\pi_{\ast}\tau^{\textrm{st}}_{\geq i}\Omega_{X_0}^{\textrm{log}\bullet}$$ in the left, middle and $$F^i=\tau^{\textrm{st}}_{\geq i}\Omega^{\bullet}_{Z_0}$$ in the right. As a consequence, the two spectral sequences (\[align\_derham\_1\]) and (\[align\_derham\_2\]) are naturally isomorphic.
To prove Proposition \[prop\_KEY\], we make some preparations. Let $X_0=X_1\cup_DX_2$ be a variety consisting of two smooth projective components $X_1$ and $X_2$ such that they intersect transversely along a smooth divisor $D$. Assume that $X_0$ has a log structure of semi-stable type (Definition \[defn\_semistable\_type\]). Then the normalization $X_1\cup X_2\rightarrow \underline{X_0}$ and the diagonal immersion $D\rightarrow X_1\cup X_2$ lift to log morphisms $$(X_1\cup X_2, D_1\cup D_2\oplus(1\mapsto 0))\rightarrow X_0$$ and $$(D,(1\mapsto 0)^{\oplus 2})\rightarrow (X_1\cup X_2, D_1\cup D_2)$$ over the base $({\textrm{Spec}}(k),1\mapsto 0)$. These log morphisms induce morphisms of sheaves on $X_0$ $$\begin{aligned}
\label{align_log_diff1}
\Omega^i_{X_0^{\textrm{log}}}\rightarrow \Omega_{X_1}^i(\textrm{log}D)\oplus \Omega_{X_2}^i(\textrm{log}D)\end{aligned}$$ and $$\begin{aligned}
\label{align_log_diff2}
\Omega_{X_1}^i(\textrm{log}D)\oplus \Omega_{X_2}^k(\textrm{log}D)\rightarrow\Omega^k_{(D,(1\mapsto 0)^{\oplus 2})/({\textrm{Spec}}(k),1\mapsto 0)}.\end{aligned}$$ for each $i$. By the definition of log cotangent sheaf, $$\Omega_{(D,(1\mapsto 0)^{\oplus 2})/({\textrm{Spec}}(k),1\mapsto 0)}\simeq\Omega_D\oplus (\mathbb{Z}^{\oplus 2}/\mathbb{Z}\otimes_{\mathbb{Z}}{\mathscr{O}}_{D})/\alpha(m)\otimes m-d\alpha(m)\otimes 1.$$ Thanks to the log structure of $(D,(1\mapsto 0)^{\oplus 2})$, $\alpha(m)\otimes m-d\alpha(m)\otimes 1$ are null relations. Therefore $$\Omega_{(D,(1\mapsto 0)^{\oplus 2})/({\textrm{Spec}}(k),1\mapsto 0)}\simeq\Omega_D\oplus {\mathscr{O}}_D.$$ This isomorphism induces the forgetful morphism $$\Omega_{(D,(1\mapsto 0)^{\oplus 2})/({\textrm{Spec}}(k),1\mapsto 0)}\rightarrow \Omega_D$$ and the log residue morphism $$\Omega_{(D,(1\mapsto 0)^{\oplus 2})/({\textrm{Spec}}(k),1\mapsto 0)}\rightarrow {\mathscr{O}}_D.$$ Therefore $$\Omega^k_{(D,(1\mapsto 0)^{\oplus 2})/({\textrm{Spec}}(k),1\mapsto 0)}\cong \bigwedge^k(\Omega_D\oplus {\mathscr{O}}_D)\simeq \Omega_D^k\oplus \Omega_D^{k-1}$$ and by local calculation the restriction morphism $$\Omega_{X_1}^k(\textrm{log}D)\rightarrow\Omega^k_{(D,(1\mapsto 0)^{\oplus 2})/({\textrm{Spec}}(k),1\mapsto 0)}$$ is equivalent to $$(\iota,\textrm{res}_D):\Omega_{X_1}^k(\textrm{log}D)\rightarrow \Omega_D^k\oplus \Omega_D^{k-1}$$ $$\beta+\gamma \frac{dz}{z}\mapsto (\beta,\gamma).$$ Here we use a local chart of $X_1$ where $D=\{z=0\}$ and $\beta$, $\gamma$ does not contain $dz$. This phenomenon is interesting in itself. The residue map $\textrm{res}_D$ is a part of the restriction map of log cotangent sheaves. It makes log geometry a convenient and natural language in such a situation.\
Assume locally $X_0$ is embedded into the affine space with a system of local coordinates $(z_1,z_2,\cdots,z_n)$ such that $X_1=\{z_1=0\}$, $X_2=\{z_2=0\}$. Since $$\frac{dz_1}{z_1}+\frac{dz_2}{z_2}=0$$ on $X_0$, a log form on $X_0$ is of the form $$\beta+\gamma_1\frac{dz_1}{z_1}=\beta-\gamma_1\frac{dz_2}{z_2}.$$ Therefore two $k$-forms $\beta_1+\gamma_1\frac{dz_1}{z_1}$ on $X_2$ and $\beta_2+\gamma_2\frac{dz_2}{z_2}$ on $X_2$ glue to a log $k$-form on $X_0$ if and only if $$\beta_1|_D=\beta_2|_D$$ and $$\gamma_1|_D+\gamma_2|_D=0.$$ This proves
\[lem\_log\_cotangent\_normalization\] For each $k\geq 0$ there is a short exact sequence of sheaves $$0\rightarrow\Omega_{X^{\textrm{log}}}^k\rightarrow \Omega_{X_1}^k(\textrm{log}D)\oplus \Omega_{X_2}^k(\textrm{log}D)\stackrel{\varphi}{\rightarrow} \Omega_D^{i}\oplus \Omega_D^{i-1}\rightarrow 0$$ where $\varphi$ is defined by $$\begin{pmatrix}
\iota & \textrm{res}_D \\
-\iota & \textrm{res}_D
\end{pmatrix}.$$ Here $\Omega_D^{-1}$ is defined to be $0$.
The following well-known lemma will be used several times in the sequel.
\[lem\_Bott\] Let $\mathbb{P}^n$ be the projective space over $k$. The following vanishing results hold:
1. $$\begin{aligned}
H^q(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^p)=0, p\neq q
\end{aligned}$$
2. If $i\neq0$, then $$\begin{aligned}
H^q(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^p(i))=0,
\end{aligned}$$ for $q=0$, $i\leq p$ or $q=n$, $i\geq p-n$ or $q\neq0,n$.
3. Let $H$ be a hyperplane in $\mathbb{P}^n$, then $$\begin{aligned}
H^q(\mathbb{P}^n,\Omega_{\mathbb{P}^n}^p(\textrm{log}H))=
\begin{cases}
k, & p=q=0 \\
0, & \textrm{otherwise}.
\end{cases}
\end{aligned}$$
\[lem\_proj\_bundle\] Let $Z$ be a smooth variety and $\pi:P\rightarrow Z$ be a projective bundle of relative dimension $r$. Let $D\subset P$ be a relative hyperplane. Then for each $i\geq 0$ there is a canonical isomorphism $$\Omega^i_Z\simeq R\pi_\ast\Omega^i_P(\textrm{log}D)$$ in $D(Z)$.
The exact sequence $$0\to \pi^\ast\Omega_Z\to \Omega_P(\textrm{log}D)\to \Omega_{P/Z}(\textrm{log}D)\to0$$ induces a decreasing filtration $$F^p=\pi^{\ast}\Omega^p_{Z}\wedge\Omega_P^{i-p}(\textrm{log}D)\subset\Omega_P^{i}(\textrm{log}D)$$ such that $$F^p/F^{p+1}\simeq\pi^{\ast}\Omega^p_{Z}\otimes\Omega_{P/Z}^{i-p}(\textrm{log}D).$$ Therefore we have a spectral sequence $$E_1^{pq}=R^q\pi_{\ast}(\pi^{\ast}\Omega^p_{Z}\otimes\Omega_{P/Z}^{i-p}(\textrm{log}D))\Rightarrow R^{p+q}\pi_\ast(\Omega^{i}_P(\textrm{log}D)).$$ By Lemma \[lem\_Bott\], we see that $$\begin{aligned}
E_1^{pq}\simeq \Omega^p_{Z}\otimes R^q\pi_{\ast}(\Omega_{P/Z}^{i-p}(\textrm{log}D))=
\begin{cases}
\Omega^i_{Z}, & p=i, q=0 \\
0, & \textrm{otherwise}.
\end{cases}
\end{aligned}$$ This proves the lemma.
\[lem\_blowup\] Let $Z_0$ be a smooth projective variety and $Y_0$ be a smooth closed subvariety of $Z_0$. Denote $\pi:X_1\rightarrow Z_0$ be the blowup along $Y_0$ with exceptional divisor $D$. Then for each $k\geq0$, there is a distinguished triangle in $D^b(Z_0)$ induced by natural morphisms: $$\Omega^k_{Z_0}\stackrel{u}{\rightarrow} R\pi_\ast\Omega_{X_1}^k\oplus\Omega_{Y_0}^k\stackrel{v}{\rightarrow} R\pi_\ast\Omega_D^k\rightarrow\Omega^k_{Z_0}[1].$$ In other words, we have the short exact sequence $$\begin{aligned}
\label{align_isomorphism_1}
0\rightarrow\Omega^k_{Z_0}\rightarrow \pi_\ast\Omega_{X_1}^k\oplus\Omega_{Y_0}^k\rightarrow \pi_\ast\Omega_D^k\rightarrow 0\end{aligned}$$ and the isomorphism $$\begin{aligned}
\label{align_isomorphism_2}
R^i\pi_\ast\Omega_{X_1}^k\rightarrow R^i\pi_\ast\Omega_D^k\end{aligned}$$ for each $i>0$.
Denote the following automorphism of $\pi_\ast\Omega_{X_1}^k\oplus\Omega_{Y_0}^k$ by $\phi$: $$(a,b)\mapsto (a,a-b),$$ By composing with $\phi$, the exactness of the sequence (\[align\_isomorphism\_1\]) is reduced to the following isomorphisms $$\Omega^k_{Z_0}\cong \pi_\ast\Omega_{X_1}^k;\quad \Omega_{Y_0}^k\cong \pi_\ast\Omega_D^k.$$ For $k=0$, these are obvious. For $k\geq 1$, their truth can be easily seen by considering the local model of a blow-up along a smooth center: we assume that $X_1$ is the blow up of $Z_0={{\mathbb A}}^n$ along $Y_0={{\mathbb A}}^r$ defined by the intersection of some coordinate hyperplanes. Then the map $$\pi: D\to Y_0$$ is the projection $${{\mathbb A}}^r\times {{\mathbb P}}^{s}\to {{\mathbb A}}^r.$$ Thus, it is trivial to get $\pi_{*}\Omega^k_{D}=\Omega^k_{Y_0}, k\geq 0$ by this description. For the first isomorphism, we use the following estimation: $$\pi^*\Omega^k_{Z_0}\subset \Omega^k_{X_1}\subset \pi^*\Omega^k_{Z_0}(kD).$$ From this, it follows that $$\Omega^k_{Z_0}\subset \pi_* \Omega^k_{X_1}\subset \Omega^k_{Z_0}\otimes \pi_*{\mathscr{O}}_{X_1}(kD)= \Omega^k_{Z_0},$$ and hence $\pi_* \Omega^k_{X_1}=\Omega^k_{Z_0}$.\
The proof of (\[align\_isomorphism\_2\]) is divided into two parts. First we show that the natural map $$R^i\pi_\ast\Omega_{X_1}^k|_{D}\rightarrow R^i\pi_\ast\Omega_D^k$$ is an isomorphism for each $i>0$. Considering the long exact sequence associated to $$0\rightarrow{\mathscr{O}}_D(1)\otimes\Omega^{k-1}_D\rightarrow\Omega^k_{X_1}|_D\rightarrow\Omega^k_D\rightarrow0$$ where ${\mathscr{O}}_D(1)$ is the tautological bundle of the projective bundle $D\rightarrow Y_0$, we see that it sufficient to prove that $$\begin{aligned}
\label{align_vanish_1}
R^i\pi_\ast({\mathscr{O}}_D(1)\otimes\Omega^{k}_D)=0,\quad i>0.\end{aligned}$$ Notice that the short exact sequence $$0\rightarrow\pi^{\ast}\Omega_{Y_0}\rightarrow\Omega_D\rightarrow\Omega_{D/Y_0}\rightarrow 0$$ induces a decreasing filtration $$F^p=\pi^{\ast}\Omega^p_{Y_0}\wedge\Omega_D^{k-p}\subset\Omega_D^{k}$$ such that $$F^p/F^{p+1}\simeq\pi^{\ast}\Omega^p_{Y_0}\otimes\Omega_{D/Y_0}^{k-p}.$$ Therefore we have a spectral sequence $$E_1^{pq}=R^q\pi_{\ast}(\pi^{\ast}\Omega^p_{Y_0}\otimes\Omega_{D/Y_0}^{k-p}\otimes{\mathscr{O}}_D(1))\Rightarrow R^{p+q}\pi_\ast({\mathscr{O}}_D(1)\otimes\Omega^{k}_D).$$ Since $D\rightarrow Y_0$ is a projective bundle, we obtain that $$E_1^{pq}=\Omega^p_{Y_0}\otimes R^q\pi_{\ast}(\Omega_{D/Y_0}^{k-p}\otimes{\mathscr{O}}_D(1))$$ for $p+q\geq 1$ and $p,q\geq 0$, thanks to the Lemma \[lem\_Bott\]. This proves (\[align\_vanish\_1\]) and thus $$R^i\pi_\ast\Omega_{X_1}^k|_{D}\rightarrow R^i\pi_\ast\Omega_D^k$$ is an isomorphism for each $i>0$.\
Next we show that the canonical morphism $$R^i\pi_\ast\Omega_{X_1}^k\rightarrow R^i\pi_\ast(\Omega_{X_1}^k|_{D})$$ is an isomorphism for each $i>0$. By the long exact sequence associated to $$0\rightarrow\Omega_{X_1}^k\otimes{\mathscr{O}}_{X_1}(-D)\rightarrow\Omega_{X_1}^k\rightarrow\Omega_{X_1}^k|_{D}\rightarrow 0,$$ we see that it sufficient to show the vanishing $$\begin{aligned}
\label{align_vanish2}
R^i\pi_{\ast}(\Omega_{X_1}^k\otimes{\mathscr{O}}_{X_1}(-D))=0\end{aligned}$$ for each $i>0$.\
Notice that the short exact sequence $$0\rightarrow\pi^{\ast}\Omega_{Z_0}\rightarrow\Omega_{X_1}\rightarrow\Omega_{X_1/Z_0}\simeq\Omega_{D/Y_0}\rightarrow 0$$ induces a decreasing filtration $$F^p=\pi^{\ast}\Omega^p_{Z_0}\wedge\Omega_{X_1}^{k-p}\subset\Omega_{X_1}^{k}$$ such that $$F^p/F^{p+1}\simeq\pi^{\ast}\Omega^p_{Z_0}\otimes\Omega_{D/Y_0}^{k-p}.$$ Therefore we have a spectral sequence $$E_1^{pq}=R^q\pi_{\ast}(\pi^{\ast}\Omega^p_{Z_0}\otimes\Omega_{D/Y_0}^{k-p}\otimes{\mathscr{O}}_{X_1}(-D))\Rightarrow R^{p+q}\pi_\ast(\Omega^{k}_{X_1}\otimes{\mathscr{O}}_{X_1}(-D)).$$ Since $D\rightarrow Y_0$ is a projective bundle, we obtain that $$E_1^{pq}=\Omega^p_{Z_0}\otimes R^q\pi_{\ast}(\Omega_{D/Y_0}^{k-p}\otimes{\mathscr{O}}_D(1))$$ for $p+q\geq 1$ and $p,q\geq 0$, thanks again to the Lemma \[lem\_Bott\]. This proves (\[align\_vanish2\]) and thus $$R^i\pi_\ast\Omega_{X_1}^k\rightarrow R^i\pi_\ast(\Omega_{X_1}^k|_{D})$$ is an isomorphism for each $i>0$. So we finish the proof of (\[align\_isomorphism\_2\]).
Now we are ready to prove Proposition \[prop\_KEY\].
By Lemma \[lem\_log\_cotangent\_normalization\] and \[lem\_proj\_bundle\], we have a distinguished triangle $$R\pi_\ast\bigwedge^i\Omega^{\textrm{log}}_{X}\rightarrow R\pi_\ast\Omega_{X_1}^i(\textrm{log}D)\oplus\Omega_{Y_0}^i\stackrel{R\pi_\ast\varphi}{\rightarrow} R\pi_\ast\Omega_D^{i}\oplus R\pi_\ast\Omega_D^{i-1}\rightarrow R\pi_\ast\Omega_{X^{\textrm{log}}}^i[1]$$ in $D^b(Z_0)$. This triangle fills in the following diagram in $D^b(Z_0)$ $$\begin{aligned}
\label{align_33}
\xymatrix{
R\pi_\ast\Omega^i_{D} \ar[r] & 0 \ar[r] & R\pi_\ast\Omega_D^i[1] \ar[r]^{\textrm{Id}} & R\pi_\ast\Omega_D^i[1]\\
R\pi_\ast\Omega^{i}_{X_1}\oplus\Omega_{Y_0}^i \ar[u] \ar[r] & R\pi_\ast\Omega_{X_1}^i(\textrm{log}D)\oplus\Omega_{Y_0}^i \ar[u] \ar[r] & R\pi_\ast\Omega_D^{i-1} \ar[u] \ar[r] & R\pi_\ast\Omega^{i}_{X_1}[1]\oplus\Omega_{Y_0}^i[1] \ar[u]\\
R\pi_\ast\wedge^i\Omega^{\textrm{log}}_{X_0} \ar[u]^{p} \ar[r] & R\pi_\ast\Omega_{X_1}^i(\textrm{log}D)\oplus\Omega_{Y_0}^i \ar[u]^{\textrm{Id}} \ar[r]^-{R\pi_\ast\varphi} & R\pi_\ast\Omega_D^{i}\oplus R\pi_\ast\Omega_D^{i-1} \ar[u]^{\textrm{pr}} \ar[r] & R\pi_\ast\wedge^i\Omega^{\textrm{log}}_{X_0}[1] \ar[u]\\
R\pi_\ast\Omega^i_{D}[-1] \ar[r] \ar[u] & 0 \ar[r] \ar[u] & R\pi_\ast\Omega^i_{D} \ar[u] \ar[r]^{\textrm{Id}} & R\pi_\ast\Omega^i_{D}, \ar[u]
}\end{aligned}$$ which is generated from the centered commutative square $$\xymatrix{
R\pi_\ast\Omega_{X_1}^i(\textrm{log}D)\oplus\Omega_{Y_0}^i \ar[r] & R\pi_\ast\Omega_D^{i-1} \\
R\pi_\ast\Omega_{X_1}^i(\textrm{log}D)\oplus\Omega_{Y_0}^i \ar[u]^{\textrm{Id}} \ar[r]^-{R\pi_\ast\varphi} & R\pi_\ast\Omega_D^{i}\oplus R\pi_\ast\Omega_D^{i-1}. \ar[u]\\
}$$
In the diagram (\[align\_33\]), $p$ is induced (non-canonically) by the above commutative square. The second horizontal line is the direct sum of the distinguished triangles $$R\pi_\ast\Omega^{i}_{X_1}\rightarrow R\pi_\ast\Omega_{X_1}^i(\textrm{log}D)\rightarrow R\pi_\ast\Omega_D^{i-1}\rightarrow R\pi_\ast\Omega^{i}_{X_1}[1]$$ and $$\Omega_{Y_0}^i\stackrel{\textrm{Id}}{\rightarrow} \Omega_{Y_0}^i\rightarrow 0\rightarrow\Omega_{Y_0}^i[1].$$ The horizontal lines of (\[align\_33\]) are distinguished triangles. The second and third vertical lines are also distinguished. By the $3\times3$ lemma of triangulated categories, the first vertical line induces a distinguished triangle $$R\pi_\ast\wedge^i\Omega^{\textrm{log}}_{X_0}\rightarrow R\pi_\ast\Omega_{X_1}^k\oplus\Omega_{Y_0}^i\rightarrow R\pi_\ast\Omega_D^k\rightarrow R\pi_\ast\wedge^i\Omega^{\textrm{log}}_{X_0}[1].$$ Comparing with Lemma \[lem\_blowup\], we see that there is a quasi-isomorphim $$R\pi_\ast\wedge^i\Omega_{X_0}^{\textrm{log}}\simeq\Omega_{Z_0}^i.$$ Note that this isomorphism may not be the natural one induced by the morphism $\pi$. However, we obtain as a consequence of the abstract quasi-isomorhism that $$R^k\pi_\ast\wedge^i\Omega_{X_0}^{\textrm{log}}\simeq 0,\quad k>0.$$ It remains to show that the natural morphism of sheaves $$\begin{aligned}
\label{align_KEY0}
\Omega_{Z_0}^i \rightarrow \pi_\ast\wedge^i\Omega_{X_0}^{\textrm{log}}\end{aligned}$$ is an isomorphism.\
Let us consider the cohomologies at place 0 of the diagram (\[align\_33\]), $$\begin{aligned}
\label{align_330}
\xymatrix{
\pi_\ast\Omega^{i}_{X_1}\oplus\Omega_{Y_0}^i \ar[r] & \pi_\ast\Omega_{X_1}^i(\textrm{log}D)\oplus\Omega_{Y_0}^i \ar[r] & \pi_\ast\Omega_D^{i-1}& \\
\pi_\ast\wedge^i\Omega^{\textrm{log}}_{X_0} \ar[u]^{p^0} \ar[r] & \pi_\ast\Omega_{X_1}^i(\textrm{log}D)\oplus\Omega_{Y_0}^i \ar[u]^{\textrm{Id}} \ar[r]^-{\pi_\ast\varphi} & \pi_\ast\Omega_D^{i}\oplus R\pi_\ast\Omega_D^{i-1} \ar[u]^{\textrm{pr}} \ar[r] & 0\\
0 \ar[u]\ar[r] & 0 \ar[r] \ar[u] & \pi_\ast\Omega^i_{D} \ar[u] \ar[r]^{\textrm{Id}} & \pi_\ast\Omega^i_{D} \ar[u]
}.\end{aligned}$$ The two vertical sequences in the middle are short exact sequences. Therefore, by the snake lemma, there is an exact sequence $$0\to \pi_\ast\wedge^i\Omega^{\textrm{log}}_{X_0}\stackrel{p^0}{\to}\pi_\ast\Omega^{i}_{X_1}\oplus\Omega_{Y_0}^i\stackrel{\delta}{\to}\pi_\ast\Omega^i_{D}$$ where $\delta$ is the boundary map which is identical to the one in (\[align\_isomorphism\_1\]). Hence by (\[align\_isomorphism\_1\]) we see that the natural map (\[align\_KEY0\]) is an isomorphism.
**Acknowledgment:** We would like to thank Luc Illusie for several valuable e-mail communications, and Weizhe Zheng for another argument to Theorem \[thm\_decom\_lifting\] which is also included in the note. Warm thanks go to Christian Liedtke for his interest and comments. The first named author would like to thank Kang Zuo for his interest and constant support. The second named author would like to express his deep gratitude to Xiaotao Sun and Jun Li for their constant encouragement throughout the work.
[^1]: This work was partially supported by National Natural Science Foundation of China (Grant No. 11622109, No. 11626253) and the Fundamental Research Funds for the Central Universities.
|
---
abstract: 'In this paper, we propose a bid optimizer for sponsored keyword search auctions which leads to better retention of advertisers by yielding attractive utilities to the advertisers without decreasing the revenue to the search engine. The bid optimizer is positioned as a key value added tool the search engine provides to the advertisers. The proposed bid optimizer algorithm transforms the reported values of the advertisers for a keyword into a correlated bid profile using many ideas from cooperative game theory. The algorithm is based on a characteristic form game involving the search engine and the advertisers. Ideas from Nash bargaining theory are used in formulating the characteristic form game to provide for a fair share of surplus among the players involved. The algorithm then computes the nucleolus of the characteristic form game since we find that the nucleolus is an apt way of allocating the gains of cooperation among the search engine and the advertisers. The algorithm next transforms the nucleolus into a correlated bid profile using a linear programming formulation. This bid profile is input to a standard generalized second price mechanism (GSP) for determining the allocation of sponsored slots and the prices to be be paid by the winners. The correlated bid profile that we determine is a locally envy-free equilibrium and also a correlated equilibrium of the underlying game. Through detailed simulation experiments, we show that the proposed bid optimizer retains more customers than a plain GSP mechanism and also yields better long-run utilities to the search engine and the advertisers.'
author:
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title: Cooperative Game Theoretic Bid Optimizer for Sponsored Search Auctions
---
Bid Optimizer, Sponsored Search, Cooperative Game Theory, Nash bargaining, Nucleolus.
Introduction
============
Sponsored search auctions have been studied extensively in the recent years due to the advent of targeted advertising and its role in generating large revenues. With a huge competition in providing the sponsored search links, the search engines face an imminent problem which can be called as the *retention problem*. If an advertiser (or alternatively bidder) does not get satisfied because of not getting the right number of clicks or the anticipated payoff, he could drop out of the auction and try sponsored links at a different search engine.
Motivation: Retention of Advertisers in Sponsored Search Auctions
-----------------------------------------------------------------
Our motivation to study the retention problem is driven by the compulsions faced by both the search engine and the advertisers.
From the advertisers’ perspective, choosing their maximum-willingness-to-pay such that they get an attractive slot subject to their budget constraints is a challenging problem. The search engines can use various mechanisms for the sponsored search auction as described in [@agt:noam; @monograph] but the most popular mechanism is the generalized second price (GSP) auction since it is simple and yields better revenue to the search engine. In the most simple version of GSP, where there are $k$ slots and $n$ advertisers (for simplicity assume $k \leq n$), the allocation and payment rule are as follows. The allocation rule is that $n$ advertisers are ranked in descending order based on their bids, with ties broken appropriately, and top $k$ advertisers’ advertisements are displayed. The payment rule is that every advertiser needs to pay bid amount of the advertiser who is just below his slot and last advertiser is charged the highest bid that has not won any slot. If the non-truthful GSP auction [@EdelOstros] is used by the search engine, the bidders will have an incentive to shade their bids. The bidders would not want to use complicated and computationally intensive bidding strategies as the bidding process is done many times (typically thousands of times) in a day. These advertisers generally build their own software agents or employ third party software agents, which adjust and readjust the bid values on behalf of these advertisers. The bidders typically specify their maximum willingness-to-pay for their keywords for any given day. Hence each keyword has a specific set of bidders bidding on it for the whole day. This scenario constitutes a repeated game between all the bidders bidding for that keyword. In this game, bidders who cannot plan their budget effectively may experience less utilities and thus may drop out of the system.
We now turn to the search engine’s perspective of the retention problem. When the bidders try to know each others’ valuations by submitting and resubmitting bids, they may find a set of strategy profiles which may yield all of them better payoffs. This may lead to collusion among the bidders. Folk theorems [@mwg] suggest that players may be able to increase individual profits by colluding thereby decreasing the search engine’s revenue. Even though the bidders in the keyword auctions are competitors, this collusion against the search engine could be stable. Vorobeychik and Reeves [@motivation] studied this phenomenon and illustrated a particular collusive strategy which is better for all the bidders (hence worst for the search engine) and can be sustainable over a range of settings. Feng and Zhang [@feng] showed that dynamic price competition between competing advertisers can lead to collusion among them. However, in this dynamic scenario, when the discounted payoffs of the bidders under the collusive strategy are considered, the stability of collusion depends inversely on the number of bidders [@mwg]. That is, the lower the number of bidders in the system, the higher is the stability of the collusion. This motivates us to study the bidder retention problem for the search engine.
Also, due to exponential growth in the space of online advertising and intense competition among the search engine companies, the switching cost for the advertisers to change from one search engine to another is almost zero [@agt:noam]. Hence, it is imperative for the search engine companies to retain their advertisers to safeguard their market share. Driven by this, the search engine companies have introduced many value added tools, such as bid optimizer, to maximize the bang-per-buck for the bidders. In what follows, we describe the bid optimizer’s role in solving the retention problem.
Bid Optimizers
--------------
A *bid optimizer* is a software agent provided by the search engine in order to assist the advertisers. The bidders are required to provide to the bid optimizer a target budget for the day and a maximum willingness-to-pay. Bid optimizers, currently provided by the search engines, promise to maximize the revenue of advertisers by adjusting the bid amount in each round of the auction based on the projected keyword traffic and remaining budget.
It can be seen that the decisions made by the bid optimizer are crucial to both the search engine and the set of advertisers, who choose to use the bid optimizer. Hence, the objective of a typical bid optimizer is to strike a balance between reduction in revenue of the search engine company versus increase the retention of advertisers. This objective is achieved by providing enhanced utilities to the advertisers, thus ensuring retention of customers, thereby sustaining high levels of revenue to the search engine company in the long run. Designing such intelligent bid optimizers is the subject of this paper.
There are some problems involved in designing bid optimizers.
1. For the search engine, maximizing its short-term revenue (that is, its payoff in a one-shot game) seems to be a viable option. But here, the lower valuation bidders are denied slots due to allocative efficiency concerns. For the bidders, as shown by Cary *et al* [@cary], where all the high valuation bidders use a particular greedy strategy, it has been proved that none of the bidders except the top $k$ bidders get the slots after a certain number of rounds of the auction. The above phenomenon can permanently drive away low valuation bidders from the search engine.
2. Dropping out of the search engine to get better utilities in another search engine is a possible option for the bidders. The low valuation bidders drop out after not getting slots for a certain period of time. The higher valuation bidders can observe this trend and shade their maximum-willingness-to-pay or collude to get better utilities. This may result in the search engine losing revenue. This is a threat to the search engine from the bidders. However, if a large number of bidders remain in the system, the collusion is not stable. The intuition for this is that, high valuation bidders cannot reduce their bids sharply, since they will have the fear of undercutting the lower valuation bidders present in the system and thus losing out on their slots.
Hence, we propose that retaining more number of bidders solves all the problems discussed above. The dependence of the search engine and the bidders on each other for mutual benefit motivates us to use a cooperative approach in general. The above threat model naturally directs us towards using a Nash bargaining model in particular. Our solution can be seen as associating the bid optimizer to a keyword rather than bidders as done by the existing bid optimizers. The overall model of the bid optimizer is depicted in Figure \[fig:blk1\].
Contributions and Outline of the Paper
--------------------------------------
=9.5cm =14.5cm
In this paper, we propose a bid optimizer that uses many ideas from cooperative game theory. The bid optimizer is shown in Figure \[fig:blk1\].
- The inputs to the bid optimizer are the willingness-to-pay values (or valuations) of the bidders.
- The output of the bid optimizer is a correlated bid profile, which, when input to a standard GSP auction mechanism, yields utilities to the search engine and the advertisers satisfying the goals set forth in the paper.
- The bid optimizer first formulates a characteristic form game involving the search engine and the advertisers. The value for each coalition is defined based on a novel Nash Bargaining formulation with the search engine as one player and a virtual player aggregating all advertisers in that coalition as the other player. The idea of using Nash Bargaining is to ensure a fair share for the search engine and the advertisers.
- The nucleolus of the above characteristic form game is selected as the utility profile for the search engine and the advertisers. The choice of nucleolus is based on key considerations such as, bidder retention, stability, and efficiency.
- The utility profile represented by the nucleolus is mapped to a correlated bid profile that satisfies individual rationality, retention, stability and efficiency. A linear programming based algorithm is suggested for this purpose.
We carry out experiments to demonstrate the viability and efficacy of the proposed bid optimizer. We show, using a credible bidder drop out model, that the proposed bid optimizer has excellent bidder retention properties and also yields higher long-run revenues to the search engine, when compared to the plain GSP mechanism.
The outline of the paper is as follows. Section \[MODEL\] presents the details of the bid optimizer and introduces the model. In Section \[NBFORMULATION\], we present a bid optimization algorithm which uses the Nash Bargaining approach for ensuring the retention of bidders in the system. We then map this fair share for the aggregated bidder to a correlated bid profile in Section \[CorrBidProfileMapping\]. We analyze the properties of our method in Section \[ANALYSIS\]. We present our experimental results in Section \[EXPERIMENTS\] and conclude the paper in Section \[SUMMARY\].
Our Approach to Bid Optimization
================================
In this section, we present our algorithm for bid optimization. The algorithm can be divided into three phases as shown in Figure \[fig:blk1\]: (1) Characteristic form game definition using Nash bargaining, (2) Computing the utility vectors for the players and (3) Inverse mapping of the utility vector into a correlated bid profile. These are discussed in the following sections. The notation in the remainder of the paper is presented in Table \[tb:Notation\].
Characteristic Form Game
------------------------
### The Model {#MODEL}
The sponsored search auction scenario we consider has $n$ bidders competing for $k$ slots of a keyword. We assume that the probability that a bidder $i$ gets clicked on the $j$th slot (or the click-through rate $CTR_{ij}$) is independent of the bidder $i$,that is, $CTR_{ij} = \beta_j$ and we also assume that $ \beta_1
\geq \beta_2 \geq \ldots \geq \beta_k$. Each bidder $i$ specifies his maximum willingness-to-pay $ \overline{s}_{i} $ to the bid optimizer. The bid optimizer takes as input all the $
\overline{s}_{i} $’s of the bidders and suggests them a correlated bid profile. This bid optimization algorithm needs to be invoked only when the number of bidders in the system or their willingness-to-pay change. We also assume that the bid of the player $i$ could be any real number in $[0,\overline{s}_{i}]$.
Given the above model, we define a bargaining problem [^1] between the search engine and the aggregated bidder and analyze its properties which will help us in formulating a characteristic form game.
----------------------------------------------------------------------------------------
**Notation** **Explanation**
-------------------- -------------------------------------------------------------------
$ A $ Auctioneer
$ B $ Aggregated bidder
$ n $ Total number of players
$ k $ Total number of slots. We assume $k < n$
$ N $ Set of bidders $\lbrace 1, 2, \ldots, n \rbrace$
$ K $ Set of slots $\lbrace 1, 2, \ldots, k \rbrace$
$ Maximum willingness-to-pay of advertiser $i$
\overline{s}_{i} $
$ S_i $ Strategy set of bidder $i, \>\>
[0,\overline{s}_{i}] $
$ S $ Set of all bid profiles $S_1 \times S_2 \times \ldots \times S_n$
$ s $ Bid profile $ (s_1, s_2, \ldots, s_n) \in S $
$ u_i(s) $ Utility of bidder $i$ on bid profile $s$
$ U_A(s) $ Utility of the auctioneer in the
$\>$ Nash bargaining formulation for bid profile $s$
$ U_B(s) $ Utility of the aggregated bidder in the
$\>$ Nash bargaining formulation for bid profile $s$
$ $\max_{ s \in S} U_A(s)$
\overline{U}_A $
$ $\max_{ s \in S} U_B(s)$
\overline{U}_B $
$ \beta_j $ Click through rate of any bidder in the $j^{th}$ slot
----------------------------------------------------------------------------------------
: Notation[]{data-label="tb:Notation"}
\
### Characterization of the Nash Bargaining Solution {#NBS}
The motivation for a cooperative approach is the dependence of the search engine and bidders on each other for their mutual benefit. Given this, the motivation behind choosing a bargaining approach is that the amount of short-term loss (or in other words, the investment of the search engine) for the auctioneer should be chosen based on the bidders present in the system. The Nash bargaining approach provides a framework for this amount to be chosen by the search engine by considering all the bidders as one aggregate agent whose bargaining power depends on all the maximum willingness-to-pay of all the bidders present in the system.
The utility of the aggregated bidder is the sum of the utilities of all the bidders over all possible allocations of slots (outcomes). Now, the bargaining utility space becomes the two dimensional Cartesian space which consists of the utility of auctioneer on one axis and the aggregate bidder’s utility on the other axis. Hence a bargaining solution on this space provides a good compromise for the search engine from its maximum possible revenue and thus gives the required investment of the search engine.
The bargaining space is defined in two dimensional Cartesian space, with utility of auctioneer $U_A(s)$ along the $x-$axis and the utility of aggregated bidder $U_B(s) = \sum_{i=1}^{n} u_i(s)$ along the $y-$axis. Let $\overline{U}_A$ and $\overline{U}_B$ be the maximum possible utilities of the auctioneer and the aggregated bidder respectively. It can be clearly seen that the value $\overline{U}_A$ is attained for the bid profile $ s =
(\overline{s}_1, \overline{s}_2, \ldots , \overline{s}_n)$ for which the corresponding $ U_B(\overline{s}) = \sum_{i=1}^{n}
\left( \sum_{j=1}^{k} \beta_j y_{ij}(\overline{s}) \right)
(\overline{s}_i - \overline{s}_{i+1}) = U_{B}^{'}$ (say). Similarly, the bid profile $s= (0, \ldots , 0)$ yields the utility pair $(0, \overline{U}_B)$. Since it is theoretically possible that all the bidders can collude and bid $(0, 0, \ldots, 0)$, we choose the point $(0,0)$ in this Nash bargaining space as the disagreement point. Ramakrishnan *et. al* studied this problem in [@ramkey:paper] and characterized the solution $
(U_{A}^{*}, U_{B}^{*})$ to this Nash bargaining(NBS) as $$(U_{A}^{*}, U_{B}^{*}) = (\overline{U}_A ,U_B^{'}) \>\>\> if \> \overline{U}_A \leq \frac{\overline{U}_B}{2}$$ $$\hspace{1.5 cm} = \left(\frac{\overline{U}_B}{2}, \frac{\overline{U}_B}{2} \right) \>\>\> otherwise$$
### Definition of the Characteristic Form Game {#NBFORMULATION}
We use the above model to define Nash bargaining solution $NBS(N) = U_{A}^{*} + U_{B}^{*}$ where $N$ is the set of bidders participating in the auction.
Let $ N = \lbrace 1, 2, \ldots, n \rbrace$ be the set of all bidders and let $0$ represent the search engine. The characteristic form game $\nu : 2^{N \cup \lbrace 0 \rbrace}\rightarrow \Re $ for each coalition $C \subseteq N \cup \lbrace 0 \rbrace $ is now defined as $$\nu(C) = NBS(C) \>\>\> if \> 0 \in C$$ $$\hspace{0.25 cm} = 0 \hspace{0.25 cm} otherwise \hspace{0.34 cm}$$ where $NBS(C)$ is defined as above. If the search engine is not a part of the coalition, its worth is zero since the players cannot gain anything without the search engine displaying their ads. Otherwise, we associate the sum of utilities in the corresponding Nash bargaining bid profile for that coalition with the search engine as the worth of each coalition. This characteristic function $\nu$ defines the bargaining power of each coalition with the search engine.
Computing a Utility Vector for the Players: Use of Nucleolus {#IdealProperties}
------------------------------------------------------------
Since there is an aggregation of the bidders’ revenue taking place in the NBS, we map the utility of the aggregated bidder in the Nash bargaining solution to a correlated bid profile. The NBS gives an aggregate amount of investment the search engine has to make on all the bidders. This investment increases the utility of the aggregated bidder. This utility has to be distributed to the bidders in a way that our goal of retention is reached. Ideally, we would like the allocation to have the following properties.
- The bidders must not have incentive for not participating in the bid optimizer (individual rationality-IR).
- It must retain as many bidders as possible(retention).
- The bidders must not have the incentive to shade their maximum willingness to pay (incentive compatibility).
- It should be stable both in the one-shot game of GSP and in the cooperative analysis (stability).
- It should divide the entire worth of the grand coalition among all the bidders (efficiency).
There are several solution concepts in cooperative game theory that one could employ here,for example, the core, the Shapley value, the nucleolus, etc. We believe the nucleolus is clearly the best choice that satisfies a majority of the above properties. Since nucleolus is defined as the unique utility vector which makes the unhappiest coalition as less unhappy as possible [@straffin:book], and given that the nucleolus is always in a non-empty core, it is the utility vector that retains the most number of bidders if the core is empty and is the most stable one retaining all the bidders if the core is non-empty. We compute the nucleolus by solving a series of linear programs [@mwg; @myerson:book] and obtain the utility vector $( x_1, x_2, \ldots, x_n)$ for the $n$ players and the search engine ($x_0$).
Mapping the Utility Vector to a Correlated Bid Profile {#CorrBidProfileMapping}
------------------------------------------------------
### Obtaining a locally envy-free bid profile for each valid coalition
To satisfy the stability criterion in the non-cooperative sense, and ensure truthful participation of all the bidders in the proposed bid optimizer, we aim to find out locally envy-free bids for each of the $ n \choose k $ possible sets of winning bidders. For finding these bids, consider a subset of $(k+1)$ bidders and allocate slots to the bidders in this subset in the sorted order of their willingness-to-pay values to satisfy the requirement for the locally envy-free equilibrium. Now, the bids can be calculated as follows. The $(k+1)^{th}$ bidder bids the reserve price (assumed to be $0$ here without loss of generality). The bid of the $k^{th}$ bidder (who pays $\overline{s}_{k+1}$) is now calculated by solving for $b_k$ in $ \beta_k ( \overline{s}_k -
b_{k+1} ) = \beta_{k-1} ( \overline{s}_k - b_k) $ to satisfy the envy-freeness. Once we obtain $b_k$, we proceed recursively by replacing the $b_{k+1}$ by $b_k$ and $k$ by $(k-1)$ in the above equation to get $b_{k-1}$ and so on till we get the bids of all the $k$ players. Note that the bid of the first player does not have a role here as long as it is greater than the next highest bid. Thus we obtain *a* set of bids which are in locally envy-free equilibrium.
### Obtaining a correlated bid profile
The solution given by the nucleolus provides a utility for each bidder. This cannot be used directly in the GSP auction of the search engine. Towards this end, we map the nucleolus to a correlated bid profile which defines the required rotation among the bidders for occupying the slots. This correlated bid profile is what is finally suggested by the bid optimizer, which retains the maximum number of advertisers without hurting the search engine.
The characterization of a correlated bid profile corresponds to assigning the probabilities associated with each of the bid profiles associated with the bidders. There exist several algorithms in general, for finding the correlated bid profile. But we would like to exploit the structure of the problem and obtain a simpler solution without going into the complex details about modifying the ellipsoid algorithm as done in most of the work in this area. See [@papadi:correlated] for example. Any correlated strategy we consider here has a subset of size $k$ bidders bidding their corresponding LEF (locally envy-free) bids (obtained in the previous section) and all other bidders bidding the reserve price. Considering only these $n \choose k$ strategy profiles corresponding to each subset of size $k$ bidders winning the slots would suffice since they exhaust all the possible outcomes of the underlying GSP auction.
The probability distribution which yields the utilities suggested by the nucleolus to the players is any distribution which satisfies the constraints that it is a probability distribution, it is individually rational for each player and it must yield the payoffs suggested by the nucleolus to the bidders subject to their budget constraints. This can be obtained by solving a linear program as follows.
$$\min \hspace{0.5 cm} \sum_{i \in N \cup \lbrace 0 \rbrace} z_i + \sum_{i \in N} \overline{s}_i \left( x_{i} - \left( \sum_{\substack{C \subseteq N ,\> \mid C \mid = k \\ i \in C ,\> j = y_{iC} }} p_{C} \beta_j (\overline{s}_{i} - b_{j}) \right) \right) \hspace{5.6 cm}$$
subject to
$$\begin{aligned}
\forall i \in N \>\>\>\>\>\> z_i & \geq & \left( \sum_{\substack{C \subseteq N ,\> \mid C \mid = k \\ i \in C ,\> j = y_{iC} }} p_{C} \beta_j (\overline{s}_{i} - b_{j}) \right) - x_{i} \\
%
\forall i \in N \>\>\>\>\>\> z_0 & \geq & \left( \sum_{\substack{C \subseteq N ,\> \mid C \mid = k }} p_{C} \sum_{\substack{ i \in C ,\> j = y_{iC}}}\beta_j b_{j+1} \right) - x_0 \\
%
\forall i \in N \>\>\>\>\>\> z_0 & \geq & x_0 - \left( \sum_{\substack{C \subseteq N ,\> \mid C \mid = k }} p_{C} \sum_{\substack{ i \in C ,\> j = y_{iC}}}\beta_j b_{j+1} \right) \\
%
\forall i \in N \>\>\>\>\>\> z_i & \geq & x_{i} - \left( \sum_{\substack{C \subseteq N ,\> \mid C \mid = k \\ i \in C ,\> j = y_{iC} }} p_{C} \beta_j (\overline{s}_{i} - b_{j}) \right) \\
%
p_{C} \beta_j (\overline{s}_{i} - b_{j}) & \geq & 0 \>\>\> \forall C \subseteq N \>\> \forall i \in C \>\> \forall j \in K \\
%
\sum_{C \subset N} p_{C} & = & 1 \\
%
\forall C \subset N \>\>\>\>\>\> p_{C} & \geq & 0
%
\end{aligned}$$
where $y_{iC} $ denotes the slot that player $i$ wins in a locally envy-free allocation if only the set $C$ of players were to win all the slots.
The linear program maps the utility vector suggested by the nucleolus into a correlated bid profile. The objective function minimizes the difference between the utility suggested by nucleolus and the expected utility in the correlated bid profile for each player. The minimization of difference leads to two constraints for each player. This is because for any two variables $x$ and $y$, $$\min \mid x - y \mid$$ is the same as $$\min z$$ $ \hspace{2 cm} $subject to $$z \geq x - y$$ $$z \geq y - x$$
In the minimization, the higher valuation bidders are given a preference over the lower valuation bidders. This is done by weighting each player’s difference from the nucleolus in the objective function by their valuation. This is a heuristic to ensure that the error in the inverse mapping of the utility vector to a correlated bid profile is biased towards the higher valuation bidders so that they voluntarily participate in the bid optimizer. Since the only problem to Individual rationality is when the higher valuation bidders shade their willingness-to-pay, this weighing gives the incentive for them to reveal their true valuations.
In the objective function, we minimize the *difference* (this is done by the first $4$ constraints of the linear program) between the utility vector and the obtained expected utility in the above linear program since the restriction of the bid profiles to the set of locally envy-free equilibria may not have a feasible correlated bid profile. The minimization is done in such a way that the higher valuation bidders obtain relatively higher utility (due to the weights given to the difference in the objective function) than the lower valuation bidders in case the optimal value of the objective function is non-zero. This is a heuristic to ensure that the error in the inverse mapping of the utility vector to a correlated bid profile is biased towards the higher valuation bidders so they voluntarily participate in the bid optimizer.
Properties of the Proposed Solution {#ANALYSIS}
-----------------------------------
The properties of the proposed solution are as follows:
- The proposed solution has the bidders participating voluntarily in the bid optimizer for the following reasons. (i) The auctioneer is benefited since he has a guaranteed revenue of at least what is suggested by the nucleolus. (ii) The high valuation bidders are benefited since they are offered the same slots at a relatively lower price. Also, since the nucleolus tries to retain the grand coalition intact, it will be individually rational for the high valuation bidders to participate in the bid optimizer rather than to deviate and bid higher. (iii) The lower valuation bidders are benefited because they get more slots and hence more clicks and their campaign is more effective. Thus, the utility of every player increases and the individual rationality (IR) condition is satisfied.
- The bids suggested are in a locally envy-free equilibrium of the game and also are in the core since the nucleolus is always in core if the core is non-empty. This indicates that the proposed solution is strategically stable. In other words, no one can profitably deviate unilaterally from the solution proposed by the bid optimizer.
- Retention and efficient division of the worth of the grand coalition are guaranteed by the nucleolus since it is the allocation which tries to retain the grand coalition.
- Truthfulness is difficult to satisfy, given that the GSP mechanism is non-truthful. But note that the lower valuation bidders have no incentive to shade their bids. If they do so, they may lose their slots or run into negative utilities. Hence there is a problem only when the higher valuation bidders do not participate in the bid optimizer or they understate their willingness-to-pay. The higher valuation bidders cannot understate their valuations by a large amount since they have a threat of losing their slots to lower valuation bidders who are retained in the system. Also, the higher valuation bidders are given more benefits to participate in the bid optimizer and it is individually rational for them to participate in the bid optimizer.
Hence this solution satisfies all the properties which were mentioned in Section \[IdealProperties\].
Experimental Results {#EXPERIMENTS}
====================
This section presents simulation based experimental results to explore the effectiveness of the approach presented in the paper. First we start with a model for the drop outs of bidders.
Bidder drop out model
---------------------
The bidders drop out if they do not get enough slots (or alternatively clicks) consistently over a period of time. The conditional probability that a bidder drops out given that he did not get a slot in a round may vary from bidder to bidder. Also, the positions of slots occupied by the bidders in the previous few rounds of auction could play an important role in the dropping out of a bidder.
To model this behavior of the bidder dropping out based on the outcomes of the previous auctions and giving more importance to recent outcomes, we propose a discounted weighting of the outcomes of the previous auctions to compute the probability that a bidder will continue in the next round of the auction. This model also captures the myopic human behavior that the bidder’s choice is dependent on only the recent outcomes. That is, the history the bidder looks into, before taking a decision to continue or not for the next round is limited. The amount of history however depends on the bidder in the form of his discounting factor. Let $x_{-i} \in \lbrace 0, 1 \rbrace$ denote the outcome of the $i^{th}$ previous round. A “$1$“ denotes that the bidder received a click in the $i^{th}$ previous auction with a zero indicating otherwise. We propose that the probability that the bidder will participate in the next round is given by $\frac{\sum_{i=1}^{\infty} \gamma^i x_{-i}}{\sum_{i=1}^{\infty} \gamma^i} = (1 - \gamma) \sum_{i=1}^{\infty} \gamma^ix_{-i}$ where $\gamma$ is the discount factor of the bidder. To see how the myopic nature of the bidders is captured, suppose that the bidder’s discount factor $\gamma = 0.95$. The discount factor for the $101^{st}$ round will be $0.006$ which is negligible. Hence the bidder’s decision is dependent on at most $100$ previous auctions. Thus, the discount factor decides the nature of the bidder.
Experimental Setup
------------------
Given a fixed set of CTRs, the valuations of the bidders are chosen close enough to each other, to analyse our model in competitive environment. The retention problem, is fundamental in competitive environment, as search engine needs to retain the bidders and allow them compete in further auctions. The results indicate that the proposed bid optimizer not only retains a higher number of bidders than the normal GSP but also yields better cumulative revenue to the search engine in the long run.
=9 cm =7 cm
Cumulative Revenue of the Search Engine
---------------------------------------
First we consider the cumulative revenue of the search engine for comparing the non-cooperative bidding and using cooperative bid optimizer. We consider $10$ advertisers with $5$ slots to be allocated.We run successive auctions and find the cumulative revenue of the search engine after each auction using the two approaches. We run this experiment until the change in the average cumulative revenue after each auction becomes acceptably small. Figure \[fig:exp1\] shows a comparison of the cumulative revenue of the search engine under the proposed approach with that of a standard GSP auction.
It can be seen in Figure \[fig:exp1\] that though initially the non-cooperative approach (GSP) yields more revenue, after a few runs, the cooperative bid optimizer, with all the solution vectors, starts outperforming. Initially the GSP outcome is better, as the advertisers are bidding their maximum willingness to pay, and hence the search engine gets high levels of revenue. However, as the utilities of the advertisers are less in the case of the non-cooperative approach, they start dropping out of the auction and hence in a long run, the cumulative revenue starts declining compared to the cooperative bid optimizer. This is the adverse effect of the dropping out of the advertisers leading to the retention problem.
=10 cm =8 cm
Number of Bidders Retained in the System
----------------------------------------
After each auction, we compute the average number of advertisers retained, to analyze the retention dynamics in the system. Figure \[fig:exp3\] gives a comparison between using the cooperative bid optimizer and the GSP approach.
In Figure \[fig:exp3\], it can be observed that there are considerably more number of advertisers retained in the system when using the solution vector suggested by the cooperative bid optimizer in comparison to the GSP approach. This explains the reason for the dip in the non-cooperative cumulative revenue of the search engine.
Through experimentation, we are able to demonstrate revenue increase for the search engine in the long run and also reduction of the retention problem considerably. It should be noted that even though the raise is not substantial, its impact on retaining and attracting the advertisers and thereby other indirect advantages are immense.
Summary and Future Work {#SUMMARY}
=======================
We have proposed a bid optimizer for sponsored keyword search auctions which leads to better retention of advertisers by yielding attractive utilities to the advertisers without decreasing the long-run revenue to the search engine. The bid optimizer is a value added tool the search engine provides to the advertisers which transforms the reported values of the advertisers for a keyword into a correlated bid profile. The correlated bid profile that we determine is a locally envy-free equilibrium and also a correlated equilibrium of the underlying game. Through detailed simulation experiments, we have shown that the proposed bid optimizer retains more customers than a plain GSP mechanism and also yields better long-run utilities to the search engine and the advertisers.
The experiments were carried out with a model that captures the phenomenon of customer drop outs and showed that our approach produces a better long run utility to the search engine and all the advertisers. The proposed bid optimizer is beneficial for both the bidders and the search engine in the long run.
We considered GSP auction, which is popularly run in most of the search engines, in our analysis. However, it would be interesting to look at the effects of other auction mechanisms like VCG auctions on the overall process.
The other important components like budget optimization and ad scheduling are involved in sponsored search auctions. We would further like to combine these components with our cooperative bid optimizer.
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Appendix {#appendix .unnumbered}
========
Nash [@Nash] proposed that there exists a unique solution function $f(F,v)$ for every two person bargaining problem, that satisfies the following 5 axioms - *[Pareto strong efficient, Individual Rationality, Symmetry, Scale Covariance, and Independence of Irrelevant Alternatives.]{} The solution function is $$f(F, v) \in {\rm argmax}_{ (x_1,x_2) \in F}
((x_1-v_1)(x_2-v_2))$$*
where, $x_1 \geq v_1$ and $x_2 \geq v_2$ and the point $v = (v_1, v_2)$ is known as the *[point of disagreement]{}. There are several possibilities for choosing the disagreement point $v$. The three popular choices are those based on (1) a minimax criterion, (2) focal equilibrium, and (3) rational threats. As part of this paper, we use the rational threats to identify disagreement point $v$ [@myerson:book]. For more details please refer to the books [@myerson:book] [@straffin:book].*
[^1]: Refer Appendix for the definition of Nash bargaining problem
|
---
abstract: 'The models of $n\bar{n}$ transitions in the medium based on unitary $S$-matrix are considered. The time-dependence and corrections to the models are studied. The lower limits on the free-space $n\bar{n}$ oscillation time are obtained as well.'
author:
- |
V.I.Nazaruk\
Institute for Nuclear Research of RAS, 60th October\
Anniversary Prospect 7a, 117312 Moscow, Russia.\*
title: '$n\bar{n}$ transitions in medium'
---
-0.6in
11.30.Fs; 13.75.Cs
Keywords: diagram technique, infrared divergence, time-dependence
\*E-mail: [email protected]
Introduction
============
In the standard calculations of $ab$ oscillations in the medium \[1-3\] the interaction of particles $a$ and $b$ with the matter is described by the potentials $U_{a,b}$ (potential model). ${\rm Im}U_b$ is responsible for loss of $b$-particle intensity. In particular, this model is used for the $n\bar{n}$ transitions in a medium \[4-10\] followed by annihilation: $$n\rightarrow \bar{n}\rightarrow M,$$ here $M$ are the annihilation mesons.
In \[9\] it was shown that one-particle model mentioned above does not describe the total $ab$ (neutron-antineutron) transition probability as well as the channel corresponding to absorption of the $b$-particle (antineutron). The effect of final state absorption (annihilation) acts in the opposite (wrong) direction, which tends to the additional suppression of the $n\bar{n}$ transition. The $S$-matrix should be unitary.
In \[11\] we have proposed the model based on the diagram technique which does not contain the non-hermitian operators. Subsequently, this calculation was repeated in \[12\]. However, in \[13\] it was shown that this model is unsuitable: the neutron line entering into the $n\bar{n}$ transition vertex should be the wave function, but not the propagator, as in the model based on the diagram technique. For the problem under study this fact is crucial. It leads to the cardinal error for the process in nuclei. The $n\bar{n}$ transitions in the medium and vacuum are not reproduced at all. If the neutron binding energy goes to zero, the result diverges (see Eqs. (18) and (19) of Ref. \[11\] or Eqs. (15) and (17) of Ref. \[12\]). So we abandoned this model \[13\]. (In the recent manuscript \[14\] the previous calculations \[11,12\] have been repeated. The model and calculation are the same as in \[11,12\]. Unfortunately, several statements are erroneous \[15\], in particular, the conclusion based on an analogy with the nucleus form-factor at zero momentum transfer (for more details, see \[15\]).)
In \[16\] the model which is free of drawbacks given above has been proposed (model [**b**]{} in the notations of present paper). However, the consideration was schematic since our concern was only with the role of the final state absorption in principle. In Sect. 2 this model as well as the model with bare propagator are studied in detail. The corrections to the models (Sect. 3) and time-dependence (Sect. 4) are considered as well. In addition, we sum up the present state of the investigations of this problem (Sect. 5).
The basic material is given in Sects. 2 and 5.
Models
======
First of all we consider the antineutron annihilation in the medium. The annihilation amplitude $M_a$ is defined as $$<\!f0\!\mid T\exp (-i\int dx{\cal H}(x))-1\mid\!0\bar{n}_{p}\!>=
N(2\pi )^4\delta ^4(p_f-p_i)M_a.$$ Here ${\cal H}$ is the Hamiltonian of the $\bar{n}$-medium interaction, $\mid\!0\bar{n}_{p}\!>$ is the state of the medium containing the $\bar{n}$ with the 4-momentum $p=(\epsilon
,{\bf p})$; $<\!f\!\mid $ denotes the annihilation products, $N$ includes the normalization factors of the wave functions. The antineutron annihilation width $\Gamma $ is expressed through $M_a$: $$\Gamma \sim \int d\Phi \mid\!M_a\!\mid ^2.$$
For the Hamiltonian ${\cal H}$ we consider the model $$\begin{aligned}
{\cal H}={\cal H}_a+V\bar{\Psi }_{\bar{n}}\Psi _{\bar{n}},\nonumber\\
H(t)=\int d^3x{\cal H}(x)=H_a(t)+V,\end{aligned}$$ where ${\cal H}_a$ is the effective annihilation Hamiltonian in the second quantization representation, $V$ is the residual scalar field. The diagrams for the model (4) are shown in Fig. 1. The first diagram corresponds to the first order in ${\cal H}_a$ and so on.
[{height=".25\textheight"}]{}
Consider now the process (1). The neutron wave function is $$n(x)=\Omega ^{-1/2}\exp (-ipx).$$ Here $p=(\epsilon ,{\bf p})$ is the neutron 4-momentum; $\epsilon ={\bf p}^2/2m+U_n$, where $U_n$ is the neutron potential. The interaction Hamiltonian has the form $$H_I=H_{n\bar{n}}+H,$$ $$H_{n\bar{n}}(t)=\int d^3x(\epsilon _{n\bar{n}}\bar{\Psi }_{\bar{n}}(x)\Psi _n(x)+H.c.)$$ Here $H_{n\bar{n}}$ is the Hamiltonian of $n\bar{n}$ conversion \[6\], $\epsilon _{n\bar{n}}$ is a small parameter with $\epsilon _{n\bar{n}}=1/\tau $, where $\tau $ is the free-space $n\bar{n}$ oscillation time; $m_n=m_{\bar{n}}=m$. In the lowest order in $H_{n\bar{n}}$ the amplitude of process (1) is [*uniquely*]{} determined by the Hamiltonian (6): $$M=\epsilon _{n\bar{n}}G_0M_a,$$ $$G_0=\frac{1}{\epsilon -{\bf p}^2/2m-U_n+i0},$$ ${\bf p}_{\bar{n}}={\bf p}$, $\epsilon _{\bar{n}}=\epsilon $. Here $G_0$ is the antineutron propagator. The corresponding diagram is shown in Fig. 2a. The annihilation amplitude $M_a$ is given by (2), where ${\cal H}={\cal H}_a+V\bar{\Psi }_{\bar{n}}\Psi _{\bar{n}}$. Since $M_a$ contains all the $\bar{n}$-medium interactions followed by annihilation including antineutron rescattering in the initial state, the antineutron propagator $G_0$ is bare. Once the antineutron annihilation amplitude is defined by (2), the expression for the process amplitude (8) [*rigorously follows*]{} from (6). For the time being we do not go into the singularity $G_0\sim 1/0$.
[![[**a**]{} $n\bar{n}$ transition in the medium followed by annihilation. The antineutron annihilation is shown by a circle. [**b**]{} Same as [**a**]{} but the antineutron propagator is dressed (see text)](f2final.eps "fig:"){height=".25\textheight"}]{}
One can construct the model with the dressed propagator. We include the scalar field $V$ in the antineutron Green function $$G_d=G_0+G_0VG_0+...=\frac{1}{(1/G_0)-V}=-\frac{1}{V}=-\frac{1}{\Sigma },$$ $\Sigma =V$, where $\Sigma $ is the antineutron self-energy. The process amplitude is $$M=\epsilon _{n\bar{n}}G_dM_b,$$ $G_dM_b=G_0M_a$ (see Fig. 2b). The block in the square braces shown in Fig.1 corresponds to the vertex function $M_b$. The models shown in Figs. 2a and 2b we denote as the models [**a**]{} and [**b**]{}, respectively.
In both models the interaction Hamiltonians $H_I$ and unperturbed Hamiltonians are the same. If $\Sigma \rightarrow 0$, the model [**b**]{} goes into model [**a**]{}. In this sense the model [**a**]{} is the limiting case of the model [**b**]{}.
We consider the model [**b**]{}. For the process width $\Gamma _b$ one obtains $$\Gamma _b=N_1\int d\Phi \mid\!M\!\mid ^2=\frac{\epsilon _{n\bar{n}}^2}{\Sigma ^2}N_1\int d\Phi
\mid\!M_b\!\mid ^2=\frac{\epsilon _{n\bar{n}}^2}{\Sigma ^2}\Gamma ',$$ $$\Gamma '=N_1\int d\Phi \mid\!M_b\!\mid ^2,$$ where $\Gamma '$ is the annihilation width of $\bar{n}$ calculated through the $M_b$ (and not $M_a$). The normalization multiplier $N_1$ is the same for $\Gamma _b$ and $\Gamma '$. The vertex function $M_b$ is unknown. (We recall the antineutron annihilation width $\Gamma $ is expressed through the amplitude $M_a$.) For the estimation we put $$M_b=M_a, \quad \Gamma '=\Gamma.$$ This is an uncontrollable approximation.
The time-dependence is determined by the exponential decay law: $$W_b(t)=1-e^{-\Gamma _bt}\approx \Gamma _bt=\frac{\epsilon _{n\bar{n}}^2}{\Sigma ^2}\Gamma t.$$ Equation (15) illustrates the result sensitivity to the value of parameter $\Sigma $.
On the other hand, for $n\bar{n}$ transitions in nuclear matter the standard calculation gives the inverse $\Gamma $-dependence \[6-9\] $$W_{{\rm stan}}(t)=2\epsilon _{n\bar{n}}^2t\frac{\Gamma /2}{({\rm Re}U_{\bar{n}}-U_n)^2+
(\Gamma /2)^2}\approx \frac{4\epsilon _{n\bar{n}}^2t}{\Gamma },$$ where $U_{\bar{n}}$ is the antineutron optical potential. The wrong $\Gamma $-dependence is a direct consequence of the inapplicability of the model based on optical potential for the calculation of the total process probability \[9\]. (The above-mentioned model describes the probability of finding an antineutron only.)
Comparing with (15), one obtains $$r= \frac{W_b}{W_{{\rm stan}}}=\frac{\Gamma ^2}{4\Sigma ^2}=25,$$ where the values $\Gamma =100$ MeV and $\Sigma ={\rm Re}U_{\bar{n}}-U_n=10$ MeV have been used. Strictly speaking, the parameter $\Sigma $ is uncertain. We have put $\Sigma =
{\rm Re}U_{\bar{n}}-U_n=10$ MeV only for estimation.
The model [**b**]{} leads to an increase of the $n\bar{n}$ transition probability. The lower limit on the free-space $n\bar{n}$ oscillation time $\tau ^b_{{\rm min}}$ increases as well: $$\tau ^b_{{\rm min}}=(3.5-7.5)\cdot 10^{8}\; {\rm s}.$$ This limit exceeds the previous one (see, for example, Refs. \[5-8\]) by a factor of five. If $\Sigma \rightarrow 0$, $W_b$ rises quadratically. So Eq. (18) can be considered as the estimation from below.
We return to the model shown in Fig. 2a. We use the basis $(n,\bar{n})$. The results do not depend on the basis. A main part of existing calculations have been done in $n-\bar{n}$ representation. The physics of the problem is in the Hamiltonian. The transition to the basis of stationary states is a formal step. It has a sense only in the case of the potential model $H=H_{{\rm pot}}={\rm Re}U_{\bar{n}}-U_n-i\Gamma /2=$const., when the Hamiltonian of $\bar{n}$-medium interaction is replaced by the effective mass $H\rightarrow H_{{\rm pot}}=
m_{{\rm eff}}$ because the Hermitian Hamiltonian of interaction of the stationary states with the medium is unknown. Since we work beyond the potential model, the procedure of diagonalization of mass matrix is unrelated to our problem.
The amplitude (8) diverges $$M=\epsilon _{n\bar{n}}G_0M_a\sim \frac{1}{0}.$$ (See also Eq. (21) of Ref. \[13\].) These are infrared singularities conditioned by zero momentum transfer in the $n\bar{n}$ transition vertex. (In the model [**b**]{} the effective momentum transfer $q_0=V=\Sigma $ takes place.)
For solving the problem the field-theoretical approach with finite time interval \[17\] is used. It is infrared free. If $H=H_{{\rm pot}}$, the approach with finite time interval reproduces all the results on the particle oscillations, in particular, the $n\bar{n}$ transition with $\bar{n}$ in the final state. (Recall that our purpose is to describe the process (1) by means of Hermitian Hamiltonian.)
For the model [**a**]{} the process (1) probability was found to be \[10,13\] $$W_a(t)\approx W_f(t)=\epsilon _{n\bar{n}}^2t^2, \quad \Gamma t\gg 1,$$ where $W_f$ is the free-space $n\bar{n}$ transition probability. Owing to annihilation channel, $W_a$ is practically equal to the free-space $n\bar{n}$ transition probability. If $t\rightarrow \infty $, Eq. (20) diverges just as the modulus (19) squared does. If $\Sigma \rightarrow 0$, Eq. (15) diverges quadratically as well.
The explanation of the $t^2$-dependence is simple. The process shown in Fig. 2a represents two consecutive subprocesses. The speed and probability of the whole process are defined by those of the slower subprocess. If $1/\Gamma \ll t$, the annihilation can be considered instantaneous. So, the probability of process (1) is defined by the speed of the $n\bar{n}$ transition: $W_a\approx W_f\sim t^2$.
Distribution (20) leads to very strong restriction on the free-space $n\bar{n}$ oscillation time \[10,13\]: $$\tau ^a_{{\rm min}}=10^{16}\; {\rm yr}.$$
Corrections
===========
We show that for the $n\bar{n}$ transition in medium the corrections to the models and additional baryon-number-violating processes (see Fig. 3) cannot essentially change the results. First of all we consider the incoherent contribution of the diagrams 3. In Fig. 3a a meson is radiated before the $n\bar{n}$ transition. The interaction Hamiltonian has the form $$H_I=\int d^3xg\Psi ^+_n\Phi \Psi _n+H_{n\bar{n}}+H.$$ In the following the background neutron potential is omitted. The neutron wave function is given by (5), were $p=(p_0,{\bf p})$ and $p_0=m+{\bf p}^2/2m$.
For the process amplitude $M_{3a}$ one obtains $$M_{3a}=gG\epsilon _{n\bar{n}}GM^{(n-1)},$$ $$G=\frac{1}{p_0-q_0-m-({\bf p}-{\bf q})^2/2m+i0},$$ where $q$ is the 4-momentum of meson radiated, $M^{(n-1)}$ is the amplitude of antineutron annihilation in the medium in the $(n-1)$ mesons. As with model [**a**]{}, the antineutron propagator $G$ is [*bare*]{}; the $\bar{n}$ self-energy $\Sigma =0$. (The same is true for Figs. 3b-3d.)
[![Corrections to the models ([**a**]{} and [**b**]{}) and additional baryon-number-violating processes ([**c**]{} and [**d**]{})](f3final.eps "fig:"){height=".3\textheight"}]{}
If $q\rightarrow 0$, the amplitude $M_{3a}$ increases since $G\rightarrow G_s$, $$G_s=\frac{1}{p_0-m-{\bf p}^2/2m}\sim \frac{1}{0}.$$ (The limiting transition $q\rightarrow 0$ for the diagram 3a is an imaginary procedure because in the vertex $n\rightarrow n\Phi $ the real meson is escaped and so $q_0\geq
m_{\Phi }$.) The fact that the amplitude increases is essential for us because for Fig. 2a $q=0$. Due to this $G_0\sim 1/0$ and $W_a\gg W_b$.
Let $\Gamma _{3a}$ and $\Gamma ^{(n)}$ be the widths corresponding to the Fig. 3a and annihilation width of $\bar{n}$ in the $(n)$ mesons, respectively; $\Gamma =
\sum_{(n)}\Gamma ^{(n)}$. Taking into account that $\Gamma ^{(n)}$ is a smooth function of $\sqrt{s}$ and summing over $(n)$, it is easy to get the estimation: $$\Gamma _{3a}\approx 5\cdot 10^{-3}g^2\frac{\epsilon _{n\bar{n}}^2}{m^2_{\Phi }}\Gamma
\approx \frac{\epsilon _{n\bar{n}}^2}{m^2_{\Phi }}\Gamma .$$
The time-dependence is determined by the exponential decay law: $$W_{3a}(t)\approx \Gamma _{3a}t=\frac{\epsilon _{n\bar{n}}^2}{m^2_{\Phi }}\Gamma t.$$ Comparing with (15) we have: $W_{3a}/W_b=V^2/m^2_{\Phi }\ll 1$. So for the model [**b**]{} the contribution of diagram 3a is negligible.
For the model [**a**]{} the contribution of diagram 3a is inessential as well. Indeed, using Eqs. (27) and (20) we get $$\frac{W_{3a}(t)}{W_a(t)}=\frac{\Gamma }{m^2_{\pi }t},$$ where we have put $m_{\Phi }=m_{\pi }$. Consequently, if $$m^2_{\pi }t/\Gamma \gg 1,$$ and $$\Gamma t\gg 1$$ (see (20)) then the contribution of diagram 3a is negligible. For the $n\bar{n}$ transition in nuclei these conditions are fulfilled since in this case $\Gamma \sim 100$ MeV and $t=T_0=1.3$ yr, where $T_0$ is the observation time in proton-decay type experiment \[18\].) In fact, it is suffice to hold condition (30) only because it is more strong.
In the calculations made above the free-space $n\bar{n}$ transition operator has been used. This is impulse approximation which is employed for nuclear $\beta $ decay, for instance. The simplest medium correction to the vertex (or off-diagonal mass, or transition mass) is shown in Fig. 3b. In this event the replacement should be made: $$\epsilon _{n\bar{n}}\rightarrow \epsilon _m=\epsilon _{n\bar{n}}(1+\Delta \epsilon ),$$ $\Delta \epsilon =\epsilon _{3b}/\epsilon _{n\bar{n}}$, where $\epsilon _{3b}$ is the correction to $\epsilon _{n\bar{n}}$ produced by the diagram 3b. For the model [**a**]{} the limit becomes $$\tau ^a_{{\rm min}}=(1+\Delta \epsilon )10^{16}\; {\rm yr}.$$ Obviously, the $\Delta \epsilon $ cannot change the order of magnitude of $\tau
_{{\rm min}}$ since the $n\rightarrow \bar{n}$ operator is essentially zero-range one. The free-space $n\bar{n}$ transition comes from the exchange of Higgs bosons with the mass $m_H>10^5$ GeV \[5\]. Since $m_H\gg m_W$ ($m_W$ is the mass of $W$-boson), the renormalization effects should not exceed those characteristic of nuclear $\beta $ decay which is less than 0.25 \[19\]. So the medium corrections to the vertex are inessential for us. The same is true for the model [**b**]{}.
Consider now the baryon-number-violating decay $n\rightarrow \bar{n}\Phi $ \[20\] shown in Fig. 3c. It leads to the same final state, as the processes depicted in Figs. 2, 3a and 3b. Denoting $\mid {\bf q}\mid=q$, for the decay width $\Gamma _{3c}$ one obtains $$\Gamma _{3c}=\frac{\epsilon _{\Phi }^2}{(2\pi )^2}\int dq\frac{q^2}{q_0}G^2\Gamma (q),$$ $q_0^2=q^2+m_{\Phi }^2$. The parameter $\epsilon _{\Phi }$ corresponding to the vertex $n\rightarrow \bar{n}\Phi $ is unknown and so no detailed calculation is possible.
The baryon-number-violating conversion $n\rightarrow \bar{\Lambda }$ in the medium \[20\] shown in Fig. 3d cannot produce interference, since it contains $K$-meson in the final state. For the rest of the diagrams the significant interferences are unlikely because the final states in $\bar{n}N$ annihilation are very complicated configurations and persistent phase relations between different amplitudes cannot be expected. This qualitative picture is confirmed by our calculations \[21\] for $\bar{p}$-nuclear annihilation. It is easy to verify the following statement: if the incoherent contribution of the diagrams 3a-3c to the total nuclear annihilation width is taken into account, the lower limit on the free-space $n\bar{n}$ oscillation time $\tau _{{\rm min}}$ becomes even better.
To summarise, the contribution of diagrams 3 is inessential for us.
Time-dependence
===============
The non-trivial circumstance is the quadratic time-dependence in the model [**a**]{}: $W_a\sim
t^2$. The heart of the problem is as follows. The processes depicted by the diagrams 2b and 3 are described by the exponential decay law. In the first vertex of these diagrams the momentum transfer (Figs. 3a-3c), or effective momentum transfer (Figs. 2b, 3d) takes place. The diagram 2a contains the infrared divergence conditioned by zero momentum transfer in the $n\bar{n}$ transition vertex. This is unremovable perculiarity. This means that the standard $S$-matrix approach is inapplicable \[10,13,17\]. In such an event, the other surprises can be expected as well. From this standpoint a non-exponential behaviour comes as no surprise to us. It seems natural that for non-singular and singular diagrams the functional structure of the results is different, including the time-dependence. The opposite situation would be strange.
The fact that for the processes with $q=0$ the $S$-matrix problem formulation $(\infty,
-\infty)$ is physically incorrect can be seen even from the limiting case $H=0$: if $H_I=
H_{n\bar{n}}$ (see (6)), the solution is periodic. It is obtained by means of non-stationary equations of motion and not $S$-matrix theory. To reproduce the limiting case $H\rightarrow
0$, i.e. the periodic solution, we have to use the approach with finite time interval.
If the problem is formulated on the interval $(t,0)$, the decay width $\Gamma $ cannot be introduced since $\Gamma =\sum_{f\neq i}\mid S_{fi}(\infty,-\infty)\mid ^2/T_0$, $T_0\rightarrow \infty $. This means that the standard calculation scheme should be completely revised. (We would like to emphasise this fact.) The direct calculation by means of evolution operator gives the distribution (20).
Formally, the different time-dependence is due to $q$-dependence of amplitudes. We consider Eq. (23), for instance. If $q$ decreases, the amplitude $M_{3a}$ increase; in the limit $q\rightarrow 0$ it is singular (see (25)). The point $q=0$ corresponds to realistic process shown in Fig. 2a. The $t^2$-dependence of this process is the consequence of the zero momentum transfer.
The more physical explanation of the $t^2$-dependence is as follows. In the Hamiltonian (22) corresponding to Fig. 3a we put $H=H_{n\bar{n}}=0$. Then the virtual decay $n\rightarrow
n\Phi $ takes place. The first vertex of the diagram 3a dictates the exponential decay law of the overall process shown in Fig. 3a. Similarly, in the Hamiltonian (6) corresponding to Fig. 2a, we put $H=0$. Then the free-space $n\bar{n}$ transition takes place which is quadratic in time: $W_f(t)=\epsilon _{n\bar{n}}^2t^2$. The first vertex determines the time-dependence of the whole process at least for small $\Gamma $. We also recall that even for proton decay the possibility of non-exponential behaviour is realistic \[22-24\].
Discussian and summary
======================
The sole physical distinction between models [**a**]{} and [**b**]{} is the zero antineutron self-energy in the model [**a**]{}; or, similarly, the definition of antineutron annihilation amplitude. However, it leads to the fundamentally different results.
If $\Sigma \rightarrow 0$, $W_b(t)$ diverges quadratically. This circumstance should be clarified; otherwise the model [**b**]{} can be rejected. The calculation in the framework of the model [**a**]{} gives the finite result, which justifies our approach from a conceptual point of view and consideration of the model [**a**]{} at least as the limiting case. In reality the model [**a**]{} seems quite realistic in itself. Indeed, we list the main drawbacks of the model [**b**]{}.
1\) The approximation $M_b=M_a$ is an uncontrollable one. The value of $\Sigma $ is uncertain. These points are closely related.
2\) The diagram 2b means that the annihilation is turned on upon forming of the self-energy part $\Sigma =V$ (after multiple rescattering of $\bar{n}$). This is counter-intuitive since at low energies \[25,26\] $$\sigma _a>2.5\sigma _s,$$ where $\sigma _a$ and $\sigma _s$ are the cross sections of free-space $\bar{n}N$ annihilation and $\bar{n}N$ scattering, respectively. The inverse picture is in order: in the first stage of the $\bar{n}$-medium interaction the annihilation occurs. This is obvious for the $n\bar{n}$ transitions in the gas. The model [**a**]{} reproduces the [*competition*]{} between scattering and annihilation in the intermediate state \[27\].
3\) The time-dependence is a more important characteristic of any process. It is common knowledge that the $t$-dependence of the process probability in the vacuum and medium is identical (for example, exponential decay law (15)). In the model [**a**]{} the $t$-dependencies in the vacuum and medium coincide: $W_a\sim t^2$ and $W_f\sim t^2$. The model [**b**]{} gives $W_b\sim t$, whereas $W_f\sim t^2$. There is no reason known why we have such a fundamental change.
4\) If $H=U_{\bar{n}}$, the model [**a**]{} reproduces all the well-known results on particle oscillations \[10\] in contrast to the model [**b**]{}.
The model [**a**]{} is free of drawbacks given above. The physics of the model is absolutely standard. For instance, for the processes shown in Fig. 3 the antineutron propagators are bare as well.
However, there is fundamental problem in the model [**a**]{}: the singularity of the amplitude (19). The approach with finite time interval gives the finite result, which justifies the models [**a**]{} and [**b**]{} at least in principle. Nevertheless, the time-dependence $W_a\sim t^2$ and limit (21) seem very unusual. The corresponding calculation contains too many new elements. Due to this we view the results of the model [**a**]{} with certain caution. Besides, due to the zero momentum transfer in the $n\bar{n}$-transition vertex, the model is extremely sensitive to the $\Sigma $. The process under study is [*unstable*]{}. The small change of antineutron self-energy $\Sigma =0
\rightarrow \Sigma =V\neq 0$, or, similarly, effctive momentum transfer in the $n\bar{n}$ transition vertex converts the model [**a**]{} to the model [**b**]{}: $W_a\rightarrow W_b$ with $W_b\ll W_a$. (For the processes with non-zero momentum transfer the result is little affected by small change of $q$.) Although we don’t see the specific reasons for similar scenario, it must not be ruled out. This is a point of great nicety.
Finally, the values $\tau ^b_{{\rm min}}=(3.5-7.5)\cdot 10^{8}$ s and $\tau ^a_{{\rm min}}=
10^{16}$ yr are interpreted as the estimations from below (conservative limit) and from above, respectively. Further investigations are desirable.
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|
---
abstract: 'Digital circles not only play an important role in various technological settings, but also provide a lively playground for more fundamental number-theoretical questions. In this paper, we present a new recursive algorithm for the construction of digital circles on the integer lattice $\mathbb{Z}^2$, which makes sole use of the signum function. By briefly elaborating on the nature of discretization of circular paths, we then find that this algorithm recovers, in a space endowed with $\ell^1$-norm, the defining constant $\pi$ of a circle in $\mathbb{R}^2$.'
address: |
Unité de Neurosciences, Information et Complexité (UNIC)\
CNRS, 1 Ave de la Terrasse, 91198 Gif-sur-Yvette, France
author:
- 'Michelle Rudolph-Lilith'
title: 'On a recursive construction of circular paths and the search for $\pi$ on the integer lattice $\mathbb{Z}^2$'
---
digital circle ,discrete geometry ,discretization ,integer lattice ,Manhattan distance ,recursive algorithms ,pi
97N70 ,68R10 ,52C05 ,11H06
Introduction {#S_Intro}
============
![\[Fig\_1\] Construction of digital circles using different algorithms (Horn [@Horn76]; Michener [@FoleyVanDam82]; Second-Order Midpoint [@FoleyEA90]; DCS [@BhowmickBhattacharya08]; 4-Connected [@BarreraEA15]; Signum: see text; for a thorough comparative study and some historical notes, see [@BarreraEA15]). Shown are examples of digital circles (black) approximating circles of radii 5, 7, 9 and 11 (gray). With the exception of the 4-connected and signum algorithm, most of the digital circle algorithms cited in the literature do not yield valid paths on $\mathbb{Z}^2$ (black dotted; see Definition \[Def\_ValidPath\]). ](Figure_1){width="\linewidth"}
The analytical characterization and algebraic representation of circles have a long history, dating back many thousands of years. With the emergence of digital computing devices utilizing grid-based interfaces in the past century, the fascination with circles and their algorithmic generation saw another drive which significantly contributed to the evolution of discrete mathematical domains such as digital calculus and digital geometry [@KletteRosenfeld04; @Chen14]. The interest in digital circles transcends, however, beyond application-focused paradigms. For instance, in number theory, the still unsolved Gauss’s Circle Problem (e.g., see [@Huxley96]) or the distribution of square numbers in discrete intervals [@BhowmickBhattacharya08] are inherently linked to the representation of the Euclidean circle on integer lattices. In physics, a related, though perhaps controversial point is the fevered search for a quantum theory of space (and time), i.e. a discrete makeup of our world, which does ultimately lead to the rejection of the ideal real number line in favour of a discrete and finite (or effinite, see [@Gauthier02]) mathematical underpinning of the very construct of reality. However, despite many advances in the past decades, a rigorous and applicable framework of a discrete finite, perhaps even ultra-finite, or effinite mathematics is still largely missing, not at least due to the combinatorial complexity inherent to such approaches.
A great number of algorithms for the generation of digital circles is known in the literature (for reviews, see [@Andres94; @BarreraEA15]). In complexity, these algorithms range from the incremental discretization of the implicit or parametric representation of the Euclidean circle [@Bresenham77; @Doros79; @Kulpa79; @McIlroy83; @Kim84; @NakamuraAizawa84; @BiswasChaudhuri85; @Pham92], the discretization of differential equations [@WuRokne87; @Holin91], sophisticted spline and polygonal approximations [@PieglTiller89; @Goldapp91; @HosurMa99; @BhowmickBhattacharya05], to algorithms which utilize number-theoretical concepts [@BhowmickBhattacharya08]. Although all incremental algorithms utilize decision (or cost) functions, the concrete form of the latter, as well as their specific implementation, can lead to quite different representations of digital circles with the same radius (Fig. \[Fig\_1\]). Moreover, with the exception of the 4-connected algorithm [@BarreraEA15] and the signum algorithm presented here, most of the used digital circle algorithms do not yield valid circular paths on the underlying 2-dimensional integer lattice. Here, a valid path is defined by
\[Def\_ValidPath\] Denoting with $\boldsymbol{x} = (x,y) \in \mathbb{Z}^2$ a point on the 2-dimensional integer lattice, a valid path $\mathcal{P}$ is defined as a set of points $\{ \boldsymbol{x}_n \}$ such that $\forall \boldsymbol{x}_n \in \mathcal{P}$, there exist at most two $\boldsymbol{x}_m , \boldsymbol{x}_{m'} \in \mathcal{P}$ with $m \neq m' \neq n$ such that $\lVert \boldsymbol{x}_n - \boldsymbol{x}_m \rVert_1 = 1$ and $\lVert \boldsymbol{x}_n - \boldsymbol{x}_{m'} \rVert_1 = 1$, where $\lVert \boldsymbol{x} \rVert_1 = |x|+|y|$ denotes the $\ell^1$-norm on $\mathbb{Z}^2$. For a valid closed path, there exist, for each $\boldsymbol{x}_n$, exactly two such $\boldsymbol{x}_m , \boldsymbol{x}_{m'} \in \mathcal{P}$ with the aforementioned properties.
In this paper, we will present a simple recursive algorithm, the signum algorithm, which generates a valid circular path on a 2-dimensional integer lattice $\mathbb{Z}^2$ (Section \[S\_SignumAlgorithm\]). Although this algorithm can not be viewed as the computationally most efficient digital circle algorithm, it allows for easy generalization to higher dimensions, thus providing a viable algorithm for constructing spheres and, generally, hyperspheres of integer radii in $\mathbb{R}^3$ and $\mathbb{R}^n$, respectively. In Section 3, we then briefly elaborate on the discretization of circles in $\mathbb{R}^2$, and present some findings which show that the numerical value of $\pi$ can be recovered in the asymptotic limit using solely the Manhattan distance ($\ell^1$-norm), thus providing an interesting link between Euclidean geometry and geometrical constructions on $\mathbb{Z}^2$.
The signum algorithm {#S_SignumAlgorithm}
====================
In order to construct a valid path on $\mathbb{Z}^2$ which approximates a circle of integer radius $r$ in $\mathbb{R}^2$, we follow an approach similar to that used in most of the known digital circle algorithms, namely utilizing a cost function to assign points on $\mathbb{Z}^2$ to the digital circle. For reasons of symmetry and notational simplicity, we restrict throughout the paper to constructing a quarter circle in the upper right quadrant starting from the horizontal axis, and assume the origin of the circle $\boldsymbol{o} = (0,0)$.
Recursive construction of a valid circular path on $\mathbb{Z}^2$ {#SS_Recursion}
-----------------------------------------------------------------
![\[Fig\_2\] Recursive construction of a circular path on $\mathbb{Z}^2$ in the upper right quadrant, approximating $S^1 \subset \mathbb{R}^2$ (left; see text for explanation), and examples of digital circles of various integer radii ($r=5,10,15,20,25,30$) constructed by using the signum algorithm (right). ](Figure_2){width="\linewidth"}
Let $\mathcal{S}^1$ denote a circular path on $\mathbb{Z}^2$, and $S^1$ a circle on $\mathbb{R}^2$. Given $\boldsymbol{x}_n = (x_n,y_n) \in \mathcal{S}^1$ with $x_n, y_n \in \mathbb{Z}, n \in \mathbb{N}$, there are only two possibilities for the unassigned neighbouring point $\boldsymbol{x}_{n+1}$ along the circular path (Fig. \[Fig\_2\], left), namely $$\label{Eq_xn1}
\boldsymbol{x}_{n+1} = (x_{n+1},y_{n+1}) =
\left\{
\begin{array}{l}
\boldsymbol{x}_{n+1}^{(1)} = (x_n-1,y_n), \\[0.2em]
\text{or, } \boldsymbol{x}_{n+1}^{(2)} = (x_n,y_n+1).
\end{array}
\right.$$ In order to decide between $\boldsymbol{x}_{n+1}^{(1)}$ and $\boldsymbol{x}_{n+1}^{(2)}$, we utilize a cost function based on a minimum criterion. To that end, consider the intersections $\boldsymbol{s}^{(1)}$, $\boldsymbol{s}^{(2)}$ on $S^1$ of lines through $\boldsymbol{o}$ and $\boldsymbol{x}_{n+1}^{(1)}$, $\boldsymbol{x}_{n+1}^{(2)}$, respectively. The line segments $\overline{\boldsymbol{s}^{(1)} \boldsymbol{x}_{n+1}^{(1)}}$ and $\overline{\boldsymbol{s}^{(2)} \boldsymbol{x}_{n+1}^{(2)}}$ have a respective Euclidean length of $$\label{Eq_dn11}
d_{n+1}^{(1)}
= \left| r - \lVert \boldsymbol{x}_{n+1}^{(1)} \rVert_2 \right|
= \left| r - \sqrt{(x_n-1)^2 + y_n^2} \right|$$ and $$\label{Eq_dn12}
d_{n+1}^{(2)}
= \left| r - \lVert \boldsymbol{x}_{n+1}^{(2)} \rVert_2 \right|
= \left| r - \sqrt{x_n^2 + (y_n+1)^2} \right| ,$$ where $\lVert \boldsymbol{x} \rVert_2 = \sqrt{x^2+y^2}$ denotes the $\ell^2$-norm (Euclidean norm) in $\mathbb{R}^2$. With this, the minimization criterion is then given by $$\label{Eq_xn1a}
\boldsymbol{x}_{n+1} =
\left\{
\begin{array}{ll}
\boldsymbol{x}_{n+1}^{(1)} & \text{ if } d_{n+1}^{(1)} \leq d_{n+1}^{(2)} \\[0.2em]
\boldsymbol{x}_{n+1}^{(2)} & \text{ if } d_{n+1}^{(1)} > d_{n+1}^{(2)} .
\end{array}
\right.$$ We note that the equal sign in the case $\boldsymbol{x}_{n+1} = \boldsymbol{x}_{n+1}^{(1)}$ is convention to account for the unlikely scenario that $d_{n+1}^{(1)} = d_{n+1}^{(2)}$. If $d_{n+1}^{(1)}$ and $d_{n+1}^{(2)}$ are equal, both $\boldsymbol{x}_{n+1}^{(1)}$ and $\boldsymbol{x}_{n+1}^{(2)}$ are equally valid neighbours of $\boldsymbol{x}_{n}$, and we choose, without loss of generality, $\boldsymbol{x}_{n+1}^{(1)}$.
To construct the associated cost function, we define $$s_n := \operatorname{sgn}(\Delta_n)$$ with $$\Delta_n := d_{n+1}^{(1)} - d_{n+1}^{(2)}
= \left| r - \sqrt{(x_n-1)^2 + y_n^2} \right| - \left| r - \sqrt{x_n^2 + (y_n+1)^2} \right|,$$ and $$\label{Eq_Sgn}
\operatorname{sgn}(x) =
\left\{
\begin{array}{ll}
-1 & \text{ if } x \leq 0 \\
1 & \text{ if } x > 0
\end{array}
\right.$$ denoting the signum function. Please note that (\[Eq\_Sgn\]) slightly deviates from the commonly used notion of the signum function in that it assigns to $x=0$ a value $\operatorname{sgn}(0) = -1$ instead of $\operatorname{sgn}(0) = 0$. This redefinition allows to accommodate the unlikely case $d_{n+1}^{(1)} = d_{n+1}^{(2)}$ in (\[Eq\_xn1a\]), and, again, does not lead to loss of generality. With this, (\[Eq\_xn1a\]) takes the form $$\label{Eq_xn1b}
\boldsymbol{x}_{n+1} = (x_{n+1},y_{n+1})
\left\{
\begin{array}{ll}
\boldsymbol{x}_{n+1}^{(1)} = (x_n-1, y_n) & \text{ if } s_n = -1 \\[0.2em]
\boldsymbol{x}_{n+1}^{(2)} = (x_n, y_n+1) & \text{ if } s_n = 1 .
\end{array}
\right.$$ Utilizing the signum function (\[Eq\_Sgn\]), we can then rewrite (\[Eq\_xn1b\]) in algebraic form as $$\label{Eq_xn1c}
\left\{
\begin{array}{l}
x_{n+1} = \frac{1}{2} ( 1 - s_n ) ( x_n - 1 ) + \frac{1}{2} ( 1 + s_n ) x_n \\[0.2em]
y_{n+1} = \frac{1}{2} ( 1 - s_n ) y_n + \frac{1}{2} ( 1 + s_n ) ( y_n + 1 ) .
\end{array}
\right.$$
We observe that, by construction, the circular path $\mathcal{S}^1$ intersects in the considered upper right quadrant with the horizontal and vertical axis at $(r,0)$ and $(0,r)$, respectively. As the Manhattan distance between these two intersection points counts the number of points on $\mathbb{Z}^2$ along a valid circular path $\mathcal{S}^1$, each quadrant will contribute $2r$ points to $\mathcal{S}^1$. With this, after simplification of (\[Eq\_xn1c\]), we can then formulate the following
\[Prop\_SignumAlgorithm\] A valid circular path $\mathcal{S}^1 \subset \mathbb{Z}^2$ approximating a circle $S^1 \subset \mathbb{R}^2$ with radius $r \in \mathbb{N}$ and origin $\boldsymbol{o} = (0,0)$ in the upper right quadrant is a set $\{ \boldsymbol{x}_n \}$ of $2r$ points $\boldsymbol{x}_n = (x_n,y_n)$ with $x_n, y_n \in \mathbb{Z}$ obeying the algebraic recursions $$\label{Eq_S1Algorithm}
\left\{
\begin{array}{l}
x_0 = r , x_{n+1} = x_n + \frac{1}{2} s_n - \frac{1}{2} \\[0.2em]
y_0 = 0 , y_{n+1} = y_n + \frac{1}{2} s_n + \frac{1}{2} ,
\end{array}
\right.$$ where $n \in [0,2r-1], n \in \mathbb{N}$, and $$\label{Eq_Cost}
s_n = \operatorname{sgn}(\Delta_n)$$ with $$\Delta_n = \left| r - \sqrt{(x_n-1)^2 + y_n^2} \right| - \left| r - \sqrt{x_n^2 + (y_n+1)^2} \right|$$ denoting the cost function.
As the proposed algorithm makes solely use of the signum function, we will, for notational convenience, refer to as *signum algorithm* in the remainder of this paper. Furthermore, we note that $$\Delta_0 = 1-|r-\sqrt{r^2+1}| \geq 2 - \sqrt{2} > 0,$$ $\forall r \geq 1$, thus $s_0 = 1$. Figure \[Fig\_2\] (right) shows representative examples of digital circles of various integer radii, constructed using the signum algorithm.
Proposition \[Prop\_SignumAlgorithm\] provides a recursive algorithm for constructing digital circles of integer radii on $\mathbb{Z}^2$. Starting at $\boldsymbol{x}_0 = (r,0)$, this algorithm yields $2r$ successive points forming a valid circular path in the upper right quadrant on the integer lattice. This contrasts, for instance, the most widely used Bresenham [@Bresenham77] and Midpoint [@FoleyEA90] algorithms, which deliver only about 70% of the points necessary for a valid circular path on $\mathbb{Z}^2$ (see Fig. \[Fig\_1\]). Moreover, in contrast to many known digital circle algorithms, the computational implementation of the signum algorithm does not require decision trees or case distinctions, but solely relies on the signum function to generate a valid path. Such an algebraic formulation has the advantage of being mathematical tractable and allowing for rigorous manipulations. Specifically, due to the special properties of the signum function $\operatorname{sgn}(x): \mathbb{R} \rightarrow \{-1,1\}$, the cost function (\[Eq\_Cost\]) can further be simplified, as shown in the next section.
Finally, we note that the geometrical basis and algebraic representation of the signum algorithm allows for direct generalization to higher dimensions. Specifically, for each given $(1/2^n)^{\text{th}}$ hypersphere in $\mathbb{R}^n$ (the generalization of the quarter circle in $\mathbb{R}^2$), Eq. (\[Eq\_xn1\]) must be extended to encompass $n$ possible neighbours for each given point along a valid “hypercircular path”. Generalizing the Euclidean distance of the associated line segments, Eqs. (\[Eq\_dn11\]) and (\[Eq\_dn12\]), to $\mathbb{R}^n$ will then yield a number of minimization criteria corresponding to (\[Eq\_xn1a\]) which can be expressed by utilizing the signum function alone, and lead to a recursive algorithm constructing a $(n-1)$-dimensional hypercircular “path” of integer radius on the $n$-dimensional integer lattice $\mathbb{Z}^n$.
Simplification of the cost function {#SS_CostFunction}
-----------------------------------
The computational complexity of the digital circle algorithm presented in Proposition \[Prop\_SignumAlgorithm\] is carried by the argument of the cost function, which requires to evaluate the square root of integer numbers. However, as we show below, due to the properties of the signum function, $\Delta_n$ can be significantly simplified. To that end, we first formulate
\[Lemma\_sgn\] The signum function $\operatorname{sgn}(x): \mathbb{R} \rightarrow \{-1,1\}$ with $$\label{Eq_sgn}
\operatorname{sgn}(x) =
\left\{
\begin{array}{ll}
-1 & \text{ if } x \leq 0 \\
1 & \text{ if } x > 0
\end{array}
\right.$$ is subject to the following property: $$\label{Eq_sgn1}
\operatorname{sgn}(x-y) = \operatorname{sgn}(f(x)-f(y))$$ for all $x,y \in \mathbb{R}: x,y \geq 0$ and strict monotonically increasing functions $f(x): \mathbb{R} \rightarrow \mathbb{R}$. Moreover, $\forall x \in \mathbb{R}: x \neq 0$ and $a \in \mathbb{R}$ $$\label{Eq_sgn2}
\operatorname{sgn}(ax) = \left\{
\begin{array}{ll}
\operatorname{sgn}(x) & \text{ if } a > 0 \\
-\operatorname{sgn}(x) & \text{ if } a < 0.
\end{array}
\right.$$
Eqs. (\[Eq\_sgn1\]) and (\[Eq\_sgn1\]) are self-evident from the definition of the signum function (\[Eq\_sgn\]).
Utilizing Lemma \[Lemma\_sgn\], we can now formulate
The cost function $s_n$ in Proposition \[Prop\_SignumAlgorithm\] is equivalent to $$\label{Eq_CostSimplified}
s_n = -\operatorname{sgn}\left(
a_n + \frac{r}{\sqrt{2}} \left( \sqrt{(a_n-1)^2+c_n^2} - \sqrt{(a_n+1)^2+c_n^2} \right)
\right),$$ where $a_n = x_n + y_n$ with $n \in [0,2r-1], n \in \mathbb{N}$ obeys the recursion $$\label{Eq_anRec}
a_0 = r , a_{n+1} = a_n + s_n$$ and $c_n = r-n-1$. Furthermore, for $r > 4$, the cost function can be approximated by $$\label{Eq_CostApproximated}
s_n = -\operatorname{sgn}\left( a_n^2 + c_n^2 + 1 - 2r^2 \right).$$
First we will show (\[Eq\_CostSimplified\]). To that end, we observe that $f(x)=x^2$ for $x \geq 0$ obeys the condition of Lemma \[Lemma\_sgn\], thus $$\begin{aligned}
s_n
& = & \operatorname{sgn}\left( \left| r - \sqrt{(x_n-1)^2 + y_n^2} \right|^2 - \left| r - \sqrt{x_n^2 + (y_n+1)^2} \right|^2 \right) \\
& = & \operatorname{sgn}\left( -2(x_n+y_n) - 2r \left( \sqrt{(x_n-1)^2+y_n^2} - \sqrt{x_n^2+(y_n+1)^2} \right) \right) \\
& = & -\operatorname{sgn}\left( x_n+y_n + r \left( \sqrt{(x_n-1)^2+y_n^2} - \sqrt{x_n^2+(y_n+1)^2} \right) \right) \\
& = & -\operatorname{sgn}\Big( a_n + \frac{r}{\sqrt{2}} \Big( \sqrt{a_n^2-2a_n+b_n^2-2b_n+2} \\
& & \hspace*{28mm} - \sqrt{a_n^2+2a_n+b_n^2-2b_n+2} \Big) \Big),\end{aligned}$$ where in the last two steps Eq. (\[Eq\_sgn2\]), $a_n := x_n + y_n$ and $b_n := x_n - y_n$ were used. Observing that $b_n$ obeys the recursion $$b_0 = r, b_{n+1} = b_n - 1,$$ hence takes the explicit form $b_n = r-n$, and defining further $$\label{Eq_cn}
c_n := \sqrt{b_n^2 - 2b_n + 1} = r-n-1,$$ we arrive at Eq. (\[Eq\_CostSimplified\]).
To show (\[Eq\_CostApproximated\]), we first note that $a_n \geq r, \forall n \in [0, 2r-1]$, with the minimum taken at $n=0$. The maximum is reached for a point on the circular path which, when connected to the origin by a line in $\mathbb{R}^2$, takes an angle with the horizontal axis closest to $\pi/4$. As $a_n = x_n + y_n$ is, in the upper right quadrant, equivalent to the Manhattan distance of $(x_n,y_n)$, we can approximate $$\max_{n} a_n \approx r \cos\left(\frac{\pi}{4}\right) + r \sin\left(\frac{\pi}{4}\right) = \sqrt{2} r.$$ As any point on the circular path $\mathcal{S}^1$ does, by construction, reside at most $\sqrt{2}$ away from the closest point on $S^1$, we can securely assume that $a_n \leq \sqrt{2}(r+1), \forall n \in [0, 2r-1]$. Thus, $$\begin{array}{rcccl}
r & \leq & a_n & \leq & \sqrt{2}(r+1) \\
r^2 & \leq & a_n^2 & \leq & 2(r^2+2r+1).
\end{array}$$ Similarly, with (\[Eq\_cn\]), $c_n$ takes its minimum of $0$ at $n=r-1$, and its maximum of $r$ for $n=2r-1$. With this, we have the following inequality $$r^2+1 \leq a_n^2+c_n^2+1 \leq 3r^2+4r+3,$$ from which $$\frac{2 a_n}{a_n^2+c_n^2+1} < 1$$ $\forall r \geq 4$ follows. With this, we can rewrite (\[Eq\_CostSimplified\]), using again Lemma \[Lemma\_sgn\], and obtain $$s_n = -\operatorname{sgn}\Big(
a_n^2 - \frac{r^2}{2} (a_n^2+c_n^2+1) \Big(
\sqrt{1 + \tfrac{2a_n}{a_n^2+c_n^2+1}} - \sqrt{1 - \tfrac{2a_n}{a_n^2+c_n^2+1}}
\Big) \Big).$$ Observing that $$\left( \sqrt{1-x} - \sqrt{1+x} \right)^2
= \sum\limits_{k=1}^{\infty} \binom{2(k-1)}{k-1} \frac{4}{2^{2k} k} x^{2k}$$ $\forall x \in \mathbb{R}: |x| \leq 1$, we then expand, for $r \geq 4$, the argument of $s_n$ in a power series. This yields $$s_n = -\operatorname{sgn}\left(
a_n^2 - \frac{r^2}{2} (a_n^2+c_n^2+1) \sum\limits_{k=1}^{\infty} \binom{2(k-1)}{k-1} \frac{4}{2^{2k} k} \left(\frac{2a_n}{a_n^2+c_n^2+1}\right)^{2k}
\right).$$ For large $r$, the sum in the last equation converges rapidly, and we can approximate $s_n$ by taking only the leading term $k=1$ into consideration, thus showing (\[Eq\_CostApproximated\]).
We note that, whereas (\[Eq\_Cost\]) and (\[Eq\_CostSimplified\]) provide exact expressions for the cost function $s_n$, Eq. (\[Eq\_CostApproximated\]) provides an approximation which, for $r \gg 1$, yields the same result as the exact expressions. However, using (\[Eq\_CostApproximated\]) will significantly lower the computational cost of constructing a digital circle, as here only integer operations are involved. Finally, we remark that both the exact alternative form of the cost function (\[Eq\_CostSimplified\]) and its approximation (\[Eq\_CostApproximated\]) are no longer given in terms of the coordinates $(x_n,y_n)$ of points along the circular path $\mathcal{S}^1$, but instead are functions of the Manhattan distance $a_n = |x_n| + |y_n|$ of each point $(x_n,y_n) \in \mathcal{S}^1$ to the center of the circle. The resulting finite sequence itself is subject to a recursion, see Eq. (\[Eq\_anRec\]), and will be used in the next section to recover the numerical value of $\pi$ from a digital circle $\mathcal{S}^1 \subset \mathbb{Z}^2$.
The search for $\pi$ on $\mathbb{Z}^2$ {#S_PiZ2}
======================================
By construction, each digital circle algorithm delivers, for any given radius $r$, a set of points on $\mathbb{Z}^2$ which, for increasing $r$, approximates with increasing precision $S^1 \subset \mathbb{R}^2$ when each pair of nearest neighbouring points is connected with a straight line in $\mathbb{R}^2$ (see Fig. \[Fig\_1\]), eventually yielding $S^1$ for $r \rightarrow \infty$. However, if we restrict to $\mathbb{Z}^2$ with its $\ell^1$-norm, all valid circular paths will remain finitely distinct from $S^1$ even in the asymptotic case, as each path is bound to the lattice. To make matters worse, if we consider the distance of each point along the circular path to the origin, then we find that it is no longer constant. This, although being a known characteristic with amusing consequences of geometric spaces endowed with $\ell^1$-norm [@Krause87], it is in direct conflict with the very original definition of a circle as put forth in Euclid’s *Elements* (Book I, §19). If we adhere to Euclid’s circle definition in such a discrete space with $\ell^1$-norm, on the other hand, the discrete circle takes, in the continuum limit, the shape of a square rotated by $\pi/4$. Thus, in other words, a *digital* circle and a *discrete* circle are two distinct geometrical objects.
Reconciling digital and discrete circles {#SS_DigitalDiscreteCircles}
----------------------------------------
Digital geometry defines a “digital circle” simply as a discrete approximation (or digitized model) of a circle in $\mathbb{R}^2$ obtained by searching for points on $\mathbb{Z}^2$ which are closest to $S^1$. Naturally, the form of each model will carry consequences for its underlying relationship to the circle on $\mathbb{R}^2$. We can thus interrogate the geometric properties of each model in $\mathbb{Z}^2$ and $\mathbb{R}^2$, specifically, explore the relationship between properties of the digital circle $\mathcal{S}^1 \subset \mathbb{Z}^2$, i.e. a circular path in a discrete space endowed with $\ell^1$-norm, and the properties of $S^1 \subset \mathbb{R}^2$, i.e. a circle in a continuous space endowed with $\ell^2$-norm. We will focus here on the defining constant of circles, $\pi$, and show below that the parametric and polar discretizations of the circle lead to an overestimate for $\pi$, measured both numerically and analytically, whereas the signum algorithm introduced in Section \[S\_SignumAlgorithm\] allows to recover its correct value in a somewhat surprising fashion.
Before outlining the details of this interrogation, we note that, firstly, an alternative, and mathematically more rigorous, definition of a circle in $\mathbb{R}^2$ is given by its parametric representation. Specifically, a circle $S^1 \subset \mathbb{R}^2$ is the set of all points $(x,y) \in \mathbb{R}^2$ which satisfy the algebraic relation $$\label{Eq_circleP}
x^2 + y^2 = r^2,$$ where $r \in \mathbb{R}: r > 0$ is called the radius of the circle. Recalling Proposition \[Prop\_SignumAlgorithm\], a digital circle $\mathcal{S}^1 \subset \mathbb{Z}^2$ is the set of all points $(x,y) \in \mathbb{Z}^2$ satisfying a specific recursive algebraic relation corresponding to Eq. (\[Eq\_S1Algorithm\]) in the upper right quadrant.
Secondly, although differences exist in the mathematical representation of the algorithmic search for points on $\mathbb{Z}^2$ closest to $S^1$, each digital circle algorithm utilizes the Euclidean norm in one form or another in its minimization criterion. The same holds for the signum algorithm presented here. However, the resulting cost function (\[Eq\_CostSimplified\]) and its approximation (\[Eq\_CostApproximated\]) are given in terms of $a_n = x_n + y_n$, which corresponds, in the upper right quadrant, to the Manhattan distance of the point $(x_n,y_n) \in \mathcal{S}^1$ to the origin. Taking both arguments together, it could be contended that the “digital circle” constructed by the signum algorithm is not only a digital model of $S^1 \subset \mathbb{R}^2$, but a valid discrete model of a circle in $\mathbb{Z}^2$, a space endowed with $\ell^1$-norm, with properties which, in the asymptotic limit, translate into those of $S^1$.
$\pi$ in discretized circles {#SS_piND}
----------------------------
To illustrate this crucial latter point, we will consider the defining constant of a circle in $\mathbb{R}^2$ (or hyperspheres in $\mathbb{R}^n $ in general), namely $\pi$, and ask whether $\pi$ can be obtained in a discrete space endowed with $\ell^1$-norm. To that end, we first recall how $\pi$ is obtained on $\mathbb{R}^2$ by calculating the circumference of the circle. Given the parametric representation of $S^1 \subset \mathbb{R}^2$, Eq. (\[Eq\_circleP\]), we have $y=\pm\sqrt{r^2-x^2}$ and for the circumference $\mathcal{C}$, using the arc length, $$\label{Eq_S}
\mathcal{C}
= 2 \int\limits_{-r}^r \sqrt{1+\left(\frac{dy}{dx}\right)^2}
= 2 \int\limits_{-r}^r \text{d}x \,
\sqrt{1+\frac{x^2}{r^2-x^2}}
= 2 \pi r.$$
Equation (\[Eq\_S\]) can be viewed as a definition of $\pi$ in terms of the ratio between the circumference of a circle and the (Euclidean) distance of each point on $S^1$ to the center, i.e. $$\label{Eq_pi}
\pi := \frac{\mathcal{C}}{2r}$$ for $r > 0$. Remaining for a moment in $\mathbb{R}^2$, but replacing the Euclidean distance $r$ by the Manhattan distance $a(x,y)=|x|+|y|$ of each point $(x,y) \in S^1$ to the center, we can define $$\label{Eq_pi}
\pi(x,y) := \frac{\mathcal{C}}{2 a(x,y)} = \frac{4r}{a(x,y)},$$ where we used the fact that the circumference of a circle in a space with $\ell^1$-norm is $\mathcal{C}=8r$. As mentioned above, as $a(x,y)$ changes depending on the point along the circle (see Fig. \[Fig\_3\], top left), $\pi(x,y)$ will be a function of $(x,y) \in S^1$, with values ranging between 4 and $2\sqrt{2}$ (see Fig. \[Fig\_3\], top right), and the value of $\pi$ residing in between these bounds. Using the parametric representation of a circle, $$\label{Eq_xy}
\left\{
\begin{array}{l}
x = r \cos(\varphi) \\
y = r \sin(\varphi)
\end{array}
\right.$$ with $0 \leq \varphi \leq 2\pi$, we have $$\label{Eq_aphi}
a(x,y) \equiv a(r,\varphi) = |r \cos(\varphi)| + |r \sin(\varphi)|.$$ With this, we can calculate the average of (\[Eq\_pi\]) over all points on $S^1$ (due to symmetry, it is sufficient to restrict to the upper right quadrant), which yields $$\label{Eq_picont}
\overline{\pi}
= \frac{2}{\pi} \int\limits_0^{\pi/2} \text{d}\varphi \, \frac{4}{\cos(\varphi)+\sin(\varphi)}
= \frac{8}{\pi} \sqrt{2} \text{ arctanh}\left( \frac{1}{\sqrt{2}} \right)
\sim 3.17406.$$ Note that the obtained value is independent of $r$. More interestingly, however, is the fact that the obtained value is close, but not identical, to $\pi$.
![\[Fig\_3\] Relative Manhattan distance $a(\varphi)/r$ (top left; see Eq. (\[Eq\_aphi\])) and associated $\pi$-values (top right; see Eq. (\[Eq\_pi\])) along points on $S^1 \subset \mathbb{R}^2$. Parametric discretization of the circle $S^1 \subset \mathbb{R}^2$ (bottom left; see text for explanation) and the resulting arithmetic mean of the $\pi_n$ values associated with each point on $S^1$ (see Eq. (\[Eq\_ApinND\])) as function of the radius $r$ (bottom right; black: $a_n(r)$ given by Eq. (\[Eq\_xnyndigi\]), light grey: $a_n(r)$ given by Eq. (\[Eq\_xnyndigiS\]), dark grey: $a_n(r)$ given by Eq. (\[Eq\_xnyndigiR\])). The asymptotic value $\overline{\pi}$ for $r \rightarrow \infty$, Eq. (\[Eq\_pidigi\]), differs from $\pi$ in all cases. ](Figure_3){width="\linewidth"}
The same holds true if we perform a parametric discretization of $S^1 \subset \mathbb{R}^2$ by introducing $2r$ discrete angles$$\label{Eq_xnyn}
\left\{
\begin{array}{l}
x_n = r \cos(\varphi_n) \\
y_n = r \sin(\varphi_n)
\end{array}
\right.$$ with $$\varphi_n = \frac{n}{2r} \, \frac{\pi}{2},$$ $n \in [0,2r-1], n \in \mathbb{N}$ (Fig. \[Fig\_3\], bottom left). In this case, remaining with the $\ell^1$-norm, we have $$\label{Eq_xnyndigi}
a(x_n,y_n) \equiv a_n(r) = |r \cos(\varphi_n)| + |r \sin(\varphi_n)|.$$ Defining, similar to (\[Eq\_pi\]), $\pi$-values associated with each point along the now discretized circle according to $$\label{Eq_piND}
\pi_n(r) := \frac{\mathcal{C}}{2 a_n(r)} = \frac{4r}{a_n(r)},$$ we consider the arithmetic mean $A(\pi_n)$ of all $\pi_n(r)$, i.e. $$\label{Eq_ApinND}
A(\pi_n) = \frac{1}{2r} \sum\limits_{n=0}^{2r-1} \pi_n(r),$$ and obtain $$\label{Eq_ApinND1}
A(\pi_n)
= 2 \sum\limits_{n=0}^{2r-1} \frac{1}{a_n(r)}
= \frac{2}{r} \sum\limits_{n=0}^{2r-1} \frac{1}{\cos\left(\frac{n\pi}{4r}\right)+\sin\left(\frac{n\pi}{4r}\right)}.$$ To simplify the last equation, we first rewrite the denominator under the sum using $$\sin(x) \pm \cos(y) = 2 \sin\left(\frac{1}{2}(x \pm y) \pm \frac{\pi}{4}\right) \cos\left(\frac{1}{2}(x \mp y) \mp \frac{\pi}{4}\right)$$ ([@GradshteynRyzhik07], relation 1.314.9$^*$). With this, (\[Eq\_ApinND1\]) takes the form $$\begin{aligned}
A(\pi_n)
& = & \frac{\sqrt{2}}{r} \sum\limits_{n=0}^{2r-1} \frac{1}{\sin\left(\frac{n\pi}{4r}+\frac{\pi}{4}\right)} \\
& = & \frac{2 \sqrt{2}}{r} \sum\limits_{n=0}^{2r-1} \sum\limits_{k=0}^{\infty} \frac{(-1)^{k+1} (2^{2k-1}-1) B_{2k}}{(2k)!} \left(\frac{\pi}{4}\right)^{2k-1} \left( \frac{n}{r}+1 \right)^{2k-1},\end{aligned}$$ where, due to $\frac{\pi}{4} \leq (\frac{n\pi}{4r}+\frac{\pi}{4}) < \frac{3 \pi}{4}$ for all $r$, in the last step we used the power expansion of $1/\sin(x) \equiv \csc(x)$ in terms of Bernoulli numbers $B_n$. Splitting off the inner sum the $k=0$ term, and executing the sum over $n$, yields $$\begin{aligned}
A(\pi_n)
& = & \frac{4 \sqrt{2}}{\pi} \sum\limits_{n=0}^{2r-1} \frac{1}{n+r} \\
& + & \frac{2 \sqrt{2}}{r} \sum\limits_{n=0}^{2r-1} \sum\limits_{k=1}^{\infty} \frac{(-1)^{k+1} (2^{2k-1}-1) B_{2k}}{(2k)!} \left(\frac{\pi}{4}\right)^{2k-1} \left( \frac{n}{r}+1 \right)^{2k-1} \\
& = & \frac{4 \sqrt{2}}{\pi} \big( \Psi(3r) - \Psi(r) \big) \\
& + & 2 \sqrt{2} \sum\limits_{k=1}^{\infty} \frac{(-1)^{k+1} (2^{2k-1}-1) B_{2k}}{(2k)!} \left(\frac{\pi}{4}\right)^{2k-1} \frac{1}{r^{2k}} \\
& & \hspace*{15mm} \times \big( \zeta(1-2k,r) - \zeta(1-2k,3r) \big),\end{aligned}$$ where $\Psi(x)$ denotes the digamma function and $\zeta(n,x)$ the Hurwitz zeta function. Exploiting $$\zeta(-n,x) = - \frac{B_{n+1}(x)}{n+1}$$ (see [@Apostol95], Theorem 12.13), which holds for $n \geq 0$ and links the Hurwitz zeta to Bernoulli polynomials $$B_{n}(x) = \sum\limits_{k=0}^n \binom{n}{k} B_{n-k} x^k,$$ we can further simplify $A(\pi_n)$ to $$\begin{aligned}
A(\pi_n)
& = & \frac{4 \sqrt{2}}{\pi} \big( \Psi(3r) - \Psi(r) \big) \\
& + & 2 \sqrt{2} \sum\limits_{k=1}^{\infty} \frac{(-1)^{k+1} (2^{2k-1}-1) B_{2k}}{(2k)! \, 2k} \left(\frac{\pi}{4}\right)^{2k-1} \frac{1}{r^{2k}} \big( B_{2k}(3r) - B_{2k}(r) \big).\end{aligned}$$ Observing that $B_{n}(x)$ are polynomials of degree $n$ in $x$, and recalling that our assessment aims at the asymptotic limit $r \rightarrow \infty$, the last equation yields $$\begin{aligned}
A(\pi_n)
& = & \frac{4 \sqrt{2}}{\pi} \big( \Psi(3r) - \Psi(r) \big) \\
& + & 2 \sqrt{2} \sum\limits_{k=1}^{\infty} \frac{(-1)^{k+1} (2^{2k-1}-1) B_{2k}}{(2k)! \, 2k} \left(\frac{\pi}{4}\right)^{2k-1} ( 3^{2k}-1 ) + \mathcal{O}\left( \tfrac{1}{r} \right).\end{aligned}$$ Performing now carefully the asymptotic limit $r \rightarrow \infty$, we finally obtain $$\begin{aligned}
\label{Eq_pidigi}
\overline{\pi}
& := & \lim_{r \rightarrow \infty} A(\pi_n) \nonumber \\
& = & \frac{2 \sqrt{2}}{\pi} \left( 2 \ln(3) + \ln(\tfrac{9}{8}) + \ln(8) - 2 \ln(16 (2-\sqrt{2}) + 2 \ln(\tfrac{16}{9} (\sqrt{2}+2)) \right) \nonumber \\
& = & \frac{4 \sqrt{2}}{\pi} \left( \ln(2+\sqrt{2}) - \ln(2-\sqrt{2}) \right) \nonumber \\
& \sim & 3.17406.\end{aligned}$$ Thus, in the case of the performed parametric discretization of $S^1 \subset \mathbb{R}^2$ given in polar coordinates, the numerical value of $\overline{\pi}$, defined as the arithmetic mean of the $\pi$-values associated with each point along the discretized circle in a space with $\ell^1$-norm, converges to (\[Eq\_pidigi\]), expectedly in accordance with its continuum counterpart (\[Eq\_picont\]).
We note, however, that the discretization performed above does, in general, not yield points $(x_n,y_n) \in \mathbb{Z}^2$. To ensure the latter, we must replace Eq. (\[Eq\_xnyndigi\]) with $$\label{Eq_xnyndigiS}
a_n(r) = \big\lfloor \, |r \cos(\varphi_n)| \, \big\rfloor + \big\lfloor \, |r \sin(\varphi_n)| \, \big\rfloor$$ or $$\label{Eq_xnyndigiR}
a_n(r) = \left\lfloor \, |r \cos(\varphi_n)| + \tfrac{1}{2} \, \right\rfloor + \left\lfloor \, |r \sin(\varphi_n)| + \tfrac{1}{2} \, \right\rfloor,$$ where the former “snaps” the points along $S^1$ to integer coordinates on $\mathbb{Z}^2$ inside the circle, i.e. $$\label{Eq_xnynS}
\left\{
\begin{array}{l}
x_n = \lfloor r \cos(\varphi_n) \rfloor \\[0.2em]
y_n = \lfloor r \sin(\varphi_n) \rfloor
\end{array}
\right.$$ in the upper right quadrant, whereas the latter associates each point on $S^1$ to the nearest lattice points on $\mathbb{Z}^2$ by rounding independently each coordinate, i.e. $$\label{Eq_xnynD}
\left\{
\begin{array}{l}
x_n = \lfloor r \cos(\varphi_n) + \frac{1}{2} \rfloor \\[0.2em]
y_n = \lfloor r \sin(\varphi_n) + \frac{1}{2} \rfloor
\end{array}
\right.$$ in the upper right quadrant. However, even with these modifications and steps towards a valid discretization, or digital model, of the circle $S^1 \subset \mathbb{R}^2$ in $\mathbb{Z}^2$, the obtained values for $\overline{\pi}$ differ numerically from $\pi$ (see Fig. \[Fig\_3\], bottom right).
$\pi$ on the digital circle {#SS_piND}
---------------------------
![\[Fig\_4\] Values of $\overline{\pi}(r)$, defined as the arithmetic mean of all $\pi_n(r)$ associated with each point on a digital circle, as function of $r$ for various digital circle algorithms (top left). Calculation of $\pi_n(r)$ in a digital circle constructed using the signum algorithm (top right; see text for explanation), and resulting $\overline{\pi}(r)$ (bottom left). As for the 4-connected algorithm (see top left), also the signum algorithm yields for large $r$ a numerical value converging to $\pi$ (see Conjecture \[Conj\_Api\]). Interestingly, considering the harmonic mean of $\pi_n(r)$ yields a value proportional to $\pi$ as well (bottom right; see Proposition \[Prop\_Hpi\]). ](Figure_4){width="\linewidth"}
The above outlined parametric discretization constitutes, in one form or the other, the basis for most published digital circle algorithms. Naturally, values of $\overline{\pi}$, defined as the asymptotic limit of the arithmetic mean of $\pi_n(r)$, see Eq. (\[Eq\_ApinND\]), will, expectedly, deviate from $\pi$ (see Fig. \[Fig\_4\], top left). However, and somewhat surprisingly, this appears to be not true for the 4-connected algorithm and the signum algorithm (Fig. \[Fig\_4\], bottom left) introduced here. Focusing on the latter, the numerical assessment of the arithmetic mean of the reciprocal Manhattan distance $a_n = |x_n| + |y_n|$ associated with each recursively generated point $(x_n,y_n) \in \mathbb{Z}^2$ (Fig. \[Fig\_4\], top right) according to (\[Eq\_S1Algorithm\]) suggests that, in this case, the correct value for $\pi$ is obtained in the asymptotic limit for $r \rightarrow \infty$ (Fig. \[Fig\_4\], bottom left). Specifically, we can formulate the following
\[Conj\_Api\] The arithmetic mean $$A(\pi_n)
= \frac{1}{2r} \sum\limits_{n=0}^{2r-1} \frac{\mathcal{C}}{2a_n(r)}
= 2 \sum\limits_{n=0}^{2r-1} \frac{1}{a_n(r)},$$ of the finite sequence $$\pi_n(r) = \frac{\mathcal{C}}{2a_n(r)} = \frac{4r}{a_n(r)},$$ where $a_n = |x_n| + |y_n|$ denotes the $\ell^1$-norm of each point $(x_n,y_n) \in \mathbb{Z}^2$ on the digital circle $\mathcal{S}^1 \subset \mathbb{Z}^2$ constructed recursively by (\[Eq\_S1Algorithm\]), converges to $\pi$ in the asymptotic limit $r \rightarrow \infty$, i.e. $$\lim_{r \rightarrow \infty} A(\pi_n) = \pi.$$
The attempt of a rigorous proof of this conjecture can be found in [@Rudolph16]. We also note that the same convergence is found in the case of the 4-connected algorithm ([@BarreraEA15]; see Fig. \[Fig\_4\], left).
Although the recovery of $\pi$ in the case of a valid path describing a digital circle in $\mathbb{Z}^2$, a space with $\ell^1$-norm, is somewhat unexpected, an even more surprising result is obtained when considering the reciprocal of the harmonic mean $H(\pi_n)$, which is proportional to the arithmetic mean of $1/\pi_n \sim a_n(r)$ itself. Specifically, we have
\[Prop\_Hpi\] The harmonic mean $$\label{Eq_Hpi}
H(\pi_n) = \left( A\left(\frac{a_n(r)}{4r}\right) \right)^{-1}$$ of the sequence of $\pi_n$ values associated with each point $(x_n,y_n) \in \mathbb{Z}^2$ along a digital circle $\mathcal{S}^1 \subset \mathbb{Z}^2$ constructed recursively through (\[Eq\_S1Algorithm\]) obeys, in the asymptotic limit $r \rightarrow \infty$, the identity $$\label{Eq_Hpi1}
\lim_{r \rightarrow \infty} \frac{1}{H(\pi_n)} = \frac{\pi}{16} + \frac{1}{8}.$$
To show (\[Eq\_Hpi1\]), we first calculate the area $\mathcal{A}(r)$ enclosed by the digital circle $\mathcal{S}^1$ (as above, for notational and symmetry reasons, we will restrict to the quarter circle in the upper right quadrant). To that end, we first construct two associated valid paths $\mathcal{P}_{\text{inner}} \subset \mathbb{Z}^2$ and $\mathcal{P}_{\text{outer}} \subset \mathbb{Z}^2$ by taking the floor and ceiling of each coordinate $(x,y)$ along the circle $S^1 \subset \mathbb{R}^2$. Both paths enclose areas $\mathcal{A}_{\text{inner}}(r)$ and $\mathcal{A}_{\text{outer}}(r)$, respectively (see Fig. \[Fig\_5\]). By construction, each point along the circular path $\mathcal{S}^1$ will reside inside or on the circumference of $\mathcal{A}_{\text{outer}}(r)$, and outside or on the circumference of $\mathcal{A}_{\text{inner}}(r)$, thus $$\mathcal{A}_{\text{inner}}(r) \leq \mathcal{A}(r) \leq \mathcal{A}_{\text{outer}}(r) .$$ Moreover, noting that we consider a quarter circle, and recalling the approximation of the area of a circle $4 \mathcal{A} = \pi r^2$ in $\mathbb{R}^2$ by a Riemannian sum, we have $$\mathcal{A}_{\text{inner}}(r) < \frac{1}{4} \pi r^2 < \mathcal{A}_{\text{outer}}(r)$$ with $$\lim_{r \rightarrow \infty} \mathcal{A}_{\text{inner}}(r)
= \lim_{r \rightarrow \infty} \mathcal{A}_{\text{outer}}(r)
= \frac{1}{4} \pi r^2,$$ thus $$\label{Eq_Alimit}
\lim_{r \rightarrow \infty} \mathcal{A}(r) = \frac{1}{4} \pi r^2.$$
![\[Fig\_5\] Construction of paths $\mathcal{P}_{\text{inner}} \subset \mathbb{Z}^2$ (left) and $\mathcal{P}_{\text{outer}} \subset \mathbb{Z}^2$ (right) which are enclosed or do enclose the digital circle $\mathcal{S}^1$ (middle), respectively, along with their associated respective areas $\mathcal{A}_{\text{inner}}(r)$, $\mathcal{A}_{\text{outer}}(r)$ and $\mathcal{A}(r)$ in the upper right quadrant (see text for explanation). ](Figure_5){width="\linewidth"}
We next construct recursively the (quarter circle) area enclosed by $\mathcal{S}^1$ through a finite recursive sequence $\mathcal{A}_n(r)$. To that end, we note that $y_{n+1} \neq y_n$ only for $s_n = 1$, whereas $x_{n+1} \neq x_n$ only for $s_n = -1$ (see Proposition \[Prop\_SignumAlgorithm\]). Starting at $\boldsymbol{x}_0 = (r,0)$, we have $$\label{Eq_An}
\mathcal{A}_0 = 0, \mathcal{A}_{n+1} = \mathcal{A}_n + \frac{1}{2} (s_n + 1) x_n$$ with $n \in \mathbb{N}, n \in [0, 2r-2]$. If $s_n = 1$, $\mathcal{A}_n$ is updated by the next horizontal “strip” according to $\mathcal{A}_{n+1} = \mathcal{A}_n + x_n$, whereas $A_{n+1} = A_n$ in the case of $s_n = -1$. This recursively “constructs” the area under $\mathcal{S}^1$ as we go along the circular path $\mathcal{S}^1$. For $n=2r-2$ in (\[Eq\_An\]), we obtain the full area, i.e. $$\label{Eq_A}
\mathcal{A}(r) = \mathcal{A}_{2r-1}.$$
It remains to evaluate $\mathcal{A}_{2r-1}$. To that end, we first rewrite the recursion (\[Eq\_An\]) in explicit form: $$\begin{aligned}
\mathcal{A}_n
& = & \mathcal{A}_0 + \frac{1}{2} \sum\limits_{k=0}^{n-1} (s_k + 1) x_k \\
& = & \mathcal{A}_0 + \frac{1}{2} \sum\limits_{k=0}^{n-1} s_k x_k + \frac{1}{2} \sum\limits_{k=0}^{n-1} x_k \\
& = & \mathcal{A}_0 + \frac{1}{2} s_0 x_0 + \frac{1}{2} \sum\limits_{k=1}^{n-1} s_k x_k + \frac{1}{2} x_0 + \frac{1}{2} \sum\limits_{k=1}^{n-1} x_k \\
& = & x_0 + \frac{1}{2} \sum\limits_{k=1}^{n-1} s_k x_k + \frac{1}{2} \sum\limits_{k=1}^{n-1} \left( x_0 + \frac{1}{2} S_{k-1} - \frac{k}{2} \right) \\
& = & \frac{1}{2} (n+1) x_0 - \frac{1}{8} n (n-1) + \frac{1}{2} \sum\limits_{k=1}^{n-1} s_k x_k + \frac{1}{4} \sum\limits_{k=1}^{n-1} S_{k-1} ,\end{aligned}$$ $n \in [0,2r-1]$, where in the penultimate step we utilized the explicit form of $x_n$, $$\label{Eq_xnExpl}
x_n = x_0 + \frac{1}{2} S_{n-1} - \frac{n}{2},$$ which can easily be deduced from (\[Eq\_S1Algorithm\]) with $$\label{Eq_Sn}
S_n := \sum\limits_{k=0}^n s_k .$$ Applying again (\[Eq\_xnExpl\]), we obtain $$\mathcal{A}_n = \frac{1}{8} n ( 1 - n + 4r ) + \frac{1}{4} \sum\limits_{k=1}^{n-1} S_{k-1} + \frac{1}{2} r S_{n-1} + \frac{1}{4} \sum\limits_{k=1}^{n-1} s_k S_{k-1} - \frac{1}{4} \sum\limits_{k=1}^{n-1} k s_k ,$$ where $x_0 = r$ and $s_0 = 1$ were used. This yields, with (\[Eq\_A\]), $$\label{Eq_An1}
\mathcal{A}(r)
= \frac{1}{4} \left( (r+1)(2r-1) + \sum\limits_{k=1}^{2r-2} S_{k-1} + 2r S_{2r-2}
+ \sum\limits_{k=1}^{2r-2} s_k S_{k-1} - \sum\limits_{k=1}^{2r-2} k s_k \right).$$
We first evaluate $S_{2r-2}$. Due to its definition (\[Eq\_Sn\]), $S_n$ is subject to the recursion $$\label{Eq_SnRec}
S_0 = s_0 = 1, S_{n+1} = S_n + s_{n+1}$$ with $n \in [0, 2r-2]$, which yields $S_{2r-2} = S_{2r-1} - s_{2r-1}$. Due to symmetry of the lower and upper half of the quarter circle, the number of steps to the left ($s_n = -1$) and upwards ($s_n = 1$) must, by construction, be equal, hence $S_{2r-1} = 0$. Moreover, again due to symmetry, $s_{2r-1} = -1$, which yields $$\label{Eq_AnA}
S_{2r-2} = 1.$$
The second last term (\[Eq\_An\]) can be similarly treated, using arguments from symmetry. Specifically, we have $$\begin{aligned}
s_n & = & - s_{2r-1-n} \\
S_n & = & S_{2r-1-(n+1)} \end{aligned}$$ $\forall n \in [0,r]$. Thus, $$\begin{aligned}
\label{Eq_AnB}
\sum\limits_{k=1}^{2r-2} s_k S_{k-1}
& = & \sum\limits_{k=1}^{r-1} s_k S_{k-1} + \sum\limits_{k=r}^{2r-2} s_k S_{k-1} \nonumber \\
& = & \sum\limits_{k=1}^{r-1} s_k S_{k-1} + \sum\limits_{k=1}^{r-1} s_{2r-1-k} S_{2r-1-(k+1)} \nonumber \\
& = & \sum\limits_{k=1}^{r-1} s_k S_{k-1} - \sum\limits_{k=1}^{r-1} s_k S_k \nonumber \\
& = & \sum\limits_{k=1}^{r-1} s_k ( S_{k-1} - S_k ) \nonumber \\
& = & -\sum\limits_{k=1}^{r-1} s_k^2 = -\sum\limits_{k=1}^{r-1} 1 = -(r-1),\end{aligned}$$ where in the last step we used again (\[Eq\_SnRec\]) and the fact that $s_n^2 = 1$ for all $n$.
Finally, reordering terms in the last sum in (\[Eq\_An\]) yields $$\begin{aligned}
\label{Eq_AnC}
\sum\limits_{k=1}^{2r-2} k s_k
& = & \sum\limits_{k=1}^{2r-2} s_k + \sum\limits_{k=2}^{2r-2} s_k + \ldots + \sum\limits_{k=2r-2}^{2r-2} s_k \nonumber \\
& = & (S_{2r-2} - S_0) + (S_{2r-2} - S_1) + \ldots + (S_{2r-2} - S_{2r-3}) \nonumber \\
& = & (2r-2) S_{2r-2} - \sum\limits_{k=0}^{2r-3} S_k \nonumber \\
& = & 2r - 2 - \sum\limits_{k=1}^{2r-2} S_{k-1},\end{aligned}$$ where in the last step we used (\[Eq\_AnA\]) and changed the summation index in the remaining sum. Taking (\[Eq\_AnA\]), (\[Eq\_AnB\]) and (\[Eq\_AnC\]), we obtain for (\[Eq\_An\]) $$\mathcal{A}(r) = \frac{1}{2} \left( r^2 + 1 \right) + \frac{1}{2} \sum\limits_{k=1}^{2r-2} S_{k-1},$$ which yields $$\label{Eq_sumSn}
\sum\limits_{k=1}^{2r-2} S_{k-1} = 2 \mathcal{A}(r) - r^2 - 1.$$
We can now calculate the arithmetic mean of $a_n$, specifically $$\begin{aligned}
A\left(\frac{a_n}{4r}\right)
& = & \frac{1}{2r} \sum\limits_{k=0}^{2r-1} \frac{a_n}{4r} \\
& = & \frac{1}{8r^2} \sum\limits_{k=0}^{2r-1} (r + S_{k-1}).\end{aligned}$$ Here we made use of the explicit form of $a_n$, which can easily be deduced from (\[Eq\_anRec\]) as $a_n = a_0 + S_{n-1}$ with $a_0 = r$. Together with (\[Eq\_sumSn\]), we then have $$\begin{aligned}
A\left(\frac{a_n}{4r}\right)
& = & \frac{1}{8r^2} \left( \sum\limits_{k=0}^{2r-1} r + \sum\limits_{k=0}^{2r-1} S_{k-1} \right) \\
& = & \frac{1}{4r^2} \mathcal{A}(r) + \frac{1}{8} - \frac{1}{8r^2},\end{aligned}$$ which yields in the asymptotic limit for $r \rightarrow \infty$, using (\[Eq\_Alimit\]), $$\lim_{r \rightarrow \infty} A\left(\frac{a_n}{4r}\right) = \frac{\pi}{16} + \frac{1}{8}.$$ Finally, noting that $\pi_n = \frac{a_n}{4r}$, and that the harmonic mean is the reciprocal dual of the arithmetic mean, we have proven Proposition \[Prop\_Hpi\].
Concluding Remarks {#S_Conclusion}
==================
The results presented in the last section hint at some deeper number-theoretical peculiarities of digital circles, beyond their defining conception as mere digital, or digitized, models of circles in $\mathbb{R}^2$. When considering digital circles rigorously in a discrete space with $\ell^1$-norm, a direct link can be drawn to their continuous ideal $S^1$. We exemplified this point by showing that $\pi$ can be recovered in the asymptotic limit of infinite radius by simply averaging over the $\pi$-values associated with each point along a valid discrete circular path in a space with $\ell^1$-norm (Conjecture \[Conj\_Api\]). Equally interesting is the finding that also the harmonic mean of this sequence of $\pi$-values yields, in the asymptotic limit, a value linear in $\pi$ (Proposition \[Prop\_Hpi\]). Although the fundamental inequality linking the arithmetic and harmonic means of a given sequence is not violated, $$\lim_{r \rightarrow \infty} A(\pi_n) = \pi > \frac{16}{\pi + 2} = \lim_{r \rightarrow \infty} H(\pi_n),$$ the construction of the sequences of $\pi_n$ and their reciprocals suggest an identity linking $\pi$ and its reciprocal.
Finally, the recursive signum algorithm for constructing a valid digital path in $\mathbb{Z}^2$ (Proposition \[Prop\_SignumAlgorithm\]) approximating $S^1 \subset \mathbb{R}^2$ allows for the construction of a recursive sequence yielding the area inside a circular path, as demonstrated in the proof of Proposition \[Prop\_Hpi\]. To what extent this approach might be exploitable for gaining deeper insights into the Gauss’s Circle Problem remains to be explored.
Acknowledgments {#acknowledgments .unnumbered}
===============
Research supported in part by CNRS. The author wishes to thank LE Muller II, J Antolik, D Holstein, JAG Willow, S Hower and OD Little for valuable discussions and comments.
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abstract: 'We give a simple characterization of chaos for weighted composition $C_0$-semigroups on $L^p_\rho(\Omega)$ for an open interval $\Omega\subseteq{\mathbb{R}}$. Moreover, we characterize chaos for these classes of $C_0$-semigroups on the closed subspace $W^{1,p}_*(\Omega)$ of the Sobolev space $W^{1,p}(\Omega)$ for a bounded interval $\Omega\subset{\mathbb{R}}$. These characterizations simplify the characterization of chaos obtained in [@ArKaMa13] for these classes of $C_0$-semigroups.'
author:
- 'T. Kalmes'
title: 'A simple characterization of chaos for weighted composition $C_0$-semigroups on Lebesgue and Sobolev spaces'
---
Introduction
============
The purpose of this article is to give a simple characterization of chaos for certain weighted composition $C_0$-semigroups on Lebesgue spaces and Sobolev spaces over open intervals. Recall that a $C_0$-semigroup $T$ on a separable Banach space $X$ is called [*chaotic*]{} if $T$ is hypercyclic, i.e. there is $x\in X$ such that $\{T(t)x;\,t\geq 0\}$ is dense in $X$, and if the set of periodic points, i.e. $\{x\in X;\,\exists t>0: T(t)x=x\}$, is dense in $X$.
The study of chaotic $C_0$-semigroups has attracted the attention of many researchers. We refer the reader to Chapter 7 of the monograph by Grosse-Erdmann and Peris [@GEPe11] and the references therein. Some recent papers in the topic are [@AlBaMaPe13; @ArPe12; @BaMo11; @EmGoGo12; @Ru12].
For $\Omega\subseteq{\mathbb{R}}$ open and a Borel measure $\mu$ on $\Omega$ admitting a strictly positive Lebesgue density $\rho$ we consider $C_0$-semigroups $T$ on $L^p(\Omega,\mu), 1\leq p<\infty,$ of the form $$T(t)f(x)=h_t(x) f(\varphi(t,x)),$$ where $\varphi$ is the solution semiflow of an ordinary differential equation $$\dot{x}=F(x)$$ in $\Omega$ and $$h_t(x)=\exp\big(\int_0^t h(\varphi(s,x))ds\big)$$ with $h\in C(\Omega)$. Such $C_0$-semigroups appear in a natural way when dealing with initial value problems for linear first order partial differential operators. While a characterization of chaos for such $C_0$-semigroups was obtained for open $\Omega\subseteq{\mathbb{R}}^d$ for arbitrary dimension $d$ in [@Ka07], evaluation of these conditions in concrete examples is sometimes rather involved. In contrast to general dimension the case $d=1$ allows for a significantly simplified characterization, see [@ArKaMa13]. However, this characterization of chaos still depends on the knowledge of the solution semiflow $\varphi$ which might be difficult to determine in concrete examples.
In section 2 we give, under mild additional assumptions on $F$ and $h$, a characterization of chaos which only depends on the ingredients $F$, $h$, and $\rho$, without refering to the semiflow $\varphi$.
In section 3 we use this result to obtain a similarly simple characterization of chaos for the above kind of $C_0$-semigroups acting on the closed subspace $$W^{1,p}_*[a,b]=\{f\in W^{1,p}[a,b];\,f(a)=0\}$$ of the Sobolev spaces $W^{1,p}[a,b]$, where $(a,b)\subseteq{\mathbb{R}}$ is a bounded interval. It was shown in [@ArKaMa13] that such $C_0$-semigroups cannot be hypercyclic, a fortiori chaotic, on the whole Sobolev space $W^{1,p}(a,b)$.
In order to illustrate our results, several examples are considered.
Chaotic weighted composition $C_0$-semigroups on Lebesgue spaces
================================================================
Let $\Omega\subseteq{\mathbb{R}}$ be open and let $F:\Omega\rightarrow{\mathbb{R}}$ be a $C^1$-function. Hence, for every $x_0\in\Omega$ there is a unique solution $\varphi(\cdot,x_0)$ of the initial value problem $$\dot{x}=F(x),\; x(0)=x_0.$$ Denoting its maximal domain of definition by $J(x_0)$ it is well-known that $J(x_0)$ is an open interval containing $0$. We make the general assumption that $\Omega$ is [*forward invariant under*]{} $F$, i.e.$[0,\infty)\subset J(x_0)$ for every $x_0\in\Omega$, that is $\varphi:[0,\infty)\rightarrow\Omega$. This is true, for example, if $\Omega=(a,b)$ is a bounded interval and if $F$ can be extended to a $C^1$-function defined on a neighborhood of $[a,b]$ such that $F(a)\geq 0$ and $F(b)\leq 0$ (cf. [@Amann Corollary 16.10]).
From the uniqueness of the solution it follows that $\varphi(t,\cdot)$ is injective for every $t\geq 0$ and $\varphi(t+s,x)=\varphi(t,\varphi(s,x))$ for all $x\in\Omega$ and $s,t\in J(x)$ with $s+t\in J(x)$. Moreover, for every $t\geq 0$ the set $\varphi(t,\Omega)$ is open and for $x\in\varphi(t,\Omega)$ we have $[-t,\infty)\subset J(x)$ as well as $\varphi(-s,x)=\varphi(s,\cdot)^{-1}(x)$ for all $s\in [0,t]$. Since $F$ is a $C^1$-function it is well-known that the same is true for $\varphi(t,\cdot)$ on $\Omega$ and $\varphi(-t,\cdot)$ on $\varphi(t,\Omega)$ for every $t\geq 0$.
Moreover, let $h\in C(\Omega)$ and define for $t\ge 0$ $$h_t:\Omega\rightarrow{\mathbb{C}}, h_t(x)=\exp(\int_0^t h(\varphi(s,x))ds).$$ For $1\leq p<\infty$ and a measurable function $\rho:\Omega\rightarrow (0,\infty)$ let $L^p_\rho(\Omega)$ be as usual the Lebesgue space of $p$-integrable functions with respect to the Borel measure $\rho d\lambda$, where $\lambda$ denotes Lebesgue measure. If $\Omega$ is forward invariant under $F$ the operators $$T(t):L^p_\rho(\Omega)\rightarrow L^p_\rho(\Omega), (T(t)f)(x):=h_t(x)f(\varphi(t,x))\;(t\geq 0)$$ are well-defined continuous linear operators defining a $C_0$-semigroup $T_{F,h}$ on $L^p_\rho(\Omega)$ if $\rho$ is $p$-[*admissible for*]{} $F$ [*and*]{} $h$, i.e. if there are constants $M\geq 1$, $\omega\in{\mathbb{R}}$ with $$\forall\,t\geq 0,x\in\Omega:\,|h_t(x)|^p\rho(x)\leq M e^{\omega t}\rho(\varphi(t,x))\exp(\int_0^t F'(\varphi(s,x))ds),$$ (see [@ArKaMa13]). Because $|h_t(x)|^p=\exp(p\int_0^t{\mbox{Re}\,}h(\varphi(s,x))ds)$ it follows that $\rho=1$ is $p$-admissible for any $p$ if ${\mbox{Re}\,}h$ is bounded above and $F'$ is bounded below, i.e. in this case the above operators define a $C_0$-semigroup $T_{F,h}$ on the standard Lebesgue spaces $L^p(\Omega)$. Under mild additional assumptions on $F$ and $h$ the generator of this $C_0$-semigroup is given by the first order differential operator $Af=F f'+hf$ on a suitable subspace of $L^p(\Omega)$ (see [@ArKaMa13 Theorem 15]).
In [@ArKaMa13 Theorem 6 and Proposition 9] it is characterized when the $C_0$-semigroup $T_{F,h}$ is chaotic on $L^p_\rho(\Omega)$. However, this characterization depends on a more or less explicit knowledge of the semiflow $\varphi$.
Our aim is to prove the following characterization of chaos for $T_{F,h}$ on $L^p_\rho(\Omega)$ valid under mild additional assumptions on $F$ and $h$ and which is given solely in terms of $F$, $h$, and $\rho$. Throughout this article, we use the following common abbreviation $\{F=0\}:=\{x\in\Omega;\,F(x)=0\}$.
\[simplified chaos\] For $1\leq p<\infty$ let $\Omega\subset{\mathbb{R}}$ be an open interval which is forward invariant under $F\in C^1(\Omega)$, and let $h\in C(\Omega)$ be such that $F'$ and ${\mbox{Re}\,}h$ are bounded and
- There is $\gamma\in{\mathbb{R}}$ such that $h(x)=\gamma$ for all $x\in\{F=0\}$.
- With $\alpha:=\inf\Omega$ and $\omega:=\sup\Omega$ the function $$\Omega\rightarrow{\mathbb{C}},y\mapsto\frac{{\mbox{Im}\,}h(y)}{F(y)}$$ belongs to $L^1((\alpha,\beta))$ for all $\beta\in\Omega$ or to $L^1((\beta,\omega))$ for all $\beta\in\Omega$.
Then for every $\rho$ which is $p$-admissible for $F$ and $h$ the following are equivalent.
- $T_{F,h}$ is chaotic in $L^p_\rho(\Omega)$.
- $\lambda(\{F=0\})=0$ and for every connected component $C$ of $\Omega\backslash\{F=0\}$ $$\int_C\exp(-p\int_x^w\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\rho(w)d\lambda(w)<\infty$$ for some/all $x\in C$.
In order to prove Theorem \[simplified chaos\], we define for $x\in\Omega$, $p\geq 1$, and $t\geq 0$ $$\begin{aligned}
\rho_{t,p}(x)&=&\chi_{\varphi(t,\Omega)}(x)|h_t(\varphi(-t,x))|^p\exp(\int_0^{-t}F'(\varphi(s,x))ds)\,\rho(\varphi(-t,x))\\
&=&\chi_{\varphi(t,\Omega)}(x)\exp(p\int_0^t{\mbox{Re}\,}h(\varphi(s,\varphi(-t,x)))ds)\exp(\int_0^{-t}F'(\varphi(s,x))ds)\,\rho(\varphi(-t,x))\end{aligned}$$ as well as $$\begin{aligned}
\rho_{-t,p}(x)&=&|h_t(x)|^{-p}\exp(\int_0^t F'(\varphi(s,x))ds)\,\rho(\varphi(t,x))\\
&=&\exp(-p\int_0^t{\mbox{Re}\,}h(\varphi(s,x))ds)\exp(\int_0^t F'(\varphi(s,x))ds)\,\rho(\varphi(t,x)).\end{aligned}$$ Then $\rho_{0,p}=\rho$, $\rho_{t,p}\geq 0$ for every $t\in{\mathbb{R}}$, and for fixed $x\in\Omega$ the mapping $t\mapsto\rho_{t,p}(x)$ is Lebesgue measurable. Moreover, it follows that $$\begin{aligned}
\label{Nummer1}
\rho_{-(t+s),p}(x)&=&\exp(-p\int_0^{t+s}{\mbox{Re}\,}h(\varphi(r,x))\,dr)\nonumber\\
&&\cdot\exp(\int_0^{t+s}F'(\varphi(r,x))dr)\rho(\varphi(t,\varphi(s,x)))\nonumber\\
&=&\exp(\int_0^s F'(\varphi(r,x))-p{\mbox{Re}\,}h(\varphi(r,x))dr)\nonumber\\
&&\cdot\exp(-p\int_0^t{\mbox{Re}\,}h(\varphi(r,\varphi(s,x)))dr)\\
&&\cdot\exp(\int_0^t F'(\varphi(r,\varphi(s,x)))dr)\rho(\varphi(t,\varphi(s,x)))\nonumber\\
&=&\exp(\int_0^s F'(\varphi(r,x))-p{\mbox{Re}\,}h(\varphi(r,x))dr)\rho_{-t,p}(\varphi(s,x))\nonumber\end{aligned}$$ and analogously $$\begin{aligned}
\label{Nummer2}
\rho_{(t+s),p}(x)&=&\chi_{\varphi(t+s,\Omega)}(x)\exp(p\int_{-(t+s)}^0{\mbox{Re}\,}h(\varphi(r,x))-\frac{1}{p}F'(\varphi(r,x))dr)\nonumber\\
&&\cdot\rho(\varphi(-(t+s),x))\\
&=&\chi_{\varphi(s,\Omega)}(x)\exp(p\int_{-s}^0{\mbox{Re}\,}h(\varphi(r,x))-\frac{1}{p}F'(\varphi(r,x))dr)\rho_{t,p}(\varphi(-s,x)).\nonumber\end{aligned}$$
The following lemma will be used in the proof of the first auxilary result. We cite it for the reader’s convenience. For a proof see [@Ka09 Lemma 7].
\[aux\] Let $\Omega\subseteq{\mathbb{R}}$ be open, let $F\in C^1(\Omega)$ be such that $\Omega$ is forward invariant under $F$, and let $h\in C(\Omega)$ be real valued. Moreover, for fixed $1\leq p<\infty$ let $\rho$ be $p$-admissible for $F$ and $h$. For $[a,b]\subset\Omega\backslash\{F=0\}$ set $\alpha:=a$ and $\beta:=b$ if $F_{|[a,b]}>0$, respectively $\alpha:=b$ and $\beta:=a$ if $F_{|[a,b]}<0$.
Then there is a constant $C>0$ such that $$\forall\,x\in [a,b]:\,\frac{1}{C}\leq\rho(x)\leq C$$ as well as $$\forall\,t\in{\mathbb{R}}, x\in[a,b]:\,\frac{1}{C}\rho_{t,p}(\alpha)\leq\rho_{t,p}(x)\leq C\rho_{t,p}(\beta).$$
\[series\] Let $\Omega\subseteq{\mathbb{R}}$ be open and forward invariant under $F\in C^1(\Omega)$, let $h\in C(\Omega)$ be such that $F'$ and ${\mbox{Re}\,}h$ are bounded. Moreover, let $\rho$ be $p$-admissible for $F$ and $h$, $1\leq p<\infty$. Then the following are equivalent.
- For all $x\in\Omega\backslash\{F=0\}$ there is $t_0>0$ such that $\sum_{k\in{\mathbb{Z}}}\rho_{k t_0,p}(x)<\infty$.
- For all $x\in\Omega\backslash\{F=0\}: \int_{\mathbb{R}}\rho_{t,p}(x)d\lambda(t)<\infty$.
- For all $x\in\Omega\backslash\{F=0\}$ and $t_0>0: \sum_{k\in{\mathbb{Z}}}\rho_{k t_0,p}(x)<\infty.$
In order to show that i) implies ii) fix $x\in\Omega\backslash\{F=0\}$ and choose $t_0>0$ according to i) for $x$. We distinguish two cases. If $x$ belongs to $\cap_{t\geq 0}\varphi(t,\Omega)$ it follows by equation (\[Nummer2\]) and the boundedness of ${\mbox{Re}\,}h$ and $F'$ $$\begin{aligned}
&&\int_{[0,\infty)}\rho_{t,p}(x)d\lambda(t)=\sum_{k=0}^\infty\int_{[0,t_0]}\rho_{k t_0+s,p}(x)d\lambda(s)\\
&=&\sum_{k=0}^\infty\int_{[0,t_0]}\chi_{\varphi(s,\Omega)}(x)\exp(p\int_{-s}^0{\mbox{Re}\,}h(\varphi(r,x))-\frac{1}{p}F'(\varphi(r,x))dr)\\
&&\rho_{kt_0,p}(\varphi(-s,x))d\lambda(s)\\
&\leq& C\sum_{k=0}^\infty\int_{[0,t_0]}\chi_{\varphi(s,\Omega)}(x)\rho_{k t_0,p}(\varphi(-s,x))d\lambda(s)\\
&=& C\sum_{k=0}^\infty\int_{[0,t_0]}\rho_{k t_0,p}(\varphi(-s,x))d\lambda(s)\\
&\leq &\begin{cases}\tilde{C}\sum_{k=0}^\infty\rho_{k t_0,p}(x)&, F(x)>0\\ \tilde{C}\sum_{k=0}^\infty\rho_{k t_0,p}(\varphi(-t_0,x))&, F(x)<0,\end{cases}
\end{aligned}$$ where $C$ and $\tilde{C}$ depend on $t_0$ and where in the last step we used lemma \[aux\] for $F$ and ${\mbox{Re}\,}h$. Since by equation (\[Nummer2\]) together with the boundedness of $F'$ and ${\mbox{Re}\,}h$ we also have with suitable $D>0$ that for all $k\geq 0$ $$\rho_{k t_0,p}(\varphi(-t_0,x))\leq D\rho_{(k+1)t_0,p}(x),$$ the above shows the existence of $\hat{C}>0$ such that $$\int_{[0,\infty)}\rho_{t,p}(x)d\lambda(t)\leq \hat{C}\sum_{k=0}^\infty\rho_{k t_0,p}(x).$$
If $x$ does not belong to $\cap_{t\geq 0}\varphi(t,\Omega)$ then $\int_{[0,\infty)}\rho_{t,p}(x)d\lambda(t)=\int_{[0,r]}\rho_{t,p}(x)d\lambda(t)$ for some $r>0$. Combining lemma \[aux\] for $F$ and ${\mbox{Re}\,}h$ with equation (\[Nummer2\]), the boundedness of $F'$ and ${\mbox{Re}\,}h$ gives for suitable $C>0$ $$\begin{aligned}
&&\int_{[0,\infty)}\rho_{t,p}(x)d\lambda(t)\\
&=&\int_{[0,r]}\chi_{\varphi(t,\Omega)}(x)\exp(p\int_{-t}^0{\mbox{Re}\,}h(\varphi(s,x))-\frac{1}{p}F'(\varphi(s,x))ds)\rho_{0,p}(\varphi(-t,x))d\lambda(t)\\
&\leq& C\int_{[0,r]}\chi_{\varphi(t,\Omega)}(x)\rho_{0,p}(\varphi(-t,x))d\lambda(t)<\infty.
\end{aligned}$$ Thus, if i) holds then $\int_{[0,\infty)}\rho_{t,p}(x)d\lambda(t)<\infty$ for all $x\in\Omega\backslash\{F=0\}$.
Moreover, by equation (\[Nummer1\]) we obtain for every $x\in\Omega\backslash\{F=0\}$ together with the boundedness of $F'$ and ${\mbox{Re}\,}h$ $$\begin{aligned}
&&\int_{(-\infty,0]}\rho_{t,p}(x)d\lambda(t)=\sum_{k=0}^\infty\int_{[-t_0,0]}\rho_{-(k t_0-s),p}(x)d\lambda(s)\\
&=&\sum_{k=0}^\infty\int_{[-t_0,0]}\exp(\int_0^{-s} F'(\varphi(r,x))-p{\mbox{Re}\,}h(\varphi(r,x)))\rho_{-k t_0,p}(\varphi(-s,x))d\lambda(s)\\
&\leq& C\sum_{k=0}^\infty\int_{[-t_0,0]}\rho_{-k t_0,p}(\varphi(-s,x))d\lambda(s)\\
&\leq& \begin{cases}\tilde{C}\sum_{k=0}^\infty \rho_{-k t_0,p}(\varphi(t_0,x))&, F(x)>0\\ \tilde{C}\sum_{k=0}^\infty \rho_{-k t_0,p}(x)&, F(x)<0,\end{cases}\end{aligned}$$ where $C$ and $\tilde{C}$ again depend on $t_0$ and where in the last step we again used lemma \[aux\] for $F$ and ${\mbox{Re}\,}h$. Equation (\[Nummer1\]) and the fact that $F'$ and ${\mbox{Re}\,}h$ are bounded yield the existence of $D>0$ such that for all $k\geq 0$ $$\rho_{-k t_0,p}(\varphi(t_0,x))\leq D\rho_{-(k+1)t_0,p}(x).$$ So the above gives $$\int_{(-\infty,0]}\rho_{t,p}(x)d\lambda(t)\leq\hat{C}\sum_{k=0}^\infty \rho_{-kt_0,p}(x)$$ for some $\hat{C}>0$. Hence, i) implies ii).
In order to show that ii) implies iii) we fix $t_0>0$ and $x\in\Omega\backslash\{F=0\}$ and distinguish again two cases. If $x$ does not belong to $\cap_{t\geq 0}\varphi(t,\Omega)$ there is $t_1>0$ such that $\rho_{t,p}(x)=0$ for all $t>t_1$. Therefore, $\sum_{k=0}^\infty\rho_{k t_0,p}(x)<\infty$.
In case of $x\in\cap_{t\geq 0}\varphi(t,\Omega)$ it follows from equation (\[Nummer2\]) together with the boundedness of $F'$ and ${\mbox{Re}\,}h$ that for some $C>0$ $$\begin{aligned}
&&\int_{[0,\infty)}\rho_{t,p}(x)d\lambda(t)=\sum_{k=0}^\infty\int_{[0,t_0]}\rho_{k t_0+t,p}(x)d\lambda(t)\\
&=&\sum_{k=0}^\infty\int_{[0,t_0]}\exp(p\int_{-t}^0{\mbox{Re}\,}h(\varphi(r,x))-\frac{1}{p}F'(\varphi(r,x))dr)\rho_{k t_0,p}(\varphi(-t,x))d\lambda(t)\\
&\geq&\sum_{k=0}^\infty C\int_{[0,t_0]}\rho_{k t_0,p}(\varphi(-t,x))d\lambda(t)\\
&\geq&\begin{cases}\tilde{C}\sum_{k=0}^\infty\rho_{k t_0,p}(x)&, F(x)<0\\ \tilde{C}\rho_{k t_0,p}(\varphi(-t_0,x))&, F(x)<0,\end{cases}\end{aligned}$$ where we used lemma \[aux\] in the last step. By equation (\[Nummer2\]) and the boundedness of $F'$ and ${\mbox{Re}\,}h$ we have $\rho_{k t_0,p}(\varphi(-t_0,x))\geq D\rho_{(k+1)t_0,p}(x)$ for suitable $D>0$ such that the above gives $$\label{Nummer3}
\int_{[0,\infty)}\rho_{t,p}(x)d\lambda(t)\geq\hat{C}_1\sum_{k=0}^\infty\rho_{k t_0,p}(x)$$ for some $\hat{C}_1$.
Additionally, applying lemma \[aux\] for $F$ and ${\mbox{Re}\,}h$ we also obtain from the boundedness of $F'$ and ${\mbox{Re}\,}h$ together with equation (\[Nummer1\]) $$\begin{aligned}
&&\int_{(-\infty,0]}\rho_{t,p}(x)d\lambda(t)=\sum_{k=0}^\infty\int_{[-t_0,0]}\rho_{-(k t_0-t),p}(x)d\lambda(t)\\
&=&\sum_{k=0}^\infty\int_{[-t_0,0]}\exp(\int_0^{-t}F'(\varphi(r,x))-p{\mbox{Re}\,}h(\varphi(r,x))dr)\rho_{-k t_0,p}(\varphi(-t,x))d\lambda(t)\\
&\geq & C\sum_{k=0}^\infty\int_{[-t_0,0]}\rho_{-k t_0,p}(\varphi(-t,x))d\lambda(t)\\
&\geq & \begin{cases}\tilde{C}\sum_{k=0}^\infty\rho_{-k t_0,p}(x)&, F(x)>0\\ \tilde{C}\sum_{k=0}^\infty\rho_{-k t_0,p}(\varphi(-t_0,x))&, F(x)<0\end{cases}\\
&\geq &\hat{C}_2\sum_{k=0}^\infty \rho_{-k t_0,p}(x).\end{aligned}$$ Hence, together with (\[Nummer3\]), iii) follows from ii), and as iii) obviously implies i), the lemma is proved.
The applicability of the previous lemma depends on an explicit knowledge of $\varphi$. The next lemma shows that the integrals appearing in the previous result can be expressed in terms of $F$, $h$, and $\rho$.
\[integral\] Let $\Omega\subseteq{\mathbb{R}}$ be open and forward invariant under $F\in C^1(\Omega)$, $h\in C(\Omega)$ and let $\rho$ be $p$-admissible for $F$ and $h$, $1\leq p<\infty$. Then for every $x\in\Omega\backslash\{F=0\}$ we have $$\int_{\mathbb{R}}\rho_{t,p}(x)d\lambda(t)=\frac{1}{|F(x)|}\int_{C(x)}\exp(p\int_w^x\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\rho(w)d\lambda(w),$$ where $C(x)$ denotes the connected component of $\Omega\backslash\{F=0\}$ containing $x$.
Fix $x\in\Omega\backslash\{F=0\}$ and let $C(x)$ be as in the lemma. Observe that $\varphi(t,x)\in C(x)$ for all $t\in J(x)$ and that $\varphi(J(x),x)=C(x)$, where $J(x)$ is the domain of the maximal solution $\varphi(\cdot,x)$ of the initial value problem $\dot{y}=F(y), y(0)=x$. Obviously, $$\int_{\mathbb{R}}\rho_{t,p}(x)d\lambda(t)=\int_{[0,\infty)}\rho_{t,p}(x)d\lambda(t)+\int_{[0,\infty)}\rho_{-t,p}(x)d\lambda(x).$$ We set $C^+(x)=\{\varphi(t,x);\,t\geq 0\}$. Applying the Transformation Formula for Lebesgue integrals we obtain with equation (\[Nummer1\]) $$\begin{aligned}
&&\int_{[0,\infty)}\rho_{-t,p}(x)d\lambda(t)\\
&=&\int_{[0,\infty)}\exp(\int_0^t F'(\varphi(r,x))-p{\mbox{Re}\,}h(\varphi(r,x)) dr)\rho(\varphi(t,x))d\lambda(t)\\
&=&\int_{[0,\infty)}\exp(\int_0^t\frac{F'(\varphi(r,x))-p{\mbox{Re}\,}h(\varphi(r,x))}{F(\varphi(r,x))}\partial_1\varphi(r,x) dr)\rho(\varphi(t,x))d\lambda(t)\\
&=&\int_{[0,\infty)}\exp(\int_x^{\varphi(t,x)}\frac{F'(y)-p{\mbox{Re}\,}h(y)}{F(y)}dy)\rho(\varphi(t,x))d\lambda(t)\end{aligned}$$ $$\begin{aligned}
&=&\int_{(0,\infty)}\frac{\exp(\int_x^{\varphi(t,x)}\frac{F'(y)-p{\mbox{Re}\,}h(y)}{F(y)} dy)}{|F(\varphi(t,x))|}\rho(\varphi(t,x))|\partial_1\varphi(t,x)|d\lambda(t)\\
&=&\int_{C^+(x)}\frac{\exp(\int_x^w \frac{F'(y)-p{\mbox{Re}\,}h(y)}{F(y)}dy)}{|F(w)|}\rho(w)d\lambda(w)\\
&=&\int_{C^+(x)}\frac{\exp(\int_w^x \frac{p{\mbox{Re}\,}h(y)-F'(y)}{F(y)}dy)}{|F(w)|}\rho(w)d\lambda(w).\end{aligned}$$ Moreover, denoting $\alpha=\sup\{t\geq 0;\,x\in\varphi(t,\Omega)\}$ we have $-\alpha=\inf J(x)$. With $C^-(x)=\varphi((-\alpha,0],x)$ it follows $C(x)=C^+(x)\cup C^-(x)$, $C^+(x)\cap C^-(x)=\{x\}$, and $$\begin{aligned}
&&\int_{[0,\infty)}\rho_{t,p}(x)d\lambda(t)=\int_{[0,\alpha)}\rho_{t,p}(x)d\lambda(t)\\
&=&\int_{[0,\alpha)}\exp(\int_{-t}^0 p{\mbox{Re}\,}h(\varphi(r,x))-F'(\varphi(r,x))dr)\rho(\varphi(-t,x))d\lambda(t)\\
&=&\int_{(-\alpha,0]}\exp(\int_t^0 p{\mbox{Re}\,}h(\varphi(r,x))-F'(\varphi(r,x))dr)\rho(\varphi(t,x))d\lambda(t)\\
&=&\int_{(-\alpha,0]}\exp(\int_t^0\frac{p{\mbox{Re}\,}h(\varphi(r,x))-F'(\varphi(r,x))}{F(\varphi(r,x))}\partial_1\varphi(r,x)dr)\rho(\varphi(t,x))d\lambda(t)\\
&=&\int_{(-\alpha,0]}\exp(\int_{\varphi(t,x)}^x\frac{p{\mbox{Re}\,}h(y)-F'(y)}{F(y)}dy)\rho(\varphi(t,x))d\lambda(t)\\
&=&\int_{(-\alpha,0]}\frac{\exp(\int_{\varphi(t,x)}^x\frac{p{\mbox{Re}\,}h(y)-F'(y)}{F(y)}dy)}{|F(\varphi(t,x))|}\rho(\varphi(t,x))|\partial_1\varphi(t,x)|d\lambda(t)\\
&=&\int_{C^-(x)}\frac{\exp(\int_w^x\frac{p{\mbox{Re}\,}h(y)-F'(y)}{F(y)}dy)}{|F(w)|}\rho(w)d\lambda(w).\end{aligned}$$ Combining these equations yields $$\begin{aligned}
&&\int_{\mathbb{R}}\rho_{t,p}(x)d\lambda(t)\\
&=&\int_{C(x)}\frac{\exp(\int_w^x\frac{p{\mbox{Re}\,}h(y)-F'(y)}{F(y)}dy)}{|F(w)|}\rho(w)d\lambda(w)\\
&=&\int_{C(x)}\frac{\exp(p\int_w^x\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\exp(\log|F(w)|-\log|F(x)|)}{|F(w)|}\rho(w)d\lambda(w)\\
&=&\frac{1}{|F(x)|}\int_{C(x)}\exp(p\int_w^x\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\rho(w)d\lambda(w)\end{aligned}$$ which proves the lemma.
\[connected component\]
The last step in the above proof shows that for $x\in\Omega\backslash\{F=0\}$ and all $v\in C(x)$ we have for every $1\leq p<\infty$ $$\begin{aligned}
&&\int_{\mathbb{R}}\rho_{t,p}(x)d\lambda(t)\\
&=&\frac{1}{|F(x)|}\int_{C(x)}\exp(p\int_w^x\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\rho(w)d\lambda(w)\\
&=&\frac{|F(v)|}{|F(x)|}\exp(p\int_v^x\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\frac{1}{|F(v)|}\int_{C(x)}\exp(p\int_x^v\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\rho(w)d\lambda(w)\\
&=&\frac{|F(v)|}{|F(x)|}\exp(p\int_v^x\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\int_{\mathbb{R}}\rho_{t,p}(v)d\lambda(t).\end{aligned}$$ Thus, under the hypotheses of Lemma \[integral\] the following are equivalent for every connnected component $C$ of $\Omega\backslash\{F=0\}$ and all $1\leq p<\infty$.
- $\exists\,x\in C:\,\int_{\mathbb{R}}\rho_{t,p}(x)d\lambda(t)<\infty$,
- $\forall\,x\in C:\,\int_{\mathbb{R}}\rho_{t,p}(x)d\lambda(t)<\infty$,
- $\exists\,x\in C:\,\int_C\exp(-p\int_x^w\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\rho(w)d\lambda(w)<\infty$,
- $\forall\,x\in C:\,\int_C\exp(-p\int_x^w\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\rho(w)d\lambda(w)<\infty$.
We have now everything at hand to prove Theorem \[simplified chaos\].\
[*Proof of Theorem \[simplified chaos\].*]{} By [@ArKaMa13 Theorem 6 and Proposition 9] $T_{F,h}$ is chaotic on $L^p_\rho(\Omega)$ if and only if $\lambda(\{F=0\})=0$ as well as for every $m\in{\mathbb{N}}$ for which there are $m$ different connected components $C_1,\ldots,C_m$ of $\Omega\backslash \{F=0\}$, for $\lambda^m$-almost all choices of $(x_1,\ldots,x_m)\in\Pi_{j=1}^m C_j$ there is $t>0$ such that $$\sum_{j=1}^m\sum_{l\in{\mathbb{Z}}}\rho_{lt,p}(x_j)<\infty.$$ By lemma \[series\], this holds precisely when $\lambda(\{F=0\})=0$ and when for $\lambda$-almost every $x\in\Omega\backslash\{F=0\}$ $$\int_{\mathbb{R}}\rho_{t,p}(x)d\lambda(t)<\infty.$$ Thus, applying Remark \[connected component\], Theorem \[simplified chaos\] follows.$\square$.\
\[remark\]
a\) Inspection of the proof of Theorem \[simplified chaos\] yields the following. Under the hypothesis of Theorem \[simplified chaos\], the following are equivalent for $\rho$ $p$-admissible for $F$ and $h$.
- $T_{F,h}$ is chaotic in $L^p_\rho(\Omega)$.
- $\lambda(\{F=0\})=0$ and for all $x\in\Omega\backslash\{F=0\}$ there is $t_0>0$ such that $\sum_{k\in{\mathbb{Z}}}\rho_{k t_0,p}(x)<\infty$.
- $\lambda(\{F=0\})=0$ and $\sum_{k\in{\mathbb{Z}}}\rho_{k t_0,p}(x)<\infty$ for all $x\in\Omega\backslash\{F=0\}$ and all $t_0>0$.
- $\lambda(\{F=0\})=0$ and $\int_{\mathbb{R}}\rho_{t,p}(x)d\lambda(t)<\infty$ for all $x\in\Omega\backslash\{F=0\}$.
- $\lambda(\{F=0\})=0$ and for every connected component $C$ of $\Omega\backslash\{F=0\}$ $$\int_C\exp(-p\int_x^w\frac{{\mbox{Re}\,}h(y)}{F(y)}dy)\rho(w)d\lambda(w)<\infty$$ for some/all $x\in C$.
b\) If $h=0$ and if $F\in C^1(\Omega)$ is as usual then the $p$-admissibility of $\rho$ does not depend on $p$. If moreover $F'$ is bounded the following are then equivalent.
- $T_F=T_{F,0}$ is chaotic in $L^p_\rho(\Omega)$ for some/all $p\in[1,\infty)$.
- $\lambda(\{F=0\})=0$ and for every connected component $C$ of $\Omega\backslash\{F=0\}$ we have $$\int_C\rho(w)d\lambda(w)<\infty.$$
a\) Let $\Omega\in\{(0,\infty),{\mathbb{R}}\}$ and let $F(x)=1$. Then $\Omega$ is forward invariant under $F$. Moreover, let $h\in C(\Omega)$ be such that ${\mbox{Re}\,}h$ is bounded. It follows from the definition, that $\rho=1$ is $p$-admissible for $F$ and $h$ for every $1\leq p<\infty$ so that $T_{1,h}$ is a well defined $C_0$-semigroup on $L^p(\Omega)$, the so-called perturbed translation semigroup. If $h$ is bounded the generator of $T_{1,h}$ in $L^p(\Omega)$ is given by $$A_p:W^{1,p}(\Omega)\rightarrow L^p(\Omega), A_pf(x)=f'+hf,$$ where $f'$ denotes the distributional derivative of $f$ (see e.g. [@ArKaMa13 Theorem 15]).
If ${\mbox{Im}\,}h\in L^1(0,\beta)$, resp. ${\mbox{Im}\,}h\in L^1(-\infty,\beta)$ for all $\beta\in\Omega$ or if ${\mbox{Im}\,}h\in L^1(\beta,\infty)$ for all $\beta\in\Omega$, by Theorem \[simplified chaos\] this $C_0$-semigroup is chaotic on $L^p(\Omega)$ if and only if $$\int_\Omega\exp(-p\int_1^w{\mbox{Re}\,}h(y)dy)d\lambda(w)<\infty.$$
b\) Consider again $\Omega\in\{(0,\infty),{\mathbb{R}}\}$ and let $F(x)=1$. Moreover, let $\rho$ be $p$-admissible for $F$ and $h=0$ (which does not depend on $p$ by Remark \[remark\] b)). We then obtain the classical translation semigroup and Remark \[remark\] a) gives the well-known characterizations of chaos for this semigroup due to Matsui, Yamada, and Takeo [@MaYaTa03; @MaYaTa04] and deLaubenfels and Emamirad [@deLaEm01], respectively.
c\) Consider $\Omega=(0,1)$ and let $F(x)=-x$. Then $\Omega$ is forward invariant for $F$. Additionally, let $h\in C(0,1)$ be such that ${\mbox{Re}\,}h$ is bounded. It follows again from the definition that $\rho=1$ is $p$-admissible for $F$ and $h$ for every $1\leq p<\infty$. Thus, we obtain a well-defined $C_0$-semigroup $T_{-id,h}$ on $L^p(0,1)$. If $h$ is bounded the generator of this semigroup in $L^p(\Omega)$ is given by $$A_p:\{f\in L^p(0,1);xf'(x)\in L^p(0,1)\}\rightarrow L^p(\Omega), A_pf(x)=-xf'(x)+h(x)f(x),$$ where $f'$ denotes again the distributional derivative of $f$ (see e.g. [@ArKaMa13 Theorem 15]).
If $x\mapsto\frac{{\mbox{Im}\,}h(x)}{x}\in L^1(0,\beta)$ for all $\beta\in (0,1)$ or if $x\mapsto\frac{{\mbox{Im}\,}h(x)}{x}\in L^1(\beta,1)$ for all $\beta\in (0,1)$, by Theorem \[simplified chaos\] this $C_0$-semigroup is chaotic on $L^p(\Omega)$ precisely when for some $x\in(0,1)$ $$\int_{(0,1)}\exp(-p\int_x^w\frac{{\mbox{Re}\,}h(y)}{-y}dy)d\lambda(w)<\infty.$$ Because of $$\exp(p\int_x^w\frac{{\mbox{Re}\,}h(y)}{y}dy)=\Big(\frac{w}{x}\Big)^{p{\mbox{Re}\,}h(0)}\exp(p\int_x^w\frac{{\mbox{Re}\,}h(y)-{\mbox{Re}\,}h(0)}{y}dy)$$ this generalizes a result of Dawidowicz and Poskrobko [@DaPo] who showed that in case of a real valued $h\in C[0,1]$ for which $x\mapsto\frac{h(x)-h(0)}{x}\in L^1(0,1)$ the above semigroup is chaotic on $L^p(0,1)$ if and only if $h(0)>-1/p$.
d\) Consider $\Omega=(0,1)$ and $F(x)=-x^3\sin(\frac{1}{x})$. Because we have $\lim_{x\rightarrow 0}F(x)= 0$ and $\lim_{x\rightarrow 1}F(x)\leq 0$ it follows that $\Omega$ is forward invariant under $F$ and since $F'$ is bounded $\rho=1$ is $p$-admissible for $F$ and $h=0$ for every $1\leq p<\infty$. Thus, $T_F$ is a well-defined $C_0$-semigroup on $L^p(0,1)$. By [@ArKaMa13 Theorem 15] its generator is $$\begin{aligned}
&&A_p:\{f\in L^p(0,1);\,-x^3\sin(\frac{1}{x})f'(x)\in L^p(0,1)\}\rightarrow L^p(0,1),\\
&&A_pf(x)=-x^2\sin(\frac{1}{x})f'(x)\end{aligned}$$ where $f'$ denotes the distributional derivative of $f$. By Remark \[remark\] it follows that this $C_0$-semigroup is chaotic on $L^p(0,1)$ for every $1\leq p<\infty$.
Weighted composition $C_0$-semigroups on Sobolev spaces
=======================================================
For a bounded interval $(a,b)$, let $F\in C^1[a,b]$ with $F(a)=0$ be such that $(a,b)$ is forward invariant under $F$, and let $h\in W^{1,\infty}[a,b]$ be such that
- $\forall\,x\in\{F=0\}:\,h(x)=h(a)\in{\mathbb{R}}$,
- the function $[a,b]\rightarrow{\mathbb{R}}, y\mapsto\frac{h(y)-h(a)}{F(y)}$ belongs to $L^\infty[a,b]$.
In [@ArKaMa13] it is shown that under the above hypothesis the operator $$A_p:\{f\in W^{1,p}[a,b];\, Ff''\in L^p[a,b]\}\rightarrow W^{1,p}[a,b], A_pf=Ff'+hf,$$ where the derivatives are taken in the distributional sense, is the generator of a $C_0$-semigroup $S_{F,h}$ on $W^{1,p}[a,b]\, (1\leq p<\infty)$ which is given by $$\forall\,t\geq 0, f\in W^{1,p}[a,b]:\,S(t)f(x)=h_t(x)f(\varphi(t,x)).$$ Moreover, it is shown in [@ArKaMa13] that this $C_0$-semigroup $S_{F,h}$ is never hypercyclic on $W^{1,p}[a,b]$. In particular, $S_{F,h}$ cannot be chaotic on $W^{1,p}[a,b]$.
Because of $F(a)=0$, the closed subspace $$W^{1,p}_*[a,b]:=\{f\in W^{1,p}[a,b];\,f(a)=0\}$$ of $W^{1,p}[a,b]$ is invariant under $S_{F,h}$ such that the restriction of $S_{F,h}$ to $W^{1,p}_*[a,b]$ defines a $C_0$-semigroup on $W^{1,p}_*[a,b]$ which we denote again by $S_{F,h}$. Its generator is given by $$A_{p,*}:\{f\in W^{1,p}_*[a,b];\, Ff''\in L^p[a,b]\}\rightarrow W^{1,p}[a,b], A_{p,*}f=Ff'+hf,$$ see [@ArKaMa13]. Using Theorem \[simplified chaos\] we derive the following characterization of chaos for $S_{F,h}$ on $W^{1,p}_*[a,b]$.
\[simplified in Sobolev\] Let $(a,b)$ be a bounded interval, $F\in C^1[a,b]$ with $F(a)=0$ such that $(a,b)$ is forward invariant under $F$. Moreover, let $h\in W^{1,\infty}[a,b]$ be such that
- $\forall\, x\in\{F=0\}:\,h(x)=h(a)\in{\mathbb{R}}$,
- the function $[a,b]\rightarrow{\mathbb{C}}, y\mapsto\frac{h(y)-h(a)}{F(y)}$ belongs to $L^\infty[a,b]$.
Then, for the $C_0$-semigroup $S_{F,h}$ on $W^{1,p}_*[a,b]$ the following are equivalent.
- $S_{F,h}$ is chaotic.
- $\lambda(\{F=0\})=0$ and for every connected component $C$ of $(a,b)\backslash\{F=0\}$ $$\int_C\exp(-p\int_x^w\frac{F'(y)+ h(a)}{F(y)}dy)d\lambda(w)<\infty$$ for some/all $x\in C$.
Observe that by the boundedness of $F'$ on $[a,b]$ $\rho=1$ is $p$-admissible for $F$ and $F'+h(a)$ for any $1\leq p<\infty$. Under the above hypothesis 1) and 2) it is shown in [@ArKaMa13 Theorem 20 and Proposition 24] that the $C_0$-semigroups $S_{F,h}$ on $W^{1,p}_*[a,b]$ and $T_{F,F'+h(a)}$ on $L^p[a,b]$ are conjugate, i.e. there is a homeomorphism $\Phi:L^p[a,b]\rightarrow W^{1,p}_*[a,b]$ such that $S_{F,h}(t)\circ\Phi=\Phi\circ T_{F,F'+h(a)}(t)$ for every $t\geq 0$. By the so-called Comparison Principle (see e.g. [@GEPe11 Proposition 7.7]) it follows that $S_{F,h}$ is chaotic on $W^{1,p}_*[a,b]$ if and only if $T_{F,F'+h(a)}$ is chaotic on $L^p[a,b]$. Thus, an application of Theorem \[simplified chaos\] proves the theorem.
a\) We consider $(a,b)=(0,1)$ and $F(x)=-x$. Then, $(0,1)$ is forward invariant under $F$. For every $h\in W^{1,\infty}[0,1]$ with $h(0)\in{\mathbb{R}}$ and $$[0,1]\rightarrow{\mathbb{C}},y\mapsto\frac{h(y)-h(0)}{y}\in L^\infty[0,1]$$ the operator $$A:\{f\in W^{1,p}_*[a,b];\, xf''(x)\in L^p[a,b]\}\rightarrow W^{1,p}[a,b], Af(x)=-xf'(x)+h(x)f(x),$$ generates a $C_0$-semigroup on $W^{1,p}_*[0,1], 1\leq p<\infty$. By Theorem \[simplified in Sobolev\] this semigroup is chaotic on $W^{1,p}_*[0,1]$ if and only if for some $x\in (0,1]$ $$\int_{[0,1]}\Big(\frac{w}{x}\Big)^{p(h(0)-1)}d\lambda(w)=\int_{[0,1]}\exp(-p\int_x^w\frac{-1+ h(0)}{-y}dy)d\lambda(w)<\infty$$ which holds precisely when $p(h(0)-1)>-1$, i.e. when $h(0)>1-\frac{1}{p}$ (see also [@ArKaMa13 Theorem 27]).
b\) Let again $(a,b)=(0,1)$. We consider $F(x)=-x(1-x)$ so that $(0,1)$ is forward invariant under $F$. For each $h\in W^{1,\infty}[0,1]$ with $h(0)=h(1)\in{\mathbb{R}}$ and $$[0,1]\rightarrow{\mathbb{C}},y\mapsto\frac{h(y)-h(0)}{y(1-y)}\in L^\infty[0,1]$$ the operator $$\begin{aligned}
&&A:\{f\in W^{1,p}_*[a,b];\, x(1-x)f''(x)\in L^p[a,b]\}\rightarrow W^{1,p}[a,b],\\
&&Af(x)=-x(1-x)f'(x)+h(x)f(x),\end{aligned}$$ generates a $C_0$-semigroup on $W^{1,p}_*[0,1], 1\leq p<\infty$. Since for any $x\in (0,1)$ the function $$w\mapsto\exp(-p\int_x^w\frac{F'(y)-h(0)}{F(y)}dy)=w^{-p(1+h(0))}(1-w)^{-p(1-h(0))}(1-x)^{p(1-h(0))}x^{p(1+h(0))}$$ does not belongs to $L^1(0,1)$ for any value of $h(0)$ it follows from Theorem \[simplified in Sobolev\] that this semigroup is not chaotic.
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[Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany]{}
[*E-mail address: [email protected]*]{}
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abstract: 'This paper is to introduce a new software called CBwaves which provides a fast and accurate computational tool to determine the gravitational waveforms yielded by generic spinning binaries of neutron stars and/or black holes on eccentric orbits. This is done within the post-Newtonian (PN) framework by integrating the equations of motion and the spin precession equations while the radiation field is determined by a simultaneous evaluation of the analytic waveforms. In applying CBwaves various physically interesting scenarios have been investigated. In particular, we have justified that the energy balance relation is indeed insensitive to the specific form of the applied radiation reaction term. By studying eccentric binary systems it is demonstrated that circular template banks are very ineffective in identifying binaries even if they possess tiny residual orbital eccentricity. In addition, by investigating the validity of the energy balance relation we show that, on contrary to the general expectations, the post-Newtonian approximation should not be applied once the post-Newtonian parameter gets beyond critical value $\sim{\textcolor{black}{ 0.08}}-0.1$. Finally, by studying the early phase of the gravitational waves emitted by strongly eccentric binary systems—which could be formed e.g.in various many-body interactions in the galactic halo—we have found that they possess very specific characteristics which may be used to identify these type of binary systems.'
author:
- |
, Gergely Debreczeni[^1], István Rácz[^2] and Mátyás Vasúth[^3]\
WIGNER RCP, RMKI\
H-1121 Budapest, Konkoly Thege Miklós út 29-33.\
Hungary
title: 'Gravitational waves from spinning eccentric binaries[^4]'
---
Introduction
============
The advanced versions of our current ground based interferometric gravitational wave observatories such as LIGO [@advligo] and Virgo [@advvirgo] are to do the first direct detection soon after their restart in 2015. It is also expected that these detectors observe yearly tens of gravitational wave signals emitted during the final inspiral and coalescence of compact binaries composed by neutron stars and low mass black holes. The identification of these type of sources is attempted to be done by making use of the matched filtering technique [@matchedfilter] where templates are deduced by using various type of theoretical assumptions. Among the physical quantities characterizing the waveforms and the evolution of binaries the time dependence of the wave amplitude and the orbital phase are of critical importance. In determining the amplitude and the phase, along with several other astrophysical properties of the sources, we need to apply templates which are sufficiently accurate and cover the largest possible parameter domain associated with the involved binaries.
The main purpose of the present paper is to introduce a new computational tool, called *CBwaves*, by the help of which the construction of gravitational wave templates for the generic inspiral of compact binaries can be done in a fast and accurate way. our principal aim was to follow generic configurations of spinning and eccentric binaries with arbitrary orientation of spins and arbitrary value of the eccentricity. In carrying out this program the post-Newtonian framework [@Blanchet06] has been applied—by making use of the analytic setup [[@Kidder; @MoraWill; @IyerWill; @WillPNSO; @WangWill; @GopaIyer; @Arun08; @FBB; @FBB2]]{}—, i.e.by integrating the .5PN accurate equations of motion and the spin precession equations of the orbiting bodies while the radiation field is determined by a simultaneous evaluation of the analytic waveforms which involves contributions up to PN order. The equations of motion are integrated by a fourth order Runge-Kutta method numerically. The most important input parameters are the initial separation, the masses, the spins, along with their orientations, of the involved bodies and the initial eccentricity of the orbit. The waveforms are calculated in time domain and they can also be determined in frequency domain by using the implemented FFT. Moreover, CBwaves does also provide the expansion of the radiation field in $s=-2$ spin weighted spherical harmonics.
In the post-Newtonian approach various order of corrections are added to the Newtonian motion where the fundamental scale of the corrections are determined by the post-Newtonian parameter $\epsilon\sim(v/c)^2\sim Gm/(rc^2)$, where $m$, $v$ and $r$ are the total mass, orbital velocity and separation of the binary system. Within this framework the damping terms are considered to be responsible for the change of the motion of the sources in consequence of the radiation of gravitational waves to infinity. Gravitational radiation reaction—sometimes called to be Newtonian radiation reaction—appears first at 2.5PN order, i.e., this correction is of the order $\epsilon^{\frac52}$. It is known for long that the relative acceleration term appearing in the radiation reaction expressions is not unique. It is however argued by various authors (see, e.g.[@IyerWill]) that the energy balance relation has to be insensitive to the specific form of the applied radiation reaction term. By making use of our CBwaves code the effect of the two standard radiation reaction terms could be compared. We have found that the energy balance relations are indeed insensitive to the specific choices of the parameters in the radiation reaction terms provided that a suitable coordinate transformation is applied while switching between gauge representations.
[It is expected that whenever the time scales of both the precession and shrinkage of the orbits of the investigated binaries are long, when they are compared to the orbital period, the adiabatic approximation is appropriate. This, in the particular case of quasi-circular inspiral orbits, means that they are expected to be correctly approximated by nearly circular ones with a slowly shrinking radius (see for e.g. Section IV in [@Kidder]). We investigated the validity of the adiabatic approximation by monitoring the rates of inspiral of the adiabatic approximation and of a corresponding time evolution yielded by CBwaves. ]{}
that a large fraction of binaries emitting gravitational wave signals detectable by our detectors are expected to have orbits with non-negligible eccentricity. Immediate examples are black hole binaries which may be formed by tidal capture in globular clusters or galactic nuclei [@wen; @kocsis]. Although a circularization of these orbits will happen by radiation reaction as the circularization process is slow a tiny residual eccentricity may remain. it is important to in what extent such a residual eccentricity may affect the detection performance of matched-filtering. that circular template banks are very ineffective in identifying binaries. It is also important to be emphasized that by using templates based on the circular motion of binaries exclusively signal-to-noise-ratio (SNR), which in turn downgrades the performance of our detectors.
The expectations concerning the applicability of the post-Newtonian expansion are based on the assumption that the time scales of precession and shrinkage are both long compared to the orbital period until the very late stage of the evolution. On contrary to this claim there are more and more evidences showing that the post-Newtonian approximation should not taken seriously once the post-Newtonian parameter reaches the parameter domain $\epsilon\sim
0.08-0.1$. By investigating the validity of the energy balance relation we show that indeed it gets to be violated as soon as the post-Newtonian parameter gets to be close to the critical upper bound $0.1$. Note that these findings are in accordance with the claims of [@janna] where the relative significance of the higher order contributions was monitored. What is really unfavorable is that this sort of loss of accuracy is getting to be more and more significant right before reaching the frequency ranges of our current ground based GW detectors.
It is that in consequence of many-body interactions strongly eccentric black hole binaries are formed in the halo of the galactic supermassive black hole. The early phase of the gravitational waves emitted by these type of strongly eccentric binary systems possesses burst type character. It is true that the amplitude and the frequency of the gravitational waves emitted by such systems change significantly during the inspiral due to the circularization effect. Nevertheless, we have found that the frequency-domain waveforms of the early phase of highly eccentric binary systems possess very specific characteristics which may suffice to determine the physical parameters of the system.
This paper is organized as follows. In Section \[motion\] some of the basics of the analytic setup will be summarized. In Section \[CBwaves\] a short introduction of the CBwaves software is given. Section \[results\] is to present our main results. Subsection \[non-spin\] deals with non-spinning circular waveforms, while in subsection \[rad-reac\] the gauge dependence of the radiation reaction is examined. A short discussion in subsection \[dom-val\] on the range of applicability of the PN approximation is followed by the investigation of eccentric motions in subsection \[ecc-motion\] introducing the found universalities in the evolution of eccentricity, along with our justification of the loss of SNR whenever tiny residual eccentricities are neglected by applying circular waveforms exclusively. Subsection \[spinning-binaries\] is to introduce our finding regarding spinning and eccentric binaries while section \[Summary\] contains our final remarks.
The motion and radiation of the binary system {#motion}
=============================================
As mentioned above in detecting low mass black hole or neutron star binaries we need to have template banks built up by sufficiently accurate waveforms which cover the largest possible parameter domain associated with the involved binaries. In determining the motion of the bodies and the yielded waveforms the analytic setup [[@Kidder; @MoraWill; @IyerWill; @WillPNSO; @WangWill; @GopaIyer; @Arun08; @FBB; @FBB2]]{} is applied.
In the post-Newtonian formalism [@Blanchet06] the spacetime is assumed to be split into the near and wave zones. The field equations for the perturbed Minkowski metric is solved in both regions. In the near zone the energy-momentum tensor is nonzero and retardation is negligible (post-Newtonian expansion), while in the wave zone the vacuum Einstein equation are solved (post-Minkowskian expansion). In the overlap of these regions the solutions are matched to each other. As a result of the process the radiation field far from the source is expressed in terms of integrals over the source (the source multipole moments). For the special case of compact binary systems the source integrals are evaluated with the point mass assumption and this hypothesis leads to various regularization issues and ambiguities already at 3PN, see e.g. [@Blanchet04; @DJS01]. the analytic setup of the present work was chosen such that the motion of the binary is taken into account only up to .5PN order while the radiation field up to PN order.
In harmonic coordinates the radiation field $h_{ij}$ far from the source is decomposed as [@Kidder] $$h_{ij}=\frac{2G\mu }{c^{4}D}\left(
Q_{ij}+P^{0.5}Q_{ij}+PQ_{ij}+PQ_{ij}^{SO}+P^{1.5}Q_{ij}+P^{1.5}Q^{SO}_{ij}
{\textcolor{black}{ + P^{2}Q_{ij}+P^{2}Q_{ij}^{SS}}} \right), \label{Wform}$$where $D$ is the distance to the source and $\mu=m_{1}m_{2}/(m_{1}+m_{2})$ is the reduced mass of the system. We have collected the relevant terms up to 1.5PN order. $Q_{ij}$ is the quadrupole (or Newtonian) term, $P^{0.5}Q_{ij}$, $PQ_{ij}$ $P^{1.5}Q_{ij}$ are higher order relativistic corrections, $PQ^{SO}_{ij}$ $P^{1.5}Q^{SO}_{ij}$ are spin-orbit terms. Note that the 1.5PN order contributions to the waveform due to wave tails—which depend on the past history of the binary [@W93; @Blanchet08]— neglected. The detailed expressions of the contributions are given in [@Kidder; @WW], and for they are also summarized in Appendix A.
For a plane wave traveling in the direction $\mathbf{\hat{N}}$, which is a unite spatial vector pointing from the center of mass of the source to the observer, the transverse-traceless (TT) part of the radiation field is given as [@Maggiore] $$h_{ij}^{TT}=\Lambda _{ij,kl}\,h_{kl}\,, \label{TTproj}$$ where $$\Lambda _{ij,kl}(\mathbf{\hat{N}})=P_{ik}P_{jl}-\frac{1}{2}P_{ij}P_{kl},\ \ \ {\rm and}\ \ \
P_{ij}(\mathbf{\hat{N}})=\delta _{ij}-N_{i}N_{j}\,. \label{TTproj2}$$
Following [@LSCCommon] an orthonormal triad, called the radiation frame, is chosen as $$\begin{aligned}
\mathbf{\hat{N}} &=&\mathbf{(}\sin \iota \cos \phi ,\sin \iota \sin
\phi,\cos \iota \mathbf{),} \label{trafo1} \\
\mathbf{\hat{p}} &=&(\cos \iota \cos \phi ,\cos \iota \sin \phi
,-\sin \iota), \label{trafo2} \\
\mathbf{\hat{q}} &=&(-\sin \phi ,\cos \phi ,0) \label{trafo3}\end{aligned}$$where the polar angles $\iota $ and $\phi $, determining the relative orientation of the radiation frame with respect to the source frame $(\mathbf{\hat{x}},\mathbf{\hat{y}},\mathbf{\hat{z}})$ as indicated on Fig.\[SourceFrame\]. The basis vectors of the source frame $\mathbf{\hat{x}}$ and $\mathbf{\hat{y}}$ are supposed to span the initial orbital plane such that $\mathbf{\hat{x}}$ and $\mathbf{\hat{z}}$ are parallel to the separation vector $\mathbf{r}=\mathbf{x_{1}}-\mathbf{x_{2}}$ and the Newtonian part of the angular momentum $\mathbf{L}_{N}=\mu \mathbf{r}\times\mathbf{\dot r}$, both at the beginning of the orbital evolution, respectively. According to the particular relations given by Eqs.(\[trafo1\])-(\[trafo3\]) the source and radiation frames can be transformed into each other simply by two consecutive rotations. Rotating first the radiation frame around the $\mathbf{\hat{z}}$ axis by angle $-\phi$ which is followed by a rotation around the $\mathbf{\hat{y}}$ axis by the angle $-\iota$.
![The relative orientation of the main vectors characterizing the binary with respect to the source frame $(\mathbf{\hat{x}},\mathbf{\hat{y}},\mathbf{\hat{z}})$, along with the polar angles $\iota $ and $\phi $ of $\mathbf{\hat{N}}$, are shown. This figure also indicates the relative orientation of the total and the orbital angular momentum vectors, $\mathbf{J}$ and $\mathbf{L}$, and that of the individual spin vectors $\mathbf{S}_1$ and $\mathbf{S}_2$ with respect to $\mathbf{J}$ by indicating the angles $\vartheta_i=\cos^{-1}(\mathbf{J} \cdot
\mathbf{S}_i/\|\mathbf{J}\|\|\mathbf{S}_i\|)$, where $i$ takes the values $1,2$.[]{data-label="SourceFrame"}](Fig/SourceFrame.pdf){width="9cm"}
The polarization states can be given, with respect to the orthonormal radiation frame $(\mathbf{\hat{N},\hat{p},\hat{q})}$, as [@FC] $$h_{+}=\frac{1}{2}\left(
\hat{p}_{i}\hat{p}_{j}-\hat{q}_{i}\hat{q}_{j}\right)
h_{ij}^{TT},\quad h_{{\times }}=\frac{1}{2}\left( \hat{p}_{i}\hat{q}_{j}+\hat{q}_{i}\hat{p}_{j}\right) h_{ij}^{TT}. \label{pstatesgeneral}$$ In the applied linear approximation the strain produced by the binary system at the detector can be given as the combination $$h(t)=F_{+}h_{+}(t)+F_{\times }h_{\times }(t),$$where the antenna pattern functions $F_{+}$ and $F_{\times }$ are given as $$\begin{aligned}
F_{+} &=&-\frac{1}{2}\left( 1+\cos ^{2}\theta \right) \cos 2\varphi
\cos
2\psi -\cos \theta \sin 2\varphi \sin 2\psi , \\
F_{\times } &=&\frac{1}{2}\left( 1+\cos ^{2}\theta \right) \cos
2\varphi \sin 2\psi -\cos \theta \sin 2\varphi \cos 2\psi\end{aligned}$$for ground-based interferometers, with Euler angles $\theta ,\varphi
$ and $\psi$ relating the radiation frame and the detector frame $(\mathbf{x},\mathbf{y},\mathbf{z})$ as it is indicated on Fig.\[DetectorFrame\]. Note that in spite of the fact that the polarization states do depend on the choice made for the triad elements $\mathbf{\hat{p}}$ and $\mathbf{\hat{q}}$, due to the compensating rotation in the polarization angle $\psi$, the strain measured at a detector remains intact. The binary is called to be optimally oriented, i.e.the detector sensitivity is maximal, if $\theta =0$ or $\pi $, and the angle of inclination $\iota =0$.
![The relative orientation of the detector frame $(\mathbf{{x}},\mathbf{{y}},\mathbf{{z}})$—the $\mathbf{{x}}$ and $\mathbf{{y}}$ axes are aligned with the detector arms—and the radiation frame $(\mathbf{\hat{N},\hat{p},\hat{q})}$, along with the polar angles $\theta $ and $\varphi$, are shown. []{data-label="DetectorFrame"}](Fig/DetectorFrame.pdf){width="7cm"}
In determining the radiation field far from the source, i.e.the evaluation of all the general expressions in Eq.(\[Wform\]), one needs to know the precise motion of the bodies composing the binary system. In the generic case the orbit of the binary system acquires precession (due to spin effects) and shrinking (due to radiation reaction) during the time evolution. In the adiabatic approach, applied in the post-Newtonian setup, it is assumed that the time scales of precession and shrinkage are both long compared to the orbital period until the very late stage of evolution. The acceleration of the reduced one-body system follows from the conservation of the energy momentum (the geodesic equation in the perturbed spacetime with harmonic coordinates) [@Kidder], $$\mathbf{a}=\mathbf{a}_{N}+\mathbf{a}_{PN}+\mathbf{a}_{SO}+\mathbf{a}_{2PN}+\mathbf{a}_{SS}+\mathbf{a}_{RR}^{BT} {\textcolor{black}{+ \mathbf{a}_{PNSO} +
\mathbf{a}_{3PN} + \mathbf{a}_{RR1PN} + \mathbf{a}_{RRSO} +
\mathbf{a}_{RRSS} }}, \label{accel}$$where $\mathbf{a}_{N}$, $\mathbf{a}_{PN}$, $\mathbf{a}_{SO}$, $\mathbf{a}_{2PN}$, $\mathbf{a}_{SS}$ and $\mathbf{a}_{RR}^{BT}$ are the Newtonian, first post-Newtonian, spin-orbit, second post-Newtonian, spin-spin and radiation reaction parts of the acceleration, respectively [@TOO; @FBB] The analytic form of these contributions are given in terms of the kinematic variables in Appendix B \[see equations (B.1)-(B.)\].
Using the CBwaves software {#CBwaves}
==========================
The software package contains man pages, a readme file and it consists of a single executable file. The source, the i686 and x86\_64 binary packages can be downloaded from the homepage of the RMKI Virgo Group [@cbwavesdownload]. The parameters necessary for the determination of the initial conditions are passed by a human readable and editable configuration file. In order to automatize and make the mass production of this type of configuration files possible a generator script is also included. The comprehensive list of all the possible configuration file parameter and their detailed explanation can be found in the man page. The most important input parameters are the initial separation $\mathbf{r}=\mathbf{x}_1-\mathbf{x}_2=r\hat{\mathbf{n}}$, the masses $m_i$, the magnitude $s_i$ of the specific spin vector $\mathbf{s}_i$ and the initial eccentricity $e$. Note that instead of the individual spin vectors $\mathbf{S}_1$ and $\mathbf{S}_2$ the specific spin vectors $\mathbf{s}_i$—defined by the relations $\mathbf{S}_i=\mathbf{s}_i m_i^2$—are applied. The magnitude $s_i$ of the specific spin vectors $\mathbf{s}_i=(s_{ix},s_{iy},s_{iz})$ are $s_i=\sqrt{s_{ix}^2+s_{iy}^2+s_{iz}^2}$ and it is assumed that $0<s_i<1$ for a black hole while $0<s_i<0.7$ for most neutron star models [@Kidder].
Since simple analytic expressions are available, the determination of the initial values of various parameters is performed with high precision in case of circular orbits. The situation is different for eccentric orbits, where the initial speed of the bodies is determined iteratively by successive approximation to ensure that the orbit possesses the required eccentricity after the first half orbit in its orbital evolution. The currently implemented approximation is set to yield initial data for the eccentricity with $0.01 \%$ precision but the accuracy can be increased to any desirable value of precision.
It is also worth to be mentioning that in order to make the submission of the software to research clusters straightforward we provide a Condor [@condor] job description file generator script, as well. All these above listed features make this code an easy-to-use, fully fledged gravitational wave generator, which already produced some very interesting and promising results discussed in the following sections.
Results
=======
Since analytic formulas are available for the motion and radiation of circular non-spinning binaries, it was straightforward to start our investigation with these systems, and focus our attention to more complicated configurations in the succeeding subsections.
Non-spinning, circular waveforms {#non-spin}
--------------------------------
We started our studies by constructing a non-spinning, circular waveform template bank to serve as a reference for the forthcoming investigations. For an immediate example of such a circular orbit with non-spinning bodies and for the pertinent emitted waveform with a slow rise in the amplitude and frequency see Fig.\[nonspin-circular\].
-- --
-- --
The detection pipelines based on the matched-filter methods are primarily interested in the frequency domain representation and phase/amplitude evolution of the waveforms both of which can easily be obtained by making use of CBwaves. The spectral distribution of such templates for various masses and distances are shown on Fig.\[circspectra\] with respect to the design sensitivity curve of future interferometric gravitational wave detectors such as Advanced LIGO [@advligo], Advanced Virgo [@advvirgo] and the Einstein Telescope [@etsens]. Note that the time interval while the corresponding source will be visible by these detectors increase significantly as the sensitivity is improved.
-0.3cm
[c]{}
-0.3cm
Recall that the simplest analytic models, like the stationary phase approximation (see e.g. [@SPA]), provide estimates for the amplitude evolution of the gravitational wave spectra as a function of frequency proportional to $\sim f^{-7/6}$ . Note that the trustful part of the spectrum of the 1.4 $M_{\odot}$ - 1.4 $M_{\odot}$ NS binary system (shown on Fig. \[circspectra\]) yields a fit of the form $|\tilde{h}(f)| = 1.019\cdot
10^{-19}*f^{-1.205}$, where the value $1.205$ in the exponent is a bit larger but it is in 3$\%$ agreement with the analytically derived value $7/6$.
{#non-spin-adiabb}
$$\label{adiab}
{\textcolor{black}{\frac{d{r}_{ad}}{dt}=\frac{{dE}/{dt}}{{dE}/{dr}}\,,}}$$ [On Fig. \[adiabradius\] the corresponding ratios are plotted for both using the approximation considered in Section IV in [@Kidder] and the approximation involving all the PN terms implemented in CBwaves. According to the graphs on Fig.\[adiabradius\] it is visible that in both of the monitored cases the adiabatic approximation yields almost the same (less then $5\%$) decrease of the rate of inspiral than the corresponding time evolution if we used the highest possible PN orders implemented in CBwaves. The graphs of these ratios $({d{r}_{inst}}/{dt})/({d{r}_{ad}}/{dt})$ make also transparent the slight improvements related to use of higher PN orders.]{}
![[]{data-label="adiabradius"}](Fig/adiabatic.pdf){width="75.00000%"}
The gauge freedom in the radiation reaction term {#rad-reac}
------------------------------------------------
The radiation reaction is determined by assuming that the energy radiated to infinity is balanced by the an equivalent loss of energy of the binary system. It is know for long that the relative acceleration term appearing in the radiation reaction expressions is not unique. Indeed, it was already shown in [@IyerWill] that at 2.5PN order there is a two-parameter family of freedom in specifying the radiation reaction terms such that for any choice of these two parameters the loss of energy and angular momentum is in accordance with the quadrupole approximation of energy and angular momentum fluxes. This freedom corresponds to possible coordinate transformations at 2.5PN and it represent a residual gauge freedom that is not fixed by the energy balance method. In spite of the two-parameter freedom in the literature two specific choices of the radiation reaction terms are applied. One of them was derived from the Burke-Thorne radiation reaction potential [@Kidder; @BT1; @BT2] $$\mathbf{a}_{RR}^{BT} = {\frac{8}{5}}\eta
{\frac{G^{2}m^{2}}{c^{5}r^{3}}}\left\{
\dot{r}\mathbf{\hat{n}}\left[ 18v^{2}+{\frac{2}{3}}{\frac{Gm}{r}}-25\dot{r}^{2}\right] -\mathbf{v}\left[
6v^{2}-2{\frac{Gm}{r}}-15\dot{r}^{2}\right] \right\}\,, \label{aRRBT}$$ while the other is the Damour-Deruelle radiation reaction formula [@DDPLA; @MoraWill] $$\mathbf{a}_{RR}^{DD} ={\frac{8}{5}}\eta
{\frac{G^{2}m^{2}}{c^{5}r^{3}}}\left\{ \dot{r}\mathbf{\hat{n}}\left[
3v^{2}+{\frac{17}{3}}{\frac{Gm}{r}}\right] - \mathbf{v}\left[
v^{2}+3{\frac{Gm}{r}}\right] \right\}\,.$$ Despite of the explicit functional differences in these two radiation reaction terms—based on the above recalled argument ending up with the gauge freedom in determining the motion of the bodies—it is held (see, e.g.[@IyerWill]) that the energy balance relation has to be insensitive to the specific form of the applied radiation reaction term. To check the validity of this claim we implemented both of these radiation reaction terms in CBwaves.
We have found that whenever a suitable coordinate transformation of the form $$\label{trafo}
\mathbf{x}'=\mathbf{x} + \delta_{\mathbf{x}_{2.5PN}}\,,$$ (for the precise form of the correction term $\delta_{\mathbf{x}_{2.5PN}}$ see equations (22) and (23) of [@ZengWill]) and all the related implications are taken into account in the total energy expression $$E_{tot}=E_{N}+E_{PN}+E_{SO}+E_{2PN}+E_{SS}+E_{RR}\,, \label{Energy0}$$ where $E_{RR}$ stands for the radiation reaction term, i.e.the energy associated with the emitted gravitational wave, in accordance with the claims in [@IyerWill], the energy balance relations are insensitive to the specific choices of the parameters in the radiation reaction terms.
instead of the energy balance relation it is more informative to consider the evolution of the fractional energy of the system $$\frac{E_{tot}(f)}{E_0}\,, \label{FracEnergy0}$$ where $E_0$ denotes the initial value of the total energy $E_0=E_{tot}(f_{low})$. Thus, the fractional energies $E^{BT}_{tot}/E_0$ and $E^{DD}_{tot}/E_0$ correspond to the alternative use of the radiation reaction terms in Eq.(\[Energy0\]) proposed by Burke-Thorne and by Damour-Deruelle. On Fig.\[RRformula\] the frequency dependence of the fractional energies $E^{BT}_{tot}/E_0$ and $E^{DD}_{tot}/E_0$ and that of the relative difference $\delta
E=(E^{BT}-E^{DD})/E^{BT}$ are plotted. It was found that $\delta E$ is less than $0.0001\%$ even at the frequency, around 380 Hz, where the energy balance relation becomes inaccurate as the error of both of the fractional energies $E^{BT}_{tot}/E_0$ and $E^{DD}_{tot}/E_0$ exceeds $~12 \%$ there for the simplest possible case where neither of the involved bodies have spin.
![The frequency dependence of the relative difference $\delta E=(E^{BT}-E^{DD})/E^{BT}$ is shown (yellow). In addition, the frequency dependence of the fractional energies $E^{BT}_{tot}/E_0$ and $E^{DD}_{tot}/E_0$ for a non-spinning binary on circular orbit with masses $m_1=m_2=1.4M_{\odot}$ are plotted. The location where the deviation of the fractional energies exceeds the value $10 \%$ is indicated by an upward pointing arrow. At that point $\epsilon\sim 0.05$.[]{data-label="RRformula"}](Fig/RRformula.pdf){width="75.00000%"}
Domain of validity of the PN approximation {#dom-val}
------------------------------------------
It is widely held within the post-Newtonian community that the applied approximations are reliable up to the frequency of the innermost stable circular orbit given as $f_{isco} =
c^3/(6\sqrt{6}\pi G\,m)$, where $c$ is the speed of light, $m = m_1
+ m_2$ is the total mass of the system while $G$ stands for the gravitational constant. As it was mentioned above all these expectations on the range of applicability of the post-Newtonian expansion is based on the assumption of adiabaticity, i.e.it is assumed that the time scales of precession and shrinkage are both long compared to the orbital period until the very late stage of the evolution.
On contrary to these expectations there are more and more indications that the post-Newtonian approximations leaves its range of applicability once the post-Newtonian parameter $\epsilon\sim
(v/c)^2\sim Gm/(rc^2)$ reaches the values $\epsilon\sim 0.08-0.1$. In particular, the graphs on Figs.\[circspectra\], on Fig.\[RRformula\], as well as, on the left panel of Figs.\[eccevolfig\] clearly justify that as soon as the value of the PN parameter gets to be close to $0.08-0.1$ a significant violation of the energy balance relation— via the fractional energy ${E_{tot}(f)}/{E_0}$ on these plots—starts to show up regardless whether the motion of the binary is as simple as being circular or complicated with the inclusion of spin(s) and/or eccentricity.
[As our conclusions are on contrary to the conventional expectations it is important to emphasize that the observed violation of the energy balance relation was found to be robust with respect to the variation of the parameters of the investigated binary systems. In addition, although the results are numerical ones, on the one hand, the convergence rate of the code was justified to be fourth order and, on the other hand, all the shown results are insensitive to the size of the applied time steps in the sense that the included figures yielded with the use of $dt=1/(256
kHz)$ are already identical to those with $dt=1/(16 kHz)$.]{}
Note [also that our]{} conclusions are in accordance with the claims of [@janna] where simply the relative significance of higher order contributions were monitored. It is of convincing that in [@janna] by inspecting merely the anomalous growth of the higher order PN contributions the same range of applicability with $\epsilon\lessapprox 0.08-0.1$ had been found.
It is worth to be mentioned here that the observed violation of the energy balance relation gets to be transparent only on the plots made in the frequency domain while by inspecting merely the corresponding time domain plots (see Fig.\[circspectra2\]) one might be ready to conclude that the energy balance relation holds almost the entire orbital evolution. In this respect it is important to be emphasized that from data analyzing point of view it is the frequency domain behavior what . In addition, it is really unfavorable that the observed loss of accuracy is getting to be more and more significant as we are approaching the sensitivity ranges of our current ground based GW detectors.
Eccentric motions {#ecc-motion}
-----------------
Due to radiation reaction the motion of eccentric binaries are expected to be circularized. By making use of CBwaves we examined the basic features of this circularization process and compared our findings to the pertinent results of the literature.
Before reporting about our pertinent results it is worth to be mentioned that there are several attempts (see, e.g. [@gpv3; @yunes; @MGS]), aiming to provide analytic expressions for the instantaneous value of the eccentricity. On contrary to our expectations neither of these analytic expressions were found to be satisfactory except for certain very narrow parameter intervals and in most of the cases these analytic expressions yielded completely inconsistent values everywhere else. We have found, however, that the simplest possible geometric definition of the eccentricity, referring to the main characteristic parameters of the orbit—i.e., the minimal and maximal separations of the bodies—, always yields completely satisfactory result. This geometric, although not instantaneous, eccentricity is defined as $$e = \frac{r_{max}-r_{min}}{r_{max}+r_{min}},$$ where $r_{max}$ and $r_{min}$ denote the maximum and minimum distances between the two masses, i.e.the distances at the succeeding ‘turning’ points.
### Frequency modulation
Let us start by investigating the evolution of a highly eccentric binary system with masses $m_1 = 1.4\ M_{\odot}$ and $m_2 = 4.4\
M_{\odot}$, and with initial eccentricity $e_{flow} = 0.8$ at initial frequency $f_{low} \approx 18\ \mathrm{Hz}$. It is clearly visible that due to non-negligible eccentricity the waveform suffers simultaneous amplitude and frequency modulations. On Fig.\[nospineccfig\] a short interval of the orbital evolution and pertinent waveform is shown for this eccentric binary system.
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![The orbital evolution (left panel) and the emitted waveform (right panel) of a non-spinning, eccentric binary system with masses $m_1 = 1.4\ M_{\odot}$ and $m_2 = 4.4\ M_{\odot}$, and with initial eccentricity $e_{flow} = 0.8$. The evolution starts at frequency $f_{low} \approx 18\ \mathrm{Hz}$. The color shading for the red and green curves is to indicate the passing of time.[]{data-label="nospineccfig"}](Fig/nospin-excentric-orbit.pdf "fig:"){width="45.00000%"} ![The orbital evolution (left panel) and the emitted waveform (right panel) of a non-spinning, eccentric binary system with masses $m_1 = 1.4\ M_{\odot}$ and $m_2 = 4.4\ M_{\odot}$, and with initial eccentricity $e_{flow} = 0.8$. The evolution starts at frequency $f_{low} \approx 18\ \mathrm{Hz}$. The color shading for the red and green curves is to indicate the passing of time.[]{data-label="nospineccfig"}](Fig/nospin-excentric.pdf "fig:"){width="50.00000%"}
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### Evolution of the eccentricity
The gravitational wave detection pipelines are mainly using circular waveform templates. This approach is based on the assumption that the binaries are circularized quickly so that by the time the emitted GWs enter the lower part of the frequency band of the detectors the orbits are—with good approximation—circular. Therefore it is crucial to check the validity of this assumption within the post-Newtonian approximation implemented in CBwaves. The relevant figures are presented by the two panels of Fig.\[eccevolfig\]. On the left panel the evolution of the eccentricity as a function of frequency for such a binary NS system is shown. For various total masses a comparison to the analytic formula $$\begin{aligned}
e=e_0\cdot\chi^{-\frac{19}{18}}\cdot\Big(
1+\frac{3323}{1824}\,e_0^2\,(1-\chi^{-\frac{19}{9}}) +
\frac{15994231}{6653952}\,e_0^4\,\Big(1-\frac{66253974}{15994231}\,\chi^{-\frac{19}{9}}
+ \nonumber \\
+ \frac{50259743}{15994231}\,\chi^{-\frac{38}{9}}\Big) +
\frac{105734339801}{36410425344}\,e_0^6\,\Big(1-\frac{1138825333323}{105734339801}\,
\chi^{-\frac{19}{9}} + \nonumber \\
+ \frac{2505196889835}{105734339801}\,\chi^{-\frac{38}{9}}
- \frac{1472105896313}{105734339801}\,\chi^{-\frac{19}{3}}\Big)\Big)
\label{eccequation},\end{aligned}$$ derived in [@yunes]—see Eq.(3.11) therein—, where $\chi$ = $f/f_0$, i.e.the ratio of the instantaneous and initial frequency, while $e_0$ is the value of eccentricity at $f=f_0$, is also indicated on both Figs.\[eccevolfig\] and \[eccevolscan\].
Our numerical findings justify that the evolution of the eccentricity as a function of frequency—at least in the early phase—can be perfectly explained by the analytic estimates of [@yunes]. However, it is important to note that towards the end of the evolution in spite of the fact that the post-Newtonian expansion parameter is still much below the critical upper bound $\sim 0.08-0.1$, where the PN approximation is supposed to be valid, there is a non-negligible difference between the numerical values and the analytical estimates. The explanation of this would deserve further investigations.
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![ The evolution of the eccentricity as a function of the frequency of a binary neutron star system with masses $m_1 = m_2 = 1.4\ M_{\odot}$, and with initial eccentricity $e_{flow} = 0.4$ at initial frequency $f_{low} \approx ![ The evolution of the eccentricity as a function of the frequency of a binary neutron star system with masses $m_1 = m_2 = 1.4\ M_{\odot}$, and with initial eccentricity $e_{flow} = 0.4$ at initial frequency $f_{low} \approx
0.5\ \mathrm{Hz}$. The fractional energy of the system (including radiated energy) and the value of the post-Newtonian expansion parameter $v^2 \approx m/r$ are shown. in order to indicate the range of the validity of the applied approximation in the late inspiral phase. [*Right panel:*]{} The evolution of the eccentricity as a function of the frequency for binary systems with various total masses. Here the post-Newtonian expansion parameter $v^2 \approx 0.5\ \mathrm{Hz}$. The fractional energy of the system (including radiated energy) and the value of the post-Newtonian expansion parameter $v^2 \approx m/r$ are shown. in order to indicate the range of the validity of the applied approximation in the late inspiral phase. [*Right panel:*]{} The evolution of the eccentricity as a function of the frequency for binary systems with various total masses. Here the post-Newtonian expansion parameter $v^2 \approx
m/r$ are also shown for each total mass in order to indicate the range of the validity of the PN approximation. []{data-label="eccevolfig"}](Fig/ecc-evol-energy-crop.pdf "fig:"){width="48.00000%"} m/r$ are also shown for each total mass in order to indicate the range of the validity of the PN approximation. []{data-label="eccevolfig"}](Fig/ecc-evol-crop.pdf "fig:"){width="48.00000%"}
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Notice, finally, that—as it is clearly transparent on the right panel of Fig.\[eccevolfig\]—the frequency dependence of the eccentricity is insensitive to the total mass or to the mass ratio. This behavior is one of the universal properties of the investigated eccentric binary systems.
-- --
-- --
### Signal losses caused by eccentricity
It is well know that the matched-filter method is the optimal one when searching for known signals in noisy data [@matchedfilter]. In practice this involves the set up of a so called template bank (a collection of theoretical gravitational waveforms), which could contain several hundreds of thousands of templates in the parameter space of the physical model. Then each element of this template bank is matched against the data. If the density of these templates is high enough (see, e.g.[@spacing]) and the templates are based on the same physical model as the signal, then it is straightforward to find the signal provided that it has sufficient strength (amplitude). The situation is different when the physical model behind the template bank differs from that of the signal the later is unknown or eccentric binary systems circular template bank.
$$\label{cs}
{\textcolor{black}{\mathcal{O}_{s,t} = \frac{ (s | t ) }{ \sqrt{(s | s )(t | t )} }\,,}}$$ $${\textcolor{black}{(a | b ) = 2 \int_{f_{min}}^{f_{max}}\frac{ \tilde{a}^*(f)\tilde{b}(f) + \tilde{a}(f)\tilde{b}^*(f)}{S_n(f)}df\,.}}$$
In ideal cases (when the ) the overlap is 1. It often occurs that some template bank give rise to higher value of overlaps with the signal than the ones with parameters. Because of this it is useful to the so called [*fitting factor*]{} [@ffactor] which is the maximum of the overlaps of all the element of the template bank with the expected signal. While for parameter estimation the overlap is the important quantity, for detection purposes the fitting factor has more relevance.
Since a considerably large fraction of binaries retain at least a tiny residual eccentricity during evolution it is important to ask in what extent this eccentricity may affect the detection performance of matched-filtering.
As a result of our pertinent investigations the overlap
It is clearly visible that tiny residual eccentricities lead to considerably large SNR loss. This loss of SNR is also found to be larger for smaller mass binaries. Accordingly the circular template banks are found to be very ineffective in identifying binaries even with negligible residual orbital eccentricity.
Spinning binary systems {#spinning-binaries}
-----------------------
Whenever at least one of the bodies possesses spin—due to the precession of the orbital plane—the emitted gravitational wave acquires a considerable large amplitude modulation [even in the simplest possible case with zero initial eccentricity]{}. Waveforms of single spinning and double spinning binary systems of this type are shown on Fig.\[spinningfig\]. On the left panel only one of the bodies possesses spin with specific spin vector $s_{1x} = 1.0 $, and the spin vector is perpendicular to the orbital angular momentum, while on the right panel both of the bodies possess spin with specific spin vectors $s_{1x} = 1.0$ and $s_{2y}=1.0$. It is straightforward to recognize the yielded amplitude modulations on both panels.
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![The waveforms emitted by binary systems with masses $m_1 = m_2 = 3.4\ M_{\odot}$ and with specific spin vector(s) on the left panel for a single spin with $s_{1x} = 1.0$, while on the right panel for a double spinning binary with $s_{1x} = 1.0$ and $s_{2y} = 1.0$ are shown. In both cases the initial frequency is $f_{low} \sim 18\,\mathrm{Hz}$ and it is straightforward to recognize the amplitude modulation caused by the rotation of the orbital plane.[]{data-label="spinningfig"}](Fig/spin-circular.pdf "fig:"){width="48.00000%"} ![The waveforms emitted by binary systems with masses $m_1 = m_2 = 3.4\ M_{\odot}$ and with specific spin vector(s) on the left panel for a single spin with $s_{1x} = 1.0$, while on the right panel for a double spinning binary with $s_{1x} = 1.0$ and $s_{2y} = 1.0$ are shown. In both cases the initial frequency is $f_{low} \sim 18\,\mathrm{Hz}$ and it is straightforward to recognize the amplitude modulation caused by the rotation of the orbital plane.[]{data-label="spinningfig"}](Fig/doublespin-circular.pdf "fig:"){width="48.00000%"}
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Generic waveforms for spinning and eccentric binaries
-----------------------------------------------------
As indicated above the CBwaves software is capable to determine the evolution and the waveforms of completely general spinning and eccentric binaries. The simultaneous effect of amplitude and frequency modulation gets immediately transparent. Such generic orbits and waveforms are shown on Figs.\[doublespineccearly\] and \[doublespineccfinal\] for the early phase and late inspiral evolution of the system, respectively.
The early phase of the gravitational wave emitted by strongly eccentric binary systems possesses burst type character. Fitting analytic formula to all the possible waveforms of these type seems not to be feasible. Nevertheless, they can be investigated with the help of the CBwaves software, making thereby possible the construction of detection pipelines, algorithms and their efficiency studies for these types of events.
The evolution of eccentricity for spinning binaries
---------------------------------------------------
[As it was emphasized already CBwaves was developed to accurately evolve and determine the waveforms emitted by generic spinning binary configurations moving on possibly eccentric orbits. Based on these capabilities it seems to be important to investigate how the evolution of eccentricity may be affected by the presence of spin or spins of the constituents. This short subsection is to present our pertinent results which can be summarized by claiming that the evolution of eccentricity is highly insensitive to the presence of spin. On Fig.\[spinning-eccentric-fig\] the time dependence of the eccentricity is plotted for a NS-NS binary. It is clearly visible that the evolutions of the eccentricity relevant for spinning NS-NS binaries with randomly oriented spins (they are indicated by thin color lines) remain always very close to the evolution relevant for the same type of binary (it is indicated by a black solid line) with no spin at all. Although the evolution shown on Fig.\[spinning-eccentric-fig\] is relevant for systems with specific masses $m_1=m_2=1.4\,M_\odot$ and initial eccentricity $e=0.4$ the qualitative behavior is not significantly different, i.e., the evolution of eccentricity is found to be insensitive to the presence of spin(s), for other binaries with various masses and for other values of the initial eccentricity.]{}
![[]{data-label="spinning-eccentric-fig"}](Fig/spinecctest.pdf){width="70.00000%"}
### Strongly eccentric binary systems
It is expected that strongly eccentric black hole binaries in the halo of the galactic supermassive black hole are formed in consequence of many-body interactions [@exc]. The amplitude and the frequency of the gravitational waves emitted by such systems changes significantly during the inspiral due to the circularization effect. In Fig.\[doublespineccearly\] the orbital evolution and the time dependence of the amplitude of the waveform—in a relatively early phase of its evolution—are shown for a binary system with masses $m_1 = 24\,M_{\odot}, m_2 = 8\,M_{\odot}$ and with specific spin vectors $s_{1x} = 0.7, s_{1z}=0.7$ and $s_{2x} =
0.7, s_{2z}=-0.7$.
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![Orbital motion (left panel) and emitted waveform (right panel) of a strongly eccentric, double spinning, precessing binary system in the early phase of the orbital evolution with masses $m_1 = 24\,M_{\odot}, m_2 = 8\,M_{\odot}$ and with specific spin vectors $s_{1x} = 0.7, s_{1z}=0.7$ and $s_{2x} = 0.7, ![Orbital motion (left panel) and emitted waveform (right panel) of a strongly eccentric, double spinning, precessing binary system in the early phase of the orbital evolution with masses $m_1 = 24\,M_{\odot}, m_2 = 8\,M_{\odot}$ and with specific spin vectors $s_{1x} = 0.7, s_{1z}=0.7$ and $s_{2x} = 0.7,
s_{2z}=-0.7$.[]{data-label="doublespineccearly"}](Fig/doublespin-ecc-early-orbit-every.pdf "fig:"){width="48.00000%"} s_{2z}=-0.7$.[]{data-label="doublespineccearly"}](Fig/doublespin-ecc-early-h.pdf "fig:"){width="48.00000%"}
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By the circularization of the orbit the features of the waveform changes significantly and it becomes more and more similar to that of a simple circular, spinning binary system. The result of this process is shown—in the very final phase of the inspiral process—on Fig.\[doublespineccfinal\] for the system as on Fig.\[doublespineccearly\].
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![In the final phase of the inspiral the orbital evolution (left panel) and emitted waveform (right panel) are shown for the highly eccentric, double spinning, precessing binary system as depicted on Fig.\[doublespineccearly\].[]{data-label="doublespineccfinal"}](Fig/doublespin-ecc-late-orbit.pdf "fig:"){width="48.00000%"} ![In the final phase of the inspiral the orbital evolution (left panel) and emitted waveform (right panel) are shown for the highly eccentric, double spinning, precessing binary system as depicted on Fig.\[doublespineccearly\].[]{data-label="doublespineccfinal"}](Fig/doublespin-ecc-late-h.pdf "fig:"){width="48.00000%"}
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From data analyzing respects it is of crucial importance that the frequency-domain waveforms of highly eccentric binary systems possess very specific characteristics which may be used to determine the physical parameters of the system. For an immediate examples see Fig.\[burstfftmass\] on which various frequency-domain waveforms are depicted for binary systems each with fixed initial eccentricity, $e_{flow} = 0.8$, and initial frequency, $f_{low} =
0.5$ Hz. [The masses are chosen such that $m_1=1 M_{\odot}$ while $m_2$ takes either of the values $1 M_{\odot}, 6 M_{\odot}, 11 M_{\odot}, 16 M_{\odot}$, respectively.]{} Note that the waveforms shown on Fig.\[burstfftmass\] (and also on Fig.\[burstfftecchigh\]) are normalized such that each waveform possesses equal power. The various waveforms differ only in amplitude—which translates to effective distance—and in the ratio of power present in the head (twice of the orbital frequency) and the tail of the frequency distribution.
![The specific characteristics of the power normalized frequency-domain waveforms of binaries with initial eccentricity $e_{flow} = 0.8$, initial frequency $f_{low} = 0.5\,\mathrm{Hz}$ and [with masses $m_1=1 M_{\odot}$ and $m_2 = 1 M_{\odot}, 6 M_{\odot}, 11 M_{\odot}, 16 M_{\odot}$, respectively,]{} are shown. []{data-label="burstfftmass"}](Fig/burst-fft-masses.pdf){width="70.00000%"}
as the initial eccentricity of the system is increased the time domain waveform—as expected—approaches an idealized Dirac-delta-like function while the frequency-domain waveform becomes more and more broadband. An consequence of this is that the waveforms spread over several—although not adjacent—frequency bands which provides a unique imprint to them. This means that despite being present on several frequencies it remains frequency limited, which may help in constructing sensible detection pipelines robust against transient noises. From the fractional power present at various frequency bands the eccentricity of a system may be deduced, which is of great importance from parameter estimation point of view. Waveforms for moderately and highly eccentric binary systems are shown on the left and right panels of Fig.\[burstfftecchigh\], respectively.
[c]{}
Summary {#Summary}
=======
Our main aim in writing up this paper was to introduce our general purpose gravitational waveform generator software, called CBwaves, which is capable to simulate waveforms emitted by binary systems on closed or open orbits with possessing spin(s) and/or orbital eccentricity. It was done by direct integration of the equation of motion of the post-Newtonian expansion thereby the software can easily be extended up to any desired order of (known) precision. The source of current version, which is accurate up to 3.5PN order, may be downloaded from [@cbwavesdownload].
With the help of CBwaves we have investigated some of the characteristics of the orbital evolution of various binary configurations and the emitted waveforms. The relevance of the results obtained for the next generation of interferometric gravitational wave detectors was also underlined. While currently allowable parametric density of a general configuration template bank practically limits the direct use of the yielded waveforms in the detection pipelines, these generic waveforms will be useful in various parameter estimation studies.
As already emphasized with the help of CBwaves various physically interesting scenarios have been investigated. The main results covered by the present paper are as follows:
- [Investigating the validity of the adiabatic approximation it was found that the adiabatic approximation yields almost the same, at 3.5PN level, less then $2\%$ faster decrease than the time evolution. ]{}
- We have justified that the energy balance relation is indeed insensitive to the specific form of the applied radiation reaction term.
- it is demonstrated that circular template banks are effective in identifying binaries possess only a tiny residual orbital eccentricity.
- some of the discrepancies, between the analytic and numerical description of the evolution of eccentric binaries, along with some universal properties characterizing their evolution[, and its insensitivity for the inclusion of spins]{}, were pointed out.
- By inspecting the energy balance relation it was shown that, on contrary to the general expectations, the post-Newtonian approximations loose its accuracy once the post-Newtonian parameter gets beyond its critical value $\epsilon\sim 0.08-0.1$.
- By studying gravitational waves emitted during the early phase of the evolution of strongly eccentric binary systems it was found that they possess very specific characteristics which may make the detection of these type of binary systems to be feasible.
From data analyzing point of view it is crucial to know which of the involved parameters are essential. In accordance with the discussion on pages 6-7 in [@BCV] some of the relative angles between the vectors determining the initial configurations are of this type and they are relevant in determining the motion of the involved bodies and in computing the emitted waveform. Nevertheless, whatever are the initial conditions the lengths and the relative angles of the vectors comprising it remain intact under a rigid rotation of the binary as a whole in space around any spatial vector of free choice. Consequently, the emitted waveform should not change more than dictated by the simultaneous replacement of the direction of observation. In the particular case when the base vector of such a rigid rotation is also distinguished by the dynamics—such as the orbital angular momentum $\mathbf{L}$ in [@BCV], or the total angular momentum $\mathbf{J}=\mathbf{L}+\mathbf{S}_1+\mathbf{S}_2$ in our case as $\dot{\mathbf{J}}\approx 0$ holds up to 2PN order—and the binary is optimally oriented, i.e.the direction of observation $\hat{\bf N}$ is chosen to be parallel to $\mathbf{J}$ and it is orthogonal to the plane spanned by the detector arms (see Figs.\[SourceFrame\] and \[DetectorFrame\]), only a simple phase shift, with angle $\hat \varphi$, is expected to show up in the waveform as a response to the rigid rotation by angle $\hat \varphi$ around $\mathbf{J}$. It is straightforward to see that in this particular case the considered rigid rotation of the source could be replaced by a rigid rotation of the detector arms by angle $-\hat \varphi$. Fig.\[doublespinrotecc\] is to provide justification of these expectations on which the rotation angle dependence of the emitted waveforms and the compensation by appropriate phase shifts are shown in case of a binary system with $m_1$ = $m_2$ = 10 $M_{\odot}$, $s_1$ = 0.7 , $s_2$ = 0 and with initial eccentricity $e$ = 0.2.
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![The $\hat \varphi$ (rotation angle) dependence of the waveform emitted by the binary system with $m_1$ = $m_2$ = 10 $M_{\odot}$, $s_1$ = 0.7 , $s_2$ = 0 and with eccentricity $e$ = 0.2 is shown for different amount of initial rotation around the total angular momentum vector. The full waveform (top), the early stage (middle left) the late phase (middle right), the phase shifted early stage (bottom left) and the phase shifted late stage (bottom right) of the inspiral process. (Note that the small initial discrepancies on the last but one plot has no physical relevance as it is an artifact of inherent inaccuracy of the method used in determining the phase shift.)[]{data-label="doublespinrotecc"}](Fig/jroteccsinglespin.pdf "fig:"){width="95.00000%"}
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Note finally that there are various physically interesting problems which may be investigated with the help of CBwaves. The most immediate ones include a systematic study of the effect of the [*spin supplementary conditions*]{} [@SSC1; @SSC2; @SSC3] and the investigation of the time evolution of open binary systems, in particular, that of the spin flip phenomenon. The results of the corresponding studies will be published elsewhere.
Acknowledgments
===============
This research was supported in parts by a VESF postdoctoral fellowship to the RMKI Virgo Group for the period 2009-2011, and by the Hungarian Scientific Research Fund (OTKA) Grant No. 67942.
Appendix A {#appendix-a .unnumbered}
==========
The radiation field
-------------------
The gravitational waveform generated by a compact binary system is expressed by a sum of contributions originating from different PN orders. The particular form of the contributions listed in Eq.(\[Wform\]) can be found in [@Kidder], but for conveniences they are also summarized below. Accordingly, the quadrupole term and higher order relativistic corrections read as $$\begin{aligned}
Q^{ij} &=&2\left[ v^{i}v^{j}-{\frac{Gm}{r}}n^{i}n^{j}\right] , \\
P^{0.5}Q^{ij} &=&{\frac{\delta m}{cm}}\left\{ 3{\frac{Gm}{r}}\left[
2n^{(i}v^{j)}-\dot{r}n^{i}n^{j}\right] (\mathbf{\hat{N}\cdot
\hat{n}})+\left[
{\frac{Gm}{r}}n^{i}n^{j}-2v^{i}v^{j}\right] (\mathbf{\hat{N}\cdot v})\right\} , \\
PQ^{ij} &=&{\frac{1}{3c^{2}}}(1-3\eta)\left\{ 4{\frac{Gm}{r}}\left[ 3\dot{r}n^{i}n^{j}-8n^{(i}v^{j)}\right] (\mathbf{\hat{N}\cdot \hat{n}})(\mathbf{\hat{N}\cdot v})+2\left[ 3v^{i}v^{j}-{\frac{Gm}{r}}n^{i}n^{j}\right] (\mathbf{\hat{N}\cdot v})^{2}\right. \\
&&\left. +{\frac{Gm}{r}}\left[ (3v^{2}-15\dot{r}^{2}+7{\frac{Gm}{r}})n^{i}n^{j}+30\dot{r}n^{(i}v^{j)}-14v^{i}v^{j}\right]
(\mathbf{\hat{N}\cdot \hat{n}})^{2}\right\}
+{\frac{4}{3}}{\frac{Gm}{r}}\dot{r}(5+3\eta
)n^{(i}v^{j)} \nonumber \\
&&+\left[ (1-3\eta )v^{2}-{\frac{2}{3}}(2-3\eta
){\frac{Gm}{r}}\right] v^{i}v^{j}+{\frac{Gm}{r}}\left[ (1-3\eta
)\dot{r}^{2}-{\frac{1}{3}}(10+3\eta
)v^{2}+{\frac{29}{3}}{\frac{Gm}{r}}\right] n^{i}n^{j}, \nonumber \\
P^{1.5}Q^{ij} &=&{\frac{\delta m}{mc^{3}}}(1-2\eta )\Biggl\{{\frac{1}{4}}{\frac{Gm}{r}}\Biggl[(45\dot{r}^{2}-9v^{2}-28{\frac{Gm}{r}})n^{i}n^{j}+58v^{i}v^{j}-108\dot{r}n^{(i}v^{j)}\Biggr](\mathbf{\hat{N}\cdot
\hat{n}})^{2}(\mathbf{\hat{N}\cdot v}) \\
&&+{\frac{1}{2}}\left[ {\frac{Gm}{r}}n^{i}n^{j}-4v^{i}v^{j}\right] (\mathbf{\hat{N}\cdot v})^{3}+{\frac{Gm}{r}}\Biggl[{\frac{5}{4}}(3v^{2}-7\dot{r}^{2}+6{\frac{Gm}{r}})\dot{r}n^{i}n^{j}-{\frac{1}{6}}(21v^{2}-105\dot{r}^{2}
\nonumber \\
&&+44{\frac{Gm}{r}})n^{(i}v^{j)}-{\frac{17}{2}}\dot{r}v^{i}v^{j}\Biggr](\mathbf{\hat{N}\cdot \hat{n}})^{3}+{\frac{3}{2}}{\frac{Gm}{r}}\left[
10n^{(i}v^{j)}-3\dot{r}n^{i}n^{j}\right] (\mathbf{\hat{N}\cdot \hat{n}})(\mathbf{\hat{N}\cdot v})^{2}\Biggr\} \nonumber \\
&&+{\frac{\delta m}{mc^{3}}}{\frac{1}{12}}{\frac{Gm}{r}}(\mathbf{\hat{N}\cdot \hat{n}})\Biggl\{n^{i}n^{j}\dot{r}\left[ \dot{r}^{2}(15-90\eta
)-v^{2}(63-54\eta )+{\frac{Gm}{r}}(242-24\eta )\right] \nonumber \\
&&-\dot{r}v^{i}v^{j}(186+24\eta )+2n^{(i}v^{j)}\left[
\dot{r}^{2}(63+54\eta
)-{\frac{Gm}{r}}(128-36\eta )+v^{2}(33-18\eta )\right] \Biggr\} \nonumber \\
&&+{\frac{\delta m}{mc^{3}}}(\mathbf{\hat{N}\cdot v})\Biggl\{{\frac{1}{2}}v^{i}v^{j}\left[ {\frac{Gm}{r}}(3-8\eta )-2v^{2}(1-5\eta )\right]
-n^{(i}v^{j)}{\frac{Gm}{r}}\dot{r}(7+4\eta ) \notag \\
&&-n^{i}n^{j}{\frac{Gm}{r}}\left[ {\frac{3}{4}}(1-2\eta )\dot{r}^{2}+{\frac{1}{3}}(26-3\eta ){\frac{Gm}{r}}-{\frac{1}{4}}(7-2\eta )v^{2}\right]
\Biggr\}\,, \nonumber\end{aligned}$$where $\mathbf{r}=\mathbf{x_{1}}-\mathbf{x_{2}}$, $\mathbf{v}={d\mathbf{r}/dt} $, $\mathbf{\hat{n}}={\mathbf{r}/r}$, $m=m_{1}+m_{2}$, $\delta m=m_{1}-m_{2}$, $\eta =\mu /m$ and the derivative with respect to time is indicated by an overdot. $$\begin{aligned}
{\textcolor{black}{ P^{2}Q^{ij} }}&=& \frac{1}{c^4}\biggr[{1 \over 60 }
(1-5\eta+5\eta^2) \biggl\{ 24({\bf \hat N \cdot v})^4 \biggl[ 5 v^i
v^j - {m \over r} {\hat n}^i {\hat n}^j \biggr]
\nonumber \\
&& +{m \over r} ({\bf \hat N \cdot \hat n})^4 \biggl[ 2 \left( 175
{m \over r} - 465 \dot r^2 + 93 v^2 \right) v^i v^j
+ 30 \dot r \left( 63 \dot r^2 - 50{m \over r} - 27 v^2 \right)
{\hat n}^{(i}v^{j)}
\nonumber \\
&& + \left(1155 {m \over r} \dot r^2 - 172 \left({m \over
r}\right)^2 - 945 \dot r^4 - 159 {m \over r} v^2 + 630 \dot r^2 v^2
- 45 v^4 \right) {\hat n}^i {\hat n}^j \biggr]
\nonumber \\
&& +24 {m \over r} ({\bf \hat N \cdot \hat n})^3 ({\bf \hat N \cdot
v}) \biggl[ 87 \dot r v^i v^j + 5 \dot r \left( 14 \dot r^2 - 15 {m
\over r} - 6v^2 \right) {\hat n}^i {\hat n}^j
\nonumber \\
&& + 16 \left( 5 {m \over r} - 10 \dot r^2 + 2v^2 \right) {\hat
n}^{(i} v^{j)} \biggr] +288 {m \over r} ({\bf \hat N \cdot \hat n})
({\bf \hat N \cdot v})^3 \biggl[ \dot r {\hat n}^i {\hat n}^j - 4
{\hat n}^{(i} v^{j)} \biggr]
\nonumber \\
&& +24 {m \over r} ({\bf \hat N \cdot \hat n})^2 ({\bf \hat N \cdot
v})^2 \biggl[ \left( 35 {m \over r} - 45 \dot r^2 + 9 v^2 \right)
{\hat n}^i {\hat n}^j
- 76 v^i v^j + 126 \dot r {\hat n}^{(i} v^{j)}
\biggr] \biggr\}
\nonumber \\
&& + {1 \over 15} ({\bf \hat N \cdot v})^2 \biggl\{
\biggl[ 5 ( 25-78\eta+12\eta^2 ) {m \over r}
- (18 - 65 \eta + 45 \eta^2 ) v^2
\nonumber \\
&&
+ 9 ( 1 - 5 \eta + 5 \eta^2 ) \dot r^2
\biggr] {m \over r} {\hat n}^i {\hat n}^j
+3\biggl[ 5 ( 1 - 9\eta + 21\eta^2 ) v^2
-2 ( 4 - 25 \eta + 45 \eta^2 ) {m \over r}
\biggr] v^i v^j
\nonumber \\
&&
+ 18 ( 6 - 15 \eta - 10 \eta^2 ) {m \over r} \dot r {\hat
n}^{(i} v^{j)} \biggr\} +{1 \over 15}({\bf \hat N \cdot \hat
n})({\bf \hat N \cdot v}){m \over r} \biggl\{
\biggl[ 3 ( 36-145\eta+150\eta^2 ) v^2 \nonumber \\
&&
-5 ( 127 - 392 \eta + 36 \eta^2 ) {m \over r}
-15( 2 - 15 \eta + 30 \eta^2 ) \dot r^2
\biggr] \dot r {\hat n}^i {\hat n}^j \nonumber \\
&&
+ 6 (98 - 295 \eta - 30 \eta^2 ) \dot r v^i v^j
+2\biggl[ 5 ( 66 - 221\eta + 96 \eta^2 ) {m \over r}\nonumber \\
&&
-9 ( 18 - 45\eta - 40 \eta^2 ) \dot r^2
- ( 66 - 265\eta +360 \eta^2 ) v^2
\biggr] {\hat n}^{(i} v^{j)}
\biggr\}
\nonumber \\
&& +{1 \over 60}({\bf \hat N \cdot \hat n})^2 {m \over r} \biggl\{
\biggl[ 3 (33- 130\eta + 150\eta^2) v^4
+ 105( 1 - 10 \eta + 30 \eta^2 ) \dot r^4
\nonumber \\
&&
+ 15 (181-572 \eta + 84 \eta^2) {m \over r} \dot r^2
- (131-770 \eta + 930\eta^2) {m \over r} v^2
\nonumber \\
&&
- 60 ( 9- 40 \eta + 60\eta^2) v^2 \dot r^2
- 8 (131-390 \eta + 30\eta^2) \left( {m \over r} \right)^2
\biggr] {\hat n}^i {\hat n}^j
\nonumber \\
&&
+ 4 \biggl[ (12+ 5\eta - 315\eta^2) v^2
-9 (39- 115\eta - 35\eta^2) \dot r^2
+5 (29- 104\eta + 84\eta^2) {m \over r}
\biggr] v^i v^j
\nonumber \\
&&
+ 4 \biggl[15 ( 18- 40\eta - 75\eta^2) \dot r^2
-5 (197- 640\eta + 180\eta^2) {m \over r }
\nonumber \\
&&
+3 (21- 130\eta + 375\eta^2) v^2
\biggr] \dot r {\hat n}^{(i} v^{j)}
\biggr\}
\nonumber \\
&& + {1 \over 60} \biggl\{ \biggl[ (467+780\eta-120\eta^2) {m
\over r} v^2
- 15( 61- 96\eta+ 48\eta^2) {m \over r} \dot r^2
\nonumber \\
&&
- (144-265\eta-135\eta^2) v^4
+ 6( 24- 95\eta+ 75\eta^2) v^2 \dot r^2
\nonumber \\
&&
- 2(642+545\eta ) \left( {m \over r} \right)^2
- 45( 1- 5\eta+ 5\eta^2) \dot r^4
\biggr] {m \over r} {\hat n}^i {\hat n}^j
\nonumber \\
&&
+ \biggl[ 4 ( 69+ 10\eta-135\eta^2) {m \over r} v^2
- 12( 3+ 60\eta+ 25\eta^2) {m \over r} \dot r^2
\nonumber \\
&&
+ 45( 1- 7\eta+ 13\eta^2) v^4
- 10( 56+165\eta- 12\eta^2) \left( {m \over r} \right)^2
\biggr] v^i v^j
\nonumber \\
&&
+4\biggl[ 2 ( 36+ 5\eta- 75\eta^2) v^2
- 6 ( 7- 15\eta- 15\eta^2) \dot r^2
+ 5 ( 35+ 45\eta+ 36\eta^2) {m \over r}
\biggr] {m \over r} \dot r {\hat n}^{(i} v^{j)}
\biggr\}\biggr] \, . \nonumber\end{aligned}$$
The analogous expressions for spin contributions are $$\begin{aligned}
PQ_{SO}^{ij}\!\!\!\!&=&\!\!\!\! {\frac{2G}{cr^{2}}}(\mathbf{\Delta
\times \hat{N}}
)^{(i}n^{j)}, \label{PQSO} \\
P^{1.5}Q_{SO}^{ij}
\!\!\!\!&=&\!\!\!\!{\frac{2G}{c^{2}r^{2}}}\Biggl\{n^{i}n^{j}\left[ (
\mathbf{\hat{n}\times v})\mathbf{\cdot }(12\mathbf{S}+6{\frac{\delta
m}{m}} \mathbf{\Delta })\right] -n^{(i}\left[ \mathbf{v\times
}(9\mathbf{S}+5{\frac{
\delta m}{m}}\mathbf{\Delta })\right] ^{j)} \label{P15QSO} \\
&&+\left[ 3\dot{r}(\mathbf{\hat{N}\cdot
\hat{n}})-2(\mathbf{\hat{N}\cdot v}) \right] \left[
(\mathbf{S}+{\frac{\delta m}{m}}\mathbf{\Delta })\mathbf{ \times
\hat{N}}\right] ^{(i}n^{j)}-v^{(i}\left[ \mathbf{\hat{n}\times }(2
\mathbf{S}+2{\frac{\delta m}{m}}\mathbf{\Delta })\right] ^{j)} \notag \\
&&+\dot{r}n^{(i}\left[ \mathbf{\hat{n}\times
}(12\mathbf{S}+6{\frac{\delta m }{m}}\mathbf{\Delta })\right]
^{j)}-2(\mathbf{\hat{N}\cdot \hat{n}})\left[ (
\mathbf{S}+{\frac{\delta m}{m}}\mathbf{\Delta })\mathbf{\times
\hat{N}} \right] ^{(i}v^{j)}\Biggr\}, \nonumber \\
{\textcolor{black}{ P^{2}Q^{ij}_{SS} }} \!\!\!\!&=&\!\!\!\! {\textcolor{black}{ -
\frac{6G}{c^2\mu r^3} \left\{ n^i n^j \left[ ({\bf S_1 \cdot S_2}) -
5 ({\bf \hat n \cdot S_1})({\bf \hat n \cdot S_2}) \right] +
2n^{(i}S_1^{j)} ({\bf \hat n \cdot S_2}) + 2n^{(i}S_2^{j)} ({\bf
\hat n \cdot S_1}) \right\}. }} \label{P2QSS}\end{aligned}$$where $\mathbf{S}=\mathbf{S}_{1}+\mathbf{S}_{2}$ and $\mathbf{\Delta}=m(\mathbf{S}_{2}/m_{2}-\mathbf{S}_{1}/m_{1})$.
In black hole perturbation theory and in numerical simulations the radiation field is frequently given in terms of spin weighted spherical harmonics. As the injection of numerical templates also requires this type of expansion [@LSCCommon] CBwaves does contain a module evaluating some of the spin weighted spherical harmonics. The relations we have applied in generating the components read as $$MH_{lm}=\oint {}^{-2}Y_{lm}^{\ast }(\iota ,\phi )\left( rh_{+}-irh_{{\times }}\right) d\Omega ,$$where, for example,$$\begin{aligned}
^{-2}Y_{2\pm 2} &=&\sqrt{\frac{5}{64\pi }}\left( 1\pm \cos \iota
\right)
^{2}e^{\pm 2i\phi }, \\
^{-2}Y_{2\pm 1} &=&\sqrt{\frac{5}{16\pi }}\sin \iota \left( 1\pm
\cos \iota
\right) e^{\pm i\phi }, \\
^{-2}Y_{20} &=&\sqrt{\frac{5}{32\pi }}\sin ^{2}\iota \ .\end{aligned}$$$h_{+}^{(lm)}$ and $h_{{\times }}^{(lm)}$ are defined as$$rh_{+}^{(lm)}(t)-irh_{{\times }}^{(lm)}(t)=MH_{lm}(t)\,.$$Note that these modes of $rh_{+}$ and $rh_{{\times }}$ are used for injections [@LSCCommon].
Appendix B {#appendix-b .unnumbered}
==========
Equations of motion
-------------------
The various order of relative accelerations, as listed in Eq.(\[accel\]), can be given as $$\begin{aligned}
\mathbf{a}_{N} &=&-{\frac{Gm}{r^{2}}}\mathbf{\hat{n}}, \\
\mathbf{a}_{PN} &=&-{\frac{Gm}{c^{2}r^{2}}}\left\{
\mathbf{\hat{n}}\left[ (1+3\eta )v^{2}-2(2+\eta
){\frac{Gm}{r}}-{\frac{3}{2}}\eta \dot{r}^{2}\right]
-2(2-\eta )\dot{r}\mathbf{v}\right\} , \\
\mathbf{a}_{SO} &=&{\frac{G}{c^{2}r^{3}}}\left\{ 6\mathbf{\hat{n}}[(\mathbf{\hat{n}}\times \mathbf{v})\mathbf{\cdot }(\mathbf{S}
+{\mbox{\boldmath$\sigma$}})]
-[\mathbf{v}\times (4\mathbf{S}+3{\mbox{\boldmath $\sigma$}})]+3\dot{r}[\mathbf{\hat{n}}\times (2\mathbf{S}+{\mbox{\boldmath $\sigma$}})]\right\} , \label{aSO}\\
\mathbf{a}_{2PN} &=&-{\frac{Gm}{c^{4}r^{2}}}\biggl\{\mathbf{\hat{n}}\biggl[{\frac{3}{4}}(12+29\eta )({\frac{Gm}{r}})^{2}+\eta (3-4\eta )v^{4}+{\frac{15}{8}}\eta (1-3\eta )\dot{r}^{4} \\
&&-{\frac{3}{2}}\eta (3-4\eta )v^{2}\dot{r}^{2}-{\frac{1}{2}}\eta (13-4\eta ){\frac{Gm}{r}}v^{2}-(2+25\eta +2\eta ^{2}){\frac{Gm}{r}}\dot{r}^{2}\biggr] \nonumber \\
&&-{\frac{1}{2}}\dot{r}\mathbf{v}\left[ \eta (15+4\eta
)v^{2}-(4+41\eta
+8\eta ^{2}){\frac{Gm}{r}}-3\eta (3+2\eta )\dot{r}^{2}\right] \biggr\}, \nonumber \\
\mathbf{a}_{SS} &=&-{\frac{3G}{c^{2}\mu r^{4}}}\biggl\{\mathbf{\hat{n}}(\mathbf{S}_{1}\cdot \mathbf{S}_{2})+\mathbf{S}_{1}(\mathbf{\hat{n}}\cdot \mathbf{S}_{2})+\mathbf{S}_{2}(\mathbf{\hat{n}}\cdot \mathbf{S}_{1})-5\mathbf{\hat{n}}(\mathbf{\hat{n}}\cdot \mathbf{S}_{1})(\mathbf{\hat{n}}\cdot \mathbf{S}_{2})\biggr\}\,, \label{aSS} \\
\mathbf{a}_{RR}^{BT} &=& {\frac{8}{5}}\eta
{\frac{G^{2}m^{2}}{c^{5}r^{3}}}\left\{
\dot{r}\mathbf{\hat{n}}\left[ 18v^{2}+{\frac{2}{3}}{\frac{Gm}{r}}-25\dot{r}^{2}\right] -\mathbf{v}\left[
6v^{2}-2{\frac{Gm}{r}}-15\dot{r}^{2}\right] \right\}\,, \label{aRRBT2}\end{aligned}$$ where ${\mbox{\boldmath
$\sigma$}}=(m_{2}/m_{1})\mathbf{S}_{1}+(m_{1}/m_{2})\mathbf{S}_{2}$. Note also that the above form of $\mathbf{a}_{SO}$ tacitly presumes the use of the covariant spin supplementary condition, $S_{A}^{\mu
\nu }{u_{A}}_{\nu }=0$, where $u_{A}^{\mu }$ is the four-velocity of the center-of-mass world line of body $A$, with $A=1,2$. Finally, as discussed above the term $\mathbf{a}_{RR}^{BT}$ refers to the radiation reaction expression derived from a Burke-Thorne type radiation reaction potential [@IyerWill; @ZengWill].
$$\begin{aligned}
{\textcolor{black}{ \mathbf{a}_{PNSO} }}&=& \frac{G}{c^4r^3} \bigg\{
\mathbf{\hat{n}} \bigg[(\mathbf{\hat{n}\times v})\mathbf{\cdot
}\mathbf{S} \bigg(-30 \eta \dot{r}^2 + 24 \eta
v^2 - \frac{G m}{r} (38 + 25 \eta) \bigg) \nonumber \\
&& + \frac{\delta m}{m} (\mathbf{\hat{n}\times v})\mathbf{\cdot
}\mathbf{\Delta} \bigg(-15 \eta \dot{r}^2 + 12 \eta v^2 - \frac{G
m}{r} (18 + \frac{29}{2} \eta) \bigg) \bigg]
\nonumber\\
&& + \dot{r} \mathbf{v} \bigg[ (\mathbf{\hat{n}\times
v})\mathbf{\cdot }\mathbf{S} (-9 + 9 \eta) + \frac{\delta m}{m}
(\mathbf{\hat{n}\times v})\mathbf{\cdot
}\mathbf{\Delta} (-3 + 6 \eta) \bigg] \nonumber \\
&& + \mathbf{\hat{n}} \times \mathbf{v} \bigg[\dot{r} ({\bf v\cdot
S}) (-3 + 3 \eta) - 8 \frac{G m}{r} \eta ({\bf \hat n \cdot S}) -
\frac{\delta m}{m} \bigg(4 \frac{G
m}{r} \eta ({\bf \hat n \cdot \mathbf{\Delta}}) + 3 \dot{r} ({\bf v \cdot \mathbf{\Delta}}) \bigg) \bigg] \nonumber\\
&& + \dot{r} \mathbf{\hat{n}} \times \mathbf{S} \bigg[-\frac{45}{2}
\eta \dot{r}^2 + 21 \eta v^2 - \frac{G m}{r} (25 + 15 \eta) \bigg]
\nonumber \\
&&+ \frac{\delta m}{m} \dot{r} \mathbf{\hat{n}} \times
\mathbf{\mathbf{\Delta}} \bigg[- 15 \eta \dot{r}^2 + 12 \eta v^2 -
\frac{Gm}{r} (9 + \frac{17}{2} \eta)\bigg] \nonumber \\
&& + \mathbf{v} \times \mathbf{S} \bigg[\frac{33}{2} \eta \dot{r}^2
+ \frac{G m}{r}(21 + 9 \eta) - 14 \eta v^2 \bigg] \nonumber \\
&& + \frac{\delta m}{m} \mathbf{v} \times \mathbf{\mathbf{\Delta}}
\bigg[9 \eta \dot{r}^2 - 7 \eta v^2 + \frac{G m}{r} (9 + \frac{9}{2}
\eta) \bigg] \bigg\} \, ,\end{aligned}$$ $$\begin{aligned}
{\textcolor{black}{ \mathbf{a}_{3PN} }}&=& \frac{Gm}{c^3r^2} \bigg\{
\mathbf{\hat{n}} \bigg[ \left [16+\left(\frac{1399}{12}
-\frac{41}{16}\pi^{2} \right )\eta +\frac{71}{2}\eta^{2}\right
]{\left (\frac{Gm}{r}\right )}^{3} +\eta\left
[\frac{20827}{840}+\frac{123}{64}\pi^{2}-\eta^{2}\right ] {\left
(\frac{Gm}{r}\right )}^{2}v^{2}
\nonumber\\
& & -\left [1+\left (\frac{22717}{168}+\frac{615}{64}\pi^{2}\right
)\eta +\frac{11}{8}\eta^{2}-7\eta^{3}\right ] {\left
(\frac{Gm}{r}\right )}^{2}{\dot{r}}^{2}
\nonumber\\
& & -\frac{1}{4}\eta (11-49\eta+52\eta^{2})v^{6} +\frac{35}{16}\eta
(1-5\eta+5\eta^2 ){\dot{r}}^{6} - \frac{1}{4}\eta\left
(75+32\eta-40\eta^{2}\right )\frac{Gm}{r}v^{4}
\nonumber\\
& & - \frac{1}{2}\eta\left (158-69\eta-60\eta^{2}\right
)\frac{Gm}{r}{\dot{r}}^{4} +\eta\left (121-16\eta-20\eta^{2}\right
)\frac{Gm}{r}v^{2}{\dot{r}}^{2}
\nonumber\\
& & + \frac{3}{8}\eta\left (20-79\eta+60\eta^{2}\right
)v^{4}{\dot{r}}^{2} -\frac{15}{8}\eta\left
(4-18\eta+17\eta^{2}\right )v^{2}{\dot{r}}^{4} \bigg]
\nonumber\\
&& + \dot{r} \mathbf{v} \bigg[ \left [4+\left
(\frac{5849}{840}+\frac{123}{32}\pi^{2}\right )\eta
-25\eta^{2}-8\eta^{3}\right ]{\left (\frac{Gm}{r}\right )}^{2}
+\frac{1}{8} \eta\left(65-152\eta-48\eta^{2}\right )v^{4}
\nonumber\\
& & +\frac{15}{8}\eta\left (3-8\eta-2\eta^{2}\right ){\dot{r}}^{4}
+\eta\left (15+27\eta+10\eta^{2}\right )\frac{Gm}{r}v^{2}
\nonumber\\
& & -\frac{1}{6}\eta\left (329+177\eta+108\eta^{2}\right
)\frac{Gm}{r}\dot{r}^{2} -\frac{3}{4}\eta\left
(16-37\eta-16\eta^{2}\right )v^{2}\dot{r}^{2} \bigg] \bigg\} \, ,\end{aligned}$$ $$\begin{aligned}
{\textcolor{black}{ \mathbf{a}_{RR1PN} }}&=& {\frac{8}{5}}\eta
{\frac{G^{2}m^{2}}{c^{7}r^{3}}}\bigg\{\dot{r}\mathbf{\hat{n}}\bigg[
\left(\frac{87}{14}-48\eta\right)v^{4} -
\left(\frac{5379}{28}-\frac{136}{3}\eta\right)v^{2}\frac{Gm}{r}
+\frac{25}{2}(1+5\eta)v^2\dot{r}^2 \nonumber\\
&&+ \left(\frac{1353}{4}-133\eta\right)\dot{r}^2\frac{Gm}{r}
-\frac{35}{2}(1-\eta)\dot{r}^4 +
\left(\frac{160}{7}+\frac{55}{3}\eta\right)\left(\frac{Gm}{r}\right)^2\bigg]
\nonumber\\
&& -\mathbf{v}\bigg[ -\frac{27}{14}v^{4} -
\left(\frac{4861}{84}+\frac{58}{3}\eta\right)v^{2}\frac{Gm}{r}
+\frac{3}{2}(13-37\eta)v^2\dot{r}^2 \nonumber\\
&&+ \left(\frac{2591}{12}+97\eta\right)\dot{r}^2\frac{Gm}{r}
-\frac{25}{2}(1-7\eta)\dot{r}^4 +
\frac{1}{3}\left(\frac{776}{7}+55\eta\right)\left(\frac{Gm}{r}\right)^2\bigg]
\bigg\}\,,\end{aligned}$$ $$\begin{aligned}
{\textcolor{black}{ \mathbf{a}_{RRSO} }}&=& -\frac{G^2\eta m}{5c^7r^4}
\biggl \{ \frac{{\dot r}{\bf \hat{n}}}{\mu r} \left [ \left (
120v^2+280{\dot r}^2+453\frac{Gm}{r} \right ) {\bf {L}}_{\rm N}
\cdot {\bf \mathbf{S}} \right .
\nonumber \\
&& \left . + \left ( 120v^2+280{\dot r}^2+458\frac{Gm}{r} \right )
{\bf {L}}_{\rm N} \cdot {\mbox{\boldmath $\sigma$}} \right ]
\nonumber \\
&& + \frac{\bf v}{\mu r} \left [ \left ( 87v^2-675{\dot
r}^2-\frac{901}{3}\frac{Gm}{r} \right ) {\bf {L}}_{\rm N} \cdot {\bf
\mathbf{S}} + 4\left ( 18v^2-150{\dot r}^2 - 66\frac{Gm}{r} \right )
{\bf {L}}_{\rm N} \cdot {\mbox{\boldmath $\sigma$}} \right ]
\nonumber \\
&& - \frac{2}{3}{\dot r}{\bf v} \times {\bf \mathbf{S}} \left (
48v^2 + 15{\dot r}^2+364\frac{Gm}{r} \right ) + \frac{1}{3}{\dot
r}{\bf v} \times {\mbox{\boldmath $\sigma$}} \left ( 291v^2
-705{\dot r}^2-772\frac{Gm}{r} \right )
\nonumber \\
&& +\frac{1}{2}{\bf \hat{n}}\times {\bf \mathbf{S}} \left (
31v^4-260v^2{\dot r}^2+245{\dot r}^4 -\frac{689}{3}v^2\frac{Gm}{r} +
537{\dot r}^2\frac{Gm}{r} +\frac{4}{3}\frac{G^2m^2}{r^2} \right )
\nonumber \\
&& +\frac{1}{2}{\bf \hat{n}}\times {\mbox{\boldmath $\sigma$}} \left
( 115v^4-1130v^2{\dot r}^2+1295{\dot r}^4
-\frac{869}{3}v^2\frac{Gm}{r} + 849{\dot r}^2\frac{Gm}{r} +
\frac{44}{3}\frac{G^2m^2}{r^2} \right ) \biggr \} \,,\end{aligned}$$ $$\begin{aligned}
{\textcolor{black}{ \mathbf{a}_{RRSS} }}&=& \frac{G^2}{c^7r^5} \biggl \{
{\bf \hat{n}} \biggl [ \biggl ( 287{\dot r}^2 - 99v^2 +
\frac{541}{5}\frac{Gm}{r} \biggr ){\dot r} (\mathbf{S}_{1}\cdot
\mathbf{S}_{2}) \nonumber \\
&& - \biggl ( 2646{\dot r}^2 - 714 v^2 +\frac{1961}{5}\frac{Gm}{r}
\biggr ){\dot r} (\mathbf{\hat{n}}\cdot \mathbf{S}_{1})
(\mathbf{\hat{n}}\cdot \mathbf{S}_{2})
\nonumber \\
&& + \biggl ( 1029{\dot r}^2 - 123 v^2 + \frac{629}{10}\frac{Gm}{r}
\biggr ) \biggl((\mathbf{\hat{n}}\cdot \mathbf{S}_{1})
(\mathbf{\hat{v}}\cdot \mathbf{S}_{2}) + (\mathbf{\hat{n}}\cdot
\mathbf{S}_{2}) (\mathbf{\hat{v}}\cdot \mathbf{S}_{1}) \biggr)\nonumber \\
&& - 336 {\dot r} (\mathbf{\hat{v}}\cdot \mathbf{S}_{1})
(\mathbf{\hat{v}}\cdot \mathbf{S}_{2}) \biggr ] + {\bf v} \biggl [
\biggl ( \frac{171}{5} v^2 - 195 {\dot r}^2 - 67\frac{Gm}{r} \biggr
) (\mathbf{S}_{1}\cdot \mathbf{S}_{2})\nonumber \\
&& - \biggl ( 174 v^2 -1386 {\dot r}^2 - \frac{1038}{5}\frac{Gm}{r}
\biggr ) (\mathbf{\hat{n}}\cdot \mathbf{S}_{1})
(\mathbf{\hat{n}}\cdot \mathbf{S}_{2})
\nonumber \\
&& - 438 {\dot r} \biggl( (\mathbf{\hat{n}}\cdot \mathbf{S}_{1})
(\mathbf{\hat{v}}\cdot \mathbf{S}_{2}) + (\mathbf{\hat{n}}\cdot
\mathbf{S}_{2}) (\mathbf{\hat{v}}\cdot \mathbf{S}_{1}) \biggr) + 96
(\mathbf{\hat{v}}\cdot \mathbf{S}_{1}) (\mathbf{\hat{v}}\cdot
\mathbf{S}_{2}) \biggr ]
\nonumber \\
&& + \biggl ( \frac{27}{10} v^2 - \frac{75}{2} {\dot r}^2 -
\frac{509}{30}\frac{Gm}{r} \biggr ) \biggl((\mathbf{\hat{v}}\cdot
\mathbf{S}_{2}) {\bf S}_1 + (\mathbf{\hat{v}}\cdot \mathbf{S}_{1})
{\bf S}_2 \biggr)
\nonumber \\
&& + \biggl ( \frac{15}{2} v^2 + \frac{77}{2} {\dot r}^2 +
\frac{199}{10}\frac{Gm}{r} \biggr ) {\dot r}
\biggl((\mathbf{\hat{n}}\cdot \mathbf{S}_{2}) {\bf S}_1 +
(\mathbf{\hat{n}}\cdot \mathbf{S}_{1}) {\bf S}_2 \biggr) \biggr \}
\, .\end{aligned}$$
In general $\mathbf{a}_{SO}$ and $\mathbf{a}_{SS}$ are not confined to the orbital plane thereby they yield a precession of th plane and, in turn, a modulation of the observed signal. In addition, spin vectors themselves precess according to their evolution equations $$\begin{aligned}
\mathbf{\dot{S}}_{i} &=&\frac{G}{c^{2}r^{3}}\left\{ \frac{4+3\zeta
_{i}}{2}\mathbf{L}_{N}-\mathbf{S}_{j}+3\left( \mathbf{\hat{n}\cdot
S}_{j}\right) \mathbf{\hat{n}}\right. \nonumber\\
&&\left. {\textcolor{black}{+\frac{G^2\mu m}{c^5r^2}\left[\frac23\left(
\mathbf{v\cdot S}_{j}\right)+30\dot{r}\left( \mathbf{\hat{n}\cdot
S}_{j}\right)\right]\mathbf{\hat{n}}}} \right\} \times
\mathbf{S}_{i}\,, \label{Sprec}\end{aligned}$$ where $\mathbf{L}_{N}=\mu \mathbf{r}\times \mathbf{v}$ is the Newtonian angular momentum and $\zeta _{i}=m_{j}/m_{i}$, with $i,j=1,2$, $i\neq j$.
The terms in the equations of motion, Eq.(\[accel\]), up to 2PN order can be deduced from a generalized Lagrangian which depends only on the relative acceleration. From this Lagrangian the energy $E$ and total angular momentum $\mathbf{J}$ of the system can be computed which are known to be conserved up to 2PN order [@Kidder], i.e.in the absence of radiation reaction. The conserved energy is given as $$E=E_{N}+E_{PN}+E_{SO}+E_{2PN}+E_{SS} {\textcolor{black}{+E_{3PN}+E_{PNSO}
}}, \label{Energy}$$$$\begin{aligned}
E_{N} &=&\mu \left\{ {\frac{1}{2}}v^{2}-{\frac{Gm}{r}}\right\} , \\
E_{PN} &=&\frac{\mu }{c^{2}}\left\{ {\frac{3}{8}}(1-3\eta )v^{4}+{\frac{1}{2}}(3+\eta )v^{2}{\frac{Gm}{r}}+{\frac{1}{2}}\eta {\frac{Gm}{r}}\dot{r}^{2}+{\frac{1}{2}}\left({\frac{Gm}{r}}\right)^{2}\right\} , \\
E_{SO} &=&{\frac{G}{c^{2}r^{3}}}\mathbf{L}_{N}\mathbf{\cdot }
{\mbox{\boldmath$\sigma$}}, \\
E_{2PN} &=&\frac{\mu }{c^{4}}\biggl\{{\frac{5}{16}}(1-7\eta +13\eta
^{2})v^{6}-{\frac{3}{8}}\eta (1-3\eta ){\frac{Gm}{r}}\dot{r}^{4}+{\frac{1}{8}}(21-23\eta -27\eta ^{2}){\frac{Gm}{r}}v^{4} \\
&&+{\frac{1}{8}}(14-55\eta +4\eta ^{2})\left( {\frac{Gm}{r}}\right)
^{2}v^{2}+{\frac{1}{4}}\eta (1-15\eta ){\frac{Gm}{r}}v^{2}\dot{r}^{2}-{\frac{1}{4}}(2+15\eta )\left( {\frac{Gm}{r}}\right) ^{3} \\
&&+{\frac{1}{8}}(4+69\eta +12\eta ^{2})\left( {\frac{Gm}{r}}\right) ^{2}\dot{r}^{2}\biggr\}, \\
E_{SS} &=&{\frac{G}{c^{2}r^{3}}}\left\{ 3\left( \mathbf{\hat{n}\cdot S_{1}}\right) \left( \mathbf{\hat{n}\cdot S_{2}}\right) - \left(
\mathbf{S_{1}\cdot S_{2}}\right) \right\}, \\
{\textcolor{black}{ E_{3PN} }}&=&\frac{\mu}{c^6}\bigg\{ \left [\frac{3}{8}+
\frac{18469}{840}\eta\right ]{\left (\frac{Gm}{r}\right )}^{4}
+\left [\frac{5}{4}-\left (\frac{6747}{280}-\frac{41}{64}\pi^{2}\right )\eta
-\frac{21}{4}\eta^{2}+\frac{1}{2}\eta^{3}\right ]{\left (\frac{Gm}{r}\right )}^{3}v^{2}
\nonumber\\
& &
+\left [\frac{3}{2}+\left (\frac{2321}{280}-\frac{123}{64}\pi^{2}\right )\eta
+\frac{51}{4}\eta^{2} +\frac{7}{2}\eta^{3}\right ]{\left (\frac{Gm}{r}\right )}^{3}\dot{r}^{2}
\nonumber\\
& &
+\frac{1}{128}\left (35-413\eta+1666\eta^{2}-2261\eta^{3}\right
)v^{8}
+\frac{1}{16}(135-194\eta+406\eta^{2}-108\eta^{3}){\left (\frac{Gm}{r}\right )}^{2}v^{4}
\nonumber\\
& &
+\frac{1}{16}(12+248\eta-815\eta^{2}-324\eta^{3}){\left (\frac{Gm}{r}\right )}^{2}v^{2}\dot{r}^{2}
-\frac{1}{48}\eta(731-492\eta-288\eta^{2}){\left (\frac{Gm}{r}\right )}^{2}\dot{r}^{4}
\nonumber\\
& &
+\frac{1}{16}(55-215\eta+116\eta^{2}+325\eta^{3})\frac{Gm}{r}v^{6}
+\frac{1}{16}\eta(5-25\eta+25\eta^{2})\frac{Gm}{r}\dot{r}^{6}
\nonumber\\
& &
-\frac{1}{16}\eta(21+75\eta-375\eta^{2})\frac{Gm}{r}v^{4}\dot{r}^{2}
-\frac{1}{16}\eta(9-84\eta+165\eta^{2})\frac{Gm}{r}v^{2}\dot{r}^{4}
\bigg\}, \\
{\textcolor{black}{ E_{PNSO} }}&=&\frac{G\mu}{2c^3r^2}(\mathbf{\hat{n}\times
v})\mathbf{\cdot }\bigg\{ \mathbf{\Delta}\frac{\delta
m}{m}\left[(1-5\eta)v^2 + (2+\eta)\frac{Gm}{r}\right] -
3\mathbf{S}\left[(1+\eta)v^2 +\eta {\dot r}^2-\frac{2Gm}{3r}\right]
\bigg\}.\label{EnergyPNSO}\end{aligned}$$while the conserved total angular momentum is $$\mathbf{J}=\mathbf{L}+\mathbf{S}\,, \label{TotAngMom}$$where $$\mathbf{L}=\mathbf{L}_{N}+\mathbf{L}_{PN}+\mathbf{L}_{SO}+\mathbf{L}_{2PN}
{\textcolor{black}{+\mathbf{L}_{3PN} }}, \label{AngMom}$$and $$\begin{aligned}
\mathbf{L}_{PN} &=&{\frac{\mathbf{L}_{N}}{c^{2}}}\left\{ {\frac{1}{2}}v^{2}(1-3\eta )+(3+\eta ){\frac{Gm}{r}}\right\} , \\
\mathbf{L}_{SO} &=&{\frac{\mu }{c^{2}m}}\Biggl\{{\frac{Gm}{r}}\mathbf{\hat{n}\times }\left[ \mathbf{\hat{n}\times }\left( 2\mathbf{S}
+{\mbox{\boldmath$\sigma$}}\right) \right]
-{\frac{1}{2}}\mathbf{v\times }\left(
\mathbf{v\times } {\mbox{\boldmath$\sigma$}} \right) \Biggr\}, \\
\mathbf{L}_{2PN} &=&{\frac{\mathbf{L}_{N}}{c^{4}}}\biggl\{{\frac{3}{8}}(1-7\eta +13\eta ^{2})v^{4}-{\frac{1}{2}}\eta (2+5\eta ){\frac{Gm}{r}}\dot{r}^{2} \\
&&+{\frac{1}{2}}(7-10\eta -9\eta ^{2}){\frac{Gm}{r}}v^{2}+{\frac{1}{4}}(14-41\eta +4\eta ^{2})\left( {\frac{Gm}{r}}\right) ^{2}\biggr\} ,\\
{\textcolor{black}{ \mathbf{L}_{3PN} }}
&=&{\frac{\mathbf{L}_{N}}{c^{6}}}\biggl\{ \left [ \frac{5}{2}-\left
(\frac{5199}{280}-\frac{41}{32}\pi^{2}\right
)\eta-7\eta^{2}+\eta^{3}\right ]{\left (\frac{Gm}{r}\right )}^{3}
\nonumber\\
& &
+\frac{1}{16}(5-59\eta+238\eta^{2}-323\eta^{3} )v^{6}
+\frac{1}{12}(135-322\eta+315\eta^{2}-108\eta^{3}){\left
(\frac{Gm}{r}\right )}^{2}v^{2}
\nonumber\\
& &
+\frac{1}{24} (12-287\eta-951\eta^{2}-324\eta^{3}){\left
(\frac{Gm}{r}\right )}^{2}\dot{r}^{2}
+\frac{1}{8}(33-142\eta+106\eta^{2}+195\eta^{3})\frac{Gm}{r}v^{4}
\nonumber\\
& &
-\frac{1}{4} \eta (12-7\eta-75\eta^2)\frac{Gm}{r}v^{2}\dot{r}^{2}
+\frac{3}{8}\eta(2-2\eta-11\eta^{2})\frac{Gm}{r}\dot{r}^{4}
\biggr\}.\end{aligned}$$Notice that at the applied level of PN approximation there is no spin-spin contribution to $\mathbf{J}$.
The leading order radiative change of the conserved quantities $E$ and $\mathbf{J}$ is governed by the quadrupole formula [@IyerWill]. To lowest 2.5PN order the instantaneous loss of energy $E$ is given as [@Kidder] $$\begin{aligned}
\frac{dE_{{\textcolor{black}{N}}}}{dt}=-\frac{8}{15}\frac{G^3m^2\mu^2}{c^5r^4}\left(12v^2
- 11\dot{r}^2 \right)\,,\end{aligned}$$ while the radiative angular momentum loss is $$\begin{aligned}
\frac{d\mathbf{J}_{{\textcolor{black}{N}}}}{dt}=-\frac{8}{5}\frac{G^2m\mu}{c^5r^3}\mathbf{L}_{N}\left(2v^2
- 3\dot{r}^2 + 2\frac{Gm}{r}\right)\,.\end{aligned}$$ $$\begin{aligned}
\label{dEdt}
{\textcolor{black}{ \frac{dE_{RR}}{dt} }} &=&
\frac{dE_N}{dt}+\frac{dE_{PN}}{dt}+\frac{dE_{SO}}{dt}+\frac{dE_{2PN}}{dt}
+\frac{dE_{SS}}{dt}+\frac{dE_{2.5PN}}{dt}+\frac{dE_{PNSO}}{dt}\,.\end{aligned}$$ $$\begin{aligned}
{\textcolor{black}{
\frac{dE_{PN}}{dt} }} &=& - {2 \over 105} {G^3m^2 \mu^2 \over
c^7r^4}\biggl\{ (785-852\eta)v^4 -160(17-\eta) {Gm \over r}v^2 +
8(367-15\eta){Gm \over r} \dot r^2 \nonumber \\
&& -2(1487-1392\eta)v^2 \dot r^2 + 3(687-620\eta) \dot r^4 +
16(1-4\eta)\left({Gm \over r}\right)^2 \biggr\},\end{aligned}$$ $$\begin{aligned}
{\textcolor{black}{ \frac{dE_{SO}}{dt} }} = - {8 \over 15} {G^3m \mu \over
c^7r^6} \Bigg\{ {\bf L_N \cdot} \Bigg[ {\bf S} (78 \dot r^2 -80v^2
-8 {Gm \over r}) \mbox{} + {\delta m \over m}{\bf \Delta} (51 \dot
r^2 - 43v^2 + 4{Gm \over r} \Biggr] \Biggr\},\end{aligned}$$ $$\begin{aligned}
{\textcolor{black}{ \frac{dE_{2PN}}{dt} }} &=&-{8 \over
15}\frac{G^3\,m^2\,\mu^2}{c^9\,r^4} \left\{
\frac{1}{42} (1692 - 5497\eta
+ 4430\eta^2)v^6
-\frac{1}{14} (1719 - 10278\eta
+ 6292\eta^2) v^4\dot{r}^2\right.\nonumber \\
&&- \left.\frac{1}{21} (4446 - 5237\eta
+ 1393\eta^2)\frac{Gm}{r}\,v^4
+\frac{1}{14} (2018 - 15207\eta
+ 7572\eta^2)v^2\dot{r}^4\right.\nonumber \\
&&+ \left.\frac{1}{7} (4987 - 8513\eta
+ 2165\eta^2)\frac{Gm}{r}\,v^2\dot{r}^2\right.\nonumber \\
&&+\left. \frac{1}{756} (281473 + 81828\eta
+ 4368\eta^2)\left(\frac{Gm}{r}\right)^2\,v^2\right.\nonumber \\
&&- \left.\frac{1}{42} (2501 - 20234\eta
+ 8404\eta^2)\dot{r}^6
-\frac{1}{63} (33510 - 60971\eta
+ 14290\eta^2)\frac{Gm}{r}\,\dot{r}^4\right.\nonumber \\
&&- \left.\frac{1}{252} (106319 + 9798\eta
+ 5376\eta^2)\left( \frac{Gm}{r}\right)^2\,\dot{r}^2\right.\nonumber \\
&&+\left. \frac{2}{63} (-253 + 1026\eta
- 56\eta^2) \left( \frac{Gm}{r}\right)^3\right\},\end{aligned}$$ $$\begin{aligned}
{\textcolor{black}{ \frac{dE_{SS}}{dt} }} &=& - {4 \over 15} {G^3m \mu \over
c^7r^6} \biggl\{ - 3 ({\bf \hat n \cdot S_1})({\bf \hat n \cdot S_2}
) \left( 168 v^2 - 269 \dot r ^2 \right) + 3 ({\bf S_1 \cdot S_2})
\left( 47 v^2- 55 \dot r ^2 \right) \nonumber \\ && + 71 ( {\bf v
\cdot S_1})({\bf v \cdot S_2} ) - 171 \dot r \left[ ({\bf v \cdot
S_1})({\bf \hat n \cdot S_2}) + ({\bf \hat n \cdot S_1})( {\bf v
\cdot S_2}) \right] \biggr\} ,\end{aligned}$$ $$\begin{aligned}
{\textcolor{black}{ \frac{dE_{2.5PN}}{dt} }} &=& -\frac{32}{5} \frac{G^3 \mu^2
m^2}{c^{10} r^4} \dot{r}\eta\bigg( -\frac{12349}{210} \frac{Gm}{r}
v^4 +\frac{4524}{35} \frac{Gm}{r}
v^2\dot{r}^2 -\frac{2753}{126} \frac{G^2\, m^2}{r^2} v^2 \nonumber\\
&&-\frac{985}{14} \frac{Gm}{r} \dot{r}^4 + \frac{13981}{630}
\frac{G^2 m^2}{r^2} \dot {r}^2 -\frac{1}{315} \frac{G^3 m^3}{r^3}
\bigg)\,,\end{aligned}$$ $$\begin{aligned}
\label{dEdtPNSO}
{\textcolor{black}{ \frac{dE_{PNSO}}{dt} }} &=& -\frac{8}{105} \frac{G^3 \mu^2
m}{c^{10} r^5} \bigg\{ {(\mathbf{\hat{n}\times v})\mathbf{\cdot
}\mathbf{S}}\left[{\dot r}^4\left(
{3144}\eta- {2244}\right)+\frac{G^2m^2}{r^2}
\left({972}
+{166}\eta\right)\right.\nonumber\\
&&+\frac{Gm}{r}{\dot r}^2\left({170}\eta-{2866}\right)+{\dot
r}^2v^2\left(
{3519}-{5004}\eta\right)\nonumber\\
&& \left.+\frac{G m}{r}v^2\left({3504}-140\eta\right)
+v^4\left({1810}\eta-{1207}\right)\right]\nonumber\\
&&+{(\mathbf{\hat{n}\times v})\mathbf{\cdot
}\mathbf{\Delta}}\frac{\delta
m}{m} \left[{\dot r}^4\left(
{2676}\eta-\frac{7941}{4}\right) + \frac{G^2m^2}{r^2}\left(
126\eta-{109}\right)\right.\nonumber\\
&&+\frac{Gm}{r}{\dot r}^2\left({1031}\eta-\frac{6613}{2}\right)+{\dot r}^2v^2\left(
{2364} - {3621}\eta\right) \nonumber\\
&&\left.+\frac{G m}{r} v^2\left(\frac{4785}{2}-455\eta\right)
+v^4\left({1040}\eta-\frac{2603}{4} \right)\right] \bigg\} \,.\end{aligned}$$
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[^1]: email: [email protected]
[^2]: email: [email protected]
[^3]: email: [email protected]
[^4]: This paper is dedicated to the memory of our colleague and friend Péter Csizmadia a young physicist, computer expert and one of the best Hungarian mountaineers who disappeared in China’s Sichuan near the Ren Zhong Feng peak of the Himalayas October 23, 2009. We started to develop CBwaves jointly with Péter a couple of moth before he left for China.
|
---
abstract: 'Two mutually coupled chaotic diode lasers exhibit stable isochronal synchronization in the presence of self feedback. When the mutual communication between the lasers is discontinued by a shutter and the two uncoupled lasers are subject to self-feedback only, the desynchronization time is found to scale as $A_d\tau$ where $A_d>1$ and $\tau$ corresponds to the optical distance between the lasers. Prior to synchronization, when the two lasers are uncorrelated and the shutter between them is opened, the synchronization time is found to be much shorter, though still proportional to $\tau$. As a consequence of these results, the synchronization is not significantly altered if the shutter is opend/closed faster than the desynchronization time. Experiments in which the coupling between two chaotic-synchronized diode lasers is modulated with an electro-optic shutter are found to be consistent with the results of numerical simulations.'
author:
- Ido Kanter
- Noam Gross
- Einat Klein
- Evi Kopelowitz
- Pinhas Yoskovits
- Lev Khaykovich
- Wolfgang Kinzel
- Michael Rosenbluh
title: Synchronization of mutually coupled chaotic lasers in the presence of a shutter
---
Chaotic systems are characterized by an irregular motion which is sensitive to initial conditions and tiny perturbations. Nevertheless, two chaotic systems can synchronize their irregular motion when they are coupled [@Pikovsky]. When the coupling is switched off, any tiny perturbation drives the two trajectories apart. The separation is exponentially fast, and it is described by the largest Lyapunov exponents of a single system.
In this Letter we show that the trajectory dynamics of coupled chaotic systems which also poses time-delayed self-feedback, is different. In a system with self-feedback, which has also been investigated in the context of secure communication with chaotic lasers [@MCPF], the time scale for the separation of the trajectories is found to be much longer than the coupling time. On the other hand, when the coupling is switched on, resynchronization occurs on a faster time scale. We investigate this phenomenon numerically and show first experiments which support our numerical simulations. The demonstrated difference between de- and re-synchronization can be used to improve the security of public-channel communication with chaotic lasers [@MCPF].
Semiconductor (diode) lasers subjected to delayed optical feedback are known to displays chaotic oscillations. Two coupled semiconductor lasers exhibit chaos synchronization. Different coupling setups such as unidirectional or mutual coupling and variations of the strength of the self and coupling feedback result in different synchronization states: the two lasers can synchronize in a leader-laggard or anticipated mode [@shore99; @locquet01], as well as in two different synchronization states; achronal synchronization in which the lasers assume a fluctuating leading role, or isochronal synchronization where there is no time delay between the two lasers’ chaotic signals [@IsoPaper; @elsasser01; @MCPF; @exception; @Liu1].
In this Letter we focus on a symmetric setup, the time delay between the lasers is denoted by $\tau_c$ and the time delay of the self-feedback is denoted by $\tau_d$. In the event of $\tau_c=\tau_d=\tau$ and for a wide range of the mutual coupling strength, $\sigma$, and the strength of the self-feedback, $\kappa$, the stationary solution is isochronal synchronization [@IsoPaper; @MCPF; @Gross2006]. The quantity with which we measure the degree of synchronization between the two lasers is the time-dependent cross correlation, $\rho$, defined as
$$\label{eqrho}
\rho(\Delta t) = \frac{\sum_i(I_{A}^{i}-<I_{A}^{i }>)\cdot{(I_{B}^{i+\Delta t }-<I_{B}^{i+\Delta t}>})}{\sqrt[]{\sum_i(I_{A}^{i}-<I_{A}^{i }>)^{2}\cdot\sum_i(I_{B}^{i+\Delta t }-<I_{B}^{i+\Delta t}>)^{2}}}
\nonumber$$
where $I_{A}$ and $I_{B}$ are the time dependent intensities of lasers A and B and the summation is over times indicated by $i$. Isochronal synchronization is defined by the cross correlation, $\rho$, having a dominant peak at $\Delta t=0$.
We control the mutual coupling between the lasers by a shutter, located at a distance $c\tau/2$ from each one of the lasers, where $c$ is the speed of light. When the shutter is open the two lasers are mutually coupled with strength $\sigma$ and with self-feedback $\kappa$. When the shutter is closed the self-feedback, $\kappa_{e}$, is increased to a value of $\sigma+\kappa$ so that the total feedback in the open/closed states remains a constant. This is required so as to prevent a sudden drop in the overall feedback, which would typically destroy the synchronization immediately [@alhers98].
The two quantities of interest in this letter are the desynchronization time, $t_d$, and the resynchronization time, $t_r$. The desynchronization time is defined as the average required time for the correlation to decay to $C_d\rho(0)$ where $C_d<1$. The time is measured from the moment the shutter is closed and $\rho(0)$ is the average correlation in the isochronal phase. The resynchronization or recovery time is measured after the shutter has remained closed for a long period and the two chaotic lasers are uncorrelated, and $\kappa_e=\kappa+\sigma$. The shutter is then opened and the self-coupling strength is reduced to $\kappa$. The resynchronization time is defined as the average time required, from the shutter opening, for $\rho$ to increase from zero to $C_r\rho(0)$, where $C_r$ is a constant $\le 1$.
To numerically simulate the system we use the Lang-Kobayashi equations [@Kobayashi] that are known to describe a chaotic diode laser. The dynamics of laser $A$ are given by coupled differential equations for the optical field, $E$, the time dependent optical phase, $\Phi$, and the excited state population, $n$;
$$\begin{aligned}
\frac{dE_{A}}{dt}=\frac{1}{2}G_{N}n_{A}E_{A}(t)+\frac{C_{sp}\gamma[N_{sol}+n
_{A}(t)]}{2E_{A}(t)} ~~~~~~~~~~~~~~~~~~~~~~~~~~~ & &
\nonumber\\
+ \kappa E_{A}(t-\tau _{d})cos[\omega _{0} \tau + \Phi
_{A}(t)-\Phi_{A} (t-\tau
_{d})]~~~~~~~~~~~~~~~~~\nonumber\\
+ \sigma E_{B}(t-\tau _{c})cos[\omega _{0} \tau _{c}
+\Phi_{A}(t)-\Phi
_{B}(t-\tau _{c})]~~~~~~~~~~~~~~\nonumber\\
\frac{d\Phi_{A}}{dt}=\frac{1}{2}\alpha G_{N}n_{A}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ & &
\nonumber\\
-\kappa \frac{E_{A}(t-\tau_{c})}{E_{A}(t)}sin[\omega _{0} \tau +\Phi
_{A}(t)-\Phi_{A}
(t-\tau _{d})] ~~~~~~~~~~~~~~~~~ & & \nonumber\\
-\sigma \frac{E_{B}(t-\tau _{c})}{E_{A}(t-\tau _{c})}sin[\omega _{0}
\tau _{c} +\Phi_{A}(t)-\Phi _{B}(t-\tau
_{c})]~~~~~~~~~~~~~~~\nonumber\\
\frac{dn_{A}}{dt}=(p-1)J_{th}-\gamma n_{A}(t)-[\Gamma +
G_{N}n_{A}(t)]E_{A}^{2}(t)~~~~~~~~~~~~~~~~~~~ & &
\nonumber\\
\nonumber\end{aligned}$$
and likewise for laser B. The values and meaning of the parameters are those used in Ref. [@alhers98; @IsoPaper; @MCPF]. The tunable parameters, both in the simulations and in the experiment, are $\kappa$, $\sigma$ and $p$ the pump parameter which is the ratio of the actual laser injection current to the threshold current.
Figure 1 displays the calculated desynchronization time as a function of $\tau$ for $\kappa=\sigma=50ns^{-1}$, $\rho(0)> 0.99$, p=1.2, and $C_d=0.8$ and $0.9$. Each data point is averaged over $50$ samples, and $\rho(0)$ is measured by averaging a sliding window (sliding length is $1$ ns) over a length $\tau$, while the solid lines are obtained by a linear fit, $t_d = A_d\tau +
constant$. The calculation shows that $t_d$ scales linearly with $\tau$ with a slope which increase as $C_d$ decreases, and is near $7$ and $9$ for $C_d=0.9$ and $0.8$, respectively. A similar linear scaling of the desynchronization time was obtained for all values of $p$ in the range from $1$ to $1.5$, where for a given $C_d$, $t_d$ decreases with $p$. The simulation assumes an ideal shutter with instantaneous closing and opening times, and also assumes a discontinuous decrease of $\sigma$ to zero while $\kappa$ increases to $\kappa+\sigma$. We find that the linear scaling as well as the slope is robust to the following two experimentally necessitated perturbations: (a) a non-ideal shutter which closes gradually over a period of $10-20$ nanoseconds; (b) an imperfectly closed shutter, allowing residual mutual coupling of a few percent of $\sigma$ while in the closed state.
In the inset of Figure 1, the average $\rho$ as a function of time for $\tau=50$ $ns$ is presented for a case where the shutter was closed at t=0. It is clear that the decay of $\rho$ does not consist of a typical exponential decay. The striking result is that the correlation coefficient is almost a constant for a long initial period (first $\sim 250$ $ns$ for the parameters of figure 1) and then crosses over to an exponential decay for very long times. Because the event of closing or opening the shutter takes a time $\tau/2$ to propagate to the lasers it is reasonable to expect that $t_d$ > $\tau/2$. It is surprising, however, that $t_d$ scales linearly with $\tau$ with a prefactor which is significantly greater than $1$. It is also not obvious from the simulation (in which $\rho(0) > 0.99$),if such behavior can be observed in an experiment where $\rho(0)\le 0.9$.
The nearly constant value of $\rho$ after the coupling between the lasers is terminated calls for a theoretical explanation. Let us discuss the synchronization for the case $\kappa\sim\sigma$ where both lasers are driven by an almost identical delayed signal which is the sum of the self-feedback and the coupling beam. When the mutual coupling is switched off and replaced with stronger self-feedback, the system still feels its synchronized state for a period of length $\tau$, since the system is coupled to its history delayed by a time $\tau$. The only difference caused by the closing of the shutter is that the lasers no longer communicate with each other and each laser is coupled only to its own state. With time, a small difference in the driving signals develops. This small difference is amplified, since the system is chaotic, and this occurs stepwise in time intervals of length $\tau$. For each interval there is a constant distance between the two trajectories, which increases for the following interval. Only the envelope of these steps is described by the largest Lyapunov exponents, but not the dynamics itself. Hence desynchronization is very slow, and its time scale should be proportional to $\tau$. On the other hand, when the exchanged beam is switched on again, both lasers receive immediately an identical feedback signal (for $\kappa=\sigma$) and synchronize very fast, independent of $\tau$. We thus expect that the desynchronization time is insensitive to the values of $\kappa$ and $\sigma$ and should scale with $\tau$. In contrast the resynchronization time should be very sensitive to the difference, $\kappa-\sigma$.
The experimental setup, which confirms many of these numerical perdictions, is shown schematically in Figure 2. We use two semiconductor lasers emitting at $660$ $nm$ and operated close to their threshold currents. The temperature of each laser is stabilized to better than $0.01K$. The lasers are subjected to similar optical feedback and are mutually coupled by injecting a fraction of each one’s output power to the other. A fast electro-optic modulator, with measured closing/opening time of 15 $ns$, is introduced in the middle of the coupling optical path to enable closing and reopening of the mutual coupling. The optical setup is designed so as to compensate the sudden drop in the overall feedback power when the shutter is closed and the mutual coupling feedback drops to near zero. We thus use the shutter as a polarization beam splitter which divides the output power of the laser into two parts: one used for the self-feedback and the other for the mutual coupling channels. The opening and closing of the shutter merely changes the ratio of powers in the channels, but maintains the overall feedback power at a constant level. Without this precaution the sudden drop in feedback power would destroy synchronization immediately. This setup, however, does not prevent the change in phase of the laser field when the shutter changes its state. This residual effect decreases the level of synchronization by a small amount (as can be seen in Figure 3). The shutter also does not close hermetically and the leakage power to the mutual coupling channel in the closed state is measured to be $\sim$ 7% of the shutter open value.
The feedback strength of each laser is adjusted using a $\lambda/$2 wave plate and an optical diode (see Figure \[schema\]) and is set to a value which leads to a reduction of about 5$\%$ in the solitary laser’s threshold current [@Gross2006]. The lengths of the self-feedback and coupling optical paths are set to be equal to obtain stable isochronal synchronization [@MCPF]. Two sets of measurements are reported here, corresponding to two self-feedback optical paths with $\tau$ = $12.7$ $ns$ and $\tau$ = $23.6$ $ns$.
Two fast photodetectors (response time $<$ 500 $ps$) are used to monitor the laser intensities which are simultaneously recorded with a digital oscilloscope (500 MHz ,1 GS/sec). The correlation coefficient, $\rho$, is calculated by dividing the intensity traces into 10 $ns$ segments (each segment containing 10 points) and $\rho$ is calculated between matching segments and then averaged.
The measured correlation coefficient, $\rho$, is shown in Figure 3, in which the shutter is closed at t=0. The coupling power decays to its closed level in about 15 $ns$, limited by the speed of the shutter. The observed decay time is only slightly shorter than the decay time obtained in simulations for the same $\tau$, and as in the simulations, $\rho$ initially maintains a high and nearly constant value for $50-100$ $ns$, which is much longer than $\tau$. The four data curves shown in the figure correspond to different experimental parameters, indicated by the value of $p$ and to the two different values of $\tau$.
Figure 3 shows that $t_d$ increases with $\tau$ and the inset of Figure 3, depicting $t_d$ as a function $p$, indicates that $t_d$ is a decreasing function of $p$. The inset of Figure 3 also demonstrates that the different decay curves all collapse to a single decay curve, independent of $p$, when scaled by a factor of $1.7$ which is very close to the ratio of $\tau_1 / \tau_2=23.6/12.7
\sim 1.86$. The numerical simulations for larger $\tau$ also exhibit such data collapse when scaled by $\sim \tau_1/\tau_2$ resulting in scaled decay curves which are independent of $p$[@prep]. This result and the linear scaling of $t_d$ for a given $p$ indicate $$t_d(\tau,p) \propto \tau g(p)$$ where $g(p)$ is a function characteristic of the specific diode laser used. For small $\tau$ finite size effects are expected as a result of the positive constant in the linear scaling shown in Figure 1. Indeed, for $\tau=12.7$ and $23.6$ $ns$, the numerical results indicate that the average ratio $t_d(\tau_1,p)/t_d(\tau_2,p)
\sim 1.75$, which is in surprisingly good agreement with the experimental result of $1.7$.
The simulations also indicate that a transition from the low frequency fluctuation (LFF) regime to the fully developed coherent collapse regime [@FCDC] occurs at $p \sim 1.35$, which is close to the experimental value of $p$ (inset of Figure 3) where $t_d$ becomes almost independent of $p$[@prep]. Though the decay time obtained from the simulations is longer that the decay time observed in the experiment, this is not surprising, since in the simulations the systems are initially correlated to a very high level ($\rho(0)>0.99$) while in the experiments the initial correlation is $\rho(0)\le 0.9$
We now turn to examine the scaling of the resynchronization time or the recovery time as a function of $\tau$. In the simulations we start with two uncoupled systems with self-feedback $\kappa_e=\kappa+\sigma$. When the shutter is opened at $t=0$, $\kappa_e$ is reduced discontinuously to $\kappa$ and the mutual coupling is set to $\sigma$. For all examined cases, with $\kappa
\ne \sigma$, our calculations indicate that the resynchronization time also scales linearly with $\tau$. This scaling is exemplified in Figure 4 for $C_r=0.9$, $\kappa=60 ns^{-1}$ and $\sigma=40
ns^{-1}$.
The inset of Figure 4 displays the resynchronization time as a function of $\kappa-\sigma$ for a given $\tau$. It appears that this difference, rather than the coupling strength itself, is what controls the resynchronization time, and $t_r$ scales almost linearly and symmetrically with $\kappa-\sigma$. For $\kappa=\sigma$, $t_r$ is very fast and simulations indicate that it is independent of $\tau$ (limited by the fixed size of the sliding window) as expected. In all examined cases, the prefactor of the linear scaling of the resynchronization time was found to be $\ll
A_d$, indicating $t_r < t_d$. We also observed similar behavior, i. e. $t_r<t_d$, in the experiment, though quantitative determination of the experimental resynchronization time is complicated by ringing in the high voltage electronics used to turn on the modulator and by the fact that the modulator response time is as long as 15 $ns$, which is comparable to $t_r$.
Although experimentally we cannot accurately measure the resynchronization time, we show, in Figure 5, a demonstration of the persistence of the synchronization between the lasers upon repeated closing-opening operations of the shutter. Shown is the typical behavior of $\rho$ while the shutter is closed for $\sim40$ $ns$ and then reopened. The other parameters of the experiment are $\tau=12.6$ $ns$ and $p \sim 1.08$. The cross correlation coefficient $\rho$ is not affected, by the closing/reopening of the shutter and the changing of $\kappa_e$ and $\sigma$, since $t_d
>40$ $ns$.
The results reported above for re/de-synchronization times, which were also obtained recently for chaotic maps [@prep], demonstrate the possibility of establishing a reliable chaos based communication channel even while the communication between the lasers is interrupted by relatively long intervals. We expect that these effects will play an important role in advanced secure communications using mutually chaotic lasers.
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P. Rees, P.S. Spencer, I. Pierce, S. Sivaprakasam, and K.A. Shore, Phys. Rev. A [**68**]{}, 033818 (2003).
R. Vicente, S. Tang, J. Mulet, C.R. Mirasso and J.M. Liu, Phys. Rev. E [**70**]{}, 046216 (2004).
N. Gross, W. Kinzel, I. Kanter, M. Rosenbluh and L. Khaykovich, Opt. Comm. (in press) and nlin.CD/0604068.
V. Alhers, U. Parlitz, and W. Lauterborn, Phys. Rev. E [**58**]{}, 7208 (1998);
R. Lang, and K. Kobayashi, IEEE J. Quantum Electron. QE-16, 347 (1980).
T. Heil, I. Fischer, and W. Elsasser, Phys. Rev. A 58, R2672 (1998)
W. Kinzel et. al. (unpublished).
|
---
bibliography:
- 'biblio.bib'
title: The inference of gene trees with species trees
---
Version dated:
[**The inference of gene trees with species trees**]{}
[Gergely J. Szöllősi$^1$, Eric Tannier$^{2,3,4}$, Vincent Daubin$^{2,3}$, Bastien Boussau$^{2,3}$]{}\
[ *$^1$ELTE-MTA “Lendület” Biophysics Research Group, Pázmány P. stny. 1A., 1117 Budapest, Hungary;\
$^2$Laboratoire de Biométrie et Biologie Evolutive, Centre National de la Recherche Scientifique, Unité Mixte de Recherche 5558, Université Lyon 1, F-69622 Villeurbanne, France;\
$^3$Université de Lyon, F-69000 Lyon, France;\
$^4$Institut National de Recherche en Informatique et en Automatique Rhône-Alpes, F-38334 Montbonnot, France;*]{}\
[**Corresponding author:**]{} Bastien Boussau, Laboratoire de Biométrie et Biologie Evolutive, Centre National de la Recherche Scientifique, Unité Mixte de Recherche 5558, Université Lyon 1, F-69622 Villeurbanne, France; Université de Lyon, F-69000 Lyon, France; E-mail: [email protected]\
This project was supported by the French Agence Nationale de la Recherche (ANR) through Grant ANR-10-BINF-01-01 “Ancestrome”. GJSz was supported by the Marie Curie CIG 618438 “Genestory” and the Albert Szent-Györgyi Call-Home Researcher Scholarship A1-SZGYA-FOK-13-0005.
|
---
author:
- 'Beuermann, K.'
- 'Burwitz, V.'
- 'Reinsch, K.'
- '1 Schwope, A.'
- 'Thomas, H.-C. [^1]'
date: 'Received 3 September 2019; accepted 2 January 2020'
title: |
Neglected X-ray discovered polars: II: The peculiar\
eclipsing binary HY Eridani
---
Introduction
============
Of the more than 1200 cataclysmic variables in the final 2016 edition of the @ritterkolb03 catalog, 114 are confirmed polars (or AM Herculis binaries), which contain a late-type main sequence star and an accreting magnetic white dwarf in synchronous rotation. The name “polar” was coined by @krzeminskiserkowski77 to describe the high degree of circular polarization, which became one of the hallmarks of the class. Another is the large portion of the bolometric luminosity emitted in high states of accretion in form of soft and hard X-ray emission, which led to the discovery of the majority of the known systems. Many individual polars are characterized by idiosyncrasies, which distinguish them from their peers and provide special insight into the physics of polars. Unresolved questions relate, for example, to the physics of accretion [@bonnetbidaudetal00; @busschaertetal15; @bonnetbidaudetal15], various aspects of close-binary evolution [@webbinkwickramasinghe02; @liebertetal05; @kniggeetal11], and the generation and structure of the magnetic field of the white dwarf (WD) [@beuermannetal07; @wickramasingheetal14; @ferrarioetal15].
Our optical programs for identifying high-galactic latitude ROSAT X-ray sources [@thomasetal98; @beuermannetal99; @schwopeetal02] have led to the discovery of 27 new polars. Twenty sources have been described in previous publications. In this series of three papers, we present results on the remaining seven. Paper I [@beuermannetal17] describes V358Aqr (=RXJ2316–05), a system that experiences giant flares on its secondary star. Here we present a comprehensive analysis of the eclipsing polar (=RXJ0501–03) based on data collected over three decades. Our early conference paper [@burwitzetal99] represents the only previous account of the system in the literature. The third paper of this series will contain shorter analyses of RXJ0154$\!-\!59$, RXJ0600$\!-\!27$, RXJ0859+05, RXJ0953+14, and RXJ1002$\!-\!19$, of which three have not been addressed previously either.
Observations {#sec:obs}
============
X-ray data {#sec:obsx}
----------
, located at RA(2000)=$05^\mathrm{h}01^\mathrm{m}46\farcs4$, DEC(2000)= $-03\degr59\arcmin20\arcsec$ ($l,b\!=\!203.5,-26.1$) was discovered as a very soft X-ray source in the RASS [@bolleretal16][^2] and identified by us spectroscopically with an eclipsing polar [@beuermannthomas93; @beuermannetal99; @burwitzetal99]. Follow-up pointed ROSAT observations were performed 1992 and 1993 with the Position Sensitive Proportional Counter (PSPC) as the detector and 1995 and 1996 with the High Resolution Imager (HRI). These data were originally published by @burwitzetal99 and reanalyzed for the present study. We also analyzed the previously unpublished data taken in 2002 with XMM-Newton equipped with the EPIC camera. On all occasions, HY Eri was encountered with an X-ray flux that corresponds to a high or near high state[^3] (Table \[tab:xray\]).
----------------------- ------------ -------------- ------- --------- ------
\[-5ex\]
\[-1.0ex\] Dates Instrument Band Total State Ref.
(keV) (ks)
\[0.5ex\]
\[-1ex\] 24–26Aug1990 RASS PSPC $0.1\!-\!2$ 0.5 high (1)
24Feb1992 ROSAT PSPC $0.1\!-\!2$ 2.5 high (1)
15–22Feb1993 ROSAT PSPC $0.1\!-\!2$ 1.9 high (1)
8–16Sep1995 ROSAT HRI $0.1\!-\!2$ 16.5 high (1)
26Feb–19Mar1996 ROSAT HRI $0.1\!-\!2$ 28.8 high (1)
24Mar2002 XMM MOS+pn $0.2\!-\!10$ 6.6 high (2)
\[1.0ex\]
----------------------- ------------ -------------- ------- --------- ------
: Time-resolved X-ray observations of
\
\[tab:xray\]
----------------------------- -------- --------- -------- ------- --------- ------
\[-5ex\]
\[-1.0ex\] Dates Number Band Expos. Total State Tel.
nights (s) (h)
\[0.5ex\]
\[-1ex\] Feb 1994, Jan 1996 3 V 20/60 11.7 high (1)
3 – 5 Feb 2001 2 R,Gunni 60 1.1 high (2)
17 – 22 Jan 2010 5 WL 10/60 10.6 high (3)
8 – 18 Nov 2010 3 WL 10/60 4.2 interm. (3)
Feb – Oct2011 3 WL 10/60 3.6 interm. (3)
Aug 2014 – Jan 2015 10 WL 15/60 8.0 low (3)
24/27 Oct 2016 2 grizJHK 35 5.1 interm. (4)
Sep 2017 – Jan 2019 23 WL 15/60 34.8 low (5)
Nov 2017 2 grizJHK 35 4.7 low (4)
Feb 2018 1 grizJHK 35 3.0 low (4)
\[1.0ex\]
----------------------------- -------- --------- -------- ------- --------- ------
: Journal of time-resolved optical photometry
\
\[tab:phot\]
----------------------- ------------ -------- ------ -------- ------- ------- --------
\[-5ex\]
\[-1.0ex\] Dates Band Resol. Num- Expos. Total State Instr.
(Å) (Å) ber (min) (h)
\[0.5ex\]
\[-1ex\] 13-17Dec1993 3500–9500 25 50 10.0 10.6 high (1)
14Nov1995 3800–9119 25 24 2.0 1.1 high (2)
15Nov1995 3869–5109 8 17 10.0 3.2 high (2)
20Nov2000 6340–10400 10 8 0.5 0.5 high (3)
20Nov2000 6340–10400 10 1 10.0 (3)
31Dec2008 3800–9200 10 40 6.5 5.6 low (4)
\[1.0ex\]
----------------------- ------------ -------- ------ -------- ------- ------- --------
: Time-resolved spectroscopy and circular spectropolarimetry
\
\[tab:spec\]
Optical photometry {#sec:phot}
------------------
Orbital $BVRI$ light curves and $V$-band eclipse light curves were measured in 1994 and 1996 with the ESO/Dutch 0.9m telescope [@burwitzetal99]. Extensive white-light (WL) photometry, performed between 2010 and 2019 with the 1.2m MONET/N and MONET/S telescopes at the McDonald Observatory and the South African Astrophysical Observatory, respectively, allowed us to establish an alias-free long-term ephemeris. Seven-color $grizJHK$ photometry was performed with the MPG/ESO 2.2m telescope equipped with the GROND[^4] photometer in 2016, 2017, and 2018. A log of the observations is provided in Table \[tab:phot\]. We measured magnitudes relative to the dM1-2 star SDSS050146.02-040042.2 (referred to as C1), which is located 43 E and 4 N of and has Sloan AB magnitudes $g\!=\!16.91, r\!=\!16.38, i\!=\!16.23$, and $z\!=\!16.17$. Its color, $g\!-\!i\!=\!0.68$, is similar to the low-state Sloan color of , $g\!-\!i\!=\!0.71$. is separated by only 62 from the center of a galaxy with Sloan $r\!=\!18.44$. All accepted eclipse light curves were taken in sufficiently good seeing to escape spillover from the galaxy.
{height="89.0mm"} {height="89.0mm"}
Optical spectroscopy {#sec:obsspec}
--------------------
Follow-up time-resolved spectrophotometric observations of in its high state were performed in 1993 and 1995, using the ESO 1.5m telescope with the Boller & Chivens spectrograph and the ESO/MPG 2.2 m telescope with EFOSC2, respectively. In the latter run, grisms G1 and G3 yielded low- and medium-resolution spectra with FWHM resolutions of 25Å and 8Å that covered the entire optical band and the blue band, respectively. Using the photometrically established eclipse ephemeris, a 10min exposure in the near-total eclipse was taken with the ESO/VLT UT1 equipped with FORS1 on 20 November 2000. Grating G300I provided coverage of the red part of the spectrum, which is dominated by the secondary star of . The magnetic nature of the WD was studied in 2008 by phase-resolved low-resolution circular spectropolarimetry performed with the ESO/VLT UT2 and FORS1. Grism G300V provided coverage from 3800–9200Å. Table \[tab:spec\] lists the wavelength ranges, number of spectra, exposure times, and total times spent on source.
{height="89.0mm"} {height="91.0mm"}
Synthetic white light photometry {#sec:obswl}
--------------------------------
As described in PaperI, we performed synthetic photometry in order to tie the WL measurements obtained with the MONET telescopes into the standard $ugriz$ system. We defined a MONET-specific WL AB magnitude $w$, which has its pivot wavelength $\lambda_\mathrm{piv}\!=\!6379$Å in the red part of the Sloan $r$ band. For a star with the colors of comparison star C1, the color difference is $w\!-\!r\!=\!+0.07$. Hence, unity relative WL flux corresponds to $r\!=\!16.38$ and $w\!=\!16.45$. For WL measurements of other stars, $w$ is a measured quantity and is related to Sloan $r$ by $r\!=\!w-(w\!-\!r)_\mathrm{syn}$. For a wide range of incident spectra, the synthetic color $|(w\!-\!r)_\mathrm{syn}|\!\lesssim\!0.1$. Hence, $w\!\simeq\!r$ is typically correct within 0.1 mag, except for very red stars.
Optical light curves {#sec:res}
====================
Orbital light curves {#sec:olc}
--------------------
In Fig. \[fig:lc\], we show the $V$-band and WL light curves in the high states of 11 January 1996 and 19 January 2010, respectively, the WL light curve in the intermediate state of 14-18 November 2010, and the $riz$ low-state light curves of 24-25 November 2017. Orbital phase $\phi\!=\!0$ is defined by the eclipse ephemeris provided in Sect. \[sec:ephem\]. reached orbital maxima of $V\!=\!16.8$ and $w\!=\!17.1$ in the high states and $w\!=\!18.7$ in the intermediate state. The 2017 peak magnitudes were $z\!=\!19.1$ and $i\!=\!19.7$, while $r$ stayed at $21$ throughout the orbit outside eclipse. In all states, the light curves exhibit the signatures of emission from two accretion regions, being shaped by cyclotron beaming. The same holds for the light curves in the right panel of Fig. \[fig:meanspec\]. Borrowing from the insight provided by the low-state spectropolarimetry (Sect. \[sec:cyc\]), we identify, for instance, the double-humped $z$-band light curve in the lower left panel of Fig. \[fig:lc\] as cyclotron emission in the 4th harmonic from two accretion regions best viewed at phases $\varphi\!\simeq\!0.35$ and $0.85$. The light curves in WL are less easily interpreted because of the lack of wavelength resolution. We loosely refer to the emission regions best seen around $\phi\!=\!0.85$ and $\phi\!\simeq\!0.35$ as “pole1" and “pole2" or “spot1” and “spot2”, respectively. In the high state, the emissions from both poles become an inextricable conglomerate. The available evidence suggests that is a permanent two-pole emitter.
Eclipse light curves {#sec:eclipse}
--------------------
We collected a total of 41 eclipses of the hot spot on the WD by the secondary star, 13 in various high states, eight in intermediate states, and 20 in low states. Ingress and egress of the hot spot take place in typically less than 20s. In Fig \[fig:ecl\], we show the eclipse light curves in the three states at the original time resolutions of 13s (exposure and readout) for the high and intermediate states and and 22s for the low state. A characteristic feature of the high state is the delayed eclipse of the accretion stream. This component disappears, when the accretion rate drops.
------------------ -------------- --- ------- ----------- --------- ------- ------- --------
\[-5ex\]
\[-1.5ex\] Cycle BJD(TDB) n Error $O\!-\!C$ State Expos Band Instr.
2400000+ (s) (s) (s)
\[0.5ex\]
\[-1ex\] -54428 49390.622004 1 20.0 7.6 high 20 V (1)
-48518 50093.728987 1 15.0 -14.8 high 60 V (1)
-48511 50094.561847 1 15.0 -8.2 high 30 V (1)
-48510 50094.680642 1 15.0 -23.2 high 30 V (1)
-33598 51868.747892 1 21.5 9.8 high 60 Spec (2)
-33229 51912.647277 1 15.0 -7.9 high 60 R (3)
-33213 51914.550874 1 10.0 0.1 high 60 R (3)
-33212 51914.669824 1 10.0 -1.5 high 60 Gunni (3)
-5483 55213.563377 1 2.1 1.3 high 10 WL (4)
-5474 55214.634061 1 3.2 -2.0 high 10 WL (4)
-5465 55215.704814 1 1.2 0.8 high 10 WL (4)
-5449 55217.608337 1 1.2 2.3 high 10 WL (4)
-5439 55218.797986 1 2.6 -1.4 high 10 WL (4)
-3000 55508.963566 1 2.2 -1.1 interm. 10 WL (4)
-2959 55513.841263 1 1.8 -4.1 interm. 10 WL (4)
-2958 55513.960283 1 1.2 0.3 interm. 10 WL (4)
-2916 55518.957004 1 1.5 2.0 interm. 10 WL (4)
-2129 55612.585630 1 2.8 -1.3 interm. 10 WL (4)
-17 55863.848336 1 2.1 0.2 interm. 10 WL (4)
0 55865.870789 1 1.1 -1.7 interm. 10 WL (4)
8675 56897.927726 3 5.0 15.6 low 20 WL (4)
9892 57042.713137 1 4.2 19.5 low 15 WL (4)
15315 57687.882864 1 10.0 56.2 interm 33 iz (5)
18137 58023.613860 1 1.7 78.9 low 15 WL (6)
18279 58040.507455 1 2.8 77.7 low 15 WL (6)
18635 58082.860538 1 10.0 85.6 low 33 iz (5)
18800 58102.490416 1 4.6 84.0 low 15 WL (6)
19496 58185.292928 2 3.8 86.9 low 15 WL (6)
21070 58372.550511 4 3.0 109.1 low 15 WL (6)
21322 58402.530740 3 4.2 111.0 low 15 WL (6)
21835 58463.561939 2 2.9 116.4 low 15 WL (6)
22237 58511.387633 1 6.9 127.2 low 15 WL (6)
\[1.0ex\]
------------------ -------------- --- ------- ----------- --------- ------- ------- --------
: Observed mid-eclipse times of
\
\
\[tab:ephem\]
The eclipse was modeled by the occultation of a circular disk of uniform surface brightness, which represents either the WD or the hot spot on the WD. The parameters of the fit are the mid-eclipse time, the FWHM, and the duration of ingress or egress. We improved the statistics and reduced the timing error in the low state, when the star became as faint as $w\!=\!20.7$ outside eclipse, by fitting up to $n\!=\!4$ eclipses on a barycentric time scale if taken nearby in time. The resulting mid-eclipse times are listed in Table \[tab:ephem\]. In the cases with $n\!>\!1$, the cycle number of the best-defined eclipse is quoted. The measured FWHM of the eclipse is the same for the different accretion states with a mean value of $\Delta\,t_\mathrm{ecl}\!=\!910.6\!\pm\!1.5$s or $\Delta\,\phi\!=\!0.08859\!\pm\!0.00015$ in phase units. This is the longest relative eclipse width of all polars. We did not detect the ingress and egress of the WD photosphere because our WL observations are dominated by cyclotron emission and the measurements in the GROND $gr$ filters lacked the required time resolution. The quoted mid-eclipse times refer to the hot spot on the WD and may deviate from true inferior conjunction of the secondary by up to $\sim\!30$s or $\sim\!0.003$ in phase.
The relative WL flux in the totality is the same in the high, intermediate, and low states, with a mean of $0.0060\pm0.0005$ or $w\!=\!22.0\!\pm\!0.1$. Using a color correction $w\!-\!r\!\simeq\!-0.3$, appropriate for the secondary star, we find $r\!\simeq\!22.3$, which compares favorably with the result of our spectrophotometry in eclipse reported Sect. \[sec:vlt\]. Hence, the residual WL flux in eclipse is largely that of the secondary star.
Eclipse ephemeris {#sec:ephem}
-----------------
We corrected the UTC eclipse times to Barycentric Dynamical Time (TDB), using the tool provided by @eastmanetal10[^5], which accounts also for the leap seconds. The complete set of eclipse times is presented in Table \[tab:ephem\]. The 2010 and 2011 data and the scanty earlier data define the alias-free linear ephemeris $$T_\mathrm{ecl}\!=\!\mathrm{BJD(TDB)}~~2455865.87081(1) + 0.118969076(2)\,E,~~
\label{eq:ephem1}$$ with $\chi^2\!=\!19.6$ for 18 degrees of freedom (solid line, cycle numbers $E\!\le\!0$). The $O\!-\!C$ offsets from the ephemeris of Eq. \[eq:ephem1\] are displayed in Fig. \[fig:ecl\], right panel. This ephemeris became increasingly invalid after 2011 and the more recent data are well represented by a cubic ephemeris for $E\!>\!-6000$ (solid curve). Currently, $O\!-\!C$ relative to the linear ephemeris of Eq. (1) has exceeded 2min, which is much too large to be explained by wandering of the accretion spot. The mid-eclipse times of 2017 to 2019 follow the linear ephemeris $$T_\mathrm{ecl}\!=\!\mathrm{BJD(TDB)}~2455865.86951(14) + 0.118969198(8)\,E,~~~
\label{eq:ephem2}$$ implying a period change relative to Eq. (1) of $10.5\!\pm\!0.7$ms. The mean rate of the period variation between 2011 and 2018 is $\dot P\!\simeq\!{5\times 10^{-11}}$ss$^{-1}$. The period change started approximately, when the system entered a prolonged low state after 2011. This is likely a coincidence, however, because it was also in a low state in 2008 and temporarily in an intermediate state in 2016.
------------------------ ---------- ----------------- ---------------------- ----------------- -----------------
\[-3.0ex\]
\[-1.0ex\] Observation Detector $T_\mathrm{bb}$ $N_\mathrm{H}$ $F_\mathrm{bb}$ $F_\mathrm{th}$
(eV) ($10^{20}$/cm$^2$s)
\[0.5ex\]
\[-1ex\] RASS 1990 PSPC 30 : 3.0 fixed 9 :
ROSAT 1992/93 PSPC $37\!\pm\!6$ $8.0\!\pm\!2.0$ $43\!\pm\!30$
$44\!\pm\!3$ 6.0 fixed 13.0
$50\!\pm\!3$ 3.0 fixed 1.7
XMM 2002 MOS1+pn $45\!\pm\!6$ $1.4\,+\!2.0,-\!1.4$ 0.5 0.08
$40\!\pm\!2$ 3.0 fixed 1.5 0.09
\[1.0ex\]
------------------------ ---------- ----------------- ---------------------- ----------------- -----------------
: Parameters of X-ray spectral fits for
\
\[tab:xspec\]
X-ray light curves and spectra {#sec:xray}
==============================
The ROSAT soft X-ray light curves taken between 1990 and 1996 [@burwitzetal99] show a structured bright phase that extends from $\phi\!\simeq\!0.4$ to $\phi\!\simeq\!0.8$ with low-level emission over the remainder of the orbit. Binned versions of these data are shown in Fig. \[fig:lc\], right panels. We also included the previously unpublished light curve measured with XMM-Newton and the MOS and pn detectors of the EPIC camera in 2002. The lower right panel of Fig. \[fig:lc\] shows the mean count rates of the two MOS detectors and the pn-detector, respectively, with the former adjusted by a factor of 3.5 upward. The X-ray bright phases in the ROSAT and XMM-Newton light curves show some similarity with the WL optical light curve of November 2010, suggesting that the observed intense X-ray emission originates from pole1, phase-modulated by rotation of the WD and possibly by internal absorption. The low-level emission around $\phi\!=\!0.3$ may stem from pole2. The bright phase reached count rates around 1.0PSPCctss$^{-1}$, 0.4HRI ctss$^{-1}$, and 1EPIC-pn ctss$^{-1}$, suggesting that was in some form of high or intermediate-to-high state during these observations.
![*Top:* Mean flux-calibrated low-resolution spectra of in the high states of 1993 and 1995. For comparison, the eclipse spectrum of 2000 (blue curve) and the mean out-of-eclipse spectrum in the 2008 low state (red curve) are added on the same ordinate scale. *Bottom:* Mean radial velocities of the broad emission lines of [H$\beta$]{} and [HeII$\lambda$4686]{} (open circles) and of the line peaks (cyan dots) derived from medium-resolution spectra taken on 15 November 1995.[]{data-label="fig:spec"}](36626f3a.eps){height="89.0mm"}
![*Top:* Mean flux-calibrated low-resolution spectra of in the high states of 1993 and 1995. For comparison, the eclipse spectrum of 2000 (blue curve) and the mean out-of-eclipse spectrum in the 2008 low state (red curve) are added on the same ordinate scale. *Bottom:* Mean radial velocities of the broad emission lines of [H$\beta$]{} and [HeII$\lambda$4686]{} (open circles) and of the line peaks (cyan dots) derived from medium-resolution spectra taken on 15 November 1995.[]{data-label="fig:spec"}](36626f3b.eps){height="89.0mm"}
The ROSAT and XMM-Newton spectra (not shown) are characterized by an intense soft X-ray and an underlying hard X-ray component, of which the latter is well covered only in the XMM-Newton run. We fitted the bright-phase spectra with the sum of a blackbody of temperature k$T_\mathrm{bb}$ and a thermal component with the temperature fixed at 10keV, both attenuated by a column density $N_\mathrm{H}$ of cold interstellar matter of solar composition[^6]. The fit parameters are listed in Table \[tab:xspec\]. The values of k$T_\mathrm{bb}$ and $N_\mathrm{H}$ differ substantially, indicating either true variability or systematic uncertainties caused by the poor energy resolution of the ROSAT PSPC, the lack of spectral coverage of the XMM-Newton detectors below 0.2keV, or the inadequacy of fitting a multi-temperature source by a single blackbody [@beuermannetal12]. In any case, is more strongly absorbed than other polars. For the combined PSPC fit to the 1992 and 1993 data, $N_\mathrm{H}$ exceeds the total galactic column density $N_\mathrm{H,gal}\!\simeq\!{5.2\times 10^{20}}$H-atomscm$^{-2}$ at the position of [@hi4pi16][^7]. The galactic extinction at the position of , $A_\mathrm{V}\!=\!0.172$ [@schlegeletal98], and the $N_\mathrm{H}\!-\!A_\mathrm{V}$ conversion factor of @predehlschmitt95 yield $N_\mathrm{H}\!\simeq\!{3\times 10^{20}}$H-atoms cm$^{-2}$, which corresponds to the galactic dust layer. Forcing the fits to this value, the ROSAT and XMM-Newton fits yield similar blackbody temperatures and a bolometric soft X-ray flux of $F_\mathrm{X}\!\simeq\!{1.5\times 10^{-11}}$[ergcm$^{-2}$s$^{-1}$]{}.
Optical spectroscopy and spectropolarimetry {#sec:oss}
===========================================
High-state spectra {#sec:spec}
------------------
In Fig. \[fig:spec\], we show the mean low-resolution spectra taken on 13–17 December 1993 and 14 November 1995, when was in high states. They are characterized by a blue continuum, strong Balmer, HeI, and HeII emission lines, the Balmer jump in emission, and weak broad cyclotron lines at the red end. Phase-resolved radial velocities were measured from medium-resolution blue spectra taken on 15 November 1995 (not shown). The Balmer and HeII emission lines were single peaked with asymmetric bases, extending to beyond $\pm\!1000$[kms$^{-1}$]{}. We measured the mean central wavelengths of the broad components and the positions of the line peaks of [H$\beta$]{}, [H$\gamma$]{}, and [HeII$\lambda$4686]{}. The former has a velocity amplitude of $520\pm24$[kms$^{-1}$]{} and reaches maximum positive radial velocity at $\phi\!=\!0.91\pm0.02$. This phasing is consistent with the plasma motion in the magnetically guided stream that leads to pole1 and away from the observer at an azimuth of $\psi\sim35^\circ$, measured from the line connecting the two stars. The line peak displays a small radial velocity with zero crossing near $\phi\!=\!0.80$. This component could arise from the ballistic accretion stream near L$_1$.
![*Top:* Eclipse light curve of the [H$\alpha$]{} emission line flux in [ergcm$^{-2}$s$^{-1}$]{} and of the underlying continuum flux observed on 20 November 2000. *Center:* Eclipse spectrum (black curve) thermal hydrogen spectrum adjusted to fit the [H$\alpha$]{} line flux (blue or red curve, see text). The ordinate is in units of $10^{-16}$ ergscm$^{-2}$s$^{-1}$Å$^{-1}$. *Bottom:* Difference spectra on the same color coding fitted by a dM6 star (red) and a dM5 star (blue, shifted upward by 0.05 units).[]{data-label="fig:eclspec"}](36626f4a.eps){height="89.0mm"}
![*Top:* Eclipse light curve of the [H$\alpha$]{} emission line flux in [ergcm$^{-2}$s$^{-1}$]{} and of the underlying continuum flux observed on 20 November 2000. *Center:* Eclipse spectrum (black curve) thermal hydrogen spectrum adjusted to fit the [H$\alpha$]{} line flux (blue or red curve, see text). The ordinate is in units of $10^{-16}$ ergscm$^{-2}$s$^{-1}$Å$^{-1}$. *Bottom:* Difference spectra on the same color coding fitted by a dM6 star (red) and a dM5 star (blue, shifted upward by 0.05 units).[]{data-label="fig:eclspec"}](36626f4b.eps){height="89.0mm"}
![*Top:* Eclipse light curve of the [H$\alpha$]{} emission line flux in [ergcm$^{-2}$s$^{-1}$]{} and of the underlying continuum flux observed on 20 November 2000. *Center:* Eclipse spectrum (black curve) thermal hydrogen spectrum adjusted to fit the [H$\alpha$]{} line flux (blue or red curve, see text). The ordinate is in units of $10^{-16}$ ergscm$^{-2}$s$^{-1}$Å$^{-1}$. *Bottom:* Difference spectra on the same color coding fitted by a dM6 star (red) and a dM5 star (blue, shifted upward by 0.05 units).[]{data-label="fig:eclspec"}](36626f4c.eps){height="89.0mm"}
Spectrum of the secondary star in eclipse {#sec:vlt}
-----------------------------------------
On 20 November 2000, we acquired a spectrum of the secondary star during the WD eclipse, using the ESO VLT UT1 with FORS1 and grism G300I (Table \[tab:spec\]). The 600s exposure was preceded by three and followed by five 30-s exposures. The run started shortly after the ingress of the accretion spot into eclipse and extended until after its egress. The dotted vertical lines in Fig. \[fig:eclspec\], top panel) indicate the duration of the eclipse. The 600 s exposure covered the phase interval $\phi\!=\!-0.0149$ to 0.0435, beginning after the stream component subsided and ending just before the spot starts to reappear at $\phi\!=\!0.0438$. The figure shows the light curves of the [H$\alpha$]{}line flux and of the continuum near [H$\alpha$]{} integrated over 30Å. The [H$\alpha$]{} emission stays finite in the eclipse.
The center panel of Fig. \[fig:eclspec\] shows the eclipse spectrum dereddened with the galactic extinction $A_\mathrm{V}\!=\!0.172$ [@schlegeletal98], where we have assumed that is located outside the principal dust layer of the galactic disk [@jonesetal11]. The secondary has a dereddened AB magnitude $i\!=\!20.99$ with an estimated systematic error of 0.10 mag from the absolute flux calibration of the spectrophotometry. The spectrum shows the TiO features characteristic of a late dM star and a strong [H$\alpha$]{} emission line with a line flux of ${6.8\times 10^{-16}}$[ergcm$^{-2}$s$^{-1}$]{}. The line is centered approximately at the rest wavelength and has a velocity dispersion of $\sim\!1000$[kms$^{-1}$]{}. It may arise from a tenuous uneclipsed section of the accretion stream. Regardless of its origin, the line emission will be accompanied by a thermal continuum that we need to define and subtract before a spectral type can be assigned to the secondary star. To this end, we calculated spectra of an isothermal hydrogen plasma of finite optical depth, added line broadening, and normalized the spectra to fit the observed [H$\alpha$]{} line flux. The free parameters of the model are the electron temperature $T_\mathrm{e}$, the pressure $P$, and the geometrical thickness $x$ of the emitting plasma. In the center panel of Fig. \[fig:eclspec\], we show two examples that bracket the permitted range of the flux of the sought-after continuum. The blue spectrum for $T_\mathrm{e}\!=\!10000$K, $P\!=\!10$dynecm$^{-2}$, $x\!=\!10^8$cm features a low thermal continuum and the red curve for $T_\mathrm{e}\!=\!20000$K, $P\!=\!200$dynecm$^{-2}$, $x\!=\!10^8$cm a high one. The bottom panel of Fig. \[fig:eclspec\] shows the observed spectrum with either one of the model spectra subtracted. Employing a set of dereddened spectra of SDSS dM stars, we find that the observed spectrum corrected with the low thermal continuum is best fitted with the dM5 star SDSSJ101639.10+240814.2 adjusted by a factor of 287 (rms spectral flux deviation 0.0055 in the ordinate unit of Fig. \[fig:eclspec\], bottom panel). The corresponding spectrum for the high thermal continuum is equally well fitted with the dM6 star SDSSJ155653.99+093656.5 adjusted by a factor of 289 (rms deviation 0.0053). The two cases tap the full range of thermal continua permitted by the eclipse spectrum and spectral types outside the dM5–6 slot quickly fail to provide an acceptable fit. For the dM5 case, the dereddened spectrum of the secondary corrected for the small thermal contribution has $i\!=\!21.03$ and colors $r\!-\!i\!=\!1.73$ and $i\!-\!z\!=\!0.95$. For the dM6 case with its larger thermal component, we find $i\!=\!21.22$ with $r\!-\!i\!=\!2.01$ and $i\!-\!z\!=\!1.10$ [^8]. The angular radius $R_2/d$ of the secondary star and its brightness are related by the surface brightness $$S = m+5\,\mathrm{log}\,\left[(R_2/d)(10\,\mathrm{pc}/R_\odot)\right],
\label{eq:surf}$$ where $m$ is the magnitude of the star in the selected band, $R_2$ its radius, and $d$ its distance. We calibrated the surface brightness $S_\mathrm{i}$ in the $i$-band as a function of color, using the extensive data set of @mannetal15 that combines Sloan *griz* photometry and stellar parameters. The desired relations are $$\begin{aligned}
S_\mathrm{i} & = & 5.34+1.554\,(r-i) \qquad \mathrm{with} \qquad \sigma(S_\mathrm{i})\!=\!0.086 \\
S_\mathrm{i} & = & 5.33+2.829\,(i-z) \qquad \mathrm{with} \qquad \sigma(S_\mathrm{i})\!=\!0.072,\end{aligned}$$ valid for $r\,-\,i\!=\!1.0-2.6$ and $i\,-\,z\!=\!0.6\!-\!1.3$, respectively. The quoted standard deviations describe the average spread of $S_\mathrm{i}$ around the fit. With the colors of the secondary star quoted above, we obtained mean values from Eqs. (4) and (5) of $S_\mathrm{i}(\mathrm{dM5})\!=\!8.02$ and $S_\mathrm{i}(\mathrm{dM6})\!=\!8.44$. From Eq. (3), the distance $d$ in pc is given by $$\mathrm{log}\,d=(i-S_\mathrm{i})/5+1+\mathrm{log}(f_\mathrm{back}R_2/R_\odot)
\label{eq:dist}$$ where $i$ is the magnitude of the continuum-corrected eclipse spectrum, $S_\mathrm{i}$ the respective surface brightness, $R_2$ the volume equivalent radius of the Roche lobe, and $f_\mathrm{back}\!\simeq\!0.961$ the reduction factor for the backside view of the lobe[^9]. The dM5–dM6 differences in $S_\mathrm{i}$ and in the $i$-band magnitude compensate in part, leading to similar distances with a ratio $d_\mathrm{dM5}/d_\mathrm{dM6}\!=\!1.11$. We employ Eq. \[eq:dist\] in Sect. \[sec:system\], using the radii of the secondary star derived from our dynamic models of . The error in the mean dM5–dM6 distance includes the $\pm\,0.10$ mag uncertainty in the flux calibration, the $\pm\,0.08$ mag scatter in $S_\mathrm{i}$, and half the difference of the dM5 and dM6 distances. Added quadratically, the error in $d$ is $\pm\,8.1$% plus any error that arises from $R_2$.
We estimated the effective temperature of the secondary star from the color dependence of [$T_\mathrm{eff}$]{} in the data of @mannetal15, which yields [$T_\mathrm{eff}$]{}$\,\simeq\!3070$K and 2900K for the dM5 and the dM6 case, respectively. With the best-fit system parameters of Sect. \[sec:system\], the luminosity of the secondary star becomes ${2.6\!-\!2.0\times 10^{31}}$[ergs$^{-1}$]{} for a spectral type range of dM$5\!-\!6$.
Circular spectropolarimetry {#sec:circ}
===========================
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We studied the magnetic field of the WD in by phase-resolved circular spectropolarimetry performed on 31 December 2008, when the system was accreting at a low level. A total of 40 sets of ordinary and extraordinary spectra were obtained with the ESO VLT UT2 equipped with FORS1 and GrismG300V, covering two consecutive orbital periods (Table \[tab:spec\]). The pipeline reduction combines two spectra each for two positions of the $\lambda/4$ plate in order to account for possible cross-talk from linear polarization. This procedure yielded set1 of 20 calibrated intensity and circular polarization spectra, each covering a phase interval $\Delta
\phi\!=\!0.10$. Spectral features faithfully repeated in both orbits and we phasefolded the intensity and polarization spectra for further analysis. In case of the intensity spectra, the phase resolution can be improved to $\Delta \phi\!=\!0.05$ by adding the individual ordinary and extraordinary spectra to form set2 of 40 provisionally calibrated intensity spectra. Set1 was employed for a quantitative analysis of the spectral flux, set2 for tracing the motion of the cyclotron line peaks and measuring the [H$\alpha$]{} radial velocities. Both were corrected for the secondary star, using the dM5 representation of the eclipse spectrum of Fig. \[fig:eclspec\] extrapolated into the blue spectral region and set1 was dereddened with $A_\mathrm{V}\!=\!0.172$ [@schlegeletal98]. [H$\alpha$]{}emission is much weaker than in the 2000 eclipse spectroscopy of Sect. \[sec:vlt\] and we do not correct for the probably tiny contribution of the associated thermal continuum. Grayscale representations of the corrected phasefolded intensity and circular polarization spectra are shown in Fig. \[fig:gray\], repeated twice for visual continuity. Rows \#1, \#11, and \#21 represent the eclipse. The orbital mean spectrum outside eclipse (rows \#3-10, $\phi\!=\!0.14-0.91$) is shown in the left panel of Fig. \[fig:meanspec\] (red curve); an appropriate model spectrum of the magnetic WD is added for comparison (black curve). The two blue circles are the mean out-of-eclipse $g$ and $r$-band fluxes of our 2017 and 2018 photometry, corrected for the secondary. They show that the 2008, 2017, and 2018 observations were performed in similar states of low accretion. The right panels of Fig. \[fig:meanspec\] show the light curves for the blue and red wavelength intervals extracted from the set1 spectra. The red ones ($7400-7600$Å and $7900-8100$Å) are shaped by the beamed optically thin cyclotron emission of poles 1 and 2. The blue one ($4000-4800$Å) represents the sum of the photospheric and spot emissions of the WD. Disentangling the two components proved infeasible. The [H$\alpha$]{} line emission is discussed in Sect. \[sec:nelhy\].
Cyclotron spectroscopy {#sec:cyc}
----------------------
The cyclotron lines of pole1 are visible from $\phi\!=\!0.6\!-\!1.1$ with positive circular polarization and those of pole2 in the remainder of the orbit with negative circular polarization. Overlaps between the two line systems occur in rows \#3 ($\phi\!=\!0.14-0.23$) and and \#7 ($\phi\!=\!0.54\!-\!0.63$), where the spectra show the signatures of both poles. We extracted the cyclotron lines from the individual intensity spectra by removing the underlying WD continuum. The cyclotron line profiles were then subjected to least squares fits using the theory of @chanmugamdulk81 for an isothermal plasma. The free parameters of the model are the plasma temperature k$T$, the field strength $B$, the viewing angle $\theta$ against the field direction, the thickness parameter $\Lambda$ and a remnant optically thick continuum represented by a second-order polynomial (dashed lines in Fig. \[fig:cyc\]). The thickness parameter $\Lambda$ of cyclotron theory is related to the column density $x_\mathrm{s}$ of the cooling region by $\Lambda\!=\!4\pi
ex_\mathrm{s}/\mu_\mathrm{e}m_\mathrm{u}B$, where $e$ is the elementary charge, $\mu_\mathrm{e}$ the number of electrons per nucleon in the plasma, $m_\mathrm{u}$ is the atomic mass unit, and $B$ the field strength.
----------------- ----- --------- ----------- ---------- ------- ------------------------- -------------- ---------------
\[-2ex\]
\[-1.5ex\] Pole Row $\phi $ Harmonics $B_{sp}$ k$T$ $\langle \theta\rangle$ log$\Lambda$ $\dot m$
\# fitted (MG) (keV) () (g/cm$^{2}$s)
\[1.0ex\]
\[-1ex\] 1 2 0.07 \(4) 5 6 27.50 2.25 63 5.84 0.0095
1 10 0.87 \(4) 5 6 27.37 2.20 49 5.81 0.0090
1 9 0.78 \(4) 5 6 27.51 2.25 59 5.84 0.0102
\[1.0ex\] 2 6 0.48 \(4) 5 6 28.83 1.70 68 5.55 0.0048
2 4/5 0.33 \(4) 5 6 28.88 1.65 62 5.50 0.0040
\[1.0ex\]
----------------- ----- --------- ----------- ---------- ------- ------------------------- -------------- ---------------
: Physical parameters of spots 1 and 2 derived from least squares fits of the model of an isothermal plasma to the cyclotron line profiles.
\[tab:cyc\]
Because of trade-off effects between k$T$ and log$\Lambda$, reliable values of these two parameters can not be determined without additional information. We took recourse to the results of the two-fluid radiation-hydrodynamic cooling theory of @fischerbeuermann01 and explain our approach for the case of the spectrum of row \#6 in the lower left panel of Fig. \[fig:cyc\]. Good fits to that spectrum were obtained along a narrow valley in the $\Lambda\!-\!\mathrm{k}T$ plane that follows log$\Lambda\!=\!6.64\!-\!4.73\,$log(k$T$) and extends to parameters quite inappropriate for a post-shock region dominated by cyclotron cooling. Cooling theory provides a second relation between log$\Lambda$ and log(k$T$) that runs nearly orthogonal to that of the line fits. For the case of the \#6 spectrum, it reads log$\Lambda\!=\!4.97\!+\!2.50$log(k$T$)[^10]. The intersection of the two relations defines the most probable values of log$\Lambda$ and k$T$, yielding k$T\!=\!1.70$keV and log$\Lambda\!=\!5.55$. Table \[tab:cyc\] lists the corresponding fits to all spectra that can uniquely be assigned to either pole1 or pole2. The emerging picture is that of two emission regions with similar field strengths $B_\mathrm{sp}\!\simeq\!27.5$MG and 28.8MG, thickness parameters log$\Lambda\!\simeq\!5.8$ and 5.5, and temperatures of k$T\!\simeq\!2.2$keV and 1.7keV for poles 1 and 2, respectively. The third parameter derived from the cyclotron fits is the mean viewing angle $\langle \theta\rangle$ between the line of sight and the direction of the accreting field line averaged over the spot. Closest approach to the field line occurs for pole1 at $\phi\!\simeq\!0.87$ in row \#10 and for pole2 at $\phi\!\simeq\!0.33$ in rows \#4 and \#5 with minimum viewing angles of $\langle \theta_\mathrm{min,1} \rangle\!\simeq\!49\degr$ and $\langle \theta_\mathrm{min,2}
\rangle\!\simeq\!62\degr$ for spots1 and 2, respectively. Since the cyclotron lines widen and weaken rapidly with decreasing $\theta$, the quoted angles may somewhat overestimate the true mean values.
Table \[tab:cyc\] also lists the mass flow densities $\dot m$ that are delivered also by two-fluid cooling theory. They fall far below the $\sim\!1$gcm$^{-2}$s$^{-1}$ of a bremsstrahlung-dominated emission region and are only about an order of magnitude away from the transition to the non-hydrodynamic regime of the bombardment solution [@woelkbeuermann92; @fischerbeuermann01].
Complementary information on $\theta$ is obtained from the orbital motion of the cyclotron line peaks and of the circular polarization extrema. The peak wavelengths measured from the spectra of sets2 and 1, respectively, are shown in the right panel of Fig. \[fig:gray\]. Near $\phi\!=\!0.1$ and 0.6, both line systems overlap and are difficult to disentangle. As in Paper I, the motion of the cyclotron lines was modeled, using a parameterized form of the frequencies of optically thin harmonics in units of the cyclotron frequency as functions of k$T$ and $\theta$. We defined the field vectors in the two spots as $\vec{B_\mathrm{sp,1}}$ and $\vec{B_\mathrm{sp,2}}$ and obtained field strengths and directions by least-squares fitting the phase-dependent motion of the line peaks of the fifth and sixth harmonics (red curves in Fig. \[fig:gray\]). As input we used the plasma temperatures of Table \[tab:cyc\] and an inclination of $i\!=\!80\degr$ from Table \[tab:system\]. The results are presented in Table \[tab:harm\], where we list the field strengths $B$, the azimuth angles $\psi_\mathrm{f}$, and the colatitudes $\delta_\mathrm{f}$ of the accreting field lines. The results are quoted for two accretion geometries with pole1 either in the “southern” hemisphere below the orbital plane (1S) or in the “northern” one above it (1N). Note that the fit does not provide information on the location of the spots and the orientation of the magnetic axis. The angle between the field vectors $\vec{B_\mathrm{sp,1}}$ and $\vec{B_\mathrm{sp,1}}$ is not far from 180, at least in the 1N–2S case. Combined with the fact, that both spots display circular polarization of opposite sign, the data suggest a field structure that is dominated by a dipole and possibly octupole rather than a quadrupole.
----------------- ----------------- --------------------- ------------------- ---------------- -------------------------------------------------------- --------------------- --
\[-2ex\]
\[-1.5ex\] Pole $B_\mathrm{sp}$ $\delta_\mathrm{f}$ $\psi_\mathrm{f}$ $\theta_{min}$ $\angle (\vec{B_\mathrm{sp,1}},\vec{B_\mathrm{sp,2}})$ $\Delta(\phi)$
(MG) () () () ()
\[1.0ex\]
\[-1.0ex\] 1N 27.2 36 49 45
2S 28.6 136 224 56 \[-1.5ex\][170.9]{} \[-1.5ex\][0.487]{}
\[1.0ex\] 1S 27.3 126 57 46
2N 28.4 28 227 52 \[-1.5ex\][153.2]{} \[-1.5ex\][0.472]{}
\[1.0ex\]
----------------- ----------------- --------------------- ------------------- ---------------- -------------------------------------------------------- --------------------- --
: Geometry of the accreting field lines derived from least squares fits to the orbital motion of the cyclotron lines (Fig. \[fig:gray\]).
\[tab:harm\]
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Zeeman spectroscopy {#sec:zee}
-------------------
The blue continuum in the left panel of Figs. \[fig:gray\] represents the photospheric emission of the WD including a strong spot component. To facilitate the Zeeman analysis, we transformed the set-1 spectra to the rest system of the WD, employing the preferred dynamical model of Sect. \[sec:system\]. There is little orbital variation in the Zeeman lines except near the transitions between the visibility of poles 1 and 2. We adopted the averages of rows \#8–10 ($\phi\!=\!0.63-0.91$) or \#3–6 ($\phi\!=\!0.14-0.52$) as representative of the hemispheres that include pole1 or 2, respectively. The dominant field strengths are 27–28MG in pole1 and 29–30MG in pole2. The 4000–5250Å section of the pole-1 spectrum is shown in Fig. \[fig:zee\] (red curve). Unfortunately, the circular polarization spectra, which contain information on the direction of the magnetic field vector $\vec{B}$, are too noisy to be of any use, limiting our ability to distinguish between different magnetic field structures that fit the intensity spectra similarly well.
Our spectral synthesis program employs an improved set of the model Zeeman spectra calculated by @jordan92 and previously used by @euchneretal02 [@euchneretal05; @euchneretal06] and @beuermannetal07. The present version includes the Balmer lines up to [H$\delta$]{} and consists of the log$g\!=\!8$ intensity spectra for 16 effective temperatures from 8 to 100kK, integer field strengths $B$ from 1 to 100MG, and 17 viewing angles $\theta\!=\!0\degr$ to $180$, uniformly distributed in cos$\theta$. The model spectra were calculated for a Stark broadening factor $C\!=\!0.1$ [@jordan92]. Interpolation in $T_\mathrm{eff}$ and in $\theta$ is unproblematic, while interpolation in $B$ is impracticable. The spectra were smoothed to match the observed resolution of 10Å FWHM. At this resolution, the 1-MG spacing is just about adequate and misfits stay small.
We considered a magnetic model that includes the zonal multipole components of degree $\ell\!=\!1\!-\!3$, that is the aligned dipole, quadrupole, and octupole, inclined by a common angle $\alpha$ against the line of sight. For the relative field strengths $r_\mathrm{dip}\!=\!1\!-\!r_\mathrm{oct}$, $r_\mathrm{qua}$, and $r_\mathrm{oct}$, the combined polar field strength is $B_\mathrm{p}\!=\!(1\!+\!r_\mathrm{qua})B_0$, with $B_0$ a scaling factor. We divided the visible disk of the WD into 6561 limb-darkened projected area elements, and collected them into 17$n$ field-strength and viewing-angle bins (k,l), $k\!=\!1...n$ for the nearest integer field strength $B_\mathrm{k}$ and $l\!=\!1...17$ for the nearest cos$\theta_\mathrm{l}$ value. The unreddened spectral flux at the Earth is $$f_\mathrm{\lambda}(B_0,r_\mathrm{oct},\alpha) = (R_\mathrm{wd}/d)^2\sum_\mathrm{k=1}^\mathrm{n}\sum_\mathrm{l=1}^{17}\,a_\mathrm{k,l}\,F_\mathrm{\lambda}(B_\mathrm{k},T_\mathrm{k,l},\theta_\mathrm{l}),
\label{eq:tomo}$$ where $a_\mathrm{k,l}$ is the integrated limb-darkened fractional projected-area of bin (k,l), $F_\mathrm{\lambda}(B_\mathrm{k},T_\mathrm{k,l},\theta_\mathrm{l})$ the data bank spectrum for that bin at the interpolated temperature $T_\mathrm{k,l}$, and $(R_\mathrm{wd}/d)^2$ the dilution factor, with $R_\mathrm{wd}$ the WD radius and $d$ the distance. The best values of $B_0$, $r_\mathrm{qua}$, $r_\mathrm{oct}$, and $\alpha$ were determined in a grid search, the $T_\mathrm{k,l}$ and the dilution factor by a formal least squares fit at each grid point. Obtaining a stable fit, requires a severely restricted number of independent temperatures. The spot emits about 2/3 of the blue flux from $\sim\!5$% of the area, requiring a minimum of two temperatures, naturally identified as a high spot temperature $T_\mathrm{sp}$ and a low photospheric temperature $T_\mathrm{ph}$. Some fits benefit from a minor third component, such as a warm polar cap with $T_\mathrm{cap}$. We extended the fit over the wavelength interval $4000\!-\!5200$Å, excluding sections around the Balmer emission lines, some HeI lines, and an unidentified line complex around 4300Å. A formal $\chi^2$ was calculated for 85 resolution elements of 10Å width, using relative flux errors of 1.8% for pole1 and 2.4% for pole2, measured from the scatter among the set-1 intensity spectra.
---------------- ------------------ ------------------ ------- ---------------------- ----------------------- ------------------------------------ ------------------------------------ ---------------------------------------- ------------------------------------ -------------------------------- ----------------- ------------------- ------------------ ------------------ ------------------ ------------------------------------- ------------------- ------------------- ---------- ------
\[-5ex\]
\[-1.5ex\] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)
Pole $r_\mathrm{qua}$ $r_\mathrm{oct}$ $B_0$ $\alpha$ $\delta_\mathrm{\mu}$ $\langle \delta_\mathrm{f}\rangle$ $\langle \theta_\mathrm{f}\rangle$ $\langle \vartheta_\mathrm{sp}\rangle$ $\langle \beta_\mathrm{sp}\rangle$ $\langle f_\mathrm{sp}\rangle$ $B_\mathrm{sp}$ $B_\mathrm{pole}$ $B_\mathrm{dip}$ $B_\mathrm{qua}$ $B_\mathrm{oct}$ $B_\mathrm{min}\!-\!B_\mathrm{max}$ $T_\mathrm{phot}$ $R_\mathrm{wd}/d$ $\chi^2$ Note
(MG) () () () () () () (MG) (MG) (MG) (MG) (MG) (MG) (kK) ($10^9$cm/kpc)
\[1.0ex\]
\[-1ex\]
\[0.5ex\] 1N $-0.05$ $ 0.00$ 38.5 37 113 44 52 52 31 0.054 $27\!-\!28$ $36.6$ $38.5$ $-1.9$ $0.0$ $19\!-\!37$ 9.2(0.7) 1.07(11) 88.6 (1)
2S 0.05 $ 0.00$ 38.5 $\hspace{-1.6mm}-18$ 63 126 69 50 32 0.055 $29\!-\!30$ $40.4$ $38.5$ $1.9$ $0.0$ $19\!-\!40$ 9.3(0.7) 1.06(11) 99.4 (1)
\[1.0ex\]
\[0.5ex\] 1S $-0.25$ $-0.77$ 22.8 61 141 113 40 34 1 0.041 $27\!-\!28$ $17.1$ $40.4$ $-5.7$ $-17.6$ $17\!-\!35$ 9.2(0.8) 1.15(12) 84.6 (2)
2N 0.25 $-0.77$ 22.8 $-41$ 39 37 45 11 0 0.044 $29\!-\!30$ $28.5$ $40.4$ $ 5.7$ $-17.6$ $27\!-\!35$ 9.2(0.8) 1.14(12) 90.0 (3)
\[1.5ex\]
---------------- ------------------ ------------------ ------- ---------------------- ----------------------- ------------------------------------ ------------------------------------ ---------------------------------------- ------------------------------------ -------------------------------- ----------------- ------------------- ------------------ ------------------ ------------------ ------------------------------------- ------------------- ------------------- ---------- ------
\
\[tab:zee\]
Stark broadening {#sec:stark}
----------------
An accepted theory of Stark broadening in the presence of a magnetic field does not exist. @jordan92 opted to equate the broadening of the individual Stark components of a Balmer line to the mean Stark shift of all components multiplied by a factor $C\!\simeq\!0.1$ and @putneyjordan95 considered values of $C\!=\!0.1$ and 1.0 for stars with vastly different field strengths. It is necessary, therefore, to consider the appropriate level of the line broadening for a given application. To this end, we adopted an approximate post factum procedure that changes the line strengths, while avoiding recalculation of the data base. We expressed the line profiles in terms of an optical depth $\tau_\mathrm{\lambda}$, setting $F_\mathrm{\lambda}\!=\!F^\mathrm{c}_\mathrm{\lambda}\,\mathrm{e}^{-\tau_\mathrm{\lambda}}$, with $F^\mathrm{c}_\mathrm{\lambda}$ the continuum flux. We then replaced $F_\mathrm{\lambda}$ in Eq. \[eq:tomo\] by $F^\mathrm{\,new}_\mathrm{\lambda}\!=\!F^\mathrm{c}_\mathrm{\lambda}\,\mathrm{e}^{-\tau_\mathrm{\lambda}\,\eta}$ and included $\eta$ as an additional free parameter in the grid search. For our best multipole model, we obtained $\chi^2\!=\!85.5$ at $\eta\!=\!1$ ($C\!=\!0.1$), the best fit with $\chi^2_\mathrm{min}\!=\!84.8$ was attained at $\eta\!=\!1.25$, and the 90% confidence level with $\chi^2_\mathrm{min}\!=\!87.5$ was reached for $\eta\!=\!0.80$ and 2.50, with [$\chi^2$]{} quickly rising at still lower and higher $\eta$. Hence, our fit favors line strengths somewhat larger than nominal ($C\!=\!0.10$, $\eta\!=\!1.0$). This result applies to our simultaneous multi-temperature fits to spectral flux and line strengths and may not be generally valid. We adopted $\eta\!=\!1.25$ for the present paper and obtained the systematic errors at the 90% confidence level for $\eta\!=\!0.8-2.5$. The large errors re-emphasize the need for an effort to calculate the Stark shifts .
Magnetic geometry of the accreting WD {#sec:magg}
-------------------------------------
The right-hand panels of Fig. \[fig:cyc\] show two selected magnetic WD geometries. For simplicity, both have rotational pole, magnetic pole, and the accretion spots on the same meridian (here the paper plane). The secondary star is located far to the left. The footpoints of the common magnetic axis are displaced from the respective viewing directions by $\alpha_1$ and $\alpha_2$, with $\alpha_1\!+\!\alpha_2\!=\!180\degr\!-\!2i$, where $i$ is the inclination and $\alpha$ is counted positive away from and negative toward the rotational pole. In case A, both spots are located between magnetic pole and viewing direction and can accrete from the nearby orbital plane. To reach spot2, the plasma must travel halfway around the WD before it attaches to a near-polar field line. In case B, spot1 can accrete from the nearby orbital plane. Spot 2, however, is located between magnetic and rotational pole and the field line leads over the rotational pole in the general direction of the secondary star. Although energetically unfavorable, this non-standard path may be active and it is not clear whether the trip over the pole or the travel around the WD should be dismissed as the less likely way to feed spot2.
In perusing parameter space, we found that all good fits require field strengths larger than the spot field $B_\mathrm{sp}$ and are rather insensitive to a lack of small field strengths. We started from a pure dipole model that fits the pole1 and pole2 spectra with $B_0\!=\!37.5$ and 40.5MG, respectively. Adding a small quadrupole component, leads to a common $B_0\!=\!38.5$MG. The parameters of this quasi-dipole fit are listed in Table \[tab:zee\]. As expected for an inclination of 80 (Sect. \[sec:system\] and Table \[tab:system\]), $\alpha_1\!+\!\alpha_2\!\simeq\!20$, confirming the presence of a common magnetic axis for the separately performed fits. The colatitudes $\delta_\mathrm{f}$ of the accreting field lines agree reasonably well with those of the 1S$-$2N geometry in Table \[tab:cyc\], considering the uncertainties of about 5. Superficially, the fit seems close to perfect were it not for the disturbing fact that the geometry probably prevents accretion in both spots. The ribbon-like spots are offset from the respective magnetic pole by $\vartheta_\mathrm{sp}\!\sim\!50\degr$, the field lines in the spots reach out to only 1.7$R_\mathrm{wd}$, and both field lines curve away from the orbital plane. Hence, the quasi-dipole model provides no convincing accretion geometry. Increasing the quadrupole component provides no remedy. For $r_\mathrm{qua}$ up to $\pm 0.40$, none of the seemingly good fits matches the requirements set up by the cyclotron fits. The same holds for moderate octupole components $r_\mathrm{oct}$ up to $\pm0.40$, some of which predict “spots” in the form of near-equatorial ribbons connected by tightly closed field lines.
The situation changes fundamentally for larger octupole components with $r_\mathrm{oct}\!\la\!-0.45$. With decreasing $r_\mathrm{oct}$, the best-fit values of $B_0$ in the primary minima of poles 1 and 2 converge and coincide for $r_\mathrm{oct}\!=\!-0.77$ and $r_\mathrm{qua}\!=\!\mp0.25$, respectively. Table \[tab:zee\] lists the fit parameters. As required, $\alpha_1\!+\!\alpha_2\!\simeq\!20$and the spot-averaged colatitudes $\langle \delta_\mathrm{f}\rangle$ and viewing angles $\langle \theta_\mathrm{f}\rangle$ agree reasonably well with the 1S$-$2N cyclotron results of Table \[tab:cyc\]. Both spots are located closer to the magnetic poles than in the quasi-dipole case and the field lines are close to radial with inclinations $\langle
\beta_\mathrm{f}\rangle\!\simeq\!0$, indicating that the field lines reach far out. The multipole model represents a convincing solution, provided the case B path to spot2 is active. In passing, we note that enforcing case A accretion at spot2 by increasing its colatitude fails because a corresponding $\chi^2$ minimum does not exist. Switching hemispheres, the 1N geometry is a mirror image of 1S, but a $\chi^2$ minimum at the parameters expected for 2S does not exist either. The general caveat holds that a multipole model with tesseral harmonics may provide a different answer.
The two left panels of Fig. \[fig:zee\] show the magnetic field distributions of the WD for the multipole model of Table \[tab:zee\] at the phases of the best visibility of spots 1S and 2N. The yellow and green bands indicate the spot field strengths of 27 and 28 MG for pole1 and 29 and 30 MG for pole2 and the black portions the viewing-angle selected spots. Although both aspects belong to the same field model, the spot geometries differ significantly. So do the full ranges of the field strengths over the visible face of the WD (bottom panels, see also Col. 17 of Table \[tab:zee\]). The right panel of Fig. \[fig:zee\] shows the spectrum (red curve) and the Zeeman fit (superposed black curve), which faithfully reproduces most Zeeman lines in the spectral regions that are free of atomic emission lines. The contributions by the spot and the photosphere+cap are also shown individually. The Zeeman lines are prominent in the spot component because of the small spread in field strength, but are washed out in the photospheric component. The fit to the pole-2 spectrum (not shown) excels at $\lambda\!>\!4500$Å, but is inferior at shorter wavelengths, possibly because it is composed of only nine independent magnetic spectra (Table \[tab:zee\], Col. 17).
WD parameters and their errors {#sec:wdtemp}
------------------------------
The multipole Zeeman fit to the observed pole1 and pole2 spectra yielded temperatures for the photosphere, cap, and spot of $T_\mathrm{ph}\!=\!9.2\,\pm\,0.7$kK (Table \[tab:zee\]) and $T_\mathrm{cap}\!=\!14.2\,\pm\,1.0$kK and $T_\mathrm{sp}\!=\!78\,\pm\,8$kK (Table 10). In most fits, the flux contributed by the cap is a minor entity. The corresponding angular radius of the WD is $R_\mathrm{wd}/d\!=\!{(1.15\,\pm\,0.12)\times 10^{9}}$[cmkpc$^{-1}$]{}. All quoted errors refer to the 90% confidence level and include besides the statistical also the systematic error caused by the remaining uncertainty in the level of Stark broadening (Section \[sec:stark\]). Quadratically adding the error from an estimated 10% uncertainty in the flux calibration of the spectropolarimetry gives $R_\mathrm{wd}/d\!=\!{(1.15\,\pm\,0.14)\times 10^{9}}$[cmkpc$^{-1}$]{}. The error budget of $R_\mathrm{wd}$ includes in addition the 8.1% uncertainty in the mean dM5–dM6 distance from Sect. \[sec:vlt\], raising the error of $R_\mathrm{wd}$ to 14.1%. Despite its large error, the measured radius proves helpful in determining the system parameters in Sect. \[sec:system\].
{height="34.0mm"} {height="34.5mm"}
{height="34.0mm"} {height="34.5mm"}
The photospheric temperature of the WD in is lower than in practically all other well-studied polars [@townsleygaensicke09]. The low temperature is directly related to the low observed equivalent width $W_\mathrm{obs}$ of the Zeeman lines in . The equivalent width in the theoretical Zeeman spectra has its peak at $T_\mathrm{eff}\!=\!12$kK and drops rapidly toward lower and higher temperatures. $W_\mathrm{obs}$ falls by a factor of 3.2 below the peak value for nominal Stark broadening. Fitting $W_\mathrm{obs}$ and the spectral slope simultaneously, requires $T_\mathrm{ph}$ significantly below 12kK and $T_\mathrm{sp}\!\gg\!12$kK. The fit deteriorates with rising $T_\mathrm{ph}$ and becomes unacceptably bad at 11kK, even allowing for a variation in the level of Stark broadening.
System parameters {#sec:sys}
=================
Narrow emission lines as tracers of the motion of the secondary star {#sec:nel}
--------------------------------------------------------------------
Polars feature narrow emission lines of hydrogen, helium and metals that are thought to originate on the irradiated face of secondary star. In some polars, however, the radial velocity amplitudes of individual lines differ. Helium lines with their high ionization potential show lower amplitudes than hydrogen, while the low-ionization near infrared CaII lines have the greatest and rather stable amplitudes [@schwopeetal00; @schwopeetal11; @schwopechristensen10]. Obviously, the distribution of the emission differs between individual lines, with the helium lines probably originating, in part, from structures outside the chromosphere of the star, such as coronal prominences. Modeling is straightforward as long as the emission originates from locations geometrically close to the surface of the star, which appears to be the case for the low-ionization metal lines and, in some polars, for hydrogen lines. Describing the line emission over the secondary star requires either a dedicated theoretical model or an empirical ansatz that derives the distribution from unfolding highly resolved observed line profiles.
Models of the irradiated secondary star {#sec:irr}
---------------------------------------
We calculated the radial velocity amplitude $K_2'$ of the narrow emission line, considering a Roche-lobe filling star that is irradiated by a source at the position of the WD. Each surface element receives an incident flux $f_\mathrm{in}\!\propto\!\mathrm{cos}\,\vartheta /\delta^2$, with $\vartheta$ the angle of incidence and $\delta$ the distance from the source. In response, it emits a line flux $f_\mathrm{line}$ that varies with $f_\mathrm{in}$, but may depend on additional parameters. For a given model, we calculated synthetic emission line spectra and determined the lever arm of the line emission region $a_\mathrm{irr}(q)\!=\!a_2\,K_2'/K_2$ as a function of the mass ratio $q\!=\!M_2/M_1$, with $a_2\!=\!1/(1+q)$. Below, we quote polynomial approximations for $a_\mathrm{irr}(q)$[^11].
The irradiation model of @beuermannthomas90 (henceforth BT90), equates the line intensity emitted from a surface element to a fraction of the total incident flux, $f_\mathrm{line}\!\propto\!\mathrm{cos}\,\vartheta /\delta^2$. The emitted intensity drops from a maximum at $L_1$, where $\vartheta\!\simeq\!40$, down to zero at the terminator of the irradiated region, where $\vartheta\!=\!90$. Hence, BT90 favors emission from regions near $L_1$. The modified version BT90m uses $f_\mathrm{line}\!\propto\!(\mathrm{cos}\,\vartheta)^{m}/\delta^2$, with $m$ a heuristic free parameter. This modification was motivated by the study of irradiated WD atmospheres by @koenigetal06 [their Sect.4.2], who found that the narrow emission cores of [Ly$\alpha$]{} increased drastically when $\vartheta$ approached 90and the incident energy was deposited increasingly higher up in the atmosphere. There is no simple way to relate $m$ to physics, however. An entirely different approach was taken in @beuermannreinsch08 [their Sects. 5.4, 6.1, and Fig. 10], where we determined the distribution of the CaII$\lambda8498$ emission as a function of $\vartheta$ empirically by unfolding the high-resolution line profiles of the intermediate polar EX Hya. A parameterized form of the CaII emission model was implemented in our model BR08. It provides an internally consistent description of the emission of CaII$\lambda8498$ and numerous other metal lines that share the motion of CaII. The model gives larger weight to surface elements near the terminator, equivalent to moderate limb brightening. To put the models into perspective, we note that BR08 corresponds approximately to BT90m with $m\!\simeq\!0.3$. BT90 and BR08 bracket about the full range of possible irradiation scenarios of the atmosphere of the secondary. Only models with still more pronounced limb brightening, equivalent to BT90m with $m\,<\,0.3$, would yield a still larger value of $a_\mathrm{irr}$ (and a still smaller primary mass). An estimate of the remaining systematic errors is given in Section \[sec:system\].
Narrow emission lines in {#sec:nelhy}
-------------------------
In its 2008 low state, displayed emission lines of hydrogen, HeI, and MgI$\lambda5170$. The near infrared CaII triplet was not detectable against the cyclotron background. In [H$\alpha$]{}, the broad component with a FWHM of 23Å and the unresolved narrow component can be separated at our 10Å spectral resolution, while this becomes infeasible for the higher Balmer lines, which are embedded in complex Zeeman absorption troughs. MgI$\lambda5170$ is not disturbed and is the only metal line that is sufficiently strong for a radial velocity study. The left panels in Fig. \[fig:vrad\] show pseudo-trailed spectra of [H$\alpha$]{} and MgI$\lambda5170$ with the phase-dependent continuum subtracted. The gray scale is inverted compared with Fig. \[fig:gray\]. We used spectral set2 with 20 spectra per orbit for [H$\alpha$]{} and set1 with 10 spectra per orbit for the weaker MgI line. The two orbits were folded and the data shown twice for better visibility.
The upper right panel in Fig. \[fig:vrad\] shows the orbital flux variations of MgI$\lambda5170$ (blue) and of the narrow [H$\alpha$]{}component (green). Maximum flux occurs near $\phi\!=\!0.5$, when the illuminated hemisphere is in view. Half an orbit later, the Mg line disappears and the narrow [H$\alpha$]{} component can no longer be discriminated against the underlying broad component (yellow). The upper half of the light curve is skewed, as is illustrated in the second orbit, where the [H$\alpha$]{} light curve is compared with its own mirror image relative to $\phi\!=\!0.5$ (dashed curve). The skew is, at least in part, due to statistical fluctuations between the two orbits. Apart from this, the light curve is well described by the irradiation model BR08 (solid curve). The lower right panel of Fig. \[fig:vrad\] shows the radial velocity curves of MgI$\lambda5170$ (blue), of the narrow component of [H$\alpha$]{} (green), and of the broad [H$\alpha$]{} emission (crosses). Although MgI$\lambda5170$ is weaker than [H$\alpha$]{}, the radial velocities have similar errors because the former were derived from single-Gaussian and the latter from the more uncertain double-Gaussian fits. MgI$\lambda5170$ has a radial velocity amplitude of $K_2'\!=\!139\pm10$[kms$^{-1}$]{} with a blue-to-red zero-crossing phase of $\phi_0\!=\!0.02\pm0.02$, the narrow [H$\alpha$]{} line has $K_2'\!=\!125\pm9$[kms$^{-1}$]{} with $\phi_0\!=\!-0.06\pm0.02$. The broad component has a velocity amplitude $K_\mathrm{broad}\!=\!220
\pm14$[kms$^{-1}$]{}with $\phi_0\!=\!-0.13\pm0.02$ and $\gamma\!=\!-21\pm9$[kms$^{-1}$]{} relative to the narrow component. All errors refer to the 90% confidence level.
We investigated the origin of the lines by calculating Doppler tomograms, using the maximum entropy method MEM [@spruit98; @marshschwope16]. Given the small number of phase intervals, the tomograms are sensitive to noise and have been slightly smoothed with a velocity filter corresponding to 0.3 spectral resolution elements. The resulting tomograms are shown in the center panels of Fig. \[fig:vrad\], with the outline of the Roche lobe of the secondary for our best-fitting dynamical model overplotted (Table \[tab:system\], line 4). The bulk of the emission can be uniquely allocated to the illuminated face of the secondary star and the vicinity of the inner Lagrangian point $L_1$. The rainbow color scale ranges from blue for the highest intensity down to red. Overall, the [H$\alpha$]{} tomogram is tilted toward the leading hemisphere of the secondary, with the asymmetry related to the finite negative $\phi_0$ and the existence of the underlying broad component. The latter with its best visibility at $\phi\!=\!0.60$ is represented by the tail that extends to $V_\mathrm{X}\!=\!-700$[kms$^{-1}$]{} and $V_\mathrm{Y}\,=\!+450$[kms$^{-1}$]{}. This direction differs significantly from that of the standard ballistic stream seen in many polars in their high states, which moves in velocity space from $L_1$ to large negative $V_\mathrm{X}$ at nearly constant $V_\mathrm{Y}$. A tail at similarly odd velocities was seen in the HeII$\lambda4686$ tomogram of AM Her [@staudeetal04] and tentatively interpreted in terms of a non-standard accretion stream that couples from the secondary immediately to a polar field line of the WD. This is an intriguing proposition in view of our suggestion in Sect. \[sec:magg\] that pole 2 of is fed by such a scenario. In view of these idiosyncrasies, we should be wary of interpreting the narrow [H$\alpha$]{}line in as of purely chromospheric origin.
The MgI$\lambda5170$ line is free of the complications by a broad component, the tomogram looks more regular, and the phase of zero radial velocity is consistent with inferior or superior conjunction of the secondary star. The bottom center panel of Fig. \[fig:vrad\] shows the enlarged central portion of the tomogram. The emission is centered on the illuminated part of the star, with the peak intensity occurring at a $Y$-velocity that agrees with $K_2'\!=\!139\!\pm\!10$[kms$^{-1}$]{} obtained from the radial-velocity analysis (small white cross).
---------------- ------------------ ------- ------ ------- ----------------- ----------------- --------------- ----------------- ------------------- ------- ------------------- ------------------- ----------------- --------------------- -------------------------------- --------- -----------------
\[-5ex\]
\[-1.5ex\] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)
Row $K_2^{'}$ $f_3$ $i$ $q$ $M_2$ $R_2$ $K_2/K_2^{'}$ $K_\mathrm{L1}$ $K_\mathrm{term}$ $M_1$ $R_\mathrm{1,He}$ $R_\mathrm{1,CO}$ $d$ $R_\mathrm{1,sp}$ $R_\mathrm{1}/R_\mathrm{1,sp}$ $R_1/d$ $T_\mathrm{ph}$
([kms$^{-1}$]{}) () ([$M_\odot$]{}) ([$R_\odot$]{}) ([$M_\odot$]{}) (pc) ($10^9$cm) ($10^9\!$cm/kpc) (kK)
\[0.5ex\]
\[-1ex\]
\[0.5ex\] 1 129 1.000 79.1 0.656 0.308 0.320 1.83 57 186 0.469 1.116 1167$\,\pm\,$95 1.343$\,\pm\,$0.189 0.83 0.956 10.19
2 139 1.000 79.8 0.615 0.305 0.318 1.76 67 196 0.497 1.065 1160$\,\pm\,$94 1.334$\,\pm\,$0.188 0.80 0.918 10.42
3 149 1.000 80.4 0.577 0.304 0.317 1.71 76 207 0.526 0.969 1154$\,\pm\,$93 1.327$\,\pm\,$0.187 0.73 0.840 10.92
\[1.0ex\]
\[0.5ex\] 4 139 1.120 80.5 0.568 0.235 0.290 1.70 72 192 0.413 1.217 1054$\,\pm\,$85 1.217$\,\pm\,$0.172 1.00 1.150 9.20
\[1.0ex\]
\[0.5ex\] 5 139 1.035 80.0 0.600 0.281 0.309 1.74 68 195 0.469 1.117 1127$\,\pm\,$91 1.296$\,\pm\,$0.183 0.86 0.991 10.00
6 149 1.135 81.4 0.524 0.224 0.285 1.64 83 202 0.427 1.190 1037$\,\pm\,$84 1.192$\,\pm\,$0.168 1.00 1.148 9.20
7 129 1.095 79.7 0.617 0.247 0.297 1.77 62 183 0.401 1.244 1081$\,\pm\,$88 1.244$\,\pm\,$0.175 1.00 1.150 9.20
8 139 1.190 81.1 0.541 0.201 0.275 1.66 75 189 0.371 1.313 1002$\,\pm\,$81 1.152$\,\pm\,$0.162 1.14 1.310 8.60
\[1.0ex\]
---------------- ------------------ ------- ------ ------- ----------------- ----------------- --------------- ----------------- ------------------- ------- ------------------- ------------------- ----------------- --------------------- -------------------------------- --------- -----------------
\
\[tab:system\]
Mass-radius relation of the secondary star {#sec:massrad}
------------------------------------------
Deriving stellar masses requires that we adopt a mass-radius relation $R_2(M_2)$ for the Roche-lobe filling secondary star. We used theoretical models by @baraffeetal98 [@baraffeetal15 henceforth BCAH and BHAC] for main sequence stars of solar composition evolved to 1Gyr. This is the approximate cooling age of the WD in , discounting compressional heating by accretion, and the minimal age of the secondary. For ease of application, we represented the radii by power laws $R_2(M_2)$[^12] Since secondary stars in CVs are known to be more or less bloated compared with field stars, we considered radii expanded over those of the BHAC models by the following processes: (i) magnetic activity and spot coverage [@chabrieretal07; @moralesetal10; @kniggeetal11; @parsonsetal18]; (ii) tidal and rotational deformation of Roche-lobe filling stars [@renvoizeetal02]; and (iii) inflation by magnetic braking that drives the star out of thermal equilibrium [@kniggeetal11]. Effect (i) describes the radius excess that compensates for the reduced radiative efficiency caused by starspots. @kniggeetal11 and @parsonsetal18 found a mean excess of 5% for stars with mass below $0.35$[$M_\odot$]{}. @moralesetal10 and @kniggeetal11 argued that high-latitude spots may mimic a larger radius in certain eclipsing binaries, accounting for 3% of the excess. Proceeding conservatively, we accept this argument and adopted a bloating factor $f_1\!=\!1.020$. A Roche-lobe filling star in a short period binary can not escape effect (ii), which increases the radius by a factor $f_2\!=\!1.045$ independent of $q$ [@renvoizeetal02; @kniggeetal11]. Effect (iii) is described by a free factor $f_\mathrm{3}$ that may range from unity up to about 1.30. The adopted stellar radii $R_2\!=\!f_1f_2f_3\,R_\mathrm{BHAC}$ are fully consistent with those employed by @kniggeetal11 in their evolutionary sequences. Using the models of stars with solar composition evolved to ages of 5 or 10 Gyr instead of 1 Gyr, the dynamical solution yields WD masses lower by 3% or 5%, respectively. For a metal-poor secondary with $[M/H]\!=\!-1$, the masses would be higher by 8% at 1 Gyr, but correspondingly lower again for larger ages.
Component masses and distance {#sec:system}
-----------------------------
For a given irradiation model and a mass-radius relation of the secondary star, we obtained the system parameters that match the radial-velocity amplitude $K_2'$ and the eclipse duration $\Delta\,t_\mathrm{ecl}$. We adopted $K_2'\!=\!139\!\pm\!10$[kms$^{-1}$]{} of the MgI$\lambda5170$ line and BR08 as the standard. Results are presented in Table \[tab:system\] and Fig. \[fig:m1m2\]. The derived parameters include the masses and radii of the components, the distance $d$ obtained from the angular radius of the secondary star (Sect. \[sec:vlt\]) and the spectroscopic radius $R_\mathrm{1,sp}$ of the WD obtained from $d$ and the angular radius of the WD (Sect. \[sec:wdtemp\]). The listed model radii in Cols. (12) and (13) of Table \[tab:system\] refer to He-core and CO-core WDs with thick hydrogen envelopes[^13] evolved to $T_\mathrm{eff}\!=\!10$kK [@paneietal07; @renedoetal10; @althausetal13][^14]. The radii for an effective temperature of 9kK are only minimally smaller.
To start with, we take the secondary to be an unbloated main sequence star with $f_3\!=\!1.0$, an assumption that yields the maximum primary mass, but disregards the spectroscopic information from Section \[sec:wdtemp\]. For the Mg line with $K_2'\!=\!139\!\pm\!10$[kms$^{-1}$]{} and BR08, the component masses are $M_1\!=\!0.497\!\pm\!0.029$[$M_\odot$]{} and $M_2\!=\!0.305\!\pm\!0.002$[$M_\odot$]{} (Table \[tab:system\], lines 1 to 3, 90% confidence errors). Interestingly, the [H$\alpha$]{} value $K_2'\!=\!125\!\pm\!9$[kms$^{-1}$]{} and BT90, yield practically the same primary mass, $M_1\!=\!0.487\!\pm\!0.026$[$M_\odot$]{}, but this combination lacks the internal consistency that exists between the MgI line and the metal-line calibrated model BR08. Cross-combining velocity amplitude and irradiation model gives an indication of the remaining systematic error: [H$\alpha$]{} and BR08 give $M_1\!=\!0.458$[$M_\odot$]{}, MgI and BT90 give $M_1\!=\!0.528$[$M_\odot$]{}, or combined $M_1\!=\!0.493\!\pm\!0.035$[$M_\odot$]{}. Hence the so-defined systematic error is of the same size as the statistical error of 0.029[$M_\odot$]{}. In summary, the assumption of a main sequence secondary identifies the primary either as a He-core WD or a CO-core WD very close to its minimum mass of $0.53\pm0.02$[$M_\odot$]{}[@moehleretal04; @kaliraietal08]. The fault with the main sequence assumption is the neglect of the spectroscopic evidence of Section \[sec:wdtemp\] on the angular radius and the effective temperature of the WD. The implied model radius of the primary in either Col. (12) or Col. (13) of Table \[tab:system\], lines $1\!-\!3$, falls far short of the spectroscopically determined radius $R_\mathrm{1,sp}$ in Col. (15), calculated from $R_\mathrm{wd}/d\!=\!{(1.15\,\pm\,0.14)\times 10^{9}}$[cmkpc$^{-1}$]{}(Sect. \[sec:wdtemp\]) and the $R_2$-dependent distance $d$ in Col. (14), where we have added the errors quadratically. The employed angular radius of the WD belongs to the best-fit photospheric temperature $T_\mathrm{ph}\!=\!9.2\,\pm\,0.7$kK. Had the WD the radius of Col. (12) or (13), the observed spectral flux would demand that its temperature would be that in Col. (18). As noted in Section \[sec:wdtemp\], a decent Zeeman spectral fit can not be achieved for $T_\mathrm{ph}\!>\!10$kK, further reducing the probability that the primary in is a low-mass CO WD.
![Dynamic models of in the $M_1\!-\!M_2$ plane. The irradiation model is BR08, $K_2'$ ranges from 111 to 159 [kms$^{-1}$]{} in steps of 2[kms$^{-1}$]{}, and the bloating factor $f_3$ ranges from 1.000 to 1.300 in steps of 0.005. The models with main sequence (MS) secondaries are marked by blue dots. Red dots denote the models that match both, the measured radial-velocity amplitude of MgI$\lambda5170$ and the spectroscopically measured WD radius within their 90% confidence errors.[]{data-label="fig:m1m2"}](36626f10.eps){height="89.0mm"}
In a second step, we considered models with bloated secondary stars. We calculated a grid of models with radial velocity amplitudes $K_2'\!=\!110$ to 160[kms$^{-1}$]{} and expansion factors $f_3\!=\!1.0$ to 1.3 in steps of 2[kms$^{-1}$]{} and 0.005, respectively. We identified $K_2'$ with the radial velocity amplitude of the MgI$\lambda5170$ line and converted it to $K_2$, using the BR08 model. The resulting component masses are depicted in Fig. \[fig:m1m2\], where each model is represented by a dot. The main sequence models considered above are marked in blue. Models that comply with the MgI amplitude $K_2'\!=\!139\!\pm\!10$[kms$^{-1}$]{}, are located between the two dashed lines, extending from the upper right to the lower left. Along this path, the bloating factor $f_3$ of the secondary star increases, the mass of the WD decreases, its radius increases, and its temperature decreases. Models with He-core WDs, whose radii agree within the uncertainties with the spectroscopically determined WD radius, $R_\mathrm{1,He}/R_\mathrm{1,sp}\!=\!1.00\!\pm\!0.14$ (90% confidence error, Section \[sec:wdtemp\]) are located between the two dashed lines that run from the upper left to the lower right, and models that match both conditions are marked by red dots. The optimal dynamical model in line 4 of Table \[tab:system\] corresponds to the intersection of the two solid lines and the WD parameters at this point correspond to those of the optimal multipole Zeeman fit in Table \[tab:zee\]. Lines $5\!-\!8$ of Table \[tab:system\] contain the model parameters for the four cardinal points of the red-dotted region, marked by the four black dots. Col. (3) of the table lists the bloating factor $f_3$. Bloating ranges from a minimal 3.5% to 19%, indicating that the secondary is only moderately expanded as may be expected for a polar that experiences reduced magnetic braking [@WickramasingheWu94; @webbinkwickramasinghe02]. As discussed in Section \[sec:disc\], minimal bloating that goes along with a low accretion rate is also required to explain the low photospheric temperature of the WD in terms of compressional heating [@townsleygaensicke09]. At the optimal position and at two of the cardinal points, the model and observed WD radii in Cols. (12) and (15) agree. At the two other cardinal points, they disagree by the permitted $\pm\!14$% (column 16). The 90% confidence region for the combined dynamical and spectroscopic fit is defined by a quasi-ellipse that is inscribed to the red-dotted quadrilateral and passes through the cardinal points (not shown). It limits the component masses to $M_1\!=\!0.413\,+\!0.058,-0.044$[$M_\odot$]{} and $M_2\!=\!0.235\!+\!0.048,-0.038$[$M_\odot$]{} (90% confidence errors), or $M_1\!=\!0.42\!\pm\!0.05$[$M_\odot$]{} and $M_2\!=\!0.24\!\pm\!0.04$[$M_\odot$]{}, with $q$ between 0.51 and 0.63. The combined dynamical and spectroscopic fit identifies the primary in as a low-mass WD, consistent with having a helium core. The mass of the secondary is normal for a CV with an orbital period of 2.855h. It may be fully convective or retain a radiative core, depending on its prehistory [@kniggeetal11].
[@l@rl@r]{}\
Parameter & Value && Error\
\[0.5ex\]\
Orbital period [$P_\mathrm{orb}$]{} (s) & 10278.9 &&\
Eclipse duration FWHM $\Delta\,t_\mathrm{ecl}$ (s) & 910.6 && 1.5\
Primary mass $M_1$ ([$M_\odot$]{}) & 0.42 && $0.05$\
Primary radius $R_1$ at $T_\mathrm{ph}$ ($10^{9}$cm)& 1.22 && $0.10$\
Secondary mass $M_2$ ([$M_\odot$]{}) & 0.24 && $0.04$\
Secondary radius $R_2$ ([$R_\odot$]{}) & 0.29 && $0.02$\
Mass ratio $q$ & 0.57 && $0.06$\
Binary separation $a$ ($10^{10}$cm) & 6.16 && $0.28$\
Inclination $i$ () & 80.6 && $0.9$\
Distance $d$ (pc) & 1050 && $110$\
$T_\mathrm{eff}$ of secondary (K) &3000 && $100$\
Luminosity of secondary $L_\mathrm{bol}$ ($10^{31}$erg/s) &2.3 && 0.6\
$T_\mathrm{eff}$ of photosphere of WD $T_\mathrm{ph}$ (kK) &9.2 && 0.7\
$T_\mathrm{eff}$ of pole cap of WD $T_\mathrm{cap}$ (kK) &14.2 && 1.0\
Weighted mean of $T_\mathrm{ph}$ and $T_\mathrm{cap}$ (kK) &10.1 && 0.9\
$T_\mathrm{eff}$ of spot1 of WD $T_\mathrm{sp}$ (kK) & 78.0& $^{(1)}$ & 8.0\
Magnetic field strength in spot1 $B_\mathrm{sp,1}$ (MG) & 27.4 && 0.2\
Magnetic field strength in spot2 $B_\mathrm{sp,2}$ (MG) & 28.7 && 0.3\
\
\
\[tab:sum\]
The observed parameters of are summarized in Tab. 10. At a distance $d\!=\!1050\,\pm\,110$pc and a galactic latitude $b\!=\!-26.1$, it is located close to 500pc below the galactic plane. The second Gaia data release [@gaia18a] yielded a parallax $\pi\!=\!-0.1\!\pm\!0.8$mas at a mean $g\,=\,20.3$, or a distance of $d\!>\!830$pc [@bailerjonesetal18], consistent with all entries in Table \[tab:system\].
Luminosity and accretion rate {#sec:sed}
-----------------------------
The spectral energy distribution (SED) in Fig. \[fig:sed\] provides an overview of the long-term variability of . It shows the spectrum of 1993 from Fig. \[fig:spec\] (black curve), that of 2008 from Fig. \[fig:cyc\] (red), and the eclipse spectrum from the center panel of Fig. \[fig:ecl\] (blue). The photometric data were accessed via the Vizier SED tool provided by the CDS[^15]. They include the SDSS[^16], the UKIDSS, and the Wide-field Infrared Survey [WISE; @cutrietal13] as cyan blue dots, the Two Micron All Sky Survey [2MASS; @skrutskietal06] as green dots, the Galaxy Evolution Explorer [GALEX; @martinetal05; @bianchietal11] as blue dots, and the catalogs PPMXL [@roeseretal10] and NOMAD [@zachariasetal05] as red dots. The range of the MONET WL measurements of Fig. \[fig:lc\] is indicated by the four open triangles that refer to the observations of January 2010, November 2010, 2017/18, and the remnant flux inside the eclipse. They delineate the same range of flux levels as the independent photometric and spectrophotometric observations. The secondary star (+) represents a lower flux limit in the red part of the SED.
![Spectral energy distribution of from the far ultraviolet to the near infrared based in part on publicly available photometric data accessed via the Vizier SED tool available at the CDS in Strasbourg.[]{data-label="fig:sed"}](36626f11.eps){height="89mm"}
To a first approximation, the high state is characterized by a roughly flat SED from the infrared to the FUV, with $F_\mathrm{\nu}\!\simeq\!0.5$mJy and an integrated energy flux of $F_\mathrm{UV,opt}\!\simeq\!{1.5\times 10^{-11}}$[ergcm$^{-2}$s$^{-1}$]{}. At a distance of 1kpc, the high-state UV–optical luminosity amounts to $L_\mathrm{UV,opt}\!\simeq\!{2.0\times 10^{33}}$[ergs$^{-1}$]{} for emission into 4$\pi$. The 2002 UV and visual photometry with the XMM-Newton optical monitor demonstrates that the simultaneous X-ray observation was also taken in a near-high state. The bolometric X-ray flux measured in the ROSAT and XMM-Newton observations is not well established. In Sect. \[sec:xray\] we quoted a conservative high-state level of the bolometric soft X-ray flux of $F_\mathrm{X}\!\simeq\!{1.5\times 10^{-11}}$[ergcm$^{-2}$s$^{-1}$]{}, although the X-ray flux may have been significantly higher at times. Hence, as a conservative estimate the total high-state luminosity, not accounting for the hard X-ray region, was $L_\mathrm{X,UV,opt}\!\simeq\!{4\times 10^{33}}$[ergs$^{-1}$]{}. For the best-fit model in Table \[tab:system\], line 4 and Table 10 with a WD of 0.42[$M_\odot$]{}, this luminosity requires an accretion rate of $\dot{M}\!\simeq\!{9\times 10^{16}}$gs$^{-1}\!=\!{1.4\times 10^{-9}}\,$[$M_\odot$]{}yr$^{-1}$, On the other hand, the integrated orbital mean cyclotron flux in the 2008 low state, including the extrapolation into the unobserved infrared regime, amounted to $F_\mathrm{opt,IR}\!\simeq\!{3\times 10^{-14}}$[ergcm$^{-2}$s$^{-1}$]{}. The implied remnant accretion rate was a mere $\dot{M}\!\simeq\!{1.4\times 10^{-12}}\,$[$M_\odot$]{}yr$^{-1}$, three orders of magnitude lower than in the high state.
Discussion {#sec:disc}
==========
belongs to the rather small group of about a dozen eclipsing polars that may be considered well studied [@ritterkolb03 final edition 7.24, 2016]. When looked at superficially, it is a rather uninspiring member of the class, being distant and faint, and lapsing into prolonged states of low accretion[^17]. The present study, however, reveals as a system that is peculiar in several respects and in part unique among polars.
We identified it as a permanent two-pole accretor with both poles being active for accretion rates differing by three orders of magnitude. Two nearly opposite accretion spots with similar field strengths seem to suggest a dipolar field, but that is probably an illusion. Our combined quasi-dipole fit to the spectra of both poles is formally good and yields field directions in both accretion spots that are consistent with the results of our cyclotron line fits, but nevertheless the magnetic geometry is characterized by tightly closed field lines in both spots that seem to preclude accretion. In an extensive grid search, we discovered an alternative close-to-perfect fit for a field structure with a polar field strength of the dipole $B_\mathrm{dip}\!=\!40.4$MG and relative field strengths of quadrupole and octupole $B_\mathrm{qua}/B_\mathrm{dip}\!=\!\mp\,0.14$ and $B_\mathrm{oct}/B_\mathrm{dip}\!=\!-\,0.44$. It possesses open field lines in both spots, allowing the southern spot1 to accrete from the nearby orbital plane, while the field in the northern spot2 leads over the rotational pole to a place somewhere near the secondary star. If that path is active, it obviates the need for the plasma accreted at spot2 to travel halfway around the WD until it attaches to a near-polar field line. We caution, however, that models that include the full zoo of tesseral harmonics may favor still different magnetic geometries. Furthermore, Zeeman fits based on intensity spectra alone may not yield a unique result. The simultaneous fit to intensity and circular polarization spectra offers better perspectives as demonstrated by @euchneretal02 [@euchneretal05; @euchneretal06] and @beuermannetal07.
The Zeeman spectra of 2008 were obtained in a state of low accretion and yielded a mean mass-flow rate $\dot m$ close to that of the bombardment solution of @woelkbeuermann92. The spot fields obtained from the Zeeman and cyclotron fits differ by about 3%, suggesting that the cyclotron emission originates at a height above the photosphere of roughly 0.01[$R_\mathrm{wd}$]{}, provided both originate at the same position on the WD. This altitude is in the same ballpark as the shock height of $\sim\!0.006$[$R_\mathrm{wd}$]{} predicted by two fluid cooling theory [@fischerbeuermann01] for the mass flow rate listed in Table \[tab:cyc\].
The property by which deviates most drastically from other polars, is the low primary mass of $M_\mathrm{wd}\!=\!0.42\!\pm\!0.05$[$M_\odot$]{}, based on the simultaneous dynamical and Zeeman spectral analysis. The dynamical analysis alone limits the primary mass to $M_1\!<\!0.53$[$M_\odot$]{}. The secondary mass of $M_2\!=\!0.24\!\pm\!0.04$[$M_\odot$]{} appears normal for a CV of 2.855h orbital period. Only two other polars in the Ritter&Kolb catalog [@ritterkolb03 final edition 7.24, 2016] have reported WD masses below 0.50[$M_\odot$]{}. @gaensickeetal00 derived the angular radius of the WD in V1043 Cen from far-ultraviolet spectroscopy and the radius and mass from a distance estimate of 200pc. At the Gaia distance of 172pc [@gaia18a], however, the radius of the WD is ${9.5\times 10^{8}}$cm and, at a temperature of 15000K, the implied mass of 0.57[$M_\odot$]{} is consistent with a carbon-oxygen interior. @schwopemengel97 identified a narrow emission line in EPDra, which they assigned to the irradiated face of the secondary star. The velocity amplitude $K_2'\!=\!210\,\pm\,25$[kms$^{-1}$]{}, led to a primary mass of $0.43\!\pm\!0.07$[$M_\odot$]{}. Since the narrow line was not detected over the entire orbit, an independent confirmation is desirable. Hence, with the possible exception of EP Dra, may be the only polar with good evidence of a low-mass primary.
A long-standing discrepancy exists between the large number of CVs with low-mass primary stars predicted by binary population synthesis models and the fact that none has been definitely detected so far [e.g., @zorotovicetal11]. @schreiberetal16 employed an empirical version of the concept of consequential angular momentum loss (eCAML) that enhances the standard AML caused by magnetic braking and gravitational radiation. The considered CAML is related to nova outbursts and affects primarily low-mass CVs, removing them preferentially from the population. @nelemansetal16 considered asymmetric nova explosions that could provide a kick to the WD and enhanced the mass transfer rate by the ensuing ellipticity of the orbit. @schenkeretal98 considered a Bondi-Hoyle type frictional interaction of the secondary with the nova envelope that transfers orbital angular momentum to the shell and may ultimately drive the system over the stability limit. The AML adopted by @schreiberetal16 varies as $C/M_1$, with C a free parameter. For appropriate $C$, it drives low-mass CVs into catastrophic mass transfer and leaves CVs with massive primaries unaffected. The frictional AML experienced by the secondary star moving in the nova shell is proportional to the ejected mass, which varies approximately as $R_1^4/M_1$. It is furthermore proportional to the duration of the interaction, that is, inversely proportional to the expansion velocity $V_\mathrm{ex}$ of the envelope. The mean expansion velocity can exceed 1000[kms$^{-1}$]{} for high-mass WDs, but is much lower for nova events on low-mass WDs, $V_\mathrm{ex}\!\simeq\!100\!-\!200$ for 0.6[$M_\odot$]{} and “a few tens [kms$^{-1}$]{}” for 0.4[$M_\odot$]{}[@sharaetal93]. A frictional AML that rises steeply with decreasing mass is, therefore, a plausible proposition. The model may explain the lack of low-mass CVs provided friction proves sufficiently effective or an alternative mechanism as the one of @nelemansetal16 can be identified.
With a mass ratio $q\!=\!0.57\!\pm\!0.06$, stays just below the stability limit at $q\!\simeq\!0.65$ [@nelemansetal16; @schreiberetal16], ensuring stable mass transfer in the absence of frictional AML. The added eCAML may render it unstable, if sufficient angular momentum can be extracted from the orbit. The long time scales in low-mass CVs imply, however, that instability is delayed from the time mass transfer started, at least by the waiting time until the first nova outburst and possibly longer. This delay holds similarly for all nova-related CAML descriptions. Nova outbursts in polars with a low-mass primary and typically a rather low accretion rate are especially rare because the ignition mass $\Delta\!M_\mathrm{ig}$ increases (i) with decreasing $\dot M$ and (ii) with decreasing $M_1$ approximately as $R_1^4/M_1$ [@townsleybildsten04 their Fig.8]. For nova outbursts on a He-core WD of 0.4[$M_\odot$]{}, @sharaetal93 found $\Delta\,M_\mathrm{ig}\!\simeq\!{9\times 10^{-4}}$[$M_\odot$]{}. Such a large ignition mass implies that nova outbursts in have a recurrence time in excess of $10^7$yrs, given the long-term mean accretion rate of of ${5\times 10^{-11}}$[$M_\odot$yr$^{-1}$]{}(see below). Hence, may have accreted already for 10 million years without having experienced a nova outburst that lured it on to destruction. This time span amounts to $\sim\,5$% of the typical ${2\times 10^{8}}$ yrs it takes non-magnetic CVs to reach the period gap [@kniggeetal11]. Hence, most CVs with a low-mass primary, must have met their fate much earlier than if the eCAML model correctly describes the evolution of CVs. The existence of suggests that it is either young and still doomed to destruction or is somehow peculiar, in having managed to escape that outcome.
By its mass, the primary in is consistent with a helium WD, but residing in a close binary, it could well be a hybrid star with a mixed He-CO composition [@pradamoroni09; @zenatietal18]. Such WD are created by massive mass loss of the progenitor star that interrupts He-burning after the star has passed the tip of the RGB. The hydrogen-deficient core mass is about 0.46[$M_\odot$]{}, when stars with an initial mass $M_\mathrm{i}\!\la\!2$[$M_\odot$]{} incur the He flash, but for initial masses of $2.2\!-\!2.6$[$M_\odot$]{}, He burning starts at a core mass of $0.32\!-\!0.34$[$M_\odot$]{} and subsequent rapid mass loss in a common envelope event could create a WD with the mass of the primary in that has a mixed He–CO composition.
With a period of 2.855h, is nominally located in the 2.1-3.1h period gap [@kniggeetal11], but the high-state accretion rate $\dot{M}\!\simeq\!{1.4\times 10^{-9}}$[$M_\odot$yr$^{-1}$]{}(Sect. \[sec:sed\]) demonstrates that has not yet entered its period gap, if it ever will. @WickramasingheWu94 and @webbinkwickramasinghe02 argued that polars experience a lower angular momentum loss by magnetic braking than non-magnetic CVs because trapping of the wind from the secondary star in the WD magnetosphere reduces the braking efficiency. They predicted that the gap disappears for primaries with sufficiently high magnetic moments, $\mu\!\ga\!{4\times 10^{34}} $Gcm$^3$. The WD in has $\mu\!<\!{3.6\times 10^{34}} $Gcm$^3$, based on $R_\mathrm{wd}\!\simeq\!{1.22\times 10^{9}}$cm and a polar field strength $B_\mathrm{p}\!\simeq\!40$MG of the dipole component (Table \[tab:zee\]). This may or may not suffice to suppress magnetic braking and dispose of the gap. Even if were about to enter the gap at a period of 2.85h, the delayed entry compared with the standard upper edge of the gap of 3.1h is easily explained by a reduced level of magnetic braking. Given, for example, a braking efficiency reduced by a factor of two over that advocated by @kniggeetal11 [their Fig.14, lower panel], the upper edge of the gap would shift down to 2.85h. A low metalicity of the secondary would add to a delayed entry into the gap [@webbinkwickramasinghe02]. If such a scenario applies to , the star may still reside above its own gap and await entering it at some time in the future, if at all.
The telling argument against strong magnetic braking is the low photospheric effective temperature of the WD of kK. The theory of compressional heating relates [$T_\mathrm{eff}$]{} to the mean accretion rate $\langle \dot M\rangle$ averaged over the Kelvin-Helmholtz time scale of the non-degenerate envelope, which is of the order of $10^6$yr for a low-mass WD [@townsleygaensicke09]. If the star succeeds in establishing an equilibrium between heating and cooling, its quiescent luminosity is $L_\mathrm{q}\!=\!{6\times 10^{-3}}\,L_{\odot}\,\langle\dot
M\rangle_{-10}\,(M_1/M_{\odot})^{0.4}$ [@townsleygaensicke09 their Eq. 1], where $\langle\dot M\rangle_{-10}$ is the accretion rate in units of $10^{-10}$[$M_\odot$yr$^{-1}$]{}. Provided compressional heating dominates over the congenital heat reservoir, we may equate $L_\mathrm{q}$ to $4\pi R_1^2\sigma T_\mathrm{ph}^4$. With $R_1\!=\!{6.56\times 10^{8}}(M_1/M_{\odot})^{-0.60}$ for $M_1\!=\!0.4\!-\!0.6$[$M_\odot$]{} [@althausetal13], we obtain $T_\mathrm{ph}\!=\!16.7\langle \dot
M\rangle_{-10}^{1/4}(M_1/M_{\odot})^{0.40}$kK. For $M_1\!=\!0.42$[$M_\odot$]{} (Tab. 10), we find that $\langle\dot
M\rangle\!=\!{5\times 10^{-11}}$ [$M_\odot$yr$^{-1}$]{} suffices to keep the temperature of the WD at $9\!-\!10$kK. In its high state, reached more than $10^{-9}$[$M_\odot$yr$^{-1}$]{}, implying that the long-term mean duty cycle must be heavily weighted toward states of low accretion. The moderate mass loss of the secondary, the reduced magnetic braking, and the moderate inflation suggested by our best-fit seem to be in line with the evolution of polars .
experienced a highly significant and so far unexplained change of its orbital period by 10.5ms between 2011 and 2018. Similar period variations have been observed in other post common-envelope binaries (PCEB). Attempts to explain the observations involve either (i) the action of additional bodies encircling the binary, causing an apparent period variation, or (ii) solar-cycle like variations in the internal constitution of the secondary star that change its quadrupole moment and the gravitational pull on the primary, leading to a genuine period variation [@applegate92; @voelschowetal18; @lanza19 and references therein]. Process (i) is the likely explanation for at least part of the period variations in NN Ser [@beuermannetal13; @boursetal16], but utterly fails in others like QS Vir [@boursetal16], for which no stable planetary model was found, even considering retrograde and highly inclined orbits (Stefan Dreizler, private communication). The finding of @boursetal16 that PCEB with convective secondaries of spectral type later than M5.5 largely lack period variations seems to favor magnetic cycles as the driving mechanism. The mechanism of @applegate92 and the variants of @voelschowetal18 and @lanza19 are appealing because they are based on physical processes known to exist in late-type stars, but most authors agree that they are too feeble to produce the observed amplitudes of the period variations in many CVs [@voelschowetal18; @lanza19]. It has also been argued that PCEB may be a natural habitat of circumbinary planets [@voelschowetal14]. The wealth of period variations observed in PCEB and RSCVN binaries may well have more than a single physical cause. The data presently available for are not sufficient to draw definite conclusions on the origin of the observed period variation.
In summary, is a rare example of the polar subgroup of magnetic CVs that harbors a low-mass primary, either a helium WD or a hybrid He-CO WD. The system may have passed through the common-envelope phase after severe mass loss on the giant branch or during initial He-burning [@pradamoroni09; @zenatietal18]. The key experiment to prove the low mass of the WD would be a direct measurement of its radius. Unfortunately, our Sloan $griz$ photometry of 2017 lacked the time resolution to measure the radius of the WD from the finite ingress and egress times of the eclipse light curves in $g$ or possibly $r$. With still in a prolonged state of low accretion as of early 2019, this task could be accomplished by high-speed photometry at a large telescope. If the low mass of the WD is confirmed, a dedicated evolutionary study could establish the origin and future evolution of .
We dedicate this paper to the lasting memory of Hans-Christoph Thomas, who analyzed part of the early data described here before his untimely death on 18 January 2012. We thank the anonymous referee for the careful reading of the paper and valuable comments that helped to improve this work. Part of the data were collected with the telescopes of the MOnitoring NEtwork of Telescopes, funded by the Alfried Krupp von Bohlen und Halbach Foundation, Essen, and operated by the Georg-August-Universität Göttingen, the McDonald Observatory of the University of Texas at Austin, and the South African Astronomical Observatory. We made use of the ROSAT Data Archive of the Max-Planck-Institut für extraterrestrische Physik (MPE) at Garching, Germany. Part of the analysis is also based on observations of REJ0501-03 obtained with XMM-Newton on 2002-03-24, Obs-Id 0109460601. The observations at ESO in the years 2000, 2001, 2008, 2016, 2017, and 2018 were collected at the La Silla and Paranal sites under the programme IDs 66.D-0128, 66.D-0513, 082.D-0695, 098.A-9099, and 0100.A-9099. In establishing the spectral energy distribution in Fig. \[fig:sed\], we accessed various data archives via the VizieR Photometric viewer operated at the CDS, Strasbourg, France (http://vizier.u-strasbg.fr/vizier/sed/). We quoted distances for and EPDra from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia).
[^1]: Deceased 18 January 2012
[^2]: http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/588/A103
[^3]: Photometrically, “high” and “low” states refer to two brightness levels between which polars oscillate in their long-term light curves. Spectroscopically, a “high” state is usually characterized by intense HeII$\lambda4686$ line emission, which is absent in a “low” state. Physically, “high” refers to accretion rates adequate to drive the standard secular evolution of CVs, while in a “low” state, accretion ceases either completely or is reduced to a trickle. “Intermediate” refers to temporary states in between.
[^4]: Gamma-ray Burst Optical/Near-infrared Detector.
[^5]: http://astroutils.astronomy.ohio-state.edu/time/
[^6]: Using XSPEC version 12.9.1n with TBabs\*(bbody+apec)
[^7]: http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=J/A+A/594/A116
[^8]: We folded the SDSS spectrum over the Sloan $i$ filter curve for airmass zero, obtaining $i\!=\!14.89$ for SDSSJ101639+240814 and $i\!=\!15.07$ for SDSSJ155653+093656. Adding the distance moduli yields the quoted magnitudes. The $i$-band magnitudes of the SDSS spectra represent the appropriate reference for the present purpose, although they differ by +0.05 and $-0.05$, respectively, from the DR15 photometry. The quoted colors $r\!-\!i$ and $i\!-\!z$ are those of the SDSS photometry.
[^9]: The cross section of the secondary as seen along the line connecting the two stars is taken as elliptical with axes $y_4$ and $z_6$, yielding $f_\mathrm{back}\!=\!\sqrt(y_4\,z_6)/(r^*)_2\!\simeq\!0.961$, in Kopal’s (1959) notation, with $(r^*)_2$ the equivalent volume filling radius of the Roche lobe.
[^10]: The second $\Lambda$-k$T$ relation is based on equations for the post-shock plasma temperature k$T$ and the column density $x_\mathrm{s}$ of the post-shock cooling flow presented in Figs. 5 and 6 and Eqs.(19) and (20) of @fischerbeuermann01. Elimination of the variable $\dot m\,B_7^{-2.6}$, with $\dot m$ the mass-flow density in gcm$^{-2}$s$^{-1}$ and $B_7$ the field strength in units of $10^7$G, yields the desired relation between k$T$ and $\Lambda(x_\mathrm{s})$, which is well fitted by a power law valid for k$T$ up to about 3keV. For the present purpose, the relations in question were re-calculated for a plasma of solar composition and a WD mass $M_1\!=\!0.4$[$M_\odot$]{} as suggested by our dynamic models in Sect. \[sec:system\].
[^11]: Model BR08: $a_\mathrm{irr}\!=\!0.806155-0.959167\,q+0.356060\,q^2$; Model BT90: $a_\mathrm{irr}\!=\!0.786470-0.978076\,q+0.368182\,q^2$; both for $q\!=\!0.35\!-\!0.80$.
[^12]: Baraffe et al. (2015) main sequence mass-radius relations, $M_2/M_\odot\!= \!0.15\!-\!0.40$[$M_\odot$]{}, $MH\!=\!0$, 1Gyr: $R_2/R_\odot\!=\!0.736(M_2/M_\odot)^{0.761}$; 5Gyr: $R_2/R_\odot\!=\!0.766(M_2/M_\odot)^{0.780}$; 10Gyr: $R_2/R_\odot\!=\!0.787(M_2/M_\odot)^{0.792}$; $MH\!=\!-1$, 1Gyr: $R_2/R_\odot\!=\!0.735(M_2/M_\odot)^{0.793}$.
[^13]: The envelope mass varies between $M_\mathrm{H}\!=\!{8\times 10^{-5}}$ at $M_1\!=\!0.52$[$M_\odot$]{} and ${3\times 10^{-4}}$[$M_\odot$]{} at $0.3$[$M_\odot$]{}, staying below the respective ignition masses.
[^14]: http://evolgroup.fcaglp.unlp.ar/TRACKS/DA.html
[^15]: Centre de Données astronomiques de Strasbourg, http://vizier.u-strasbg.fr/vizier/sed/
[^16]: Data Release 15, http://www.sdss.org/dr15
[^17]: A long-term light curve is available from the Catalina Sky Survey at http://nesssi.cacr.caltech.edu/catalina/CVservice/CVtable.html
|
---
abstract: 'The compound SmB$_6$ is the best established realization of a topological Kondo insulator, in which a topological insulator state is obtained through Kondo coherence. Recent studies have found evidence that the surface of SmB$_6$ hosts ferromagnetic domains, creating an intrinsic platform for unidirectional ballistic transport at the domain boundaries. Here, surface-sensitive X-ray absorption (XAS) and bulk-sensitive resonant inelastic X-ray scattering (RIXS) spectra are measured at the Sm N$_{4,5}$-edge, and used to evaluate electronic symmetries, excitations and temperature dependence near the surface of cleaved samples. The XAS data show that the density of large-moment atomic multiplet states on a cleaved surface grows irreversibly over time, to a degree that likely exceeds a related change that has recently been observed in the surface 4f orbital occupation.'
author:
- Haowei He
- Lin Miao
- Edwin Augustin
- Janet Chiu
- Surge Wexler
- 'S. Alexander Breitweiser'
- Boyoun Kang
- 'B. K. Cho'
- 'Chul-Hee Min'
- Friedrich Reinert
- 'Yi-De Chuang'
- Jonathan Denlinger
- 'L. Andrew Wray'
title: 'Irreversible proliferation of magnetic moments at cleaved surfaces of the topological Kondo insulator SmB$_6$'
---
[^1]
The topological Kondo insulator (TKI) state is a variant of the topological insulator state [@pred1; @ColemanReview; @FuOriginal; @ZahidMooreReview], in which a topologically ordered insulating electronic band structure is obtained from Kondo physics. The realization of a TKI state in mixed-valent SmB$_6$ was strongly indicated by early theoretical investigations [@pred1; @pred2], and has now been established through direct measurement of the topological surface states via angle resolved photoemission [@MadhabARPES; @FengARPES; @ShiSpinTexture; @Kondo110K; @ReinertChargeFluctuations; @JonathanSmB6PEreview] and transport studies [@TKItransport; @TKIsurfaceFM]. Strong evidence has recently been found suggesting that the surface of polished SmB$_6$ samples can also host ferromagnetic domains [@TKIsurfaceFM], a property that is theoretically associated with exotic axion electrodynamics, an inverse spin-galvanic effect, and ballistic one dimensional transport channels at domain boundaries [@magTI1; @MagTI2; @MagTI3; @ZahidMooreReview]. Moreover, surface sensitive X-ray photoemission (XPS) measurements have shown that the surface 4f occupation evolves irreversibly towards 4f$^5$ as a function of time following cleavage in ultra high vacuum (UHV) [@surfaceValencePhMag]. Here, multiplet-dominated X-ray absorption spectroscopy (XAS) and resonant inelastic X-ray scattering (RIXS) measurements in the vacuum ultraviolet (VUV) regime are used as a symmetry-sensitive probe to map the Sm N$_{4,5}$-edge excitations and show that a similarly large change in the density of large-moment samarium sites accompanies this time evolution. This evolution is consistent with expectations for the transition from a Kondo insulating state to magnetism, and represents a means for incrementally tuning the strength of the surface magnetic instability.
Measurements were performed at the beamline 4.0.3 (MERLIN) RIXS endstation (MERIXS) [@YiDeDetector; @YiDeSRN] at the Advanced Light Source (ALS), Lawrence Berkeley National Laboratory. Large single crystals of SmB$_6$ were grown by the Al flux method as in Ref. [@surfaceValencePhMag], cleaved at low temperature, and maintained at a UHV pressure of approximately 3$\times$10$^{-10}$ Torr. The photon beam had a grazing 30$^o$ or angle of incidence to the cleaved \[001\] sample face, and scattered photons were measured at 90$^o$ to the incident beam trajectory. XAS was measured using the total electron yield (TEY) method, and the expected penetration depth of measurements is roughly d$\lesssim$2 nm for XAS [@universalCurve] and d$\sim$10-30 nm for RIXS [@XrayPath]. To minimize sensitivity to surface inhomogeneity, the beam profile on the sample was configured as a very broad strip with dimensions of roughly $10 \times 600 \mu m^2$ (similar results from additional cleaves are shown in the online Supplimental Material (SM) [@SM]).
Atomic multipet simulations were performed with typical renormalization values for the multipolar Slater-Condon interaction parameters [@SM]. Similar multiplet models that focus on Sm f-electrons, disregarding the itinerant 5d electron gas, have been remarkably successful in reproducing XAS and XPS features of SmB$_6$ [@multipletSmB6_PES2015; @JonathanSmB6PEreview]. The multiplet ground states have f-electron angular momentum quantum numbers of (4f$^5$) J=5/2 and (4f$^6$) J=0, representing the presence or absence of a hole in the J=5/2 4f bands. The atomic multiplet picture is expected to be most accurate as a description of the resonance states, which are dominated by extremely strong angular momentum coupling between the 4d core hole and 5d electrons on the scattering site (a $\sim$20eV combined energy scale). Multiplet state energetics in the VUV are defined in terms of coherent local moment symmetries. In contrast to the previous study of 4f charge density at the SmB$_6$ surface [@surfaceValencePhMag], atomic multiplet measurements in the VUV are sensitive to the *coherent multi-particle symmetry* of electrons in hybridized electronic orbits involving both the scattering site and neighboring atoms [@WrayFrontiers; @EdwinProc]. This multiplet symmetry can be thought of as the ‘nominal valence’ state defining local moment degrees of freedom, and can deviate significantly from the atomically resolved charge density.
The samarium N$_{4,5}$-edge XAS spectrum of a pristine SmB$_6$ surface that has been recently cleaved and maintained at T$<$100K within UHV is shown in Fig. \[fig:ChangingKondoLattice\](a) (blue curve). To facilitate comparison of features, the non-resonant background has been aligned beneath the resonance at $h\nu$=123 eV, and the curves have been set to have the same integrated area. The spectrum contains a large number of features, and the higher energy features have broader line shapes, as is the typical trend for core hole lifetimes at a multiplet-split resonance [@HaverkortAutoIon; @KotaniIdea; @KotaniIdea2; @WrayNiO; @WrayFrontiers; @WrayRIXSinterference; @EdwinProc]. Relatively sharp line shapes are observed in the incident energy range from 126-133, suggesting that there may be a charge transfer threshold at h$\nu$$\sim$133 eV [@HaverkortAutoIon; @EdwinProc].
The spectrum changes pronouncedly as the sample is aged. Difference spectra in Fig. \[fig:ChangingKondoLattice\](b) show that as time progresses, spectral intensity shifts into higher energy features at $h\nu>136$ eV, and the sharper 126-133 eV features shift into a new spectral pattern that bears little resemblance to that seen initially. Even though the time interval between each pair of successive curves is approximately the same (45 to 60 min), the magnitude of the change is significantly larger for the interval from 4.5 to 5.5 hr, in which the sample was heated to room temperature. Cycling back to low temperature (black curve) resulted in only small quantitative changes. Later scans did not reveal continued changes, however the aging trend can be taken further still by exposing the sample surface to air (see red curve), suggesting that aging the surface is promoting changes in the f-electron count that resemble oxidation. The fractional change in feature intensities after thermal cycling is dramatic, and ranges from 10-50$\%$ throughout most of the spectrum.
To identify the physical significance of the surface evolution, a multiplet simulation in Fig. \[fig:ChangingKondoLattice\](c) shows the features expected in Sm N$_{4,5}$-edge XAS from J=0 and J=5/2 sites. Dashed drop-lines highlight an excellent qualitative match between the regions that lose intensity during the aging process and the J=0 features, while the J=5/2 features correlate with a gain (or reduced loss) of intensity. Only one feature is clearly anomalous in this analysis. The lowest energy XAS peak at h$\nu$$=$126.6 eV is not reproduced by either the J=0 or the J=5/2 multiplet calculation, and has temperature dependence consistent with a J=5/2 symmetry attribution.
Similar XAS measurements have also been performed as a function of time on an electron doped sample with the composition Sm$_{0.98}$La$_{0.02}$B$_6$ (see Fig. \[fig:ChangingKondoLatticeLa\]). In this case, the sample was maintained at T$<$100K for a much longer 46 hr period, and the measurements confirm earlier observations that the aging process at low temperature proceeds on a time scale longer than 1 day [@surfaceValencePhMag]. Heating to T=250K produced small changes that were fully reversed upon cycling back to T$<$100K. This is in contrast to the more rapid measurement on undoped SmB$_6$, and suggests that the physical end-point of low temperature aging is the same as the rapidly achieved end-point of room temperature aging.
Lanthanum is expected to act as a net electron donor, entering an ionization state much closer to 3+ as compared to Sm. Doping into the Sm 4f orbitals of roughly 0.3e$^-$/La atom is attributed from susceptibility studies [@tempAndValence]. The initial Sm$_{0.98}$La$_{0.02}$B$_6$ XAS spectrum is nearly identical to the slightly aged t=4.5 hr spectrum of undoped SmB$_6$, and the final aged (cycled) curve is qualitatively identical to XAS from the fully aged undoped SmB$_6$ sample. The fact that J=5/2 features in the base (t$\sim 3$ hr) spectrum of Sm$_{0.98}$La$_{0.02}$B$_6$ are more prominent than in the low temperature spectrum of SmB$_6$ is at odds with the identification of La as an electron donor, and suggests some variability in the nature of the cleaved surface (this is confirmed in Fig. S3 of the SM [@SM]).
Measuring resonant inelastic scattering at the photon energies used for XAS reveals a wide range of excitations, some of which are quite sharp in energy, as seen in Fig. \[fig:RIXSanalysis\](a). Prominent features at E=0.9, 2.5, 3.7 and 5.2 eV appear to be Raman-like (non-dispersive), however the upper bound of intensity on the energy loss axis disperses with a slope similar to 1 as a function of incident energy, starting from $\sim$3 eV at an incident energy of h$\nu$=126.6 eV, as expected for scattering scenarios that involve interplay with degrees of freedom that have similar energetics with or without a core hole present. This suggests that a truly accurate model of the RIXS excitations must incorporate significant non-local physics, going beyond the degrees of freedom on a single scattering site. The RIXS spectrum and optical conductivity [@allenOptical1977; @RussianOC; @newOC] both picks up intensity from roughly 1.5eV, suggesting the onset of a large density of itinerant continuum states, however the optical data contain no candidates for *any* of the energetically sharp features seen by RIXS.
In spite of not fulfilling this requirement, the atomic multiplet model is still expected to be accurate for certain relatively localized excitations, and to yield accurate matrix elements of the ‘direct RIXS’ scattering process [@AmentRIXSReview; @WrayRIXSinterference]. One can attribute a tentative correspondence between energy loss regions with large intensity in the J=5/2 simulation and the intensity seen by RIXS (see highlighted regions in Fig. \[fig:RIXSanalysis\](a-b)). The feature at E$\sim$0.9 eV gives a particularly close correspondence, and has no competing interpretation within the J=0 simulation (Fig. \[fig:RIXSanalysis\](c)). This E$\sim$0.9 eV mode is actually a collection of closely spaced features, with additional peaks visible in the $h\nu=$127.8 and 130.8 eV curves. Lower energy excitations visible in the simulations are not resolved from the elastic line, which is strong due to the broad off-angle tail of specular reflection in the VUV [@WrayFrontiers]. The lack of easily identifiable J=0 derived features may indicate that single-atom excitations on 4f$^6$ sites are shorter lived (i.e. broader), possibly because they can easily delocalize into 4f$^5$**X** states, where ‘**X**’ indicates an electron that has entered a more delocalized band symmetry. The anomalous $h\nu=126.6$ eV resonance, which does not occur in our multiplet simulations, resonates primarily with the 0.9 eV feature, corroborating identification of the $h\nu=126.6$ eV resonance with scattering from a J=5/2 site.
A sufficiently large increase in the density of J=5/2 sites in the mixed-valent samarium lattice is expected to destabilize the Kondo insulating state and induce magnetism, as has been seen in high pressure studies [@pressureNf; @pressure2; @pressureNickButch]. The amplitude of the surface change can be evaluated from the XAS data in Fig. \[fig:ChangingKondoLattice\], if we adopt the approximation that the spectra can be broken down into linear combinations of pure J=5/2 and J=0 curves (see derivations in the SM [@SM]). The accuracy of this approximation is supported in the present case by the observation of a fairly stable isospectral (constant intensity) point at 136.7-137eV for all warming curves.
Monovalent J=5/2 and J=0 XAS spectra algebraically obtained from comparing SmB$_6$ XAS measurements at the pristine (t=3 hr) and aged (cycled) surface are plotted in Fig. \[fig:toMagnetism\](b). Shaded regions represent error margins as described in the caption, and are associated with different extrapolated changes in the population of J=5/2 and J=0 sites over the aging period. The outer shaded boundaries represent extreme scenarios that lead to clear anomalies in the extrapolated monovalent curves, such as duplicated features and negative XAS intensity [@SM]. The best fit J=0 XAS curve is obtained with the assumption that the J=5/2 population grows by a remarkable 60-80$\%$. Growth estimates beneath 40$\%$ lead to multiple significant artifacts in the extrapolated XAS spectrum. Based on the extrapolated XAS curves, we can assess that a single-atom multiplet calculation provides many matrix elements of the N$_{4,5}$-edge resonance process, but is of marginal use for understanding the RIXS excitations, and omits XAS features that appear to be associated with coupling to itinerant states.
High pressure studies have found that a new magnetic ground state [@pressureNf; @pressure2; @pressureNickButch] is realized in bulk SmB$_6$ beyond a crossover point thought to occur when the fractional density of 4f$^5$ sites reaches roughly $n_{5/2}\sim0.65$ [@pressureNf; @pressureNickButch]. Combined examination of the extrapolated J=5/2 and J=0 curves enables an algebraic derivation of the t=3 hr J=5/2 site density as $n_{5/2}$=0.39$\pm$0.05 [@SM], with a surface evolution plotted in Fig. \[fig:toMagnetism\](a). In distinguishing $n_{5/2}$ from the nominal bulk 4f occupancy, which is thought to be $n_f\lesssim 5.5$ [@pressureNf; @pressureNickButch; @JonathanSmB6PEreview; @XASvalenceTdep; @JonathanARPES_DMFT], this picture suggests that the transition is one from a singlet-dominated low temperature regime, with fewer observed large-moment sites than the 4f occupancy alone would suggest, to an aged surface in which local moments are less screened. This scenario matches the attributed behavior as the crystal is driven into a magnetic state under pressure [@pressureNf; @pressureNickButch]. We note that the nature of the magnetic order achieved under pressure is not definitively understood, however a proximate ferromagnetic state can be achieved from 1$\%$ Fe doping at ambient pressure [@Fe1pctMag]. The air-exposed end point of $n_{5/2}\sim0.9$ is consistent with the proposed interpretation of an earlier soft X-ray M-edge XAS measurement [@CdopingAndXAS].
A comprehensive interpretation of the mechanism behind the observed irreversible surface changes is beyond the scope of the present study. Such a theory may need to address multiple factors including the correlated electronic structure, the intrinsically polar and anisotropic nature of cleaved surfaces, and the presence of multiple known surface reconstructions [@JonathanSmB6PEreview; @STM1; @flashed]. The bulk of the sample also evolves towards a lower 4f occupancy with increasing temperature [@pressureNf; @JonathanSmB6PEreview; @XASvalenceTdep; @JonathanARPES_DMFT; @Kondo110K]. However, the bulk-sensitive RIXS spectrum undergoes no easily visible changes as a function of temperature [@SM], suggesting that surface aging is a far more dramatic effect. The La doped sample is thought to have roughly identical bulk temperature dependence [@tempAndValence].
These resonant X-ray absorption results show that the density of large-moment Sm sites in the top $\lesssim$2 nm of cleaved SmB$_6$ more than doubles as the surface ages in UHV. The increase can be accelerated by heating to room temperature, and is taken further through exposure to air. Lower energy excitations of SmB$_6$ are mapped with RIXS, providing a window into dynamics and energetics of the valence electrons, which will serve as a reference for future theoretical and spectroscopic investigations. The large surface changes seen by electron yield XAS are distinct from the lack of extraordinary temperature dependence at depths of 10-30 nm, as evaluated from RIXS spectra. The increased density of large-moment sites on aged samples provides a plausible explanation for the recent observation of ferromagnetic domains at a polished SmB$_6$ surface [@TKIsurfaceFM]. More generally, the apparent sensitivity of the surface evolution to temperature and the gas environment provides mechanisms for control of the surface properties, to achieve a desired interplay between surface magnetic moments and the topologically ordered bulk electronic structure.
**Acknowledgements:** The Advanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. C.H.M. was supported by the DFG (through SFB 1170 “ToCoTronics", projects C06). We are grateful for productive discussions with H. Dehghani and J. Hoffman.
\[
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[^1]: Corresponding author
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---
abstract: 'We present a decaying dark matter scenario where the daughter products are a single massless relativistic particle and a single, massive but possibly relativistic particle. We calculate the velocity distribution of the massive daughter particle and its associated equation of state and derive its dynamical evolution in an expanding Universe. In addition, we present a model of decaying dark matter where there are many massless relativistic daughter particles together with a massive particle at rest. We place constraints on these two models using supernovae type Ia observations. We find that for a daughter relativistic fraction of 1% and higher, lifetimes of at least less than 10 Gyrs are excluded, with larger relativistic fractions constraining longer lifetimes.'
author:
- Gordon Blackadder
- 'Savvas M. Koushiappas'
bibliography:
- 'decayingDM.bib'
title: 'Dark matter with two- and many-body decays and supernovae type Ia'
---
\[sec:level1\]Introduction
==========================
Decaying dark matter has become a matter of considerable interest over the last few years. It has been conjectured to answer specific questions related to cosmological large scale structure and unexpected observations of high-energy neutrinos, the positron fraction, and gamma-ray measurements. Further motivation comes from particle physics models beyond the standard model as well as a basic desire to better understand the nature of dark matter.
Issues surrounding structure formation center on two problems. First is the so-called “cuspy core” issue where observation suggests galaxy cores have a constant dark matter density whereas simulations predict a cuspy density profile rising in the center [@Kaplinghat:2005aa; @Gong:2008aa; @Strigari:2007aa; @DeLope-Amigo:2009aa; @Peter:2010aa; @Wang:2012aa; @Wang:2013aa]. Second is the problem of missing satellites where dark matter-only numerical simulations predict a large number of dark matter halos present in the potential well of a host halo, while observations support the existence of roughly a factor of 5-10 fewer (e.g., in the Milky Way halo – for a more in depth analysis of these two problems see [@Weinberg:2013aa]). Both of these problems have been shown to be potentially solved by N-body simulations of decaying dark matter [@Wang:2014aa].
From the observational point of particle astrophysics there are some rather interesting problems that may relate to decaying dark matter. For example, recently, IceCube reported on the observation of very high-energy neutrinos ($\sim$PeV energies) whose observational properties (energies, directions, and flavors) are not consistent with what one would expect from the known backgrounds at the $4\sigma$ level [@IceCube-Collaboration:2013aa; @2014PhLB..733..120E]. The now well-known problem of excess energetic positrons as reported by PAMELA [@Adriani:2011ab] and AMS-02 [@Aguilar:2013aa] may also have a decaying dark matter explanation (e.g., [@Ibarra:2014aa; @Cirelli:2012aa]) as well as observations of gamma-ray lines and diffuse background measurements [@Yuksel:2008aa; @Bell:2010aa; @Buchmuller:2012aa].
From the theoretical particle physics point of view, there are many dark matter candidates that arise in the context of decays in physics beyond the standard model, such as sterile neutrinos [@1994PhRvL..72...17D], hidden photinos [@Morrissey:2009aa], gravitino dark matter [@Moroi:1995aa] (all of which are discussed in detail in @Essig:2013aa), as well as cryptons [@Ellis1990257], moduli dark matter[@Asaka:1998aa], axinos [@Kim200218], and quintessinos [@Bi:2004aa] (all of which are covered in [@Chen:2004aa]). Indeed as is pointed out by @Ibarra:2013aa there is no *a priori* reason to believe that dark matter particles should be absolutely stable.
One of the draw backs to considering specific dark matter candidates with particular decay channels and known outcomes is that while they can be tightly constrained such results are not widely applicable. Many general decaying dark matter models have been derived that try to make only a few assumptions about decay products.
In this paper we derive two rather general models. In the first we assume that the parent dark matter particle decays over time to a single, massless and relativistic particle and a single, massive and possibly relativistic particle (with velocity determined by momentum conservation). The velocity of the massive particle falls as the Universe expands and therefore there is a distribution of velocities as different heavy daughter particles will have been created at different times. The only assumption made in determining the evolution of the velocity distribution (besides the existence of such a two-body dark matter decay) is that the particles are noninteracting (i.e., no standard model interactions and no interactions among themselves). We also present a second model in which the assumption that there is only one massless particle is relaxed. This necessarily means that the velocity of the massive particle is indeterminate and so it will be assumed to be stationary. We then use recent supernovae type Ia data to constrain these rather general models.
Type Ia supernovae are good candidates for constraining cosmological models. As standard candles their luminosity is well correlated with their observed brightness profiles. Therefore the only parameter affecting the observed luminosity flux is their luminosity distance – the main idea behind the Hubble diagram and the revolutionary discovery of the accelerated Universe. Luminosity distance is a function of all known energy budget contributions to the Universe and their dynamics (including relativistic and nonrelativistic components). This fact is what motivates the use of supernovae type Ia as a probe of the possibility of a decaying dark matter scenario. Of course, as supernovae type Ia are late-universe standard candles \[i.e., located at $z \sim {\cal{O}}(1)$\], it is expected that their constraining power will be concentrated towards long-living decaying dark matter particles (of order the age of the Universe).
The paper is structured as follows. Section \[sec:level2\] derives the two-body decay while Sec. \[sec:level3\] looks at the many-body decay. Section \[sec:level4\] details the data against which the models will be compared and considers some of the other relevant physics required to calculate the cosmological effects of decaying dark matter. The results are given in Sec. \[sec:level5\] alongside a detailed discussion on where these constraints fit in the bigger picture.
\[sec:level2\]Two-Body Decay
============================
In this section we consider two-particle decay, with a parent dark matter particle (labeled with a subscript 0) with mass $m_{0}$ moving at rest relative to the expansion of the Universe, a massless and relativistic daughter particle (with subscript 1) and a second daughter particle (subscript 2) with mass $m_{2}$. The second particle may or may not be relativistic at the time of its creation (see Fig. \[fig:TWOpict\]).
First, consider the 4-momenta of the particles at the time of decay, $$\begin{aligned}
p_{\mu, 0} &=& (m_{0}c^{2}, \textbf{0}) \\
p_{\mu, 1} &=& (\epsilon m_{0}c^{2},\boldsymbol{p_{1}}) \\
p_{\mu, 2} &=& ((1-\epsilon) m_{0}c^{2},\boldsymbol{p_{2}}) \end{aligned}$$ Here $\epsilon$ denotes the fraction of the energy of the parent particle that has been transferred to the massless daughter particle. Energy and momentum conservation implies $$\epsilon = \frac{\widetilde{m} \beta_{2}}{\sqrt{1-\beta_{2}^{2}}},
\label{momentum2body}$$ and $$(1-\epsilon)^{2}= \widetilde{m}^{2} + \frac{\widetilde{m}^{2}\beta_{2}^{2}}{1-\beta_{2}^{2}}
\label{energy2body}$$ where $\widetilde{m} = m_{2}/m_{0}$ and $\beta_{2} = v_{2}/c$. Note that throughout the rest of the derivation we shall use natural units such that $c=1$.
These two expressions give a unique relationship between $\epsilon$ and $\widetilde{m}$ and $\epsilon$ and $\beta_{2}$, $$\epsilon = \frac{1}{2}(1 -\widetilde{m}^{2})
\label{epsilon2body}$$ $$\beta_{2}^{2} = \frac{\epsilon^{2}}{(1 - \epsilon)^{2}}
\label{beta2body}$$ Note that $\epsilon = 0$ when $m_{2} = m_{0}$, and the maximum value is $\epsilon = 1/2$ when $\widetilde{m} = 0$. As $\epsilon$ approaches the value of 1/2, $\beta_{2}$ approaches the value of 1 which corresponds to the second particle being relativistic with a boost given by $$\gamma_{2} = \frac{1}{\sqrt{1 - \epsilon^{2} / (1-\epsilon)^{2}}}.$$ We will now consider the evolution of the densities of these three particle species.
![A pictorial of a two-body decay from a massive, stationary parent particle to a massless, relativistic particle and a massive, possibly relativistic particle.[]{data-label="fig:TWOpict"}](Twobody.pdf){width="25.00000%"}
The parent
-----------
The rate of change of the parent particle is straightforward. The density decreases over time due to the expansion of the Universe and due to the decay. If the decay rate is $\Gamma = 1/\tau$ where $\tau$ is the lifetime of the particle, the time evolution of the parent particle is given by $$\frac{d\rho_{0}}{dt} + 3\frac{\dot{a}}{a}\rho_{0} = -\Gamma \rho_{0}$$ or $$\rho_{0}(a) = {\cal{A}} \, \frac{e^{-\Gamma t(a)}}{ a^3},
\label{eq:parent2body}$$ where ${\cal{A}}$ is some normalization constant and $t(a)$ is the age of the Universe at $a$. We choose to normalize the density of the heavy parent particle at the epoch of recombination (at a scale factor of $a_*$) using cosmic microwave background (CMB) data (see Sec. IV). We will further simplify matters by assuming that no decays occur in the early Universe before recombination. Under these assumptions the normalization constant is $${\cal A} = {\rho_{\mathrm{c}}}\Omega_{\mathrm{cdm}} \, e^{ \Gamma t(a_*)},$$ where, ${\rho_{\mathrm{c}}}$ is the present value of the critical density, $\Omega_{\mathrm{cdm}}$ is the matter density as measured at the present epoch by CMB experiments, and $t(a_*)$ is the age of the Universe at the epoch of recombination (taken to be the age of the Universe that corresponds to approximately a redshift of $z \approx 1090$ [@Planck-Collaboration:2013ab]).
The massless daughter
---------------------
The evolution of the massless daughter particle’s density is governed by the decay rate of the parent particle and by the expansion of the Universe including the effect of redshifting, i.e., $$\frac{d\rho_{1}}{dt} + 4 \frac{\dot{a}}{a} \rho_{1} = \epsilon \Gamma \rho_{0}$$ Using Eq. (\[eq:parent2body\]) we can write this as $$\rho_{1}a^{4} = \epsilon{\cal A} \int_{a_*}^a \Gamma e^{-\Gamma t} a dt$$ and with integration by parts and $d(e^{-\Gamma t}) = -\Gamma e^{-\Gamma t} dt$ we get $$\rho_{1}(a) = \frac{\epsilon \cal A}{a^{4}} \left[ \int_{a_{*}}^{a} e^{-\Gamma t(a^{\prime})} da^{\prime} - a^{\prime} \, e^{-\Gamma t(a^{\prime})} \bigg|_{a_{*}}^{a}\right],$$ where the lower bound of the integrals have been evaluated at $a_{*}$ in keeping with the boundary conditions. Evaluating the last term gives $$\rho_{1}(a) = \frac{\epsilon \cal A}{a^{4}} \left[ \int_{a_{*}}^{a} e^{-\Gamma t(a^{\prime})} da^{\prime} - a \, e^{-\Gamma t(a)} + a_{*} \, e^{-\Gamma t(a_{*})}\right]
\label{eq:rho1a}$$
The massive daughter and its equation of state
----------------------------------------------
Consider the change in the comoving abundance of the massive daughter particles at some time $t_{D}$ (or scale factor $a_D$). This is related to the comoving abundance of the parent particle by $$\frac{d n_2}{dt_D} = - \frac{d n_0}{dt_D} = -\frac{d (a^{3}\rho_{0}/m_{0})}{dt_{D}} = \frac{\Gamma {\cal{A}} \, e^{-\Gamma t_{D}}}{m_{0}}.
\label{eq:n2numberdensity}$$ In other words, the change in the number of massive daughter particles is equal to minus the change in the number of parent particles over the same time interval; for every parent that decays one massive daughter is created.
The momentum of the massive daughter particle at some later time $t>t_D$ when the scale factor is $a > a_D$ will then be inversely proportional to $a$ (i.e., the longer the particle has been around the slower it moves), $$p_2(a) =\frac{m_2 \beta_2 }{\sqrt{1 - \beta_2^2}} \, \left( \frac{a_{D}}{a} \right).
\label{decayvelocity2particle}$$ For small values of $\epsilon$, $\beta_2 \rightarrow 0$, and we recover the nonrelativistic redshifting of velocity ($v(a) \propto a^{-1}$).
The ratio of the energy of massive daughter particles at $a$ to the rest mass energy of the parent particle is $$\begin{aligned}
\frac{E_2(a, a_{D})}{m_{0}}&=& \widetilde{m} \sqrt{\frac{\beta_{2}^{2}}{1-\beta_{2}^{2}}\left(\frac{a_{D}}{a}\right)^{2}+ 1} \\
&=& \sqrt{1-2 \, \epsilon} \, \left[ \frac{\beta_2^2}{1 - \beta_2^2} \left(\frac{a_D}{a} \right)^2 + 1 \right]^{1/2}
\label{eq:particle2energy}\end{aligned}$$
Equation (\[eq:particle2energy\]) shows that at early times ($a_D \approx a$) the energy of the daughter particle is as expected $(1-\epsilon)m_{0}$, while at later times ($a_D \ll a$) it falls to $\sqrt{1-2\epsilon}m_{0}$. For small values of $\epsilon$ this effect is negligible, but as $\epsilon $ approaches the value of $\epsilon \approx 1/2$ this effect becomes significant as we will discuss further below.
It is now relatively straightforward to derive the energy density of massive daughter particles that were created at time $t_D$. First, we calculate the energy density at time $a > a_D$ as $$\begin{aligned}
\rho_{2}(a) = \frac{1}{a^{3}} \int _{a_{*}}^{a} E_{2}(a, a_{D})dn_{2}(a_{D})\end{aligned}$$ Substituting from Eq. (\[eq:n2numberdensity\]) and using $dt_D = da_D / (a_D H_D)$, where $H_D$ is the expansion parameter at the epoch of decay, we get that the total energy density of daughter particles at a redshift $a > a_D$ is $$\begin{aligned}
\rho_2(a) &=& \frac{ {\cal{A}} \, \Gamma \sqrt{1 - 2 \epsilon}}{a^3} \int_{a_*}^a {\cal{J}}(a, a_{D}) da_D \label{eq:rho2evolution} \\
{\cal{J}}(a, a_{D}) & \equiv & \frac{e^{-\Gamma t(a_D)}}{a_D H_D} \sqrt{ \frac{\beta_2^2}{1 - \beta_2^2} \left( \frac{a_D}{a} \right)^2 + 1 } \nonumber $$ Note that the integral of Eq. (\[eq:rho2evolution\]) must be solved iteratively as $H_D$ \[and consequently $t(a_{D})$\] depends on the value of $\rho_2$ at each decaying epoch $a_D$ \[$H_D^2(a_{D}) = H_0^2 \sum_i \Omega_i(a_{D})$, where $\Omega_i = \rho_i / {\rho_{\mathrm{c}}}$, and $i$ runs over all the constituents of the Universe, including the massive daughter particle (i=2) with density $\rho_2$\].
Equation (\[eq:rho2evolution\]) shows that when $a\approx a_{D}$ and for large values of $\beta_{2}$ the density falls off as $a^{4}$ but as $a$ increases sufficiently ($ a \gg a_D$) the density falls with $a^{3}$. This is the case for a particle that is born relativistically at decay, but becomes nonrelativistic at late times (an important feature in decaying dark matter physics that has been relatively absent from the literature).
One convenient way of expressing the cosmological evolution of the massive daughter particle is the equation of state, $$w_{2}(a) = \frac{1}{3} \langle v_2(a)^{2}\rangle.$$ This useful quantity can be derived from some basic thermodynamic assumptions and the results of the previous section.
The velocity of a massive daughter particle, whose parent decayed at $a_D$, has velocity $v$ at epoch $a$ given by $$v_2^2(a,a_D) = \frac{ (a_D/a)^2 \beta_2^2 }{1 + \beta_2^2 [ ( a_D/a)^2 - 1]}$$ The averaged (over all particles) velocity is derived by integrating over all particles that were created at or before $a$. $$\langle v^{2}(a) \rangle = \left[ \int_{a_*}^a v^{2}(a,a_D) \frac{dn_{D}}{dt_D} \, dt_D \right] \left[ \int_{a_*}^a \frac{dn_{D}}{dt_D} \, dt_D \right]^{-1}$$ We can use Eq. (\[eq:n2numberdensity\]), and by expressing the integral over time as an integral over the scale factor in the first term we get $$\begin{aligned}
w_2(a) &=& \frac{1}{3}\frac{ \Gamma \beta_2^2}{e^{- \Gamma t_* } - e^{- \Gamma t} }\nonumber \\
& \times & \int_{a_*}^{a} \frac{ e^{-\Gamma t(a_D)} d\ln a_D}{H_D [(a/a_D)^2 ( 1 - \beta_2^2 ) + \beta_2^2]} \label{eq:w2a}\end{aligned}$$ This expression must also be solved iteratively as $H_D$ is a function of $\rho_2(a)$ and its evolution with $a$ (thus $w_2(a)$). Figure \[fig:w2\] shows the evolution of the equation of state of the massive daughter as a function of scale factor for various values of $\epsilon$ and $\tau$. For the highly relativistic case where $\epsilon$ approaches the value of $1/2$ the massive daughter behaves at early times in a fashion similar to radiation (i.e., with a value of $w_2 \approx 1/3$). At later times (depending on the decay time scale) the value of $w_2$ decreases, thus the massive daughter behaves in a nonrelativistic manner, with an equation of state that approaches $w_2 \approx 0$ as expected.
Note that Eq. (\[eq:w2a\]) and Fig. \[fig:w2\] serve as a sanity check on the validity of the calculation presented here; however, in practice it is easier to implement \[eq:rho2evolution\] rather than \[eq:w2a\] (as we discuss in the next section).
![The evolution of the equation of state of the massive daughter particle in two-body decays \[Eq. (\[eq:w2a\])\]. As the value of $\epsilon$ approaches $\epsilon \rightarrow 1/2$ the value of $w_{2}$ tends toward $w_2 \rightarrow 1/3$. Increasing the lifetime prolongs the period of time during which $w_{2}$ has an elevated value. Note that the current age of the Universe corresponds to the solid green curve with $\log_{\mathrm{10}}(\tau / {\mathrm{Gyr}}) = 10.14$.[]{data-label="fig:w2"}](w2a.pdf)
\[sec:level3\]Many-Body Decay
=============================
In this section we discuss the relevant physics of many-body decay in which the daughter products consist of many relativistic particles and a single massive particle (see Fig. \[fig:MANYpict\]). By loosening the constraint on the number of relativistic particles we lose the ability to determine the velocity of the heavy daughter. For this reason we assume the particle to be stationary. Nonzero values of velocity could have been assumed (as e.g., in [@Bell:2010aa]) but this is necessarily arbitrary.
Parts of the derivation for the two-particle decay are identical to the many-body case. For simplicity the parent will still be referred to with subscript $0$, the massless daughter particles, even though there are many of them, with subscript $1$ and the heavy daughter with subscript $2$. Equation \[eq:parent2body\] for the parent particle density is the same as is the formula for the relativistic particles in Eq. (\[eq:rho1a\]). Note however that this density refers to many relativistic particles created in each decay.
Note also that in the many-body decay $\epsilon$ has a slightly different form. In the two-body decay $\epsilon$ was defined as the fraction of the energy of the parent particle that was transferred to the massless particle and a formula was derived in terms of $m_{0}$ and $m_{2}$. We maintain this definition but in the case where there are many massless, relativistic particles and a single, stationary, massive particle formula is more simply derived as being $$\epsilon = \frac{m_{0} - m_{2}}{m_{0}}$$ where we see that $\epsilon$ is allowed to take any value between $0$ and $1$ (in contrast to the two-body case where the limit was $1/2$).
![A pictorial of a many-body decay from a massive, stationary parent particle to many massless, relativistic particles and a massive, stationary daughter particle.[]{data-label="fig:MANYpict"}](Manybody.pdf){width="25.00000%"}
The only particle density that has a different form in the many-body case is the heavy daughter which, without having to consider its kinetic energy in the derivation, is much more straightforward. The evolution of the density of the massive daughter is governed by $$\frac{d\rho_{2}}{dt} + 3 \frac{\dot{a}}{a} \rho_{2} = (1 - \epsilon)\Gamma\rho_{0}\\$$ whose solution is $$\rho_{2} =\frac{{\cal{A}} ( 1 - \epsilon)}{a^3} \left[ e^{-\Gamma t_*} -e^{-\Gamma t} \right]
\label{eq:manyrho2}$$ where it has been assumed that $\rho_2 = 0$ at $t = t_*$. The evolutions of the parent $\rho_0$ and the relativistic by-products $\rho_1$ are governed by the same expressions as in the two-body, namely, Eqs. (\[eq:parent2body\]) & (\[eq:rho1a\]) respectively.
\[sec:level4\]Decaying dark matter and Supernovae Type Ia
=========================================================
In the previous two sections we derived the dynamical evolution of two decaying dark matter models. Here we will explore the constraints on these models that come from the cosmological information encoded in the observed brightness of supernovae type Ia (SNIa).
We use SNIa from the Union2.1 catalog of 580 supernovae [@Suzuki:2012aa]. The recently published Joint Light-curve Analysis (JLA) catalog (from the SNLS-SDSS collaborative effort) [@Betoule:2014aa] features a greater number of supernovae as well as an improved photometric calibration of two of the largest supernova surveys. However, as of the time of writing the JLA collaboration has not yet published a data release that will allow the straightforward propagation of statistical and systematic uncertainties and for this reason the Union2.1 data is used in this paper (as was the case with the Planck 2013 data release [@Planck-Collaboration:2013ab]). The Union2.1 data set includes 580 supernovae up to a redshift of about $z \approx 1.4$ and excludes those with redshift below $z=0.015$ in order to minimize any error due to peculiar velocities.
The physically important quantity in SNIa is the luminosity distance as a function of redshift of each supernova (essentially a Hubble diagram). The luminosity distance is related to the redshift, in a flat Universe where the scale factor relates to the redshift via $a= 1 / (1 + z)$, by $$d_{L} (z) = \frac{c(1+z)}{H_{0}} \int_{0}^{z} {\cal{F}}^{-1/2} (z^\prime) \, \,dz^{\prime}
\label{eq:lumdist}$$ where $$\begin{aligned}
{\cal{F}}(z^\prime) &=& \Omega_0(z^\prime) + \Omega_1(z^\prime) + \Omega_2(z^\prime) \nonumber \\
&+& \Omega_\nu(z^\prime) + \Omega_\gamma(z^\prime) + \Omega_{b}(z^{\prime}) + \Omega_\Lambda ,
\label{eq:Ffortwo}\end{aligned}$$ For each species $i=\{\gamma, \nu, 0, 1, 2, b, \Lambda\}$, $\Omega_i = \rho_i / \rho_{\mathrm{crit}}$, and $\rho_{\mathrm{crit}}$ is the critical density of the Universe. The values of $\rho_0$ and $\rho_1$ are given by Eqs. (\[eq:parent2body\]) & (\[eq:rho1a\]) respectively. The evolution of $\rho_2$ is given by Eq. (\[eq:rho2evolution\]) for the two-body scenario and by Eq. (\[eq:manyrho2\]) respectively for the many-body decay.
Note that the redshift dependence of each dimensionless cosmological parameter $\Omega_i$ above is due to the fact that the abundance of each species changes with time due to decay (for $i=0, 1, 2$) as well as the expansion of the Universe. This is especially important for $i=2$ where the redshift evolution contains both the effects of production (by decay) and the dynamical evolution of the population, some of which may or may not be relativistic.
The distance modulus is simply a manipulation of the luminosity distance (where $d_{L}$ is measured in parsecs): $$\mu(z) = 5\log_{10}d_{L}(z)-5,
\label{modulus}$$ which is the parameter that is constrained by observations.
We obtain goodness of fit constraints to decaying dark matter models parametrized by $\epsilon$ and $\tau$ in the following way. We compute the luminosity distance to the $j$th SNIa with redshift $z_j$, and the subsequent distance modulus $\mu(z_{j, {\mathrm{DDM}}})$. We then compare that with the observed absolute magnitude and redshift of each SNIa. The sum of the squares of their variance weighted difference is the $\chi^2$ distribution of that particular dark matter decaying scenario. $$\chi^{2} = \sum_{j=1}^{580} \left\{ \frac{1}{\sigma_{j}^{2}} \left[ \mu(z_{j, {\mathrm{DDM}}}) - \mu(z_j) \right]^2 \right\}$$ where $\sigma_{j}$ is the uncertainty of the distance modulus measured for each supernova [@Suzuki:2012aa] . This is then compared to a $\chi^2$ distribution with 578 degrees of freedom (580 SNIa minus 2 degrees of freedom, corresponding to $\epsilon$ and $\tau$) and assign a goodness of fit confidence.
In order to properly compute the luminosity distance in a decaying dark matter scenario via Eq. (\[eq:lumdist\]), we need knowledge of the rest of the cosmological energy budget (in addition to matter and radiation derived from the parent dark matter decay, either in the two-body scenario or the many-body scenario).
The photon density $\rho_\gamma$ is derived from the present photon temperature $T_{\gamma,0} = 2.7255$K [@2009ApJ...707..916F] using $\rho_{\gamma}(a) = 4\sigma (T_{\gamma,0}/a)^{4}/c$, where $\sigma$ is the Stefan-Boltzmann constant. This temperature is inflated by electron-positron annihilation, a heating that did not affect the neutrino temperature which leads to the well-known result for massless neutrinos, $T_{\nu}(a) = \left(4 / 11\right)^{1/3} T_{\gamma}(a)$ and an energy density given by $\rho_{\nu}(a) = N_{\mathrm{eff}} \left( 7 / 8 \right) \left(4/ 11 \right)^{4/3} \rho_{\gamma}(a)$, where the effective neutrino number density, $N_{\mathrm{eff}}$, takes the standard value $N_{\mathrm{eff}}=3.046$ [@Mangano:2002aa].
However, this standard treatment of neutrinos is slightly inaccurate because they are both relativistic and massive and therefore we follow Sec. 3.3. in @2011ApJS..192...18K that provides the following expression for the energy density of massive neutrinos: $$\rho_{\nu}(a) = \frac{7}{8} \left(\frac{4}{11}\right)^{4/3}N_{\mathrm{eff}}\rho_{\gamma}(a) f(y)$$ where $$f(y) \equiv \frac{120}{7\pi^{4}} \int_{0}^{\infty} dx \frac{x^{2} \sqrt{x^{2} + y^{2}}}{e^{x}+1}$$ This form of the neutrino density takes into account the transition from relativistic to nonrelativistic expansion. A fitting formula gives the approximation $$f(y) \approx [1 + (Ay)^{p}]^{1/p},
\label{eq:neuApprox}$$ where $A = 180 \, \zeta(3) / 7\pi^{4} \approx 0.3173$, $p=1.83$ and $\zeta$ is the Riemann zeta function, which is what we use for the remainder of this paper.
As the decaying dark matter formalism that we derived in Sec. \[sec:level2\] is normalized to the value of dark matter at the epoch of the CMB we choose to use cosmological parameters from CMB experiments. We use the cosmological parameters derived from the combination of Planck [@Planck-Collaboration:2013ab] and low-l WMAP [@Hinshaw:2013aa] likelihoods with the high-l Atakama Cosmology Telescope [@Das:2014aa] and South Pole Telescope [@Reichardt:2012aa] likelihoods (which were combined in [@Planck-Collaboration:2013ab] and called in short [*Planck + WP + highL*]{}). These are: $\Omega_{\mathrm{CDM}} h^2 = 0.12025$, $\Omega_{\mathrm b} h^2 = 0.022069$, $h = 0.6715$, $z_* = 1090.43$ and $w = -1 $. This cosmological model is consistent with the Union2.1 supernovae sample that we use here (see Fig. 19 in [@Planck-Collaboration:2013ab], and we use it as a benchmark over which we can test the SNIa constraints on the two-body and many-body decaying dark matter scenarios [^1]).
\[sec:level5\]Results and Discussion
====================================
![ Goodness of fit contour plots for the two-body decaying dark matter scenario in the $\epsilon -\tau$ parameter space. Color density corresponds to the value of the goodness of fit. The two contours depict the $3 \sigma$ and $5 \sigma$ values. The constraining power of supernovae is evident for lifetimes greater than $10^{10}$ years and values of the daughter relativistic fraction ($\epsilon$) greater than roughly $1\%$.[]{data-label="fig:tworesult"}](TWO_goodness_of_fit_100x100_planckwphighl.pdf)
![ Goodness of fit contour plots for the many-body decaying dark matter scenario in the $\epsilon -\tau$ parameter space. Color density corresponds to the value of the goodness of fit. The two contours depict the $3 \sigma$ and $5 \sigma$ values. The constraining power of supernovae is evident for lifetimes greater than $10^{10}$ years and values of the daughter relativistic fraction ($\epsilon$) greater than roughly $1\%$.[]{data-label="fig:manyresult"}](MANY_goodness_of_fit_100x100_planckwphighl.pdf "fig:")\
Figures \[fig:tworesult\] and \[fig:manyresult\] show the derived SNIa constraints on the two-body and many-body decaying dark matter scenarios, respectively. The color density corresponds to the value of the goodness of fit confidence, while the two curves depict the $3\sigma$ and $5\sigma$ contours in the $\epsilon - \tau$ parameter space. It is evident that the constraining power of SNIa is concentrated in large values of $\epsilon$ ($\epsilon > 10^{-2}$, and lifetimes of less than $\tau \sim 10^{10}$ years; the latter is not surprising as SNIa are fairly recent in cosmological history, and therefore only dark matter that decays appreciably at the sample epoch of the SNIa we are using here can be constrained).
Both plots look very similar and indeed share the same features (note that $\epsilon$ only extends to $1/2$ in the two-body case whereas it can rise as far as $1$ in the many-body scenario). At short lifetimes, the contours are approximately vertical. The supernovae to which we are comparing only extend back to a redshift of $z \lesssim1.5$ and for very short lifetimes essentially all of the dark matter has decayed by this epoch rendering differences between small $\tau$ and even smaller $\tau$ irrelevant. Moving vertically up from small lifetimes to large lifetimes we see that the confidence level decreases. The longer the lifetime the smaller the difference between decaying dark matter and $\Lambda$CDM and so we essentially return to the base $\Lambda$CDM model found by Planck. Conversely moving horizontally from small $\epsilon$ to high $\epsilon$ the confidence level increases. As more and more radiation is added to the model the further away it is from the true Universe as traced by SNIa. At intermediate and high lifetimes we observe diagonal contours across the plots indicating that the effect of an increase in the lifetime (reducing the amount of additional radiation) can be offset by an increase in $\epsilon$. Finally observe that in the two-body case, but not the many-body, there is an upward inflection in the contour lines at high $\epsilon$. The reason for this can easily be seen by consulting Fig. \[fig:w2\] where $w_{2}$ varies greatly with changes in $\epsilon$ between $0.3$ and $0.5$.
It is important to also mention, however, that the choice of a cosmological model that sets the initial conditions can have a strong effect on the derived constraint, or turning the problem around, the results obtained here are rather sensitive on the choice of the cosmological model. For example, cosmological models that allow $w \neq -1$ have much more constraining power on decaying dark matter than models within the standard paradigm of $w=-1$. In addition, if we use the WMAP9-only derived cosmological model [@Hinshaw:2013aa], the constraining power of SNIa is less, scaling roughly by changing the $3\sigma$ contour into a $1\sigma$ contour. On the other hand, using the Planck-only cosmological parameters the constraining power of SNIa are stronger (perhaps a manifestation of the apparent tension between Planck and SNIa [@Planck-Collaboration:2013ab]). The choice of the aforementioned cosmological model of using Planck data together with low-$\ell$ WMAP and high-$\ell$ ACT/SPT data ([*Planck + WP + highL*]{}) is however consistent with the Union2.1 supernovae we consider here and we feel this is the most appropriate and self-consistent choice of cosmological parameters in the normalization of the decaying dark matter models we explore here.
The derived constraints from SNIa on the two-body and many-body decaying dark matter scenarios are complementary to other approaches to the problem which we show in Figs. (\[fig:twocompare\]) and (\[fig:manycompare\]).
For example, in a recent paper, @Hasenkamp:2013aa derived a two-body decaying scenario with one daughter particle assumed to be of negligible mass and relativistic, and a second massive, possibly relativistic, daughter. They use a different, indeed complementary approach, to ruling out parameter space. They assume that the relativistic energy produced by decaying dark matter manifests itself as additional effective neutrinos, justified by findings such as in @Dunkley:2011aa. In addition, the density of the decaying parent and daughters is allowed to vary between models they explore, and is constrained by present limits on nonrelativistic and relativistic dark matter measurements. The density of the parent particle is allowed to vary between models in order to obtain the same amount of additional relativistic energy regardless of the other specified parameters, and they derive limits based on the current observed cold and hot dark matter densities. However, @Hasenkamp:2013aa make a number of simplifying assumptions. More specifically, they assume a sudden transition from radiation to matter domination, that the massive daughter particle is relativistic unless it’s momentum is equal to or less than its mass, and that all the particles decay at a time equal to the lifetime $\tau$. This last assumption is obviously quite a simplification from the exponential decay and so the authors derive a correction factor to alter the density with two values, one when the lifetime is within radiation domination and one during matter domination. This approximation progressively improves for observational times significantly greater than the lifetime.
Within this framework @Hasenkamp:2013aa looked at many different scenarios. For example they consider the contour where the number of additional neutrino degrees of freedom is 1 \[labeled as Hasenkamp & Kersten (2013a) in Fig. \[fig:twocompare\]\]. They also considered a bound based on demanding that the amount of decaying dark matter could not exceed the total amount of dark matter that is observed \[in their paper this was referred to as the non-domination constraint and in Figure \[fig:twocompare\] is labeled as Hasenkamp & Kersten (2013b)\]. Their results are complementary as they rule out parameter space at small values of $\epsilon$ and $\tau$ (the lower left area of the plot) while the results presented here, along with most previous findings, have ruled out space in the large values of $\epsilon$, small $\tau$ region (the lower right region). We note however that we do not make the assumption of instantaneous decay and a relativistic cutoff of the heavy daughter as in @Hasenkamp:2013aa. Instead we allow both decay and relativistic behavior to be monotonically continuous functions, thus providing additional insight to the effects of decaying dark matter.
Another model of two-body decaying dark matter is considered in @Yuksel:2008aa in the specific case where the massless particle is a photon. However this model does not consider the (equal and opposite) momentum of the heavy daughter, merely constraining the decay by comparing the resulting photon density against the isotropic diffuse photon background and a Milky Way $\gamma$-ray line search. This model is further complicated as it only constrains the product $m_{\chi}\tau$, where $m_{\chi}$ is the mass of the parent particle, against the energy of the photon.
Two-body decays in decaying dark matter were shown to be a possible solution to problems in structure formation in papers by @Kaplinghat:2005aa and @Strigari:2007aa that looked at very early decays with lifetimes of less than one year and later decays ($z<1000$), respectively. They showed that dynamical dark matter could have positive implications for constant density cores in halos reducing the quantity of small scale substructure.
A pair of papers, @Peter:2010aa and @Wang:2013aa, analyzed structure formation data while looking at two-body decay scenarios where there was only a slight mass splitting between the parent and heavy daughter (a decay with small $\epsilon$) giving the massive particle a nonrelativistic velocity. The authors parametrized the decay in terms of the recoil “kick” velocity $v_{k}$ of the heavy daughter particle, which is given as $v_{k}/c \simeq (m_{0} - m_{2})/m_{0}$ where $m_{0}$ is the mass of the parent particle and $m_{2}$ is the mass of the heavy daughter. For the small values considered in these papers $\epsilon \simeq v_{k}/c$. Both of these constraints are shown in Fig. \[fig:twocompare\].
![Summary of the two-body decay constraints presented here as compared to other studies. In all cases, parameter space is ruled out (at various levels of confidence) below the contour line. The results obtained in this work appear to rule out parameter space more aggressively than previous studies, but note that a direct comparison is not straightforward (for caveats see text). Note that both Wang et al. 2012 [@Wang:2012aa] and Wang et al. 2013 [@Wang:2013aa] considered only small mass splittings. This results in an abrupt cutoff in their corresponding contour lines at lower values of $\epsilon$. Similarly @Hasenkamp:2013aa considered only shorter lifetimes causing their contours to end abruptly in the parameter space.[]{data-label="fig:twocompare"}](TWO_compare_planckwphighl.pdf)
![Summary of the many-body decay constraints presented here as compared to other studies. The results obtained in this work appear to rule out parameter space more aggressively than previous studies, but note that a direct comparison is not straightforward (for caveats see text).[]{data-label="fig:manycompare"}](MANY_comparison_planckwphighl.pdf)
In @Peter:2010aa N-body simulations of dark matter halos are compared to observations of dwarf-galaxies, groups, and clusters to rule out regions of parameter space (note that this paper quotes a value of $\tau$ for different decay models which is the half-life of the decay as opposed to the lifetime used through out this paper). As this study was based on a suite of N-body simulations it is difficult to assign a numerical value of confidence. Instead, decaying dark matter parameters were allowed, if a few of the realizations of the satellite populations produced at least the minimum number of satellites expected in a Milky Way-like halo.
Lyman-$\alpha$ forest data was used to constrain a dynamical dark matter model in @Wang:2013aa. Decaying dark matter affects structure growth and thus the authors used SDSS 1D Ly$\alpha$ data to measure large-scale structure growth [@McDonald:2006aa].They looked at kick velocities up to $2\times 10^{7}$m/s though without considering relativistic effects. A related paper by @Wang:2012aa projected how dynamical dark matter might be constrained by weak lensing results from future experiments such as Euclid [@Refregier:2010aa] and LSST [@LSST-Science-Collaboration:2009aa]. Recently @Wang:2014aa produced the most sophisticated N-body simulations of galaxy formation assuming dark matter decay. They showed that problems associated with large-scale structure formation such as the missing satellites problems are largely solved for particular values of the lifetime of the decaying dark matter particle and the recoil kick velocity of the daughter.
@Hasenkamp:2014aa recently explored a two-body decay with lifetimes less than 1500 years, in particular decays occurring before and during big-bang nucleosynthesis. What they found is that there was no difference in the value of $N_{\mathrm{eff}}$ or $m^{\mathrm{eff}}_{\mathrm{hdm}}$ for the massive relativistic daughter compared to thermally produced $\nu_s$HDM but that the temperature at which such particles became nonrelativistic differed by a factor of $\sim 2$. Such a difference could have an observable impact on the CMB. This presents an interesting avenue of future work as there is a possibility of connecting the effects on the CMB to late Universe probes (longer lifetimes), such as the work presented here.
When considering a many-body decay, @Gong:2008aa modified CosmoMC to include dynamical dark matter, and ruled out parameter space by making comparison to the distance modulus of 182 supernovae and the position of the first peak in the WMAP3 angular power spectrum (comparison shown in Fig. \[fig:manycompare\]). Since this paper was written there has been much improvement in the quantity and quality of the data, in particular with the 580 supernovae in the Union2.1 catalog [@Suzuki:2012aa] and in the Planck 2013 results [@Planck-Collaboration:2013aa].
Further constraints are placed on many-body decay by @Zhang:2007aa by assuming that some portion $f_{\chi}$ of the decay products are released as electromagnetically interacting particles and that some portion of that, $f$, is then deposited in baryonic gas thus affecting both reionization and recombination. Unfortunately this model can only constrain the product $ff_{\chi}$ against $\Gamma$ and it is difficult to map in the $\epsilon - \tau$ parameter space.
A later paper by @DeLope-Amigo:2009aa aimed to update the results of @Gong:2008aa and @Zhang:2007aa. However they only considered the specific case where all the energy from decay was transferred to relativistic energy, the $\epsilon = 1$ scenario. When this was true they found that in the case where the fraction $f$ of energy then deposited in baryonic gas was negligible then the Integrated Sachs-Wolf effect [@1967ApJ...147...73S] constrained the lifetime to be over $100$ Gyr at $2\sigma$ confidence. For non-negligible deposition they found $(f\Gamma)^{-1} \gtrsim 5.3 \times 10^{8}$Gyr.
Specific decay models, where the decay is assumed to produce particular standard model particles, are more highly constrained than the general models above. @Ibarra:2014aa sets competitive limits on the lifetime of the parent particle by making comparison to the recent AMS-02 data release [@Aguilar:2013aa]. They assumed decay products such as $b\bar{b}$, $e^{+}e^{-}$, $\mu^{+}\mu^{-}$, $\tau^{+}\tau^{-}$ and $W^{+}W^{-}$ and set lower limits on the lifetime in the region of $10^{7} \sim 10^{11}$ Gyr. @Essig:2013aa looked at the same decays and found similar constraints when making comparison to recent gamma-ray and x-ray data from the Fermi Gamma-ray Space Telescope [@The-Fermi-LAT-Collaboration:2012aa], INTEGRAL [@Bouchet:2008aa], EGRET [@Strong:2004aa], and HEAO-1 [@Gruber:1999aa]. Similar constraints were also found by @Cirelli:2012aa using Fermi, H.E.S.S. [@2012ApJ...750..123A], and PAMELA [@2009Natur.458..607A; @2009PhRvL.102e1101A; @2010PhRvL.105l1101A; @2011PhRvL.106t1101A].
It is worth noting that the recently measured high-energy neutrino detections at IceCube [@IceCube-Collaboration:2013aa] have been hypothesized as originating from decaying dark matter (see for example @Ema:2013aa) though more data will be needed before limits on such decaying models can be set.
An interesting extension to specific decays was investigated in @Bell:2010aa where they considered a three-body decay in which the daughters consisted of two electrons, two photons or two neutrinos plus a heavy daughter that possessed a kick velocity. As there were three particles there was no derivable value of this velocity so they assumed it moved in the range of $[5-90]$ km/s in order to derive lower lifetime limits very approximately in the region of $10^{2}$ - $10^{8}$ Gyr.
Of course there could also be decay into nonstandard model particles such as gravitinos, gauginos, or sneutrinos (see for example @Ibarra:2013aa).
As shown in Figs. \[fig:twocompare\] and \[fig:manycompare\], the results presented here rule out regions of contour space at clearly defined levels of confidence and do so at much higher levels than previously achieved. We underline that the comparison is somewhat opaque. In the two-body case the model developed here is more highly developed with its sophisticated treatment of the possibly relativistic, heavy daughter particle. On the other hand our results rely only on comparisons to supernovae while the other plotted results were compared against other, and in many cases, several other data sets. It would therefore be of interest to implement the derived two-body and many-body decay scenarios to a multitude of cosmological probes [@BKinprep], as well as generic particle physics models (e.g., dynamical dark matter [@2012PhRvD..85h3523D; @2012PhRvD..85h3524D]).
In summary, we developed a sophisticated model of two- and many-body dark matter decays. In the case of the former it takes into account the gradual slowing, from relativistic to nonrelativistic, of the heavy particle without having to choose an arbitrary cutoff for what counts as a relativistic velocity. The level of confidence at which areas of the decaying dark matter parameter space is strongly constrained by SNIa shows that cosmological probes may in fact strongly constrain decaying dark matter scenarios.
We acknowledge useful conversations with Alex Geringer-Sameth, Jasper Hasenkamp, Deivid Ribeiro and Andrew Zentner. We thank the referees for the constructive feedback that helped improve the content of the paper. SMK is supported by DOE DE-SC0010010, NSF PHYS-1417505 and NASA NNX13AO94G. G.B. is partially supported by NSF PHYS-1417505. S.M.K. thanks the Aspen Center for Physics for hospitality where part of this work was completed.
[^1]: We can readily provide results upon request for many of the cosmological models discussed in [@PlanckSuppl].
|
---
abstract: 'We consider the problem of identifying the most influential nodes for a spreading process on a network when prior knowledge about structure and dynamics of the system is incomplete or erroneous. Specifically, we perform a numerical analysis where the set of top spreaders is determined on the basis of prior information that is artificially altered by a certain level of noise. We then measure the optimality of the chosen set by measuring its spreading impact in the true system. Whereas we find that the identification of top spreaders is optimal when prior knowledge is complete and free of mistakes, we also find that the quality of the top spreaders identified using noisy information doesn’t necessarily decrease as the noise level increases. For instance, we show that it is generally possible to compensate for erroneous information about dynamical parameters by adding synthetic errors in the structure of the network. Further, we show that, in some dynamical regimes, even completely losing prior knowledge on network structure may be better than relying on certain but incomplete information.'
author:
- Şirag Erkol
- Ali Faqeeh
- Filippo Radicchi
title: Influence maximization in noisy networks
---
Introduction
============
In a social network where an opinion diffuses according to an irreversible spreading process, a fundamentally important role for the ultimate success of the opinion is played by the nodes that act as initiators or seeds of the spreading process. For example, the popularity of memes in social media is often determined by just a few early adopters [@ratkiewicz2011truthy]. Given the high sensitivity of the outcome of spreading processes to initial configurations, a very interesting problem regards the identification of the initial configuration, among the many possible, that maximizes the extent of diffusion. The problem is traditionally named as influence maximization. It has been first considered by Domingos and Richardson [@Domingos01], and slightly later generalized by Kempe [*et al.*]{} [@Kempe03]. Roughly speaking, influence maximization consists in an optimization problem based on a few assumptions and subjected to one constraint. The function that one wants to maximize is the size of the outbreak, i.e., the number of nodes that will end up acquiring the opinion that is diffusing in the system. The assumptions in the formulation of the problem regard the structure of the network and the type of spreading that is taking place on the network. This information is generally assumed as a prior knowledge, and it is actively used for finding solutions to the optimization problem. The only constraint used in the optimization problem is the number of seeds. Only initial configurations consisting of a given number of active nodes are considered as potential solutions to the optimization problem.
As most optimization problems, influence maximization is NP-complete [@Kempe03]. Exact solutions can be found only in very small networks. Suboptimal solutions can be achieved with approximated or greedy optimization techniques [@Kempe03; @Chen09]. These approaches are generally effective, but they are designed for the analysis of small to medium networks. The identification of influential spreaders in large networks is allowed only through the use of heuristic techniques where dynamics is [*de facto*]{} neglected, and the solution to the optimization problem is approximated relying on network centrality metrics [@Kitsak10; @Bauer12; @klemm2012measure; @Chen12; @Chen13; @PhysRevE.90.032812; @Morone15]. This approach finds its rationale in interpreting the high sensitivity of the outcome of a spreading process to the initial conditions as a consequence of the heterogeneity of the underlying network. However, geometry alone is not enough to provide a sufficiently accurate description of the state of a dynamical system running on a network [@radicchi2017maximum]. The identity of the most influential nodes in a network generally changes from type to type of spreading process, and, even for the same type of process, it may depend on its dynamical regime [@radicchi2017fundamental].
Most of the studies we mentioned above rely on one strong assumption: prior knowledge of system structure and dynamics is complete and free of errors. When dealing with a real application of influence maximization, we should however recognize that this assumption is at least optimistic. The presence/absence of a connection in a social network is generally established from the result of some experimental observation, and it is therefore potentially affected by experimental errors [@Newman17]. Similarly, we may be aware of the type of process that drives spreading, but we may be unsure about the exact value of the rates at which spreading occurs. There are techniques that allow to accurately estimate spreading rates from empirical observations of spreading events [@saito2008prediction]. However, these techniques rely on the assumption that structural information is complete and free of mistakes. Further, in influence maximization, one aims at controlling the fate of a future or ongoing spreading process, so posterior estimates of the rates are not very helpful.
Several previous studies have considered the reliability of network centrality metrics when computed from noisy or incomplete structural information [@dall2005statistical; @Leskovec2006]. In the context of influence maximization, we are aware of previous tests of robustness of some centrality metrics in noisy structural data [@Kitsak10]. However, to the best of our knowledge, there are no previous studies that attempted to understand how incomplete or erroneous information about both structure and dynamics affects our ability to solve the problem of influence maximization. Please note that we may naively expect that noise doesn’t dramatically modify the overall trend of a geometric centrality metric, as it was shown in Ref. [@Kitsak10]. However, the distortions that noise can create in the solutions of an optimization problem such as influence maximization are far less predictable. The current paper aims at filling this gap of knowledge.
Methods
=======
Network structure
-----------------
We assume that the network where the spreading process takes place and where we aim at solving the problem of influence maximization is given by $N$ total nodes and $M$ edges. The network is unweighted and undirected. Structural information about the network is fully specified by the adjacency matrix $A$, whose generic element $A_{ij} = A_{ji} =1$ if nodes $i$ and $j$ are connected, and $A_{ij} = A_{ji} =0$, otherwise. Note that, in the spreading process, only the state associated to the nodes of the network can change. Edges do not have states that evolve in time, but serve only as static media for spreading.
Spreading dynamics
------------------
In this paper, we focus our attention on a spreading model that is very popular in studies about the identification of influential spreaders in networks: the Independent Cascade Model (ICM) [@Kempe03]. The ICM is very similar to the traditional Susceptible-Infected-Recovered (SIR) model [@PastorSatorras15]. During ICM dynamics, nodes in the network can be found in three different states: S, I, or R. Generally, in an initial configuration of the dynamics, all nodes are set in state S, except for a subset of seeds $Q$ that are set in state I. At each discrete stage of the dynamics, two rules are applied in sequence: (i) every node in state I infects, with probability $p$, each of its neighbors in state S; (ii) all nodes in state I, that attempted to infect their neighbors at step (i), recover and change their state from I to R. Rules (i) and (ii) are iterated until no infected nodes are longer present in the network. The size of the outbreak is given by the total number of nodes that are in state R at the end of the dynamics. This number is a stochastic variable that may change its value from instance to instance of the model. Observed values depend on the network structure encoded by the adjacency matrix $A$, the value of the spreading probability $p$, and the set of seeds $Q$ that initiated the spreading process. In our numerical study, we measure the spreading intensity for given $A$, $p$, and $Q$ in terms of the average value of outbreak size, namely $O$, over $20$ independent simulations of the spreading process.
Influence maximization
----------------------
Solving the problem of influence maximization for a spreading process on a network means finding the set of seeds that maximizes the average value of the outbreak size $O$. The maximization is performed for a fixed size $|Q|$ of the set of seeds. The optimization relies on prior knowledge about the structure of the network and the dynamical rules of the spreading process. Information on the structure is provided by the adjacency matrix $A$. Information about the dynamics consists in knowing that spreading is regulated by the ICM and the value of the probability of spreading is $p$. Even with full knowledge of system structure and dynamics, identifying the set of optimal seeds is a NP-complete problem, and thus exact solutions are achievable only in extremely small networks [@Kempe03]. Greedy optimization, however, allows to provide suboptimal solutions that are granted to be within $63\%$ of the optimum [@Kempe03]. This is the consequence of the fact that the size of the outbreak $O$ is a submodular function of the set of seeds [@Kempe03; @Nemhauser1978]. According to greedy optimization, the set of influential spreaders $Q$ is constructed sequentially by adding one node at a time. At every stage, the best node to be added to the set of seeds is the one that leads to the maximal value of $O$, in a spreading process initiated by all nodes that are already part of the seed set plus the node under consideration. In the original version of the algorithm by Kempe [*et al.*]{} [@Kempe03], greedy steps rely on direct simulations of the dynamical process. This algorithm is very general, and can be used for other types of dynamical models, not just the ICM. For the specific case of the ICM, the greedy algorithm can be further speeded up by taking advantage of the mapping between static properties of the SIR and bond percolation [@grassberger1983critical]. The mapping allows to use configurations of the percolation model to infer information about final configurations arising from ICM dynamics. Each node is associated with a score whose value is proportional to the average size of independent clusters (i.e., clusters that do not contain previously identified seed nodes) which the node belongs to. Now the set $Q$ is constructed by adding the nodes to it, one by one, starting from the nodes with the highest score. This method was designed, implemented and validated by Chen [*et al.*]{} [@Chen09]. The results of the current paper are based on our re-implementation of the algorithm by Chen [*et al.*]{} We rely on $R = 1,000$ independent simulations of the bond percolation model to compute the scores associated with the nodes. Please note that as long as $R$ is a finite number, scores associated with nodes are subjected to finite-size fluctuations. As a consequence, the solution to the optimization problem provided by the algorithm by Chen [*et al.*]{} may depend on the specific set of bond percolation simulations considered. As the results of the SM show, we verified that, while the identity of the nodes in the solutions provided by the algorithm is not always the same, the average sizes $O$ of the outbreak associated with those solutions are very similar. This finding suggests that the exact detection of the set of top spreaders is statistically irrelevant for the outcome of the spreading process, in the sense that there are many nearly-optimal solutions to the problem of influence maximization.
Modeling errors in system dynamics and structure
------------------------------------------------
The process of selection of top spreaders relies on prior knowledge about the structure of the underlying network and the details of the dynamical process. This means that the set $Q$ depends on the information at our disposal regarding the structure of the network, i.e., the adjacency matrix $A$. $Q$ also depends on our prior knowledge about the dynamical process that is taking place on the network, that is the ICM model with spreading probability equal to $p$. It is common practice to assume prior information complete and exact. This is equivalent to assuming that the inputs $A$ and $p$ of the algorithm for the identification of top spreaders are equal to their true values, namely $A_{true}$ and $p_{true}$, respectively. However, in practical situations, this may not be the case. Prior knowledge may be affected by errors, so that the actual information used to solve the problem of influence maximization is given by $A_{err}$ and $p_{err}$, respectively. There are potentially many different ways to model errors that deteriorate dynamical and structural information of the system. Here, we opt for simple, yet realistic, models.
Errors that affect our prior knowledge of the spreading dynamics are simply obtained by setting $p_{err} \neq p_{true}$. Essentially, we assume to know that spreading occurs following the rules of the ICM, but we pretend that we don’t know the exact value of the spreading probability. Instead of using raw values for the spreading probabilities, we rescale them as $\phi_{true} = p_{true} / p_c$ and $\phi_{err} = p_{err} / p_c$, where $p_c$ is the critical value of the spreading probability for the ICM model on the true network $A_{true}$. This transformation is used only to simplify the presentation of the results. It allows us in fact to use the same reference value for all networks to distinguish between different dynamical regimes of spreading. The critical value of the spreading probability is computed directly from numerical simulations where ICM is initiated from a randomly selected seed node [@Radicchi16]. Depending on whether $\phi_{true}$ is larger, equal or smaller than one, we say that the network is respectively in the supercritical, critical or subcritical regime of spreading. Similarly, the value of $\phi_{err}$ tells us what type of dynamical regime is hypothesized for the selection of influential spreaders. From previous studies, we know that the identity of the nodes that populate the sets of top spreaders is highly dependent on the regime of spreading [@radicchi2017fundamental]. We expect therefore that the error in the estimation of the spreading probability may strongly affect our ability to properly predict top spreaders.
Errors in the structure of the network are generated artificially according to a model similar to the one considered in Ref. [@Newman17]. The total number of nodes is unaffected by noise, so that we can indicate it as $N$. Errors happen at the level of pairs of nodes. This means that the total number of edges $M_{err}$ in the altered structure generally differs from $M_{true}$, i.e., the number of edges in the true network. We consider two potential sources of errors. The first source is responsible for making true edges disappear from our prior knowledge of the network. Specifically, given the true adjacency matrix $A_{true}$, every pair of connected nodes is disconnected with probability $0 \leq
\epsilon_{del} \leq 1$ in $A_{err}$. The number of true edges that are deleted equals zero for $\epsilon_{del} = 0$, and equals $M_{true}$ for $\epsilon_{del}=1$. The second source of error generates false edges. Every pair of nodes that is not connected according to $A_{true}$ appears as connected in $A_{err}$ with probability $\epsilon_{add} \, M_{true} / [N(N-1)/2-M_{true}]$, where $0 \leq
\epsilon_{add} \leq 1$. For $\epsilon_{add} = 0$, no false edges are added to the true network. For $\epsilon_{add} = 1$ instead, the expected total number of false edges equals $M_{true}$. The definition of the noise parameters $\epsilon_{del}$ and $\epsilon_{add}$ are such that both parameters are confined in the range $[0,1]$, and their maximum values correspond to an expected alteration (i.e., deletion or addition) of $100\%$ of the true number of edges.
Please note that removing true edges is generally not the inverse operation of adding false edges. For instance, even the addition of a very small number of edges among pairs of non-connected nodes is able to decrease substantially the average path length of graphs that do not originally satisfy the small-world property [@watts1998collective]. For example, in networks with strong spatial embedding, as some of those we analyze in this paper, random edges likely behave as shortcuts between spatially far regions of the system. On the other hand, the removal of a small fraction of true edges doesn’t change dramatically the average path length of the graph. Also, we do not expect the two sources of structural errors to be equally likely in real networks. Their proportion may depend strongly on the type of network considered, and on the way the network is actually constructed from empirical observations [@Newman17]. As the two sources of structural errors cannot be treated on the same footing, in our analysis we always consider them separately, i.e., we always work with the condition $\epsilon_{del} \,
\epsilon_{add} =0$ satisfied.
We stress that our choice for the noise model affecting prior structural information is heavily inspired by Ref. [@Newman17]. We find the model simple enough, yet able to naturally describe sources of uncertainty in empirically constructed social networks. Alternative models of structural noise could be considered and studied using the same exact methods as those described here. For example, a model consisting in shuffling true edges with a certain probability would provide a way to introduce structural noise without altering the degree of the nodes in the network. In general, we believe that the choice of the noise model should depend on the specific question that one wants to address, or the specific system that one is considering. The set of questions that we are considering here can be fully addressed by the particular choice we made.
Measuring performance
---------------------
Given the inputs $A_{err}$ and $\phi_{err}$, we make use of the algorithm by Chen [*et al.*]{} to identify the set $Q_{err}$ of top spreaders. As the algorithm for the identification of top spreaders is based on a finite number of numerical simulations and the output of the algorithm is subjected to finite-size fluctuations, we apply the algorithm $V = 10$ times to find $V$ potentially different sets $Q_{err}$. For each of them, we use numerical simulations of the ICM model relying on $A_{true}$ and $p_{true}$ to evaluate performance. In particular, as the algorithm by Chen [*et al.*]{} naturally ranks nodes according to the order in which they are added to the set of top spreaders, we explicitly use this information to quantify the performance of the set $Q_{err}$ as follows. The size of the set $Q_{err}$ is indicated with $|Q_{err}|$. Nodes have been added to the set according to the sequence $q_1, q_2, \ldots,
q_{|Q_{err}|}$. Define $Q_{err}^{(r)} = \cup_{v=1}^{r} q_v$ as the set of nodes with rank up to $r$. Please note that, by definition, $Q_{err}^{(|Q_{err}|)} \equiv Q_{err}$. The overall performance of the set $Q_{err}$ is computed according to the equation $$P (Q_{err}) = \frac{1}{N \, |Q_{err}|} \, \sum_{r=1}^{|Q_{err}|} \,
O(Q^{(r)}_{err}) \; ,
\label{eq:performance}$$ where $O(Q^{(r)}_{err})$ is the average size of the outbreak in the ICM when the set of seeds is given by $Q^{(r)}_{err}$. For given $A_{err}$, the overall performance is finally obtained by taking the average over all $V$ realizations of the sets of top spreaders. To account for the stochastic nature of structural noise, we repeat the entire procedure on $G = 10$ instances of $A_{err}$ and quantity the spreading performance of top spreaders as the average value over these independent realizations. Please note that the sum on the r.h.s. of Eq. (\[eq:performance\]) allows us to estimate not only the overall performance of the set $Q_{err}$, but also the way the set is constructed. The pre-factor appearing in the r.h.s. of Eq. (\[eq:performance\]) is used only to confine values of our performance metric to the interval $[0,1]$. Our focus here is not on measuring the effectiveness of the algorithm used to determine the spreaders, rather on the importance that prior information has in the selection of the top spreaders. Other metrics could be used in place of the one defined in Eq. (\[eq:performance\]). For instance, metrics that consider the identity of the nodes in $Q_{err}$ with respect to those found in the true set of top spreaders. We believe, however, that this second type of metric may be misleading as the difference in terms of outbreak size between the optimal solution and a slightly-less optimal solution may be very small despite a high dissimilarity in terms of the nodes that define the two solutions. In the SM, we actually verified that the identity of the nodes in the set $Q_{err}$ may be sensitive to the choice of the noise parameters and the specific run of the identification algorithm; instead, the size of the outbreak $O$ is not much affected by fluctuations.
Results
=======
![Influence maximization in presence of structural and dynamical noise. The true network structure analyzed here is given by the email communication network of Ref. [@PhysRevE.68.065103]. (a) Relative size of the outbreak $O/N$ as a function of the number of seeds found by greedy optimization. The true spreading probability of the ICM is such that $\phi_{true} =
0.5$. Prior dynamical knowledge used by the greedy algorithm is not affected by noise, i.e., $\phi_{err} = \phi_{true} = 0.5$. The different curves correspond to different level of noise that affect prior structural information. We consider various combinations of the parameters $\epsilon_{del}$ and $\epsilon_{add}$. (b) Same as in panel a, but for $\phi_{true} =
1.0$ and $\phi_{err} =0.5$. (c) Same as in panel a, but for $\phi_{true} =
\phi_{err} =1.0$. (b) Same as in panel a, but for $\phi_{true} =
1.5$ and $\phi_{err} =1.0$. []{data-label="fig:1"}](fig1.pdf){width="45.00000%"}
In Figure \[fig:1\], we display results obtained for the email communication network originally considered in Ref. [@PhysRevE.68.065103]. In the majority of the cases, the performance of the set of top spreaders appears robust against structural noise. However, as Figure \[fig:1\] shows, the overall ability to properly select top spreaders may be seriously affected by the level of noise associated with prior information both at the structural and the dynamical levels. Major issues seem to arise when $\phi_{true} > 1$, but $\phi_{err} < 1$ (see Fig. \[fig:1\]d).
![Performance of the top spreaders in the presence of structural and dynamical noise. We consider the same network as in Fig. \[fig:1\]. (a) We compute Eq. (\[eq:performance\]) for the set of top spreaders of size $|Q_{err}|
= 100$, and we plot the value of the performance as a function of the noise level in prior structural information. Performance is measured for $\phi_{true} = 0.5$. The shaded part of the plot serves to report results valid for $0 \leq \epsilon_{del} \leq 1 $ and $\epsilon_{add}
=0$. The non-shaded part of the graph represents instead results for $0 \leq \epsilon_{add} \leq 1 $ and $\epsilon_{del}
=0$. (b) Same as in panel a, but for $\phi_{true} = 1.0$. (c) Same as in panel a, but for $\phi_{true} = 1.5$. (d) Same as in panel a, but for $\phi_{true} = 2.0$. []{data-label="fig:2"}](fig2.pdf){width="45.00000%"}
To better characterize the observed trend, we set $|Q_{err}| = 100$ and quantify the performance defined in Eq. (\[eq:performance\]) for different levels of noise. The results of this analysis are presented in Figure \[fig:2\]. The various panels of the figure refer to different dynamical regimes identified by different values of $\phi_{true}$. In every panel, we present three curves, each representing a specific value of $\phi_{err}$. Each curve stands for spreading performance $P(Q_{err})$ as a function of the structural noise parameters $\epsilon_{del}$ and $\epsilon_{add}$. Please consider that, although the two sources of structural noise are never considered active simultaneously, we present them in the same plot for the sake of compactness. The general observed behavior can be summarized as follows. Maximal performance is reached at $\epsilon_{del} = \epsilon_{add} =
0$, only for $\phi_{err} = \phi_{true}$. The two sources of structural noise affect the choice of the top spreaders differently. Consider first the case $\epsilon_{del} =0$, but $0 \leq \epsilon_{add}
\leq 1$. Roughly, we see that $P(Q_{err})$ is a monotonic function of $\epsilon_{add}$, decreasing if $\phi_{err} \geq \phi_{true}$, and increasing, otherwise. In the region $\epsilon_{add} = 0$ and $0 \leq \epsilon_{del}
\leq 1$, instead, $P(Q_{err})$ is not a monotonic function of the structural error. Further, the trend changes depending on whether the system is in the subcritical or supercritical regime: if $\phi_{true} \leq 1$, $P(Q_{err})$ is concave; if $\phi_{true} \geq 1$, $P(Q_{err})$ is convex. In this second regime, it becomes possible to obtain higher performance by adding further structural mistakes. If $\phi_{err} < \phi_{true}$, best performance is achieved for $\epsilon_{del} =1$, essentially using no prior knowledge of the network structure, i.e. seeds are sampled at random.
![Best values of the structural errors in the presence of dynamical uncertainty. We consider the same network as in Fig. \[fig:1\]. (a) We set $\epsilon_{add}
= 0$, and, for given dynamical parameters $\phi_{true}$ and $\phi_{err}$, we determine $\hat{\epsilon}_{del} $, i.e., the value of the error parameter $\epsilon_{del} $ that leads to the maximum performance in the prediction of top spreaders. Best estimates of $\hat{\epsilon}_{del} $ are reported in the cells of the table. The intensity of the background color is proportional to the value of $\hat{\epsilon}_{del} $. (b) Same as in panel a, but for the other source of structural error. We set here $\epsilon_{del}
= 0$ and focus on $\hat{\epsilon}_{add} $, i.e., the value of the error parameter $\epsilon_{add} $ that allows to identify the best performing set of top spreaders. []{data-label="fig:3"}](fig3.pdf){width="45.00000%"}
We analyzed the phenomenon systematically. For different combinations $\phi_{true}$ and $\phi_{err}$, we computed $\hat{\epsilon}_{del}$ and $\hat{\epsilon}_{add}$, i.e., the values of the parameters of structural noise where $P(Q_{err})$ reaches its maximum. As Figure \[fig:3\]a shows, for $\phi_{err} \simeq \phi_{true}$, $\hat{\epsilon}_{del} \simeq 0$. However, as soon as the error on the dynamical parameter increases, this error may be compensated by making further mistakes in the structure. When structural noise is allowed through the addition of edges instead, noise is helpful only in the supercritical regime (Fig. \[fig:3\]b).
![Performance of the top spreaders on a spatially embedded network in presence of structural and dynamical noise. Same analysis as in Fig. \[fig:2\], but for a different network. Here, the true network structure is given by the US power grid network of Ref. [@watts1998collective]. []{data-label="fig:4"}](fig4.pdf){width="45.00000%"}
So far we reported results only for a specific network. However, our main findings are not very sensitive to this choice, in the sense that our qualitative results are very similar for all real networks we analyzed (see Supplementary Material, SM). The only major difference arises for networks characterized by strong spatial embedding (hence, a specific modular structure identified by very loose intermodule connections [@Faqeeh16]). There, the strong asymmetry between the operations of altering the network structure by adding or removing random edges is apparent. This fact is for instance visible in Figure \[fig:4\], where we report the analogue of Figure \[fig:2\] for the power grid network originally considered in Ref. [@watts1998collective]. From the figure, we see that $P(Q_{err})$ doesn’t behave smoothly around $\epsilon_{del} = \epsilon_{add} = 0$. However, the general findings valid for the two different sources of structural noise are almost identical to those valid for networks with no spatial embedding.
The same qualitative results hold for other values of the size of the set of seed nodes $|Q_{err}|$, as long as its value is large enough compared to the size of the network $N$. In the SM, we report results valid for $|Q_{err}| = 10$ for the email network considered here in the main text for which $N=1,133$. In such a case, the clear pattern of Figure \[fig:2\] becomes much noisier. For networks of smaller size, we observe that the pattern is already clear even for $|Q_{err}| = 10$.
![Average degree of the set of top spreaders. We consider the same network as in Fig. \[fig:1\]. (a) Average degree of the set of $|Q_{err}| = 100$ of top spreaders identified for $\phi_{err} = 0.5$ and different values of the structural errors $\epsilon_{del}$ and $\epsilon_{add}$. As in Figure \[fig:2\], we use left part of the plot, highlighted with a gray-shaded background, to report results valid for $0 \leq
\epsilon_{del} \leq 1 $ and $\epsilon_{add} =0$. The non-shaded part of the graph represents instead results for $0 \leq \epsilon_{add} \leq 1 $ and $\epsilon_{del} =0$. (b, c and d) Same as in panel a, but for $\phi_{err} = 1.0$, $\phi_{err} = 1.5$, and $\phi_{err} = 2.0$, respectively. []{data-label="fig:5"}](fig5.pdf){width="45.00000%"}
One may explain our results with the following naive argument. For simplicity, let us consider only the case where noise randomly deletes true edges. In our prior knowledge, every true edge becomes invisible with probability $\epsilon_{del}$. Further, we believe that the ICM has spreading probability $\phi_{err}$. In summary, prior knowledge forces us to think that the effective spreading probability on a random edge that we are considering in the true, but unknown, network is $(1 - \epsilon_{del}) \phi_{err}$, rather than the actual true value $\phi_{true}$. Best predictions should be obtained for $(1 - \epsilon_{del}) \phi_{err}
\simeq \phi_{true}$. If $\phi_{err} > \phi_{true}$, one can correct the mistake by choosing appropriately $\epsilon_{del} \in [0, 1]$. If $\phi_{err} < \phi_{true}$, there is no way to satisfy the previous equation. The best performance would be naively expected for $\epsilon_{del} = 0$, as this value corresponds to the noise level that minimizes the difference between effective and true spreading probability. However, this is not what we observe in our numerical results, where best performance is actually achieved for $\epsilon_{del} = 1$. The apparent paradox can be solved by accounting for structural correlations. As it is well known, true top spreaders in the ICM depend on the critical regime [@radicchi2017fundamental]. In the subcritical regime, central nodes are generally better locations for seeds. In the supercritical regime instead, peripheral nodes are selected first. In both regimes, seeds are generally placed on nodes that are not directly connected, as a source of spreading that is too redundant is generally not optimal. If the probability $\epsilon_{del}$ of random deletion of edges is not very high, then the ranking based on the degree centrality of the nodes is basically unaffected. However, pairs of truly connected nodes appear as disconnected in the noisy version of the network regardless of their degree. As a result, for $\phi_{err} < 1$ many high-degree nodes are chosen as seeds. However, they may behave poorly in terms of seed set, as they constitute a source of spreading that is too redundant to be optimal. A visual intuition of this structural explanation is provided in Figure \[fig:5\]. There, we consider the true value of the average degree of the set of top spreaders identified using noisy information. Each panel corresponds to a different value of $\phi_{err}$. Please note that the value of the average degree measured at $\epsilon_{del} = \epsilon_{add} = 0$ is the one that corresponds to the true set of optimal seeds for the dynamical regime $\phi_{true} = \phi_{err}$. As expected, for the subcritical regime, the identified set of top spreaders has high average degree and the structural noise doesn’t affect much the value of this variable, except when $\epsilon_{del} \simeq 1$. In the supercritical regimes instead, best performance is achieved for sets with low values of the average degree, comparable with the average degree of the network. Structural noise changes dramatically the set of seeds, especially in the region $0 < \epsilon_{del} <
1$. However, in the regime of very large noise, the average degree of the seed set is basically equal to the average degree of the network as nodes are chosen using (almost) no structural information.
Conclusions
===========
In this paper, we considered a simple yet practically relevant scenario. We assumed that prior information used in the solution of the influence maximization problem is affected by some noise, and we studied how the quality of the solution found using noisy information deteriorates as a function of the noise intensity. Our main finding is that the quality of the solution always decreases monotonically with noise, if structural and dynamical noise are considered independently. However, when both sources of noise act simultaneously, one of them can compensate the disruptive effect of the other. In essence, noise affecting dynamical information may be suppressed by additional noise at the structural level, or [*vice versa*]{}. This fact is particularly apparent when structural noise is such that random edges of the original network disappear with a certain probability. As this is a plausible model of error that may affect our knowledge of the true network structure [@Newman17], our results may be important in real-world applications. More in general, the approach presented here may be used to understand how incomplete and/or erroneous information at the level of network structure and dynamics affects our ability to solve optimization problems in a meaningful way.
The authors thank C. Castellano for critical reading of the manuscript. ŞE and FR acknowledge support from the National Science Foundation (CMMI-1552487). AF and FR acknowledge support from the US Army Research Office (W911NF-16-1-0104). AF acknowledges support from the Science Foundation Ireland (16/IA/4470).
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---
abstract: 'The outcomes of measurements on entangled quantum systems can be nonlocally correlated. However, while it is easy to write down toy theories allowing arbitrary nonlocal correlations, those allowed in quantum mechanics are limited. Quantum correlations cannot, for example, violate a principle known as macroscopic locality, which implies that they cannot violate Tsirelson’s bound. This work shows that there is a connection between the strength of nonlocal correlations in a physical theory, and the structure of the state spaces of individual systems. This is illustrated by a family of models in which local state spaces are regular polygons, where a natural analogue of a maximally entangled state of two systems exists. We characterize the nonlocal correlations obtainable from such states. The family allows us to study the transition between classical, quantum, and super-quantum correlations, by varying only the local state space. We show that the strength of nonlocal correlations - in particular whether the maximally entangled state violates Tsirelson’s bound or not - depends crucially on a simple geometric property of the local state space, known as strong self-duality. This result is seen to be a special case of a general theorem, which states that a broad class of entangled states in probabilistic theories - including, by extension, all bipartite classical and quantum states - cannot violate macroscopic locality. Finally, our results show that there exist models which are locally almost indistinguishable from quantum mechanics, but can nevertheless generate maximally nonlocal correlations.'
address:
- '$^1$ Fakult[ä]{}t für Physik und Astronomie, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany'
- '$^2$ Institute for Physics and Astronomy, Potsdam University, 14476 Potsdam, Germany'
- '$^3$ Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, United Kingdom'
- '$^4$ Department of Mathematics, Royal Holloway, University of London, Egham Hill, Egham TW20 0EX, United Kingdom'
- '$^5$ H.H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom'
author:
- |
Peter Janotta$^1$, Christian Gogolin$^{1,2,3}$, Jonathan Barrett$^{4,5}$\
and Nicolas Brunner$^5$
title: 'Limits on non-local correlations from the structure of the local state space'
---
Introduction
============
Nonlocality is a key feature of quantum mechanics. By performing measurements on separated systems in an entangled state, one can obtain correlations that are stronger than those of any local model, as witnessed by the violation of Bell inequalities [@bell64]. On the other hand, sets of nonlocal correlations are known that are stronger than those of quantum mechanics, but which do not allow for instantaneous signalling. This led Popescu and Rohrlich [@PR] to raise the question of why nonlocality seems to be limited in nature.
In recent years, new insights have been gained into this question by studying the information theoretic properties of super-quantum correlations. For instance these correlations lead to implausible reductions for all communication complexity problems, such that they can be solved with only constant communication [@vanDam; @brassard; @BS]. The principle of *information causality* [@IC] is satisfied by quantum correlations, but can be violated if certain super-quantum correlations are available — similarly the principle of *macroscopic locality* [@ML]. Various multi-player games have been described, for which super-quantum correlations would provide an advantage over quantum correlations [@noah; @GYNI].
The above studies focused on the information theoretic power of correlations without any reference to the physical theories they emerge from. Recent works revealed interesting connections between the structure of quantum mechanics and the nonlocal correlations that can be generated by quantum systems. Barnum et al. [@Beigi09], for example, considered a theory that is locally equivalent to quantum mechanics but whose non-locality is only limited by the no-signalling principle. Despite this theory being less restrictive than quantum mechanics, the set of bipartite correlations that can be obtained is identical to that of quantum states. This implies that, despite the fact that quantum correlations are clearly a global property of joint systems, their limitation does not result from the lack of joint states, but rather from the structure of the local state spaces. Meanwhile, Acín et al. [@Acin10] have shown that this result does not extend to three or more parties.
In this paper, we show that the connection between local state spaces and the limitation of bipartite nonlocal correlations is actually a more general phenomenon. In particular, if local state spaces have a property known as *strong self-duality*, then the correlations obtainable from maximally entangled states must be compatible with the principle of macroscopic locality. It follows that they must also respect Tsirelson’s bound. A precise definition of strong self-duality is given later, but in the quantum case it corresponds roughly to the fact that the same rank one projector represents both a pure state and the outcome of a measurement which identifies that state.
By way of illustration, we introduce along the way a family of models, where each model is defined by the local state space for a single system, and the state space is taken to be a regular polygon with $n$ vertices (see ). For two such systems, there is a natural analogue of a maximally entangled state. The family includes the classical case of two trits ($n=3$); systems generating the super-quantum correlations introduced by Popescu and Rohrlich ($n=4$); and systems producing quantum correlations ($n \rightarrow \infty$). Thus the family allows us to study the transition between these theories, and the bipartite correlations that can be produced by a maximally entangled state, by modifying only the local state space. For high $n$ the local state spaces are almost indistinguishable from a quantum system. Nevertheless it turns out that these models show dramatically different correlations — and thereby have fundamentally different information theoretic capabilities — depending on the parity of $n$. This is explained by the fact that those with odd $n$ are *strongly self-dual*, while those with even $n$ only *weakly self-dual*.
One way of viewing the polygon models is that moving from $n\rightarrow \infty$ to $n=3$, there is a progressive weakening of the superposition principle. A weakened superposition principle means that states can only be superposed in certain combinations. In a similar spirit, a different range of models was introduced in Ref. [@UR], with each model defined by a relaxation of the uncertainty relations of quantum mechanics. Here too, a transition from quantum correlations to Popescu-Rohrlich correlations was observed.
This paper is organized as follows. Section \[operationalmodels\] gives a brief, not too technical, introduction to a mathematical formalism in which a very broad range of probabilistic theories can be expressed, including quantum theory and classical probability theory. Section \[sec:afamilyofmodels\] introduces the polygon models, and by investigating the properties of bipartite correlations, sheds some light on the relation between these and the local state space structure. Section \[sec:selfdualityandtsirelsonsbound\] returns to the general case and contains the proof of the main theorem, which establishes a rigorous limit on the nonlocal correlations obtainable from a broad class of bipartite states in general probabilistic theories. In particular, states obtainable by norm-preserving local transformations from what we call *inner product states* cannot violate the principle of macroscopic locality. Section \[polygonsrevisited\] provides a formal definition of strong and weak self-duality, and discusses consequences of the main theorem for the correlations in bipartite polygon systems. Section \[sec:Correlationsoutsideofq1\] presents a strongly self-dual system in which a non-maximally entangled state gives rise to correlations that cannot be obtained from any inner product state. Finally, section \[discussion\] discusses some open questions.
Operational models {#operationalmodels}
==================
Systems and measurements
------------------------
This section describes briefly the framework of generalized probabilistic theories [@barrett], using the notation and conventions of Ref. [@Barnum]. The aim is to be able to describe theoretical models other than the classical and quantum theories, and for these two to be included as special cases.
We start by taking an operational point of view. A *state* of a system is a mathematical object that defines the outcome probabilities for all the measurements that can possibly be performed on this system. The *state space* $\Omega$ of a system is the set of states that it can be prepared in.
By defining the operations of summation and multiplication by a real number on states, we can identify $p \omega_1 + (1-p) \omega_2$ as the probabilistic mixture obtained by preparing $\omega_1$ with probability $p$ and $\omega_2$ with probability $1-p$. The state space $\Omega$ is now a convex set, embedded in a real vector space $V$. For simplicity, assume that $\Omega$ is compact and finite dimensional. States that can be represented by convex combinations of other states are *mixed* states. The extremal points of the state space $\Omega$ cannot be written in such a form, and are *pure* states. For a quantum system, for example, $\Omega$ is the set of density operators on a Hilbert space, and the pure states are the rank one projectors. For a qubit, $\Omega$ is particularly easy to visualize, since it corresponds to the Bloch ball, with pure states on the surface of the ball. For a (finite-dimensional) classical system, $\Omega$ is the set of probability distributions over some finite sample space.
A measurement outcome is represented by an *effect*, that is a map $e\colon \Omega \to [0,1]$, where $e(\omega)$ is the probability of obtaining the outcome $e$ when the measurement is performed on a system in the state $\omega$. Probabilities of measurement outcomes should respect probabilistic mixtures of states, meaning that $e[p\,\omega_1 + (1-p) \, \omega_2] = p\, e(\omega_1) + (1-p)\, e(\omega_2)$, i.e., the effects are affine maps. A special effect is the *unit effect* $u$, which is uniquely defined such that $u(\omega)=1$ for all $\omega \in \Omega$. The unit effect represents a measurement with a single outcome that is certain to occur regardless of what the state is. An arbitrary measurement is a set of effects $\{e_i\}$ summing to the unit effect $\sum_i e_i = u$. This ensures that outcome probabilities of measurements sum to one.
The set of proper effects $E(\Omega) = \{e : 0 \leq e(\omega) \leq 1\ \forall \omega \in \Omega\}$ is the convex hull of the unit effect, the zero effect and a set of extremal effects. For a quantum system, if states are density operators on a Hilbert space, then effects can be identified with positive semidefinite operators on the Hilbert space, in such a way that outcome probabilities are given by the usual trace rule. Measurements correspond to positive operator-valued measures. For a classical system, effects can be identified with fuzzy indicator functions on the sample space, i.e., maps from the sample space into $[0,1]$.
Unnormalized states
-------------------
It is frequently useful to work with unnormalized states. Given a state space $\Omega$ and effect space $E(\Omega)$, let $V$ be the linear span of $\Omega$. The linear span of $E(\Omega)$ is then the dual space $V^*$. Both $V$ and $V^*$ are real vector spaces. In the case of a quantum system, for example, $V$ is the linear span of the density operators, which is the set of all Hermitian operators on the corresponding Hilbert space. Similarly, $V^*$ is the linear span of the positive semidefinite operators, which is also the set of all Hermitian operators.
An unnormalized state is an element of $V$ of the form $r\,\omega$, with $r>0$ and $\omega\in\Omega$. The set of all unnormalized states is a cone denoted $V_+$. Similarly, an unnormalized effect is an element of $V^*$ of the form $r\,e$ for $r>0$ and $e \in E(\Omega)$. The set of unnormalized effects is the *dual cone* to $V_+$, denoted $V_+^*$. The cone $V_+$ and the dual cone $V_+^*$ are related via $$V_+^* = \{ e \in V^*: e(\omega) \geq 0, \forall \omega \in V_+\}.$$ In the case of a quantum system, both $V_+$ and $V_+^*$ can be identified with the set of positive semidefinite operators on the Hilbert space. In general a cone $V_+$ can have a very different structure than its dual cone $V_+^*$, e.g., they may have a different number of extremal rays.
Bipartite states {#bipartitestates}
----------------
Given two systems $A$ and $B$, an operational model needs to specify the set $\Omega^{AB}$ of available joint states, in addition to the individual state spaces $\Omega^A$ and $\Omega^B$. In general, one can imagine many weird and wonderful ways in which two systems might combine to form a joint system. By imposing two quite natural conditions, however, one can narrow down these possibilities significantly.
The first condition is the *no-signalling principle*, which says that it should not be possible to send messages instantaneously by performing measurements on the separate parts of a joint system. The second is that of *local tomography*. Given a single system, call a measurement *informationally complete* if its outcome probabilities are sufficient to determine uniquely the state of the system. The principle of local tomography states that if an informationally complete measurement is performed separately on each of the subsystems of a composite system, then the joint outcome probabilities are sufficient to determine uniquely the state of the joint system.
These two conditions together are sufficient to ensure that the linear space $V^{AB}$ in which the joint state space $\Omega^{AB}$ and the cone of associated unnormalized states are embedded can be taken to be $V^A\otimes V^B$ (see for example Ref. [@Barnum] and the references therein). If simultaneous measurements are performed on systems $A$ and $B$, then the joint probability for outcomes $e$ and $f$ is given by $(e\otimes f)(\omega^{AB})$.
It is convenient to define the unit effect of the joint state space as $u^{AB} = u^A \otimes u^B$ such that a joint state is normalized if $$\label{jointstatenorm}
(u^A \otimes u^B)(\omega^{AB}) = 1,$$ where $u^A$ and $u^B$ are the unit effects for systems $A$ and $B$ respectively. Naturally, probabilities are positive, so a joint state must satisfy $$\label{eq:tensormax}
(e^A\otimes e^B)(\omega^{AB}) \geq 0$$ for all $e^A\in E(\Omega^A)$, $e^B\in E(\Omega^B)$.
The *maximal tensor product* of $\Omega^A$ and $\Omega^B$, denoted $\Omega^A {\otimes_{\mathrm{max}}}\Omega^B$, is the set of all $\omega^{AB} \in V^A\otimes V^B$ such that and are satisfied.
It is easy to check that the no-signalling principle is indeed satisfied for such an $\Omega^{AB}$. Consider two measurements on $A$, corresponding to sets of effects $x = \{ e_1,\ldots, e_m \}$ and $x' = \{ e'_1,\ldots, e'_n \}$. The marginal probability for an outcome $f$ of a measurement on $B$ is $$\sum_{i=1}^m (e_i\otimes f)(\omega^{AB}) = (u^A\otimes f)(\omega^{AB}) = \sum_{j=1}^n (e'_j\otimes f) (\omega^{AB}),$$ i.e., it is independent of whether $x$ or $x'$ is performed on $A$.
Intuitively, the maximal tensor product is the set of all non-signalling joint states that can be written down for two systems, given the individual state spaces $\Omega^A$ and $\Omega^B$. A particular theory or model need not assume that every element of the maximal tensor product is an allowed state for the joint system. In general, a model will specify a joint state space $\Omega^{AB}$ which is a subset of $\Omega^A{\otimes_{\mathrm{max}}}\Omega^B$.
Straightforwardly generalizing the notions well known from quantum theory, one calls a state a *product state* if it can be written in the form $\omega^A \otimes \omega^B$ for some states $\omega^A \in \Omega^A$ and $\omega^B \in \Omega^B$. States that can be written as probabilistic mixtures of product states are *separable*, while states that are not separable are *entangled*.
This work mostly considers correlations obtained from product measurements on bipartite states. The general formalism, however, does not assume that all measurements on composite systems are product measurements. As in the case of single systems, outcomes of measurements on a composite system correspond to effects, where these are maps $\Omega^{AB}\rightarrow [0,1]$. The set of all such effects is written $E(\Omega^{AB})$, and may include entangled, as well as product, effects. However, $E(\Omega^A{\otimes_{\mathrm{max}}}\Omega^B)$ only contains separable effects.
Quantum theory provides a useful example of many of the concepts above. In this case, $\Omega^{AB}$ is the set of density operators on the Hilbert space $H^{AB} = H^A \otimes H^B$. Recall that $V^A$ and $V^B$ are real vector spaces of Hermitian operators on $H^A$ and $H^B$ respectively. The set of Hermitian operators on $H^{AB}$ can be identified with $V^A\otimes V^B$, so the joint quantum states are indeed elements of $V^A\otimes V^B$. The density operators on $H^{AB}$ are a proper subset of $\Omega^A{\otimes_{\mathrm{max}}}\Omega^B$. Elements of $\Omega^A{\otimes_{\mathrm{max}}}\Omega^B$ which are not density operators are (normalized) *entanglement witnesses*. An entanglement witness $w$ is locally positive, meaning that for all product measurements, $(e^A\otimes e^B)(w) \geq 0$. But $w$ is not a density operator, since there are entangled measurement outcomes $e$ with $e(w) < 0$.
A family of models {#sec:afamilyofmodels}
==================
Polygon systems {#polygonsystems}
---------------
This section defines a family of models such that the state spaces $\Omega$ of single systems are regular polygons with $n$ vertices. It is convenient to represent both states and effects by vectors in $\mathbb{R}^3$ such that $e(\omega)$ is the usual Euclidean inner product. For fixed $n$, let $\Omega$ be the convex hull of $n$ pure states $\{\omega_i\}$, $i=1,...,n$, with $$\label{eq:localpolygons}
\omega_i = \begin{pmatrix}
r_n \cos(\frac{2 \pi i}{n})\\
r_n \sin(\frac{2 \pi i}{n})\\
1
\end{pmatrix} \in \mathbb{R}^3 ,$$ where $r_n= \sqrt{\sec(\pi/n)}$.
The unit effect is $$u = \begin{pmatrix}
0\\
0\\
1
\end{pmatrix}.$$ In the case of even $n$, the set $E(\Omega)$ of all possible measurement outcomes is the convex hull of the zero effect, the unit effect, and $e_1,\ldots, e_n$, with $$\label{eff_even}
e_i = \frac{1}{2} \, \begin{pmatrix}
r_n \cos(\frac{(2 i-1) \pi}{n})\\
r_n \sin(\frac{(2 i-1) \pi}{n})\\
1
\end{pmatrix} .$$ Let $\bar{e_i} = u - e_i$, hence a possible dichotomic measurement is $\{e_i, \bar{e_i} \}$. When this measurement is performed on a system in the state $\omega_j$, the probabilities for the two outcomes are given by $e_i\cdot \omega_j$ and $\bar{e_i}\cdot \omega_j$, and satisfy $e_i\cdot \omega_j + \bar{e_i}\cdot \omega_j = 1$. Observe that for even $n$, $\bar{e_i} = e_{(i+n/2) \mathrm{mod} \ n}$.
The case of odd $n$ is slightly different. In this case, define $$\label{eff_odd}
e_i = \frac{1}{1 + {r_n}^2} \, \begin{pmatrix}
r_n \cos(\frac{2 \pi i}{n})\\
r_n \sin(\frac{2 \pi i}{n})\\
1
\end{pmatrix}$$ and again let $\bar{e_i} = u - e_i$, so that a possible dichotomic measurement is $\{e_i, \bar{e_i} \}$. This time, however, $\bar{e_i}$ does not equal $e_j$ for any $j$. The set $E(\Omega)$ of all possible measurement outcomes is the convex hull of the zero effect, the unit effect, $e_1\ldots,e_n$, and $\bar{e_1},\ldots,\bar{e_n}$. As can be seen in in such theories there are effects that are extremal in $E(\Omega)$ (namely the $\bar{e_i}$) but not ray extremal, i.e., they do not lie on an extremal ray of the cone $V_+^*$. This also happens in quantum mechanics, but only if the dimension of the Hilbert space is larger than two. For example the effect ${\mathds{1}}- | \psi \rangle\langle \psi |$ for any rank one projector $| \psi \rangle\langle \psi |$ is then extremal in the set of proper effects, but not ray extremal.
A two-dimensional illustration of the state and effect spaces is given in and a three-dimensional illustration in .
The $n=3$ case corresponds to a classical system with three pure states. Think of it as a trit. The three pure states are $\omega_1$, $\omega_2$ and $\omega_3$, and correspond to the three different possible values of the trit. The state space $\Omega$ is a triangle. A generic point in $\Omega$ is a mixture of the three pure states and corresponds to a probability distribution over the three trit values. Notice that in this case, $e_1+e_2+e_3 = u$, hence a possible measurement is a three-outcome measurement with outcomes $e_1,e_2$ and $e_3$. This is the obvious measurement that simply reads off the value of the trit. Below we shall consider bipartite states of polygon systems. Given two trits, the only possible joint states are separable, and it is not possible to produce nonlocal correlations. The case $n=4$ corresponds to a single system in a toy theory known as ‘box world’, which has been discussed elsewhere in the literature (see for instance Ref. [@barrett]). The state space is a square. As shown below, a notable feature of box world is that given two of these systems, it is possible to construct joint states that are more nonlocal than quantum states. In fact, an entangled state of two of the $n=4$ systems can produce maximally nonlocal correlations known as *PR box* correlations [@PR], which have been much explored in the literature [@vanDam; @brassard; @IC; @noah].
As $n \rightarrow\infty$, the state space tends to a disc of radius one. This makes it similar to a quantum mechanical qubit, whose state space is the Bloch ball. The disc can be thought of as the equatorial plane of the Bloch ball. We will refer to this case, somewhat loosely, as the quantum case.
Bipartite states of polygon systems {#sec:defpolybox}
-----------------------------------
We shall not attempt a complete characterization of the set of all possible non-signalling states $\Omega^A {\otimes_{\mathrm{max}}}\Omega^B$ for each value of $n$. Instead, this section describes a particular joint state of two polygon systems, which is the natural analogue of a maximally entangled state of two qubits. The next section examines the nonlocal correlations that can be obtained from performing measurements on these maximally entangled polygon systems.
Recall that a joint state is an element of $V^A\otimes V^B$, hence in the case of two polygon systems, a joint state is an element of $\mathbb{R}^3 \otimes \mathbb{R}^3 = \mathbb{R}^9$. It is convenient to represent the joint state as a $3\times3$ matrix such that $(e_i \otimes e_j)(\omega^{AB})$ can be calculated by simply left and right multiplying this matrix with the representations of the effects $e_i$ and $e_j$ in $\mathbb{R}^3$. Define $$\begin{aligned}
\label{definitionphi}
\mathrm{odd \ n:}\quad\phi^{AB} &=& \left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right), \nonumber\\
\mathrm{even \ n:}\quad\phi^{AB} &=& \left(\begin{array}{ccc} \cos(\pi/n) & \sin(\pi/n) & 0 \\ -\sin(\pi/n) & \cos(\pi/n) & 0 \\ 0 & 0 & 1 \end{array}\right). \end{aligned}$$
The state $\phi^{AB}$ is the natural analogue of a quantum mechanical maximally entangled state for the following reasons. First, it can be verified (see, e.g., Ref. [@steering]) that except for $n=3$, $\phi^{AB}$ is an entangled pure state, where pure means that it is extremal in the maximal tensor product, hence cannot be written as a mixture of other non-signalling states. The $n=3$ case corresponds to two classical trits, with $\phi^{AB}$ the maximally correlated state, i.e., if the trit values are $1$, $2$, $3$, then $\phi^{AB}$ corresponds to $P(11)=P(22)=P(33)=1/3$. Second, $\phi^{AB}$ is constructed so that if a measurement is performed on the $A$ system, and outcome $e_i$ obtained, then the updated (or collapsed) state for the $B$ system is $\omega_i$. The marginal probability for Alice to obtain outcome $e_i$ is the same for all $i$. Compare this with the case of two spin-1/2 particles in the state $1/\sqrt{2} (|00\rangle + |11\rangle)$, where $|0\rangle$ and $|1\rangle$ are the eigenstates of spin-$z$. If a spin measurement in direction $\vec{m}$ in the $xz$-plane is performed on system $A$, then the probability of obtaining the up outcome is $1/2$, and if the up outcome is obtained, then the collapsed state of the $B$ system is spin up in direction $\vec{m}$. These quantum predictions are recovered by $\phi^{AB}$ in the limit $n \rightarrow \infty$.
The following sections investigate the nonlocal correlations that can be produced by performing measurements on two systems in the state $\phi^{AB}$. For this it is useful to have an expression for the joint probability of obtaining outcome $e^A_i$ on system $A$ and $e^B_j$ on system $B$. This is easy to calculate from . For even $n$, $$\label{eq:nevencorrelations}
(e^A_i \otimes e^B_j)(\phi^{AB}) = \frac{1}{4}\left( 1+r_n^2 \cos(\alpha_i-\beta_j)\right),$$ where $\alpha_i = \frac{2\pi i}{n}$ and $\beta_j = \frac{(2j-1)\pi}{n}$, and as before, $r_n = \sqrt{\sec(\pi/n)}$. For odd $n$ $$\label{eq:noddcorrelations}
(e^A_i \otimes e^B_j)(\phi^{AB}) = \frac{1}{(1+r_n^2)^2}\left( 1+r_n^2 \cos(\alpha_i-\beta_j)\right),$$ where $\alpha_i = \frac{2\pi i}{n}$ and $\beta_j = \frac{2\pi j}{n}$. Notice the cosine dependence, which is reminiscent of quantum mechanical correlations.
The Clauser-Horne-Shimony-Holt inequality {#sectionchsh}
-----------------------------------------
One commonly used measure of the degree of nonlocality that a bipartite system exhibits is the maximal violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality [@chsh]. The CHSH inequality involves two parties, conventionally called Alice and Bob. Each chooses between two dichotomic measurements. Let Alice’s choice of measurement be $x$, and Bob’s $y$, with $x,y \in \{0,1\}$. Denote the measurement outcomes $a,b \in \{0,1\}$. A set of correlations is characterized by the joint probability distribution $P(a,b|x,y)$. The strength of the correlations is quantified by the CHSH parameter $$\label{chshparameter}
S = | E_{0,0}+ E_{0,1} + E_{1,0} - E_{1,1}| ,$$ where $E_{x,y} = P(0,0|x,y)+P(1,1|x,y)-P(0,1|x,y)-P(1,0|x,y)$. As CHSH showed, local correlations must satisfy $S\leq 2$. In quantum mechanics, correlations can violate this inequality, but must respect Tsirelson’s bound $S\leq 2 \sqrt{2}$ [@cirelson80].
By inspection, the algebraic maximum of $S$ is $4$, and it is easy to see that it is attained by the following correlations: $$\label{PR1}
P(a,b|x,y) =
\begin{cases}
\frac{1}{2} & \text{if $ a \oplus b = xy$} \\
0 & \text{otherwise}.
\end{cases}$$ Here, $\oplus$ denotes addition modulo $2$. These correlations were described by Popescu and Rohrlich, who pointed out that they are maximally nonlocal, yet still respect the no-signalling principle [@PR]. Since they cannot occur in quantum mechanics, they are imagined to be produced by a fictitious device, which is often referred to as a *PR box*. As discussed in the introduction, PR boxes have been explored in the literature and are known to be particularly powerful for certain kinds of information theoretic problem, especially communication complexity problems [@vanDam; @brassard; @BS; @IC; @ML; @noah; @GYNI].
It is interesting to see how the maximal CHSH value obtainable from polygon systems in the state $\phi^{AB}$ varies as the number of vertices $n$ of the polygon increases. The $n=4$ case is particularly simple. The optimal choice of measurements to violate the CHSH inequality is $$\begin{aligned}
x = 0:& \{ e_1^A,e_3^A \}, & x = 1:& \{ e_2^A,e_4^A \}, & y = 0:& \{ e_2^B,e_4^B \}, & y = 1:& \{ e_1^B,e_3^B \}, \end{aligned}$$ and it can be verified from that the correlations obtained give $S=4$. In other words, the maximally entangled state of two $n=4$ systems can act as a PR box. It follows that this state has the same information theoretic power that PR boxes are known to have.
For general $n$, assume that Alice’s measurement choices are of the form $\{ e^A_i, \bar e^A_i\}$ and Bob’s of the form $\{ e^B_j,\bar e^B_j\}$. A lengthy but straightforward calculation gives the following analytic expressions. For even $n$, $$\label{CHSHeven}
S = r_n^2 \sum_{x,y=0,1} (-1)^{xy} \cos\left(\alpha_x-\beta_y\right),$$ where as before, $\alpha_x = \frac{2\pi i_x}{n}$ and $\beta_y = \frac{(2j_y-1)\pi}{n}$. For odd $n$, $$\label{CHSHodd}
S = \frac{2}{\left(1+r_n^2 \right)^2} \, \bigg| (r_n^2-1)^2 + 2\,r_n^2 \sum_{x,y=0,1}(-1)^{xy} \cos(\alpha_x-\beta_y) \bigg|,$$ where $\alpha_x = \frac{2\pi i_x}{n}$ and $\beta_y = \frac{2\pi j_y}{n}$. Maximizing these expressions over all possible choices for the angles $\alpha_i$ and $\beta_j$ gives the maximal violation achievable by local measurements on the maximally entangled state $\phi^{AB}$. A detailed analysis of these expressions can be found in \[app:optimalchshsvalue\]. shows the maximal CHSH value for the maximally entangled state of polygon systems as a function of $n$.
The most important feature of is that the correlations of even $n$ systems can always reach or exceed Tsirelson’s bound, while the correlations of odd $n$ systems are always below Tsirelson’s bound. Thus Tsirelson’s bound appears as a natural separation between the correlations of these two different kinds of polygon state spaces. Sections \[sec:selfdualityandtsirelsonsbound\] and \[polygonsrevisited\] show why this is. Section \[sec:selfdualityandtsirelsonsbound\] shows that for odd $n$, the maximally entangled state $\phi^{AB}$ belongs to a broad class of states we call *inner product states*, and that all correlations obtainable from measurements on inner product states satisfy Tsirelson’s bound. Section \[polygonsrevisited\] goes further, and relates this to a fundamental geometric difference between polygons with even $n$ and odd $n$. In , the difference is seen in the fact that for odd $n$, the effect cone $V_+^*$ coincides with the state cone $V_+$, whereas for even $n$, the effect cone is isomorphic to the state cone but rotated through some angle. We have only considered correlations obtainable from the maximally entangled state $\phi^{AB}$. In principle there could be joint states other than the maximally entangled state which show stronger violations for some Bell inequalities. While this seems unlikely for the CHSH inequality, other Bell inequalities are known to be maximized by non-maximally entangled states in quantum mechanics [@nonmax].
The Braunstein-Caves inequalities {#sectionbc}
---------------------------------
The Braunstein-Caves (or *chained*) Bell inequalities [@chained] are similar to the CHSH inequality, but involve $N$ measurement settings on each system, rather than two. Let Alice’s choice of measurement be $x$, and Bob’s $y$, with $x,y \in \{1,\ldots, N \}$. Let the outcomes be $a,b \in \{0,1\}$. Local correlations satisfy $$S_N = \bigg|\sum_{j=1}^{N-1} (E_{j,j}+E_{j,j+1}) + E_{N,N} -E_{N,1} \bigg| \leq 2N-2,$$ where as before $E_{x,y} = P(0,0|x,y)+P(1,1|x,y)-P(0,1|x,y)-P(1,0|x,y)$. In the case $N=2$, this is equivalent to the CHSH inequality, up to relabelling of measurement settings.
The algebraic maximum of $S_N$ is $2N$. This maximum can be attained by performing measurements on the maximally entangled state of even $n$ polygon systems with $n=2N$. This state is thus tailor made for violating the Braunstein-Caves Bell inequalities. To see this, let Alice’s and Bob’s measurement choices be given by $$\begin{aligned}
x &= i: \{ e^A_i, \bar e^A_i \}, \quad i=1,\ldots, N, \\
y &= j: \{ e^B_j, \bar e^B_j \}, \quad j=1,\ldots, N,\end{aligned}$$ and note that (i) $E_{j,j}=1$ for $j=1,...,N$, (ii) $E_{j,j+1}=1$ for $j=1,...,N-1$ and (iii) $E_{N,1} = -1$. In the case $n\rightarrow \infty$, maximal violation of the Braunstein-Caves inequality is achieved in the limit of infinitely many settings. This is also true for a quantum mechanical maximally entangled state, as shown in Ref. [@barrettkentpironio].
In general, given a set of correlations $P(a,b|x,y)$, they can be written as a mixture $$P(a,b|x,y) = q {P^{\text{NL}}}(a,b|x,y) + (1-q) {P^{\text{L}}}(a,b|x,y),$$ where $0\leq q \leq 1$, $P^{NL}(a,b|x,y)$ is a set of nonlocal correlations and $P^L(a,b|x,y)$ a set of local correlations. Suppose, however, that the correlations $P(a,b|x,y)$ return the maximum value $S_N$ for an appropriate Braunstein-Caves inequality. Then $q(S_N) +(1-q)(S_N-2)\geq S_N$, hence $q=1$. Therefore, the fact that the maximally entangled state of even $n$ polygon systems returns the maximum value for the appropriate Braunstein-Caves inequality indicates that there is no local part in the correlations with $N=n/2$ measurement settings. This was pointed out in the case of quantum systems in Ref. [@barrettkentpironio; @epr2]. As a further curiosity, if we did have access to these systems, they could be used for secure key distribution, using the protocol of Ref. [@BHK].
Distillation {#sec:distill}
------------
So far, we have only considered correlations that can be produced by measuring a single copy of a bipartite polygon system. There remains the possibility that stronger correlations could be produced by performing local measurements on multiple bipartite pairs, and locally processing the data (there is a further possibility, involving entangled measurements across multiple copies on each side, which we do not discuss).
Consider the bipartite state $\phi^{AB}$ of two even $n$ polygon systems, and suppose that Alice and Bob are choosing from the measurements $$x = 0: \{ e_1^A,\bar e_1^A \}, \quad x = 1: \{ e_2^A,\bar e_2^A \}, \quad y = 0: \{ e_1^B,\bar e_1^B \}, \quad y = 1: \{ e_2^B,\bar e_2^B \},$$ with outcomes $a,b \in \{0,1\}$ as usual. Recall that $E_{j,j}=1$ for $j=0,1$ and $E_{0,1}=1$. also gives $E_{1,0}=2\cos(\frac{2\pi}{n})-1$. The correlations produced can be written as a probabilistic combination of maximally nonlocal correlations (equivalent up to relabelling to the PR box correlations of ), and another term which describes local correlations: $$\label{NLC}
P_\epsilon(a,b|x,y) = \epsilon {P^{\text{PR}}}(a,b|x,y) + (1-\epsilon) {P^{\text{L}}}(a,b|x,y).$$ Here, $ 0 \leq \epsilon=1-\cos(\frac{2\pi}{n}) \leq 1$, ${P^{\text{PR}}}$ is given by $$\label{PR}
{P^{\text{PR}}}(a,b|x,y) =
\begin{cases}
\frac{1}{2} & \text{if $ a \oplus b = x(y \oplus 1)$} \\
0 & \text{otherwise}
\end{cases}$$ and ${P^{\text{L}}}$ is a set of local correlations given by $$\label{Pc}
{P^{\text{L}}}(a,b|x,y) =
\begin{cases}
\frac{1}{2} & \text{if $ a \oplus b = 0$,} \\
0 & \text{otherwise.}
\end{cases}$$ In Ref. [@BS], it is shown that all correlations of the form with $0<\epsilon <1$ can be distilled into stronger correlations using a protocol that involves two copies of a bipartite system. Importantly, this protocol consists only of local processing and does not involve any communication. In the asymptotic limit of infinitely many copies of a bipartite system, the correlations can be distilled to PR box correlations by iterating the protocol. Thus for any finite even $n$, the polygon systems produce correlations that can be distilled arbitrarily close to PR box correlations (since $\epsilon = 1-\cos(\frac{2\pi}{n})>0$). It is only in the limit $n \rightarrow \infty$ (the quantum case), that we get $\epsilon=0$ and thus lose the ability to distill PR box correlations.
The consequence of the above is that polygon systems with even and finite $n$ inherit the powerful communication properties of PR boxes as long as there are multiple copies of the maximally entangled state available. For instance, they collapse communication complexity [@vanDam], allow for better than classical non-local computation [@noah], violate information causality [@IC] and macroscopic locality [@ML]. Moreover, since the PR box can be considered as a unit of bipartite nonlocality [@unit; @forster2], it follows that any bipartite no-signalling probability distribution can be generated from multiple copies of polygon systems with even $n$. This is particularly surprising as in practice, an individual polygon system with even and very large $n$ would be very difficult to distinguish from one with odd $n$, and also from the quantum case, i.e. the disc that one gets in the limit $n\to\infty$. These toy theories thus show that practically indistinguishable theories can have fundamentally different limits to the non-local correlations they allow.
For polygon systems with odd and finite $n$, the situation is dramatically different, as seen in the next section.
Bounds on correlations {#sec:selfdualityandtsirelsonsbound}
======================
For even $n$ polygon systems, the maximally entangled state can produce arbitrarily strong nonlocal correlations, whereas for odd $n$ polygon systems, the nonlocality is highly constrained. The maximally entangled state of odd $n$ polygon systems cannot, for example, violate Tsirelson’s inequality. This section shows that this is a consequence of a much more general result.
We first introduce a class of bipartite states in general theories, which we call *inner product states*. The main theorem establishes a strong constraint on the nonlocal correlations that can be produced from measurements on inner product states. One consequence is that inner product states cannot violate Tsirelson’s inequality. The maximally entangled states of odd $n$ polygon systems are inner product states, hence the theorem explains what was only established by direct calculation above — that these states do not violate Tsirelson’s inequality. On the other hand, the maximally entangled states of even $n$ polygon systems are not inner product states, which is consistent with them producing arbitrary non-signalling correlations. We also show that all classical and quantum states are, in terms of non-local correlations, no stronger than an inner product state.
Inner product states {#sec:innerproductstates}
--------------------
Recall that a state cone $V_+$ is the set of unnormalized states of a system, and that these span a vector space $V$. An effect cone $V_+^*$ is the set of unnormalized measurement outcomes, and these span the vector space $V^*$. Given two systems $A$ and $B$, if the state cones $V^A_+$ and $V^B_+$ span vector spaces $V^A$ and $V^B$ respectively, then a joint state is an element of $V^A\otimes V^B$.
Call two distinct systems *similar* if their state spaces are isomorphic. Examples of similar systems are two quantum mechanical qubits, or two classical trits, or two $n$-vertex polygon systems. For the rest of this section, assume a bipartite system composed of two similar subsystems $A$ and $B$. In this case, the respective state spaces and effect spaces can be identified, so that $V^A=V^B=V$, $(V^A)^*=(V^B)^*=V^*$, $u^A=u^B=u$, and so on.
A joint state $\omega^{AB}$ is *symmetric* if $(e\otimes f)(\omega^{AB})=(f\otimes e)(\omega^{AB})$ for all measurement outcomes $e,f\in V_+^*$.
A joint state $\omega^{AB}$ is an *inner product state* if $\omega^{AB}$ is symmetric, and positive semidefinite, i.e., $(e\otimes e)(\omega^{AB}) \geq 0 \ \forall e \in V^*$.
Note that by definition of a joint state, it is always true that $(e\otimes e)(\omega^{AB}) \geq 0$ when $e\in V_+^*$, i.e., when $e$ is a valid effect. This is simply a statement of the fact that measurement outcome probabilities have to be greater than or equal to zero. The definition requires something stronger, which is that $(e\otimes e)(\omega^{AB}) \geq 0$ for any $e$ in the whole of the vector space $V^*$.
Any symmetric product state $\omega^{AB} = \omega \otimes \omega$ is an inner product state.
Consider two classical systems, each of which is a *nit*, taking values $\{1,\ldots, n\}$. A joint state is simply a joint probability distribution over nit values. Write the joint state as a matrix $P$, where $P_{ij}$ is the joint probability that $A=i$ and $B=j$. This is an inner product state iff the matrix $P$ is symmetric and positive semi-definite. In particular this includes any perfectly correlated state of the form $$\begin{aligned}
P_{ij} &=& 0 \quad\mathrm{if}\quad i\ne j\\
P_{ii} &=& q_i, \quad q_i \geq 0, \quad \sum_i q_i = 1.\end{aligned}$$
Consider two polygon systems, each corresponding to a state space with $n$ vertices. Section \[sec:defpolybox\] defined an analogue of a maximally entangled state $\phi^{AB}$. In the matrix representation of , $\phi^{AB}$ is an inner product state if and only if the matrix is symmetric and positive semi-definite. Hence $\phi^{AB}$ is an inner product state for odd $n$, whereas for even $n$, $\phi^{AB}$ is not an inner product state.
\[quantumexample\] The quantum case is slightly subtle. Given two qubits, the maximally entangled state $$\Phi^+ = {\left|\Phi^+\right\rangle}{\left\langle\Phi^+\right|},\qquad {\left|\Phi^+\right\rangle} = \frac{1}{\sqrt{2}} \left( {\left|00\right\rangle} + {\left|11\right\rangle} \right)$$ is symmetric but is not an inner product state, since if $\sigma_y$ is a Pauli spin matrix, then $(\sigma_y\otimes \sigma_y)(\Phi^+) = -1$. Consider the operator defined by $\tilde{\Phi} = ({\mathds{1}}\otimes T)(\Phi^+)$, where $T$ is the linear map that takes an operator in $V^B$ to its transpose with respect to the computational basis. The new operator $\tilde{\Phi}$ is not a valid quantum state. It is locally positive but not globally positive, hence is not a density operator. But it is in the maximal tensor product of two qubits, and it is an inner product state. In fact, $\tilde{\Phi}$ predicts perfect correlation whenever Alice and Bob perform measurements in the same direction. However, the two states are equivalent in terms of the non-local correlations they can produce (as was first shown in Ref. [@Beigi09]).
Theorem \[corr:identitystatecorrelationsareinq1\] below establishes a constraint on the nonlocal correlations that can be obtained from measurements on an inner product state. It may seem as if the definition of an inner product state is quite restrictive, given that an inner product state must be symmetric, for example, and given that the maximally entangled state $\Phi_+$ of two qubits is not included. This would diminish the interest of the theorem. However, suppose that a bipartite state $\omega^{AB}$ can be obtained from an inner product state via a transformation of one of its subsystems. Then any correlations obtained from $\omega^{AB}$ could also be obtained from an inner product state. Hence any restriction on the correlations from inner product states also applies to $\omega^{AB}$. Formally,
\[equivstatesweaker\] Consider a joint state $\omega^{AB}$, which can be written in the form $\omega^{AB} = ({\mathds{1}}\otimes \tau)(\sigma^{AB})$, for some $\tau: V_+\rightarrow V_+$ that takes normalized states to normalized states. Any correlations obtained from measurements on $\omega^{AB}$ can also be obtained from measurements on $\sigma^{AB}$.
Define the adjoint map $\tau^{\dagger}: V_+^*\rightarrow V_+^*$ such that for any effect $e\in V_+^*$ and any state $\omega\in V_+$, $$(\tau^\dagger(e))(\omega) = e(\tau(\omega)).$$ Since $\tau$ takes normalized states to normalized states, $\tau^{\dagger}(u) = u$. Given a measurement $y$ on system $B$, with outcomes $\{f_1,\ldots, f_r\}$, let $y'$ be the measurement with outcomes $\{\tau^{\dagger}(f_1), \ldots, \tau^{\dagger}(f_r)\}$. Note that from $f_1+\cdots+ f_r = u$, and $\tau^{\dagger}(u) = u$, it follows that $\tau^{\dagger}(f_1) + \cdots + \tau^{\dagger}(f_r) = u$, as must be the case for $y'$ to be a valid measurement. Then measurements $x$ and $y$ on $\omega^{AB}$ have the same joint outcome probabilities as measurements $x$ and $y'$ on $\sigma^{AB}$. Hence, if a particular set of correlations can be obtained by performing measurements on $\omega^{AB}$, those same correlations can be obtained by performing different measurements on $\sigma^{AB}$.
Further,
\[quantumstatesareinnerprod\] Given two $d$-dimensional quantum systems, any pure state $\rho^{AB}={\left|\psi\right\rangle}{\left\langle\psi\right|}$ can be written in the form $\rho^{AB} = ({\mathds{1}}\otimes \tau)(\tilde{\rho}^{AB})$, where $\tau: V_+\rightarrow V_+$ takes normalized states to normalized states, and $\tilde{\rho}^{AB}$ is an inner product state.
Using the Schmidt decomposition, every pure quantum state ${\left|\psi\right\rangle}$ can be written in the form: $$\label{eq:schmidtdecomposition}
{\left|\psi\right\rangle} = \sum_{i=1}^r \lambda_i {\left|a_i\right\rangle} \otimes {\left|b_i\right\rangle},$$ where $r$ is the Schmidt rank, $\{{\left|a_i\right\rangle}\}$ and $\{{\left|b_i\right\rangle}\}$ are orthonormal bases and the $\lambda_i$ are real and positive. A unitary transformation $U$, on system $B$, which maps $\{{\left|b_i\right\rangle}\}$ to $\{{\left|a_i\right\rangle}\}$ gives $${\left|\psi'\right\rangle} = \sum_{i=1}^r \lambda_i {\left|a_i\right\rangle} \otimes {\left|a_i\right\rangle}.$$ Now let $$\tilde{\rho}^{AB} = ({\mathds{1}}\otimes T)({\left|\psi'\right\rangle}{\left\langle\psi'\right|}),$$ where $T$ is the transpose map, acting on the $B$ system, defined with respect to the basis $\{{\left|a_i\right\rangle}\}$. Note that $\tilde{\rho}^{AB}$ is symmetric since for Hermitian operators $E$ and $F$, $$(E\otimes F)(\tilde{\rho}^{AB}) = \mathrm{Tr}[(E \otimes F)\tilde{\rho}^{AB}] = \sum_{ij} \lambda_i \lambda_j E_{ji} F_{ij}
= (F\otimes E)(\tilde{\rho}^{AB}).$$ Note also that $\tilde{\rho}^{AB}$ is positive semi-definite since for any Hermitian operator $E$, $$(E\otimes E)(\tilde{\rho}^{AB}) = \mathrm{Tr}[(E \otimes E)\tilde{\rho}^{AB}] = \sum_{ij} \lambda_i \lambda_j E_{ji} E_{ij} = \sum_{ij} \lambda_i \lambda_j |E_{ji}|^2 \geq 0.$$ Therefore $\tilde{\rho}^{AB}$ is an inner product state. The quantum state $\rho^{AB}$ can be written $\rho^{AB} = ({\mathds{1}}\otimes \tau)(\tilde{\rho}^{AB})$, where $\tau$ is the transpose map followed by $U^{-1}$, which proves the theorem.
Now any correlations that can be obtained from measurements on a bipartite classical or quantum system, pure or mixed, can also be obtained from measurements on a pure quantum state of two $d$-dimensional systems for some $d$. This follows from the fact that mixed quantum states always have a purification on a larger Hilbert space. Combining this observation with theorems \[equivstatesweaker\] and \[quantumstatesareinnerprod\] gives
\[classicalquantumcorrelationsinnerprod\] Any correlations obtained from measurements on a bipartite, pure or mixed, classical or quantum system could also be obtained from measurements on an inner product state.
Hence as far as correlations go, the fact that we consider only inner product states is not nearly so restrictive as it looks. By extension, the results apply to all classical and quantum bipartite systems.
The set $Q_1$ {#sec:thesetq1}
-------------
The problem of characterizing those correlations which could in principle be produced by performing measurements on quantum systems, and those that cannot, is an interesting one. Tsirelson’s inequality, which limits the possible violation of the CHSH inequality in quantum theory, was the first result in this direction. A great deal of progress is made in Refs. [@NPA; @momentproblem], where the problem is reduced to the following form. A hierarchy of sets $Q_1, Q_2,\ldots$ is defined, such that each $Q_k$ is a proper subset of the set of all possible bipartite non-signalling correlations, and each $Q_k$ is strictly contained in its predecessor. For given correlations $P(a,b|x,y)$, and for each $k$, it is a semi-definite programming problem to determine whether $P(a,b|x,y)$ is contained in $Q_k$. Furthermore, a given set of correlations $P(a,b|x,y)$ can be obtained from measurements on quantum systems if and only if $P(a,b|x,y)$ is contained in $Q_k$ for some $k$. Hence the sets $Q_k$ become smaller as $k$ increases, until in the limit $k\rightarrow\infty$ they converge towards the set $Q$ of quantum correlations.
The set $Q_1$, which is the largest in the hierarchy, is of further significance. In Ref. [@ML] it is shown that correlations in $Q_1$ satisfy a readily comprehensible physical principle called *macroscopic locality*. For a precise description of what this means, see Ref. [@ML], but in a nutshell, the principle states that the coarse-grained statistics of correlation experiments involving a large number of particles should admit a description by a local hidden variable model. In other words, the set of microscopic correlations that satisfy the principle of macroscopic locality are those which are compatible with classical physics in a certain limit in which the number of particle pairs being tested is large, and only coarse-grained statistics, rather than settings and outcomes for every pair, are collected. It is also known that $Q_1$ is closed under *wiring* [@ML; @Allcock], in other words it is not possible to distill correlations in $Q_1$ to correlations outside $Q_1$ by performing measurements on a number of distinct pairs of systems, and locally manipulating the data. Finally, in the specific case of binary measurement choices and outcomes, all correlations in $Q_1$ respect Tsirelson’s bound of $2\sqrt{2}$ for the CHSH scenario. The main theorem below states that correlations from measurements on inner product states are contained in the set $Q_1$.
First, we give a formal definition of $Q_1$. Suppose that Alice and Bob share two systems in a bipartite state, and let Alice choose a measurement $x$ and Bob choose a measurement $y$. Up to now, when we discussed correlations, Alice’s and Bob’s outcomes were labelled $a$ and $b$, and correlations written $P(a,b|x,y)$. For the specific purpose of defining $Q_1$, however, it is more useful to label the measurement outcomes in such a way that outcomes of distinct measurements have different labels. Hence let the index $i$ range over all possible outcomes of all of Alice’s measurement choices. For example, if Alice is choosing from $N$ possible measurements, each of which has $k$ possible outcomes, then $i$ takes values in $\{1,\ldots, kN\}$, with $i=1,\ldots, k$ the outcomes of the $x=1$ measurement, $i=k+1,\ldots, 2k$ the outcomes of the $x=2$ measurement, and so on. Let the same conventions apply to Bob’s outcome, which is denoted $j$. With a slight abuse of notation, let $x(i)$ denote the unique measurement choice of Alice for which $i$ is a possible outcome. Similarly, $y(j)$. Write $P(i,j)$ for the probability of obtaining outcomes $i$ and $j$ when the measurements $x(i)$ and $y(j)$ are performed. Let $P_A(i)$ denote the marginal probability for Alice to obtain outcome $i$ when she performs measurement $x(i)$, and $P_B(j)$ denote the marginal probability for Bob to obtain outcome $j$ when he performs measurement $y(j)$.
\[q1def\] A set of correlations $P(i,j)$ is in $Q_1$ iff there exists a positive semi-definite matrix $\gamma$ of the form $$\label{eq:q1def1}
\gamma =
\begin{pmatrix}
1 & \vec{P}_A^T & \vec{P}_B^T \\
\vec{P}_A & \tilde{Q} & \tilde{P} \\
\vec{P}_B & \tilde{P}^T & \tilde{R}\\
\end{pmatrix},$$ such that
1. $\vec{P}_A$ and $\vec{P}_B$ are the vectors of probabilities $P_A(i)$ and $P_B(j)$,
2. $\tilde{P}$ is a matrix with elements $\tilde{P}_{ij} = P(i,j)$,
3. $\tilde{Q}$ and $\tilde{R}$ are sub-matrices with diagonal elements $\tilde{Q}_{ii} = P_A(i)$ and $\tilde{R}_{jj} = P_B(j)$,
4. $\tilde{Q}_{ii'} = 0$ if $i\ne i'$, $x(i)=x(i')$,
5. $\tilde{R}_{jj'} = 0$ if $j\ne j'$, $y(j)=y(j')$.
In words, the last two conditions state that elements of $\tilde{Q}$ and $\tilde{R}$ corresponding to different outcomes of the *same* measurement must be zero. The remaining off-diagonal elements of $\tilde{Q}$ and $\tilde{R}$ can be chosen freely.
The main theorem {#sec:Q1Poly}
----------------
\[corr:identitystatecorrelationsareinq1\] Consider two similar systems, whose joint state is an inner product state. All correlations that can be obtained from local measurements lie in $Q_1$.
It is sufficient to show that for any set of correlations generated by measurements on an inner product state, there exists a matrix $\gamma$ of the form , which is symmetric, positive semi-definite, and has the feature that entries in the blocks $\tilde{Q}$ and $\tilde{R}$ corresponding to different outcomes of the same measurement are zero.
Consider correlations generated by measurements on an inner product state $\omega^{AB}$. Using the notation introduced in section \[sec:thesetq1\], let $e_i$ be the effect corresponding to Alice’s measurement outcome $i$, and $f_j$ the effect corresponding to Bob’s measurement outcome $j$. Suppose that $i$ ranges from $1,\ldots,n^A$ and $j$ from $1,\ldots,n^B$. Define a vector of effects $g = (u,e_1,\dots,e_{n^A},f_1,\dots,f_{n^B})$, and denote the entries $g_1=u, g_2=e_1,\ldots,g_{1+n^A+n^B}=f_{n^B}$. Define the $(1+n^A+n^B)\times(1+n^A+n^B)$ matrix $\tilde{\gamma}$ such that $\tilde{\gamma}_{kl} = (g_k \otimes g_l)(\omega^{AB})$. From the fact that $\omega^{AB}$ is an inner product state, it follows directly that $\tilde{\gamma}$ is a symmetric and positive semi-definite matrix [@Bhatia].
Now define a matrix $\gamma$ of the form , with $\gamma_{kl} = \tilde{\gamma}_{kl}$ for all $k,l$ except for the following elements of the sub-matrices $\tilde{Q}$ and $\tilde{R}$:
1. $\tilde{Q}_{ii} = P_A(i)$, and $\tilde{R}_{jj} = P_B(j)$.
2. $\tilde{Q}_{ii'} = 0$ if $i\ne i'$, $x(i)=x(i')$,
3. $\tilde{R}_{jj'} = 0$ if $j\ne j'$, $y(j)=y(j')$.
By construction, $\gamma$ satisfies conditions (i)-(v) of Definition \[q1def\], and symmetry of $\gamma$ follows from symmetry of $\tilde{\gamma}$. It remains to show that $\gamma$ is positive semi-definite.
To this end, let $\delta = \gamma - \tilde{\gamma}$ and note that $\delta$ is of the form $$\delta =
\begin{pmatrix} 0 & \cdots & 0 \\
\vdots & \delta_Q & \tilde{0} \\
0 & \tilde{0}^T & \delta_R \\
\end{pmatrix},$$ where $\delta_Q$ is an $n_A\times n_A$ sub-matrix, $\delta_R$ is an $n_B\times n_B$ sub-matrix, and $\tilde{0}$ is the $n_A \times n_B$ matrix with all entries $0$. Since both $\gamma$ and $\tilde{\gamma}$ are symmetric, $\delta$ is also symmetric. We will show that $\delta_Q$ and $\delta_R$ are positive semi-definite. It follows that $\delta$ is positive semi-definite. Since $\gamma = \delta + \tilde{\gamma}$, it follows that $\gamma$ is also positive semi-definite.
Note that $(\delta_Q)_{ii'} = 0$ for $x(i)\ne x(i')$. It follows that $\delta_Q$ is block diagonal, with each block corresponding to a particular measurement choice of Alice. Consider a particular block, corresponding to a measurement with, say, $r$ outcomes. It is of the form $$M = \begin{pmatrix}
e_1\otimes u - e_1 \otimes e_1 & -e_1 \otimes e_2 & \cdots & -e_1\otimes e_r \\
- e_2 \otimes e_1 & e_2 \otimes u - e_2\otimes e_2 & \cdots & -e_2\otimes e_r \\
& & \vdots & \\
-e_r \otimes e_1 & -e_r \otimes e_2 & \cdots & e_r \otimes u - e_r \otimes e_r
\end{pmatrix}
(\omega^{AB}).$$ Using $e_1 + \cdots + e_r = u$, this matrix can be decomposed into a sum of $(r^2-r)/2$ matrices $$ M = \sum_{n=2}^r \sum_{m=1}^{n-1} M^{mn},$$ where all entries of the matrices $M^{mn}$ are $0$, except for $$\begin{aligned}
(M^{mn})_{mm} &= (M^{mn})_{nn} = (e_m\otimes e_n)(\omega^{AB}) \\
(M^{mn})_{mn} &= (M^{mn})_{nm} = -(e_m\otimes e_n)(\omega^{AB}).\end{aligned}$$ Each $M^{mn}$ is manifestly positive semi-definite, hence $M$ is positive semi-definite. Since each block of $\delta_Q$ is positive semi-definite, $\delta_Q$ is also positive semi-definite. A similar argument shows that $\delta_R$ is also positive semi-definite. Therefore $\delta$ and $\gamma$ are positive semi-definite. This concludes the proof.
\[corollary\] Consider two systems, whose joint state is of the form $\omega^{AB} = ({\mathds{1}}\otimes \tau) (\sigma^{AB})$, where $\tau: V_+\rightarrow V_+$ takes normalized states to normalized states and $\sigma^{AB}$ is an inner product state. All correlations obtainable from measurements on $\omega^{AB}$ lie in $Q_1$.
This is immediate from theorem \[corr:identitystatecorrelationsareinq1\] and theorem \[equivstatesweaker\].
Theorem \[classicalquantumcorrelationsinnerprod\] then implies that all correlations from bipartite classical and quantum states lie in $Q_1$. This was known already of course from Refs. [@NPA; @momentproblem]. One could view the theorem and corollary as an independent proof of this fact.
Polygons revisited {#polygonsrevisited}
==================
It has already been observed that given two $n$-vertex polygon systems, the maximally entangled state $\phi^{AB}$, defined in section \[sec:defpolybox\], is an inner product state if and only if $n$ is odd. Theorem \[corr:identitystatecorrelationsareinq1\] states that correlations obtained from measurements on an inner product state lie in the set $Q_1$, which means in particular that they respect Tsirelson’s bound for the CHSH inequailty. This explains why Tsirelson’s bound is satisfied by the odd $n$ polygon systems, and is consistent with violation of Tsirelson’s bound by the even $n$ polygon systems.
This section relates these observations to simple geometrical properties of the state spaces of polygon systems. A quick glance at figures \[model\] and \[fig:poly3d\] reveals an obvious difference between the odd $n$ and even $n$ cases. For odd $n$, the effect cone $V_+^*$ coincides with the state cone $V_+$. For even $n$ on the other hand, the effect cone is isomorphic to the state cone, but is rotated by some non-zero angle. This simple observation lies at the heart of why it is only the maximally entangled states of odd $n$ polygon systems that are inner product states, and hence why it is only these that must satisfy Tsirelson’s bound.
The fundamental difference between the odd $n$ and even $n$ state spaces can be stated more formally as follows. First
\[def:weaklyselfdual\] A system is *weakly self-dual* iff the state and effect cones are isomorphic.
All of the polygon state spaces are weakly self-dual. The isomorphisms are simply the rotations and improper rotations around the $z$ axis by $(1 + 2k) \pi/n,\ k\in\{0,\dots,n-1\}$ if $n$ is even and by $2k \pi/n,\ k\in\{0,\dots,n-1\}$ if $n$ is odd.
The odd $n$ polygon state spaces, on the other hand, satisfy a stronger condition, whereby there are additional restrictions on the isomorphism connecting $V_+^*$ and $V_+$.
\[def:stronglyselfdual\] A system is *strongly self-dual* iff there exists an isomorphism $T: V^*_+ \to V_+$ which is symmetric and positive semi-definite, i.e., $f[T(e)] = e[T(f)]$ for all $e,f \in V^*$, and $e[T(e)] \geq 0$ for all $e \in V^*$.
Given the representation of sections \[polygonsystems\] and \[sec:defpolybox\], the identity map is an example of such an isomorphism. The odd $n$ polygon state spaces are strongly self-dual, but the even $n$ are not.
The concepts of strong and weak self-duality have appeared earlier in the literature, for example in Ref. [@teleport]. Weak self-duality is intimately related to the operational tasks of probabilistic remote state preparation (steering) and teleportation [@steering; @teleport].
Now we can relate these properties of individual systems to the bipartite maximally entangled state $\phi^{AB}$. Notice that given two similar systems, any isomorphism $T: V^*_+ \to V_+$ corresponds to a bipartite state $\omega_T^{AB}$ via $$(e \otimes f)(\omega_T^{AB}) = \frac{f[T(e)]}{u[T(u)]} .$$ The state defined is normalized by construction and is locally positive since $0 \leq f[T(e)]/u[T(u)] \leq 1$ for all $e,f\in E(\Omega)$. Intuitively, $\omega_T^{AB}$ is defined so that if Alice performs a measurement and obtains outcome $e$, then Bob’s unnormalized collapsed state, conditioned on that outcome, is $T(e)$.
In the special case that the individual systems are strongly self-dual and the isomorphism $T$ has the additional properties required by definition \[def:stronglyselfdual\], then the induced state $\omega_T^{AB}$ is symmetric and positive semi-definite, hence it is an inner product state. This is the case for the maximally entangled state $\phi^{AB}$ of odd $n$ polygon systems, defined in , where $\phi^{AB}$ corresponds to a map $T$ which is simply the identity map. It follows that for odd $n$, correlations from $\phi^{AB}$ lie in $Q_1$.
In the case that individual systems are weakly but not strongly self-dual, the maximally entangled state corresponds to an isomorphism $T$, but there is no such $T$ with the additional properties of symmetry and positive semi-definiteness, hence the maximally entangled state is not an inner product state. This is the case for the maximally entangled state $\phi^{AB}$ of the even $n$ polygon systems, defined in , where $\phi^{AB}$ corresponds to a map $T$ which is a rotation in $\mathbb{R}^3$ by $\pi/n$. This is why for even $n$, correlations from $\phi^{AB}$ need not lie in $Q_1$.
Correlations outside of $Q_1$ {#sec:Correlationsoutsideofq1}
=============================
Correlations obtained from the maximally entangled state of two odd $n$ polygon systems must be contained in $Q_1$, and this has been seen to be related to the fact that the individual systems are strongly self-dual. It is natural to ask whether the correlations obtained from *any* joint state of strongly self-dual subsystems must also lie in $Q_1$. An explicit counterexample shows that this is not the case.
Consider a strongly self-dual system with normalized extremal states $$\begin{aligned}
\omega_1&=(1, 0, 1)^T & \omega_2&=(0, 1, 1)^T & \omega_3&=(-1, 0, 1)^T\\
\omega_4&=(-1, -1, 1)^T & \omega_5&=(1, -1, 1)^T ,\end{aligned}$$ and normalized ray extremal effects $$\begin{aligned}
e_1 &= \frac{1}{2} (1, 0, 1)^T & e_2 &= \frac{1}{2} (0, 1, 1)^T & e_3 &= \frac{1}{2} (-1, 0, 1)^T\\
e_4 &= \frac{1}{3} (-1, -1, 1)^T & e_5 &= \frac{1}{3} (1, -1, 1)^T & u &= (0,0,1)^T .\end{aligned}$$ The state space for this system looks something like a house and is depicted in .
We have explicitly calculated all extremal states in the maximal tensor product of two such systems. One of these joint states can be written as $$\left(\begin{array}{ccc} -1 & -\frac{1}{4} & -\frac{1}{2} \\ \frac{1}{4} & -\frac{1}{2} & -\frac{1}{4} \\ \frac{1}{2} & -\frac{1}{4} & 1\end{array}\right) ,$$ where we have used the same representation as a $3\times 3$ matrix that was introduced in section \[sec:defpolybox\]. This state is extremal in the maximal tensor product, but is not an inner product state. With a suitable choice of measurements, correlations can be produced which violate Uffink’s quadratic inequality [@Uff02] $$\label{eq:uffink}
(E_{0,0}+E_{1,0})^2 + (E_{0,1}-E_{1,1})^2 \leq 4.$$ In particular the measurement choices $$\begin{aligned}
x = 0:& \{e_5, u-e_5\}, & x=1:& \{e_3, u-e_3\}, & y=0:& \{e_2, u-e_2\}, & y=1:& \{e_3, u-e_3\}\end{aligned}$$ give $$(E_{0,0}+E_{1,0})^2 + (E_{0,1}-E_{1,1})^2 = \frac{17}{4} > 4 .$$ However, satisfaction of Uffink’s inequality is known to be a necessary condition for membership of $Q_1$ [@QMbound]; hence these correlations cannot lie in $Q_1$.
Although these correlations violate Uffink’s inequality and lie outside of $Q_1$, they do not violate Tsirelson’s bound for the CHSH inequality. In fact, we have not been able to find a joint state of two strongly self-dual subsystems that violates the CHSH inequality beyond Tsirelson’s bound. This leads us to conjecture that Tsirelson’s bound holds for every theory with strongly self-dual subsystems.
Discussion
==========
One way of viewing the difference between classical and quantum systems is that the structure, or shape, of the space of possible states of a system is different. For example in the case of a classical trit, the state space is the space of probability distributions over trit values, which is geometrically a triangle. In the case of a qubit, the state space is the Bloch ball. This work considers a very general setting in which a whole range of probabilistic models can be defined, with the classical and quantum theories as special cases. There is little constraint on the state space, except that it is assumed to be convex, and joint systems are assumed to satisfy a no-signalling principle and a principle of local tomography. The aim is to investigate the nonlocal correlations that can be produced by measurements on entangled systems in these models, and to compare and contrast with the classical and quantum cases.
The main theorem, with its corollary, states that correlations from a broad class of bipartite states in probabilistic theories cannot be arbitrarily nonlocal — they are constrained to obey the principle of *macroscopic locality*, or equivalently to lie within the set $Q_1$, which means in particular that they satisfy Tsirelson’s bound for violation of the CHSH inequality. This theorem extends to all bipartite quantum states, which explains why quantum mechanics cannot violate macroscopic locality or Tsirelson’s bound.
The work has also revealed an intimate and intricate relationship between the shape of the state space for an individual system, and the strength of the nonlocal correlations that can be obtained from two systems in an entangled state. This is illustrated by a family of models, in each of which the state space for a single system is a regular polygon with $n$ vertices. Given two such systems, there is an analogue of a maximally entangled state. It turns out that the strength of nonlocal correlations generated by this state depends dramatically on the parity of the number of vertices $n$ of the local polygon. If $n$ is even, maximally nonlocal correlations can be generated, including those that violate macroscopic locality. If $n$ is odd, however, the maximally entangled state respects macroscopic locality. This is in turn explained by the fact that odd $n$ polygons have a geometric property known as strong self-duality, while even $n$ polygons do not.
It would be natural to think that *all* bipartite states of strongly self-dual subsystems would respect macroscopic locality, but the house-shaped counterexample shows that this is not the case. An interesting open question, therefore, is the following: What additional property of local state spaces would ensure that all bipartite states give correlations which respect macroscopic locality? One suggestion is the constraint that for any ray extremal effect, there is a unique state on which this effect will occur with certainty. This property is very attractive from a physical point of view. It allows a natural definition of the post-measurement states of these effects, such that repeating a measurement reproduces the same outcome. This extra constraint is indeed not satisfied by the house model, since the effect $e_1$ occurs with certainty for both states $\omega_1$ and $\omega_5$, but it is satisfied by odd $n$ polygon models. Another possibility that seems to be plausible is that strong self-duality together with the property that all extremal states of the local systems can be transformed into one another reversibly might limit the set of possible correlations to the ones compatible with macroscopic locality.
Finally, it is worth emphasizing that two theories which have almost identical local state spaces can lead to dramatically different nonlocal correlations. In particular, given any finite level of accuracy, it is always possible to find a polygon model with an even and sufficiently large number of vertices $n$, which is locally indistinguishable from the quantum-like case, where the state space is a disc. Nevertheless, while quantum correlations are restricted, any non-signalling correlations can be distilled in the former model by using multiple copies of the maximally entangled state.
We thank Andreas Winter, Volkher Scholz, Markus Müller and Cyril Branciard for insightful discussions. JB is supported by an EPSRC Career Acceleration Fellowship. We acknowledge financial support from the German National Academic Foundation. NB is supported by the UK EPSRC.
Optimal CHSH value {#app:optimalchshsvalue}
==================
In the main text, we gave expressions for the maximal CHSH value returned by measurements on a maximally entangled state of two $n$-vertex polygon systems. The expression for even $n$ is given in , and for odd $n$, in . The choice of angles that maximize these quantities is not unique. We will see below that we have to take two different sets of optimal angles into account.
[ccccc]{} & $\alpha^*_0$ & $\alpha^*_1$ & $\beta^*_0$ & $\beta^*_1$\
Set $1$ & $0$ & $\frac{\pi}{2}$ & $\frac{\pi}{4}$ & $-\frac{\pi}{4}$\
Set $2$ & $0$ & $\frac{\pi}{2}$ & $-\frac{3\,\pi}{4}$ & $\frac{3\,\pi}{4}$\
Note that the optimization has been performed without any restriction on the values of the angles $\alpha^*_x$ and $\beta^*_y$. However, due to the polygon structure of our model, only specific angles, corresponding to extremal effects, are admissible. Thus the optimal CHSH values are obtained by taking the extremal effects which are closest to the optimal angles.
[ccccc]{} $x$ & $\Delta\alpha_1$ & $\Delta\beta_0$ & $\Delta\beta_1$ & $S$\
$0$
------------------------------------------------------------------------
& $0$ & $\frac{\pi}{n}$ & $\frac{\pi}{n}$ & $2 \sqrt{2}$\
$1$ & $\frac{-\pi}{2 n}$ & $\frac{-\pi}{4 n}$ & $\frac{\pi}{4 n}$ & $\frac{2}{\left(1+\sec\left(\frac{\pi}{n}\right)\right)^2} \, \left[ 1 + \sec\left(\frac{\pi}{n}\right) \left( 2 \cos\left(\frac{n+3}{4 n} \, \pi\right)+6 \sin\left(\frac{n+1}{4 n} \, \pi\right) + \sec\left(\frac{\pi}{n}\right) - 2 \right)\right]$\
$2$ & $\frac{\pi}{n}$ & $\frac{\pi}{2 n}$ & $\frac{-\pi}{2 n}$ & $\sec\left(\frac{\pi}{n}\right) \, \left[ 3 \cos(\frac{n+2}{4 n} \, \pi) + \sin\left(\frac{n+6}{4n} \, \pi\right) \right]$\
$3$ & $\frac{\pi}{2 n}$ & $\frac{\pi}{4 n}$ & $\frac{-\pi}{4 n}$ & $\frac{-2}{\left(1+\sec(\frac{\pi}{n})\right)^2} \left[1 - \sec\left(\frac{\pi}{n}\right) \left(6 \cos\left(\frac{n+1}{4 n} \, \pi\right)+ 2 \sin\left(\frac{n+3}{4 n} \, \pi\right)- \sec\left(\frac{\pi}{n}\right)\right)\right]$\
$4$ & $0$ & $0$ & $0$ & $2 \sqrt{2} \, \sec(\frac{\pi}{n})$\
$5$ & $\frac{-\pi}{2 n}$ & $\frac{-\pi}{4 n}$ & $\frac{\pi}{4 n}$ & $\frac{-2}{\left(1+\sec(\frac{\pi}{n})\right)^2} \left[1 - \sec\left(\frac{\pi}{n}\right) \left(6 \sin\left(\frac{n+1}{4 n} \, \pi\right)+ 2 \cos\left(\frac{n+3}{4 n} \, \pi\right)- \sec\left(\frac{\pi}{n}\right)\right)\right]$\
$6$ & $\frac{\pi}{n}$ & $\frac{-\pi}{2 n}$ & $\frac{\pi}{2 n}$ & $\sec(\frac{\pi}{n}) \, \left[ \cos\left(\frac{n+6}{4 n} \, \pi\right) + 3 \sin\left(\frac{n+2}{4n} \, \pi\right) \right]$\
$7$ & $\frac{\pi}{2 n}$ & $\frac{\pi}{4 n}$ & $\frac{-\pi}{4 n}$ & $\frac{2}{\left(1+\sec\left(\frac{\pi}{n}\right)\right)^2} \, \left[ 1 + \sec\left(\frac{\pi}{n}\right) \left( 2 \sin\left(\frac{n+3}{4 n} \, \pi\right)+6 \cos\left(\frac{n+1}{4 n} \, \pi\right) + \sec\left(\frac{\pi}{n}\right) - 2 \right)\right]$\
The deviation from the optimal angles will be called $\Delta\alpha_0, \Delta\alpha_1, \Delta\beta_0, \Delta\beta_1$. Without loss of generality we set $\Delta\alpha_0$ to $0$. A detailed analysis reveals a total of eight classes of deviation angles characterized by the remainder $x = n \mod 8$ of the division of $n$ by $8$. For a free choice of angles both sets in lead to the same maximum value of the CHSH-coefficient. Whether the available extremal effects are closer to the angles of set 1 or set 2, however, depends on the number of vertices. It turns out that for even $n$ as well as for $x \in \{1,7\}$ this is the case for set $1$, whereas for $x \in \{3,5\}$ the smallest derivation can be achieved to set $2$. The maximal CHSH value for each polygon system is given by the following parameters for and : $$\begin{aligned}
\beta_y &= \beta^*_y + \Delta\beta_y\\
\alpha_x &= \alpha^*_x + \Delta\alpha_x\end{aligned}$$
The eight classes can clearly be seen in . The analytic expressions for the maximal CHSH value as a function of the number of vertices $n$ and the remainder $x$ are given in .
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|
---
author:
- Simon Renard
- Thomas Schwager
- Thorsten Pöschel
- Clara Salueña
date: 'Received: / Revised version: '
title: 'Vertically shaken column of spheres. Onset of fluidization'
---
Introduction
============
Granular material confined in a vertically vibrating container reveals complex effects such as surface structure formation (e.g. [@Coulomb:1773; @MeloUmbanhowarSwinney:1994]), spontaneous heap formation (e.g. [@DinkelackerHueblerLuescher:1987]), convection (e.g. [@EhrichsJaegerKarczmarKnightKupermanNagel:1994]) and others. A precondition common to all these effects is that (at least) the particles at the free surface of the granular material lose contact to their neighbours for at least a small part of the oscillation period. If a granular material is agitated in a way so that the particles at the surface separate from their neighbours we will call the material fluidized.
The conditions under which surface fluidization occurs are not clear yet. For sinusoidal vertical excitation of the container most of the literature reports that surface fluidization starts as soon as the acceleration amplitude $A\omega^2$ of the oscillation $z=A\cos\omega
t$ exceeds gravity $g$ or another critical constant. The Froude number is defined as $\Gamma=A\omega^2/g$ and, hence, it has been reported in numerous publications that surface fluidization or, respectively, effects which require fluidization, occur for $\Gamma>1$.
Few references [@BarkerMehta:1993a] report, however, that in numerical simulations surface fluidization was observed for $\Gamma\lesssim 1$. So far these results have not been confirmed experimentally.
Whereas a rigid solid body, e.g. a single sphere on a oscillating surface, would certainly start to jump if the acceleration amplitude $A\omega^2$ exceeds gravity $g$, there are arguments which may lead to a different conclusion for a shaken amount of granular material, i.e., a many body system:
- The deformation-force-law of contacting bodies may be nonlinear due to geometrical effects, even if the material deformation is small enough to assume linear material properties, i.e., Hooke’s law. For contacting ideally elastic spheres it has been derived by Hertz [@Hertz:1882] that the interaction force $F$ scales with the deformation $\xi$ as $F\sim \xi^{3/2}$. This law applies for all particle contacts (under mild conditions) provided the deformation $\xi$ is small enough [@Brilliantov]. As a result, nonlinear phenomena like the propagation of solitary waves has been shown to appear in bead chains with Hertz contact law [@CosteFalconFauve:1994PRE].
- Under the influence of gravity the particles are deformed differently even at rest, depending on their vertical position in the material. Therefore, the speed of sound in the material is not uniform but a function of the vertical coordinate. This property may lead to complicated pulse motion through the material and has been reported also for chains of contacting spheres [@SinkovitsSen:1995].
- In a polydisperse granular material the geometrical properties, i.e., the contact network, varies with the applied force. If the material is loaded the particles are deformed and more and more contacts emerge. This effect leads to a more complicated deformation – force law, $F\sim\xi^\gamma$ with $1.5 \le \gamma \le 4$ [@HerrmannStaufferRoux:1987a].
In a recent theoretical paper [@BelowG] the granular material was modeled as a vertically shaken column of viscoelastic spheres. The main result of [@BelowG] is that the column may be fluidized even if the condition $\Gamma>1$ does not hold. Instead a different condition for fluidization was derived: Assuming viscoelastic material properties the force between two contacting spheres of radius $R$ at vertical positions $z_k$ and $z_{k+1}$ reads [@BSHP] $$\label{eq:BSHP}
F_{k,k+1} = -\sqrt{R} \left(\mu\xi_{k,k+1}^{3/2}+\alpha\dot{\xi}_{k,k+1}
\sqrt{\xi_{k,k+1}}\right)\,,$$ where $\xi_{k,k+1}\equiv 2R-\left|z_k-z_{k+1}\right|$ is the compression and $\mu$ and $\alpha$ are elastic and dissipative material constants (details see [@BSHP]). The height of the column of $N$ spheres is $L=2NR$. It was found that the top sphere separates from its neighbour if $$\label{eq:flui}
\frac{A\omega^2}{g}>\Gamma\left(\frac35\right)\left|M^{2/5}
\,J_{-2/5}\left(2M\right)\right|\,,$$ where $J_{-2/5}$ is the (complex) Bessel function and $M$ abbreviates $$M\equiv\frac{(18\pi)^{1/3}}{5}\left(\frac{\mu L^5\rho^2}
{g}\right)^{1/6}\frac{\omega}{\sqrt{3\mu-2i\omega\alpha}}\,,$$ with $i=\sqrt{-1}$ and $g$ and $\rho$ being gravity and material density.
For small frequency $\omega$ the Taylor expansion of the rhs. of the condition (\[eq:flui\]) yields $$\frac{A\omega^2}{g}>1-B_2\omega^2+B_4\omega^4
\label{eq.flui.small}$$ with $$\begin{aligned}
B_2&\equiv&\frac{18^\frac{2}{3}}{45}
\left(\frac{\pi^2\rho^2L^5}{g\mu^2}\right)^\frac{1}{3}\\
B_4&\equiv&\frac{1}{18^\frac{2}{3}}\!
\left(\frac{\pi^2\rho^2L^5}{g\mu^2}\right)^\frac{2}{3}\!\!\!
\left(\!\frac{3}{100}+\frac{4\cdot 324^\frac{2}{3}}{405}
\frac{\alpha^2}{\pi^2} \! \left(\frac{g \pi^4}{\mu^4 \rho^2 L^5}
\right)^\frac{1}{3}\!\right)\end{aligned}$$ Both coefficients $B_2$ and $B_4$ are positive definite values, i.e., for [*all*]{} materials there is a range of frequency where the rhs. of the inequality (\[eq:flui\]) is smaller than one. This means that there exists always a frequency interval where the column fluidizes although the acceleration amplitude of shaking is smaller than gravity, $A\omega^2/g <1$. The full derivation of the sketched theory and the discussion of the frequency range where the condition (\[eq:flui\]) holds can be found in Ref. [@BelowG]. For our experimental system consisting of steel spheres, of density $\rho$=7700 Kg/m$^3$, Young modulus $Y$=210 GPa, Poisson ratio $\nu=0.29$, in a column of $L=$0.6 m, the Taylor expansion Eq. (\[eq.flui.small\]) holds for frequencies $\omega$ much smaller than 1000 s$^{-1}$, which covers entirely the experimentally accessible range of frequencies.
In the present paper we want to report experimental results on the onset of fluidization of a vertically shaken column of spheres, i.e., on the same system which was studied theoretically in Ref. [@BelowG]. The dynamical behaviour of a vibrated one dimensional system of spheres in the well fluidized regime was studied e.g. in [@BernuDelyonMazighi:1994; @x1]. This regime is explicitely not considered here. In [@x] a column of particles which interact as linear springs with linear damping, $\ddot{\xi}+2\mu\dot{\xi}+\Omega^2\xi=0$, was studied with Molecular Dynamics. For their sets of parameters the authors found that fluidization occurs above $\Gamma\approx 1.1$, however, the authors claim that the onset of fluidization depends on the choice of the parameters $\Omega$ and $\mu$. Within their model it might be possible to get also fluidization for $\Gamma<1$, depending on the material parameters. If one considers spheres, however, in the limit of vanishing damping the Hertz law implies that the duration of contact behaves as $t_c\sim g^{-1/5}$, whereas for the model considered in [@x] one finds a constant $t_c=\mbox{const.}=\pi/\Omega$. Hence, the linear spring model certainly does not describe the contact of spheres, not even in the case of pure elastic contact. For the same linear spring system in [@xx] the detachment effect was reported, the same effect for viscoelastic spheres was discussed in [@xxx].
Experimental setup {#sec:setup}
==================
We expect that the effect to be measured is quite small, i.e., for the critical parameters of shaking when the material starts to fluidize we expect $1-A\omega^2/g$ to be a small value. Therefore, we have to adjust the amplitude of shaking $A$ and in particular the frequency $\omega$ with good accuracy. Moreover, we have to assure that no energy originating from different frequencies than the driving frequency enters the column of spheres, e.g., due to high frequency noise.
The column of spheres was confined in a framework of three aluminum bars, which assures pure one dimensional motion of the spheres at low friction, see Fig. \[fig:frame\].
The framework with the spheres was mounted on a precisely vertically balanced linear bearing which was driven by a stepping motor via a crankshaft with adjustable eccentricity, Fig. \[fig:pleuel\].
The motor was computer controlled, i.e., in precisely 20,000 impulses the motor axis revolves once. This high angular resolution provides quasi-steady motion. The motion was smoothened additionally using a flywheel (with diameter 160 mm, thickness 32 mm, mass 4.5 kg and moment of inertia $J=0.0144 ~\mbox{kg\,m}^2$) which was fixed on the axis and a hard rubber clutch connecting the motor axis and the axis of the shaking device (see Fig. \[fig:gesamtanlage\_vorn\]). Using an acceleration sensor we checked that the finite step size per computer signal does not influence the sinusoidal motion of the column of spheres. The entire mechanical device was mounted on a massive oscillation damping table. Figure \[fig:gesamtanlage\_vorn\] sketches the device and Fig. \[foto9\] shows a photograph.
In the experiments the column consists of up to 20 ball bearing balls of radius $R=12.5$ mm. There are several possibilities to determine the onset of fluidization, i.e., for fixed amplitude $A$ the critical frequency $\omega$ when the top sphere separates from its neighbour:
- Direct observation: a comoving camera was mounted to the probe carrier and a LASER pointer was fixed at the opposite side of the column. As soon as fluidization starts at the critical frequency the emerging submillimeter gap between the top particle and its neighbour can be recognized as a flash on the screen.
- Acoustic method: When fluidization occurs there emerges a periodic acoustic signal originating from the collision of the top particles. This signal can be detected either using a microphone or the acceleration sensor which was mounted on the top of the device, see Fig. \[fig:gesamtanlage\_vorn\].
- Electric circuit: We attached a very thin wire (diameter $<0.1$mm) to the top particle and its neighbour and observed the contact of the spheres using an oscilloscope.
We checked all three methods and found that the results are very similar. We have chosen the electric method since it turned out to be the most simple and most reliable one. The direct observation requires complicated mechanical tuning and for the acoustic method one has to filter the periodic signal from other sources of sound as, e.g., the sound of the motor.
Raw Data
========
For fixed amplitude $A$ we want to determine the according smallest frequency $\omega$ for which the top sphere separates from its neighbour in each period. Figure \[fig:cond\] shows a sketch of the oscilloscope signal for subcritical (top), critical (middle) and over-critical frequency, all for the same amplitude of shaking. For subcritical frequency the top particle follows the motion of the vibrating table resulting in a sinusoidal signal of the acceleration sensor and a horizontal line for the conductivity (limited by a resistor). At over-critical frequency the conductivity drops to zero for a certain time interval in each period. Close to the critical frequency the electrical contact between the spheres is not in each period interrupted, instead we get a noisy signal.
To collect data points we tuned the adjustable crankshaft to a desired amplitude $A$. Then the motor was accelerated up to a rotation speed which is slightly less than the speed where we expect fluidization (initial ramp). At this value all spheres are permanently in contact (top in Fig. \[fig:cond\]). From this point on we increased the motor speed in very small steps until we observe separation of the top sphere in each period (bottom in Fig. \[fig:cond\]). This procedure was repeated 10 times for all amplitudes independently for 2 and 20 spheres. The critical frequency for each amplitude was determined as the average over the 10 independent measurements. Figure \[fig:acc\_ampl\_2\_20\] shows the raw data points with error bars.
The somewhat unusual representation of the data as acceleration vs. amplitude is due to the fact that our experimental input is the amplitude, the output, as stated above, is the according critical frequency and hence the acceleration.
In Figure \[fig:acc\_freq\_all\] the critical acceleration of the container ($A\omega^2$) vs. frequency $\omega$ for 2 and 20 beads is shown. One can see that even for two beads the critical acceleration, which is supposed to be a constant varies significantly with the frequency, i.e., it decreases with increasing frequency. The curve for 20 beads shows a similar behaviour, however, the critical acceleration for 20 beads is smaller than the critical acceleration for 2 beads. This is a first indication to an amplification effect similar as theoretically predicted. The experimental device reveals its own resonances, namely 7.6 Hz for two beads and 11.8 Hz for twenty beads. These resonances are caused by the mechanical properties of our experimental setup but not by the properties of the columns of beads themselves. Data points which belong to these frequencies have been eliminated for further analysis (see Fig. \[fig:acc\_ampl\_2\_20\]).
The data points are affected by large statistical errors, i.e., the data points scatter around a smooth curve which one expects to find if only systematic effects were relevant. Therefore, we have to apply a sophisticated data analysis to extract the amplification effect in order to compare it with the theoretical prediction. This procedure is described in the following sections.
Discussion of the Measuring Procedure
=====================================
The measurement is affected by different systematic and random errors:
- Due to the generation of the shaking motion via a connecting rod of finite length ($l=18$ cm) there is a systematic deviation of the motion from the ideal sinusoidal shaking. Instead of a sinusoidal oscillation the generated oscillation is $$z=A \cos \phi + \sqrt{l^2-A^2 \sin^2 \phi} -l\,.$$ For small ratios $A/l$ we can write $$z\approx A\cos\phi - \frac{A^2}{2L}\sin^2\phi$$ This leads to a constant shift in the extremal acceleration of the container $$\ddot{z}_{Extr}\approx \pm A\omega^2 - \frac{A^2}{L}\omega^2 =
A\omega^2\left(\pm 1 - \frac{A}{L}\right)$$ We see that the absolute value of the acceleration at the upper turning point is too large by $A/l$ which can be as much as $3\%$.
- Uncertainty in determining the driving frequency and amplitude are very small as described above.
- Due to the noise caused by the motor and the friction of the linear bearings the systematic (ideally sinusoidal) vertical motion of the column is superposed by a spectrum of high frequencies. These waves are source of an undesired extra energy feed which is not controllable and causes errors. Measurements of this noise using the acceleration sensor have shown that it does not depend much on the height of the column, it is determined mainly by the motor and the bearings. This property suggests to measure our effect not directly but to compare the results for a column of 20 steel spheres with the results for a “column” of two spheres (see below).
Data Analysis
=============
To eliminate systematic errors we compare the result for a column of 20 spheres with the equivalent measurement for a “column” of only two spheres. The column of two spheres serves as a reference. Actually, the second bead is fixed to the bottom and oscillates rigidly while the top bead is free. Effectively there is only one free moving sphere, for one single sphere taking off from a flat wall could not serve as a reference for the case when the top bead of a column of $N$ spheres separates from the rest. This approach is based on the assumption that the systematic errors originating from the motor noise and from the bearings affect both systems almost in the same way. With the above mentioned method (see section \[sec:setup\]) we determined the critical frequency of driving at which the topmost bead starts to jump for a given shaking amplitude. From this we calculated the critical acceleration. Due to the influence of systematic errors also the curve $a_{\rm crit}(\omega)$ for two beads, which is supposed to be constant, shows a significant dependence on the driving frequency, resp. amplitude. As already mentioned before, in Fig. \[fig:acc\_freq\_all\] one can see that the critical acceleration decreases with increasing frequency. The curve for 20 spheres shows a similar behaviour but the critical amplitude is always smaller than the critical amplitude for 2 spheres at the same driving frequency. To identify the amplification effect, i.e., to eliminate the error of the apparatus we divide the critical acceleration for 20 spheres by the critical amplitude for 2 spheres at the same frequency which yields the absolute value for the critical Froude number for 20 spheres. This method is applicable if all significant systematic errors affect the measurements by a factor which is independent on the number of beads. This precondition holds certainly for the influence of the limited length of the connecting rod and holds to a reasonable degree for the influence of the higher harmonics in the driving motion due to the discrete steps in the motor motion. Since we believe these errors to be the most significant ones our method should give reliable results.
Following this idea we fit both sets of data, for 2 and 20 spheres, to the function $$\label{eq:templf3}
f(\omega)=\left(a_2\omega^2+a_1\omega+a_0\right)^{-1}$$ and take the ratio of these curves. There are other possibilities to construct fit-functions which reflect the desired properties of monotonous decay and significant curvature, it turns out that the final result does not depend sensitively on the choice of this function.
The ratio curve, which represents the predicted amplification effect, can be described in good approximation as a decreasing parabola with maximum at $\omega=0$. Its curvature is small, in agreement with the expansion given by Eq. (\[eq.flui.small\]). Dissipative effects, represented by the material constant $\alpha$, are not observable at these frequencies, as the restitution is high for all velocities (0.90 or more): we estimated the second term of $B_4$ being of the order of 10$^{-6}$ or less. This curve, multiplied by the reference acceleration 9.81 m/s$^2$, is shown in Fig. \[fig:final\_result\]. The ratio between the fitting curves in Fig. \[fig:limited\_inv\_quadr\_fit\] is not exactly 1 at $\omega=0$, but by about $3\%$ smaller. This discrepancy is due to the above mentioned systematic errors, namely due to the higher harmonics in the motion of the stepping motor (see Fig. \[fig:oszi\]) which can have different magnitude at different load. Therefore the division of the curve for 20 beads by the reference curve alone can not completely remove this systematic error. Other systematic errors such as deviations from an ideal sinusoidal motion due to the finite length of the connecting rod have been removed by the data analysis procedure.
The influence of the higher harmonics can be observed in Fig. \[fig:oszi\] showing the output of the acceleration sensor on top of the column. The sinusoidal motion of the container with driving frequency $10$ Hz is superimposed by an oscillation of frequency of about $150$ Hz. When calculating the maximum acceleration of this motion one gets a value which is up to 20% higher than $A\omega^2$. Thus, by taking $A\omega^2$ as the maximum acceleration of the container we underestimate the actual acceleration, which, since the magnitude of the high frequency oscillation varies with different load, contributes to the difference between the maximum of the ratio curve in Fig. \[fig:final\_result\] and the theoretical value of 9.81m/s$^{2}$.
Discussion
==========
We investigated experimentally the onset of fluidization of a vertically vibrated column of spheres. It was found that one observes fluidization even if the amplitude of acceleration of the vibrating motion is smaller than the acceleration due to gravity, $A\omega^2/g
<1$. This result is in agreement with the theoretical prediction [@BelowG]. The quantitative theoretical result for the critical parameters for the onset of fluidization given by Eqs. (\[eq:flui\]) and (\[eq.flui.small\]) does not contradict the experiment but could also not be conclusively confirmed by the experiment due to insufficiencies of the experimental setup. The most important shortcoming is the existence of high frequency vibrations mainly due to the motor noise and the bearings. While qualitatively agreement is shown, due to these limitations the experimental measurements are not completely conclusive to verify the theoretical results: a quantitative check of the theoretical predictions requires a more sophisticated experimental setup, directed to reduce noise and enhance the effect. The noise can be reduced by improving mechanical isolation: enclosing the column of spheres into a massive block or wall, using very rigid bearings and separating the motor at a good distance from the column. Higher columns or more balls will help to reduce the noise, but one has to work harder on a correct vertical alignment; for this reason the beads should not be very small. The use of a softer material alone would enhance the effect, but then one would like still good restitution, smooth surface and good conductivity (for the electrical method of determination), so the choice of material is not straightforward. Also higher frequency will enhance fluidization at lower Froude numbers, but then controlling amplitude (and noise) will become much more critical.
We thank Hans Scholz for discussion and Christine Rosinska for technical aid.
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|
---
abstract: 'The [<span style="font-variant:small-caps;">Katrin</span>]{} experiment aims to measure the effective mass of the electron antineutrino from the analysis of electron spectra stemming from the -decay of molecular tritium with a sensitivity of . Therefore, a cumulative amount of about of gaseous tritium is circulated daily in a windowless source section. An accurate description of the gas flow through this section is of fundamental importance for the neutrino mass measurement as it significantly influences the generation and transport of -decay electrons through the experimental setup. In this paper we present a comprehensive model consisting of calculations of rarefied gas flow through the different components of the source section ranging from viscous to free molecular flow. By connecting these simulations with a number of experimentally determined operational parameters the gas model can be refreshed regularly according to the measured operating conditions. In this work, measurement and modelling uncertainties are quantified with regard to their implications for the neutrino mass measurement. We find that the magnitude of systematic uncertainties related to the source model is represented by $\left|\Delta m_\upnu^2\right|=\left(3.06\pm 0.24\right)\cdot\SI{e-3}{\electronvolt\squared}/\textrm{c}^4$, and that the gas model is ready to be used in the analysis of upcoming [<span style="font-variant:small-caps;">Katrin</span>]{} data.'
author:
- Laura Kuckert
- Florian Heizmann
- Guido Drexlin
- Ferenc Glück
- Markus Hötzel
- Marco Kleesiek
- Felix Sharipov
- Kathrin Valerius
bibliography:
- 'ms.bib'
title: 'Modelling of gas dynamical properties of the [[<span style="font-variant:small-caps;">Katrin</span>]{}]{} tritium source and implications for the neutrino mass measurement'
---
=1
<span style="font-variant:small-caps;">Keywords:</span> rarefied gas flow, gas dynamics, transitional flow, viscous flow, molecular flow, vacuum, hydrogen, tritium, simulation, direct neutrino mass determination
|
---
author:
- 'L. M. Cairós'
- 'J.N. González-Pérez'
bibliography:
- 'vimos.bib'
date: 'Received ...; accepted ...'
subtitle: The case of the blue compact dwarf Mrk 900
title: 'Understanding star formation and feedback in small galaxies[^1]'
---
[Low-luminosity, active star-forming blue compact galaxies (BCGs) are excellent laboratories for investigating the process of star formation on galactic scales and to probe the interplay between massive stars and the surrounding interstellar (or intergalactic) medium.]{} [We investigate the morphology, structure, and stellar content of BCG Mrk 900, as well as the excitation, ionization conditions, and kinematics of its regions and surrounding ionized gas. ]{} [We obtained integral field observations of Mrk 900 working with the Visible Multi-Object Spectrograph at the Very Large Telescope. The observations were taken in the wavelength range 4150-7400Å covering a field of view of 27$\arcsec \times$ 27$\arcsec$ on the sky with a spatial sampling of $0\farcs$67. From the integral field data we built continuum, emission, and diagnostic line ratio maps and produced velocity and velocity dispersion maps. We also generated the integrated spectrum of the major regions and the nuclear area to determine reliable physical parameters and oxygen abundances. Integral field spectroscopy was complemented with deep broad-band photometry taken at the 2.5 m NOT telescope; the broad-band data, tracing the galaxy up to radius 4 kpc, allowed us to investigate the properties of the low surface brightness underlying stellar host.]{} [We disentangle two different stellar components in Mrk 900: a young population, which resolves into individual stellar clusters with ages $\sim$5.5-6.6 Myr and extends about 1 kpc along the galaxy minor axis, is placed on top of a rather red and regular shaped underlying stellar host, several Gyr old. We find evidence of a substantial amount of dust and an inhomogeneous extinction pattern, with a dust lane crossing the central starburst. Mrk 900 displays overall rotation, although distorted in the central, starburst regions; the dispersion velocity map is highly inhomogeneous, with values increasing up to 60 km s$^{-1}$ at the periphery of the SF regions, where we also find hints of the presence of shocks. Our observational results point to an interaction or merger with a low-mass object or infalling gas as plausible trigger mechanisms for the present starburst event. ]{}
Introduction
============
Blue compact galaxies (BCGs) are low-luminosity (M$_{B}\geq$-18) and small systems (starburst radius $\leq$ 1 kpc) that form stars at unusually high rates (up to 3 M$_\odot$ yr$^{-1}$; [@ThuanMartin1981; @Cairos2001a; @Cairos2001b; @Fanelli1988; @HunterElmegreen2004]). They are also rich in gas (M$_{HI}$=10$^{8}$-10$^{9}$M$_\odot$) and present low metal abundances, as derived from their warm ionized gas (1/40 Z$_\odot\leq$ Z $\leq$1/2 Z$_\odot$; [@Thuan1999; @Salzer2002; @Izotov1999; @Kunth2000]). These characteristics make them excellent targets for investigating the process of star formation (SF) in galaxies:
First, the lack of spiral density waves and strong shear forces (the mechanisms assumed to trigger and maintain SF in spirals; [@Shu1972; @Nelson1977; @Seigar2002]) allow us to investigate the SF process in a relatively simple environment and to search for alternative trigger mechanisms [@Hunter1997]; for instance, there is evidence that feedback from massive stars could be responsible for the ongoing starburst in several BCGs [@Cairos2017a; @Cairos2017b].
Second, the impact of massive stars into the interstellar medium (ISM) of a low-mass galaxy can be dramatic: as the blast-waves created by supernova (SN) explosions propagate, they give rise to huge expanding shells [@McCray1987], which in the absence of density waves and shear forces grow to larger sizes and live longer than in typical spirals. Observations of dwarf star-forming galaxies reveal ionized gas structures stretching up to kiloparcec scales [@Hunter1990; @Marlowe1995; @Bomans1997; @Bomans2007; @Martin1998; @Cairos2001b; @Cairos2015]. In a shallow potential well these expanding shells can break out of the galaxy disk, or even the halo, and can vent the SN enriched material into the ISM and/or intergalactic medium (IGM; [@Dekel1986; @Marlowe1995; @Martin1998; @Martin1999; @MacLow1999]).
Finally, the low metal content and high star formation rate (SFR) of BCGs give us the opportunity to characterize starburst events that are taking place in conditions very similar to those of the early Universe [@Madden2006; @Madden2013; @Lebouteiller2017].
Motivated by the relevance of these topics we initiated a project focused on BCGs and, in particular, on their current starburst episode and the impact on the ISM. To this end we took integral field spectroscopy (IFS) of a sample of forty galaxies. The first results from our analysis are presented in [@Cairos2009a; @Cairos2009b; @Cairos2010; @Cairos2012; @Cairos2015]. A complete understanding of the SF process in low-mass systems and on the complex interaction of massive stars with their environment demand also detailed analyses of individual objects [@Cairos2017a; @Cairos2017b]. Here we focus on the BCG Mrk 900 and combine IFS observations with optical imaging to investigate its recent SF and evolutionary history.
Mrk 900 (NGC 7077) is a relatively luminous BCG (M$_{B}$=-17.08; this work), included in the [@Mazzarella1986] catalog of Markarian galaxies. Surface photometry in the optical and near-infrared (NIR) revealed a blue starburst on top of a redder regular host [@Doublier1997; @Doublier1999; @GildePaz2003; @GildePaz2005; @Micheva2013a; @Janowiecki2014]. [@Cairos2015] investigated the starburst component of Mrk 900 by means of IFS: the galaxy emission-line maps showed that SF occurs in various knots, aligned on a southeast–northwest axis, with the largest region displaced northwest. The distinct morphology in continuum maps, which peaks at the galaxy center are suggestive of different episodes of SF. In addition, holes and filaments in the ionized gas, together with strong low-ionization lines (\[\] $\lambda6300$ and \[\] $\lambda\lambda6717,\,6731$) revealed an important impact of the SF on the surrounding ISM and suggest the presence of shocks. For all these reasons Mrk 900 appears to be an ideal target for a thorough study of the effect of SF on the ISM of a dwarf galaxy.
[lcc]{} Parameter & Data & Reference\
\
Other names & NGC 7077, UGC 11755 &\
RA (J2000) & 21$^h$29$^m$59$\fs$6 &\
DEC (J2000) & 02$\degr$24$\arcmin$51$\arcsec$ &\
V$_{hel}$ & 1152$\pm$5 km s$^{-1}$ &\
Distance & 18.9$\pm$1.3 Mpc &\
Scale & 91 pc arcsec$^{-1}$ &\
D$_{25}$ & 49.9$\pm$0.06 $\arcsec$ (4.54 kpc) & RC3\
A$_{B}$ & 0.211 & SF11\
M$_{B}$ & -17.07 & This work\
M$_{*}$ & 9.5$\times$10$^{8}$M$\odot$& H17\
M$_{HI}$ & 1.55$\times$10$^{8}$M$\odot$& VSS01\
M$_{DYN}$ & 1.64$\times$10$^{9}$M$\odot$& VSS01\
D$_{HI}$/D$_{25}$ & 1.2 & VSS01\
Morphology & SO$^{-}$pec?; BCD & RC3; GP03\
Notes: RA, DEC, heliocentric velocity, distance, scale, and Galactic extinction are from NED (http://nedwww.ipac.caltech.edu/). The distance was calculated using a Hubble constant of 73 km s$^{-1}$ Mpc$^{-1}$, and taking into account the influence of the Virgo Cluster, the Great Attractor, and the Shapley supercluster. HI-to-optical diameter (D$_{HI}$/D$_{25}$) is measured at the 10$^{20}$ atoms cm$^{-2}$ and 25 mag arcsec$^{-2}$ isophotes. References: GP03 = [@GildePaz2003]; H17=[@Hunt2017]; RC3= [@deVaucouleurs1991]; SF11=[@Schlafly2011]; VSS01=[@VanZee2001].
Observations and data processing
================================
We carried out a spectrophotometric analysis of the BCG Mrk 900. The galaxy central starburst was studied by means of IFS, whereas deep broad-band imaging in the optical was used to derive the properties of the underlying host galaxy. Details of the observations are provided Table \[Table:log\].
----------- ------ -------- -------------- ------ -------------
Date Tel. Inst. Grism/Filter Exp. Seeing
(s) ($\arcsec$)
Aug. 2005 NOT ALFOSC B 2700 0.8
Aug. 2005 NOT ALFOSC V 2700 0.6
Aug. 2005 NOT ALFOSC R 2700 0.7
Aug. 2007 VLT VIMOS HR-Blue 4320 0.9-1.4
Aug. 2007 VLT VIMOS HR-Orange 4320 0.5-0.9
----------- ------ -------- -------------- ------ -------------
: Log of the observations
Notes: The columns list, respectively, observation date, telescope and instrument used, grism or filter, total exposure time and seeing. \[Table:log\]
Broad-band imaging
------------------
{width="1.0\linewidth"}
Broad-band observations of Mrk 900 were carried out in August 2005 with the Nordic Optical Telescope (NOT) at Observatorio del Roque de los Muchachos (ORM) in La Palma. The telescope was equipped with ALFOSC (Andalucía Faint Object Spectrograph and Camera), which provided a field of view (FoV) of 6.5$'\times$6.5$'$ and a scale of 0.188$"$ pix$^{-1}$. We collected CCD images through the $B$, $V$, and $R$ filters. We observed under excellent weather conditions: the seeing varied between 0.6 and 0.8 arcsec and the night was photometric.
Image processing was carried out using standard [iraf]{}[^2] procedures. Each image was corrected for bias, using an average bias frame, and was flattened by dividing by a mean twilight flat-field image. The average sky level was estimated by computing the mean value within several boxes surrounding the object, and subtracted out as a constant. The frames were then registered (we took a set of three dithered exposures for each filter) and combined to obtain the final frame, with cosmic ray events removed and bad pixels cleaned out. Flux calibration was performed through the observation of photometric stars from the @Landolt1992 list.
Integral field spectroscopy
---------------------------
Spectrophotometric data of Mrk 900 were collected at the Very Large Telescope (VLT; ESO Paranal Observatory, Chile), working with the *Visible Multi-Object Spectrograph* (VIMOS; [@LeFevre2003]) in its integral field unit (IFU) mode. The observations were done in Visitor Mode in August 2007 with the blue (HR-Blue) and orange (HR-Orange) grisms in high-resolution mode (dispersion of 0.51Åpix$^{-1}$ in the wavelength range of 4150–6200Å, and dispersion of 0.60Åpix$^{-1}$ in the range 5250–7400Å). A field of view (FoV) of 27$\arcsec \times$ 27$\arcsec$ on the sky was mapped with a spatial sampling of $0\farcs$67. The weather conditions were good (see Table \[Table:log\]), and galaxy exposures were taken at airmass 1.12-1.21. The spectrophotometric standard EG 274 was observed for flux calibration.
Data were processed using the ESO VIMOS pipeline (version 2.1.11) via the graphical user interface [Gasgano]{}. The observations and a complete description of the data reduction are presented in [@Cairos2015].
### Emission-line fitting {#linefitting}
The next step in the IFS data process is the measurement of the fluxes in emission lines, required to produce the bidimensional galaxy maps. We computed the fluxes by fitting Gaussians to the line profiles. The fit was performed using the task [fit]{} of [Matlab]{} with the [*Trust-region*]{} algorithm for nonlinear least squares. We ran an automatic procedure, which fits a series of lines for every spaxel, namely, H$\beta$, \[\] $\lambda4959$, \[\] $\lambda5007$, \[\] $\lambda6300$, H$\alpha$, \[\] $\lambda6584$ and \[\] $\lambda\lambda6717,\,6731$. For each line the fitting task provides the flux, the centroid position, the line width and continuum, and the corresponding uncertainties of each parameter; 1 $\sigma$ errors are computed by the task using the inverse factor from QR decomposition of the Jacobian, the degrees of freedom, and the root mean square. After carefully inspecting the data, in particular in regions with constant emission-line fluxes, we found that the errors given by the fitting task seem underestimated. Therefore, we increased the errors of the emission-line fluxes by multiplying by 1.34, a factor that matches the spatial variations of the fluxes.
Line profiles were fitted using a single Gaussian in all but the Balmer lines. Determining accurate fluxes of the Balmer emission lines is not straightforward since these fluxes can be significantly affected by stellar absorption [@McCall1985; @Olofsson1995; @GonzalezDelgado1999b]. To take this effect into account, we applied two different methods depending on the characteristics of the individual profile. If the absorption wings were clearly visible (as was often the case in H$\beta$), we used two Gaussians, one in emission and one in absorption, and derived the fluxes in absorption and emission simultaneously. In the absence of clear visible absorption wings, a reliable decomposition is impossible (this is often the case with H$\alpha$), and we assumed that the equivalent width in absorption in H$\alpha$ was the same as in H$\beta$; this is well supported by the predictions of the models [@Olofsson1995; @GonzalezDelgado1999b].
In low surface brightness (LSB) regions a spatial smoothing procedure was applied to increase the accuracy of the fit. Depending on the signal-to-noise ratio (S/N), the closest 5, 9, or 13 spaxels were averaged before the fit was carried out. In this way, we maintain the spatial resolution of the bright regions of the galaxy while obtaining a reasonable S/N for the faint parts, but with a lower spatial resolution.
### Creating the galaxy maps {#creatingmaps}
The parameters of the line fits were used subsequently to construct the 2D maps, taking advantage of the fact that the combined VIMOS data are arranged in a regular 44$\times$44 matrix.
Continuum maps in different spectral ranges were obtained by integrating the flux in specific spectral windows, selected so as to avoid strong emission lines or residuals from the sky spectrum subtraction. We also built an integrated continuum-map by integrating over the whole spectral range, but masking the spectral regions with a significant contribution of emission lines.
Line ratios maps for lines falling in the wavelength range of either grism were computed by dividing the corresponding flux maps. In the case of H$\alpha $/H$\beta$, the line ratio map was derived after registering and shifting the H$\alpha$ map to spatially match the H$\beta$ map. The shift was calculated using the difference in position of the center of the brighter regions. The shift was applied using a bilinear interpolation. In order to correct for the fact that H$\alpha$ and H$\beta$ had been observed under different seeing conditions, the H$\alpha$ map was degraded convolving with a Gaussian to match both PSFs.
Only spaxels with a flux level higher than the 3$\sigma$ level were considered when building the final maps. All maps were corrected for interstellar extinction in terms of spaxels applying Eq. \[ext\_eq\] (see Sect. \[extinction\]).
Results
=======
Broad-band morphology: a first view on the stars {#morphologyandstars}
------------------------------------------------
![Mrk 900 continuum map from the VIMOS/IFU data built by summing over the whole blue spectral range (4150–6200Å), but masking the emission lines. Contours in H$\alpha$ are overplotted. The scale is logarithmic and the units are arbitrary. The FoV is $\sim$2.5$\times$2.5 kpc$^{2}$, with a spatial resolution of about 61 pc per spaxel. North is up and east to the left; the axes represent the displacements in RA and DEC with respect to the center of the FoV, also in all the maps presented from here on.[]{data-label="Figure:mrk900_cont"}](Figure2.eps){width="0.9\linewidth"}
The NOT B and R frames of Mrk 900 are presented in Figure \[Figure:mrk900\_not\]. In the full ALFOSC FoV the irregular high surface brightness (HSB) region is very well distinguished on top of the low surface brightness (LSB) host. The HSB area is resolved into three major knots ([a]{}, [b]{}, and [c]{} in Figure \[Figure:mrk900\_not\]); the emission peaks at the position of the knot [a]{} (the clump displaced northwest), while knot [c]{} is situated roughly at the center of the redder host. The VIMOS continuum map (Fig. \[Figure:mrk900\_cont\]) essentially reproduces the same morphological pattern of the galaxy in broad-band frames, but with a notable difference: in continuum the intensity peaks at the position of knot [c]{}, and knot [a]{} is just a moderate emitter. We masked strong emission lines when constructing the VIMOS continuum map; however, broad-band filters also encompass the emission from the ionized gas[^3]. The much higher intensity of knot [a]{} in broad-band filters compared with the continuum indicates that a significant part of its light originates in the ionized gas and reveals the presence of ionizing (very young) stars.
We built the galaxy color maps (i.e., the ratio of the flux emitted in two different filters). We computed the maps from the flux calibrated broad-band images after they were aligned and matched to the same seeing. In order to improve the S/N in the outer galaxy regions, we applied a circular-averaging filter to the individual images, the radius of the filter depending on the flux level in the R band; the filtering is only implemented at large radius ($\geq 10"$), so that the spatial resolution in the central HSB regions is preserved. To avoid artifacts we did not remove foreground stars from the field.
Two major stellar components are clearly distinguished in the $(B-R)$ color map of Mrk 900 (Figure \[Figure:mrk900\_br\]). The HSB region appears distorted and much bluer (most probably the result of one or more episodes of SF), while the host galaxy shows a smooth elliptical shape with a markedly redder color. There is an apparent color gradient among the major central knots: $(B-R)\sim$0.23, 0.56, and 0.64 in knots [a]{}, [b]{}, and [c]{}, respectively, corrected from Galactic extinction following [@Schlafly2011]. We find for the galaxy host a roughly constant $(B-R)\sim$1.21, in good agreement with the color profiles reported by [@GildePaz2003] and [@Micheva2013a].
{width="0.9\linewidth"}
Integrated and surface brightness photometry {#photometry}
--------------------------------------------
The photometry of Mrk 900 is presented in Table \[tab:photometry\]. We computed the integrated magnitudes as the magnitudes within the isophote at surface brightness 26.5 mag arcsec$^{-2}$ in the B band, namely the Holmberg isophote. We corrected for Galactic extinction using the reddening coefficients from [@Schlafly2011].
Parameter B V R
------------------------------------ ---------------- ---------------- ----------------
m (mag) 14.30$\pm$0.03 13.68$\pm$0.04 13.35$\pm$0.03
M (mag) -17.08 -17.70 -18.03
$\mu_{0,host}$ (mag arcsec$^{-2}$) 21.60$\pm$0.10 20.87$\pm$0.10 20.20$\pm$0.06
$\alpha_{host}$ (pc) 726$\pm$25 754$\pm$24 673$\pm$13
m$_{host}$ (mag) 14.89 14.12 13.73
M$_{host}$ (mag) -16.49 -17.26 -17.65
: Integrated photometry and structural parameters of Mrk 900.\[tab:photometry\]
Notes. Integrated magnitudes computed from the flux within the isophote at 26.5 mag per arcsec$^{-2}$. Structural parameters of the exponential disk model best fitting the underlying host galaxy and integrated and absolute magnitudes of the host; all magnitudes are corrected from Galactic extinction following [@Schlafly2011].
We also built the galaxy surface brightness profiles (SBPs). There are many studies on surface brightness photometry of BCGs [@Papaderos1996; @Papaderos2002; @Doublier1997; @Doublier1999; @Cairos2001b; @Noeske2003; @Noeske2005; @GildePaz2005; @Caon2005; @Micheva2013a; @Micheva2013b; @Janowiecki2014]. As pointed out in most of them, the construction of the SBP of a BCG is not straightforward; the most common methods assume some sort of galaxy symmetry and do not necessarily apply to the irregular HSB region of a BCG [@Papaderos1996; @Papaderos2002; @Doublier1997; @Cairos2001a; @Noeske2003; @Noeske2005]. A simple way to overcome this problem is to construct the profile using different approaches in the high- and low-intensity regimes; in the brighter regions, a method that does not require any assumption on the galaxy morphology is used, whereas ellipses are fitted to the isophotes of the smooth host [@Cairos2001a].
Here we opted to build the light profiles using standard techniques, but bearing in mind that they provide a proper description of the galaxy structure only outside the starburst region, which is easily assessed using the color map (Figure \[Figure:mrk900\_br\]). After masking out foreground stars and nearby objects, we fitted elliptical isophotes to the images using the IRAF task [ellipse]{} [@Jedrzejewski1987]. The fit was done in two steps: first, the parameters of the ellipses were left free to vary; in a second run, and to improve the stability of the final fit, we fixed the center to the average center of the outer isophotes found in the first step. We found that at large radius (20“$\leq$ r $\leq$40”) the galaxy isophotes are well fitted with ellipses of position angle PA=40$^{\circ}$ and ellipticity $\epsilon$=0.30, in good agreement with the values PA$=41^{\circ}$ and $\epsilon=$0.27 reported by [@Micheva2013a] .
The SBPs of Mrk 900 in $B$, $V$, and $R$ are displayed in Figure \[Figure:SBP\]. The error bars on the surface brightness magnitudes were calculated taking into account the errors given by the fit and the errors on the estimation of the sky value. The profiles show the common behavior among BCGs: there is a brightness excess in the inner region, but at large radii the intensity seems well described by an exponential decay.
We fitted an exponential disk to the outer profile to obtain the structural parameters of the underlying host galaxy. We performed a weighted linear least-squares fit to the function
$$\mu(r)=\mu_{0}+1.086\frac{r}{\alpha}
\label{mu_eq}
,$$
where $\mu$(r) is the observed surface brightness at radius r, $\mu_{0}$ is the extrapolated central surface brightness, and $\alpha$ is the exponential scale of the disk. A delicate step is the selection of the fitting radial range: the lower radial limit is given by the position where the contribution of the starburst light is negligible, and the higher radius is set by the point where the sky errors are very large or the ellipses parameters unstable; the final values of the parameters are not very sensitive to the higher radius used since the larger errors make the weight of these points to be very low. The structural parameters resulting from the fit in each band (i.e., the central surface brightness, $\mu_{0,host}$, and the scale length, $\alpha_{host}$) are presented in Table \[tab:photometry\]. Adopting the disk model, we also computed the integrated and absolute magnitudes of the host galaxy.
![Surface brightness profiles of Mrk 900 in B, V, and R. For a better visualization the V and R profiles are shifted 1 and 2 magnitudes, respectively. We also plotted the straight lines describing the exponential profile of the underlying host galaxy.[]{data-label="Figure:SBP"}](Figure4.eps){width="0.8\linewidth"}
Emission-line maps: warm ionized gas {#vimoslinemaps}
------------------------------------
Emission lines in starburst galaxies trace regions of warm ionized gas, and hence the population of ionizing stars. Mrk 900 shows an irregular morphology in emission lines (see Figures \[Figure:Mrk900\_oiiiflux\]-\[Figure:Mrk900\_siiflux\]): several SF knots appear close to the galaxy center and along the minor axis (southeast–northwest). The brightest knot, which spatially coincides with knot [a]{} in the broad-band frames, is displaced about 350 pc northwest from the continuum peak (at a distance of 18.9 Mpc the spaxel element translates into 61 pc).
These SF regions are embedded in an extended envelope of diffuse ionized gas (DIG), which fills almost the whole FoV. Filaments and curvilinear features are conspicuous at faint brightness levels: the larger ones extend in the SW direction and NE direction. A hole in the warm gas emission, with a diameter of about 260 pc, is visible to the NW close to the biggest SF region.
We adopt here the common definition for the DIG, namely diffuse ionized hydrogen outside the discrete (well-defined) regions [@Reynolds1973; @Dopita2003; @Oey2007]. To make a clear-cut distinction between spaxels belonging to regions and to the DIG is not straightforward (see, e.g., the discussion in @FloresFajardo2009). However a substantial fraction of the warm ionized emission in Mrk 900 belong to the DIG according to the currently most used criteria: for instance, the high values of the low-ionization \[\] $\lambda\lambda6717,\,6731$/ line ratio (see Section \[choques\] below) and the low values of the H$\alpha$ equivalent width (Fig. \[Figure:Mrk900\_haeqw\]) in the outer galaxy regions are characteristic of the DIG following the criteria proposed in [@Blanc2009] and [@Lacerda2018], respectively.
The galaxy displays the same morphological pattern in all emission lines, but fainter SF clumps appear better delineated in the high-excitation \[\] $\lambda5007$ line than in the hydrogen recombination lines, and they are only barely visible in the low-ionization \[\] $\lambda6584$ and \[\] $\lambda\lambda6717,\,6731$ maps.
![\[\] $\lambda5007$ emission-line flux map for Mrk 900, with a cross marking the position of the continuum peak. The spatial scale in pc is indicated at the bottom left; the spatial resolution is about 61 pc per spaxel (91 pc per arcsec) and flux units are $10^{-18}$ergs$^{-1}$cm$^{-2}$, also in all the emission-line maps presented from here on. []{data-label="Figure:Mrk900_oiiiflux"}](Figure5.eps){width="0.9\linewidth"}
![H$\alpha$ emission-line flux map for Mrk 900.[]{data-label="Figure:mrk900_haflux"}](Figure6.eps){width="0.9\linewidth"}
![\[\] $\lambda6584$ emission-line flux map for Mrk 900.[]{data-label="Figure:Mrk900_niiflux"}](Figure7.eps){width="0.9\linewidth"}
![\[\] $\lambda\lambda6717,\,6731$ emission-line flux map for Mrk 900.[]{data-label="Figure:Mrk900_siiflux"}](Figure8.eps){width="0.9\linewidth"}
Interstellar extinction pattern {#extinction}
-------------------------------
The interstellar extinction in a nebulae is computed by comparing the observed Balmer decrement values to the theoretical ones [@Osterbrock2006]. Since the Balmer line ratios are well known from atomic theory, deviations from the predicted values are assumed to be due to dust; the interstellar extinction coefficient, C(H$_{\beta}$), is derived as $$\frac{F_{\lambda}}{F(H_{\beta})}=\frac{F_{\lambda,0}}{F(H_{\beta,0})}\times10^{-C(H_{\beta})[f(\lambda)-f(H_{\beta})]}
\label{ext_eq}
,$$ where $F_{\lambda}/F(H_{\beta}$) is the observed ratio of Balmer emission-line intensities relative to H$_{\beta}$, $F_{\lambda,0}/F(H_{\beta,0}$) the theoretical ratio and $f(\lambda$) the adopted extinction law. We applied Eq. (\[ext\_eq\]) to every spaxel to obtain an interstellar extinction map (Figure \[Figure:mrk900\_reddening\]).
In practice, we derived the extinction coefficient using only the H$\alpha$/H$\beta$ ratio; although H$\gamma$ also falls into the observed wavelength range, it has a poorer S/N and is also more severely affected by underlying stellar absorption compared to H$_{\alpha}$ and H$_{\beta}$ [@Olofsson1995; @GonzalezDelgado1999a]. We adopted case B recombination and low-density limit with a temperature of 10000 K for $F_{\lambda,0}/F(H_{\beta,0}$) in Eq. (\[ext\_eq\]). We used the Galactic extinction law from [@ODonnell1994]; Mrk 900 is closer in metallicity to the Large Magellanic Cloud (LMC), but differences between the Galactic and the LMC extinction curves are insignificant in the observed spectral range [@Dopita2003].
The interstellar extinction map of Mrk 900 (Figure \[Figure:mrk900\_reddening\]) reveals a clear spatial pattern. The lowest extinction values are reached at the position of the SF regions, and the highest in the inter-knot area. In addition, extinction anticorrelates with the H$\alpha$ intensity: the maximum extinction values are in the regions of fainter H$\alpha$ emission. These results are consistent with dust being destroyed or swept away by the most massive stars.
In the whole FoV, H$_{\alpha}$/H$_{\beta}\geq$3.5, which indicates a significant amount of dust even at the position of the SF regions (H$\alpha$/H$\beta$=3.5 implies an extinction coefficient C(H$\beta$)=0.23 and a color excess $E(B-V)$=0.17). Away from the SF regions, H$_{\alpha}$/H$_{\beta}$ reaches values up to 6, meaning C(H$\beta$)=0.87 and $E(B-V)$=0.64. Such large spatial variability in the extinction agrees with the results of previous analyses [@Lagos2014; @Cairos2015; @Cairos2017a; @Cairos2017b] and strengthens the conclusions of those works: applying a unique extinction coefficient to the whole galaxy, as usually done for long-slit spectroscopic observations, can lead to large errors in derived fluxes, magnitudes, and SFR.
![H$\alpha$/H$\beta$ ratio map for Mrk 900, with the contours of the H$\alpha$ flux map overplotted. []{data-label="Figure:mrk900_reddening"}](Figure9.eps){width="0.9\linewidth"}
Excitation and ionization mechanisms {#choques}
------------------------------------
Specific emission-line ratios, namely \[\] $\lambda5007$/, \[\] $\lambda6584$/, \[\] $\lambda\lambda6717,\,6731$/, and \[\] $\lambda6300$/, reveal the excitation and ionization mechanisms in a nebula [@Dopita2003; @Osterbrock2006]; IFU observations enable us to investigate the spatial variations of excitation and ionization conditions in galaxies [@Sharp2010; @Rich2011; @Rich2012; @Rich2015; @Cairos2017a; @Cairos2017b; @Mingozzi2019].
The \[\] $\lambda5007$/ distribution of Mrk 900 (Figure \[Figure:diagnostic-oiii\]) is not uniform, but shows a clear spatial pattern: the highest excitation is reached at the position of the SF regions. This is expected as a harder ionization source ($h\nu\geq$ 35.1 eV), such as very young O stars, is required to cause a strong \[\] $\lambda5007$ line. The excitation reaches its maximum at the position of the major H$\alpha$ emitter (\[\] $\lambda5007/{\mbox{H$\beta$}}{}\sim$3.0). Interestingly, there are two other zones, not co-spatial with any SF knot, with a relatively high excitation (Figure \[Figure:diagnostic-oiii\]). An enhancement on the excitation in regions not photoionized by stars can be due to the presence of shocks and/or a large amount of dust; the excitation maximum at the south is very close to the peak in interstellar extinction.
The maps for low-ionization species \[\] $\lambda6300$/, \[\] $\lambda6584$/, and \[\] $\lambda\lambda6717,\,6731$/ (Figures \[Figure:diagnostic-oi\] – \[Figure:diagnostic-sii\], respectively) show the opposite behavior, with the line ratios reaching their minimum at the center of the SF knots. This is expected in regions ionized by UV photons coming from massive stars. All three line ratios appreciably increase outward (\[\] $\lambda\lambda6717,\,6731$/ $\geq$ 0.6), which are values much higher than those predicted for stellar photoionization.
![\[\] $\lambda5007$/ emission-line ratio map with contours on H$\alpha$ overplotted. The arrows indicate areas of enhancement of the line ratio, which do not spatially coincide with any SF regions.[]{data-label="Figure:diagnostic-oiii"}](Figure10.eps){width="0.9\linewidth"}
![\[\] $\lambda6300$/ emission-line ratio map with contours on H$\alpha$ overplotted.[]{data-label="Figure:diagnostic-oi"}](Figure11.eps){width="0.9\linewidth"}
![\[\] $\lambda6584$/ emission-line ratio map with contours on H$\alpha$ overplotted.[]{data-label="Figure:diagnostic-nii"}](Figure12.eps){width="0.9\linewidth"}
![\[\] $\lambda\lambda6717,\,6731$/ emission-line ratio map with contours on H$\alpha$ overplotted.[]{data-label="Figure:diagnostic-sii"}](Figure13.eps){width="0.9\linewidth"}
From the emission-line ratio maps, we generated spaxel resolved diagnostic diagrams; we plotted in the classical diagnostic diagrams [@Baldwin1981; @Veilleux1987] the value of the line ratios at each individual element of spatial resolution. This technique permits us to explore the power sources acting in different galaxy regions [@Sharp2010; @Rich2011; @Rich2012; @Rich2015; @Leslie2014; @Cairos2017a; @Cairos2017b].
Figure \[Figure:diagnostic-spaxel\] displays the diagnostic line diagrams for Mrk 900, together with the maximum starburst line (or photoionization line) derived by [@Kewley2001]; this line traces the limit between gas photoionized by young stars and gas ionized via other mechanisms (AGN or shock excitation). We found that a considerable fraction of the spaxels fall outside of the area occupied by star photoionization in the diagrams \[\] $\lambda5007$/ [versus]{} \[\] $\lambda6300$/ and \[\] $\lambda5007$/ [versus]{} \[\] $\lambda\lambda6717,\,6731$/. The ratio \[\] $\lambda6584$/, weakly dependent on the hardness of the radiation but strongly dependent on the metallicity, is not effective in separating shocks from photoionized regions. In particular, at low metallicities (0.2 Z$\odot\leq$ Z $\leq0.4~$Z$\odot$) this diagram is degenerated, and shock-ionization and photoionization overlap [@Allen2008; @Hong2013].
In order to better visualize this result, we display the spatial position of these spaxels on the galaxy in Fig. \[Figure:map-diagnostic\]. The areas not photoionized are situated mainly at the periphery of the mapped region and in the inter-knot space, which conforms to the idea that they are shocked regions, generated in the interface between the expanding bubbles (produced by the massive stellar feedback) and the ambient ISM. This behavior has also been observed in other BCGs, for example Haro 14 and Tololo 1937-423 [@Cairos2017a; @Cairos2017b], and in dwarf irregular galaxies, for example NGC4449 [@Kumari2017] and NGC 5253 [@Calzetti1999; @Calzetti2004].
{width="0.75\linewidth"}
![Spatial localization of the spaxels in the diagnostic diagram \[\] $\lambda5007$/ vs. \[\] $\lambda\lambda6717,\,6731$/. The color-coding is the same as in Figure \[Figure:diagnostic-spaxel\]. The redder regions are those situated the above the maximum starburst line from [@Kewley2001]. []{data-label="Figure:map-diagnostic"}](Figure15.eps){width="0.8\linewidth"}
Ionized gas kinematics {#kinematics}
----------------------
The line-of-sight (LOS) velocity map in H$\alpha$, measured from the Doppler shift of the line profile centroid relative to the galaxy systemic velocity, is shown in Figure \[Figure:velocitymaps\]. A distorted, but ordered rotation field pattern is evident, with the velocity gradient aligned along the optical major axis: the regions situated northeast are moving away from us, while the southwest regions are moving toward us. Deviations from the global rotation are apparent across the whole FoV; particularly interesting is the curvilinear feature in the southwest, spatially coincident with the regions of enhancement in \[\] $\lambda6584$/ and \[\] $\lambda\lambda6717,\,6731$/ in the diagnostic maps.
The kinematics of Mrk 900 has been studied in the multi-pupil IFS analysis of 18 BCGs by [@Petrosian2002]. These authors found a homogeneous velocity field in the H$\alpha$ emission line and excluded the possibility of a strong rotational gradient in the galaxy, but this result clearly arises from their limited FoV. Their velocity map, centered roughly where the continuum peaks, covers about 9.1$\times$13 arcsec; consequently, the whole field is dominated by the starburst emission. Using synthesis observations, [@VanZee2001] investigated the kinematics of Mrk 900 in a much larger FoV; the overall pattern of their velocity map (see their Figure 6) is in good agreement with our results.
The H$\alpha$ LOS velocity dispersion, derived from the width of the Gaussian fit to the line profile after accounting for the instrumental ($\sim$89.9 km/sec) and thermal broadening ($\sim$9.1 km/s), also shows a clear spatial pattern (Figure \[Figure:velocitymaps\]). The minimum values are reached at the SF knots (5 km s$^{-1}$ $\leq$ $\sigma$ $\leq$ 15 km s$^{-1}$), while the velocity dispersion increases up to values $\sim$60 km s$^{-1}$ at the galaxy periphery and in the inter-knot region. A nonuniform spatial distribution has also been found in several BCGs for which 2D maps of the ionized gas velocity dispersion has been published [@Bordalo2009; @Moiseev2012; @Moiseev2015; @Cairos2015; @Cairos2017a; @Cairos2017b]. Based on this result, [@Moiseev2015] claimed that the large velocity dispersions (turbulent motions in the ionized gas) observed in dwarf galaxies do not reflect virial motions, but are instead related to the feedback from young massive stars.
{width="0.8\linewidth"}
Integrated spectroscopy {#integratedspec}
-----------------------
![H$\alpha$ equivalent width map (in Å) for Mrk 900, with the seven major regions of SF labeled.[]{data-label="Figure:Mrk900_haeqw"}](Figure17.eps){width="0.9\linewidth"}
![Flux-calibrated spectra of three of the seven selected regions and the nuclear region in Mrk 900, in logarithmic units.[]{data-label="Figure:spectra"}](Figure18.eps){width="1\linewidth"}
Using the bidimensional maps we identified the major regions of SF in the galaxy. To delimit their borders we used the H$\alpha$ equivalent width map; equivalent widths of hydrogen recombination lines are excellent age indicators [@Dottori1981; @Stasinska1996; @Leitherer2005; @Levesque2013]. The H$\alpha$ equivalent width map of Mrk 900 (Figure \[Figure:Mrk900\_haeqw\]) traces the H$\alpha$ flux, but the regions are better delineated and, in particular, various small clumps located at the filaments are conspicuous.
Because there is no clear-cut criterion to set the limits of the knots, we integrated over a boundary tracing the morphology of the cluster, taking into account that we are limited by the seeing. In this way we singled out seven major regions in Mrk900 (labeled in Figure \[Figure:Mrk900\_haeqw\]). We generated the integrated spectrum of each knot by adding the spectra of the corresponding spaxels, so we obtained a higher S/N spectrum compared to those of the individual fibers, more suitable to derive physical parameters and abundances. We also produced the “nuclear” spectrum, adding the signal of the spaxels around the continuum peak, and the total spectrum, adding all the spaxels in the H$\alpha$ emission line flux map.
All SF knots display a typical nebular spectrum, with strong Balmer lines and \[OIII\], \[NII\], and \[SII\] forbidden lines in emission, on top of an almost featureless continuum. As expected, the spectrum of the nuclear region presents a higher continuum and more pronounced absorption features. As an illustration, we show the spectra of three out of the seven SF regions and the nuclear region in Figure \[Figure:spectra\]. In knot 1 we detect the broad-band Wolf-Rayet (WR) bump at $\lambda4650-4690$ Å, the unresolved blend of NIII $\lambda$4640, CIII $\lambda$4650, CIV $\lambda$4658, and HeII $\lambda$4686Å lines (see Fig. \[Figure:Mrk900\_wr\]; [@Conti1991; @Schaerer1999]). The S/N of the spectra is too low for a reliable measurement of the WR bump flux, but just the detection of WR stars strongly constrains the age, duration, and initial mass function (IMF) of the SF episode [@Meynet1995; @Schaerer1998; @Guseva2000]. At the metallicity of Mrk 900 (see Section \[abundances\] below) the presence of WRs is consistent with these stars being formed in an instantaneous burst, with ages between 3 and 6.3 Myr, and an IMF with an upper mass limit of 100 M$\odot$ [@Leitherer1995; @Leitherer1999].
![Enlarged region of the spectrum around blue WR bump ($\lambda4650-4690$Å) in the knot 1. This feature is attributed to the blends of NIII $\lambda$4640, CIII $\lambda$4650, CIV $\lambda$4658, and HeII $\lambda$4686Å lines.[]{data-label="Figure:Mrk900_wr"}](Figure19.eps){width="0.9\linewidth"}
### Emission-line fluxes
We measured the observed emission-line fluxes in each of these spectra using the [iraf]{} task [splot]{}. The contribution of the Balmer lines in absorption was taking into account, as discussed in Section \[creatingmaps\]. From the observed Balmer decrement we computed the interstellar extinction coefficient, as explained in Section \[extinction\]. The reddening-corrected intensity ratios are presented in Table \[tab:fluxes\], together with the derived interstellar extinction coefficient, C(H$\beta$); for an easier comparison with other data, the number of magnitudes of extinction in V, A$_{V}$, and the color excess, $E(B-V)$, are also shown.
### Diagnostic line ratios, physical parameters, and abundances {#abundances}
The values of the diagnostic line ratios, electron densities ($N_{\rm e}$), and electron temperatures ($T_{\rm e}$) for the selected regions and the integrated spectrum of Mrk 900 are shown in Table \[tab:diagnostic\]. The values of $N_{\rm e}$ and $T_{\rm e}$ were computed using the five-level atom [fivel]{} program in the [iraf nebular]{} package [@deRobertis1987; @ShawDufour1995]. The electronic densities were estimated from the \[\] $\lambda\lambda6717,\,6731$ line ratio [@Osterbrock2006]; we found that all regions present values in the low-density regime; knot 5 has a slightly higher density, close to 100 cm$^{-3}$. Where the \[\]$\lambda$4363 line could be measured (knots 1, 3, and 5) we derived the electron temperature from the \[\]$\lambda$4363/($\lambda$4959+$\lambda$5007) line ratio.
The oxygen abundance could not be determined using the direct $T_{\rm e}$ method (not even in the three knots where \[\]$\lambda$4363 was measured) because this method requires the measurement of the \[\] $\lambda$3727+3729 line, which unfortunately falls out of the VIMOS spectral range. Alternatively, the auroral \[\] $\lambda\lambda$7320,7330 lines could be used, but for Mrk 900 these features fall very close to the edge of the VIMOS spectrum in a zone that is highly affected by sky residuals, which prevents us from obtaining reliable flux values. We therefore estimated the oxygen abundances by adopting the empirical method introduced by [@PilyuginGrebel2016], which utilizes the intensities of the strong lines \[\]$\lambda\lambda$4957,5007, \[\]$\lambda\lambda$6548,6584, and \[\]$\lambda\lambda$6717,6731. This calibrator provides (as usual) separate relations for high-metallicity and low-metallicity objects, but this degeneration can be simply broken using the N$_{2}$ line ratio (N$_{2}$=\[\]$\lambda\lambda$6548,6584)/H$\beta$). For Mrk 900, all the measured spectra (i.e., the SF regions and the nuclear and integrated spectra) lie in the upper branch of the calibrator; the division between the upper and lower branches takes place at log N$_{2}$= -0.6. The relative accuracy of the abundance derived using this method is 0.1 dex.
The oxygen abundances are similar in all the knots and the integrated spectrum, 12+log(O/H)$\sim$8.25 ($\approx$0.3 Z$\odot$[^4]); this value is slightly higher than that found by [@Zhao2010] applying the direct-$T_{\rm e}$ method, 12+log(O/H)=8.07$\pm$0.03. Nearly homogeneous chemical abundances have been found for most of the blue compact and dwarf irregular galaxies investigated so far [@Croxall2009; @Haurberg2013; @Lagos2014; @Lagos2016; @Cairos2017a; @Cairos2017b].
Discussion
==========
The results derived in the previous section make it possible to investigate the star-forming history (SFH) and the SF process in Mrk 900. The first step used to probe the SFH in a galaxy is to disentangle its stellar populations; it is already difficult to work with galaxies that cannot be resolved into stars, and this task becomes particularly tricky when we deal with starburst objects. IFS observations, supplying a broad set of observables for every element of spatial resolution, provide a novel and powerful way to approach the problem [@Cairos2017a; @Cairos2017b].
Combing our VIMOS/IFU and broad-band imaging data we distinguish at least two stellar populations in Mrk 900: a very young component (exposed in the emission-line maps) and a significantly older stellar population (traced to large galactocentric distances in the broad-band pictures). We can constrain the properties of these different stellar populations by comparing the derived observables with the predictions of evolutionary synthesis models.
We focus first on the very young stars. Only O and early B stars, with temperatures higher than 30.000 K and masses above 10 M$_{\odot}$, produce photons with energy high enough to ionize hydrogen. Models show that these stars evolve quickly: ionizing stars cannot be older than 10 Myr [@Leitherer1995; @Ekstroem2012; @Langer2012]. Hence, the star clusters generating the regions in the central parts of Mrk 900 are younger than 10 Myr.
The H$\alpha$ flux in the whole mapped area (derived from the summed spectrum) is 5.7$\pm$0.03$\times10^{-13}$erg s$^{-1}$cm$^{-2}$, consistent with the value 5.5$\pm$0.6$\times10^{-13}$erg s$^{-1}$cm$^{-2}$ reported by [@GildePaz2003] from narrow-band images. From the total luminosity L(H$\alpha$)=2.4$\times$10$^{40}$ erg s$^{-1}$, we derive a SFR=0.17$\pm$0.02 M$\odot$ yr$^{-1}$, applying the following expression from [@Hunter2010]: $${\rm SFR}(M_{\odot}~{\rm yr}^{-1})=6.9\times10^{-42}\,L_{H\alpha}({\rm erg~s}^{-1}).$$ This formula, derived in [@Kennicutt1998], was modified for a subsolar metallicity, which is the most appropriate for dwarf galaxies. The derived SFR is slightly higher than the value 0.12 M$\odot$ yr$^{-1}$ reported in [@Hunt2015][^5].
To compare this SFR with those of other galaxies, we must take into account the galaxy size (see discussion in [@HunterElmegreen2004]). Normalizing the SFR to the radio at surface brightness level 25 mag arcsec$^{-2}$ ($R_{25}$) we obtain SFR=0.0105 M$\odot$ yr$^{-1}$ kpc$^{-2}$, a substantially high SFR even among the BCG class; for a sample of 23 BCGs, [@HunterElmegreen2004] reported an average SFR of 0.0062 M$\odot$ yr$^{-1}$ kpc$^{-2}$.
The global SFR (0.17$\pm$0.02 M$\odot$ yr$^{-1}$) and the mass of atomic hydrogen (M$_{HI}$=1.55$\times$10$^{8}$ M$\odot$; [@VanZee2001]) imply a depletion time (the timescale to exhaust the current gas supply of the galaxy; [@Roberts1963]) $\tau$=0.9 Gyr. According to the classical definition, this classifies Mrk 900 as a starburst galaxy, i.e., a galaxy whose “SFR cannot be sustained for a significant fraction of the Hubble time with the available interstellar gas” [@Gallagher2005; @Heckman2005; @McQuinn2010a].
------ ------------------------------------ --------------------- --------------------- -------------- ------- -- --
Knot F(H$\alpha$) log\[L(H$\alpha$)\] SFR W(H$\alpha$) Age
(10$^{-16}$ erg cm$^{-2}$s$^{-1}$) (erg s$^{-1}$) M$\odot$ yr$^{-1}$ (Å) (Myr)
1 1602$\pm$37 39.83$\pm$0.14 0.047$\pm$0.007 214 5.5
2 95$\pm$14 38.61$\pm$0.20 0.0028$\pm$0.0006 94 6.3
3 95$\pm$11 38.61$\pm$0.18 0.0028$\pm$0.0006 78 6.5
4 32$\pm$6 38.13$\pm$0.23 0.0009$\pm$0.0002 77 6.5
5 382$\pm$186 39.21$\pm$0.51 0.011$\pm$0.006 73 6.6
6 118$\pm$6 38.70$\pm$0.15 0.0035$\pm$0.0005 78 6.5
7 147$\pm$9 38.80$\pm$0.15 0.0043$\pm$0.0006 107 6.2
------ ------------------------------------ --------------------- --------------------- -------------- ------- -- --
Note: H$\alpha$ fluxes were corrected from interstellar extinction using the values provided in Table \[tab:fluxes\] \[SF-KnotI\]
The H$\alpha$ fluxes, luminosities, and equivalent widths of the individual regions identified in Mrk 900 (see Section \[integratedspec\]) are shown in Table \[SF-KnotI\]. To constrain their properties, we compare the H$\alpha$ equivalent widths with the predictions of the [Starburst 99]{} evolutionary synthesis models [@Leitherer1999]. Adopting the models with metallicity Z=0.008 (the value closest to the metallicity derived from the emission-line fluxes) we find that we can reproduce the measured equivalent widths with an instantaneous burst of SF, a Salpeter IMF with an upper mass limit of 100 M$\odot$, and ages ranging from 5.5 to 6.6 Myr. This age range must be understood, however, as an upper limit: the measured equivalent widths can decrease as a result of absorption from A-F stars and/or dilution due to the continuum from an older stellar population [@Fernandes2003; @Levesque2013]. We corrected the H$\alpha$ measurements in Table \[SF-KnotI\] for stellar absorption (see Section \[linefitting\]), but no attempt was made to correct for the presence of the older stars; the uncertainties in the ages of the SF knots due to the contribution from an older population were estimated by [@Cairos2002; @Cairos2007] to be up to $\sim$1-1.5 Myr.
The ionized gas emission is manifestly dominated by knot 1: it generates $\sim$30% of the total H$\alpha$ flux in the observed area. With a H$\alpha$ luminosity of 6.84$\pm$0.9 $\times$10$^{39}$ergs$^{-1}$ and a diameter about 300 pc, knot 1 is classified as a giant extragalactic region (GEHR). It is, indeed, comparable in size with the 30 Doradus nebula, the largest GEHR in the Local Group. The detection of the WR bump in this knot provides an independent age estimation of 3-6 Myr [@Schaerer1998], consistent with that derived from H$\alpha$.
The morphology of the central starburst region in the continuum (Figure \[Figure:mrk900\_cont\]) strongly differs from the morphology in emission lines (Figures \[Figure:Mrk900\_oiiiflux\]-\[Figure:Mrk900\_siiflux\]). From the seven regions identified in emission lines, only knot 1 has a visible counterpart in the continuum (knot [a]{}); however, it appears only as a moderate continuum emitter. The continuum maximum (knot [c]{}), located about the center of the elliptical host, is about 130 pc southwest from its nearest region (knot 5). Such morphological patterns may indicate the presence of distinct bursts of SF and spatial migration of the SF sites [@Petrosian2002]. By carrying out similar IFU analyses in the BCGs Haro 14 and Tololo 1937-423, we identified two temporally and spatially separated bursts whose ages suggest a scenario of triggered SF [@Cairos2017a; @Cairos2017b]. In Mrk 900, however, the presence of a second episode of SF cannot be definitely confirmed, as knot [c]{} could be also the nucleus of the host component and the increase in intensity the result of a higher stellar density. The high values of the H$\gamma$ and H$\beta$ equivalent widths in absorption ($\sim$5Å) in the nuclear spectrum might indicate the presence of intermediate-age stars, but the possibility that such large equivalent widths originate in the nearby regions and/or are due to a strong dilution in the continuum peak cannot be ruled out.
We now focus on the properties of the galaxy host, very well traced in the NOT images. In the galaxy outskirts, we find no SF regions but a rather red and regularly shaped host, with elliptical isophotes. The outer regions of the SBP are well described by an exponential function; consistently, the velocity field indicates the presence of a rotating disk, although perturbed in the inner parts. The region covered by VIMOS is still heavily affected by the starburst, and we could not derive the rotation curve; however, from the velocity field we estimated a maximum velocity amplitude $\sim$80–90 km sec$^{-1}$ along the optical major axis. This value is in very good agreement with the velocity 82.0 km sec$^{-1}$ at a distance of $\sim$1 kpc reported by [@VanZee2001] from data.
A comparison of the colors of the host galaxy with the predictions of evolutionary synthesis models [@Vazdekis1996; @Vazdekis2010; @Fioc1997; @LeBorgne2004] suggests ages of several Gyr. Thus, the morphology, structure, dynamics, and age of the host galaxy are similar to those presented by dE galaxies [@LinFaber1983; @vanZee2004a; @vanZee2004b].
In summary, we identified in Mrk 900 a very young population ($\leq$6.6 Myr) resolved in an ensemble of regions and extending about 1 kpc along the galaxy minor axis. This young component presents a rather distorted appearance in emission lines, with multiple filamentary and bubble-like structures. Underlying the young stars there is an old (several Gyr) stellar population with smooth elliptical isophotes, which extends up to radius of 4 kpc.
What mechanism has ignited the actual SF burst in Mrk 900 after (most probably) several Gyr of inactivity is not evident. The nature of the starburst trigger in low-mass systems has been (and is still) widely debated [@Pustilniketal2001; @Brosch2004]. Several internal processes have been discussed, for example stochastic self propagating star formation (SSPSF, [@Gerola1980]) or torques in spiral disk clumps [@Elmegreen2012]. However, an increasing amount of observations suggests that external triggers, such as interactions and mergers [@Brinks1990; @Taylor1996; @Ostlin2001; @Pustilniketal2001; @Ekta2008; @Lelli2014] or external gas infalling [@LopezSanchez2012; @Nidever2013; @Ashley2014; @Miura2015; @Turner2015] play a major role in the ignition of the starburst in BCGs. These observational results are strengthened by numerical simulations that successfully reproduce the merger [@Bekki2008] or gas infall scenario [@Verbeke2014].
The observational evidence speaks against interactions or mergers with a massive galaxy as being responsible for the present-day SF in Mrk 900: in the [hyperleda]{} database the galaxy appears classified as “isolated” (i.e., it does not have bright neighboring objects); the LSB component shows a very regular behavior, without tidal features or any other sign of asymmetry, in the optical and NIR (Section \[photometry\] and [@Micheva2013a; @Janowiecki2014]); the distortion of the isophotes in the central area is clearly due to the superposition of the SF episode; the velocity field is relatively smooth out of the starburst regions, both in the optical (Section \[kinematics\]) and in radio [@VanZee2001]; finally, we did not find any significant metallicity variations among the individual stellar clusters (Section \[integratedspec\]).
On the other hand, interactions with low-mass companions, mergers between gas-rich dwarf galaxies or external gas infalling are all triggering scenarios consistent with our results: a centrally concentrated SF, as we observe in Mrk 900, is expected after an interaction/merger event; a collision can drive mass inflows to the galaxy central regions and ignite a nuclear starburst [@Mihos1994; @Bekki2008]; the alignment of the SF with the galaxy minor axis points to inflowing (generated in a collision) or outflowing gas, but the two scenarios are rather difficult to distinguish observationally. The presence of shocked and high velocity-dispersion regions (Section \[kinematics\]; Section \[choques\]) and the substantial amount of dust (Fig. \[Figure:mrk900\_reddening\]) is also suggestive of an interaction or accretion event.
In addition, the synthesis observations of Mrk 900 presented in [@VanZee2001] reveal an extended and distorted morphology. The galaxy neutral gas distribution shows clear deviations from symmetry in the outer regions, the most evident being two extensions departing toward the east and northeast (Fig. 6 in [@VanZee2001]). More recent work, dealing with higher sensitivity and spatial resolution observations on nearby starbursting dwarfs, have resolved such distorted morphologies into different components, for example tails, plumes, or clouds (IC 10, [@Ashley2013; @Nidever2013]; NGC 5253, [@Kobulnicky2008; @LopezSanchez2012] or NGC 1569, [@Johnson2012; @Johnson2013]), and interpreted these findings as evidence of interactions or inflowing gas. In the case of BCG NGC 5253, the infall scenario is further supported by kinematic CO observations [@Miura2015; @Turner2015].
Summary and conclusions
=======================
This work presents results on a spectrophotometric analysis of the BCG Mrk 900, carried out by combining VIMOS/IFU observations with deep (BVR) broad-band imaging. From the IFU data we built continuum, emission-line, and diagnostic line ratio maps, and generated LOS velocity and velocity dispersion maps of the central starburst region. Using the broad-band frames, which trace the underlying stellar host, we derived SBPs and color maps.
From our analysis we highlight the following results:
$\bullet$ We disentangled two stellar components in Mrk 900: an ionizing (very young) population, resolved in individual stellar clusters and extending about 1 kpc along the galaxy minor axis, and a very regular LSB stellar host, which reaches galactocentric distances up to $\sim$4 kpc and exhibits red colors, consistent with ages of several Gyr.
$\bullet$ We generated the integrated spectrum of the major seven regions identified in Mrk 900, and of its nuclear region. From these spectra we derived reliable physical parameters and oxygen abundances. We found, for all knots, similar values of the abundance, 12+log(O/H)$\sim$8.25 ($\approx$0.3 Z$\odot$), and no evidence of metallicity variations. Using evolutionary synthesis models we estimated ages of 5.5-6.6 Myr for all the ionizing clusters. We detected in the larger region (knot 1) the WR bump at $\lambda4650-4690$ Å, and therefore we demonstrated that Mrk 900 can be classified as a WR galaxy.
$\bullet$ We showed that Mrk 900 contains a substantial amount of dust, with A$_{V}\geq$0.48 in the whole mapped area. The dust distribution is inhomogeneous, with a dust lane crossing the central starburst northeast–southwest; dust lanes and patches are also distinguished in the broad-band frames and color map.
$\bullet$ Diagnostic maps and diagnostic diagrams have shown the presence of shock-dominated zones in Mrk 900; these regions are situated primarily at the periphery of the mapped (starburst) area and in the inter-knot region, conforming with the idea of shocks being generated in the interface between expanding bubbles (generated by the massive stars) and the ambient ISM.
$\bullet$ We built velocity and velocity dispersion fields from the brightest emission lines. Although deviation from circular motions is evident, the galaxy displays an overall rotation pattern. The dispersion map is inhomogeneous, and the areas of higher dispersion coincide spatially with the areas of low surface brightness.
$\bullet$ Given our observational results, we argue that an interaction with a low-mass system, a merger between gas-rich dwarf galaxies, or infalling from external gas clouds are all plausible scenarios for the ignition of the actual burst of SF in Mrk 900.
Ion
--------------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- -----------------
4340 H$\gamma$ 0.482$\pm$0.004 0.434$\pm$0.020 0.589$\pm$0.029 0.498$\pm$0.042 0.494$\pm$0.009 0.474$\pm$0.016 0.485$\pm$0.019 0.498$\pm$0.011 0.493$\pm$0.003
4363 \[OIII\] 0.021$\pm$0.002 — 0.037$\pm$0.013 — 0.045$\pm$0.012 — — — —
4472 HeI 0.038$\pm$0.002 — — — 0.034$\pm$0.008 — — — —
4861 H$\beta$ 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
4959 \[OIII\] 1.043$\pm$0.006 0.930$\pm$0.030 0.834$\pm$0.028 0.751$\pm$0.039 0.958$\pm$0.014 0.608$\pm$0.013 0.696$\pm$0.016 0.825$\pm$0.012 0.791$\pm$0.003
5007 \[OIII\] 3.091$\pm$0.016 2.771$\pm$0.074 2.534$\pm$0.062 2.288$\pm$0.088 2.864$\pm$0.031 1.794$\pm$0.030 2.047$\pm$0.034 2.509$\pm$0.030 2.401$\pm$0.007
5875 HeI 0.102$\pm$0.002 0.098$\pm$0.021 0.088$\pm$0.013 0.093$\pm$0.032 0.088$\pm$0.006 — 0.085$\pm$0.007 0.091$\pm$0.007 —
6300 \[OI\] 0.024$\pm$0.001 0.070$\pm$0.014 0.097$\pm$0.009 — 0.050$\pm$0.005 0.066$\pm$0.008 0.077$\pm$0.007 0.056$\pm$0.006 —
6548 \[NII\] 0.074$\pm$0.001 0.088$\pm$0.011 0.130$\pm$0.012 0.137$\pm$0.037 0.103$\pm$0.004 0.124$\pm$0.007 0.126$\pm$0.007 0.118$\pm$0.006 0.135$\pm$0.002
6563 H$\alpha$ 2.870$\pm$0.020 2.870$\pm$0.104 2.870$\pm$0.095 2.870$\pm$0.146 2.870$\pm$0.043 2.870$\pm$0.060 2.870$\pm$0.061 2.870$\pm$0.047 2.870$\pm$0.012
6584 \[NII\] 0.226$\pm$0.002 0.270$\pm$0.015 0.403$\pm$0.017 0.404$\pm$0.032 0.337$\pm$0.008 0.381$\pm$0.012 0.385$\pm$0.011 0.367$\pm$0.008 0.361$\pm$0.002
6678 HeI 0.030$\pm$0.001 — — — 0.022$\pm$0.005 — 0.025$\pm$0.007 — —
6717 \[SII\] 0.238$\pm$0.003 0.354$\pm$0.018 0.585$\pm$0.024 0.541$\pm$0.040 0.375$\pm$0.008 0.487$\pm$0.014 0.496$\pm$0.013 0.438$\pm$0.009 0.477$\pm$0.002
6731 \[SII\] 0.170$\pm$0.002 0.241$\pm$0.014 0.386$\pm$0.017 0.369$\pm$0.032 0.282$\pm$0.006 0.336$\pm$0.011 0.354$\pm$0.011 0.324$\pm$0.008 0.339$\pm$0.002
7136 \[ArIII\] 0.097$\pm$0.002 0.083$\pm$0.012 0.090$\pm$0.011 0.097$\pm$0.023 0.096$\pm$0.004 — 0.086$\pm$0.008 0.091$\pm$0.006 —
F$_{H\beta}$ 558$\pm$13 30$\pm$4 33$\pm$4 11$\pm$2 133$\pm$65 41$\pm$2 51$\pm$3 16$\pm$1 1997$\pm$24
C$_{H\beta}$ 0.439$\pm$0.006 0.467$\pm$0.030 0.434$\pm$0.028 0.398$\pm$0.042 0.439$\pm$0.012 0.288$\pm$0.017 0.317$\pm$0.018 0.418$\pm$0.014 0.408$\pm$0.004
W(H$\gamma$)$_{ab}$ 2.1 – 3.4 — 3.7 — 1.9 5.0 2.5
W(H$\beta$)$_{ab}$ 1.9 2.2 3.3 — 4.5 2.4 3.7 5.1 3.5
$A_{V}$ 0.945$\pm$0.012 1.012$\pm$0.065 0.938$\pm$0.058 0.860$\pm$0.092 0.948$\pm$0.027 0.623$\pm$0.038 0.685$\pm$0.038 0.904$\pm$0.030 0.882$\pm$0.007
E(B-V) 0.306$\pm$0.004 0.326$\pm$0.021 0.302$\pm$0.019 0.278$\pm$0.029 0.306$\pm$0.009 0.201$\pm$0.012 0.221$\pm$0.012 0.292$\pm$0.009 0.284$\pm$0.002
: Reddening-corrected line intensity ratios, normalized to H$\beta$, for the SF knots and the nuclear region in Mrk 900.\[tab:fluxes\]
Notes. Reddening-corrected line fluxes normalized to F(H$\beta$)=1. The reddening-corrected H$\beta$ flux (in units of 10$^{-16}$erg s$^{-1}$ cm$^{-2}$), the interstellar extinction coefficient C$_{H\beta}$, and the values of the equivalent width in absorption for H$\gamma$ and H$\beta$ (in Å) are also provided in the table. A$_{V}$=2.16$\times$C(H$\beta$) and $E(B-V)$=0.697$\times$C(H$\beta$) [@Dopita2003].
Parameter
------------------------------------------------------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- -----------------
$[\ion{O}{iii}]~\lambda5007/{\mbox{H$\beta$}}$ 3.091$\pm$0.094 2.784$\pm$0.573 2.534$\pm$0.371 2.280$\pm$0.469 2.864$\pm$0.190 1.794$\pm$0.115 2.047$\pm$0.142 2.508$\pm$0.175 2.401$\pm$0.040
$[\ion{O}{i}]~\lambda6300/{\mbox{H$\alpha$}}$ 0.008$\pm$0.001 0.021$\pm$0.004 0.034$\pm$0.004 — 0.017$\pm$0.002 0.023$\pm$0.003 0.027$\pm$0.003 0.019$\pm$0.002 —
$[\ion{N}{ii}]~\lambda6584/{\mbox{H$\alpha$}}$ 0.079$\pm$0.001 0.092$\pm$0.010 0.140$\pm$0.011 0.141$\pm$0.017 0.118$\pm$0.004 0.133$\pm$0.006 0.134$\pm$0.005 0.128$\pm$0.005 0.126$\pm$0.001
$[\ion{S}{ii}]~\lambda\lambda6717\,6731/{\mbox{H$\alpha$}}$ 0.142$\pm$0.002 0.202$\pm$0.169 0.338$\pm$0.021 0.317$\pm$0.031 0.229$\pm$0.007 0.287$\pm$0.010 0.296$\pm$0.010 0.266$\pm$0.008 0.285$\pm$0.002
N$_{e}$ (cm$^{-3}$) $<$100 $<$100 $<$100 $<$100 $\approx$100 $<$100 $<$100 $<$100 $<$100
T$_{e}$ (K) 10236 — 13450 — 13840 — —- — —
12+log(O/H)$^{2}$ 8.23 8.21 8.24 8.26 8.25 8.25 8.25 8.25 8.25
: Line ratios, physical parameters, and abundances.\[tab:diagnostic\]
Notes. Abundances are derived following [@PilyuginGrebel2016].
L.M. Cair[ós]{} acknowledges support from the Deutsche Forschungsgemeinschaft (CA 1243/1-1 and CA 1243/1-2). We thank Rafael Manso Sainz for extremely stimulating discussions and a careful reading of the manuscript. The data presented here were obtained \[in part\] with ALFOSC, which is provided by the Instituto de Astrofísica de Andalucía (IAA) under a joint agreement with the University of Copenhagen and NOTSA. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, Caltech, under contract with the National Aeronautics and Space Administration. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr).
[^1]: Based on observations made with ESO Telescopes at Paranal Observatory under program ID 079.B-0445.
[^2]: The Image Reduction and Analysis Facility (IRAF) is a software system for the reduction and analysis of astronomical data. It is distributed by the NOAO, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation
[^3]: The B-band filter is mostly affected by the H$\gamma$ and H$\beta$ lines, whereas the emission from H$\alpha$, \[SII\], and \[NII\] fall in the R band.
[^4]: With solar abundance, 12+log(O/H)=8.69 [@Asplund2009]
[^5]: The SFR in [@Hunt2015] is derived using H$\alpha$ and 24 $\mu$m following [@Calzetti2010].
|
---
abstract: 'Fatigue is a critical factor in structures as wind turbines exposed to harsh operating conditions, both in the design stage and control during their operation. In the present paper the most recognized approaches to estimate the damage caused by fatigue are discussed and compared, with special focus on their applicability for wind turbine control. The aim of this paper is to serve as a guide among the vast literature on fatigue and shed some light on the underlying relationships between these methods.'
author:
- 'J.J. Barradas Berglind'
- Rafael Wisniewski
bibliography:
- 'WindEnergyRefs.bib'
title: |
Fatigue Estimation Methods Comparison\
for Wind Turbine Control
---
=1
[^1]
[^2]
Introduction and Motivation
===========================
Fatigue has been widely and exhaustively studied from different perspectives, and the literature is vast and approached from different perspectives; thus, incorporating fatigue or wear in components of a wind turbine in a control problem may seem as a daunting task. Fatigue is regarded as a critical factor in structures such as wind turbines, where it is necessary to ensure a certain life span under normal operating conditions in a turbulent environment. These environmental conditions lead to irregular loadings, which is also the case for waves and uneven roads. The main focus of the present is on fatigue estimation methods for wind turbine control, and as such the most widely used methods are described, with special emphasis in the applicability of these techniques for control.
In general, fatigue can be understood as the weakening or breakdown of a material subject to stress, especially a repeated series of stresses. From a materials perspective, it can be also thought of as elastoplastic deformations causing damage on a certain material or structure, compromising its integrity.
Fatigue is a phenomenon that occurs in a microscopic scale, manifesting itself as deterioration or damage. Consequently, it has been of interest in different fields and has been studied extensively with different perspectives; a very detailed history of fatigue can be found in [@schutz96]. It could be argued that two major turning points in the history of fatigue came firstly with the contributions of W[ö]{}hler, who suggested design for finite fatigue life in the 1860’s [@Wohler_1860] and the so-called W[ö]{}hler curve (or S-N curve stress versus number of cycles to failure) which still sets the basis for theoretical damage estimation; and secondly with the linear damage accumulation rule by Palmgren [@Palmgren_1924] and Miner [@Miner_1945], still under use nowadays.
Fatigue Estimation for Wind Turbine Control
===========================================
Perhaps the most recognized and used measure for fatigue damage estimation is the so-called rainflow counting (RFC) method, which is used in combination with the Palmgren-Miner rule. In the wind turbine context, the impact on fatigue from a load can be described by an equivalent damage load (EDL); basically, the EDL is calculated using the Palmgren-Miner rule to determine a single, constant-rate fatigue load that will produce equivalent damage [@Sutherland99].
Load or fatigue reduction techniques for wind turbines can be roughly divided in active and passive. The former makes use of the controller, e.g., by changing the pitching angle or the generator torque, while the latter entails the design of the structure. In [@bottasso13], both strategies are combined to reduce loads in the blades. In the wind turbine control context, the control algorithm may have substantial effects on the wind turbine components; for example, controlling the pitching angle may lead to thrust load changes, which consequently affects the loads on the tower and blades [@bossanyi03]. In [@bossanyi03], [@bossanyi05], [@larsen05] reductions in loading are achieved by controlling the pitch of each blade independently; the damage of different control strategies is assessed by EDL, using S-N curves. In [@lescher06], [@nourdine10] a load reduction control strategies are proposed, where the damage is evaluated using the RFC algorithm. Model predictive control (MPC) strategies using wind preview have been proposed in [@Sol_11], [@madsen12] to reduce loads, evaluated via EDL. In [@Hamm_07], control strategies were designed, by approximating fatigue load by an analytical function based on spectral moments. The Aeolus project [@aeolus] has a simulation platform, which considers the fatigue load of wind farm for optimization as a post-processing method.
A large amount of the current control methods rely on the calculation of the damage either by EDL or RFC, which can be only used as post-processing tools; other methods are based on minimization of some norms of the stress on different components of the wind turbine, which are hoped to reduce fatigue, but they are not a reliable characterization of the damage [@Sol_11], [@Mirzaei_13]. Thus, in this paper we will introduce and compare the most recognized fatigue estimation methods, and explore different alternatives with a focus on whether they can be incorporated in control loops and thus be used in the controller synthesis directly.
Fatigue Estimation Methods
==========================
Some of the most recognized approaches to estimate the damage caused by fatigue will be discussed and compared in the sequel. From a materials perspective, an extensive survey for homogeneous materials was done in [@fatemi_98]. In the wind turbine context, [@Sutherland99] goes through the counting and spectral techniques used for wind turbine design. The perspective taken here is from a control point of view and as such we categorize the fatigue estimation methods as follows:
1. [Counting methods]{}
2. [Frequency domain or spectral methods]{}
3. [Stochastic methods]{}
4. [Hysteresis operator]{}
In all cases, we assume that the input signal is obtained from time history of the loading parameter of interest, such as force, torque, stress, strain, acceleration, or deflection [@fatiguelee_2005].
Counting Methods
----------------
Cycle counting methods are algorithms that identify fatigue cycles by combining and extrapolating information from extrema (maxima and minima) in a time series. These algorithms are used together with damage accumulation rules, which calculate the total damage as a summation of increments. The most popular method among the counting methods is the so-called rainflow counting (RFC) method, jointly with the Palmgren-Miner rule of linear damage accumulation to calculate the expected damage. The Palmgren-Miner rule is the most popular due to its simplicity; however, by applying it one assumes a fixed-load, neglecting interaction and sequence effects that might have a significant contribution to the damage, e.g., [@Agerskov] for tests with random loading.
Other cycle counting methods include: peak-valley counting (PVC), level-crossing counting (LCC), range counting (RC), and range-pairs counting (RPC); for more details see [@ASTMe1049] and [@Benasciutti_Thesis]. Here, we will focus on the RFC method, which is the most widely used and the most accurate in identifying the damaging effects caused by complex loadings, [@Dowling71]. The rainflow counting method, first introduced by Endo [@Endo_1967], has a complex sequential and nonlinear structure in order to decompose arbitrary sequences of loads into cycles, and its name comes from an analogy with roofs collecting rainwater to explain the algorithm, sometimes also referred to as pagoda roof. A figure depicting the described procedure is shown below in Figure \[fig:Int\_RFC\].
![Rainflow counting damage estimation procedure.[]{data-label="fig:Int_RFC"}](Intro_RFCalgproc2)
For many materials there is an explicit relation between number of cycles to failure and cycle amplitude, which is known as S-N or W[ö]{}hler curves, given as a line in a log-log scale as
$$\begin{aligned}
s^{k}N = K,
\label{SNcurve}\end{aligned}$$
where $k$ and $K$ are material specific parameters and $N$ is the number of cycles to failure at a given stress amplitude $s$. Then, for a time history, the total damage under the linear accumulation damage (Palmgren-Miner) rule is given as
$$\begin{aligned}
D(T) = \sum\limits_{i=1}^{N(T)}\Delta D_{i} = \sum\limits_{i=1}^{N(T)}\frac{1}{N_{i}},
\label{DamagePM1}\end{aligned}$$
for damage increments $\Delta D_{i}$ associated to each counted cycle, $N_{i}$ the number of cycles to failure associated to stress amplitude $s_{i}$, and the number of all counted cycles $N(T)$. Taking the S-N curve relationship in , we can rewrite as
$$\begin{aligned}
D(T) = \sum\limits_{i=1}^{N(T)}\frac{s_{i}^{k}}{K}.\end{aligned}$$
Different RFC algorithms have been proposed such as [@downing1982] and [@rychlik_3], with different rules but providing the same results. A way to implement the RFC algorithm is using the Rainflow toolbox introduced in [@Nieslony_09]. An example is presented below, using the wind turbine model from the standard NREL 5MW wind turbine [@Jonk_NREL5MW], running is closed-loop with standard pitch and torque controllers. The input used for the comparison is a time series of the tower bending moment extracted after the simulation of $600$ seconds. The results are presented on Figure \[fig:Ex1\_RFC\]. On the top the input stress is shown, and in the bottom part the instantaneous damage and the accumulated damage are shown. For our example, we will let $k=4$ and $K=6.25\times 10^{37}$ as in [@Hamm06_Thesis], where the value of $k$ is adequate for steel structures. For this example, the instantaneous damage was extrapolated to its causing time, such that it can be plotted in the right time scale instead of the reduced turning-point scale.
![Rainflow counting algorithm example, using the toolbox from [@Nieslony_09].[]{data-label="fig:Ex1_RFC"}](M_RFC){width="95.00000%"}
Other outputs provided by the toolbox in [@Nieslony_09] are amplitude and cycle mean histograms, as well as the so-called rainflow matrix (RFM), from which the number of counted cycles with a given amplitude and mean value are obtained from the given stress history. Since the RFM will play a role further on this paper, we will elaborate on its construction. Load signals can be discretized to a certain number of levels, allowing an efficient storage of the cycles in a so-called rainflow matrix, which is an upper triangular matrix by definition. Consequently, cycle amplitudes and mean values can be grouped in bins, such that the cycle count can be summarized as a matrix (for details see [@rychlik_2], and Chapter 2 in [@Benasciutti_Thesis]); sometimes this matrix is shown transposed. The rainflow matrix for the aforementioned example is depicted on Figure \[fig:Ex1\_RFC2\] for 10 bins, where cycle mean is on the $y-$axis, cycle amplitude in the $x-$axis and number of cycles in the $z-$axis.
![Rainflow Matrix, using the toolbox from [@Nieslony_09].[]{data-label="fig:Ex1_RFC2"}](RFM2)
Lastly, NREL has a an estimator of fatigue-life called `MLife` (currently in alpha version, an improvement on `MCrunch` [@MCrunch]), which runs the RFC algorithm of [@Nieslony_09]. `MLife` calculates fatigue life for one or several time series, incorporating the Goodman correction to the damage calculation (to account and correct for the fixed-load assumption). These calculations include short-term damage equivalent loads and damage rates, lifetime results based on time series, accumulated lifetime damage, and time until failure [@MLife].
Spectral Methods
----------------
An alternative to counting methods are the so-called spectral or frequency domain methods [@Bishop_99], which assume narrow band processes and calculate the lifetime estimate by using an empirical formula that uses the spectral moments of the input signal; the aim of these methods is to approximate the rainflow density of the RFC algorithm. This procedure is depicted on Figure \[fig:Int\_Spectral\]. It is worth mentioning that some of these methods are based on empiric formulas, being essentially black-box and may be restricted to Gaussian histories. A comparison of different spectral methods was carried out in [@Benasciutti_06].
![Spectral methods damage estimation procedure.[]{data-label="fig:Int_Spectral"}](Into_SprectralProc2)
Spectral methods are based on statistical information of the signal of interest, i.e., its spectral moments. Following from [@Bishop_99] and [@Hamm_07], the *m*$^{th}$ spectral moment of the process $x(t)$ is defined as
$$\begin{aligned}
\lambda^{x}_{m} = \frac{1}{\pi}\int\limits_{0}^{\infty}f^{m}\cdot S_{x}(f) df,\end{aligned}$$
where $S_{x}(f)$ is the power density (PSD) of the process, with the following properties
$$\begin{aligned}
\lambda^{x}_{0} = \sigma^{2}_{x}, \;\; \lambda^{x}_{2} = \sigma^{2}_{\dot{x}} \;\; \text{and} \;\; \lambda^{x}_{4} = \sigma^{2}_{\ddot{x}}.
\label{moments4}\end{aligned}$$
In other words, the variance of the process is given by $\lambda^{x}_{0}$, the variance of the process’ first derivative is then given by the second moment, and lastly the variance of the process’ second derivative is given by the fourth moment. Consequently, following the results in [@Benasciutti05] and [@Ryc93] the damage rate for narrow-banded Gaussian stress histories is given by
$$\begin{aligned}
d_{\curlywedge} = \frac{1}{2\pi}\sqrt{\frac{\lambda_{4}}{\lambda_{2}}}\frac{1}{K}\left(2\sqrt{2\lambda_{0}}\right)^{k}\Gamma\left(1+\frac{k}{2}\right),\end{aligned}$$
where $\Gamma(\cdot)$ corresponds to the gamma distribution, and $k$, $K$ are the S-N parameters used in the RFC case. In [@Benasciutti05], the authors proposed an estimate of the expected fatigue damage rate given as the narrow-band approximation augmented with a correction factor to account for the process not necessarily being narrow-band
$$\begin{aligned}
E\left[d\right] \approx d_{\curlywedge} \cdot \left(b+(1-b)\alpha_{2}^{k+1}\right) \end{aligned}$$
with
$$\begin{aligned}
b = \frac{\left(\alpha_{1}-\alpha_{2}\right)\left[1.112\left(1+\alpha_{1}\alpha_{2}-(\alpha_{1}+\alpha_{2})\right)e^{2.11\alpha_{2}}+\left(\alpha_{1}-\alpha_{2}\right)\right]}{\left(\alpha_{2}-1\right)^{2}}\end{aligned}$$
and
$$\begin{aligned}
\alpha_{1} = \frac{\lambda_{1}}{\sqrt{\lambda_{0}\lambda_{2}}}, \hspace{12pt} \alpha_{2}= \frac{\lambda_{2}}{\sqrt{\lambda_{0}\lambda_{4}}}.\end{aligned}$$
In [@Hamm_07] and [@Hamm06_Thesis], the numerical integration of the spectral density as in is avoided, since the spectral moments are computed by means of polynomial evaluation and differentiation, involving a logarithm and an inverse tangent function. This allowed the method to be incorporated in the control loop.
In order to compare the spectral method with the example presented in the previous section, the spectral moments $\lambda=(\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{4})$ of the time series were calculated using the `WAFO toolbox` [@WAFO] (through integration)
$$\begin{aligned}
\lambda=\{4.4071E^{14},-3.949E^{07},2.2904E^{11},2.1263E^{11}\},\end{aligned}$$
and then the damage was computed using the Benasciutti approximation, using the `Matlab` script in Appendix B.3. of [@Hamm06_Thesis], such that
$$\begin{aligned}
d_{B} = 4.0024E^{-12},\end{aligned}$$
which is a little off compared to the RFC case; this can be explained by the fact that we need to scale the damage rate according to the geometry of the system, which is generally unknown. However, the obtained damage rate can be normalized to be used for control purposes, for details see [@Hamm_07]. In [@Ragan_07] the RFC method is compared with the spectral method using Dirlik’s formula (which approximates the rainflow density, see [@Dirlik_85]) for fatigue analysis of several components of wind turbines, where it is concluded that spectral methods work very well in some cases, but rather poorly in others due to the narrow band assumption. However, spectral methods do have the advantage of conveniently relying on spectral information that is easier to estimate from limited data.
Stochastic Methods
------------------
In [@Sobczyk_87], a thorough survey of stochastic methods for fatigue estimation in materials is presented, including reliability-inspired approaches, evolutionary probabilistic approaches and models for random fatigue crack growth. Modeling fatigue as a stochastic process makes sense due to the random nature of fatigue, which becomes more obvious under time-varying random loading.
Due to the broadness of this class of methods, we will focus on one example of the evolutionary approach. Following [@Sobczyk_87], by introducing the hypothesis that the process is Markovian, such that future outcomes only depend on present information, disregarding the past. This way, we will have a random process with only forward transitions,
$$\begin{aligned}
E_{0} \rightarrow E_{1} \rightarrow \cdots \rightarrow E_{k} \rightarrow E_{k+1} \cdots \rightarrow E_{n}=E^{*},
\label{Eq:fatigueMC}\end{aligned}$$
where $E_{0}$ denotes a damage-free state and $E^{*}$ characterizes the ultimate damage or destruction. Letting $P_{k}(t)$ be the probability that the specimen at time $t$ is on state $E_{k}
$ (notice that the state transitions are discrete, while the time evolution is continuous), then we obtain the following system of differential equations
$$\begin{aligned}
\frac{dP_{0}(t)}{dt} &= q_{0}P_{0}(t) \nonumber\\
\frac{dP_{k}(t)}{dt} &= q_{k}P_{k}(t) + q_{k-1}P_{k-1}(t), \hspace{12pt} k \geq 1,\end{aligned}$$
or in shorter notation
$$\begin{aligned}
\frac{dP_{k}(t)}{dt} &= Q P_{k}(t), \hspace{12pt} k \geq 0,
\label{Mchain}\end{aligned}$$
which corresponds to a Markov chain (MC) with intensity or transition matrix $Q$. Markov chains are well studied and have been successfully used in control settings; however, a shortcoming of this approach is that it is assumed that the intensity matrix $Q$ is not generally known. It could be assumed that the intensities are obtained from physical experiments, but this would correspond to a certain load; so, if the load changes, the parameters will change as well. However, the elements of $Q$ could be identified, using for instance recursive maximum likelihood identification methods, in order to capture the shifts in the load introduced by the controller.
In the present, for the sake of comparison, we will make use of the equivalence in [@rychlik_2], where a method to convert between rainflow matrix to a Markov matrix is presented. As an example, we take the rainflow matrix depicted in Figure \[fig:Ex1\_RFC2\] and use the `WAFO` toolbox to convert it into a Markov matrix, and obtain its corresponding intensity matrix $Q$. Additionally, the MC is simulated for as many steps as the length of turning points of the RFC algorithm, such that the instantaneous damage can be reconstructed in the appropriate time instances. The simulation of the MC is presented on Figure \[fig:MCsimulation\], where the size of the MC corresponds to the number of bins of the RFM.
![Markov Chain simulation, using the WAFO toolbox [@WAFO].[]{data-label="fig:MCsimulation"}](MCsim)
Then, the damage evolution is scaled according to the RFM amplitudes, and afterwards the Palmgren-Miner rule is used. One of the possible realizations is compared against the RFC method on Figure \[fig:RFCvsMC\]. Note that many realizations for the damage evolution are possible, since the MC in is governed by probabilities.
![RFC versus Markov chain method damage comparison.[]{data-label="fig:RFCvsMC"}](RFC_MC)
Hysteresis Operator
-------------------
As mentioned in [@downing1982] and [@fatiguelee_2005], the purpose of the RFC method is to identify the closed hysteresis loops in the stress and strain signals. In [@tchankov1998], an incremental method for the calculation of dissipated energy under random loading is presented, where the dissipated hysteresis energy to failure is used as the fatigue life parameter; the physical interpretation is that as some of the energy is dissipated, certain damage is introduced to a material or structure.
In [@BroSpre_96] an equivalence between symmetric RFC and a Preisach hysteresis operator is provided. This is a very useful result, since it gives the opportunity to incorporate the fatigue estimation online in the control loop. Additionally, this method is strongly related to the physical behavior of the damaging process as explained in [@BroDreKre_96]. If one associates values to individual cycles or hysteresis loops, it is being assumed that the underlying process is rate independent, thus meaning that only the loops themselves are important, but not the speed with which they are traversed; in other words, what causes the damage is the cycle amplitude and not how fast it occurs. Rate independent processes are mathematically formalized as hysteresis operators, see [@KrasPok_89], [@Mayergoyz_91] [@BroSpre_96].
The aforementioned equivalence in [@BroSpre_96] between symmetric rainflow counting (RFC) and a type of Preisach operator, is given as
$$\begin{aligned}
D_{ac}(s)=\sum_{\mu<\tau}\frac{c(s)(\mu,\tau)}{N(\mu,\tau)}=\text{Var}(\mathcal{W}(s)).
\label{Cor2_13}\end{aligned}$$
where the left-hand side corresponds to the damage given by the RFC with $c(s)(\mu,\tau)$ being the rainflow count associated with a fixed string $s=(v_{0},\cdots,v_{N})$, counting between the values of $\mu$ and $\tau$, and $N(\mu,\tau)$ denotes the number of times a repetition of the input cycle $(\mu,\tau)$ leads to failure.
The right-hand side of is the variation of a special hysteresis operator, namely the Preisach operator defined as,
$$\begin{aligned}
\mathcal{W}(s) = \int_{\mu<\tau}\rho(\mu,\tau) \mathcal{R}_{\mu,\tau}(s)d\mu d\tau.
\label{preisach_op}\end{aligned}$$
with density function $\rho(\mu,\tau)$, interpreted as a gain that changes with the different values of $\mu$ and $\tau$, being a function of $N(\mu,\tau)$. To interpret the right-hand side of we will need to introduce the relay operator $\mathcal{R}_{\mu,\tau}(s) = \mathcal{R}_{\mu,\tau}(v_{0},\cdots,v_{N})=(w_{0},\cdots,w_{N})$, where its output is given by
$$\begin{aligned}
w_{i} = \left\{
\begin{array}{l l}
1, & \quad v_{i}\geq \tau,\\
0, & \quad v_{i}\leq \mu,\\
w_{i-1}, & \quad \mu<v_{i}<\tau.
\end{array} \right.\end{aligned}$$
with $\mu<\tau$ and $w_{-1} \in \{0,1\}$ given. The relevant threshold values for the relays $\mathcal{R}_{\mu,\tau}$ in the Preisach operator $\mathcal{W}(s)$ then lie within the triangle
$$\begin{aligned}
P=\left\{(\mu,\tau)\in\mathbb{R}^{2},-M\leq \mu\leq \tau\leq M\right\}.
\label{prei_plane}\end{aligned}$$
known as the Preisach plane. The variation operator $\text{Var}(\cdot)$ is a counting element defined as
$$\begin{aligned}
\text{Var}(s)=\sum^{N-1}_{i=0}\left|v_{i+1}-v_{i}\right|\end{aligned}$$
for an arbitrary input sequence $s=(v_{0},\cdots,v_{N})$; so essentially, $\text{Var}(\mathcal{W}(s))$ represents the counting between the thresholds $\mu$ and $\tau$, weighted by certain gain $\rho$. Notice as well, that the limit under the integral defining the Preisach operator in is congruent with the RFM being upper triangular.
In order to apply this fatigue estimation method to the previous example, the Preisach operator $\mathcal{W}(s)$ was approximated as a parallel connection of three relay operators
$$\begin{aligned}
\mathcal{H}(s)=\sum_{i}\nu(\mu_{i},\tau_{i})\mathcal{R}_{\mu_{i},\tau_{i}}(s),\end{aligned}$$
for $i=\{1,2,3\}$. The thresholds were set to $(\mu_{1},\tau_{1})=(-0.66M,0.66M)$, $(\mu_{2},\tau_{2})=(0.66M,0.66M)$ and $(\mu_{3},\tau_{3})=(-0.66M,-0.66M)$ corresponding to uniform discretization, where $M$ is the bound for the Preisach plane in calculated as $M=\max\left\{\min\left\{s\right\},\max\left\{s\right\}\right\}$. The initial conditions of the relays were given according to the following condition:
$$\begin{aligned}
w_{-1}(\mu_{i},\tau_{i}) = \left\{
\begin{array}{l l}
1, & \quad \mu_{i}+\tau_{i}<0, \\
0, & \quad \mu_{i}+\tau_{i}\geq 0.
\end{array} \right.\end{aligned}$$
Lastly, since the Preisach density function $\rho(\mu,\tau)$, captured by the weightings on each relay $\nu(\mu_{i},\tau_{i})$ is unknown, the individual weightings of each relay were normalized such that $\nu_{1}=\alpha$, $\nu_{2}=\alpha^2$, $\nu_{3}=\alpha^3$ for $\nu_{1}+\nu_{2}+\nu_{3}=1$. Thus the accumulated damage can be written in closed form as
$$\begin{aligned}
D_{ac}(s) = \text{Var}\left(\mathcal{H}(s)\right),
\label{eq:EXdamage}\end{aligned}$$
where we let the input signal $s$ be the tower bending moment from the previous examples.
![RFC versus Hysteresis method damage comparison.[]{data-label="fig:RFCvsHyst"}](Ex3_Combined3)
A comparison between the RFC, using the procedure described before, and the hysteresis method obtained by is shown in Figure \[fig:RFCvsHyst\]. Even though the magnitude in the damage given by the hysteresis method is off scale, this could be resolved by identifying the Preisach density, see [@Kris_01] for an identification procedure and a summary of other identification methods.
It is worth mentioning that the results in apply to symmetric RFC. As mentioned in [@BroDreKre_96] not all RFC methods are symmetric; however, for symmetric RFC the so-called Madelung rules apply, i.e., deletion pairs commute, meaning that it does not matter the order in which the sequences are deleted. However, if the primal concern is to apply this technique online, no deletion is actually possible since the estimation is done directly on measurements.
Crack Growth approaches
-----------------------
Another alternative for fatigue estimation is the crack growth approach, which can be both addressed from a deterministic view-point using Paris’ law ([@paris63]), or a stochastic perspective using for example jump processes, diffusion processes or stochastic differential equations (SDEs). However, in the crack growth approach a microscopic scale perspective is taken, thus making it difficult to transport to system level. We refer the interested readers to [@Sobczyk_FCG], [@fatemi_98] and the references therein.
Methods Comparison and Discussion
=================================
The aforementioned fatigue estimation methods share certain relations between each other. Firstly, there is an equivalence between the rainflow matrix and the Markov matrix or intensity of the Markov chain. Moreover, both have zeros below the diagonal, which is also the case for the Preisach plane $P$ in the Hysteresis method. The Spectral methods are related to RFC, since their intention is to approximate the rainflow density by spectral formulas, and they also relate to the stochastic methods in that their goal is to approximate certain density function. The hysteresis method is strongly related to the RFC, since the RFC actually identifies the closed hysteresis loops by counting cycles. A sketch of these relationships is depicted on Figure \[fig:ConnectionDiagr\].
![Relationship between the compared methods.[]{data-label="fig:ConnectionDiagr"}](ConnectionDiagram)
Furthermore, a method comparison summary is shown on Table \[tab:AdvDisadv\], where advantages and disadvantages are presented for each method previously introduced.
**Method** **Advantages** **Disadvantages**
------------ -------------------------------- ------------------------------------
Rainflow Active Standard (ASTM E1049) Post-processing
Counting Widely used Relies on linear accum. hypothesis
Algorithmic, very non-linear
Spectral Can be used for control Black-box
Based on statistical measures Narrow-band approximation
Stochastic Account for random loading Parameters generally unknown
Methods Could be used for prediction May involve PDEs, SDEs
Very abstract formulation
Hysteresis Online estimation Typically hard control problem
Strong physical interpretation Density generally unknown
Close mathematical form Approximation may be needed
: Methods advantages and disadvantages.[]{data-label="tab:AdvDisadv"}
For the next comparison part we will focus just on the MC instead of the whole stochastic methods class, which is quite broad. The accumulated damage provided by the RFC, MC and Hysteresis methods are compared in Figure \[fig:RFCHystMC\]. The damage given by the hysteresis was normalized, such that it matches the accumulated damage of the RFC. The spectral method example could not be included, since the method delivers the damage rate itself and not instantaneous measurements. For the RFC and the MC method presented here, the instantaneous damage is given every time an extrema occurs and zero elsewhere, which is exactly what the hysteresis does, i.e., hold the value between certain thresholds. All these techniques can be used as post-processing tools, however not all of them can be used in the control loop. A brief summary is presented on Table \[tab:Control\], where it is reported if the methods can be implemented directly online or indirectly, i.e., not using measurements. The spectral methods are included indirectly, since they were included in the loop through transfer functions and not based on measurements. The Markov chain could be included online if the intensity matrix is parametrized with respect to the controls, which may not be realizable.
**Method** **Online** **Indirect** **Comments**
------------ ------------ -------------- ---------------------------------------
RFC - - Only Post-processing
Spectral - X Moments obtained by transfer function
Hysteresis X - Approximation may be needed
: Methods applicability for control.[]{data-label="tab:Control"}
![Normalized accumulated damage for different estimation methods.[]{data-label="fig:RFCHystMC"}](RFC_Hyst_MC)
Conclusions
===========
The literature regarding fatigue estimation methods is vast, since fatigue is an entire discipline by itself. The aim of the present paper is to provide a guide to the most recognized methods, which were assembled in four groups. These methods were presented and compared, from a control perspective in a Wind Turbine setting by estimating the damage from a tower bending moment time-series. A chart describing their advantages and disadvantages is presented on Table \[tab:AdvDisadv\] and their applicability to control in Table \[tab:Control\]. We also attempted to shed some light on the underlying relations between them.
Summarizing, the most widely used and standardized method is the RFC, but its algorithmic nature restricts its usage primarily as a post-processing tool. The spectral methods provide an alternative by trying to emulate the rainflow density function, they are based on statistical measures that are easier to calculate, but they are black-box and restricted (mainly) to narrow-band processes. The stochastic methods can accommodate the randomness of fatigue, but their construction is abstract and complicated, often involving stochastic or partial differential equations, and their parameters may need identification. The hysteresis method can be implemented online, acting on instantaneous measurements, but its complex and non-linear nature results in hard control problems. In general, one could say that the controller will influence the loading in the wind turbine components, and thus for implementing any of these techniques in the control loop, variable load should be considered by the estimation method in some sense.
ACKNOWLEDGEMENT
===============
This work was partially supported by the Danish Council for Strategic Research (contract no. 11-116843) within the ‘Programme Sustainable Energy and Environment’, under the “EDGE” (Efficient Distribution of Green Energy) research project.
[^1]:
[^2]: *Pre-print submitted to Wind Energy.*
|
---
author:
- Roy Barbara
title: The Rational Distance Problem for Equilateral Triangles
---
§-1 **Abstract**\
Let *(P)* denote the problem of existence of a point in the plane of a given triangle T, that is at rational distance from all the vertices of T. Answer to *(P)* is positive if T has a rational side and the square of all sides are rational (see \[1\]). In \[2\], a complete solution to *(P)* is given for all isosceles triangles with one rational side. In this article, we provide a complete solution to *(P)* for all equilateral triangles.\
*In all what follows*, $\theta$ denotes an arbitrary positive real number and $T=[\theta]$ denotes the equilateral triangle with side-length $\theta$. For convenience, we say that $\theta$ is “good” (or “suitable”) if answer to *(P)* is positive for the triangle $T=[\theta]$. Clearly, the property “$\theta$ is good” is invariant by any rational re-scaling of $\theta$.\
It turns out that the *good* $\theta$ must have algebraic degree $1,\,2,\,$ or $4$, and they form a subclass of the *positive* bi-quadric numbers, that is, the positive roots of equations of the form $x^4+ux^2+v=0$, $\,u,\,v\,\in \mathbb{Q}$. The general form of such numbers is
$\sqrt{\alpha\pm\sqrt{\beta}},\;\;\;\;\; \alpha,\,\beta\;\in \mathbb{Q},\;\,\beta\geq \,0,\;\alpha\pm\sqrt{\beta}\,\geq\,0 $
that includes positive numbers of the form
$\alpha,\,\sqrt{\alpha},\,\alpha\pm\sqrt{\beta}\,\,\sqrt{\alpha}\pm\sqrt{\beta},\;\;\;\;\; \alpha,\,\beta \,\in \mathbb{Q},\;\,\alpha,\,\beta\,\geq \,0.$
**Notations and conventions**: $(x,\,y)$ and $(x,\,y,\,z)$ denote the g.c.d. $\displaystyle \big{(}\frac{x}{p}\big{)}$ denotes legendre’s symbol. A triangle with side-lengths $a,\,b,\,c \,$ is denoted by $T=[a,\,b,\,c]$. A triangle is non-degenerated if it has positive area. A radical is non-degenerated if it is irrational.\
\
13.5pt §-2 **The results**
**0** If $\theta$ is good, then, $\theta$ is bi-quadric. More precisely, $\theta^2=\alpha\pm\sqrt{\beta}$ for some $\alpha,\,\beta\in \mathbb{Q},\,\, \beta\geq\,0,$ and $\alpha$ *positive*.
**1** Suppose $\theta \notin \mathbb{Q}$ and $\theta^2 \in \mathbb{Q}$. Then,\
$\theta$ is good $\Leftrightarrow \theta $ has the form $\theta=\lambda\sqrt{p_1...p_r}$ where $\lambda \in \mathbb{Q},\,\lambda\,> \,0,\,r\ge \,1,\, p_1,...,p_r$ are distinct odd primes, $p_i$ is either $3$ or of the form $6k+1$.
**2** Suppose $\theta^2=\alpha\pm \sqrt{\beta},\,\alpha,\,\beta\,\in\mathbb{Q},\,\alpha,\,\beta\,>0,\,\sqrt{\beta}\,\notin \mathbb{Q}$. Then,\
$\theta$ is good $\;\Leftrightarrow\;$ up to a rational re-scaling of $\theta,\,\,\theta$ is described as follows:
$2\theta^2=(a^2+b^2+c^2)\pm\,4\Delta \sqrt{3}$
where $[a,\,b,\,c]$ is a non-degenerated primitive integral triangle with area $\Delta$ such that $4\Delta\,\sqrt{3}\,\notin \mathbb{Q}$.
$\Delta$ is given by Hero’s formula, $\Delta=\sqrt{s(s-a)(s-b)(s-c)},\,s=\frac{1}{2}(a+b+c)$. Equivalently, $4\Delta\sqrt{3}=\sqrt{3(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}$, and the condition $4\Delta\sqrt{3}\,\notin\,\mathbb{Q}$ means that this latter radical is non-degenerated.
§-3 **Proofs of theorems 0 and 1**\
*Proof of theorem 0:* Suppose $\theta$ good. Let $M$ be a point in the plane of triangle $T=[\theta]$, whose distances from the vertices of $T$ are all rational. the following fundamental relation is well-known (see\[3\]):
$3(a^4+b^4+c^4+\theta^4)=(a^2+b^2+c^2+\theta^2)^2\;\;\;\;\qquad\quad(\centerdot)$
Expanding $(\centerdot)$ yields a relation as $\;\,\theta^4-u\theta^2+v=0$, where $u,\,v\,\in \mathbb{Q}\;$ and $\;u=a^2+b^2+c^2\\>\,0$. Solving for $\theta^2$ yields $\;\,\theta^2=\alpha\pm\sqrt{\beta},$ with $\alpha,\,\beta\,\in \mathbb{Q}\;$ and $\;\alpha=\frac{1}{2}u\,>\,0.$
**1**: let $q\,>\,1$ be a square-free integer. Then we have:\
The equation $x^2+3y^2=qz^2$ has a solution in integers $x,\,y,\,z,$ with $z\ne 0$ if and only if any prime factor of $q$ is either $3$ or of the form $6k+1$
***Proof***: Suppose first that q has only prime factors as $3$ or $6k+1$. Since the quadratic form $x^2+3y^2,\,x,\,y\,\in \mathbb{Z},$ represents $3$ and every prime $p=6k+1$, and since the set$\{x^2+3y^2,\,x,\,y\,\in \mathbb{Z}\}$ is cosed by multiplication, we conclude that the equation $x^2+3y^2=q.z^2$ has a solution in integers $x,\,y,\,z$ with $z=1.$\
Conversely, suppose that $x^2+3y^2=q.z^2$ has a solution in integers $x,\,y,\,z,\;\,z\ne 0$. Pick such a solution with $|z|$ minimum. Clearly, $(x,\,y)=1.$ I claim that $q$ is *odd* and has no prime factor $6k-1$. For the purpose of contradiction, we consider two cases:\
**case 1**: $q$ is even. Set $q=2w, w$ odd. From $x^2+3y^2=2wz^2$, we see that $x\equiv y (mod. 2).$ As $(x,y)=1,\,x$ and $y$ must be odd, so $x^2+3y^2 \equiv 4 \,(mod. 8)$. Now, $4/2wz^2$ yields $wz^2$ even. But $w$ is odd, hence $z$ is even, so $2wz^2\equiv 0\;(mod. 8)$. We get a contradiction.\
**case 2**: $q=p.w$ for some prime $p=6k-1.\,\, x^2+3y^2=pwz^2$ yields $x^2+3y^2\equiv 0 \,(mod. p)$. As $(x,y)=1,\,p$ cannot divide $y$. Hence for some $t\in\mathbb{Z},\,\,yt\equiv 1 \,(mod.p).$ Therefore, $x^2t^2+3y^2t^2\equiv x^2t^2+3\equiv 0 (mod.p),$ so $-3\equiv (xt)^2 \,(mod. p)$. Hence $\displaystyle \big{(}\frac{-3}{p}\big{)}=+1$ contradicting $ p=6k-1$.\
**2**: Let $\theta = \lambda \sqrt{q},\,\lambda\,\in\,\mathbb{Q},\,\lambda > 0,\, q > 1$ square-free integer. We have:\
$\theta$ is good $\Leftrightarrow$ There are $a,\,b,\,e,\,r,\,s\,\in \mathbb{Q},\,e\ne\,0,$ such that
$a^2+3b^2=q\;\;\;\;\;\quad\quad\qquad\qquad\qquad (1)$
$(a+e)^2+3(b+e)^2=qr^2\;\;\;\;\;\;\qquad(2)$
$(a-e)^2+3(b+e)^2=qs^2\;\;\;\;\;\;\qquad(3)$
***Proof***: By re-scaling, we take $\theta=2\sqrt{q}$. Let $T=ABC=[\theta]$. Choose a $x-y$ axis to get the coordinates $A(0,\,\sqrt{3q}),\;B(-\sqrt{q},\,0),\;C(\sqrt{q},\,0)$.\
$\bullet$ Suppose first that $\theta$ is good: There is a point $M=M(x,\,y)$ in the plane of $T$ such that $MA,\, MB,\, MC\,\in \mathbb{Q}$. Clearly, $M\ne A,\,B,\,C$. Set $w=\displaystyle\frac{MA}{q},\,r=\frac{MB}{wq},\,s=\frac{MC}{wq}$. Then, $w,\,r,\,s\,\in \mathbb{Q}-\{0\}.$ The Pythagoras relations are:\
$\overline{MA}^2=x^2+(y-\sqrt{3q})^2=w^2q^2\;\;\;\;\; (1')$
$\overline{MB}^2=(x+\sqrt{q})^2+y^2=w^2q^2r^2\;\;\;\;\; (2')$
$\overline{MC}^2=(x-\sqrt{q})^2+y^2=w^2q^2s^2\;\;\;\;\; (3')$
Subtracting (2’) and (3’) yields $\;x=\frac{1}{4}w^2q(r^2-s^2).\sqrt{q},\;$ that is,
$x=\alpha\sqrt{q},\,\;\alpha \in \mathbb{Q}\;\;\;\;\;\;\;\;\;\;\qquad(4)$
Then (2’) gives $y^2\in\mathbb{Q}$, and then (1’) gives $ 2y\sqrt{3q}\in \,\mathbb{Q}$, hence, $y=\gamma \sqrt{3q},\,\gamma\in \mathbb{Q}$.\
For convenience, we put $\gamma=\beta+1$, obtaining
$y=(\beta+1)\sqrt{3q},\,\,\beta\in\mathbb{Q}\;\;\;\;\quad (5)$
Due to (4) and (5), equations (1’), (2’), (3’) become after dividing by $q$ :
$\alpha^2+3\beta^2=qw^2$
$(\alpha+1)^2+3(\beta+1)^2=qw^2r^2$
$(\alpha-1)^2+3(\beta+1)^2=qw^2s^2$
Set $a=\displaystyle \frac{\alpha}{w},\,b=\frac{\beta}{w},\,e=\frac{1}{w}$. Dividing by $w^2$, we get precisely relations (1), (2), (3).\
$\bullet$ Conversely suppose that relations (1), (2), (3) hold with some $a,\,b,\,e,\,r,\,s\,\in \mathbb{Q},\, e\ne 0$. Define point $M=M(x,\,y)$ in the plane of $T$ by
$\displaystyle x=\frac{a}{e}\sqrt{q},\;\;\;\; y=(\frac{b}{e}+1)\sqrt{3q}$
We may write:
$\displaystyle\overline{MA}^2=x^2+(y-\sqrt{3q})^2=q\frac{a^2}{e^2}+3q\frac{b^2}{e^2}=\frac{q}{e^2}(a^2+3b^2)=\frac{q}{e^2}.q=(\frac{q}{e})^2$
$\displaystyle\overline{MB}^2=\big{(}(\frac{a+e}{e})\sqrt{q}\big{)}^2+\big{(}(\frac{b+e}{e})\sqrt{3q}\big{)}^2=\frac{q}{e^2}\big{(}(a+e)^2+3(b+e)^2\big{)}=\frac{q}{e^2}.qr^2=(\frac{qr}{e})^2$
$\displaystyle\overline{MC}^2=\big{(}(\frac{a-e}{e})\sqrt{q}\big{)}^2+\big{(}(\frac{b+e}{e})\sqrt{3q}\big{)}^2=\frac{q}{e^2}\big{(}(a-e)^2+3(b+e)^2\big{)}=\frac{q}{e^2}.qs^2=(\frac{qs}{e})^2$
Therefore, $MA,\,MB,\,MC$ are all rational.\
\
***Proof of theorem 1***\
Let $\theta$ such that $\theta\,\notin \mathbb{Q}$ and $\theta^2\,\in\mathbb{Q}$: $\theta$ can be written as $\theta=\lambda\sqrt{q},\,\lambda\in\,\mathbb{Q},\,\lambda\,>\,0,\,q>1$ square-free integer.\
$\bullet$ Suppose first that $q$ is even or has a prime factor $6k-1$. By lemma 1, $a^2+3b^2=q,\;\,a,\,b\in \mathbb{Q}$, is impossible. *Hence*, relation (1) in lemma 2 fails, so $\theta$ is not good.\
$\bullet$ Suppose now that $q$ has only prime factors as $3$ or $6k+1$. We show that $\theta$ is good using the characterization of lemma 2:\
By lemma 1, for some $a,\,b\in \mathbb{Q}$, we have $a^2+3b^2=q$. Set $e=\displaystyle -\frac{q}{4b}=\frac{-(a^2+3b^2)}{4b}$, $r=\frac{a-b}{2b},\;\;\;s=\frac{a+b}{2b}\;\;$. We have\
$(a+e)^2+3(b+e)^2=(a^2+3b^2)+4e^2+2e(a+3b)=q+\displaystyle\frac{q^2}{4b^2}-\frac{q}{2b}(a+3b)$\
$\quad=\displaystyle\frac{q}{4b^2}\big{(}4b^2+q-2b(a+3b)\big{)}=\frac{q}{4b^2}(4b^2+a^2+3b^2-2ab-6b^2)$\
$=\displaystyle\frac{q}{4b^2}(a^2+b^2-2ab)=q\frac{(a-b)^2}{4b^2}=q.r^2 \quad \quad $ and\
$\displaystyle (a-b)^2+3(b+e)^2=(a^2+3b^2)+4e^2-2e(a-3b)=q+\frac{q^2}{4b^2}+\frac{q}{2b}(a-3b)$\
$=\displaystyle\frac{q}{4b^2}\big{(}4b^2+q+2b(a-3b)\big{)}=\frac{q}{4b^2}(4b^2+a^2+3b^2+2ab-6b^2)$\
$=\displaystyle\frac{q}{4b^2}(a^2+b^2+2ab)=q\frac{(a+b)^2}{4b^2}=q.s^2$\
\
\
§-4 ****Proof of theorem 2****\
**3**: Let $x,\,y,\,z,\,t$ be positive real numbers such that
$3(x^4+y^4+z^4+t^4)=(x^2+y^2+z^2+t^2)^2\;\;\quad\qquad(\circledcirc)$
Then, any three of $x,\,y,\,z,\,t$ satisfy the triangle inequality.
$ $\
***Proof***: Since $x,\,y,\,z,\,t$ play symmetric roles, it suffices to show that $x,\,y,\,z$ satisfy the triangle inequality. Write $(\circledcirc$) as
$t^4-(x^2+y^2+z^2)t^2+(x^4+y^4+z^4-x^2y^2-y^2z^2-z^2x^2)=0$
The discriminant $\bigtriangleup$ of this trinomial in $t^2$ must be non-negative. But, $\bigtriangleup=6(x^2y^2+y^2z^2+z^2x^2)-3(x^4+y^4+z^4)$ that factors as $\;\bigtriangleup =3(x+y+z)(-x+y+z)(x-y+z)(x+y-z)$.\
Hence, $(-x+y+z)(x-y+z)(x+y-z)\geq 0.$ The reader can easily check (using contraposition) that $x,\,y,\,z$ must satisfy the triangle inequality.\
**4**: Let $T=ABC\,=\,\big{[}\theta\big{]}$. Let $a,\,b,\,c$ be positive real numbers satisfying
$3(a^4+b^4+c^4+\theta^4)=(a^2+b^2+c^2+\theta^2)^2$
Then, there is a point $M$ in the plane of $T$ such that $\;MA=a,\,MB=b,$ and $MC=c$.
***Proof***: By lemma 3, $a,\,b,$ and $\theta$ satisfy the triangle inequality. In particular, $a+b\geq\theta$. It folows that the circle $\mathcal{C}(A,a)$ intersects the circle $\mathcal{C}(B,b)$ at two points $M_1$ and $M_2\;\,(M_1=M_2$ if $a+b=\theta)$. Set $c_1=M_1C$ and $c_2=M_2C$. By the fundamental relation $(\centerdot)$ we have $3(a^4+b^4+c_1^4+\theta^4)=(a^2+b^2+c_1^2+\theta^2)^2$ and $3(a^4+b^4+c_2^4+\theta^4)=(a^2+b^2+c_2^2+\theta^2)^2$. Therefore, $c_1^2$ and $c_2^2$ are the roots of the trinomial in $T$
$ T^2-(a^2+b^2+\theta^2)T+(a^4+b^4+\theta^4-a^2b^2-b^2\theta^2-\theta^2a^2)=0$
Since by hypothesis $c^2$ is also a root of this trinomial, we must have $c^2=c_1^2$ or $c^2=c_2^2$. Hence $c=c_1$ or $c=c_2$. Therefore, $a,\,b$ and $c$ are the distances from either point $M_1$ or $M_2\;$ to the vertices $A,\,B$ and $C\;$ of $T$.\
\
***Proof of theorem 2***:\
Let $\theta > 0$ such that $\theta^2=\alpha\pm\sqrt{\beta},\;\,\alpha,\,\beta\,\in\mathbb{Q},\;\alpha,\,\beta\,>0,\;\sqrt{\beta}\notin\mathbb{Q}$.\
$\bullet$ Suppose first that $\theta$ is good: let $P$ be a point in the plane of $T=ABC=[\theta]$ such that $PA=a,\,PB=b,\,PC=c\;$ are all rational. We have
$3(a^4+b^4+c^4+\theta^4)=(a^2+b^2+c^2+\theta^2)^2\;\;\;\qquad\quad(\centerdot)$
By lemma 3, $\;a,\,b,$ and $c\,$ satisfy the triangle inequality. Relation $(\centerdot)$ yields
$\theta^4-U\theta^2+V=0\;$ with $\;U=a^2+b^2+c^2\;$ and $V=a^4+b^4+c^4-a^2b^2-b^2c^2-c^2a^2\;\;(U,V\in\mathbb{Q})$.
Solving for $\theta^2$, we get
$2\theta^2=(a^2+b^2+c^2)\pm\sqrt{3(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}\;\;\;\quad\quad(\star)$
Since $\theta^2$ has algebraic degree $2$, then, the radical in $(\star)$ is non-degenerated. In particular, the triangle $[a,b,c]$ is non degenerated. Select a sufficiently large positive integer $N$ such that $Na,\,Nb,\,Nc$ are all integers and set $D=(Na,\,Nb,\,Nc)$. If we multiply relation $(\star)$ by $\displaystyle\frac{N^2}{D^2}$, this results in replacing in $(\star)\;\theta$ by $\displaystyle\frac{N}{D}.\theta$ and $a,\,b,\,c$ by the integers $\displaystyle\frac{Na}{D},\,\frac{Nb}{D},\,\frac{Nc}{D}$ respectively. As an outcome, we obtain *essentially* the same relation $(\star)$ where $\theta$ has been re-scaled by the rational $\displaystyle\frac{N}{D}$, and where the new symbols $a,\,b,\,c$ represent relatively prime *positive* integers, satisfying the triangle inequality.\
$\bullet$ Conversely, suppose that for some positive rational $\lambda,\;\theta_0=\lambda.\theta$ is described precisely as in theorem 2. Eliminating the radical
$4\bigtriangleup \sqrt{3}=\sqrt{6(a^2b^2+b^2c^2+c^2a^2)-3(a^4+b^4+c^4)}$
in the relation $\;2\theta_0^2=(a^2+b^2+c^2)\pm4\bigtriangleup\sqrt{3}\;\;$ leads to
$3(a^4+b^4+c^4+\theta_0^4)=(a^2+b^2+c^2+\theta_0^2)^2$
By lemma 4 there is a point $M$ in the plane of $T=[\theta_0]$ that is at distances $a,\,b,\,c$ from the vertices of $T$. Since $ a,\,b,\,c$ are integers, then, $\theta_0$ is good. Therefore, $\theta=\lambda^{-1}\theta_0$ is also good.\
\
\
We end this article with a few exercises:
1. Check which are “good” among the radicals: $\sqrt{2},\,\sqrt{3},\,\sqrt{5},\,\sqrt{6},\,\sqrt{7},\,\sqrt{10}$.
2. Show that the positive real number $\theta=\sqrt{25+12\sqrt{3}}$ is “good”.
3. Suppose that $2\theta^2=\alpha+\sqrt{\beta},\,\alpha,\,\beta\in\mathbb{Q},\,\alpha,\beta > 0,\sqrt{\beta}\notin\mathbb{Q}$, and $\alpha^2\,<\,\beta$. Show that $\theta$ is not good.
4. Produce solution-points to problem *(P)* for the triangle $T=[\sqrt{3}]$.
5. Let $\theta=\alpha+\beta\sqrt[4]{q}\,>\,0,\;\alpha,\beta\in \mathbb{Q},\,\beta\neq0,\,q\,>\,1$ square-free integer. Show that $\theta$\
is not good.
6. Suppose that $2\theta^2=\alpha\pm\sqrt{\beta}\,>0,\,\alpha,\,\beta\in\mathbb{Q},\,\alpha,\beta > 0,\sqrt{\beta}\notin\mathbb{Q}$. Write the fraction $\alpha$ in *lowest terms* as $\displaystyle \alpha=\frac{m}{n}\,(m,\,n\;$ positive integers) and suppose that $mn$ has the form $\;mn=4^l(8k+7),\,k,\,l$ non-negative integers. Then, prove that $\theta$ is not good.
[99]{}
T.G. Berry, *Points at rational distance from the vertices of a triangle*, Acta Arithmetica, LXII.4(1992).
Roy Barbara and Antoine Karam, *The Rational Distance Problem for Isosceles Triangles with one Rational Side*, Communications in Mathematics and Applications, Vol.4, Number 2, pp. 169-179 (2013). Wikipedia *or* Wolfram Mathworld, *Equilateral Triangle*.
Roy Barbara, *the rational distance problem for polygons*, Mathematical Gazette, Vol. 97, Number 538, note 97.11 (2013).
\[lastpage\]
|
---
abstract: 'Accurate measurements of osculating orbital elements are essential in order to understand and model the complex dynamic behavior of Near Earth Asteroids (NEAs). ESA’s Gaia mission promises to have great potential in this respect. In this article we investigate the prospects of constraining orbits of newly discovered and known NEAs using nearly simultaneous observations from the Earth and Gaia. We find that observations performed simultaneously from two sites can effectively constrain preliminary orbits derived via statistical ranging. By linking discoveries stored in the Minor Planet Center databases to Gaia astrometric alerts one can identify nearly simultaneous observations of Near Earth Objects and benefit from improved initial orbit solutions at no additional observational cost.'
address: 'IMCCE Observatroire de Paris, UPMC, Université Lille 1, 75014 Paris, (France)'
author:
- Siegfried Eggl
- Hadrien Devillepoix
bibliography:
- 'eggl.bib'
title: 'On the prospects of Near Earth Asteroid orbit triangulation using the Gaia satellite and Earth-based observations'
---
Introduction
============
Both, the airburst of a bolide over Chelyabinsk/Russia on Feb. 15$^{th}$, 2013 and the deep close encounter of the asteroid 2012 DA14 missing the Earth by as little as 3.5 Earth radii [@jpl-sentry-2012] highlighted once more that Near Earth Object (NEO) pose a non-negligible threat to mankind. Predicting future encounters between asteroids and the Earth should, therefore, be considered a task of high priority. Yet, current estimates project that only about 30% of the total NEO population with diameters between $100$m and $1$km have been discovered so far [@mainzer-et-al-2012]. This issue is further aggravated by the fact that discovering an asteroid does not automatically entail knowledge on whether it will collide with the Earth or not. Due to the complex interplay of gravitational and non-gravitational forces, long term predictions of NEA impact risks are a difficult task, especially when initial orbits[^1] are poorly constrained. Regarding discovery and orbit improvement, the Gaia astrometry mission [@mignard-et-al-2007] has been found to hold great potential for NEA research [@bancelin-et-al-2010; @hestroffer-et-al-2010; @tanga-mignard-2012]. Given the fact that Gaia is a space observatory with a fixed scanning law, however, consecutive observations of newly discovered objects, which are vital for initial orbit determination, are necessarily sparse. In practice this means that many objects have to be followed up from ground based sites [@thuillot-2011]. Since only directional data are available from astrometric observations, preliminary orbital elements and ephemeris are generated using statistical ranging (SR) [@virtanen-et-al-2001]. The resulting orbits are used to project a NEO’s future position on the sky plane to facilitate follow up observations. Due to down-link schedules and data processing, the delay between a discovery alert and a follow up can be as large as 48 hours. If the uncertainties in the initial orbits are large, the range of possible locations for the target NEO tends to grow very quickly. Retrieving newly discovered objects might become difficult in such cases. In this work we assess the potential of nearly simultaneous[^2] NEO observations from Gaia and the Earth to tackle this issue. A combination of Gaia data and ground based observations has already been found to greatly enhance the quality of orbit predictions [@bancelin-et-al-2011]. Yet, simultaneous TR of one object from two different locations was not considered in this work. In contrast, several authors have proposed asteroid orbit improvement based on multiple observing stations considering a combination of space-based and Earth-bound observation sites [@gromaczkiewicz-2006; @granvik-et-al-2007; @chubey-et-al-2010; @eggl-2011]. Exploring techniques to link observations from sites with a considerable parallax, [@granvik-et-al-2007] gave hints that simultaneous observations can be favorable for constraining initial orbits. [@gromaczkiewicz-2006; @chubey-et-al-2010; @eggl-2011] showed that independent orbit determination for asteroids via basic trigonometry is possible if simultaneous observations from two sites are available. We shall discuss some of the benefits and issues of simultaneous observation from Gaia and the Earth in the next sections.
Identification and Linking {#eggl:sec:idl}
==========================
Let us assume that nearly simultaneous observations from Gaia and at least one additional ground-based site have been performed. Given the substantial parallax between the two observers, the astrometric FOVs can differ significantly. Hence, determining whether an object observed from both sites is in fact the same asteroid becomes crucial. For cataloged objects ephemeris predictions are mostly accurate enough for this purpose. The correct linking of observations of newly discovered objects can be more difficult. Should a sufficient number of observations be available, orbital element bundles can be constructed for each observation site. A comparison of the orbital element probability density functions generated via orbital inversion of single night sets can then be used to identify and link the same objects in each frame. This so-called ephemeris-space multiple-address-comparison (eMAC) method has been suggested by [@granvik-muinonen-2005]. [@granvik-et-al-2007] showed that this technique works for observations with large parallaxes. If observations are too sparse to generate orbital element bundles for each site individually, one can try to find observations from both sites that are nearly simultaneous. In this case one can assume that each site has a pair of right ascension ($\alpha$) and declination ($\delta$) values for each object recorded at approximately the same time. Timing errors that lead to deviations smaller than the astrometric precision of the observing instrument are acceptable. These ($\alpha$,$\delta$) pairs yield directional unit vectors that point from the respective observer to the NEO. In the absence of strong gravitational fields, straight lines can be constructed from such astrometric data, connecting the observation sites to the target, see Figure \[eggl:fig1\]. The distance $d$ between those lines is given by $$\begin{aligned}
d=|(\vec{e}_E \wedge \vec{e}_G)\cdot({\vec{r}_G}-\vec{r}_E)|\end{aligned}$$ where $\vec{r}_E$ and $\vec{r}_G$ represent the heliocentric position of the Earth and Gaia respectively, and $\vec{e}_E$ and $\vec{e}_G$ are the corresponding line unit vectors $$\begin{aligned}
\vec{e}_{E,G} =(\cos \alpha \cos \delta, \sin \alpha \cos \delta, \sin \delta)^T_{E,G}.\end{aligned}$$ If the distance between the two lines is smaller than the sum of the radii of the astrometric uncertainty ellipses evaluated at the point of closest approach, the two observations can be attributed to the same object. Of course, this is only a necessary, not a sufficient linking condition.
![The triangulation setup allows to link objects in both observer frames without the need to construct and compare individual sets of orbital elements per site, if the observations are nearly synchronous. Astrometric right ascension ($\alpha$) and declination ($\delta$) pairs corresponding to the target’s position in the respective FOVs can be used to construct geometric lines that have the observing stations at their origin. If the distance (d) between those lines is smaller than the combined radius of the astrometric uncertainty ellipses at the point of closest approach, objects recorded by the two observers can represent the same NEO. \[eggl:fig1\]](pics/triangid.eps)
Initial Orbit Determination
===========================
The possibility to constrain the location of an asteroid at a given epoch constitutes the main advantage of synchronously recorded parallactic data sets. As discussed in the previous section, this can be implicitly achieved by performing a classical or SR based orbit determination for both sites individually. The resulting orbital element distributions can be cross-matched and outlying solutions excluded. This, however, requires a sufficient number of observations. Since no delay due to Gaia’s alert time has to be taken into account in such a scenario, simple trigonometry can be used to confine possible NEO locations for the time of observation. The advantage of TR compared to combining multiple ’one-site’ observations lies in the fact that orbital constraints can be constructed with as little as one synchronously recorded frame per site (see section \[eggl:sec:statr\]). Having an Earth-based telescope observe the same field of view as Gaia at specific times[^3] would, thus, allow for an improvement of initial orbits of newly discovered asteroids. It is questionable, however, whether telescope time would be made available for a program that stares along Gaia’s line of sight without a predefined target. Fortunately, active sky surveys such as Pan-STARRS will produce data sets that are temporally and spatially overlapping with Gaia observations. As these observations are available via the Minor Planet Center, they can be combined with Gaia astrometry to constrain possible orbit solutions. Even an independent epochal orbit determination via TR is possible, if the object is recorded in more than one consecutive frame by both observation sites [@eggl-2011]. In other words, initial orbital elements can in principle be created with only two frames per observing station.
Orbit Refinement
================
Many initial NEO orbits had to be constructed based on very few observational data points. Due to the gravitationally active near Earth environment, positioning uncertainties tend to grow rather quickly for NEOs, and followup observations become essential in order not to lose track. Of course, once the Gaia mission has been completed and all observations have been reduced with the new catalog, the quality of orbits derived via Gaia data will be substantially better than anything achievable from ground based observations [@bancelin-et-al-2011]. However, up to that point, simultaneous observations can still be useful to improve NEO orbital elements. Having an initial orbit estimate at our disposal, it is possible to predict future Gaia-FOV crossings of NEOs. Given Gaia’s relatively large FOV, even asteroids on orbits with large uncertainties should be recoverable. Earth-based observations could then be conducted simultaneously with Gaia-FOV crossings, so that positioning of the asteroid via TR becomes possible. Strictly speaking, TR is not necessary in this case, since the additional observational data from the second site does itself contain the constraints on the NEO’s orbit. However, we will use TR instead of a statistical ranging or differential corrections technique in order to study the impact of a readily accessible ranging and localization of the target in coordinate space on orbital elements.
NEA Positioning Via Triangulation {#eggl:sec:pos}
=================================
Triangulation (TR) from observatories based on different sites is a fairly common tool in meteroid orbit determination [@ceplecha-1987]. The application of TR to refine NEA orbital elements has been investigated by [@gromaczkiewicz-2006] for two satellites in the Lagrangian points $L_4$ and $L_5$ of the Sun-Earth system. The approach was extended in [@eggl-2011] to encompass free satellite positioning and Earth based observations. Their results suggest that given two observations with sufficient spatial as well as temporal resolution, accurate orbital elements can be derived without having to rely on previous data or orbit-fitting models. We will apply the method proposed in [@eggl-2011] to evaluate the potential merits of NEA TR given Gaia and Earth-based observations. Let us assume we have two sets of pairwise orthogonal angles, e.g. $(\alpha_G,\delta_G), (\alpha_E,\delta_E)$ - where the subscripts stand for Gaia and the Earth respectively - and the distance between the observing locations $d_{GE}$, all measured at the same instant. It is then possible to reconstruct the position of an observed object in a locally Euclidean frame of reference (EFOR). Let us furthermore choose the EFOR to be heliocentric. The fact, that the observed object has to be accessible from both sites - Gaia and the Earth - at the same time is one of the most limiting factors for TR. As Gaia’s scanning law prohibits observations in two $45^\circ$ cones centered at $L_2$ (Sun-Earth) with axes towards and away from the Sun [@hestroffer-et-al-2010], the accessible region for TR is rather restricted, see Figure \[eggl:fig1b\].
![Schematic of the triangulation setup projected into the Earth’s orbital plane. All non-shaded regions are inaccessible to the triangulation process. We assume that all regions that can be observed by the Gaia satellite are simultaneously accessible from the Earth.\[eggl:fig1b\]](pics/Gaia2m.eps)
Nevertheless, due to the configuration’s proper motion, most of the Earth orbit crossing NEA population brighter than H magnitude $20$ should be observable during the 5 year mission.[^4] In order to maximize the uniformity of sky coverage Gaia will perform 6-hour great circle scans, where the precessing spin axis retains a $45^\circ$ angle to the Sun at all times. A full precession cycle requires 63 days (see e.g. [@hestroffer-et-al-2010]). Having a fixed scanning law, Gaia’s $0.7^\circ \times 0.7^\circ$ field of view (FOV) cannot be altered to accommodate special observation schedules. Hence calculating rendezvous times between Near Earth Objects and Gaia’s FOV becomes necessary. This is not an easy task, however, because it requires precise knowledge on the thermal and bulk properties of the satellite as well as the exact initial positioning and attitude at $L_2$ after launch. Even-though tools like DPSC’s rendezvous simulator do exist allowing to model FOV crossings, the scanning law’s initial phase remains unknown. In the following we will, therefore, resort to a simplified statistical approach to determine the quality of positioning data achievable with TR. In order to investigate the quality of NEA positioning via TR, observations of four NEAs chosen form NASA JPL’s Sentry risk table [@jpl-sentry-2012] are modeled, see Table \[eggl:tab1\] The aim is to estimate the median quality of positioning of individual asteroids achievable during the 5 year Gaia mission. Using an $8^{th}$ order symplectic integrator [@yoshida-1993] the NEA orbits were propagated together with the 8 planets of the Solar System, the Pluto-Charon center of mass, as well as Ceres, Vesta, and Pallas. The Gaia satellite was positioned and kept exactly at $L_2$ (Earth-Sun), and no relativistic or non-gravitational forces were considered. All initial osculating elements were taken from JPL-HORIZONS system for Jan $1^{st}$, 2014. The total integration time was set to 5 years corresponding to Gaia’s planned mission duration. As the actual dates of NEA FOV crossings are dependent on the initial phase of Gaia’s scanning law, the following statistical approach has been chosen: A measurement process was simulated every $\Delta t=106.5$ minutes[^5]. Measurements were accepted when the respective asteroid was in the accessible region for TR, and it had an apparent magnitude smaller than $v=20$mag. Here, we assumed that the dominating uncertainties are due to:
- the finite angular resolution in determining $(\alpha_G,\delta_G), (\alpha_E,\delta_E)$,
- the uncertainty in the momentary distance between Earth and the Gaia satellite,
- and the observed positioning offset of the asteroid due to finite light speed (see light travel time offset in [@eggl-2011]) which depends on the asteroid’s velocity and the observers’ positions relative to the asteroid.
Furthermore it is assumed, that the Earth’s as well as Gaia’s velocities in the EFOR are well known, so that differential and absolute aberration effects can be corrected for and do not have to be considered in this setup. Gaia’s attitude noise ($10^{-8} - 10^{-7}rad$) [@keil-theil-2010] is also assumed to be correctable a posteriori using relative positioning with respect to the stellar background. Simultaneous Earth-based measurements are assumed to be able to achieve the same limits in apparent magnitude as Gaia. To assess the apparent magnitude of the asteroids investigated, their absolute magnitudes were extracted from [@jpl-sentry-2012]. A model of ideal diffusely reflective spheres has been adopted to evaluate the reflected light at any given phase angle following [@bowell-et-al-1989]. Asteroids in the TR setup’s accessible region are, of course, not automatically within Gaia’s FOV. Therefore, the Ansatz is to estimate the median accuracy an observation would produce, if the asteroid passed the FOV once during the 5 year mission taking the dominating uncertainties into account. In Table \[eggl:tab2\] uncertainty values for a best-case as well as a worst-case scenario are provided. Gaia’s performance values were extracted from the mission’s data sheets[^6] as well as from [@hestroffer-et-al-2009]. Observation residuals listed in the Minor Planet Center database[^7], suggest Earth-based astrometric precision to range from roughly $0.1"$ to around $2"$. The simulated TR results are compared to the NEAs’ actual positions calculated via numerical orbit integration. The median and median deviation of the relative error in positioning is given in Figure \[eggl:fig2\]. As expected, the best and worst case scenarios are separated by an order of magnitude, which is mainly due to the change in the quality of Earth-based observations. The quality of positioning is non uniform for the four NEAs considered. Even-though the results for 2008 UV99 are quite promising, Earth-based observations with a precision of $2"$ make a TR of 1979XB practically impossible. Yet, in most cases positioning uncertainties between 0.001 and 0.1 au seem achievable. The implications for NEA orbit determination and refinement will be discussed in the next two sections.
![Median and median deviation (error bars) values for 3D positioning errors of 4 NEAs using triangulation with Gaia and Earth-based observations. The empty symbols denote best case results and the full symbols depict simulated results for worst-case observational uncertainties, see Table \[eggl:tab2\]. \[eggl:fig2\]](pics/gaia_new_dr.eps)
Refinement of NEA Orbital Elements {#eggl:sec:orbit}
==================================
In order to get a full set of orbital elements via TR, NEA velocities have to be estimated. As suggested in [@eggl-2011] a simple one-step interpolation approach is taken here, using two consecutively triangulated observations. This is possible despite Gaia’s fixed scanning law, as there are two FOVs which are separated by an angle of $106.5^\circ$. Given an axial spin-rate of $60"s^{-1}$ the second FOV will pass over the same region as the first one with a delay of 106.5 minutes. However, Gaia’s revolving scanning law includes additional precession as well as the drift caused by the satellite’s proper motion around the Sun. Consequently, the two FOVs will only have an overlap between 0.01 and 0.04 square degrees - less than 10% of the original FOV. One could, therefore, argue that the prospects of observing an asteroid in both FOVs are rather small. To counter this argument DPSC’s Gaia CU4 rendezvous simulator was used to determine whether the investigated asteroids 1999 RQ36, 2008 UV99, 2010 AU118 and 1979 XB will be observable by Gaia and cross both FOVs. For 2008 UV99, 2010 AU118 and 1979 XB this was indeed the case for an initial phase angle of $0^\circ$. However, the actual crossing date as well the appearance in both FOVs depended on the scanning law’s initial phase. Given the TR method’s strong dependency on the initial scanning phase for individual cases, a statistical argument may allow for a better evaluation of its applicability. [@bancelin-et-al-2010] conclude that independently of initial phase angles a total of 2180 NEOs and 585 Potentially Hazardous Asteroids (PHA) will be observed by Gaia. While precise ranging data can be acquired for all of them using TR, two rapid consecutive observations would be required in order to generate a full set of orbital elements. Current estimates predict a fraction of approximately 20% of Lead/Trail measurements for solar system objects (P. Tanga, 06/2013, private communication). This population - roughly 400 NEOs - would allow for two consecutive TRs of positions, which in turn can be used to estimate the asteroids’ velocity vectors. Without observational uncertainties the accuracy of the estimated velocities for NEOs would be on the order of $10^{-5} au/D$. In order to see how the observational uncertainties in Table \[eggl:tab2\] together with the proposed velocity estimates influence an independent TR based determination of NEA orbital elements, the simulation presented in the previous section was used to generate statistics on orbital element quality for the four selected NEAs (Table \[eggl:tab1\]). Without knowing the initial phase of the Gaia satellite’s scanning law, precise predictions on the quality of triangulated state vectors are not possible. Therefore, the median quality of triangulated osculating elements using two consecutive observations with $\Delta t=106.5$ simultaneously measured from Gaia and the Earth are presented in Figure \[eggl:fig3\]. Only measurements in the method’s accessible region (Figure \[eggl:fig1b\]) during the missions lifetime were considered, and then only when the target object was brighter than Gaia’s limiting magnitude. Analogous to Figure \[eggl:fig4\] best cases are depicted by empty, worst cases by full symbols (see also Table \[eggl:tab2\]). Given a best case scenario, i.e. 0.1 arc-second precision Earth-based observations for the asteroids 2008 UV99, 2010 AU118 and 1979 XB, the expected quality of the orbital elements achievable is quite high. Especially in the case of 2008 UV99 the orbit uncertainties could be decreased by an order of magnitude compared to current values using two consecutive TRs only (cf. Table \[eggl:tab1\]). If Earth-based observations are no better than arc-second precision such as assumed for the worst case scenarios, TR does not offer any substantial orbit refinement potential.
----------------- ------- ----------- -------------------- ------------------- ------------------- --------------------- --------------------- -------------------
**Designation** **H** **Nobs.** **$\sigma_a$** **$\sigma_e$** **$\sigma_i$** **$\sigma_\omega$** **$\sigma_\Omega$** **$\sigma_M$**
1999 RQ36 20.7 298 1.2$\cdot10^{-10}$ 3.4$\cdot10^{-8}$ 4.3$\cdot10^{-6}$ 6.4$\cdot10^{-6}$ 5.6$\cdot10^{-6}$ 3.6$\cdot10^{-6}$
1979 XB 18.5 17 0.24 0.03 0.82 0.37 0.06 2.12
2010 AU118 17.9 19 0.57 0.04 3 50 14 89
2008 UV99 19.6 22 0.30 0.38 27 79 18 165
----------------- ------- ----------- -------------------- ------------------- ------------------- --------------------- --------------------- -------------------
: NEA data from NASA JPL’s Sentry risk table [@jpl-sentry-2012]. The uncertainties in the NEAs’ orbital elements are shown in the corresponding $\sigma$ columns, where $\sigma_a$ is given in \[au\] and $\sigma_{i,\omega,\Omega,M}$ are given in \[deg\]. $H$ corresponds to the absolute magnitude, and Nobs. are the number of observations. \[eggl:tab1\]
------------ ----------------------- ----------------------- --------------------------------------- ---------------------
**$\Delta \alpha_G$** **$\Delta \delta_G$** **$\Delta \alpha_E=\Delta \delta_E$** **$\Delta d_{EG}$**
best case $3\cdot 10^{-4}$ $1.8\cdot 10^{-3}$ $0.1$ $150$
worst case $5\cdot 10^{-3}$ $6\cdot 10^{-2}$ $2$ $150$
------------ ----------------------- ----------------------- --------------------------------------- ---------------------
: Dominating uncertainties for NEA orbit triangulation. $\Delta \alpha_{G,E}$ and $\Delta \delta_{G,E}$ denote the respective angular uncertainties in arc-seconds \[$"$\] of Gaia [@hestroffer-et-al-2009] and Earth-based observations. $\Delta d_{EG}$ is the predicted in-mission uncertainty in the distance between Gaia and the Earth in \[m\].\[eggl:tab2\]
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![Expected quality of orbital elements gained via two consecutive triangulations using simultaneous observations from Gaia and the Earth. Medians and median deviations of the differences in actual and acquired osculating orbital elements are given for 4 NEAs. The open symbols denote the best case, and the full symbols the worst-case scenarios given in Table \[eggl:tab2\]. The simulation time-span corresponded to Gaia’s planned mission duration (5 years). \[eggl:fig3\]](pics/gaia_new_a.eps "fig:") ![Expected quality of orbital elements gained via two consecutive triangulations using simultaneous observations from Gaia and the Earth. Medians and median deviations of the differences in actual and acquired osculating orbital elements are given for 4 NEAs. The open symbols denote the best case, and the full symbols the worst-case scenarios given in Table \[eggl:tab2\]. The simulation time-span corresponded to Gaia’s planned mission duration (5 years). \[eggl:fig3\]](pics/gaia_new_w.eps "fig:")
![Expected quality of orbital elements gained via two consecutive triangulations using simultaneous observations from Gaia and the Earth. Medians and median deviations of the differences in actual and acquired osculating orbital elements are given for 4 NEAs. The open symbols denote the best case, and the full symbols the worst-case scenarios given in Table \[eggl:tab2\]. The simulation time-span corresponded to Gaia’s planned mission duration (5 years). \[eggl:fig3\]](pics/gaia_new_e.eps "fig:") ![Expected quality of orbital elements gained via two consecutive triangulations using simultaneous observations from Gaia and the Earth. Medians and median deviations of the differences in actual and acquired osculating orbital elements are given for 4 NEAs. The open symbols denote the best case, and the full symbols the worst-case scenarios given in Table \[eggl:tab2\]. The simulation time-span corresponded to Gaia’s planned mission duration (5 years). \[eggl:fig3\]](pics/gaia_new_om.eps "fig:")
![Expected quality of orbital elements gained via two consecutive triangulations using simultaneous observations from Gaia and the Earth. Medians and median deviations of the differences in actual and acquired osculating orbital elements are given for 4 NEAs. The open symbols denote the best case, and the full symbols the worst-case scenarios given in Table \[eggl:tab2\]. The simulation time-span corresponded to Gaia’s planned mission duration (5 years). \[eggl:fig3\]](pics/gaia_new_i.eps "fig:") ![Expected quality of orbital elements gained via two consecutive triangulations using simultaneous observations from Gaia and the Earth. Medians and median deviations of the differences in actual and acquired osculating orbital elements are given for 4 NEAs. The open symbols denote the best case, and the full symbols the worst-case scenarios given in Table \[eggl:tab2\]. The simulation time-span corresponded to Gaia’s planned mission duration (5 years). \[eggl:fig3\]](pics/gaia_new_m.eps "fig:")
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Statistical Ranging Constraints {#eggl:sec:statr}
===============================
The results of the previous section paint a rather bleak picture for TR regarding independent orbit determination and refinement. Yet, the capability to produce observer-to-NEO distances provided by TR can be helpful in another way. Currently, the Gaia mission pipeline is intended to perform SR for newly discovered asteroids, if the observational data is not sufficiently plentiful to allow for standard orbit determination. Hereby, a set of possible orbit solutions compatible with the observed FOV positions are generated. The bundle of initial orbits can then be propagated to generate ephemeris for follow up observations. TR can be used to further constrain SR solutions by providing additional distance information. We pointed out in section \[eggl:sec:pos\] that observational errors prohibit an exact TR based localization of the asteroid. The uncertainties in the distance measurements would, however, be sufficiently small to constrain statistical ranging solutions. This is portrayed in Figures \[eggl:fig4\] and \[eggl:fig5\]. Locations predicted by simulated SR and TR measurements are compared. Figure \[eggl:fig4\] shows the simulated positioning results from a single Gaia FOV crossing of the asteroids 1943 Anteros, 2063 Bacchus and 2102 Tantalus. One can see that the intersection of the SR and TR based position sets contains the true orbit solution. Hence, those orbital solutions found by SR that are not compatible with TR solutions can be eliminated, see Figure \[eggl:fig5\]. Even if the uncertainties in both methods are comparable, the combination of SR and TR can provide substantially more information on the location of the object.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Simulated localizations of three asteroids achieved via statistical ranging (SR) are compared to positioning via triangulation (TR). The true orbit solutions are given by the black rectangle. The possible range of TR results has been sampled from a uniform distribution, where worst-case scenarios have been assumed as defined in Table \[eggl:tab2\]. The scaling of the top panels is ’signed logarithmic’, i.e. $-2$ corresponds to $-10^2=-100$ au. \[eggl:fig4\]](pics/1943_no_reduc_xy.eps "fig:") ![Simulated localizations of three asteroids achieved via statistical ranging (SR) are compared to positioning via triangulation (TR). The true orbit solutions are given by the black rectangle. The possible range of TR results has been sampled from a uniform distribution, where worst-case scenarios have been assumed as defined in Table \[eggl:tab2\]. The scaling of the top panels is ’signed logarithmic’, i.e. $-2$ corresponds to $-10^2=-100$ au. \[eggl:fig4\]](pics/1943_no_reduc_xz.eps "fig:")
![Simulated localizations of three asteroids achieved via statistical ranging (SR) are compared to positioning via triangulation (TR). The true orbit solutions are given by the black rectangle. The possible range of TR results has been sampled from a uniform distribution, where worst-case scenarios have been assumed as defined in Table \[eggl:tab2\]. The scaling of the top panels is ’signed logarithmic’, i.e. $-2$ corresponds to $-10^2=-100$ au. \[eggl:fig4\]](pics/2063_no_reduc_xy.eps "fig:") ![Simulated localizations of three asteroids achieved via statistical ranging (SR) are compared to positioning via triangulation (TR). The true orbit solutions are given by the black rectangle. The possible range of TR results has been sampled from a uniform distribution, where worst-case scenarios have been assumed as defined in Table \[eggl:tab2\]. The scaling of the top panels is ’signed logarithmic’, i.e. $-2$ corresponds to $-10^2=-100$ au. \[eggl:fig4\]](pics/2063_no_reduc_xz.eps "fig:")
![Simulated localizations of three asteroids achieved via statistical ranging (SR) are compared to positioning via triangulation (TR). The true orbit solutions are given by the black rectangle. The possible range of TR results has been sampled from a uniform distribution, where worst-case scenarios have been assumed as defined in Table \[eggl:tab2\]. The scaling of the top panels is ’signed logarithmic’, i.e. $-2$ corresponds to $-10^2=-100$ au. \[eggl:fig4\]](pics/2102_no_reduc_xy.eps "fig:") ![Simulated localizations of three asteroids achieved via statistical ranging (SR) are compared to positioning via triangulation (TR). The true orbit solutions are given by the black rectangle. The possible range of TR results has been sampled from a uniform distribution, where worst-case scenarios have been assumed as defined in Table \[eggl:tab2\]. The scaling of the top panels is ’signed logarithmic’, i.e. $-2$ corresponds to $-10^2=-100$ au. \[eggl:fig4\]](pics/2102_no_reduc_xz.eps "fig:")
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Possible semimajor axes (a) versus eccentricities of three asteroids are portrayed. Simulated statistical ranging (SR) results from Gaia data are shown as (+) signs. The solutions that are constrained via triangulation (TR) from an Earth-based observer are denoted by ($\times$). The cross-hair gives the location of the true solution. TR is capable of constraining SR orbital solutions considerably. \[eggl:fig5\]](pics/1943_constr_ae2.eps "fig:")\
![Possible semimajor axes (a) versus eccentricities of three asteroids are portrayed. Simulated statistical ranging (SR) results from Gaia data are shown as (+) signs. The solutions that are constrained via triangulation (TR) from an Earth-based observer are denoted by ($\times$). The cross-hair gives the location of the true solution. TR is capable of constraining SR orbital solutions considerably. \[eggl:fig5\]](pics/2063_constr_ae2.eps "fig:")\
![Possible semimajor axes (a) versus eccentricities of three asteroids are portrayed. Simulated statistical ranging (SR) results from Gaia data are shown as (+) signs. The solutions that are constrained via triangulation (TR) from an Earth-based observer are denoted by ($\times$). The cross-hair gives the location of the true solution. TR is capable of constraining SR orbital solutions considerably. \[eggl:fig5\]](pics/2102_constr_ae2.eps "fig:")
Discovery Cross-Matching
========================
Throughout the previous sections we have assumed that dedicated ground based sites are available to produce synchronized observations. This is certainly the most proliferating mode of operation, in particular regarding initial orbit determination. Yet, it is probably not the most economic in terms of observational resources requirements. Therefore, one should consider alternatives, such as using the results from large surveys. Potential discoveries of asteroids are published very rapidly at the Minor Planet Center.[^8] We suggest that new discoveries or candidates observed from ground based facilities be cross-linked with Gaia astrometric alerts. Should nearly simultaneous observations be available, new objects can be linked via the methods discussed in section \[eggl:sec:idl\], and initial orbits may be constrained. As discussed previously, observations would not have to be perfectly synchronized. Asynchronisities that cause the object to remain within the uncertainties caused by ground based astrometric precision are permissible. For fast moving objects, however, this margin is small. No accurate predictions can be given at this point on how often simultaneous observations of new discoveries will occur. Most likely they are rare events. However, discovery cross-matching comes at very little computational and no observational cost. Hence, we consider cross matching a worthwhile effort. A preliminary software procedure called GODSEND (Gaia and grOunD SurvEys for Neo Detections) is currently implemented in the framework of the Gaia-FUN-SSO network.
Conclusions
===========
Simultaneous observations of NEOs from Gaia and ground based sites are valuable. They can be used to constrain statistical ranging results and facilitate identification and linking of observed objects. Triangulation is a straight forward way to achieve such constraints, particularly when the number of available observations is small. The TR method’s potential for orbit refinement of known NEOs strongly depends on the astrometric equipment available and the quality of results can vary from case to case. Using the large parallax between Gaia and Earth-based observatories, substantial improvements of weakly constrained NEA orbits are possible, if a ground-based angular accuracy well below 1” can be achieved. In this context it might also be interesting to consider triangulation between Gaia and other contemporary space missions such as NEOSSat [@neossat-2012], especially for orbital regions that are difficult to access from the ground. Unfortunately, the actual NEO population available for a fully independent determination of orbital elements via triangulation alone may be as small as 400, of which only 100 are PHAs. Therefore, a precise rendezvous predictions using the actual initial scanning law phase are necessary to draw a clearer picture on the possible merits of independent orbit generation via triangulation. Nevertheless, synchronous observations from two sites have a substantial impact on the quality of preliminary orbital elements acquired via statistical ranging. Even if no dedicated simultaneous observations were to be conducted from Earth based sites, Gaia discoveries can be cross-matched with the Minor Planet Center survey database in order to triangulate newly found objects at no additional observational cost.\
\
**Acknowledgments** The authors would like to thank Daniel Hestroffer, David Bancelin, Paolo Tanga, Benoit Carry and Enrico Gerlach for their valuable input, as well as the DPSC for granting access to the Gaia CU4 rendezvous simulator. Furthermore, the authors would like to acknowledge the support of the European Union Seventh Framework Program (FP7/2007-2013) under grant agreement no. 282703, as well as the Gaia FUN-SSO network.
[^1]: the osculating orbital elements that are derived when a NEO is first discovered
[^2]: Observations do not have to be exactly simultaneous. Asynchronicities that do not lead to a discrepancy in the observed astrometric position greater than the instrumental precision are permissible.
[^3]: e.g. whenever the FOV approaches the ecliptic
[^4]: A potential shift of the Gaia mission’s cutoff H magnitude to $21$ is currently debated. Going to fainter magnitudes would make more NEOs accessible to triangulation assuming that most of the ground based sites can also observe the fainter objects with reasonable astrometric precision. We will continue assuming a conservative magnitude limit of $H=20$mag, however.
[^5]: See section \[eggl:sec:orbit\] for details.
[^6]: Specific data regarding the Gaia mission were acquired from <http://rssd.esa.int/gaia>, 2012.
[^7]: <http://www.minorplanetcenter.net/iau/special/residuals.txt>, 2013
[^8]: <http://www.minorplanetcenter.net/iau/NEO/ToConfirm.html>, 2013
|
---
abstract: 'Discussed are field-theoretic models with degrees of freedom described by the $n$-leg field in an $n$-dimensional “space-time” manifold. Lagrangians are generally-covariant and invariant under the internal group GL$(n,{\bf R})$. It is shown that the resulting field equations have some correspondence with Einstein theory and possess homogeneous vacuum solutions given by semisimple Lie group spaces or their appropriate deformations. There exists a characteristic link with the generalized Born-Infeld type nonlinearity and relativistic mechanics of structured continua. In our model signature is not introduced by hands, but is given by integration constants for certain differential equations.'
author:
- |
Jan J. Sławianowski\
Institute of Fundamental Technological Research,\
Polish Academy of Sciences,\
21 Świȩtokrzyska str., 00-049 Warsaw, Poland\
e-mail: [email protected]
title: |
Teleparallelism, modified Born-Infeld nonlinearity\
and space-time as a micromorphic ether.
---
[**Keywords:**]{} affinely-rigid body, Born-Infeld nonlinearity, micromorphic continuum, teleparallelism, tetrad.
Introduction
============
The model suggested here has several roots and arose from some very peculiar and unexpected convolution of certain ideas and physical concepts seemingly quite remote from each other. In a sense it unifies generalized Born-Infeld type nonlinearity, tetrad approaches to gravitation, Hamiltonian systems with symmetries (mainly with affine symmetry; so-called affinely-rigid body), generally-relativistic spinors and motion of generalized relativistic continua with internal degrees of freedom (relativistic micromorphic medium, a kind of self-gravitating microstructured “ether” generalizing the classical Cosserat continuum). The first two of mentioned topics (Born-Infeld, tetrad methods) were strongly contributed by Professor Jerzy Plebański, cf. e.g. [@16; @18]. The same concerns spinor theory [@17]. My “micromorphic ether”, although in a rather very indirect way, is somehow related to the problem of motion in general relativity; the discipline also influenced in a known way by J. Plebański [@13]. There exist some links between generalized Born-Infeld nonlinearity and the modern theory of strings, membranes and p-branes. Geometrically this has to do with the theory of minimal surfaces [@9].
Born-Infeld motive
==================
Let us begin with the Born-Infeld motive of our study. No doubt, linear theories with their superposition principle are in a sense the simplest models of physical phenomena. Nevertheless, they are too poor to describe physical reality in an adequate way. They are free of the essential self-interaction. In linear electrodynamics stationary centrally symmetric solutions of field equations are singular at the symmetry centre and their total field energy is infinite. If interpreting such centres as point charges one obtains infinite electromagnetic masses, e.g. for the electron. In realistic field theories underlying elementary particle physics one usually deals with polynomial nonlinearity, e.g. the quartic structure of Lagrangians is rather typical. Solitary waves appearing in various branches of fundamental and applied physics owe their existence to various kinds of nonlinearity, very often nonalgebraic ones. General relativity is nonlinear (although quasilinear at least in the gravitational sector) and its equations are given by rational functions of field variables, although Lagrangians themselves are not rational. In Einstein theory one is faced for the first time with a very essential non-linearity which is not only nonperturbative (it is not a small nonlinear correction to some dominant linear background) but is also implied by the preassumed symmetry conditions, namely by the demand of general covariance. Indeed, any Lagrangian theory invariant under the group of all diffeomorphisms must be nonlinear (although, like Einstein theory may be quasilinear). Nonlinearity of non-Abelian gauge theories is also due to the assumed symmetry group. In our mechanical study of affinely-rigid bodies [@20] nonlinearity of geodetic motion was also due to the assumption of invariance under the total affine group. This, by the way, established some link with the theory of integrable lattices.
The original Born-Infeld nonlinearity had a rather different background and was motivated by the mentioned problems in Maxwell electrodynamics. There was also a tempting idea to repeat the success of general relativity and derive equations of charges motion from the field equations. Unlike the problem of infinities, which was in principle solved, the success in this respect was rather limited. The reason is that in general relativity the link between field equations and equations of motion is due not only to the nonlinearity itself (which is, by the way, necessary), but first of all to Bianchi identities. The latter follow from the very special kind of nonlinearity implied by the general covariance.
As we shall see later on, some kinds of generalized Born-Infeld nonlinearities also may be related to certain symmetry demands. But for us it is more convenient to begin with some apparently more formal, geometric aspect of “Born-Infeld-ism”.
In linear theories Lagrangians are built in a quadratic way from the field variables $\Psi$. Thus, they may involve $\Psi \Psi$-terms algebraically quadratic in $\Psi,\
\partial \Psi \partial \Psi$-terms algebraically quadratic in derivatives of $\Psi$, and $\Psi \partial \Psi$-terms bilinear in $\Psi$ and $\partial
\Psi$; everything with constant coefficients. In any case, the dependence on derivatives, crucial for the structure of field equations is polynomial of at most second order in $\partial \Psi$ (linear with $\Psi$-coefficients in the case of fermion fields).
But there is also another, in a sense opposite pole of mathematical simplicity of Lagrangians. By its very geometrical nature, Lagrangian $L$ is a scalar Weyl density of weight one; in an $n$-dimensional orientable manifold it may be represented by a differential $n$-form locally given by: $\pounds= L(\Psi,\partial
\Psi)dx^{1}\wedge \ldots \wedge dx^{n}.$ But as we know, there is a canonical way of constructing such densities: just taking square roots of the moduli of determinants of second-order covariant tensors, $$\label{1}
L=\sqrt{|{\rm det}[L_{ij}]|},$$ or rather constant multiples of this expression, when some over-all negative sign may occur. In the sequel the square-rooted tensor will be referred to as the Lagrange tensor, or tensorial Lagrangian. In general relativity and in all field theories involving metric tensor $g$ on the space-time manifold $M$, $L$ is factorized in the following way: $L=\Lambda (\Psi,\partial
\Psi)\sqrt{|{\rm det}[g_{ij}]|},$ where $\Lambda$ is a scalar expression. Here the square-rooted metric $g_{ij}$ offers the canonical scalar density. In linear theories for fields $\Psi$ considered on a fixed metrical background $g,\ \Lambda$ is quadratic (in the aforementioned sense) in $\Psi$. In quasilinear Einstein theory, where in the gravitational sector $\Psi$ is just $g$ itself, one uses the Hilbert Lagrangian proportional to $R[g]\sqrt{|g|}$ ($R[g]$ denoting obviously the scalar curvature of $g$ and $|g|$ being an obvious abbreviation for the modulus of ${\rm det}[g_{ij}]$). Obviously, Lagrangians of linear and quasilinear theories also may be written in the form (\[1\]), however, this representation is extremely artificial and inconvenient. For example, for the Hilbert Lagrangian we have $
L={\rm sign}R\sqrt{|{\rm det}[R^{2/n}g_{ij}]|}, $ i.e. locally we can write $ L_{ij}=|R|^{2/n}g_{ij};\ L_{ij}=\sqrt{|R|}g_{ij}\ {\rm
if}\ n=4.$
One can wonder whether there exist phenomena reasonably and in a convenient way described just in terms of (\[1\]). It is natural to expect that the simplest models of this type will correspond to the at most quadratic dependence of the tensor $L_{ij}$ on field derivatives. Unlike the linear and quasilinear models, now the theory structure will be lucid just on the level of $L_{ij}$. Among all possible nonlinear models such ones will be at the same time quite nonperturbative but also in a sense similar to the linear and quasilinear ones.
The historical Born-Infeld model [@2] is exceptional in that the Lagrange tensor is linear in field derivatives; Lagrangian is $$\label{2}
L=-\sqrt{|{\rm det}[bg_{ij}+F_{ij}]|}+b^{2}\sqrt{|{\rm det}[g_{ij}]|},$$ where $F_{ij}=A_{j,i}-A_{i,j}$ is the electromagnetic field strength, $A_{i}$ is the covector potential and the constant $b$ is responsible for the saturation phenomenon; it determines the maximal attainable field strength. The field dynamics is encoded in the first term. The second one, independent of $F$, fixes the energy scale: Lagrangian and energy are to vanish when $F$ vanishes. Therefore, up the minus sign preceding the square root, we have $L_{ij}=bg_{ij}+F_{ij}$. For weak fields, e.g. far away from sources, $L$ asymptotically corresponds with the quadratic Maxwell Lagrangian. And all singularities of Maxwell theory are removed — static spherically symmetric solutions are finite at the symmetry centre (point charge) and the electromagnetic mass is finite. The finiteness of solutions is due to the saturation effect. $L$ has a differential singularity of the type $\sqrt{0}$ when the field is so strong that the determinant of $\left[bg_{ij}+F_{ij}\right]$ vanishes. Such a situation is singular-repulsive just as $v=c$ situation for the relativistic particle, where the interaction-free Lagrangian is given by $L=-mc^{2}\sqrt{1-v^{2}/c^{2}}$ (in three-dimensional notation). The classical Born-Infeld theory is in a sense unique, exceptional among all a priori possible models of nonlinear electrodynamics [@1; @18]. It is gauge invariant, the energy current is not space-like, energy is positively definite, point charges have finite electromagnetic masses and there is no birefringence. There exist solutions of the form of plane waves combined with the constant electromagnetic field; in particular, solitary solutions may be found [@1].
Although the amazing success of quantum field theory and renormalization techniques (even classical ones, as developed by Dirac) for some time reduced the interest in Born-Infeld theory, nowadays this interest is again growing on the basis of new motivation connected, e.g. with strings, p-branes, alternative approaches to gravitation, etc. [@5; @6; @7].
Linearity of $L_{ij}$ in field derivatives is an exceptional feature of electrodynamics among all models developed in the Born-Infeld spirit. Usually $L_{ij}$ must be quadratic in derivatives because of purely geometric reasons. For example, let us consider the scalar theory of light, neglecting the polarization phenomena. In linear theory one uses then the real scalar field $\Psi$ ruled by the d’Alembert Lagrangian $ L=g^{ij}\Psi_{,i}\Psi_{,j}\sqrt{|g|}.$ The only natural “Born-Infeld-ization” of this scheme is based on $$\label{3}
L=-\sqrt{|{\rm det}[bg_{ij}+\Psi_{,i}\Psi_{,j}]|}+b^{2}\sqrt{|{\rm det}[g_{ij}]|},$$ thus, as a matter of fact $L_{ij}$ is quadratic in field derivatives $\partial \Psi$. It is interesting that such a model gives for the stationary spherically symmetric solutions in Minkowski space the formula which is exactly identical with that for the scalar potential $\varphi=A_{0}$ in the usual Born-Infeld model, namely, the expression $ f(r)=\sqrt{Ab}\int_{0}^{r}du/\sqrt{A+u^{4}},$ where $A$ denotes the integration constant (related to the value of point charge producing the field). Let us mention, incidentally, that such a scalar Born-Infeld model was successfully applied in certain problems of nonlinear optics. Therefore, the very use of $L_{ij}$ quadratic in derivatives does not seem to violate philosophy underlying the Born-Infeld model. There are also other arguments. Born-Infeld theory explains point charges in a very nice way as “regular singularities”, but is not well-suited to describing interactions with autonomous external sources, e.g. with the charged (complex) Klein-Gordon or Dirac fields. Combining in the usual way Born-Infeld Lagrangian with expressions describing matter fields and the mutual interactions one obtains equations involving complicated nonrational expressions. It seems much more natural to use the expression (\[1\]) with $L_{ij}$, e.g. of the form: $
L_{ij}=\alpha g_{ij}+\kappa \overline{\Psi}\Psi g_{ij}+
bF_{ij}+cD_{i}\overline{\Psi}D_{j}\Psi, $ where $\alpha,\kappa,b,c$ are constants and $D_{j}\Psi=\Psi_{,j}+ieA_{j}\Psi$ (electromagnetic covariant derivatives). Here $\Psi$ denotes the complex scalar field and for weak fields we obtain the usual mutually coupled Maxwell-Klein-Gordon system. The same may be done obviously for field multiplets and for fermion fields. Such a model has a nice homogeneous structure and the field equations are rational in spite of the square-root expression used in Lagrangian.
Once accepting $L_{ij}$ quadratic in derivatives we can also think about admitting such models also for the pure electromagnetic field, e.g. $$\label{4}
L_{ij}=\alpha g_{ij}+\beta F_{ij}+ \gamma g^{kl}F_{ik}F_{lj}+\delta
g^{kr}g^{ls}F_{kl}F_{rs}g_{ij},$$ where $\alpha,\beta,\gamma,\delta$ are constants. The terms quadratic in $F$ in (\[4\]) are known as contributions to the energy-momentum tensor of the Maxwell field.
If we try to construct Born-Infeld-like models for non-Abelian gauge fields, then the quadratic structure of $L_{ij}$ is as unavoidable as in scalar electrodynamics, e.g. $$\label{5}
L_{ij}=\alpha g_{ij}+\gamma g^{kl}F^{K}{}_{ik}F^{L}{}_{lj}h_{KL}$$ is the most natural expression. Obviously, $\alpha,\gamma$ are constants, $F^{K}{}_{ij}$ are strength of the gauge fields and $h_{KL}$ is the Killing metric on the gauge group Lie algebra.
Lagrangians (\[1\]) may be modified by introducing “potentials”, i.e. scalar $\Psi$-dependent multipliers at the square-root expression, or at $L_{ij}$ itself, or finally, at the determinant. However, the less number of complicated and weakly-motivated corrections of this type, the more aesthetic and convincing is the dynamical hypothesis contained in $L$.
It is worthy of mentioning that the scalar Born-Infeld models with quadratic $L_{ij}$ have to do with the theory of minimal surfaces [@9] and with some interplay between general covariance and internal symmetry. Namely, we can consider scalar fields $\Psi$ on $M$ with values in some linear space $W$ of dimension $m$ higher than $n={\rm dim}M$. Let $W$ be endowed with some (pseudo)Euclidean metric $h\in W^{*}\otimes W^{*}$. We could as well consider pseudo-Riemannian structure as a target space, however, now we prefer to concentrate on the simplest model. If $M$ is structureless, then the only natural possibility of constructing Lagrangian invariant under rotations O$(W,\eta)$ and under Diff$M$ (generally-covariant) is to take the pull-back Lagrange tensor $L_{ij}=g_{ij}=h_{KL}\Psi^{K}{}_{,i}\Psi^{L}{}_{,j}.$ If $h$ is Euclidean, this means that we search minimal surfaces in $W$; $M$ is used as a merely parametrization. Field equations have the form $g^{ij}\nabla_{i}\nabla_{j}\Psi^{K}=0,\ K=\overline{1,m},$ where the covariant differentiation is meant in the Levi-Civita $g$-sense. We can fix the coordinate gauge by putting $\left((1/2)g^{ij}g^{ab}-g^{ia}g^{jb}\right)g_{ab,i}=0,$ e.g. making the assumption $\Psi^{i}=x^{i},\ i=\overline{1,n}$, i.e. identifying $n$ of the fields $\Psi^{K}$ with $M$-coordinates themselves. Then the gauge-free content of our field equations is given by: $g^{ij}\Psi^{\Sigma}{}_{,ij}=0,\
\Sigma=\overline{n+1,m}. $ These equations follow from the effective Lagrange tensor $$\label{6}
L^{\rm eff}{}_{ij}=h_{ij}+2h_{\Sigma (i}\Psi^{\Sigma}{}_{,j)}+h_{\Sigma \Lambda}
\Psi^{\Sigma}{}_{,i}\Psi^{\Lambda}{}_{,j}.$$ Here we easily recognize something similar to (\[4\]), i.e. second-order polynomial in derivatives with the effective background metric $h_{ij}$ in $M$. If $h$ has the block structure with respect to $(\Sigma,i)$-variables, then there are no first-order terms, just as in (\[3\]),(\[5\]). It is seen that the “almost classical” Born-Infeld form with the effective metric on $M$ may be interpreted as a gauge-free reduction of generally-covariant dynamics in $M$ with some internal symmetries in the target space $W$. One can also multiply the corresponding Lagrangians by some “potentials” depending on the $h$-scalars built of $\Psi$, however with the provisos mentioned above. Surprisingly enough, such scalar models describe plenty of completely different things, e.g. soap and rubber films, geodetic curves, relativistic mechanics of point particles, strings and p-branes, minimal surfaces and Jacobi-Maupertuis variational principles. There were also alternative approaches to gravitation based on such models [@14].
Tetrads, teleparallelism and internal affine\
symmetry
=============================================
There are various reasons for using tetrads in gravitation theory, in particular for using them as gravitational potentials, in a sense more primary then the metric tensor [@15]. First of all they provide local reference frames reducing the metric tensor to its Minkowskian shape. They are unavoidable when dealing with spinor fields in general relativity. This has to do with the curious fact that $\overline{{\rm
GL}^{+}(n,{\bf R})}$, the universal covering group of ${\rm GL}^{+}(n,{\bf R})$, is not a linear group (has no faithful realization in terms of finite matrices). Also the gauge approaches to gravitation (SL$(2,{\bf C})$-gauge, Poincaré gauge models) are based on the use of tetrad fields. And even in standard Einstein theory the tetrad formulation enables one to construct first-order Lagrangians which are well-defined scalar densities of weight one. If one uses the metric field as a gravitational potential, the Hilbert Lagrangian is, modulo the cosmological term, the only possibility within the class of essentially first-order variational principles. Unlike this, the tetrad degrees of freedom admit a wide class of nonequivalent variational principles. Some of them were expected to overwhelm singularities appearing in Einstein theory.
Let us begin with introducing necessary mathematical concepts. It is convenient to consider a general “space-time” manifold $M$ of dimension $n$ and specify to $n=4$ only on some finite stage of discussion. The principal fibre bundle of linear frames will be denoted by $\pi : FM \rightarrow M$ and its dual bundle of co-frames by $\pi^{*}: F^{*}M \rightarrow M$. The duality between frames and co-frames establishes the canonical diffeomorphism between $FM$ and $F^{*}M$. The co-frame dual to $e=(\ldots
,e_{A},\ldots)$ will be denoted by $\widetilde{e}=(\ldots,
e^{A},\ldots)$; by definition $\langle
e^{A},e_{B}\rangle=\delta^{A}{}_{B}$. When working in local coordinates $x^{i}$ we use the obvious symbols $e^{i}{}_{A},\
e^{A}{}_{i}$, omitting the tilde-sign at the co-frame. Therefore, $e^{A}{}_{i}e^{i}{}_{B}=\delta^{A}{}_{B},\
e^{i}{}_{A}e^{A}{}_{j}=\delta^{i}{}_{j}$. The structure group GL$(n,{\bf R})$ acts on $FM,\ F^{*}M$ in a standard way, i.e. for any $L\in {\rm GL}(n,{\bf R}):$ $e \mapsto
eL=(\ldots,e_{A},\ldots)L=(\ldots,e_{B}L^{B}{}_{A},\ldots),\
\widetilde{e} \mapsto \widetilde{e}L=(\ldots,e^{A},\ldots)L=
(\ldots,L^{-1}{}^{A}{}_{B}e^{B},\ldots).$ Fields of (co-)frames ((co-)tetrads when $n=4$) are sufficiently smooth cross-sections of $F^{*}M$, respectively $FM$ over $M$. They are affected by elements of GL$(n,{\bf R})$ pointwise, according to the above rule. In gauge models of gravitation one must admit local, i.e. $x$-dependent action of GL$(n,{\bf R})$. Any field $M \ni x\mapsto
L(x) \in {\rm GL}(n,{\bf R})$ acts on cross-section $M \ni x
\mapsto e_{x} \in FM$ according to the rule: $
\left(eL\right)_{x}=e_{x}L(x).$ Obviously, for any $x \in M,\
e_{x} \in \pi^{-1}(x)$ may be identified with a linear isomorphism of ${\bf R}^{n}$ onto the tangent space $T_{x}M$; similarly, $\widetilde{e}_{x} \in \pi^{*}{}^{-1}(x)$ is an ${\bf
R}^{n}$-valued form on $T_{x}M$. Therefore, the field of co-frames is an ${\bf R}^{n}$-valued differential one-form on $M$. In certain problems it is convenient to replace ${\bf R}^{n}$ by an abstract $n$-dimensional linear space $V$. The reason is that ${\bf R}^{n}$ carries plenty of structures sometimes considered as canonical (e.g. the Kronecker metric) and this may lead to false ideas.
In the sequel we shall need some byproducts of the field of frames. If $\eta$ is a pseudo-Euclidean metric on ${\bf R}^{n}$ (on $V$), then the Dirac-Einstein metric tensor on $M$ is defined as: $ h[e,\eta]=\eta_{AB}e^{A}\otimes e^{B},\
h_{ij}=\eta_{AB}e^{A}{}_{i}e^{B}{}_{j}. $ In general relativity $n=4$ and $[\eta_{AB}]={\rm diag}(1,-1,-1,-1)$. Obviously, the prescription for $e\mapsto
h[e,\eta]$ is invariant under the local action of the pseudo-Euclidean group O$(n,\eta)$. In general relativity it is the Lorentz group O$(1,3)$ that is used as internal symmetry. The metric $\eta$, or rather its signature, is an absolute element of the theory.
The field of frames gives rise to the teleparallelism connection $\Gamma_{\rm tel}[e]$; it is uniquely defined by the condition $\nabla e_{A}=0,\ A=\overline{1,n}.$ In terms of local coordinates: $\Gamma^{i}{}_{jk}=e^{i}{}_{A}e^{A}{}_{j,k}.$ Obviously, its curvature tensor vanishes and the parallel transport of tensors consists in taking in a new point the tensor with the same anholonomic $e$-components. The prescription $e
\mapsto \Gamma (e)$ is globally GL$(n,{\bf R})$-invariant, $\Gamma[eA]=\Gamma[e],\ A \in {\rm GL}(n,{\bf R}).$ The torsion tensor of $\Gamma_{\rm tel}$, $ S[e]^{i}{}_{jk}=\Gamma_{\rm
tel}{}^{i}{}_{[jk]}=
(1/2)e^{i}{}_{A}\left(e^{A}{}_{j,k}-e^{A}{}_{k,j}\right)$ may be interpreted as an invariant tensorial derivative of the field of frames. It is directly related to the non-holonomy object $\gamma$ of $e$, $
S^{i}{}_{jk}=\gamma^{A}{}_{BC}e^{i}{}_{A}e^{B}{}_{j}e^{C}{}_{k},\
\left[e_{A},e_{B}\right]=\gamma^{C}{}_{AB}e_{C}$ (as usual, $[u,v]$ denotes the Lie bracket of vector fields $u,v$).
In general relativity tetrad field is interpreted as a gravitational potential; the space-time metric $h[e,\eta]$ is a secondary quantity. When expressed through $e$, Hilbert Lagrangian may be invariantly reduced to some well-defined scalar density of weight one and explicitly free of second derivatives. Indeed, one can show that $$\label{7}
L_{\rm H}=R
[h[e]]\sqrt{|h|}=\left(J_{1}+2J_{2}-4J_{3}\right)\sqrt{|h|}+
4(S^{a}{}_{ab}h^{bi}\sqrt{|h|})_{,i},$$ where $|h|$ is an abbreviation for $|{\rm det}\left[h[e]_{ij}\right]|$ and $J_{1}=h_{ai}h^{bj}h^{ck}S^{a}{}_{bc}S^{i}{}_{jk},\
J_{2}=h^{ij}S^{a}{}_{ib}S^{b}{}_{ja},\ J_{3}=h^{ij}S^{a}{}_{ai}S^{b}{}_{bj}$ are Weitzenböck invariants built quadratically of $S$. They are invariant under the global action of O$(1,3)$ on $e$. The last term in (\[7\]) is a well-defined scalar density and the divergence of some vector density of weight one. It absorbs the second derivatives of $e$. Therefore, Hilbert Lagrangian is equivalent to the first term in (\[7\]), $$\label{8}
L_{\rm{H-tel}}:=
L_{1}+2L_{2}-4L_{3}=\left(J_{1}+2J_{2}-4J_{3}\right)\sqrt{|h|}.$$ It is invariant under the local action of O$(1,3)$ modulo appropriate divergence corrections. And resulting field equations for $e$ are exactly Einstein equations with $h[e,\eta]$ substituted for the metric tensor. In this sense one is dealing with different formulation of the same theory. Obviously, due to the mentioned local O$(1,3)$-invariance, the tetrad formulation involves more gauge variables.
The use of tetrads as fundamental fields opens the possibility of formulating more general dynamical models. The simplest modification consists in admitting general coefficients at three terms of (\[8\]), $$\label{9}
L=c_{1}L_{1}+c_{2}L_{2}+c_{3}L_{3}.$$ When the ratio $c_{1}:c_{2}:c_{3}$ is different than $1:2:(-4)$, the resulting model loses the local O$(1,3)$-invariance and is invariant only under the global action of O$(1,3)$. The whole tetrad $e$ becomes a dynamical variable, whereas in (\[8\]) everything that does not contribute to $h[e,\eta]$ is a pure gauge. Models based on (\[9\]) were in fact studied and it turned out that in a certain range of coefficients $c_{1},\ c_{2},\ c_{3}$ their predictions agree with those of Einstein theory and with experiment. One can consider even more general models with Lagrangians non-quadratic in $S$: $$\label{10}
L(S,h)=g(S,h)\sqrt{|h|},$$ where $g$ is arbitrary scalar intrinsically built of $S,h,$ e.g. some nonlinear function of Weitzenböck invariants. The resulting theories are not quasilinear any longer. There were some hopes to avoid certain non-desirable infinities by appropriate choice of $g$ (people were then afraid of singularities, nowadays they love them). To write down field equations in a concise form it is convenient to introduce two auxiliary quantities $H_{i}{}^{jk}:=\partial L/\partial S^{i}{}_{jk}
=e^{A}{}_{i}H_{A}{}^{jk}= e^{A}{}_{i}\partial L/\partial
e^{A}{}_{j,k},\ Q^{ij}:=\partial L/\partial h_{ij}$ referred to, respectively, as a field momentum and Dirac-Einstein stress. They are tensor densities of weight one. One can show that equations of motion have the form: $K_{i}{}^{j}:=\nabla_{k}H_{i}{}^{jk}+2S^{l}{}_{lk}H_{i}{}^{jk}-2h_{ik}Q^{kj}=0.
$ The covariant differentiation is meant here in the $e$-teleparallelism sense.
To the best of our knowledge, all teleparallelism models of gravitation belonged to the above described class. They are invariant under global action of O$(n,\eta)$ (i.e. O$(1,3)$ in the physical four-dimensional case). Let us observe, however, that there are some fundamental philosophical objections concerning this symmetry. The corresponding local symmetry in Einstein theory was well-motivated. It simply reflected the fact that the tetrad field was a merely nonholonomic reference frame, something without a direct dynamical meaning. It was only its metrical aspect $h[e,\eta]$ that was physically interpretable. If we once decide seriously to make the total $e$ a dynamical quantity, the global O$(e,\eta)$-symmetry evokes some doubts. Why not the total GL$(n,{\bf R})$-symmetry? Why to introduce by hands the Minkowskian metric $\eta$ to ${\bf R}^{n}$, the internal space of tetrad field? Such questions become very natural when, as mentioned above, we use an abstract linear space $V$ instead of ${\bf R}^{n}$. From the purely kinematical point of view the most natural group is GL$(V)$. It seems rather elegant to use a bare, amorphous linear space $V$ than to endow it a priori with geometrically nonmotivated absolute element $\eta \in V^{*}
\otimes V^{*}$. To summarize: when one gives up the local Lorentz symmetry O$(V,\eta)$, then it seems more natural to use the global GL$(V)$ than global O$(V,\eta)$. Then $L$ in (\[10\]) does not depend on $h$ and our field equations for Lagrangians $L(S)$ have the following general form: $$\label{11}
K_{i}{}^{j}:=\nabla_{k}H_{i}{}^{jk}+2S^{l}{}_{lk}H_{i}{}^{jk}=0.$$ If the model is to be generally-covariant, that we always assume, then some Bianchi-type identities imply that $L$ is an $n$-th order homogeneous function of $S$, $S^{i}{}_{jk}\partial
L/\partial S^{i}{}_{jk}=S^{i}{}_{jk}H_{i}{}^{jk}=nL,$ i.e. $L(\lambda S)=\lambda^{n}L(S)$ for any $\lambda >0$. This is a kind of generalized Finsler structure.
If we search for models with internal linear-conformal symmetry ${\bf R}^{+}{\rm O}(V,\eta)$$=e^{\bf R}{\rm O}(V,\eta)$, then $L$ must be homogeneous of degrees $0$ in $h$, $h_{ij}\partial
L/\partial h_{ij}=h_{ij}Q^{ij}=0,$ i.e. $L(S,\lambda h)=L(S,h)$ for any $\lambda \in {\bf R}^{+}$.
The simplest GL$(n,{\bf R})$-invariant (GL$(V)$-invariant) and generally-covariant models have the following generalized Born-Infeld structure: $L=\sqrt{|{\rm
det}[L_{ij}]|}$ with the Lagrange tensor quadratic in derivatives: $$\label{12}
L_{ij}=4\lambda S^{k}{}_{im}S^{m}{}_{jk}+4\mu S^{k}{}_{ik}S^{m}{}_{jm}+ 4\nu
S^{k}{}_{lk}S^{l}{}_{ij},$$ where $\lambda,\mu,\nu$ are real constants. One can in principle complicate them and make more general multiplying the above Lagrangian $L$, or Lagrange tensor $L_{ij}$, or the under-root expression (to some extent the same procedure) by a function of some basic GL$(V)$-invariant and generally-covariant scalars built of $S$. All such scalars are zeroth-order homogeneous functions of $S$.
The first two terms of (\[12\]) are symmetric and may be considered as a candidate for the metric tensor of $M$ built of $e$ in a globally GL$(V)$-invariant and generally-covariant way: $$g_{ij}=\lambda \gamma_{ij}+\mu \gamma_{i}\gamma_{j}=4\lambda S^{k}{}_{im}S^{m}{}_{jk}
+4\mu S^{k}{}_{ik}S^{m}{}_{jm}.$$ The best candidate is the dominant term $\gamma_{ij}$ built of $S$ according to the Killing prescription.
The mentioned scalar potentials used for multiplying (\[12\]) may be built of expressions like $\gamma_{il}\gamma^{jm}\gamma^{kn}S^{i}{}_{jk}S^{l}{}_{mn},\
\gamma^{ij}S^{k}{}_{ik}S^{m}{}_{jm},\
\Gamma^{i}{}_{j}\Gamma^{j}{}_{k}\ldots\Gamma^{l}{}_{m}\Gamma^{m}{}_{i},$ etc., where $\Gamma_{ij}:=4S^{k}{}_{lk}S^{l}{}_{ij},\
\Gamma^{i}{}_{j}=\gamma^{im}\Gamma_{mj}$ and $\gamma^{ij}$ is reciprocal to $\gamma_{ij}$, $\gamma^{ik}\gamma_{kj}=\delta^{i}{}_{j};$ we assume it does exist.
No doubts, the simplest and maximally “Born-Infeld-like” are models without such scalar potential terms, with the Lagrange tensor (\[12\]) quadratic in derivatives.
Due to the very strong nonlinearity, it would be very difficult to perform in all details the Dirac analysis of constraints resulting from the Lagrangian singularity. Nevertheless, the primary and secondary constraints may be explicitly found. If the problem is formulated in $n$ dimensions and all indices both holonomic and nonholonomic are written in the convention $K,i=\overline{0,n-1}$ (zeroth variable referring to “time”), then primary constraints, just as in electrodynamics, are given by $\pi^{0}{}_{K}=0,$ where $\pi^{i}{}_{K}$ are densities of canonical momenta conjugated to “potentials” $e^{K}{}_{i}$. Thus, there are a priori $n$ redundant variables among $n^{2}$ quantities $e^{i}{}_{K}$ and they may be fixed by coordinate conditions, like, e.g. $e^{i}{}_{K}=\delta^{i}{}_{K}$ for some fixed value of $K$ or $e^{K}{}_{i}{}^{,i}=e^{K}{}_{i,j}g^{ji}=0$ (Lorentz transversality condition). Secondary constraints are related to the field equations free of second “time” derivatives, these may be shown to be: $K_{i}{}^{0}=\nabla_{j}H_{i}{}^{0j}+2S^{k}{}_{kj}H_{i}{}^{0j}-2h_{ij}Q^{j0}=0.$ We have left the $Q$-term, because the statement, just as that about primary constraints is valid both for affinely-invariant and Lorentz invariant models. Obviously, for affine models the $Q$-term vanishes. Let us observe an interesting similarity to the empty-space Einstein equations, where secondary constraints are related to $R_{i}{}^{0}=0.$ Similarly, for the free electromagnetic field: $H^{0j}{}_{,j}={\rm
div} \overline{D}=0.$
As mentioned, discussion of the consistency of our model in terms of Dirac algorithm would be extremely difficult. Nevertheless, one can show that our field equations are not self-contradictory (this might easily happen in models invariant under infinite-dimensional groups with elements labelled by arbitrary functions). Namely, one can explicitly construct some particular solutions of very interesting geometric structure. Of course, there is still an unsolved problem “how large” is the general solution.
Analysing the structure of equations (\[11\]) one can easily prove the following
If field of frames $e$ has the property that its “legs” $e_{A}$ span a semi-simple Lie algebra in the Lie-bracket sense, $[e_{A},e_{B}]=\gamma^{C}{}_{AB}e_{C},\ \gamma^{C}{}_{AB}=const,$ det$[\gamma^{C}{}_{DA}\gamma^{D}{}_{CB}]\neq 0,$ then $e$ is a solution of (\[11\]) for any GL$(n,{\bf R})$-invariant model of $L$, in particular, for (\[12\]).
Roughly speaking, this means that semisimple Lie groups, or rather their group spaces are solutions of variational GL$(n,{\bf
R})$-invariant filed equations for linear frames. They are homogeneous, physically non-excited vacuums of the corresponding model. Fixing some point $a \in M$ we turn $M$ into semisimple Lie group. Its neutral element is just $a$ itself, $e_{A}$ generate left regular translations and are right-invariant. This gives rise also to left-invariant vector fields ${}^{a}e_{A}$ generating right regular translations, $\left[e_{A},e_{B}\right]=\gamma^{C}{}_{AB}e_{C}, \
\left[{}^{a}e_{A},{}^{a}e_{B}\right]=-\gamma^{C}{}_{AB}{}^{a}e_{C},
\ \left[e_{A},{}^{a}e_{B}\right]=0.$ The tensor $\gamma_{ij}$ becomes then the usual Killing metric on Lie group; it is parallel with respect to the teleparallelism connection $\Gamma_{\rm
tel}[e]$, $\nabla \gamma_{ij}=0.$ This means that $\left(M,\gamma,\Gamma_{\rm tel}\right)$ is a Riemann-Cartan space. For the general $e$ it is not the case. For semisimple Lie-algebraic solutions there exists such a bilinear form $\eta$ on ${\bf R}^{n}$ (on $V$) that: $\gamma[e]=h[e,\eta]$ and obviously $\eta_{AB}=\gamma^{C}{}_{DA}\gamma^{D}{}_{CB}.$ The metric field $\gamma [e]$ has $2n$ Killing vectors $e_{A},{}^{a}e_{A}$. Obviously, within the GL$(n,{\bf
R})$-invariant framework $\gamma[e]$ is a more natural candidate for the space-time metric than $h[e,\eta]$. In the special case of Lie-algebraic frames they in a sense coincide, but neither $\eta$ itself nor even its signature are a priori fixed. Instead, they are some features, a kind of integration constants of some particular solutions.
Let us mention, there is an idea according to which all fundamental physical fields should be described by differential forms (these objects may be invariantly differentiated in any amorphous manifold [@8; @21; @22; @23]). Of particular interest are the special solutions, constant in the sense that their differentials are expressed by constant-coefficients combinations of exterior products of primary fields.
The question arises as to a possible link between the above GL$(n,{\bf
R})$-framework and the ideas of general relativity. For Lie-algebraic fields of frames some kind of relationship does exist. Namely, if $\gamma_{ij}$ is the Killing metric on a semisimple $n$-dimensional Lie group, $R_{ij}$ is its Ricci tensor and $R$ — the curvature scalar, then, as one can show [@12] $$R_{ij}-\frac{1}{2}R\gamma_{ij}=-\frac{1}{8}(n-2)\gamma_{ij}.$$ Rescaling the definition of the metric tensor on $M,\
g_{ij}=a\gamma_{ij},\ a={\rm const},$ we obtain $R_{ij}-(1/2)Rg_{ij}=\Lambda g_{ij},\ \Lambda=-(n-2)/8a,$ and these are just Einstein equations with a kind of cosmological term. Therefore, at least in a neighbourhood of group-like vacuums the both models seem to be somehow interrelated.
There is however one disappointing feature of affinely-invariant $n$-leg models and their interesting and surprising Lie group solutions. Everything is beautiful for the abstract, non-specified $n$. But for our space-time $n=4$ and there exist no semisimple Lie algebras in this dimension. There are, fortunately, a few supporting arguments:
1. We can try to save everything on the level of Kaluza-Klein universes of dimension $n>4$. It is interesting that the $n$-leg field offers the possibility of deriving the very fibration of such a universe over the usual four-dimensional space-time as something dynamical, not absolute as in Kaluza-Klein theory. The fibration and the structural group would then appear as features of some particular solutions.
2. One can show that Lie-algebraic solutions exist in some sense for systems consisting of the $n$-leg and of some matter field, e.g. the complex scalar field $\Psi$. Lagrange tensor is then given by $L_{ij}=(1-a \overline{\Psi}
\Psi)\gamma_{ij}+b\overline{\Psi}_{,i} \Psi_{,j}.$ Even if $e_{A}$ span a nonsimple Lie algebra, there are $(e,\Psi)$-solutions with det$[L_{ij}]\neq 0$ and with the oscillating complex unimodulary factor at $\Psi$ [@10; @11]. The same may be done for higher-dimensional multiplets of matter fields, $L_{ij}=(1-a_{\overline{k} l}
\overline{\Psi}{}^{\overline{k}}\Psi^{l})\gamma_{ij}+
b_{\overline{k}l}\overline{\Psi}{}^{\overline{k}}{}_{,i}\Psi^{l}{}_{,j}.$
3. In dimensions “semisimple plus one” (e.g. $4$) there exist also some geometric solutions with the group-theoretical background. They are deformed trivial central extensions of semisimple Lie groups [@21].
Let us describe roughly the last point. We fix some Lie-algebraic $n$-leg field $E=(\ldots,E_{A},\ldots)=(E_{0},\ldots,E_{\Sigma},
\ldots)$, where $A=\overline{0,n-1}$, $\Sigma=\overline{1,n-1}$, and the basic Lie brackets are as follows: $\left[E_{0},E_{\Sigma}\right]=0,\
\left[E_{\Sigma},E_{\Lambda}\right]=E_{\Delta}C^{\Delta}{}_{\Sigma
\Lambda}, $ and ${\rm det}[C_{\Lambda \Gamma}]:={\rm
det}\left[C^{\Sigma}{}_{\Lambda \Delta} C^{\Delta}{}_{\Gamma
\Sigma}\right]\neq 0.$ In adapted coordinates $(\tau,x^{\mu})=(x^{0},x^{\mu})$ (where $\mu=\overline{1,n-1}$) we have $E_{0}=\partial/\partial \tau,\
E_{\Sigma}=E^{\mu}{}_{\Sigma}(x)\partial/\partial x^{\mu}.$ The dual co-frame $E=(\ldots,E^{A},\ldots)=(E^{0},\ldots,E^{\Sigma},
\ldots)$ is locally represented as: $E^{0}=d \tau,\
E^{\Sigma}=E^{\Sigma}{}_{\mu}(x)dx^{\mu},\
E^{\Sigma}{}_{\mu}E^{\mu}{}_{\Lambda}=\delta^{\Sigma}{}_{\Lambda}.$ The corresponding Lie algebra obviously is not semisimple. But we can construct new fields of frames $e$ or $'e$ given respectively by $e=\rho E,\ 'e_{0}=E_{0},\ 'e_{\Sigma}=\rho
E_{\Sigma}=e_{\Sigma},$ where $\rho$ is a scalar function such that $e_{\Sigma}\rho =E_{\Sigma}\rho=0$, i.e. in adapted coordinates it depends only on $\tau,\ \partial \rho/\partial
x^{\mu}=0$.
For any $\rho$ without critical points, both $e$ and $\ 'e$ are solutions of any GL$(n,{\bf R})$-invariant and generally covariant equations (\[11\]). In both cases $\gamma[e]=\gamma['e]$ is stationary and static in spite of the expanding (contracting) behaviour of $e,\ 'e$.
If the Lie algebra spanned by $(E_{1},\ldots,E_{n-1})$ is of the compact type, then $\gamma[e]$ is normal-hyperbolic and has the signature $(+-\ldots -)$ with respect to the nonholonomic basis $(E_{0},\ldots,E_{\Sigma},\ldots)$, thus the $\tau$-variable and coordinates $x^{i}$ have respectively time-like and space-like character. The above function $\rho$ is a purely gauge variable and in appropriately adapted coordinates: $$\label{13}
\gamma[e]=\gamma['e]=dx^{0}\otimes dx^{0}+{}_{(n-1)}\gamma_{\alpha
\beta}(x^{\kappa})dx^{\alpha}\otimes dx^{\beta},$$ where $x^{0}:=\pm \sqrt{(n-1)}{\rm ln}(x^{0}/\delta)$, $\delta$ is constant, ${}_{(n-1)}\gamma_{\alpha \beta}=4S^{\kappa}{}_{\lambda
\alpha}S^{\lambda}{}_{\kappa \beta}$. Obviously, in all formulas the capital and small Greek indices, both free and summed run over the “spatial” range $\overline{1,n-1}$ (conversely as in the usually used notation).
Another, coordinate-free expression: $\gamma[e]=(n-1)\left(d\rho /d\tau
\right)^{2}e^{0}\otimes e^{0}+\rho^{2}C_{\Lambda \Sigma}\\$$e^{\Lambda}\otimes
e^{\Sigma}.$ With such solutions $M$ becomes locally ${\bf R}_{\rm time}\times G_{\rm
space}$, $G$ denoting the $(n-1)$-dimensional Lie group with structure constants $C^{\Delta}{}_{\Lambda \Sigma}$. The above metric $\gamma$ has $(2n-1)$ Killing vectors; one time-like and $2(n-1)$ space-like ones, when $G$ is compact-type. This is explicitly seen from the formula (\[13\]), or its coordinate-free form $\gamma=(n-1)\left(d{\rm ln} \rho /d \tau \right)^{2}E^{0}\otimes E^{0}+C_{\Lambda
\Sigma}E^{\Lambda}\otimes E^{\Sigma}.$ If we introduce spinor fields, then in their matter Lagrangians we must use the Dirac-Einstein metric $h[e,\eta]$ with $\eta$ of the form: $\eta_{00}=\beta={\rm const},\ \eta_{0\Lambda}=0,\ \eta_{\Lambda
\Sigma}=C_{\Lambda \Sigma}$. This metric is subject to the cosmological expansion (contraction) known from general relativity, e.g. $h['e,\eta]$ in its spatial part expands according to the de Sitter rule. Therefore, in spite of stationary-static character of $\gamma$, the test spinor matter will witness about cosmological expansion (contraction). This may be an alternative explanation of this phenomenon. If $n=4$ there are the following Lie-algebraic-expanding vacuum solutions: ${\bf
R}\times {\rm SU}(2)$ or ${\bf R}\times {\rm SO}(3,{\bf R})$ with the normal-hyperbolic signature $(+\ -\ -\ -)$, the plus sign related to $E_{0}$. There are also solutions of the form ${\bf R}\times {\rm SL}(2,{\bf R})$, ${\bf R}\times
\overline{{\rm SL}(2,{\bf R})}$; they have the signature $(+\ +\ +\ -)$; now the time-like contribution has to do with the “compact dimension” of SL$(2,{\bf R})$, whereas the mentioned “expansion” holds in one of spatial directions. It is seen that our GL$(n,{\bf R})$-models in a sense distinguish both the normal-hyperbolic signature and the dimension $n=4$, just on the basis of solutions of local differential equations. In any case, something like the $\eta$-signature of standard tetrad description is not here introduced by hands.
Finally, let us observe that one can speculate also about another cosmological aspects of our model. In generally-relativistic spinor theory one uses the Dirac amplitude, tetrad and spinor connection (or affine Einstein-Cartan connection) as basic dynamical variables. The corresponding matter (Dirac) Lagrangian is locally SO$(1,3)$- or rather SL$(2,{\bf C})$-invariant. The same concerns gravitational Lagrangian for the tetrad and spinor connection either in Einstein or in gauge-Poincaré form. The idea was formulated some time ago that the true gravitational Lagrangian should contain a term which is only globally invariant under internal symmetries. Additional tetrad degrees of freedom were then expected to have something to do with the dark matter, at least in a part of it [@3; @4]. Our GL$(4,{\bf R})$-models would be from this point of view optimal.
Finally, let us notice that our “expanding” Lie solutions for dimensions “semisimple plus one” might be cosmologically interpreted as the motion of cosmical relativistic fluid ($e_{0}$-legs of the tetrad) with internal affine degrees of freedom ($e_{\Sigma}$-legs). This would be something like the relativistic micromorphic continuum [@20; @21].
[**Acknowledgements**]{} The author is greatly indebted to Colleagues from Cinvestav, first of all to Professor Maciej Przanowski, for invitation to the conference and for their cordial hospitality in Mexico City. The research itself and participation were also partly supported by the Polish Committee of Scientific Research (KBN) within the framework of the grant 8T07A04720.
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---
abstract: 'The SFiNCs (Star Formation in Nearby Clouds) project is an X-ray/infrared study of the young stellar populations in 22 star forming regions with distances $\la1$ kpc designed to extend our earlier MYStIX survey of more distant clusters. Our central goal is to give empirical constraints on cluster formation mechanisms. Using parametric mixture models applied homogeneously to the catalog of SFiNCs young stars, we identify 52 SFiNCs clusters and 19 unclustered stellar structures. The procedure gives cluster properties including location, population, morphology, association to molecular clouds, absorption, age ($Age_{JX}$), and infrared spectral energy distribution (SED) slope. Absorption, SED slope, and $Age_{JX}$ are age indicators. SFiNCs clusters are examined individually, and collectively with MYStIX clusters, to give the following results. (1) SFiNCs is dominated by smaller, younger, and more heavily obscured clusters than MYStIX. (2) SFiNCs cloud-associated clusters have the high ellipticities aligned with their host molecular filaments indicating morphology inherited from their parental clouds. (3) The effect of cluster expansion is evident from the radius-age, radius-absorption, and radius-SED correlations. Core radii increase dramatically from $\sim0.08$ to $\sim0.9$ pc over the age range $1-3.5$ Myr. Inferred gas removal timescales are longer than 1 Myr. (4) Rich, spatially distributed stellar populations are present in SFiNCs clouds representing early generations of star formation. An Appendix compares the performance of the mixture models and nonparametric Minimum Spanning Tree to identify clusters. This work is a foundation for future SFiNCs/MYStIX studies including disk longevity, age gradients, and dynamical modeling.'
author:
- |
K. V. Getman,$^{1}$[^1] M. A. Kuhn,$^{2,3}$ E. D. Feigelson,$^{1}$P. S. Broos,$^{1}$ M. R. Bate,$^{4}$ G. P. Garmire$^{5}$\
$^{1}$Department of Astronomy & Astrophysics, 525 Davey Laboratory, Pennsylvania State University, University Park PA 16802\
$^{2}$Instituto de Fisica y Astronomia, Universidad de Valparaiso, Gran Bretana 1111, Playa Ancha, Valparaiso, Chile\
$^{3}$Millenium Institute of Astrophysics, Av. Vicuna Mackenna 4860, 782-0436 Macul, Santiago, Chile\
$^{4}$Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter, Devon EX4 4QL, UK\
$^{5}$Huntingdon Institute for X-ray Astronomy, LLC, 10677 Franks Road, Huntingdon, PA 16652, USA
date: 'Accepted for publication in MNRAS, 2018 February 19'
title: Young Star Clusters In Nearby Molecular Clouds
---
\[firstpage\]
infrared: stars – stars: early-type – open clusters and associations: general – stars: formation – stars:pre-main sequence – X-rays: stars
Introduction {#intro_section}
============
Most stars in the Galaxy were formed in compact bound stellar clusters [@Lada2003] or distributed, unbound stellar associations [@Kruijssen2012]. Short-lived radioisotopic daughter nuclei in the Solar System meteorites indicate that our Sun formed in a modest-sized cluster on the edge of a massive OB-dominated molecular cloud complex [@Gounelle2012; @Pfalzner2015]. It is well accepted that most clusters quickly expand and disperse upon the removal of the residual molecular gas via stellar feedback so only a small fraction survive as bound open clusters [@Tutukov1978; @Krumholz2014].
But the formation mechanisms of rich clusters is still under study. Dozens of sophisticated numerical (radiation)-(magneto)-hydrodynamic simulations of the collapse and fragmentation of turbulent molecular clouds followed by cluster formation have been published [@Krumholz2014; @Dale2015]. The inclusion of stellar feedback is generally found to better reproduce important characteristics of star formation such as the stellar initial mass function (IMF)[@Bate2009; @Offner2009; @Krumholz2012], gas depletion time, and star formation efficiency [@Krumholz2014; @VazquezSemadeni2017].
Constraints can be imposed on the simulations through quantitative comparison with the detailed properties of large cluster samples. For instance, realistic simulations of stellar clusters in the solar neighborhood, such as @Bate2009 [@Bate2012], are anticipated to obey the empirical cluster mass-size relationship seen in infrared (IR) cluster catalogs @Carpenter2000 [@Lada2003], X-ray/IR catalogs @Kuhn2014 [@Kuhn2015a; @Kuhn2015b], and compilations of massive stellar clusters @Portegies10 [@Pfalzner2013; @Pfalzner2016].
Our effort called MYStIX, Massive Young Star-Forming Complex Study in Infrared and X-ray, produces a large and homogeneous dataset valuable for analysis of clustered star formation [@Feigelson2013; @Feigelson2018 <http://astro.psu.edu/mystix>]. MYStIX identified $>30,000$ young diskless (X-ray selected) and disk-bearing (IR excess selected) stars in 20 massive star forming regions (SFRs) at distances from 0.4 to 3.6 kpc. Using quantitative statistical methods, @Kuhn2014 identify over 140 MYStIX clusters with diverse morphologies, from simple ellipsoids to elongated, clumpy substructures. @Getman2014a [@Getman2014b] develop a new X-ray/IR age stellar chronometer and discover spatio-age gradients across MYStIX SFRs and within individual clusters. @Kuhn2015a [@Kuhn2015b] derive various cluster properties, discover wide ranges of the cluster surface stellar density distributions, and provide empirical signs of dynamical evolution and cluster expansion/merger.
More recently, the Star Formation in Nearby Clouds (SFiNCs) project [@Getman2017] extends the MYStIX effort to an archive study of 22 generally nearer and smaller SFRs where the stellar clusters are often dominated by a single massive star — typically a late-O or early-B — rather than by numerous O stars as in the MYStIX fields. Utilizing the MYStIX-based X-ray and IR data analyses methods, Getman et al. produce a catalog of nearly 8500 diskless and disk-bearing young stars in SFiNCs fields. One of our objectives is to perform analyses similar to those of MYStIX in order to examine whether the behaviors of clustered star formation are similar — or different — in smaller (SFiNCs) and giant (MYStIX) molecular clouds.
In the current paper, the MYStIX-based parametric method for identifying clusters using finite mixture models [@Kuhn2014] is applied to the young stellar SFiNCs sample. Fifty two SFiNCs clusters and nineteen unclustered stellar structures are identified across the 22 SFiNCs SFRs. Various basic SFiNCs stellar structure properties are derived, tabulated, and compared to those of MYStIX. In an Appendix, we compare our parametric method to the common non-parametric method for identifying stellar clusters based on the minimum spanning tree (MST) [@Gutermuth2009; @Schmeja11].
The method of finite mixture models for spatial point processes is briefly described in §\[model\_section\]. The SFiNCs stellar sample is provided in §\[sample\_section\]. The cluster surface density maps, model validation, error analysis, and membership are presented in §§\[surface\_density\_maps\_subsection\]-\[membership\_subsection\]. A multivariate analysis of SFiNCs$+$MYStIX clusters is given in §§\[ma\_section\]-\[cloud\_subsection\]. The main science results are discussed in §\[discussion\_sec\]. The Appendices discuss each SFiNCs region and compare the performance of cluster identification methods.
Cluster Identification with Mixture Models {#model_section}
==========================================
Identification and morphological characterization of SFiNCs clusters uses the parametric statistical mixture model developed in MYStIX [@Kuhn2014]. The mixture model is the sum of multiple clusters fit to the sky distribution of young stellar objects (YSOs) using maximum likelihood estimation. The number of clusters is one of the model parameters optimized in the procedure.
Briefly, this analysis investigates the distribution of a set of points $\mathbf{X}=\{\mathbf{x}_1,...,\mathbf{x}_N\}$ giving the (RA,Dec) coordinates of each YSO. Each point belongs to one of $k$ clusters, or to an unclustered component of scattered stars characterized by complete spatial randomness. Each cluster is described by the probability density function $g(\mathbf{x};\mathbf{\theta}_j)$, where $\theta_j$ is the set of parameters for the component. Mixture models of point processes commonly use a normal (Gaussian) function [@McLachlan2000; @Fraley2002; @KuhnFeigelson2017]. But, following @Kuhn2014, we choose $g$ to be a two-dimensional isothermal ellipsoid defined in their equation 4. This is the radial profile of a dynamically relaxed, isothermal, self-gravitating system where the stellar surface density having a roughly flat core and power-law halo. The effectiveness of the isothermal ellipsoidal model is shown in §\[model\_validation\] below for SFiNCs and by @Kuhn2014 [@Kuhn2017] for other young clusters. The relative contribution of each cluster is given by mixing coefficients $\{a_1,...,a_k\}$, where $0\le a_j\le1$ and $\sum_{j=1}^{k}a_j=1$. The distribution of the full sample $f(\mathbf{x})$ is then the sum of the $k$ cluster components, $\sum_{j=1}^{k+1}a_j g(\mathbf{x};\mathbf{\theta}_j)$, plus a constant representing the unclustered component.
The mixture model is fit to the (RA,Dec) point distribution for each SFiNCs field from @Getman2017 by maximum likelihood estimation. Each cluster has six parameters, $\theta=\{x_0,y_0,R_c,\epsilon,\phi,\Sigma_c\}$: coordinates of cluster center ($x_0,y_0$), isothermal core radius $R_c$ defined as the harmonic mean of the semi-major and semi-minor axes, ellipticity $\epsilon$, position angle $\phi$, and central stellar surface density $\Sigma_c$. The full model, including the unclustered surface density, has $6k+1$ parameters. The density in our chosen model never vanishes at large distances from the cluster center. However, the absence of a parameter for the outer truncation radius in our model does not strongly affect the results, because the spatial distribution of the MYStIX and SFiNCs stars is examined over a finite field of view. The core radius parameter can be better constrained than the truncation radius [e.g., @Kroupa2001], and our previous work shows that inclusion of a truncation radius parameter is not necessary to get a good fit [e.g., @Kuhn2010].
To reduce unnecessary computation on the entire parameter space, an initial superset of possible cluster components is obtained by visual examination of the YSO adaptively smoothed surface density maps. This initial guess then iteratively refined using the Nelder-Mead simplex optimization algorithm implemented in the $R$ statistical software environment. The optimal model set is chosen by minimizing the penalized likelihood Akaike Information Criterion (AIC). Additional software for post-fit analysis and visualization is provided by $R$’s [*spatstat*]{} CRAN package, a comprehensive statistical tool for analyzing spatial point patterns [@Baddeley2015]. See @Kuhn2014 for further details; the $R$ code for cluster identification using parametric mixture models is presented in Appendix of @Kuhn2014.
Figure \[fig\_fitting\_example\_be59\] shows an example of the fitting process for SFiNCs field containing cluster Berkeley 59. Panel (a) shows an initial guess for a cluster that is purposefully misplaced. Its location is corrected by the algorithm in panel (b) giving $AIC= -2340$. In the remaining panels, new components are iteratively added until $AIC$ reaches its minimum value of $AIC=-2395$ for two clusters (panel d). The attempt to add a third cluster raises the AIC (panel f), so the $k=2$ model is considered to be optimal for this field.
![An example of the mixture model fitting for the “flattened” star sample in Be 59 (see explanation for “flattened” sample in §\[sample\_section\]). The fitting consists of a sequence of main operations, starting with panel (a) and ending with panel (f). The panels show the smoothed projected stellar surface density of the YSOs in Be 59, with a color-bar in units of observed stars per pc$^{-2}$ (on a logarithmic scale). The core radii of the considered clusters are marked by the black ellipses and labeled by numbers. Panels (a), (c), and (e) display the refinement of the initial guess for the number of clusters. Panels (b), (d), and (f) display the fitting results using the initial guesses presented in panels (a), (c), and (e), respectively. The panel titles give the number of considered clusters ($k$), and, for the fitting operations, the resulting values of the Akaike Information Criterion. \[fig\_fitting\_example\_be59\]](f1.pdf){width="80mm"}
We are aware that the use of parametric mixture models for star cluster identification is unusual; most researchers use nonparametric methods. One important nonparametric procedure is the pruned Minimal Spanning Tree (MST) of the stellar celestial locations, also known as the astronomers’ friends-of-friends algorithm and the statisticians’ single linkage hierarchical clustering algorithm. In Appendix, we compare the performances of the mixture model and MST methods both for simulated situations and our SFiNCs datasets.
Sample Selection {#sample_section}
================
Using MYStIX-based data analysis methods described by @Feigelson2013, @Getman2017 perform a homogeneous reanalysis of the archived [*Chandra*]{}-ACIS X-ray, [*Spitzer*]{}-IRAC mid-IR data for the 22 nearby SFiNCs SFRs. Table \[tbl\_cluster\_sample\] lists the target fields and their estimated distances. This analysis resulted in $\simeq 15,300$ X-ray and $\sim 1,630,000$ mid-IR point sources. Further combining these X-ray and mid-IR source data with the archived 2MASS near-IR catalog and applying a decision tree classification method (based on the photometry and spatial distributions of the X-ray and IR point sources), Getman et al. identify 8,492 SFiNCs probable cluster members (SPCMs) across the 22 SFiNCs SFRs.
The SPCMs are a union of [*Chandra*]{} X-ray selected diskless and disk-bearing YSOs, [*Spitzer*]{} IR excess (IRE) disk-bearing YSOs, and published OB stars. SPCMs were considered as disk-bearing (diskless) when their infrared spectral energy distributions in 2MASS+IRAC IR bands exhibited (did not exibit) an IRE when compared to the dereddened median SED templates of IC 348 PMS stellar photospheres [@Lada2006]. A fraction of SPCMs that lack IR SEDs were classified as “PMB” (possible member). Out of 8492 SPCMs, 66%, 30%, and 4% were classified as disk-bearing, diskless, and “PMB”, respectively. The total numbers of SPCMs for each field are listed in Column 3 of Table \[tbl\_cluster\_sample\].
As in MYStIX, substantial spatial variations in X-ray sensitivity are present in the SPCM dataset due to the off-axis mirror vignetting and degradation of the point-spread function, and further complicated by the disorganized mosaics of [*Chandra*]{} fields with different exposures. To mitigate these effects, weak X-ray SPCMs with their X-ray photometric flux below the X-ray completeness limit ($F_{X,lim}$) were excluded from the cluster identification analysis. $F_{X,lim}$ is calculated as in MYStIX [@Kuhn2014]. The values of $F_{X,lim}$ flux and the number of the remaining X-ray SPCMs are given in Columns 4 and 6 of Table \[tbl\_cluster\_sample\], respectively.
The SPCM dataset is further culled of the non-X-ray [*Spitzer*]{} selected disk-bearing YSOs that lie outside the [*Chandra*]{}-ACIS-I fields. The number of the remaining [*Spitzer*]{} disk-bearing YSOs is given in Column 7. Column 8 gives the number of the OB-type SPCMs that are either X-ray sources with $F_{X} < F_{X,lim}$ or non-X-ray disk-bearing YSOs lying within the [*Chandra*]{}-ACIS-I fields. The final number of SPCMs $N_\star$ left for the cluster identification analysis is 5,164 (Column 5) constituting 61% of the original SPCM dataset. Referred to hereafter as the “flattened” sample, these stars are used for identifying SFiNCs clusters and deriving their morphological properties. However, the membership analysis (§\[membership\_subsection\]) and derivation of cluster’s absorption, SED slope, and age (§\[sample\_subsection\]) are based on the original sample of 8492 SPCMs.
[@c@c@c@c@c@c@c@c@c@[ ]{}]{} &&&&&&&&\
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Region & Distance & SPCMs && $\log F_{X,lim}$ & N$_\star$ & X-ray & IRE & OB \
& (kpc) & (stars) && (ph s$^{-1}$cm$^{-2}$) & (stars) & (stars) & (stars) & (stars)\
(1)&(2)&(3) &&(4)&(5)&(6)&(7)&(8)\
&&&&&&&&\
Be 59 & 0.900 & 626 && -6.000 & 435 & 315 & 118 & 8\
SFO 2 & 0.900 & 71 && -6.000 & 63 & 34 & 29 & 0\
NGC 1333 & 0.235 & 181 && -5.750 & 118 & 55 & 62 & 4\
IC 348 & 0.300 & 396 && -5.750 & 224 & 162 & 62 & 1\
LkH$\alpha$ 101 & 0.510 & 250 && -5.875 & 149 & 99 & 48 & 4\
NGC 2068-2071 & 0.414 & 387 && -5.750 & 234 & 120 & 113 & 3\
ONC Flank S & 0.414 & 386 && -5.375 & 237 & 133 & 104 & 1\
ONC Flank N & 0.414 & 327 && -5.875 & 217 & 151 & 64 & 4\
OMC 2-3 & 0.414 & 530 && -5.250 & 238 & 144 & 91 & 5\
Mon R2 & 0.830 & 652 && -5.750 & 280 & 134 & 144 & 5\
GGD 12-15 & 0.830 & 222 && -5.875 & 147 & 72 & 75 & 2\
RCW 120 & 1.350 & 420 && -6.250 & 278 & 157 & 121 & 1\
Serpens Main & 0.415 & 159 && -6.125 & 105 & 55 & 50 & 0\
Serpens South & 0.415 & 645 && -6.250 & 288 & 56 & 232 & 0\
IRAS 20050+2720 & 0.700 & 380 && -6.250 & 281 & 121 & 160 & 0\
Sh 2-106 & 1.400 & 264 && -6.125 & 221 & 123 & 98 & 1\
IC 5146 & 0.800 & 256 && -6.250 & 232 & 141 & 90 & 6\
NGC 7160 & 0.870 & 143 && -6.000 & 93 & 86 & 2 & 8\
LDN 1251B & 0.300 & 49 && -5.500 & 31 & 21 & 10 & 0\
Cep OB3b & 0.700 & 1636 && -5.875 & 1019 & 551 & 465 & 9\
Cep A & 0.700 & 335 && -5.750 & 164 & 81 & 83 & 0\
Cep C & 0.700 & 177 && -5.750 & 132 & 52 & 80 & 0\
&&&&&&&&\
@Feigelson2013 [their Appendix] discuss issues of incompleteness and bias in samples derived in this fashion. Most importantly, due to a well-known X-ray/mass correlation, the X-ray-selected SPCM subsamples are approximately complete above mass limits around $0.1-0.3$ M$_{\odot}$ for the range of $ F_{X,lim}$ values. For the IRE subsamples in most of the SFiNCs SFRs, the histograms of the \[3.6\]-magnitude for the SFiNCs [*Spitzer*]{} point sources peak near 17 mag equivalent to $\la 0.1$ M$_{\odot}$ sensitivity limits [@Getman2017 their Figure 3]. The IRE sub-sample is further limited by the 2MASS sensitivity limit of $K_s \sim 14.3$ mag, which translates to $\sim 0.1-0.3$ M$_{\odot}$ PMS star at distances of $300-900$ pc. A future SFiNCs paper will compensate for these incompleteness effects using the X-ray luminosity function and initial mass function following the procedure of @Kuhn2015a.
The detection of highly absorbed clusters relies heavily on the catalog of the [*Spitzer*]{} selected disk-bearing YSOs. But, in the cluster centers of five SFiNCs SFRs (LkH$\alpha$ 101, Mon R2, RCW 120, Sh 2-106, and Cep A), the IRAC point source sensitivity is reduced by the high background nebular emission from heated dust [@Getman2017 their Figure Set 6]. Unlike in the case of the X-ray sample, application of a uniform MIR flux limit seems infeasible here as it would leave only a handful of bright IRAC sources. For the detection of these clusters, our procedures depend mainly on the catalog of the [*Chandra*]{} selected YSOs.
Clusters in SFiNCs Clouds {#clusters_subsection}
=========================
The fitting of the isothermal ellipsoid mixture model to the “flattened” sample of SPCMs yields 52 clusters across the 22 SFiNCs SFRs. Table \[tbl\_cluster\_morphology\] lists their morphological properties: the celestial coordinates of the cluster center, core radius, ellipticity, orientation, and total number of YSOs estimated by integrating the cluster model across an ellipse four times the size of the core. The estimated uncertainties on these cluster parameters are explained in §\[error\_analysis\]. Also listed is a flag indicating the association with a molecular cloud (§\[cloud\_subsection\]). Note that $4 \times R_c$ corresponds roughly to the projected half-mass radius for a cluster with an outer truncation radius of $\sim 17 \times R_c$ [@Kuhn2015b]. This somewhat arbitrary decision of $4 \times R_c$, as an integration radius, was the result of our visual inspection of trial star assignments among MYStIX [@Kuhn2014] and SFiNCs clusters, taking into consideration two main factors, the typical separation between adjacent clusters and typical size of field of view. The numbers of stars within a specific radius follow the Equation 3 in @Kuhn2014, assuming that the model provides an accurate description of the distribution of stars. According to this equation, the number of stars in a cluster increases by a factor 1.6 when changing the integration radius from, say, $4 \times R_c$ to $10 \times R_c$.
Morphology of Clustered Star Formation {#surface_density_maps_subsection}
--------------------------------------
We study the spatial structure of clustering in SFiNCs molecular clouds using adaptively smoothed maps of stellar surface density. The upper panel of Figure \[fig\_cluster\_identification\] shows the map for the “flattened” sample for the OMC 2-3 field. Similar maps for the other 21 SFiNCs fields are provided in the Supplementary Materials. These maps are constructed using Voronoi tessellations, where the estimated intensity in each tile is the reciprocal of the tile area, implemented in function [*adaptive.density*]{} from the [*spatstat*]{} package [@Baddeley2015]. The color scale is in units of observed stars per pc$^{-2}$. Cluster cores are outlined by the black ellipses.
In two cases the AIC minimization led to the acceptance of very small clusters ($R_c < 0.01$ pc) with few stellar members: Cluster A in IRAS 20050+2720, and the cluster D in Cep OB3b. This arises when the sparse subcluster is very compact. The mixture model has no rigid threshold on the number of points in a cluster.
![Identification of SFiNCs clusters through the mixture model analysis, performed on the “flattened” SPCM samples within the [*Chandra*]{} ACIS fields. An example is given for the OMC 2-3 SFR, and the full set of panels for the 22 SFiNCs fields is available in the Supplementary Material. The upper panel shows the smoothed projected stellar surface density with a color-bar in units of observed stars per pc$^{-2}$ on a logarithmic scale. The figure title gives the name of a SFiNCs SFR, the number of identified clusters, and the final value of the Akaike Information Criterion. The lower panel shows smoothed map of residuals between the data and the model with a color bar in units of stars per pc$^{-2}$ on a linear scale. On both panels, the core radii of the identified SFiNCs clusters are outlined by the black ellipses. \[fig\_cluster\_identification\]](f2.pdf){width="70mm"}
As with MYStIX, SFiNCs shows wide diversity of stellar structures. For the majority of the SFiNCs SFRs (Be 59, SFO 2, IC 348, LkH$\alpha$ 101, GGD 12-15, Serpens Main, Serpens South, IRAS 20050+2720, Sh 2-106, IC 5146, LDN 1251B, Cep A, Cep C), a single main compact ($R_c<0.5$ pc) cluster is detected within the [*Chandra*]{}-ACIS-I field. For a few SFRs (NGC 7160 and ONC Flanking Fields), a single but rather loose ($R_c>0.5$ pc) stellar structure is found.
[@c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}]{} &&&&&&&&&&&\
Cluster & R.A.(J2000) & Decl.(J2000) & PosErr & $R_c$ &$\sigma R_c/R_c$ & $\epsilon$ & $\sigma \epsilon/\epsilon$ & $\phi$ & $N_{4,model}$ & $~\sigma N/N$ & Cloud \
& (deg) & (deg) & () & (pc) & & & & (deg) & (stars) & &\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) & (12)\
&&&&&&&&&&&\
Be 59 A & 0.508713 & 67.520259 & 701 & 1.320 & 0.25 & 0.39 & 0.37 & 99 & 486 & 0.37 & R\
Be 59 B & 0.562134 & 67.418764 & 1 & 0.210 & 0.20 & 0.15 & 0.46 & 8 & 152 & 0.23 & R\
SFO 2 A & 1.018036 & 68.565945 & 5 & 0.076 & 0.51 & 0.61 & 0.18 & 155 & 23 & 0.19 & C\
NGC 1333 A & 52.276137 & 31.280885 & 70 & 0.100 & 0.55 & 0.55 & 0.24 & 48 & 41 & 0.61 & C\
NGC 1333 B & 52.279194 & 31.364583 & 44 & 0.100 & 0.44 & 0.72 & 0.12 & 45 & 54 & 0.47 & C\
IC 348 B & 56.141167 & 32.158825 & 31 & 0.200 & 0.21 & 0.39 & 0.25 & 6 & 156 & 0.17 & R\
IC 348 A & 55.999718 & 32.031989 & 18 & 0.040 & 0.49 & 0.60 & 0.24 & 119 & 10 & 0.28 & C\
LkHa101 A & 67.542392 & 35.268025 & 17 & 0.210 & 0.17 & 0.11 & 0.52 & 177 & 108 & 0.12 & R\
NGC 2068-2071 A & 86.543905: & -0.138717: & ... & 0.110: & ... & 0.96: & ... & 12: & 16: & ... & C\
NGC 2068-2071 B & 86.666392 & 0.088074 & 46 & 0.290 & 0.43 & 0.50 & 0.29 & 129 & 68 & 0.28 & R\
NGC 2068-2071 C & 86.775230 & 0.375235 & 17 & 0.060 & 0.49 & 0.31 & 0.38 & 159 & 25 & 0.24 & C\
NGC 2068-2071 D & 86.808844 & 0.316468 & 23 & 0.110 & 0.50 & 0.86 & 0.05 & 76 & 28 & 0.21 & C\
ONC Flank S A & 83.862602 & -5.479723 & 167 & 1.230 & 0.49 & 0.75 & 0.10 & 28 & 1521 & 0.37 & ...\
ONC Flank N A & 83.819526 & -4.845647 & 90 & 0.804 & 0.21 & 0.32 & 0.36 & 15 & 578 & 0.28 & R\
OMC 2-3 A & 83.825638: & -5.271570: & ... & 0.120: & ... & 0.82: & ... & 17: & 65: & ... & C\
OMC 2-3 B & 83.835852 & -5.012671 & 34 & 0.100 & 0.77 & 0.88 & 0.06 & 134 & 14 & 0.45 & C\
OMC 2-3 C & 83.855702 & -5.156237 & 10 & 0.060 & 0.44 & 0.76 & 0.08 & 163 & 23 & 0.28 & C\
OMC 2-3 D & 83.883413: & -5.266815: & ... & 0.020: & ... & 0.43: & ... & 67: & 10: & ... & ...\
Mon R2 A & 91.937730 & -6.344872 & 65 & 0.200 & 0.45 & 0.77 & 0.11 & 173 & 55 & 0.34 & C\
Mon R2 B & 91.948257 & -6.378512 & 11 & 0.100 & 0.47 & 0.77 & 0.10 & 29 & 33 & 0.47 & C\
Mon R2 C & 91.961306 & -6.428142 & 6 & 0.070 & 0.95 & 0.35 & 0.38 & 114 & 19 & 0.30 & C\
GGD 12-15 A & 92.710405 & -6.194814 & 11 & 0.140 & 0.19 & 0.46 & 0.16 & 68 & 77 & 0.13 & C\
RCW 120 A & 258.037282 & -38.516262 & 7 & 0.070 & 0.37 & 0.45 & 0.33 & 144 & 15 & 0.18 & C\
RCW 120 B & 258.099513 & -38.487682 & 15 & 0.290 & 0.25 & 0.52 & 0.17 & 31 & 60 & 0.19 & R\
RCW 120 C & 258.165621 & -38.451639 & 20 & 0.240 & 0.40 & 0.48 & 0.26 & 162 & 29 & 0.33 & C\
RCW 120 D & 258.179085 & -38.376186 & 40 & 0.110 & 1.25 & 0.86 & 0.06 & 155 & 12 & 0.43 & C\
Serpens Main A & 277.464566 & 1.267515 & 35 & 0.040 & 0.66 & 0.71 & 0.19 & 126 & 12 & 0.54 & C\
Serpens Main B & 277.492003 & 1.216660 & 15 & 0.060 & 0.35 & 0.48 & 0.22 & 5 & 38 & 0.25 & C\
Serpens South A & 277.468911: & -1.962172: & ... & 0.024: & ... & 0.45: & ... & 105: & 10: & ... & C\
Serpens South B & 277.492366: & -2.138106: & ... & 0.022: & ... & 0.87: & ... & 81: & 4: & ... & C\
Serpens South C & 277.511870 & -2.048380 & 10 & 0.056 & 0.25 & 0.65 & 0.08 & 168 & 58 & 0.13 & C\
Serpens South D & 277.574336: & -2.148296: & ... & 0.022: & ... & 0.31: & ... & 2: & 7: & ... & C\
IRAS 20050+2720 A & 301.713709: & 27.343918: & ... & 0.007: & ... & 0.71: & ... & 17: & 4: & ... & ...\
IRAS 20050+2720 B & 301.741427: & 27.559032: & ... & 0.171: & ... & 0.23: & ... & 35: & 8: & ... & C\
IRAS 20050+2720 C & 301.742363: & 27.511739: & ... & 0.051: & ... & 0.65: & ... & 145: & 20: & ... & C\
IRAS 20050+2720 D & 301.772854 & 27.487538 & 13 & 0.081 & 0.57 & 0.56 & 0.25 & 178 & 63 & 0.40 & C\
IRAS 20050+2720 E & 301.915203: & 27.562103: & ... & 0.075: & ... & 0.26: & ... & 39: & 33: & ... & ...\
Sh 2-106 A & 306.812857 & 37.461188 & 31 & 0.300 & 0.35 & 0.58 & 0.25 & 104 & 15 & 0.39 & R\
Sh 2-106 B & 306.823055 & 37.376261 & 11 & 0.110 & 0.45 & 0.11 & 0.50 & 56 & 16 & 0.34 & C\
Sh 2-106 C & 306.853428: & 37.293251: & ... & 0.030: & ... & 0.39: & ... & 20: & 10: & ... & C\
Sh 2-106 D & 306.860795 & 37.382119 & 6 & 0.090 & 0.31 & 0.48 & 0.23 & 120 & 41 & 0.14 & C\
IC 5146 A & 328.140096 & 47.228756 & 16 & 0.090 & 0.67 & 0.51 & 0.31 & 165 & 12 & 0.32 & C\
IC 5146 B & 328.381131 & 47.265152 & 7 & 0.170 & 0.16 & 0.25 & 0.34 & 17 & 115 & 0.08 & R\
NGC 7160 A & 328.443151 & 62.585448 & 29 & 0.720 & 0.24 & 0.30 & 0.35 & 146 & 108 & 0.18 & R\
LDN 1251B A & 339.696683 & 75.193642 & 19 & 0.030 & 0.60 & 0.41 & 0.38 & 73 & 15 & 0.34 & C\
Cep OB3b A & 343.446066 & 62.596289 & 24 & 0.450 & 0.20 & 0.44 & 0.21 & 52 & 304 & 0.15 & R\
Cep OB3b B & 343.743449 & 62.569278 & 20 & 0.090 & 1.00 & 0.48 & 0.22 & 66 & 30 & 0.28 & ...\
Cep OB3b C & 344.167892 & 62.701711 & 22 & 0.580 & 0.14 & 0.18 & 0.44 & 1 & 520 & 0.13 & R\
Cep OB3b D & 344.279145: & 62.641799: & ... & 0.001: & ... & 0.53: & ... & 86: & 7: & ... & C\
Cep A A & 344.073598 & 62.030905 & 14 & 0.210 & 0.20 & 0.54 & 0.12 & 112 & 87 & 0.14 & C\
Cep C A & 346.441016 & 62.502827 & 23 & 0.180 & 0.25 & 0.53 & 0.16 & 50 & 64 & 0.14 & C\
Cep C B & 346.654364: & 62.530596: & ... & 0.020: & ... & 0.25: & ... & 34: & 5: & ... & C\
&&&&&&&&&&&\
In some SFRs (IC 348, Serpens Main, Serpens South, IRAS 20050+2720, Sh 2-106, Cep C), their main clusters are accompanied by minor siblings on $\sim 1$ pc scales. A few fields (OMC 2-3, Mon R2, RCW 120) harbor linear chains of clusters on $2-3$ pc scales. For the NGC 2068-2071 and Cep OB3b SFRs with wider angular coverage, multiple stellar structures are seen on $5$ pc scales.
In MYStIX, @Kuhn2014 [@Kuhn2015b] define the following four morphological classes of clusters: isolated clusters, core-halo structures, clumpy structures, and linear chains of clusters. The SFiNCs’ single compact and loose clusters are morphological analogues to the isolated MYStIX clusters. Meanwhile, SFiNCs lacks structures similar to the MYStIX clumpy structures that are seen on large spatial scales of $\ga 5-10$ pc in the rich M 17, Lagoon, and Eagle SFRs. SFiNCs’ linear chain clusters, observed on a $2-3$ pc scale, are much smaller than the MYStIX chain structures observed on $10-15$ pc scales (DR 21, NGC 2264, NGC 6334, and NGC 1893). The differences between the SFiNCs and MYStIX cluster morphologies could have several causes: (1) the intrinsic spatial scale of the sample (MYStIX SFRs are chosen to be giant molecular clouds); (2) observational fields of view (MYStIX fields are generally more distant with more mosaicked [*Chandra*]{} exposures) allowing larger-scale clumpy or linear structures; and/or (3) intrinsic cluster richness (MYStIX clusters are richer allowing the algorithm to find more secondary structures).
In Appendix, we summarize the YSO cluster distributions in a multi-wavelength astronomical context such as molecular cloud and other interstellar features.
Model Validation {#model_validation}
----------------
As discussed in the Appendix, parametric modeling such as our mixture model has advantages over nonparametric clustering methods in identifying optimal number of clusters with clear physical properties. However, these advantages accrue only if the parametric assumptions hold; in statistical parlance, the model must be correctly specified. It is therefore important to examine whether the mixture of two-dimensional isothermal ellipsoid models fits the stellar distribution in SFiNCs clusters. We perform here two model validation tests.
First, we show maps of the residuals between the data and model in the lower panels of Figure \[fig\_cluster\_identification\] computed using the $R$ function [*diagnose.ppm*]{} from the [*spatstat*]{} package. As discussed by @Kuhn2014 and @Baddeley2015, residual maps for spatial point processes offer direct information about where the model and the data agree or disagree. The residuals are shown smoothed by a Gaussian kernel width of 0.4 pc in units of stars per pc$^{-2}$; the blue and red colors indicate negative (model $>$ data) and positive (data $>$ model) residuals, respectively.
Examining the relative surface density scales of the original and residual maps, one sees for nearly all of the SFiNCs regions that the peak residuals are less than $\sim 10$% of the original peak levels indicating good matches between the data and the model. On occasion, positive residuals exceeding $30$% are present such as residual hotspots in SFO 2, RCW 120, IRAS 20050+2720, and Cep C. Some of these coincide with dusty cloudlets seen in far-infrared images of the regions and could be small embedded star groups whose contributions to the model likelihood were too small to be accepted as new clusters. For a few other cases such as RCW 120 (clusters A and C), Sh 2-106 (A and C), and IRAS 20050+2720 (B), the residuals exceed $30$%, indicating that the observed stellar distribution is not well fit with equilibrium elliptical models. Evaluation of individual residual maps appears in Appendix \[sec\_individual\_subclusters\]. Overall, we conclude that the residual maps show mostly random noise and, in most SFiNCs fields, the mixture model did not either create false or miss true clusters in an obvious fashion.
![Validation of the mixture models with random Voronoi tesselations. Two SFiNCs fields are shown here; similar panels for the full SFiNCs sample are presented in the Supplementary Materials. YSOs from the “flattened” SPCM sample are shown as gray points. Three numbers are given in each tile: the number of observed YSOs ($N_{observed}$; upper left), the number expected from the best fit mixture model ($N_{expected}$; upper right); and Pearson residual or ‘sigma’ deviation (bottom). \[fig\_chi2\_global\]](f3.pdf){width="70mm"}
Second, we compare observed and model star counts in polygonal tiles obtained from Voronoi tesselation using Poisson statistics. This method is used in astronomy [@Schmeja11] and is a variant of the quadrat counting test widely used in other fields [@Baddeley2015]. Goodness of fit can be estimated from Pearson’s $X^2$ statistic $$X^{2} = \sum_{i=1}^{n} \frac{(N_{i,observed}-N_{i,expected})^2}{N_{i,expected}},$$ where $n$ is the number of tiles, and $N_{i,observed}$ and $N_{i,expected}$ are the numbers of observed and expected YSOs in an $i$ tile, respectively. This statistic is compared to the $\chi^{2}$ distribution with $n-1$ degrees of freedom to evaluate the probability for the null hypothesis that the data are drawn from the point process model distribution. The expected values are obtained from an inhomogeneous Poisson process based on the mixture model clusters. The test is computed using the [*quadrat.test*]{} and [*dirichlet*]{} functions in [*spatstat*]{} [@Baddeley2015].
[@c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}]{} &&&&\
Region & $N_{subclust}$ & $X^2$ & dof & $P_{\chi^2}$ \
(1) & (2) & (3) & (4) & (5)\
&&&&\
Be 59 & 2& 5.1 & 9 &0.35\
SFO 2 & 1& 1.6 & 2 &0.89\
NGC 1333 & 2& 1.3 & 4 &0.29\
IC 348 & 2& 5.1 & 7 &0.71\
LkH$\alpha$ 101 & 1& 7.9 & 5 &0.33\
NGC 2068-2071 & 4&13.8 &10 &0.36\
ONC Flank S & 1& 4.9 & 9 &0.32\
ONC Flank N & 1& 8.5 & 8 &0.76\
OMC 2-3 & 4& 4.5 & 7 &0.56\
Mon R2 & 3& 7.0 & 9 &0.73\
GGD 12-15 & 1& 2.6 & 4 &0.75\
RCW 120 & 4& 4.8 & 7 &0.63\
Serpens Main & 2& 3.4 & 5 &0.73\
Serpens South & 4&10.0 & 12 &0.77\
IRAS 20050+2720 & 5&13.6 & 9 &0.27\
Sh 2-106 & 4&13.0 & 8 &0.23\
IC 5146 & 2&12.8 & 7 &0.16\
NGC 7160 & 1&5.0 &4 &0.57\
LDN 1251B & 1&5.0 &2 &0.16\
Cep OB3b & 4&30.8 &30 &0.85\
Cep A & 1&10.0 & 6 &0.25\
Cep C & 2&4.9 &5 &0.85\
&&&&\
Figure \[fig\_chi2\_global\] shows randomly tessellated stellar spatial distributions for two SFiNCs fields; the complete SFiNCs sample is shown in the Supplementary Materials. The number of points in a tile is drawn from the Poisson distribution, but the calculation of $\chi^{2}$ assumes that the distribution of points is normal. To allow the normal approximation to Poisson distribution, only SFiNCs tessellations with the number of sources in a tile $\geq 10$ are considered. Pearson residuals $X$ are shown as labels to each tile. Out of $>$190 tiles across 22 SFiNCs SFRs, only 3 tiles exhibit Pearson residuals above 2 in absolute value, an indicator of a likely departure from the fitted model. The high fraction of tiles with low Pearson residuals is a clear evidence that the SFiNCs YSOs distributions are generally well fit with the isothermal elliptical models. Table \[tbl\_chi2\_global\] shows high probability values ($P_{\chi^2} >> 0.01$) that the data are successfully drawn from the model for the full SFiNCs sample.
Cluster Parameter Uncertainties {#error_analysis}
-------------------------------
In order to estimate statistical errors on the SFiNCs cluster parameters derived in §\[clusters\_subsection\], we conduct Monte Carlo simulations. For each of the SFiNCs SFRs, we simulate 100 random sets of the spatial distributions of their “flattened” stellar samples; these distributions follow isothermal ellipsoid models with individual parameters taken from Table \[tbl\_cluster\_morphology\]. This analysis was performed using function [*simulate.ppm*]{} in [*spatstat*]{}, as well as the CRAN [*spatgraphs*]{} package [@Baddeley2015; @Rajala2015]. Figure \[fig\_mma\_simulations\] shows individual examples of the simulated SFiNCs data.
For the model fitting of the simulated data, the “initial guess” stage is imitated by choosing the positions of initial cluster models to match the positions of the simulated clusters. Likelihood maximization is further mimicked by running the model refinement stage via Nelder-Mead [Appendix of @Kuhn2014] several times. For each of the cluster parameters, the 68% confidence interval is derived from a sample distribution of 100 simulated values. The inferred confidence intervals for the cluster’s position, core radius, ellipticity, and modeled number of stars within the area four times the size of the core are reported in Table \[tbl\_cluster\_morphology\].
![Error analysis of the mixture models with simulated stellar spatial distributions (black points). Six SFiNCs fields are shown here; similar panels for the remaining SFiNCs regions are presented in the Supplementary Materials. The core radii of the simulated isothermal ellipsoid clusters are marked by the green ellipses. \[fig\_mma\_simulations\]](f4.pdf){width="80mm"}
Two cluster quantities of interest are the core radius and ellipticity. The fractional statistical error on the core radius ranges from 30-60% for weak clusters with $20<N_{4,model} < 70$ stars to $<25$% for more populous clusters (Figure \[fig\_simulation\_errors\]a). The fractional statistical error on cluster ellipticity does not correlate with cluster population but rather depends on the ellipticity value itself (Figure \[fig\_simulation\_errors\]b). The ellipticity errors are typically $>$30% for clusters with $\epsilon < 0.4$ and $<$15% for extremely elongated clusters.
For SFiNCs clusters that are either sparse ($N_{4,model} < 20$ stars) and/or strongly affected by nearby clusters, parameter values are poorly constrained by the simulations. In Table \[tbl\_cluster\_morphology\], the reported parameter values for these clusters are appended by the warning sign “:”, and they are omitted from the multivariate analyses in §\[ma\_section\] and \[cloud\_subsection\].
![Uncertainties of cluster core radii and ellipticities inferred from the simulations of SFiNCs clusters (§\[error\_analysis\]). (a) Fractional error of cluster core radius versus number of stars estimated by integrating the model component out to four times the size of the cluster core. (b) Fractional error of cluster ellipticity versus cluster ellipticity. \[fig\_simulation\_errors\]](f5.pdf){width="80mm"}
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Region & Desig & R.A.(J2000) & Decl.(J2000) & Flag & $ME$ & $J$ & $H$ & $\alpha_{IRAC}$ & $\log L_{X,tc}$ & $Age_{JX}$ & Clus \
& & (deg) & (deg) & & (keV) & (mag) & (mag) & & (erg s$^{-1}$) & (Myr) &\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11) & (12)\
&&&&&&&&&&&\
Be59 & 000033.87+672446.2 & 0.141150 & 67.412846 & 110 & 2.62 & $15.19\pm 0.05$ & $13.58\pm 0.04$ & $ -2.44\pm 0.04$ & 30.94 & ... & A\
Be59 & 000036.43+672658.5 & 0.151798 & 67.449596 & 110 & 1.88 & $13.68\pm 0.03$ & $12.64\pm 0.03$ & $ -2.69\pm 0.02$ & 30.79 & ... & A\
Be59 & 000045.20+672805.8 & 0.188345 & 67.468297 & 110 & 1.66 & $13.90\pm 0.03$ & $12.85\pm 0.03$ & $ -2.60\pm 0.04$ & ... & ... & A\
Be59 & 000046.19+672358.2 & 0.192477 & 67.399503 & 110 & 1.58 & $14.08\pm $... & $13.01\pm 0.04$ & $ -2.77\pm 0.11$ & 30.25 & ... & A\
Be59 & 000050.10+672721.4 & 0.208781 & 67.455954 & 110 & 1.77 & $15.09\pm 0.05$ & $13.89\pm 0.04$ & $ -2.58\pm 0.05$ & 29.90 & 2.5 & A\
Be59 & 000051.39+672648.8 & 0.214161 & 67.446914 & 110 & 2.12 & $14.66\pm 0.04$ & $13.33\pm 0.04$ & $ -2.31\pm 0.04$ & 30.63 & ... & A\
Be59 & 000053.45+672615.0 & 0.222725 & 67.437501 & 110 & 2.37 & $15.43\pm 0.06$ & $13.99\pm 0.05$ & $ -2.63\pm 0.10$ & 30.82 & ... & A\
Be59 & 000054.01+672119.8 & 0.225079 & 67.355504 & 110 & 2.37 & $15.41\pm 0.05$ & $14.03\pm 0.04$ & $ -2.40\pm 0.04$ & 30.81 & ... & A\
Be59 & 000055.58+672647.8 & 0.231621 & 67.446638 & 110 & 2.08 & $15.30\pm 0.05$ & $13.96\pm 0.04$ & $ -2.54\pm 0.09$ & 30.21 & 4.3 & A\
Be59 & 000056.24+672835.1 & 0.234343 & 67.476426 & 110 & 3.00 & $16.19\pm 0.09$ & $15.00\pm 0.08$ & $ -2.68\pm 0.13$ & 31.25 & ... & A\
&&&&&&&&&&&\
Cluster Membership {#membership_subsection}
------------------
As in MYStIX [@Kuhn2014], the mixture model is also used here as a “soft classifier” that, with additional decision rules, allows individual YSOs to be assigned to the clusters or to the unclustered component. For each YSO, the probability of membership is calculated based on the relative contribution of the different cluster model components to the stellar density at the location of a YSO. The following MYStIX decision rules are further adopted for the cluster membership assignment of the SFiNCs YSOs: the probability for the assigned cluster must exceed $30$% and the cluster members must lie within an ellipse four times the size of the cluster core. Stars that fail these rules have “uncertain” membership. The result of this membership assignment procedure is shown in Figure \[fig\_cluster\_assignment\_maps\].
![Cluster assignments in SFiNCs, shown for the full SFiNCs Probable Complex Member sample within the [*Chandra*]{} ACIS field for the OMC 2-3 star forming region; similar panels for the full SFiNCs sample are presented in the Supplementary Materials. SPCMs are superimposed on gray-scale far-IR images taken by $Herschel$-SPIRE at 500 $\mu$m or (for NGC 7822, IRAS 00013+6817, IRAS 20050+2720, NGC 7160, CepOB3b) $AKARI$-FIS at 160 $\mu$m. These images trace the locations of the SFiNCs molecular clouds shown on a logarithmic grayscale where denser clouds appear darker. The red ellipses show the best fit mixture model and the stars assigned to each cluster are color coded. Unclustered members are shown in magenta and unassigned stars in yellow. The [*Chandra*]{}-ACIS field of view is outlined by the green polygon. \[fig\_cluster\_assignment\_maps\]](f6.pdf){width="100mm"}
For the entire SPCM sample (8,492 stars), Table \[tbl\_cluster\_membership\] lists the cluster assignments along with several X-ray, NIR, and MIR source properties. For the SPCM sources located outside the [*Chandra*]{} ACIS-I fields their cluster assignments are unknown (indicated as “...” in Column 12). Inside the ACIS-I fields, the individual SPCMs are either assigned to specific clusters (“A”-“E” in Column 12) or unclustered population (“U”) or marked as objects with uncertain membership (“X”). The difference between the YSOs with uncertain membership (“X”) and unclustered YSOs (“U”) is that the former fail to meet the assignment criteria for any model component while the latter are assigned successfully to the unclustered model component. Out of 8,492 SPCMs, 5,214 are clustered (“A”-“E”), 1,931 are unclustered (“U”), 252 have uncertain membership (“X”), and 1,095 lie outside the ACIS-I fields and have unknown membership (“...”).
Source properties included in Table \[tbl\_cluster\_membership\] are taken from @Getman2017: X-ray median energy ($ME$) that measures absorption to the star, X-ray luminosity corrected for this absorption, NIR 2MASS $J$ and $H$-band magnitudes, slope of the $3.6-8$ $\mu$m spectral energy distribution $\alpha_{IRAC} = d \log(\lambda F_{\lambda})/d \log(\lambda)$, and stellar age, $Age_{JX}$. SFiNCs $Age_{JX}$ values are calculated by @Getman2017 following the methodology of @Getman2014a. This age estimator is based on an empirical X-ray luminosity-mass relation calibrated to well studied Taurus PMS stars and to theoretical evolutionary tracks of @Siess2000. Below, median values of these properties for member stars will be used to characterize each cluster.
Comparison with other astronomical studies
------------------------------------------
In the Appendix §\[sec\_individual\_subclusters\], we discuss each cluster found above with respect to molecular cloud maps and previous studies of young stellar clustering. Most previous studies use nonparametric techniques based either on the Minimal Spanning Tree (MST) or $k$-nearest neighbor analysis. In Appendix §\[sec\_sfincs\_vs\_g09\] we compare in detail our parametric mixture model method and the nonparametric MST-based procedure of @Gutermuth2009. The latter exhibits significant deficiencies called ’chaining’ and ’fragmentation’. This has been well-established in the statistical literature (§\[sec.stat.bkgd\]), appears in some simple simulations we perform (§\[sec\_mst\_simulations\]), and in detailed applications to our SFiNCs star distributions (§\[sec\_mma\_mst\_sfincs\]).
The main results of these comparisons can be summarized as follows:
1. For most of the richer SFiNCs clusters, the data-minus-model residual map values are small (typically $<10$%) indicating that the isothermal ellipsoid models provide good fits. Our second validation technique, the quadrat counting test presented in §\[model\_validation\], shows similar results.
2. The majority of the SFiNCs clusters are associated with clumpy and/or filamentary dusty structures seen in the far-IR images of the SFiNCs SFRs.
3. In many cases, cluster identification with nonparametric Minimal Spanning Tree (MST), either by us or by @Gutermuth2009, “chain” multiple SFiNCs clusters into unified structures. Examples include: Be 59 (A+B), NGC 1333 (A+B), NGC 2068-2071 (C+D), Mon R2 (A+B+C), RCW 120 (A+B), Serpens Main (A+B), IRAS 20050+2720 (C+D), and Sh 2-106 (D+B). Independent information on associated molecular cloud structures, when available (NGC 1333, RCW 120, Serpens Main, Sh 2-106), suggests that the MST chaining is physically unreasonable.
4. In other cases, the MST procedure fragments the SFiNCs clusters into multiple structures and/or fractionates the outer regions of larger clusters. Examples include: Be 59 (A) into 4 fragments, IC 348 (B) into 3 fragments, LkH$\alpha$ 101 (A) into 3 fragments, RCW 120 (D) into 2 fragments, Cep OB3b (A) into several fragments, and Cep C (A) into 2 fragments. Most of the MST stellar structures resulted from such fragmentations have no associations with any cloud structures.
{width="180mm"}
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[@c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}]{} &&&&&&&&&\
Cluster & $N_{4,data}$ & $N_{JH}$ & $J-H$ & $N_{ME}$ & $ME$ & $N_{\alpha_{IRAC}}$ & $\alpha_{IRAC}$ & $N_{Age_{JX}}$ & $Age_{JX}$\
& (stars) & (stars) & (mag) & (stars) & (keV) & (stars) & & (stars) & (Myr)\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10)\
&&&&&&&&&\
Be 59 A & 321 & 301 & $1.17\pm0.01$ & 248 & $1.76\pm0.02$ & 319 & $-2.31\pm 0.11$ & 92 & $1.80\pm0.23$ \
Be 59 B & 220 & 189 & $1.10\pm0.02$ & 180 & $1.74\pm0.03$ & 198 & $-2.30\pm 0.11$ & 39 & $2.20\pm0.37$\
Be 59 UnCl & 1 & ... & ... & ... & ... & ... & ... & ... & ...\
SFO 2 A & 66 & 51 & $1.11\pm0.07$ & 37 & $1.61\pm0.18$ & 66 & $-1.11\pm 0.15$ & 5 & $1.80\pm1.81$\
SFO 2 UnCl & 5 & 4 & $0.85\pm0.09$ & 5 & $1.54\pm0.09$ & 5 & $-2.76\pm 0.08$ & 3 & $1.90\pm1.63$\
NGC 1333 A & 60 & 39 & $1.03\pm0.16$ & 36 & $1.50\pm0.17$ & 60 & $-0.83\pm 0.27$ & 13 & $2.50\pm1.18$\
NGC 1333 B & 101 & 67 & $1.22\pm0.07$ & 71 & $1.63\pm0.08$ & 97 & $-1.10\pm 0.14$ & 23 & $1.70\pm0.34$\
NGC 1333 UnCl & 15 & 11 & $0.97\pm0.16$ & 9 & $1.34\pm0.24$ & 15 & $-1.65\pm 0.44$ & 4 & $2.55\pm0.80$\
IC 348 B & 280 & 259 & $0.92\pm0.01$ & 245 & $1.39\pm0.03$ & 279 & $-2.34\pm 0.06$ & 109 & $2.50\pm0.14$\
IC 348 A & 28 & 13 & $1.56\pm0.33$ & 14 & $2.06\pm0.46$ & 27 & $-0.41\pm 0.23$ & 6 & $2.30\pm0.78$\
IC 348 UnCl & 45 & 44 & $0.86\pm0.05$ & 35 & $1.34\pm0.07$ & 45 & $-2.47\pm 0.08$ & 15 & $3.80\pm0.37$\
LkH$\alpha$ 101 A & 182 & 156 & $1.29\pm0.03$ & 152 & $1.66\pm0.05$ & 167 & $-1.50\pm 0.19$ & 54 & $1.45\pm0.36$\
LkH$\alpha$ 101 UnCl & 63 & 53 & $1.15\pm0.06$ & 43 & $1.60\pm0.08$ & 56 & $-1.74\pm 0.17$ & 25 & $2.20\pm0.61$\
NGC 2068-2071 A & 23 & 15 & $1.75\pm0.18$ & 14 & $1.84\pm0.54$ & 23 & $-0.53\pm 0.63$ & 4 & $0.45\pm0.21$\
NGC 2068-2071 B & 116 & 101 & $1.28\pm0.06$ & 79 & $1.61\pm0.06$ & 116 & $-1.22\pm 0.13$ & 41 & $1.20\pm0.18$\
NGC 2068-2071 C & 33 & 18 & $1.98\pm0.16$ & 19 & $2.05\pm0.12$ & 30 & $-0.69\pm 0.40$ & 7 & $0.60\pm0.18$\
NGC 2068-2071 D & 44 & 34 & $1.34\pm0.13$ & 28 & $1.68\pm0.09$ & 44 & $-1.22\pm 0.23$ & 10 & $0.95\pm0.29$\
NGC 2068-2071 UnCl & 117 & 90 & $1.07\pm0.06$ & 85 & $1.63\pm0.06$ & 114 & $-1.80\pm 0.17$ & 37 & $2.30\pm0.44$\
ONC Flank S A & 325 & 286 & $0.77\pm0.02$ & 222 & $1.15\pm0.02$ & 316 & $-1.62\pm 0.08$ & 109 & $1.60\pm0.18$\
ONC Flank N A & 260 & 242 & $0.79\pm0.02$ & 195 & $1.15\pm0.03$ & 250 & $-1.94\pm 0.09$ & 79 & $1.70\pm0.22$\
OMC 2-3 A & 54 & 44 & $1.06\pm0.13$ & 35 & $1.54\pm0.18$ & 52 & $-0.84\pm 0.18$ & 18 & $0.90\pm0.25$\
OMC 2-3 B & 25 & 9 & $2.15\pm0.41$ & 18 & $2.23\pm0.43$ & 23 & $-0.34\pm 0.43$ & 3 & $2.10\pm1.45$\
OMC 2-3 C & 50 & 30 & $1.03\pm0.16$ & 35 & $1.38\pm0.16$ & 50 & $-0.50\pm 0.18$ & 13 & $1.20\pm0.43$\
OMC 2-3 D & 10 & 9 & $1.02\pm0.12$ & 9 & $1.34\pm0.21$ & 9 & $-2.03\pm 0.76$ & 6 & $2.05\pm0.73$\
OMC 2-3 UnCl & 234 & 203 & $0.78\pm0.02$ & 187 & $1.12\pm0.04$ & 230 & $-1.86\pm 0.09$ & 89 & $1.70\pm0.19$\
Mon R2 A & 134 & 85 & $1.34\pm0.07$ & 107 & $1.95\pm0.07$ & 119 & $-0.56\pm 0.10$ & 26 & $1.20\pm0.13$\
Mon R2 B & 127 & 51 & $1.60\pm0.08$ & 101 & $2.71\pm0.22$ & 87 & $ 0.18\pm 0.15$ & 11 & $1.50\pm0.41$\
Mon R2 C & 32 & 19 & $1.46\pm0.20$ & 16 & $1.46\pm0.21$ & 32 & $-0.28\pm 0.27$ & 5 & $0.80\pm0.20$\
Mon R2 UnCl & 247 & 214 & $1.13\pm0.02$ & 175 & $1.54\pm0.04$ & 247 & $-1.40\pm 0.10$ & 80 & $1.70\pm0.18$\
GGD 12-15 A & 108 & 56 & $1.54\pm0.13$ & 59 & $1.95\pm0.21$ & 105 & $-0.55\pm 0.10$ & 12 & $0.65\pm0.69$\
GGD 12-15 UnCl & 104 & 90 & $0.86\pm0.03$ & 79 & $1.41\pm0.03$ & 104 & $-2.36\pm 0.20$ & 33 & $2.50\pm0.51$\
RCW 120 A & 30 & 5 & $1.72\pm0.35$ & 22 & $3.11\pm0.30$ & 24 & $ 0.34\pm 0.30$ & ... & ...\
RCW 120 B & 110 & 77 & $1.30\pm0.06$ & 98 & $1.82\pm0.05$ & 103 & $-1.46\pm 0.19$ & 19 & $0.80\pm0.20$\
RCW 120 C & 56 & 41 & $1.38\pm0.07$ & 40 & $1.93\pm0.10$ & 56 & $-1.10\pm 0.15$ & 7 & $0.70\pm0.42$\
RCW 120 D & 17 & 5 & $1.68\pm0.11$ & 6 & $2.71\pm0.21$ & 17 & $-0.73\pm 0.24$ & ... & ...\
RCW 120 UnCl & 157 & 71 & $1.29\pm0.10$ & 86 & $1.98\pm0.27$ & 156 & $-1.58\pm 0.15$ & 8 & $1.25\pm0.25$\
Serpens Main A & 14 & ... & ... & 7 & $3.91\pm0.44$ & 13 & $ 1.30\pm 0.66$ & ... & ...\
Serpens Main B & 61 & 27 & $1.88\pm0.30$ & 39 & $2.04\pm0.28$ & 59 & $-0.44\pm 0.16$ & 9 & $0.60\pm0.76$\
Serpens Main UnCl & 65 & 45 & $1.25\pm0.07$ & 45 & $1.66\pm0.10$ & 65 & $-1.70\pm 0.16$ & 16 & $2.25\pm0.51$\
Serpens South A & 12 & ... & ... & ... & ... & 12 & $ 1.28\pm 0.56$ & ... & ...\
Serpens South B & 6 & ... & ... & ... & ... & 6 & $ 0.04\pm 0.39$ & ... & ...\
Serpens South C & 74 & 4 & $1.29\pm0.51$ & 33 & $3.70\pm0.19$ & 70 & $ 0.94\pm 0.19$ & ... & ...\
Serpens South D & 7 & ... & ... & ... & ... & 7 & $ 0.97\pm 0.60$ & ... & ...\
Serpens South UnCl & 199 & 31 & $1.80\pm0.21$ & 44 & $2.46\pm0.24$ & 198 & $-0.92\pm 0.14$ & 13 & $1.80\pm0.84$\
IRAS 20050+2720 A & 0 & ... & ... & ... & ... & ... & ... & ... & ...\
IRAS 20050+2720 B & 14 & 5 & $1.26\pm0.14$ & 7 & $1.79\pm0.08$ & 14 & $-0.73\pm 0.31$ & 4 & $2.70\pm0.48$\
IRAS 20050+2720 C & 29 & 8 & $1.49\pm0.06$ & 13 & $2.27\pm0.18$ & 29 & $-0.31\pm 0.35$ & 3 & $1.60\pm0.57$\
IRAS 20050+2720 D & 111 & 26 & $1.49\pm0.08$ & 64 & $2.51\pm0.11$ & 106 & $-0.18\pm 0.14$ & 9 & $1.90\pm0.69$\
IRAS 20050+2720 E & 31 & 25 & $1.26\pm0.04$ & 20 & $1.79\pm0.12$ & 31 & $-1.53\pm 0.14$ & 6 & $4.00\pm0.18$\
IRAS 20050+2720 UnCl & 130 & 82 & $1.21\pm0.04$ & 85 & $1.77\pm0.09$ & 127 & $-1.52\pm 0.10$ & 25 & $3.30\pm0.41$\
Sh 2-106 A & 25 & 19 & $1.13\pm0.15$ & 15 & $1.66\pm0.20$ & 24 & $-1.51\pm 0.36$ & ... & ...\
Sh 2-106 B & 24 & 13 & $1.72\pm0.14$ & 11 & $2.21\pm0.40$ & 24 & $-0.40\pm 0.22$ & ... & ...\
Sh 2-106 C & 5 & 4 & $1.60\pm0.29$ & 3 & $2.15\pm0.06$ & 5 & $-1.68\pm 0.78$ & ... & ...\
Sh 2-106 D & 53 & 16 & $1.64\pm0.11$ & 51 & $2.53\pm0.20$ & 23 & $-0.44\pm 0.57$ & ... & ...\
Sh 2-106 UnCl & 144 & 98 & $1.34\pm0.06$ & 76 & $1.82\pm0.12$ & 143 & $-1.41\pm 0.14$ & 4 & $0.80\pm0.36$\
IC 5146 A & 10 & 9 & $1.01\pm0.08$ & 9 & $1.13\pm0.13$ & 10 & $-1.05\pm 0.22$ & ... & ...\
IC 5146 B & 142 & 129 & $0.99\pm0.03$ & 85 & $1.58\pm0.05$ & 140 & $-1.47\pm 0.07$ & 32 & $1.55\pm0.18$\
IC 5146 UnCl & 93 & 87 & $0.92\pm0.04$ & 62 & $1.50\pm0.06$ & 93 & $-1.83\pm 0.19$ & 23 & $2.60\pm0.49$\
NGC 7160 A & 141 & 135 & $0.67\pm0.02$ & 134 & $1.25\pm0.02$ & 140 & $-2.68\pm 0.01$ & 28 & $4.05\pm0.44$\
LDN 1251B A & 14 & 8 & $1.01\pm0.29$ & 8 & $1.31\pm0.99$ & 12 & $-0.80\pm 0.85$ & ... & ...\
LDN 1251B UnCl & 34 & 29 & $0.96\pm0.08$ & 30 & $1.43\pm0.10$ & 34 & $-2.52\pm 0.22$ & 13 & $2.70\pm0.83$\
Cep OB3b A & 508 & 420 & $1.02\pm0.01$ & 344 & $1.55\pm0.02$ & 502 & $-1.66\pm 0.06$ & 161 & $2.20\pm0.21$\
Cep OB3b B & 42 & 37 & $1.08\pm0.03$ & 25 & $1.67\pm0.06$ & 42 & $-1.30\pm 0.07$ & 10 & $1.70\pm0.32$\
Cep OB3b C & 817 & 725 & $0.98\pm0.01$ & 586 & $1.47\pm0.01$ & 810 & $-1.89\pm 0.08$ & 281 & $2.40\pm0.14$\
Cep OB3b D & 0 & ... & ... & ... & ... & ... & ... & ... & ...\
Cep OB3b UnCl & 98 & 84 & $1.02\pm0.02$ & 57 & $1.48\pm0.04$ & 98 & $-1.92\pm 0.18$ & 35 & $3.40\pm0.43$\
Cep A A & 172 & 82 & $1.52\pm0.09$ & 120 & $2.29\pm0.17$ & 159 & $-1.07\pm 0.11$ & 29 & $1.40\pm0.28$\
Cep A UnCl & 98 & 89 & $1.20\pm0.02$ & 70 & $1.73\pm0.07$ & 98 & $-2.34\pm 0.15$ & 48 & $2.00\pm0.26$\
Cep C A & 86 & 43 & $1.61\pm0.13$ & 36 & $2.03\pm0.14$ & 84 & $-0.72\pm 0.13$ & 9 & $0.80\pm0.28$\
Cep C B & 4 & ... & ... & 4 & $2.88\pm0.65$ & 4 & $-0.97\pm 0.61$ & ... & ...\
Cep C UnCl & 82 & 66 & $1.14\pm0.03$ & 53 & $1.69\pm0.06$ & 76 & $-1.64\pm 0.33$ & 27 & $2.20\pm0.87$\
&&&&&&&&&\
[@c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}c@[ ]{}]{} &&&&&&&&\
Cluster & $N_{4,model}$ & $\log(R_c)$ & $\epsilon$ & $J-H$ & $ME$ & $\log(Age_{JX})$ & $\alpha_{IRAC}$ & Cloud \
& (stars) & (pc) & & (mag) & (keV) & (yr) & &\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9)\
&&&&&&&&\
Orion B & 73 & -1.31 & 0.30 & 0.87 & 1.6 & 6.04 & ... & R\
Orion C & 834 & -0.66 & 0.49 & 1.05 & 1.6 & 6.18 & -1.08 & R\
Orion D & 48 & -1.04 & 0.84 & 1.17 & 1.4 & 6.43 & -0.93 & C\
Flame A & 219 & -0.91 & 0.37 & 1.79 & 2.8 & 5.90 & -0.61 & C\
W40 A & 187 & -0.79 & 0.04 & 2.10 & 2.5 & 5.90 & -0.83 & R\
RCW36 A & 196 & -0.84 & 0.33 & 1.63 & 2.3 & 5.95 & -0.69 & C\
&&&&&&&&\
5. In several cases, very small and sparse clusters are identified by the SFiNCs mixture modeling procedure with $N_{4,data} < 10$. Examples include: Serpens South (B and D), IRAS 20050+2720 (A), Sh 2-106 (C), Cep OB3b (D), and Cep C (B). Based on the presence of cloud counterparts and/or independent identification with MST method, these seem to be real stellar groupings; MST analysis often adds nearby YSOs to these clusters. However, the SFiNCs model parameters (core radius, ellipticity, etc.) of these structures are undoubtedly unreliable.
Properties of SFiNCs and MYStIX Clusters {#ma_section}
========================================
Combined SFiNCs and MYStIX Sample {#sample_subsection}
---------------------------------
In Table \[tbl\_cluster\_other\_props\], we provide a homogeneous set of median properties for 71 SFiNCs stellar structures: 52 SFiNCs clusters and 19 unclustered stellar structures. The median values of the $J-H$, $ME$, $\alpha_{IRAC}$, and $Age_{JX}$ quantities and their bootstrap uncertainties are computed by averaging over individual stellar members of these groups listed in Table \[tbl\_cluster\_membership\]. The table also provides $N_{4,data}$, the estimated number of stars assigned to each cluster. In a future SFiNCs paper, the full intrinsic stellar populations will be estimated by correcting for incompleteness at low masses following the procedure of [@Kuhn2015a].
Some median values are omitted due to small sample limitations. For highly absorbed clusters, the measurements reported in Table \[tbl\_cluster\_other\_props\] might not be representative of their true, intrinsic property values due to the lack of the measurements for most of their extremely absorbed stellar members. For the multivariate cluster analysis given below the median values of various properties are omitted when less than 10 members are available. We also omit the ellipticity and core radius values for weak SFiNCs clusters where $N_{4,data} < 30$ stars (equivalent to $N_{4,model} < 20$ stars)[^2].
In Table \[tbl\_mystix\_clusters\] we provide a similar list of properties for 91 MYStIX clusters. Data are obtained from @Kuhn2014 [@Kuhn2015b] except for $\alpha_{IRAC}$ that we calculate here. As with the SFiNCs sample, the MYStIX dataset is culled of weak ($N_{4,model} < 20$ stars) clusters with poorly constrained cluster parameters.
Relationships Between Cluster Properties {#mv_presentation_subsection}
----------------------------------------
Figures \[fig\_pairs\_plot\_sfincs\] and \[fig\_pairs\_plot\_mystixsfincs\] summarize univariate distributions and bivariate relationships for the SFiNCs and SFiNCs$+$MYStIX samples, respectively. Similar plots for the MYStIX sample alone, but totaling 141 clusters including weak clusters ($N_{4,model} < 20$ stars), appear in @Kuhn2015b. These plots were created using functions [*pairs.panels*]{} and [*corr.test*]{} in CRAN package $psych$ [@Revelle17]. To guide the eye for identification of possible trends, a least squares linear fit (controlled by [*pairs.panels*]{}) is added to each of the bivariate scatter plots (red line) obtained using $R$’s [*lm*]{} function.
Figure \[fig\_logn\_logrc\] highlights an additional relationship between the total apparent number of SPCMs in SFiNCs clusters ($N_{4,data}$) and cluster size ($R_c$). The correlation has a significance level of $p=0.002$ using Kendall’s $\tau$ statistic. The fitted line, computed with a standard major axis procedure that treats the variables symmetrically [@Legendre1998], has slope $0.85$. However, we recall that the $N_{4,data}$ measurement is subject to strong observational selection biases involving the distance, absorption and instrumental exposure sensitivities. In a future study, the $N_{4,data}$ quantity will be corrected for the incompleteness at low masses using the well established X-ray luminosity function and initial mass function analyses following @Kuhn2015a.
![Number of observed (not corrected for the observation incompleteness at low masses) SPCMs in a SFiNCs cluster as a function of cluster size. A symmetrical linear regression fit is plotted in green. \[fig\_logn\_logrc\]](f9.pdf){width="80mm"}
For each of the pairs, the statistical significance of their correlations were evaluated by testing the null hypothesis that the Kendall’s $\tau$ coefficient is equal to zero. Here we adopt $p-$values of $0.003<p\leq0.05$ and $p\leq0.003$ as indicators of marginally and strong statistical significant correlations, respectively. The [*pairs.panels*]{} function offers a control against Type I errors (the null hypothesis is true, but is rejected) when multiple comparisons are under consideration. As we computed each correlation for three cases (SFiNCs alone, MYStIX alone (not shown), and SFiNCs+MYStIX) we adjust $p$-values for 3 tests using the False Discovery Rate procedure of @Benjamini1995.
The important relationships between cluster properties emerging from these plots are:
1. [*Cluster core radius:*]{} Core radii are clearly linked to four interrelated properties: anti-correlation with X-ray median energy $ME$ and $J-H$ color, two measures of cloud absorption; anti-correlation with [*SED slope*]{}, a measure of inner protoplanetary disk; and correlation with $Age_{JX}$, a measure of stellar age. Together, these empirical relationships provide strong evidence that clusters expand (i.e. $R_c$ increases) as they age. This result was first reported by @Kuhn2015b for the MYStIX sample alone and is discussed further below (§\[expansion\_subsection\]). For the merged MYStIX$+$SFiNCs sample, a linear regression fit that treats variables symmetrically (obtained using $R$’s $lmodel2$ function) gives the relationship $$\log R_c = -12.4(\pm 2.0) + 1.9(\pm 0.3) \times \log Age_{JX}~{\rm pc}$$ over the approximate ranges $0.08 < R_c < 1$ pc and $1 < Age_{JX} < 3.7$ Myr.
There is also a hint that the SFiNCs clusters have systematically smaller sizes (median core radius = 0.18 pc) compared to those of MYStIX (0.22 pc). An Anderson-Darling two-sample test on the core radii distributions indicates this is only a possibility ($p_{AD} = 0.04$).
2. [*Cluster ellipticity:*]{} The ellipticities of the MYStIX+SFiNCs clusters show no correlations with any of the cluster properties. However, there is a significant difference between the two samples with MYStIX clusters being rounder than SFiNCs clusters ($p_{AD} = 0.001$). The median ellipticities are 0.48 and 0.30 for the SFiNCs and MYStIX clusters, respectively.
3. [*Extinction:*]{} Since both the NIR $J-H$ and X-ray $ME$ quantities serve as surrogates for extinction, there is a strong and tight correlation between these two variables. For the merged MYStIX$+$SFiNCs sample, $J-H$ and $ME$ strongly correlate with core radius, age, and SED slope.
4. [*Age:*]{} As mentioned in item 1 above, the ages of SFiNCs$+$MYStIX clusters are significantly correlated with cluster radius and inversely correlated with extinction indicators (negative), and SED slope.
The SFiNCs and MYStIX clusters have statistically indistinguishable age ECDFs with median of 1.7 Myr. However the unclustered SFiNCs stars are systematically older than the clustered stars with median 2.3 Myr ($p_{AD} = 0.001$). This echoes a similar result for clustered vs. unclustered populations in the MYStIX sample reported by @Getman2014a (see §\[db\_populations\_subsection\] for further discussion).
5. [*SED Slope:*]{} This slope is a well-established surrogate for disk-bearing stars [@Richert2018]. As mentioned in item 1, in the SFiNCs$+$MYStIX cluster sample, it is strongly correlated with core radius, extinction[^3], and age.
The SED slopes of the SFiNCs unclustered structures are systematically lower than those of the SFiNCs clusters ($p_{AD} < 0.001$). This indicates that spatial gradients of apparent disk fraction are generally present, where the disk fraction is decreasing from the cluster centers towards the peripheries of the SFiNCs SFRs (see §\[db\_populations\_subsection\] below).
Association With Molecular Clouds {#cloud_subsection}
=================================
Figure \[fig\_cluster\_assignment\_maps\] shows all SPCMs within the [*Chandra*]{} ACIS-I fields superimposed on the [*Herschel*]{} and [*AKARI*]{} far-IR images that trace the SFiNCs molecular clouds. For MYStIX, similar maps appear in Figure 7 of @Getman2014a. SPCM stars are color-coded according to their cluster assignments. We visually identify clusters that closely associated with clouds, either embedded within a cloud or revealed and emerging from a cloud. The results are given in the last column of Tables \[tbl\_cluster\_morphology\] and \[tbl\_mystix\_clusters\].
Nearly all of the SFiNCs clusters are associated with molecular clouds; of the 52 clusters, 35 appear embedded and 12 appear revealed. These include all of the clusters in the SFiNCs SFRs close to the Sun such as NGC 1333, IC 348, NGC 2068-2071, OMC 2-3, GGD 12-15, Serpens Main, Serpens South, and Cep C. In most cases, the clusters are positioned and elongated along $\ga 1$ pc-long molecular filaments that are clearly distinguished by eye on the far-IR images. Some clusters are associated with large ($\ga 1$ pc) molecular clump and/or hub-filament systems such as Cep A, Sh 2-106, Mon R2, and IRAS 20050+2720. Other star-cloud configurations are present: for instance, the three minor clusters in RCW 120 lie projected along a molecular shell, and the main cluster in SFO 2 is embedded at the tip of a bright-rimmed cloud. All of the aforementioned clusters are marked by the flag “C” in Table \[tbl\_cluster\_morphology\].
![Cluster properties stratified by the presence or absence of molecular cloud material. Empirical cumulative distribution functions of cluster core radius, ellipticity, $J-H$ and $ME$ (as surrogates for absorption), age, and SED slope. SFiNCs clusters are in green and MYStIX clusters are in black. Clusters closely associated with molecular clouds (i.e., fully or partially embedded in molecular clouds) are shown with solid lines. Clusters emerged (revealed) from clouds are shown with dash-dotted lines. \[fig\_ecdfs\_rev\_vs\_cloud\]](f10.pdf){width="90mm"}
About a quarter of the SFiNCs clusters are revealed, near but not embedded in clouds. For instance, the three main clusters in IC 348, NGC 2068-2071, and Be 59 lie in regions of dispersed molecular material. The main clusters in ONC Flank N, RCW 120, and IC 5146 lie mainly inside ionized molecular bubbles. The two main clusters in Cep OB3b lie outside a giant molecular cloud. And no molecular material left in the vicinity of the oldest cluster, NGC 7160. These clusters are marked as “R” in Table \[tbl\_cluster\_morphology\].
Figure \[fig\_ecdfs\_rev\_vs\_cloud\] shows various physical properties derived from the best-fit mixture models for SFiNCs clusters stratified by their location with respect to clouds. Quantitative comparisons between these univariate distributions for SFiNCs and MYStIX clusters, stratified by cloud proximity, appear in Table \[tbl\_rev\_cloud\_univariate\].
The following results emerge from the cluster-cloud associations:
1. A clear difference in cloud location is seen between SFiNCs and MYStIX clusters: about two-thirds of SFiNCs sample are closely associated with clouds (probably embedded) while 60% of the MYStIX clusters are revealed.
2. For the merged SFiNCs+MYStIX sample, the revealed clusters are substantially larger (median core radius $R_c = 0.25 \pm 0.03$ pc) than the cloud-associated clusters ($0.11 \pm 0.02$ pc).
3. The cloud-associated (embedded) clusters are significantly more elongated (median $\epsilon = 0.51 \pm 0.03$) than the revealed clusters ($0.27 \pm 0.03$). Ellipticities of SFiNCs cloud-associated clusters appear systematically higher than those from the MYStIX sample ($p_{AD} = 0.009$).
4. The cloud-associated clusters are more absorbed (median $A_V$ of $8-10$ mag[^4]) than the revealed clusters ($2-4$ mag).
5. The cloud-associated clusters are systematically younger (median $Age = 1.4 \pm 0.1$ Myr) than the revealed clusters ($1.8 \pm 0.2$ Myr).
6. The cloud-associated clusters have systematically much higher SED slope (median $\alpha_{IRAC} = -0.6 \pm 0.1$) than the revealed clusters ($-1.7 \pm 0.1$).
[@c@c@c@c@c@c@c@c@c@[ ]{}]{} &&&&&&&&\
&\
&&&&&&&&\
&&&&&&&&\
Property & Sample & N & Median & Sc-R & Mc-C & Mc-R & McSc-C & McSc-R \
& & (clusters) & (Value) & & & & &\
(1)&(2)&(3) &(4)&(5)&(6)&(7)&(8)&(9)\
&&&&&&&&\
$R_c$ \[pc\] & Sc-C & 16 & $0.10\pm0.02$ & $0.000$ & $0.127$ & $0.000$ & ... & ...\
& Sc-R & 11 & $0.29\pm0.15$ & ... & $0.001$ & $0.166$ & ... & ...\
& Mc-C & 24 & $0.14\pm0.02$ & ... & ... & $0.001$ & ... & ...\
& Mc-R & 43 & $0.25\pm0.03$ & ... & ... & ... & ... & ...\
& McSc-C & 40 & $0.11\pm0.02$ & ... & ... & ... & ... & $0.000$\
& McSc-R & 54 & $0.25\pm0.03$ & ... & ... & ... & $0.000$ & ...\
Ellipticity & Sc-C & 16 & $0.56\pm0.05$ & $0.000$ & $0.009$ & $0.000$ & ... & ...\
& Sc-R & 11 & $0.32\pm0.06$ & ... & $0.233$ & $0.382$ & ... & ...\
& Mc-C & 24 & $0.41\pm0.07$ & ... & ... & $0.039$ & ... & ...\
& Mc-R & 43 & $0.26\pm0.04$ & ... & ... & ... & ... & ...\
& McSc-C & 40 & $0.51\pm0.03$ & ... & ... & ... & ... & $0.000$\
& McSc-R & 54 & $0.27\pm0.03$ & ... & ... & ... & $0.000$ & ...\
$J-H$ \[mag\] & Sc-C & 20 & $1.50\pm0.07$ & $0.000$ & $0.015$ & $0.000$ & ... & ...\
& Sc-R & 12 & $1.06\pm0.07$ & ... & $0.001$ & $0.156$ & ... & ...\
& Mc-C & 24 & $1.83\pm0.27$ & ... & ... & $0.000$ & ... & ...\
& Mc-R & 43 & $0.96\pm0.14$ & ... & ... & ... & ... & ...\
& McSc-C & 44 & $1.58\pm0.09$ & ... & ... & ... & ... & $0.000$\
& McSc-R & 55 & $0.99\pm0.09$ & ... & ... & ... & $0.000$ & ...\
$ME$ \[keV\] & Sc-C & 24 & $2.03\pm0.10$ & $0.001$ & $0.092$ & $0.001$ & ... & ...\
& Sc-R & 12 & $1.60\pm0.06$ & ... & $0.006$ & $0.203$ & ... & ...\
& Mc-C & 24 & $2.30\pm0.26$ & ...& ... & $0.000$ & ... & ...\
& Mc-R & 43 & $1.60\pm0.12$ & ... & ... & ... & ... & ...\
& McSc-C & 48 & $2.13\pm0.13$ & ... & ... & ... & ... & $0.000$\
& McSc-R & 55 & $1.60\pm0.06$ & ... & ... & ... & $0.000$ & ...\
$Age$ \[Myr\] & Sc-C & 9 & $1.20\pm0.19$ & $0.063$ & $0.916$ & $0.028$ & ... & ...\
& Sc-R & 11 & $1.80\pm0.28$ & ... & $0.035$ & $0.833$ & ... & ...\
& Mc-C & 19 & $1.41\pm0.17$ & ... & ... & $0.012$ & ... & ...\
& Mc-R & 38 & $1.80\pm0.22$ & ... & ... & ... & ... & ...\
& McSc-C & 28 & $1.41\pm0.13$ & ... & ... & ... & ... & $0.001$\
& McSc-R & 49 & $1.80\pm0.21$ & ... & ... & ... & $0.001$ & ...\
SED Slope & Sc-C & 28 & $-0.54\pm0.10$ & $0.000$ & $0.338$ & $0.000$ & ... & ...\
& Sc-R & 12 & $-1.78\pm0.16$ & ... & $0.000$ & $0.502$ & ... & ...\
& Mc-C & 23 & $-0.61\pm0.21$ & ... & ... & $0.000$ & ... & ...\
& Mc-R & 42 & $-1.71\pm0.11$ & ... & ... & ... & ... & ...\
& McSc-C & 51 & $-0.55\pm0.09$ & ... & ... & ... & ... & $0.000$\
& McSc-R & 54 & $-1.71\pm0.10$ & ... & ... & ... & $0.000$ & ...\
&&&&&&&&\
Discussion {#discussion_sec}
==========
Bias in the SFiNCs Cluster Catalog {#bias_subsection}
----------------------------------
It is important to recall a few biases pertinent to the current study of the SFiNCs clusters. Similar issues are discussed for the MYStIX survey in the Appendix of @Feigelson2013.
First, our methodology (§\[model\_section\]) is biased against identifying poor clusters. The AIC model selection statistic requires that a putative cluster significantly improve the model likelihood for the entire region (§\[model\_section\]). Thus a real cluster with only a few members may not be discriminated from the unclustered component, or a nearby cluster, in a field with hundreds of SFiNCs Probable Complex Members. Even fairly rich deeply embedded clusters may be missed because only a small fraction of the population is detected by $Chandra$ due to X-ray absorption by cloud gas. Any clustering finding procedure will encounter similar difficulties when using the SPCM sample.
Second, for highly absorbed SFiNCs clusters, due to the lack of the $J-H$, $ME$, $\alpha_{IRAC}$, and $Age_{JX}$ estimates for the vast majority of their absorbed cluster core members, the values reported in Table \[tbl\_cluster\_other\_props\] might not be representative of the intrinsic properties of these clusters. We seek to mitigate this problem by restricting the analysis in §§\[ma\_section\] and \[cloud\_subsection\] to cluster sub-samples, that have at least 10 cluster members with reliable measurements of $J-H$, $ME$, $Age_{JX}$, and SED slope. With respect to the $R_c$ and $\epsilon$ properties, the sub-samples are restricted to richer clusters ($N_{4,model} \geq 20$ stars). A result of this decision is that for instance the $\log R_{c} - \epsilon$ and $\log Age_{JX} - \alpha_{IRAC}$ relationships for the SFiNCs clusters (Figure \[fig\_pairs\_plot\_sfincs\]) employ unequal numbers of clusters.
Third, the $N_{4,data}$ and $N_{4,model}$ quantities are not reliable measures of the total intrinsic stellar populations due to different distances, $Chandra$ and $Spitzer$ exposure times, and intervening absorption. A future SFiNCs study employing the analyss of X-ray luminosity functions and Initial Mass Functions [@Kuhn2015a] will give intrinsic population estimates.
Cluster Elongations in SFiNCs and MYStIX {#elongation_subsection}
----------------------------------------
Shells, bubbles, and filamentary molecular cloud structures are ubiquitous in the Galaxy [@Churchwell2006; @Andre2014] and are often sites of star formation. Hierarchical fragmentation in molecular cloud filaments is often observed on scales ranging from several parsecs to $\leqslant 0.1$pc. Different physical mechanisms appear to trigger the cloud fragmentation including: gravitational collapse; thermal, turbulent, and magnetic pressures; angular momentum; and dynamical feedback from young stellar outflows, winds and radiation pressure.
Observations suggest that different mechanisms could dominate the cloud fragmentation in different situations [@Takahashi2013; @Contreras2016; @Teixeira2016]. It is also possible that filamentary clouds are made up of collections of velocity-coherent subfilaments [@Hacar2013]. Turbulent energy cascades are proposed to play a major role in the formation of both the subfilaments and integrated filaments [@Smith2016]. Dense, gravitationally bound prestellar cores then form by cloud fragmentation along the densest filaments; core growth through filamentary accretion is also reported [@Andre2014]. Small star clusters can then emerge in these cores through star formation mediated by turbulent core accretion [@McKeeTan2003], competitive accretion [@Bonnell2001; @Wang2010], and/or stellar mergers [@BonnellBate2005]. Molecular gas can then be expelled by feedback effects of young stars including OB ionizing radiation and winds, supernovae, protostellar accretion heating, protostellar jets and outflows [@Dale2015].
We therefore expect embedded SFiNCs clusters to inherit morphological characteristics from these star forming processes. Perhaps most interesting is our finding that the SFiNCs clusters are both (a) more elongated than the MYStIX clusters (§\[mv\_presentation\_subsection\]) and (b) more closely associated with molecular filaments and clumps (§\[cloud\_subsection\]). Furthermore, examination of Figure \[fig\_cluster\_assignment\_maps\] shows that, in most cases, cluster elongations are oriented along the axes of their filamentary molecular clouds.
The higher elongations of the SFiNCs cloud-associated clusters could be due to their more tranquil environments lacking numerous OB-type stars. O-stars, if present, dominate the stellar feedback [@Dale2015]; a single O7 star may be capable of photoionising and dispersing a $10^{4}$ M$_{\odot}$ molecular cloud in $1-2$ Myr [@Walch2012]. Since the SFiNCs environments harbor fewer and less massive OB-type stars than MYStIX regions, their gas removal timescales could be longer, allowing SFiNCs clusters to remain bound to and retain the morphological imprints of their parental molecular gas for a longer period of time.
We warn, however, that elongated cluster shapes can also result from dynamical cluster mergers and be unrelated to the original cloud morphology [@Maschberger2010; @Bate2012]. And the lower ellipticities of the more populous MYStIX clusters may reflect the mergers of numerous smaller SFiNCs-like clusters that would reduce the ellipticity of the merger product. We will examine this hypothesis in a future paper, where the total intrinsic stellar populations could be compared to the cluster ellipticity for the combined SFiNCs+MYStIX cloud-associated samples.
Cluster Sizes in SFiNCs and MYStIX {#sizes_subsection}
----------------------------------
For young stellar clusters in the solar neighborhood with a wide range of masses, from 30 M$_{\odot}$ to $>10^{4}$ M$_{\odot}$, @Kuhn2015b and @Pfalzner2016 report a clear correlation between the cluster mass and cluster radius following $M_c \propto R_c^{1.7}$. A similar slope of $\sim 1.7$ is found in the relationship of mass and radius for the sample of a thousand massive star-forming molecular clumps across the inner Galaxy [@Urquhart2014]. These findings point to the uniform star formation efficiency for stellar clusters with drastically different sizes and masses [@Pfalzner2016]. The mass-radius cluster relation could be a result from cluster growth, either through star formation or subcluster mergers [@Pfalzner2011; @Kuhn2015b] and/or from an initial cloud mass-radius relation [@Pfalzner2016].
Our finding of a clear trend of increasing cluster’s apparent population size ($N_{4,data}$) with increasing cluster radius ($R_c$) in SFiNCs (Figure \[fig\_logn\_logrc\]) is consistent with the trend of the positive correlation $M_c-R_c$ reported by @Kuhn2015a and @Pfalzner2016. The SFiNCs relationship has a shallower slope ($N_{4,data} \propto R_c^{0.85}$); however, at this moment it is unclear if the slope difference is due to an astrophysical effect or simply due to the imperfection of $N_{4,data}$ as being an apparent rather than an intrinsic quantity (§\[bias\_subsection\]).
Since the SFiNCs targets are generally closer and less populous SFRs than MYStIX, the hint of smaller cluster sizes for SFiNCs, compared to those of MYStIX (§\[mv\_presentation\_subsection\]), is in line with the presence of the aforementioned correlations ($N_{4,data}-R_c$ and $M_c-R_c$). This is also tightly linked to our findings that the revealed (older and less absorbed) clusters appear to have much larger core radii (median $R_c \sim 0.25$ pc) than the cloud-associated (younger and more absorbed) clusters (0.11 pc). This can be astrophysically linked to the effect of cluster expansion (§\[expansion\_subsection\]).
It is also important to note that the median value of $R_c \sim 0.11$ pc found in both the SFiNCs and MYStIX cloud-associated cluster samples is very similar to the typical inner width of the molecular filaments ($\sim 0.1$ pc) found in the nearby clouds of the Gould Belt [@Andre2014]. This provides an indirect link between cloud-associated clusters and their natal molecular filaments.
Cluster Expansion {#expansion_subsection}
-----------------
It is well established that upon the removal of the residual molecular gas via the feedback of newly born stars, the gravitational potential of the molecular gas weakens causing young cluster expansion and, in most cases, eventual dispersal [e.g., @Tutukov1978; @Moeckel2010; @Banerjee2017]. In addition to the gas loss, expansion can arise from other causes: two-body relaxation and binary heating [@Moeckel2012; @Parker2014; @Banerjee2017]; hierarchical cluster merging [@Maschberger2010; @Banerjee2015]; and mass loss via winds of massive stars and supernova explosions [@Banerjee2017].
For the MYStIX clusters, the $R_c - Age$, $R_c - \rho_0$ ($\rho_0$ is the volumetric stellar density), and $\rho_0 - Age$ correlations provide direct empirical evidence for cluster expansion [@Kuhn2015b]. For the sample of embedded clusters provided by @Lada2003, @Pfalzner2011 reports a clear radius-density correlation. Both the Kuhn et al. and Pfalzner et al. radius-density relationships are flatter that that expected from a pure isomorphic cluster expansion. The cause is uncertain but may arise from a non-uniform initial cluster state via inside-out star formation [@Pfalzner2011] or a cluster growth process involving hierarchical cluster mergers [@Kuhn2015b]. We note that @Getman2014b and @Getman2018 find a radial age gradient opposite from what is expected in the inside-out scenario of Pfalzner.
Figure \[fig\_pairs\_plot\_mystixsfincs\] shows that the SFiNCs and MYStIX clusters occupy the same locii on the diagrams of $R_c -$absorption, $R_c - Age$, and $R_c - \alpha_{IRAC}$ (§\[mv\_presentation\_subsection\]). The cluster radii are strongly correlated with $ME$, age, and SED slope, as well as a marginally significant correlation with $J-H$. Considering that the absorption and SED slope are excellent surrogates for age, these correlations give a strong empirical evidence of SFiNCs+MYStIX cluster expansion. This echoes the MYStIX-only result of @Kuhn2015b.
For the combined MYStIX+SFiNCs cluster sample, the $\log(R_c)-\log(Age)$ and $\log(R_c)-\alpha_{IRAC}$ regression fits (Figure \[fig\_pairs\_plot\_mystixsfincs\]) indicate that the clusters with a core radius of $R_c \sim 0.08$ pc (0.9 pc) have a typical age of 1 Myr (3.5 Myr). Assuming that the projected half-mass radius for a cluster is roughly $4 \times R_c$ [@Kuhn2015b], the MYStIX+SFiNCs clusters might grow from 0.3 pc to 3.6 pc half-mass radii over 2.5 Myr.
It remains unclear what fraction of stars remains in bound clusters after cluster expansion and ultimately at the end of star formation [@Lada2003; @Kruijssen2012]. A number of physical parameters are proposed to control the fraction of stars that remain bound in an expanding and re-virialized cluster, such as the star formation efficiency (SFE), gas removal time ($\tau _{gr}$), clusters’s initial dynamical state, mass, and size [@Lada1984; @Goodwin2009; @Brinkmann2016]. For instance, the N-body simulations of relatively small ($<100$ M$_{\odot}$) and large ($>10^4$M$_{\odot}$) clusters predict that the bound fraction of the stellar cluster increases with a higher SFE, longer $\tau_{gr}$, smaller initial cluster size, and larger initial cluster mass. A cluster with an initial sub-virial dynamical state might have higher chances to remain bound after gas expulsion.
In the past, these physical parameters have not been well constrained by empirical data. Here, for the cloud-associated (i.e., fully or partially embedded in a cloud) SFiNCs+MYStIX clusters, we can employ our age estimates to provide empirical constraints on the gas removal timescale parameter $\tau_{gr}$. Figures \[fig\_ecdfs\_rev\_vs\_cloud\] and Table \[tbl\_rev\_cloud\_univariate\] show that for the SFiNCs and MYStIX cloud-associated samples, the median ages are of $1.2-1.4$ Myr, meaning that for at least half of the SFiNCs/MYStIX cloud-associated clusters $\tau _{gr} > 1.2$ Myr. For the sub-samples with reliable SED slope measurements, their median SED slope is of $\alpha_{IRAC}(SFiNCs+MYStIX) = -0.55$. Employing the SFiNCs$+$MYtIX regression line[^5] $\alpha_{IRAC} = 24.8(\pm3.7) - 4.2 (\pm 0.6) \times \log(Age_{JX})$, the corresponding age values are of $1.1$ Myr. Therefore, we conclude that for at least about half of the SFiNCs and MYStIX cloud-associated clusters, the gas removal timescale is longer than $1$ Myr.
We thus emerge with two estimates of the cluster expansion timescale: core radii increase by an order of magnitude over $1-3.5$ Myr period, and the molecular gas is removed over $>1$ Myr. This timescale does not naturally emerge from theoretical models of star cluster evolution. Some astrophysical calculations have assumed the gas expulsion and cluster expansion occurs within $\sim 10^5$ yr [e.g., @Lada1984; @Baumgardt2007]. These appear to be excluded by our findings. Note however that our timescales are tied to $Age_{JX}$ estimates of @Getman2014a which in turn are calibrated to the pre-main sequence evolutionary tracks of @Siess2000. If, for example, we were to use the more recent tracks of @Feiden2016 that treat magnetic pressure in the stellar interior, the timescales would be a factor of two or more longer than the estimates here. See our study @Richert2018 for the effects of evolutionary tracks on pre-main sequence timescales.
SFiNCs Distributed Populations {#db_populations_subsection}
------------------------------
In the MYStIX SFRs, @Kuhn2014 and @Getman2014a show that dispersed young stellar populations that surround the compact clusters and molecular clouds are ubiquitous on spatial scales of $5-20$ pc. In the Carina Complex, where a very large [*Chandra*]{} mosaic survey is available, the population size of the dispersed stellar sample is comparable to that of the clustered stellar sample [@Townsley2011; @Feigelson2011]. In the smaller MYStIX fields, typically $10-20$% of the YSOs are in widely distributed populations [@Kuhn2015a]. @Getman2014a find that these distributed populations are nearly always have older ages than the principal MYStIX clusters. These results demonstrate that massive molecular clouds with current star formation have had (continuous or episodic) star formation for many millions of years in the past.
We find here that, for most SFiNCs SFRs (exceptions are Be 59, SFO 2, NGC 1333, ONC Flanking Fields, and NGC 7160) relatively rich populations of distributed young stars are identified (Figure \[fig\_cluster\_assignment\_maps\]). Across all 22 SFiNCs SFRs, the relative fractions of the observed (not intrinsic) clustered and distributed YSOs within the [*Chandra*]{} fields of view are 70% and 26%, respectively. (The remaining 4% are unassigned). In several fields (OMC 2-3, Serpens South, Sh 2-106, and LDN 1251B), the apparent numbers of the distributed YSOs exceed those of the clustered YSOs.
Figures \[fig\_pairs\_plot\_sfincs\] and \[fig\_pairs\_plot\_mystixsfincs\] show that across all SFiNCs SFRs the SFiNCs cluster sample (median age of 1.6 Myr) is significantly younger than the sample of SFiNCs distributed populations (median age of 2.3 Myr) with the difference at the $p<0.001$ significance level. These older dispersed populations are present on the spatial scales of $\gtrsim 2-3$ pc (in IC 348, Serpens Main, Cep A, Cep C, GGD 12-15, and Mon R2) and $\gtrsim 5-7$ pc (in NGC 2068, IC 5146, Cep OB3b, and RCW 120). Similarly, in §\[mv\_presentation\_subsection\], we find evidence for spatial gradients of the IRAC SED slope (a surrogate for age), with the slope decreasing from the cluster centers towards the peripheries of the SFiNCs SFRs.
Two interpretative issues arise. First, the reported star membership assignments in the clustered versus unclustered MYStIX/SFiNCs mixture model components are approximate and can be changed, to some degree, by varying the membership probability threshold and the limiting cluster size (§\[membership\_subsection\]). Second, the stars assigned to the SFiNCs/MYStIX “unclustered” components (and referred here as stars belonging to distributed populations) may have different astrophysical origins. Some may be ejected members of nearby clusters while others may belong to earlier generations of star formation. In cases where the distributed stars are a large fraction of the total YSO population, the interpretation that SFiNCs SFRs have star formation enduring for millions of years seems reasonable.
Conclusions {#summary_sec}
===========
The SFiNCs (Star Formation in Nearby Clouds) project is aimed at providing detailed study of the young stellar populations and star cluster formation in nearby 22 star forming regions. The input lists of young stars, SFiNCs Probable Complex Members, were obtained in our previous study [@Getman2017]. This study complements and extends our earlier MYStIX survey of richer, more distant clusters. Both efforts share consistent data sets, data reduction procedures, and cluster identification methods. The latter are based on maximum likelihood parametric mixture models (§\[model\_section\]) which differs from the nonparametric procedures used in most previous studies. Appendix gives a detailed comparison of the methodological approaches.
We identify 52 SFiNCs clusters and 19 unclustered stellar components in the 22 SFiNCs star forming regions (§\[clusters\_subsection\]). The clusters include both recently formed embedded structures and somewhat older revealed clusters. The unclustered components represent a distributed stellar populations (§\[db\_populations\_subsection\]). Model validation analyses show that the SFiNCs YSOs spatial distributions are generally well-fit with isothermal elliptical models (§\[model\_validation\]); our parametric modeling procedures are thus self-consistent.
Our parametric mixture model results include the number of significant clusters and a homogeneous suite of cluster physical parameters (Tables \[tbl\_cluster\_morphology\] and \[tbl\_cluster\_other\_props\]). These include: cluster celestial location, core radius ($R_c$), ellipticity ($\epsilon$), observed number of YSO members ($N_{4,data}$), association to molecular clouds, interstellar absorption (based on $J-H$ color and X-ray median energy $ME$), age [the $Age_{JX}$ estimate derived by @Getman2014a], and a circumstellar disk measure ($\alpha_{IRAC}$). Similar properties are compiled for the MYStIX clusters from our previous studies. These cluster properties are mostly median values obtained on the largest available samples of YSOs derived in a uniform fashion.
Together, these cluster characteristics can powerfully aid our understanding of clustered star formation. The multivariate analyses of the univariate distributions and bivariate relationships of the merged SFiNCs and MYStIX cluster samples are presented in §§\[ma\_section\] and \[cloud\_subsection\]. We emerge with the following main science results.
1. The SFiNCs sample is dominated ($75$%) by cloud-associated clusters that are fully or partially embedded in molecular clouds. In contrast, the majority ($60$%) of the MYStIX clusters are already emerged from their natal clouds.
2. The cloud-associated clusters are found to be on average younger and more absorbed than the revealed clusters (§\[cloud\_subsection\]). This was previously reported for the MYStIX-only sample by @Getman2014a.
3. The SFiNCs cloud-associated clusters are on average more elongated than the revealed SFiNCs/MYStIX and cloud-associated MYStIX clusters. Their major axes are generally aligned with the host molecular filaments. Therefore their high ellipticity is probably inherited from the morphology of their parental molecular filaments (§\[elongation\_subsection\]).
4. The cloud-associated clusters are considerably smaller than the revealed clusters. In part, this is a consequence of the effect of cluster expansion that is clearly evident from the strong $R_c - Age$, $R_c -$absorption, and $R_c - \alpha_{IRAC}$ correlations for the combined MYStIX+SFiNCs cluster sample (§§\[sizes\_subsection\] and \[expansion\_subsection\]). Core radii increase dramatically from $\sim0.08$ to $\sim0.9$ pc over the age range $1-3.5$ Myr. These confirm and extend previously reported MYStIX-only results by @Kuhn2015b.
5. For at least about half of the SFiNCs and MYStIX cloud-associated clusters, the estimated gas removal timescale responsible for cluster expansion is longer than $\sim 1$ Myr (assuming @Siess2000 timescale; §\[expansion\_subsection\]). This gives an important constraint on early star cluster evolution.
6. For the majority of the SFiNCs SFRs, relatively rich populations of distributed YSOs are identified. These probably represent early generations of star formation (§\[db\_populations\_subsection\]). For the MYStIX SFRs, similar findings on the presence of older distributed stellar populations were previously reported [@Feigelson2013; @Kuhn2014; @Getman2014a; @Kuhn2015a].
A number of studies using the combined SFiNCs and MYStIX cluster results are planned:
- @Richert2018 reexamines the longevity distribution of inner protoplanetary disks using this large sample of young clusters, updating results of @Haisch2001 and others based on smaller, less homogenous cluster samples.
- @Getman2018 use the $Age_{JX}$ chronometer to show that age spreads are common within young stellar clusters with a particular spatial age gradient: stars in cluster cores appear younger (formed later) than in cluster peripheries. This extends the earlier result on only two clusters by @Getman2014b.
- A study is planned that will correct the observed cluster populations ($N_{4,data}$) to intrinsic populations for the full Initial Mass Function, following the procedure of @Kuhn2015a. This will allow a number of astrophysical issues concerning cluster formation to be addressed with more assurance than possible in the present paper.
- Additional possible efforts include dynamical modeling of SFiNCs-like clusters (analogous to the simulation studies of @Bate2009 [@Bate2012]), measurements of gas-to-dust ratios in the molecular clouds following @Getman2017, and others.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the anonymous referee for helpful comments. The MYStIX project is now supported by the [*Chandra*]{} archive grant AR7-18002X. The SFiNCs project is supported at Penn State by NASA grant NNX15AF42G, [*Chandra*]{} GO grant SAO AR5-16001X, [*Chandra*]{} GO grant GO2-13012X, [*Chandra*]{} GO grant GO3-14004X, [*Chandra*]{} GO grant GO4-15013X, and the [*Chandra*]{} ACIS Team contract SV474018 (G. Garmire & L. Townsley, Principal Investigators), issued by the [*Chandra*]{} X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-03060. The Guaranteed Time Observations (GTO) data used here were selected by the ACIS Instrument Principal Investigator, Gordon P. Garmire, of the Huntingdon Institute for X-ray Astronomy, LLC, which is under contract to the Smithsonian Astrophysical Observatory; Contract SV2-82024. This research made use of data products from the [*Chandra*]{} Data Archive and the [*Spitzer Space Telescope*]{}, which is operated by the Jet Propulsion Laboratory (California Institute of Technology) under a contract with NASA. This research used data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has also made use of NASA’s Astrophysics Data System Bibliographic Services and SAOImage DS9 software developed by Smithsonian Astrophysical Observatory.
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Comparison of Clusters Identified with Mixture Model and MST Analysis Methods {#sec_sfincs_vs_g09}
=============================================================================
We compare our isothermal ellipsoid mixture model analysis (MMA) with the cluster identification procedure of @Gutermuth2009 [hereafter G09] based on the Minimum Spanning Tree (MST). Following a brief review of statistical issues (§\[sec.stat.bkgd\]), we apply the two methods to SFiNCs “flattened” YSO samples (§§\[sec\_mst\_to\_sfincs\] and \[sec\_mma\_mst\_sfincs\]) and to a simulated multi-cluster region (§\[sec\_mst\_simulations\]). Detail comparisons of the methods for individual SFiNCs regions are provided in §\[sec\_individual\_subclusters\].
Statistical Background {#sec.stat.bkgd}
----------------------
The use of the pruned MST for cluster identification is a well-known technique of nonparametric clustering for multivariate data. Though operationally based on the MST, it has been reinvented many times in different forms since the 1950s [@Rohlf1982]. It is mathematically identical to the astronomers’ friends-of-friends percolation algorithm [@Turner1976] and the statisticians’ single-linkage agglomerative clustering algorithm. It is important to recognize that the method has several limitations:
1. Like all nonparametric clustering methods, there is no obvious quantity (such as likelihood) to maximize and thus no clear way to choose the number of clusters present in an objective and reproducible manner.
2. There are no theorems to guide the choice of algorithm. There is no criterion to choose between different hierarchical clustering procedures (single linkage, average linkage, complete linkage, and Ward’s distances are most commonly used) or between agglomerative hierarchical and other approaches (such as $k$-means and other partitioning procedures).
3. When clustering methods are compared in simulation, single linkage agglomeration produces dendrograms and clusters that can differ widely from all other common methods [e.g. @Jain2004; @Izenman2008].
4. Single linkage clustering has the particular problem of ‘chaining’ together unrelated clusters in the presence of noise.
The strong limitations of this approach are summarized in the most widely read statistical textbook in the field by @Everitt11 (see also @Aggarwal14):
> “It has to be recognized that hierarchical clustering methods may give very different results on the same data, and empirical studies are rarely conclusive. ... Single linkage, which has satisfactory mathematical properties and is also easy to program and apply to large datasets, tends to be less satisfactory than other methods because of ‘chaining’; this is the phenomenon in which separate clusters with ‘noise’ point in between tend to be joined together."
Application of MST to SFiNCs Fields {#sec_mst_to_sfincs}
-----------------------------------
The MST analysis was performed using the physical (x,y) coordinates of the SFiNCs YSOs that are proportional to physical parsec scales, projected on the sky. Briefly stated, the MST procedure: constructs the unique Minimal Spanning Tree for the stellar two-dimensional spatial distribution; plots the cumulative distribution function of the MST branches; chooses a critical branch length at the intersection of linear regressions to the upper and lower portions of this function; removes all tree branches longer than this critical length; defines cluster members as contiguous linked data points. These steps are visualized for the SFiNCs stellar distributions in the right-hand panels of Figure \[fig\_mma\_vs\_mst\] and its associated Supplementary Materials. The MMA result is summarized in the lower-left panel. For many regions, the SFiNCs MST results based on the X-ray/IR datasets resemble the MST results of G09 that are based on purely IR stellar samples. Detailed comparison of the MMA and MST results for individual SFiNCs regions (depicted in Figure \[fig\_mma\_vs\_mst\]) are further provided in §\[sec\_individual\_subclusters\].
{width="6.5in"}
MMA and G09-MST Comparison for SFiNCs Fields {#sec_mma_mst_sfincs}
--------------------------------------------
G09 used [*Spitzer*]{} data to provide a homogeneous set of disk-bearing YSO populations across 36 nearby star forming regions. Using the MST method, they identify 39 cluster cores and derive some of their basic properties, such as cluster position, size, aspect ratio, stellar density, and extinction. Fourteen G09 regions are in common with SFiNCs. For these clusters, we compare the cluster sizes and stellar densities inferred by our MMA procedure to those derived in G09 in Table \[tbl\_sfincs\_vs\_g09\].
{width="5.6in"}
In five SFiNCs SFRs (NGC 1333, Mon R2, Serpens Main, IRAS 20050+2720, Sh 2-106), our mixture model procedure find multiple clusters within the locations of single G09 clusters. To facilitate the comparison with MST method, the SFiNCs YSO spatial distributions in these SFRs are re-fitted here with mixture models composed of a single cluster component ($k=1$) plus a single unclustered component. These fits are shown in Figure \[fig\_new\_fits\]. Notice that the systematically higher AIC values, compared to the original multi-cluster fits given in Figure \[fig\_cluster\_identification\], indicate poorer fits.
[@@c@c@c@c@c@c@c@c@c@]{} &&&&&&&&\
SFR & &&\
&&&&&&&&\
& Cluster & Size & Number & $\Sigma_{mean}$ && Name & $R_{hull}$ & $\sigma_{mean}$ \
& & (pc) & (stars) & (pc$^{-2}$) && & (pc) & (pc$^{-2}$)\
(1) & (2) & (3) & (4) & (5) && (6) & (7) & (8)\
&&&&&&&&\
SFO 2 & A & 0.32 &66 & 205 && Core & 0.34 & 89 \
NGC 1333 & AB & 0.76 &176 & 97 && Core & 0.37 & 219 \
IC 348 & A & 0.16 &28 & 348 && Core-2 & 0.23 & 148 \
IC 348 & B & 0.80 &280 & 139 && Core-1 & 0.39 & 114 \
LkH$\alpha$ 101 & A & 0.84 &182 & 82 && Core & 0.67 & 40 \
Mon R2 & ABC & 1.12 &385 & 98 && Core & 0.73 & 79 \
GGD 12-15 & A & 0.56 &108 & 110 && Core & 0.50 & 99 \
Serpens Main & AB & 0.36 &66 & 162 && Core & 0.35 & 141 \
IRAS 20050+2720 & CD & 0.44 &155 & 255 && Core-1 & 0.36 & 200 \
Sh 2-106 & BD & 0.60 &63 & 56 && Core & 0.79 & 18 \
IC 5146 & B & 0.68 &142 & 98 && Core & 0.68 & 66 \
Cep A & A & 0.84 &172 & 78 && Core-1 & 0.53 & 60 \
Cep C & A & 0.72 &86 & 53 && Core-1 & 0.40 & 77 \
Cep C & B & 0.08 &4 & 199 && Core-2 & 0.20 & 90 \
&&&&&&&&\
{width="5.5in"}
We summarize two of the cluster properties, cluster size and apparent stellar surface density, found using the SFiNCs and G09 methods in Figure \[fig\_sfincs\_vs\_g09\]. The two methods give correlated values but with wide scatter.
Some of the discrepancies may have identifiable causes. Mon R2 and Cep A are subject to bright mid-IR nebular background emission; the smaller cluster sizes inferred by G09 may arise from an IR catalog bias. In IC 348, G09 split the main cluster (SFiNCs cluster B) into two subclusters, Core 1 and Core 3, which leads to a truncated size of the main cluster. For NGC 1333, the MST fragmentation of the northern region results in a smaller cluster size. In the stellar density plot, the SFiNCs stellar densities are systematically higher than those of G09; this is expected due to the inclusion of X-ray selected disk-free YSOs as well as IR excess YSOs.
Simulation of mixture model and MST procedures {#sec_mst_simulations}
----------------------------------------------
To illustrate the difficulties of MST-derived clusters and the effectiveness of mixture modeling under some circumstances, we conducted a single series of simulations. It is a Gaussian mixture model (GMM) with a central round cluster of 200 stars, an overlapping elongated cluster with $4:1$ axis ratio of 100 stars, a sparser well-separated cluster of 20 stars, and 100 uniformly distributed unclustered stars. These are placed in a dimensionless $10 \times 10$ square window. Figure \[fig\_mst\_simulations\] (upper left panel) shows a typical example of the simulated star distribution.
{width="6.5in"}
A standard maximum likelihood GMM fitting procedure, [*R*]{} function [*Mclust*]{} [@Fraley2002], fits maximum likelihood Gaussian clusters with $1 < k < 10$, choosing the best-fit $k$ with the Bayesian Information Criterion. The methods has no user-supplied parameters. For an ensemble of 100 simulations, the GMM separates the central and elongated clusters in all of the simulations, and identified sparse 20-point cluster in 3/4 of the simulations. The upper right panel of Figure \[fig\_mst\_simulations\] gives a typical example. This failure to uniformly capture sparse clusters is consistent with the more extensive simulations of @Lee2009. In no case is the main cluster subdivided into clusters.
Each simulated star distribution was then subject to the MST procedures described by G09[^6]. The branch length distribution of the MST of the typical example is displayed in the lower left panel of Figure \[fig\_mst\_simulations\]. Two user-supplied parameters are needed. First, one must define how the linear regressions of the MST branch length distribution are calculated. Here we choose the lower 40% of the distribution to fit one line and the upper 10% to fit the other line (red lines). The lines intersect around branch length $0.5-0.6$ (green line) which serves as the threshold for pruning the MST. A third user-supplied parameter is then needed to exclude small fragments of two or more points. We choose a threshold that clusters must have at least 10 members. The clusters can then be identified.
The MST-based procedure was much less successful than the GMM procedure in recovering the simulated clusters. The elongated cluster was discriminated from the main cluster in only 5% of the cases, and the sparse separated cluster was recovered in 25% of the cases. In a third of the simulations, the main cluster was erroneously subdivided into subclusters. Different choices of the user-supplied parameters would not give much better solutions. A longer branch threshold improves the detection of sparse clusters but worsens the ability to discriminate close or overlapping clusters. A smaller cluster membership threshold improves the detection of the sparse cluster but increases the erroneous fractionation of the main cluster. In real astronomical situations, there is insufficient knowledge to tune the user-supplied parameters.
The simulation study here is limited and perhaps tuned to the success of GMMs as the clusters were assumed to have Gaussian shapes. In the SFiNCs analysis (§\[model\_section\]), we use isothermal ellipsoids rather than multivariate Gaussians as the model, and then validate that this model accurately fits the resulting clusters (§\[model\_validation\]). The MST method shows similar problems of chaining and fragmentation in the real SFiNCs datasets (see §\[sec\_individual\_subclusters\] below) as seen in the simulations here. Overall, the simulation here shows that that nonparametric clustering procedures like the pruned MST can be substantially less successful than parametric mixture model procedures in recovering star clusters in patterns that resemble the spatial distribution of SFiNCs membership samples (§\[sec\_individual\_subclusters\]).
Comments on Individual SFiNCs Regions and Clusters {#sec_individual_subclusters}
==================================================
Here we provide information on the association of the individual SFiNCs clusters found with the mixture model analysis (MMA) with known molecular structures, previously published stellar clusters, previous searches for stellar clusters, and results of the MST analysis described in §\[sec\_mst\_to\_sfincs\] and shown in Figure \[fig\_mma\_vs\_mst\].
[*Be 59 (NGC 7822)*]{} is the principal cluster responsible for the ionization of the nearby, $\sim 40$ pc diameter Cepheus Loop bubble [@Kun2008 and references therein]. The western part of Be 59 is bounded by the giant molecular cloud associated with the Cepheus Loop shell (Figure Set \[fig\_cluster\_assignment\_maps\]). Our MMA identifies two clusters (A and B): B is a rich dense central cluster and A is an excess of stars in the halo of B. Such a cluster morphology with a cluster core and an asymmetric halo, observed within the ACIS-I field on a spatial scale of a few parsecs, is reminiscent of a number of MYStIX SFRs [@Kuhn2014]. Low data-model residual values of $<10$% across both clusters provide evidence for a good model fit to the data (Figure Set \[fig\_cluster\_identification\]).
The MST method picks out the main rich cluster B, but treats B as an elongated structure chained to the densest part of the cluster A. MST also subdivides the sparser part of the cluster A into four sparse ($N\geq10$) groups (Figure \[fig\_mma\_vs\_mst\]).
[*SFO 2 (BRC 2, S 171, IRAS 00013+6817)*]{} As part of the aforementioned Cepheus Loop bubble, about 1$\degr$ north of Be 59 lies the bright-rimmed cloud SFO 2, surrounded by the ionized rim NGC 7822 facing Be 59 [@Kun2008 and references therein]. MMA identifies a single cluster A located at the tip of the bright-rimmed cloud. The data-model residuals are below $15$% at the center of the cluster (Figure \[fig\_cluster\_identification\]). There is a small residual spot at the south-eastern part of the field; but this is associated with only a few points and could be a random fluctuation. G09 applied the MST analysis to [*Spitzer*]{} YSO sample of this region (their target S 171) and obtained a similar result of a single cluster.
[*NGC 1333*]{} is an SFR within the Perseus molecular cloud complex noted for its large population of protostars and young stellar outflows [@Luhman2016 and references therein]. MMA identifies two clusters (A and B), each corresponding to different filamentary parts of the molecular cloud (Figure \[fig\_cluster\_assignment\_maps\]). Low residuals of $<3$% and $<10$% at the cores and halos of the clusters, respectively, indicate good model fits to the data (Figure Set \[fig\_cluster\_identification\]).
Both the SFiNCs MST (Figure Set \[fig\_mma\_vs\_mst\]) and the MST analysis of G09 chain A and B into a single cluster. However, the SFiNCs clusters A and B coincide with the double-cluster identified earlier by @Gutermuth2008 who applied a nearest neighbor algorithm to their [*Spitzer*]{} YSO catalog. No kinematic differences between the stellar populations of the two clusters are found in the INfrared Spectra of Young Nebulous Clusters (IN-SYNC) project by @Foster2015.
[*IC 348*]{} is the richest SFR in the nearby Perseus molecular cloud complex [@Luhman2016 and references therein]. MMA identifies two clusters. The small, heavily absorbed cluster A is associated with the dense part of a molecular filament, while the main rich and lightly absorbed cluster B lies projected against the area with dispersed molecular material (Figure \[fig\_cluster\_assignment\_maps\]). Model residuals across the clusters are $<10$% (Figure \[fig\_cluster\_identification\]).
The SFiNCs MST analysis fragments the main cluster B into three components: two sparse northern groups each with $\sim 10$ stars, and a main cluster. The MST analysis by G09 identifies the small embedded SFiNCs cluster A, but fragments cluster B into two pieces: a main rich scluster and a secondary southern sparser cluster with $\ga20$ members.
[*LkH$\alpha$ 101*]{} is a rich SFR associated with the dense molecular filament L1482, part of the giant California molecular cloud [@Andrews2008; @Lada2009]. SFiNCs MMA finds one single rich cluster A that lies projected against a dense molecular structure (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residuals are $<10$% (Figure \[fig\_cluster\_identification\]).
SFiNCs MST fragments the cluster A into three parts: one main cluster and two secondary sparser ($N \ga 10$) groups to the west of the main cluster (Figure \[fig\_mma\_vs\_mst\]). However, the MST analysis of G09 treats the cluster A as a single cluster, consistent with the MMA result.
[*NGC 2068-2071*]{} NGC 2068 and NGC 2071 are SFRs associated with the northern part of the Orion B molecular cloud, also known as L1630N [@Spezzi2015 and references therein]. MMA identifies four clusters: the richest B cluster lies projected against a dispersed cloud structure in NGC 2068; the C and D clusters are associated with clumpy and filamentary molecular structures in NGC 2071; and the A cluster is associated with filamentary molecular structure in the southern part of L1630N, termed by Spezzi et al. as the HH 24-26 area (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residuals are $\la 10$% across the B, C, and D clusters and about 20% around the A cluster (Figure Set \[fig\_cluster\_identification\]).
SFiNCs MST chains clusters C and D together. Consistent with MMA, MST identifies the cluster B as a single cluster, but fragments the elongated cluster A into two sparse groups (Figure Set \[fig\_mma\_vs\_mst\]).
@Spezzi2015 applied a nearest neighbor algorithm to their VISTA$+$[*Spitzer*]{} YSO catalog of L1630N. They define NGC 2071 (SFiNCs clusters C and D) and NGC 2068 (SFiNCs cluster B) as loose stellar clusters, and HH 24-26 (SFiNCs cluster A) as a loose stellar group. @Megeath2016 applied the ‘N10’ $k$-nearest neighbor surface density estimator to the Spitzer YSO catalog of NGC 2068-2071. They identify two main clusters: their northern cluster is a composite of the SFiNCs clusters C and D; their southern cluster coincides with SFiNCs cluster B. Their surface density map shows a density increase at the location of HH 24-26 (SFiNCs cluster A) but it is not labeled as a cluster.
[*OMC 2-3*]{} is associated with a part of the Orion A molecular filament that extends northward from the OMC 1/Orion Nebula region [@Peterson2008]. MMA identifies four clusters. Clusters A, B, and C are positioned and elongated along the main molecular filament. Cluster D represents a group of several YSOs located off the filament. The data-model residuals are $\la 5$% around the clusters and $\sim 20$% at the outskirts of the [*Chandra*]{} field.
SFiNCs MST identifies all four SFiNCs clusters. Compared to MMA, MST chains several additional YSOs to clusters C and D. Based on their N10 surface density estimator of the Orion [*Spitzer*]{} YSO catalog, @Megeath2016 do not report any stellar clusters/groups at the location of OMC 2-3.
[*ONC Flank N (NGC 1977)*]{} Located at the northern end of the Orion A molecular cloud (right to the north of OMC 2-3), this is an HII bubble ionized by a few early B-type stars [@Peterson2008]. Our MMA analysis identifies a rich, un-obscured cluster A that lies projected against the HII region. The data-model residuals are $\la 10$% across the cluster and about 20% at the outskirts of the [*Chandra*]{} field.
The SFiNCs MST analysis is consistent with SFiNCs MMA at identifying this single cluster. Based on their N10 surface density estimator of the Orion Spitzer YSO catalog @Megeath2016 do not report any stellar clusters at the location of NGC 1977.
[*ONC Flank S*]{} Located in the OMC 4 part of the Orion A molecular filament that extends southward from the OMC 1/Orion Nebula region . MMA analysis identifies a single rich, unobscured, and elongated cluster A that lies projected against the OMC 4 filament. The data-model residuals are $\la 5$% across the cluster and $<20$% at the outskirts of the [*Chandra*]{} field.
The SFiNCs MST analysis fragments the SFiNCs cluster A into three structures: a main ($N>100$ YSOs) and two secondary each with several-dozen YSOs. In addition, several small groups of $N \sim 10-20$ YSOs are identified in the outer region by MST. Based on their N10 surface density estimator of the Orion [*Spitzer*]{} YSO catalog @Megeath2016 do not report any stellar clusters/groups at the location of ONC Flank S.
[*Mon R2*]{} is a rich SFR associated with the Mon R2 molecular core that is part of the giant Monoceros R2 molecular cloud complex [@Pokhrel2016 and references therein]. MMA identifies three star clusters. The extremely absorbed cluster B is associated with the central part of the molecular clump, and the less obscured clusters A and C lie projected against the northern and southern edges of the clump, respectively (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residuals are $<$5% across the clusters (Figure \[fig\_cluster\_identification\]).
Both the SFiNCs MST (Figure \[fig\_mma\_vs\_mst\]) and the MST analysis of G09 chain the A, B, and C clusters into a single structure. The YSO catalog of G09 misses numerous X-ray selected YSOs at the center of the field due to the presence of high nebular MIR emission [Figure 6 in @Getman2017].
[*GGD 12-15*]{} is part of the giant Monoceros R2 molecular cloud complex located to the east of the Mon R2 SFR [@Pokhrel2016 and references therein]. MMA finds a single rich cluster A associated with a molecular clump (Figure Set \[fig\_cluster\_assignment\_maps\]). The data-model residuals range from less than 5% at the center to $<25$% at the halo of the cluster (Figure Set \[fig\_cluster\_identification\]).
The SFiNCs MST analysis (Figure \[fig\_mma\_vs\_mst\]) and the MST analysis of G09 are consistent with SFiNCs MMA at identifying this single cluster A.
[*RCW 120*]{} is a nearby, $\sim 4$ pc diameter HII bubble [@Figueira2017 and references therein]. MMA identifies four clusters: the primary ionizing cluster B surrounded by a dusty shell, and the secondary clusters A, C, and D associated with the filamentary and clumpy parts of the dusty shell (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residuals do not exceed 10% at the centers of the clusters (Figure \[fig\_cluster\_identification\]). However, the high ($30-50$%) negative residuals (model $>$ data) in the halos of the embedded clusters A and C indicate poor fits with the isothermal ellipsoid models for these clusters. In addition, a few spots with positive residuals of $50-100$% at the locations of clumpy molecular structures suggest that a few independent, possibly embedded, sparse ($N = 3-7$) stellar groups might have been missed by MMA.
SFiNCs MST chains the clusters A and B into a single cluster (Figure \[fig\_mma\_vs\_mst\]). MST is consistent with MMA at identifying the cluster C. MST breaks the cluster D into two sparse groups, each consisting of several YSOs. MST shows groups of $N=3-7$ YSOs at the aforementioned spots with the positive MMA residuals.
[*Serpens Main*]{} is one of a few young, active SFRs in the nearby Serpens/Aquila Molecular Complex [@OrtizLeon2017 and references therein]. MMA identifies two clusters associated with different molecular filamentary and clumpy structures, B being richer than A (Figure \[fig\_cluster\_assignment\_maps\]). Low data-model residual values of less than a few percent across both clusters show a good model fit (Figure \[fig\_cluster\_identification\]).
SFiNCs MST analysis is consistent with MMA at identifying both clusters (Figure \[fig\_mma\_vs\_mst\]). Unlike the SFiNCs MST and MMA analyses, the MST analysis of G09 chains both clusters into a single cluster. This result disagrees with their own smoothed surface density map that shows two cluster cores.
[*Serpens South*]{} is another active SFR in the Serpens/Aquila Molecular Complex [@OrtizLeon2017 and references therein]. It is one of the youngest among nearby rich SFRs. MMA identifies four clusters associated with different filamentary molecular structures (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residuals are less than $10$% across the field (Figure \[fig\_cluster\_identification\]). Parameters for the sparse clusters B and D are poorly constrained due to the small samples ($N_{4,data} = 6-7$).
SFiNCs MST chains the two main clusters A and C into a single cluster (Figure Set \[fig\_mma\_vs\_mst\]). In agreement with MMA, MST identifies the cluster B as a group of several YSOs. MST chains cluster D to a dozen of additional YSOs; some of those lie projected against an associated molecular filament. At the edges of the field, MST identifies two additional possible weak clusters ($N \ga 10$) with half YSOs lying projected against molecular clumps.
[*IRAS 20050+2720*]{} is an active SFR in the Cygnus rift molecular complex [@Poppenhaeger2015 and references therein]. MMA identifies five clusters. Clusters B, C and D are associated with molecular clumps while A and E lie projected against diffuse molecular structures (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residuals are less than $10$% across the A, C, D, and E clusters (Figure \[fig\_cluster\_identification\]). High positive residuals of $\ga 30$% between the B and C clusters suggest that independent sparse stellar groups might have been missed by MMA. Parameters are poorly constrained for the sparse cluster A.
SFiNCs MST chains the main clusters C and D into a single cluster (Figure \[fig\_mma\_vs\_mst\]). MST identifies the sparse clusters A and B as groups of several YSOs. In agreement with MMA, MST identifies the cluster E as a single rich stellar structure. In addition, MST suggests the presence of a new small stellar group of $N=6$ YSOs projected against a faint molecular clumpy structure located between the A and D clusters.
As with the SFiNCs MST analysis, the MST procedure of G09 chains clusters C and D into a single cluster. This is inconsistent with their smoothed source density map that shows two cluster cores. Their MST analysis does not identify the small cluster B. Their field of view does not include the clusters A and E.
@Gunther2012 apply the MST clustering analysis to their [*Spitzer/Chandra*]{} YSO catalog of IRAS 20050+2720. Their MST analysis finds two cluster structures: “cluster core E” and “cluster core W”. Their “cluster core E” matches the SFiNCs cluster E. Their “cluster core W” chains the three SFiNCs clusters B, C, and D into a single cluster. Günther et al. acknowledge the limitations of MST in discriminating close clusters. They further attempt a manual separation of “cluster core W” into two clusters: “If we cut the cluster core W along the dashed black line (Figure 6), we end up with two subcores.”
[*Sh 2-106*]{} is a rich SFR in the giant Cygnus-X molecular complex [@Adams2015 and references therein]. MMA identifies four clusters. The two main clusters B and D are associated with dense molecular clumps (Figure \[fig\_cluster\_assignment\_maps\]). The sparse cluster C is associated with a diffuse molecular structure, and the sparse cluster A is revealed. The data-model residuals are $<10$% across the main B and D clusters, but higher residuals of $>30$% around A and C suggest poor fits with the isothermal ellipsoid models (Figure \[fig\_cluster\_identification\]). Parameters for the sparse cluster C ($N_{4,data}=5$) are poorly constrained.
SFiNCs MST chains clusters B and D into a single cluster (Figure \[fig\_mma\_vs\_mst\]). MST identifies clusters A and C. For the latter case, MST adds many additional YSOs that are projected on a diffuse molecular structure.
The MIR YSO catalog of G09 misses the core of the SFiNCs cluster D and the eastern portion of the cluster B due to the presence of high nebular MIR emission (Figure 6 in @Getman2017). These are recovered in SFiNCs through X-ray selection. The MST procedure of G09 identifies a single cluster that spans both main SFiNCs clusters B and D. However their clustering result is likely affected by the aforementioned bias in their data.
[*IC 5146*]{} is a SFR in the Cocoon Nebula that is part of the IC 5146 molecular complex [@Johnstone2017 and references therein]. MMA identifies two clusters: the main revealed cluster B surrounded by a dusty shell, and the secondary cluster A associated with molecular clumpy structures (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residuals are less than 2% at the core of the cluster B, but reach $\la 20$% at the two halo spots where the cluster crosses parts of the surrounding dusty shell, suggesting possible contribution from YSOs embedded in the shell (Figure \[fig\_cluster\_identification\]). Higher residuals of $>30$% across the weak cluster A indicate a poor fit with the isothermal ellipsoid model for that cluster.
Consistent with MMA, SFiNCs MST identifies both of these clusters (Figure \[fig\_mma\_vs\_mst\]). Consistent with SFiNCs MST and MMA, the MST analysis of G09 identifies the main cluster B. A group of several YSOs chained together by their MST method is located at the position of the cluster A. Their smoothed surface density map also shows two clusters at the locations of A and B.
[*NGC 7160*]{} is part of the Cepheus OB2 association located inside a 100 pc diameter dusty shell, named the Cepheus Bubble [@Kun2008 and references therein]. The NGC 7160 region is relatively old and free of molecular material. Both MMA and MST identify a single revealed stellar cluster in this field (Figure \[fig\_cluster\_assignment\_maps\], Figure \[fig\_mma\_vs\_mst\]). The high data-model residuals of over 30% across the cluster indicate a poor fit with the isothermal ellipsoid model (Figure \[fig\_cluster\_identification\]).
[*LDN 1251B*]{} is a SFR in the L 1251 cloud located at the eastern edge of the giant Cepheus Flare molecular complex [@Kun2008 and references therein]. MMA identifies a single compact cluster associated with a molecular clump (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residual values are below $15$% across the cluster (Figure \[fig\_cluster\_identification\]). SFiNCs MST identifies the same single cluster, but adds to it another dozen of YSOs located outside the clump (Figure Set \[fig\_mma\_vs\_mst\]).
[*Cep OB3b*]{} is a rich SFR located at the interface between the Cepheus OB3 association and the giant Cepheus molecular cloud [@Kun2008 and references therein]. MMA identifies four clusters: the revealed A, B, and C clusters, and the cluster D embedded in the Cepheus B cloud (Figure Set \[fig\_cluster\_assignment\_maps\]). The data-model residual values vary from less than several percent at the cores to $<20$% at the halos of the rich clusters A, B, and C (Figure \[fig\_cluster\_identification\]). Parameters for the weak cluster D are unreliable.
SFiNCs MST identifies the richer clusters A, B and C (Figure \[fig\_mma\_vs\_mst\]). MST breaks the cluster C into a main structure surrounded by a few sparse groups, each including several YSOs. One of the groups is associated with the molecular clump of the Cepheus F cloud.
@Allen2012 applied the N=11 nearest neighbor algorithm to a [*Spitzer/Chandra*]{} catalog of Cep OB3b. They identify two clusters that match the SFiNCs clusters A and C.
[*Cep A*]{} is an active SFR associated with one of the few dense and massive molecular clumps of the giant Cepheus molecular cloud [@Kun2008 and references therein]. MMA finds the single, rich cluster associated with a molecular clump (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residuals are $<10$% throughout the field (Figure \[fig\_cluster\_identification\]). The SFiNCs MST procedure also identifies the cluster A (Figure \[fig\_mma\_vs\_mst\]).
Due to the presence of high nebular MIR emission at the center of the region, the MIR YSO catalog of G09 misses numerous X-ray selected YSOs that lie projected against the center of the molecular clump (Figure 6 in @Getman2017). Their MST analysis fragments the main cluster into two subclusters; however this clustering result is likely affected by the aforementioned bias in their data.
[*Cep C*]{} is an active SFR associated with the most massive clump of the giant Cepheus molecular cloud [@Kun2008 and references therein]. MMA finds two clusters associated with molecular clumps and filaments (Figure \[fig\_cluster\_assignment\_maps\]). The data-model residual values are $<7$% across the main cluster A (Figure \[fig\_cluster\_identification\]). To the west of cluster A lies a high residual spot associated with a dozen YSOs that was not classified as a significant cluster by MMA. Cluster B has unreliable parameters due to the small sample ($N_{4,data}=4$).
SFiNCs MST breaks the main cluster A into two subclusters (Figure \[fig\_mma\_vs\_mst\]). The western component corresponds to the aforementioned high residual spot. MST also identifies the cluster B, but chains additional YSOs to it. Some are projected against a molecular clump but some lie outside the clump.
Consistent with MMA, the MST analysis of G09 identifies the cores of the two SFiNCs clusters A and B.
\[lastpage\]
[^1]: E-mail: [email protected] (KVG)
[^2]: Recall that $N_{4,model}$ and $N_{4,data}$ differ mainly due to the differences between the “flattened” and entire stellar samples.
[^3]: Since the SFiNCs and MYStIX $\alpha_{IRAC}$ quantities are observed (not corrected for extinction) slopes, they may overestimate intrinsic SED slopes [@Lada2006 their Table 2] by $<3$% ($<15$%) for YSOs that are subject to source extinction of $A_V\leq5$ mag ($A_V \leq 10$ mag).
[^4]: Using the conversion from $J-H$ and $ME$ to the $V$-band extinction $A_V$ based on the SFiNCs data tabulated in @Getman2017.
[^5]: This regression line (obtained using $R$’s [*lmodel2*]{} function) is analogous to the red line shown in Figure \[fig\_pairs\_plot\_mystixsfincs\], but treating variables symmetrically.
[^6]: This analysis was performed using [*R*]{}’s [*spatstat*]{} and [*spatgraphs*]{} packages [@Baddeley2015; @Rajala2015]. Useful functions include [*ppp*]{}, [*psp*]{}, [*edgeLengths*]{} and [*spatgraph*]{}.
|
---
abstract: |
We report results of magneto-acoustic studies in the quantum spin-chain magnet NiCl$_2$-4SC(NH$_2$)$_2$ (DTN) having a field-induced ordered antiferromagnetic (AF) phase. In the vicinity of the quantum critical points (QCPs) the acoustic $c_{33}$ mode manifests a pronounced softening accompanied by energy dissipation of the sound wave. The acoustic anomalies are traced up to $T >
T_N$, where the thermodynamic properties are determined by fermionic magnetic excitations, the “hallmark” of one-dimensional (1D) spin chains. On the other hand, as established in earlier studies, the AF phase in DTN is governed by bosonic magnetic excitations. Our results suggest the presence of a crossover from a 1D fermionic to a 3D bosonic character of the magnetic excitations in DTN in the vicinity of the QCPs.
author:
- 'O. Chiatti'
- 'A. Sytcheva'
- 'J. Wosnitza'
- 'S. Zherlitsyn'
- 'A. A. Zvyagin'
- 'V. S. Zapf'
- 'M. Jaime'
- 'A. Paduan-Filho'
title: '**Character of magnetic excitations in a quasi-one-dimensional antiferromagnet near the quantum critical points: Impact on magneto-acoustic properties**'
---
The interest in quasi-1D quantum spin systems has grown considerably during the last decade. This is fostered by the progress in preparing materials with well-defined 1D spin subsystems and the possibility of analyzing the experimental data with the help of non-perturbative theories for 1D models. [@Zb] In addition, such systems often manifest quantum phase transitions at $T$=0 which are governed by parameters other than the temperature. True 1D models do not exhibit any long-range order at finite temperatures. [@Zb] Real quasi-1D antiferromagnetic (AF) materials, containing weakly coupled spin chains with gapless spectra of their low-lying excitations, are usually magnetically ordered at low temperatures. At temperatures higher than the Neél temperature, $T_N$, but of the order of the exchange constant, these systems behave as quantum spin chains, where any long-range magnetic order is destroyed by enhanced quantum fluctuations. [@Zb] One should note that quasi-1D magnets, in which the low-energy eigenstates of their 1D subsystems have spin gaps, usually do not manifest long-range magnetic ordering. [@Reg] However, an external magnetic field can close the spin gap, $\Delta$, and for $H > H_c \sim \Delta$ a quantum phase transition to a phase with gapless spin excitations takes place. A further increase of the field yields a second quantum phase transition to a spin-polarized phase at $H > H_s$. In the spin-polarized phase the low-energy excitations are also gapped. Hence, the magnetically ordered phase can be observed in the field domain where spin excitations are gapless, and the Néel temperature in such systems is field dependent. The magnetic susceptibilities of a quasi-1D spin system in mean-field approximation can be written as $$\chi_{\bf q}^{\alpha} = {(\chi_{\bf q}^{\alpha})^{(1)}(T)\over 1 -
ZJ_{\perp}({\bf q}) (\chi_{\bf q}^{\alpha})^{(1)}(T)} \ , \
\label{chi3d}$$ where the superscript $(1)$ denotes the susceptibility of one chain, $\alpha =x,y,z$, $J_{\perp}$ is the weak interchain exchange constant, $Z$ is the coordination number, and [**q**]{} is the wave vector. The quasi-1D spin system becomes ordered when the denominator becomes zero (which defines $T_N(H)$).
The low-$T$ thermodynamics of a state with long-range magnetic order is determined by bosonic excitations, magnons. Recently, several groups have observed phenomena in some AF systems that have been interpreted as Bose-Einstein condensation (BEC) of magnons, viz., as a thermodynamically large number of magnons in the same ground state. [@Seb; @Rad; @Rue] For quasi-1D spin systems at $T > T_N$ the projection of a single spin may have only a limited number of values (e.g., two values for spin-1/2 systems, three values for spin-1, etc.). That is why thermodynamic properties of, e.g., AF spin-1/2 chains are often determined by low-energy eigenstates which behave as interacting [*fermions*]{}. [@Zb] The fermionic nature of the low-energy excitations of these quantum spin chains with only short-range correlations is related to the limited number of projections of each spin. Very recently it has been shown [@MHO] that the behavior of a spin-1 spin-gapped system in magnetic fields close to field-induced quantum critical points (QCP’s) can be described by free fermions as well. The fermionic nature of these excitations is also related to the limited number of projection values at each spin-1 site. The fermionic behavior of excitations is characteristic for quantum spin chains with short-range correlations. From this perspective it is very interesting to study the behavior of a spin-gapped quasi-1D AF system in the vicinity of $T_N$, close to $H_c$ and $H_s$. Here, low-lying excitations of the quasi-1D system should change their statistical properties from fermionic, at $T > T_N(H)$, to bosonic, at $T < T_N(H)$. Hence, by varying $H$ and $T$ one may observe features in the same spin system characteristic either to fermions or bosons.
One of the best candidates for studying such a crossover in the excitation statistics is the spin-1 system dichloro-tetrakis thiourea-nickel(II), NiCl$_2$-4SC(NH$_2$)$_2$, known as DTN. Recently, some features in the magnetically ordered phase of DTN at $H \ge H_c$ and $T <T_N(H)$ were interpreted as BEC of spin degrees of freedom. [@Zapf] The bosonic character of the spin excitations in DTN in the magnetically ordered phase has been corroborated and is considered to be a well established fact. In this work, we study magnetic and magneto-acoustic characteristics of DTN near the critical values of $H_c$ and $H_s$. We show that the behavior of the observed properties outside of the AF phase can be well described by an effective [*fermionic*]{} model of low-lying spin excitations. In this way DTN manifests 1D fermionic character of spin excitations at $T > T_N(H)$. This fact, together with previous results, showing bosonic 3D behavior of magnetic excitations in DTN for $T < T_N(H)$, [@Zapf] leads us to the conclusion that a crossover from fermionic to bosonic features of the low-lying magnetic excitations takes place at $T_N$ near the quantum critical points.
Following Ref. \[\] we describe the spin-1 chain at low excitation densities using an effective free-fermion theory with two branches of low-energy states. Two branches are used because the strong single-ion “easy-plane” magnetic anisotropy $D$ observed in DTN [@Zv] splits the spin triplet of the spin-gap modes, and makes one of them ineffective at the critical fields. [@MHO] The two fermionic branches have features at $H_c$ and $H_s$, while for $D \ll T$ the contribution of the third branch is exponentially small and can be neglected in our approximation. Both critical fields are related only to the lowest branch of our model (they correspond to van Hove singularities, connected with two edges of that branch). However, the field dependence is present also in the temperature-dependent factors of both branches.
![(Color online) Low-temperature magnetization ($T = 0.6$ K) of DTN as a function of external magnetic field $H
\parallel$\[001\] (circles). [@Arm] The line is the result of the free-fermion effective theory. The inset sketches the temperature - magnetic field phase diagram of DTN. [@Zapf] The quantum critical points are $H_c \approx 2.1$ T and $H_s \approx 12.6$ T. The maximum temperature of the AF order is $T_N^{max} \approx
1.2$ K.[]{data-label="fig1"}](fig1.eps){height="5.0cm" width="45.00000%"}
In Fig. 1 the solid line shows the calculated field dependence of the magnetization of the quasi-1D spin system (1D subsystems are considered within this effective free-fermion model) for $T$ slightly above the phase boundary $T_N(H)$ (see inset in Fig. 1), where the susceptibility of the quasi-1D system diverges. For comparison we also plot experimental data taken at $T$ = 0.6 K. [@Arm] Note that $T_N(H) <$ 0.6 K in the vicinity of the quantum critical points $H_c$ and $H_s$, since there is a line of phase transitions $T_N(H)$ with $T_N(H_c)=T_N(H_s)=0$ and the system in our model is [*not*]{} in the magnetically ordered phase inside of the interval $H^\prime_c \le H \le H^\prime_s$. $H^\prime_c$ and $H^\prime_s$ are the critical fields at non-zero $T$. There is a good agreement between the effective free-fermion theory and the experimental data at $H < H^\prime_c$, $H > H^\prime_s$, and near the critical values of $H$. On the other hand, inside of the interval $H^\prime_c < H < H^\prime_s$, the real system is ordered, $T_N(H)>$ 0.6 K, and our 1D fermionic description cannot be applied.
Ultrasonic investigations are a powerful experimental technique to study various phase transitions and critical phenomena. This technique is well established as an important tool for the investigation of low-dimensional spin systems. [@L] Spin-lattice interactions are responsible for the attenuation of acoustic waves and influence the sound velocity in magnetic crystals. These interactions are connected either with a strain modulation of the exchange interactions or with a magnetostrictive coupling of a single-ion type. [@L] We have performed measurements of the relative change of the sound velocity and attenuation in DTN, using a phase-sensitive detection technique based on a standard pulse-echo method with a set-up similar to the one described in Ref. \[\]. DTN has a tetragonal crystallographic symmetry (space group I4) with two formula units in the unit cell. The investigated single crystal has a size of about $2\times 2\times
4.1$ mm$^3$. Since the as-grown surfaces of the crystal were smooth and parallel, we glued piezoelectric film transducers directly to the surfaces normal to the crystallographic \[001\] direction, without any additional sample polishing. This geometry corresponds to the longitudinal acoustic $c_{33}$ mode, with propagation direction and polarization along the spin chains. A number of ultrasonic echoes have been detected. The absolute value of the sound velocity at liquid-helium temperature has been determined as $v_l = 2640\pm
20$ m/s. Note that the measurement accuracy for a relative change of sound velocity is of the order of 10$^{-6}$. The sample-length change is relatively small for the applied temperatures and magnetic fields. [@Za1] Therefore, we did not have to take into account any length-change corrections to the sound velocity. The data have been collected using the ultrasonic signal at 78 MHz. The magnetic field was applied along the \[001\] direction, i.e., parallel to the sound-propagation direction.
![(Color online) Field dependence of the relative change of the sound velocity (top) and of the sound attenuation (bottom) of the acoustic $c_{33}$ mode in DTN at $T$ below $T_N^{max}$. The magnetic field was applied along the \[001\] axis. The ultrasonic frequency was 78 MHz. The insets show the sound velocity and attenuation in the vicinity of $H_c$ in enlarged scale.[]{data-label="fig2"}](fig2.eps){width="45.00000%"}
Figure 2 shows the magnetic-field dependence of the relative change of the sound velocity and attenuation of the $c_{33}$ mode in DTN for $T$ below the maximum of $T_N^{max} \approx 1.2$ K. There is a pronounced softening of the $c_{33}$ mode in the vicinity of both critical fields, though the anomaly at $H^\prime_s$ is approximately one order of magnitude larger than that at $H^\prime_c$. There is a relative increase of $\Delta v/v$ = 7$\times$10$^{-4}$ between the sound velocity at $H=0$ and $H
> 12.6$ T, where all spins are polarized. The relative decrease of the sound velocity reaches about 4$\times$10$^{-3}$ at 12 T and 0.3 K. The softening of the $c_{33}$ mode is accompanied by a peak in the sound attenuation. Both the sound-velocity and sound-attenuation anomalies become smaller and broader with increasing $T$. The $H$ dependence of the sound velocity in the ordered phase (far from the critical regions) resembles $c$-axis magnetostriction data. [@Za1] However, the change in the sound velocity cannot be explained by the lattice-parameter change, since the length change observed in Ref. is too small.
![(Color online) Field dependence of the relative change of the sound velocity (top) and of the sound attenuation (bottom) of the acoustic $c_{33}$ mode in DTN for $T$ above $T_N^{max}$.[]{data-label="fig3"}](fig3.eps){width="45.00000%"}
In Fig. 3, we show the field dependence of the sound velocity and attenuation of the $c_{33}$ mode in DTN at various temperatures above $T_N^{max}$. One can see some transformation of the acoustic anomalies by moving from $T < T_N^{max}$ to $T > T_N^{max}$. Here, the softening of the $c_{33}$ mode disappears in the vicinity of $H_c$; only a smooth increase in the sound velocity is detected. Close to $H_s$ one can still observe a minimum in the sound velocity and a maximum in the sound attenuation, but those anomalies are smaller in amplitude and broader than the corresponding ones measured below $T_N^{max}(H)$ (Fig. 2).
In magnetic materials the dominant contribution to the spin-lattice interactions mostly arises from the exchange-striction coupling. In our calculations we assumed that in DTN the spatial dependence of the magnetic anisotropy constant is weaker than the spatial dependence of the exchange integrals. In this case, one can expect that only longitudinal sound waves interact with the spin subsystem.
![(Color online) Attenuation (upper surface, blue) and relative change of the velocity (lower surface, red) of the longitudinal sound versus $H$ and $T^\prime$, calculated in the framework of the proposed theory (arbitrary units were used for all parameters, see text for details).[]{data-label="fig4"}](fig41.eps){width="40.00000%" height="50.00000%"}
According to Ref. , the relative renormalization of the longitudinal sound velocity can be written as $(\Delta v/v) =
- (A_1 +A_2)/ (N\omega_{\bf k})^2$, where $$\begin{aligned}
&&A_1 = 2 |G_0^z({\bf k})|^2 \langle S_0^z \rangle^2 \chi_0^z + T
\sum_{\bf q} \sum_{\alpha =x,y,z} |G_{\bf q}^{\alpha} ({\bf k})|^2
(\chi_{\bf q}^{\alpha})^2 \ ,
\nonumber \\
&&A_2 = H_0^z({\bf k}) \langle S_0^z \rangle^2 +{T\over
2}\sum_{\bf q} \sum_{\alpha =x,y,z} H_{\bf q}^{\alpha} ({\bf k})
\chi_{\bf q}^{\alpha} \ . \label{otn}\end{aligned}$$ Here, $N$ is the number of spins in the system, $\omega_{\bf k} =v
k$ is the low-$k$ dispersion relation with sound velocity $v$ in the absence of spin-phonon interactions, $\langle S^z_0 \rangle$ is the average magnetization along the direction of the magnetic field, $\chi_{\bf q}^{x,y,z}$ are non-uniform magnetic susceptibilities, and the subscript $0$ corresponds to $q=0$. In the framework of our effective free-fermion model the temperature and magnetic-field dependence of the uniform susceptibility of one spin chain can be written as $$\begin{aligned}
&&(\chi_0^z)^{(1)} = {8\over \pi T}\int_{H_c}^{H_s} {x dx\over
\sqrt{(H_s^2-x^2)(x^2-H_c^2)}} \times
\nonumber \\
&&{1 + \cosh(H/T)\cosh(x/T)\over [\cosh (H/T) +\cosh(x/T)]^2} \ ,
\label{1d}\end{aligned}$$ where we set the units for the effective $g$-factor, Bohr’s magneton, and Boltzmann’s constant equal to 1. For spin systems with AF interactions the main contribution to the summation over ${\bf
q}$ in Eqs. (2) comes from terms with $q =\pi$, $$\begin{aligned}
&&(\chi_{\pi}^z)^{(1)} = {8\over \pi}\int_{H_c}^{H_s} {dx\over
\sqrt{(H_s^2-x^2)(x^2-H_c^2)}}\times
\nonumber \\
&&{\sinh (x/T)\over \cosh (H/T) +\cosh(x/T)} \ . \label{1dPi}\end{aligned}$$ To calculate magnetic susceptibilities of the quasi-1D spin system we use Eqs. (1), (3) and (4).
The renormalization is proportional to the spin-phonon coupling constants $$\begin{aligned}
&&G_{\bf q}^{\alpha} = {1\over m}\sum_n e^{i{\bf q}{\bf
R}_{nm}}\left(e^{i{\bf k}{\bf R}_{nm}} -1\right) {\bf e}_{\bf k}
{\partial J_{mn}^{\alpha}\over
\partial {\bf R}_{m}} \ ,
\nonumber \\
&&H_{\bf q}^{\alpha} = {1\over m}\sum_n e^{-i{\bf q}{\bf
R}_{nm}}\left(e^{i{\bf k}{\bf R}_{nm}} -1\right)\left(e^{-i{\bf
k}{\bf R}_{nm}} -1\right) \times
\nonumber \\
&&{\bf e}_{\bf k}{\bf e}_{\bf -k}{\partial^2 J_{mn}^{\alpha}\over
\partial {\bf R}_{n}\partial {\bf R}_m} \ . \label{sphon}\end{aligned}$$ Here, $m$ is the mass of the magnetic ion, $J_{mn}^{\alpha}$ denote (anisotropic, generally speaking) exchange integrals, ${\bf e}_{\bf
k}$ is the polarization of the phonon with wave vector ${\bf k}$, and ${\bf R}_n$ is the position vector of the $n$-th site. [@TM] Figure 4 (lower surface) shows the $H$ and $T'$ dependence of the relative change of the longitudinal sound velocity of a quasi-1D spin system calculated in the framework of the effective free-fermion model. We fixed $H_c$ and $H_s$ and used arbitrary units for $T^\prime$ in Fig. 4 (they are not equal to the temperatures in the experiment). It is challenging to calculate $\chi_{\bf q}^{x,y}$ in the framework of the used model. Clearly they have to be smooth functions of $H$ and $T$, except at the line $T_N(H)$. $G_{\bf q}^{\alpha}({\bf k})$ and $H_{\bf q}^{\alpha}({\bf
k})$ in Eqs. (2) are also unknown for any $\alpha$ (one of the coupling constants can be estimated using Ref. ). That is why, in order to obtain the results presented in Fig. 4, we used $J_{\perp}(q)=0.18$ from Ref. , $\chi_{\bf
q=0}^{z}$ and $\chi_{\bf q=\pi}^{z}$ multiplied by some (not known) values of the spin-phonon coupling constants, and $H^z(\bf k)$ two times smaller than $G^z(\bf k)$. The temperature of the divergence in the magnetic susceptibility of the quasi-1D system is generally determined by anisotropic couplings between the spin chains (these couplings are unknown). The divergence at $T_N(H)$, which we used in our theory, does not depend on the direction of the order parameter. Such divergences are present in a quasi-1D model, when any component of the magnetic susceptibility (but with different phase-transition temperatures, $T_N(H)$) is considered. Even in this approximation our simplified theory reproduces the main features of the experimentally observed behavior. Our model reproduces the pronounced minimum at $H^\prime_s$, the almost field-independent behavior at $H > H^\prime_s$ and $H < H^\prime_c$, the larger value of $\Delta v/v$ for $H > H^\prime_s$ as compared to $H <
H^\prime_c$, and the maximum (with $\Delta v/v > 0$) in the interval between $H^\prime_c$ and $H^\prime_s$. With increasing $T'$ the features near the critical fields become weaker, the same way as it was observed in the experiment (cf. Fig. 2 and Fig. 3). At the phase boundary $T_N(H)$ the susceptibility of the quasi-1D system diverges (see above), and our theory predicts very narrow and large peaks at the critical values of $H$ (not shown). Therefore, for the sake of clarity, the curves in Fig. 4 are not plotted starting from $T'=0$. $T_N^{max}(H)$ in our units is $T'=0.02$. Concerning the other values of $\chi_{\bf q}$ (i.e., ${\bf q} \ne 0, \pi$) we affirm, as it was discussed above, that their inclusion does not affect the qualitative behaviour of the sound velocity and attenuation. Following Ref. we also calculated the attenuation coefficient for DTN, $$\begin{aligned}
&&\Delta \alpha (\equiv \Delta \alpha_k) ={1\over Nv} \bigg[2
|G_0^z({\bf k})|^2 \langle S_0^z \rangle^2 \chi_0^z
{\gamma_0^z\over (\gamma_{0}^z)^2 + \omega_{\bf k}^2}
\nonumber \\
&&+ T \sum_{\bf q} \sum_{\alpha =x,y,z} |G_{\bf q}^{\alpha} ({\bf
k})|^2 (\chi_{\bf q}^{\alpha})^2{2\gamma_{\bf q}^{\alpha}\over
(2\gamma_{\bf q}^{\alpha})^2 + \omega_{\bf k}^2} \bigg] \ , \
\label{alpha}\end{aligned}$$ where $\gamma_{\bf q}^{\alpha}$ are the relaxation rates, which can be approximated by $\gamma_{\bf q}^{\alpha} =B/T\chi_{\bf
q}^{\alpha}$, where $B$ is a material-dependent constant (see Ref. ). In our calculations we used the approximation, in which the relaxation rates do not depend on the direction and on the wave vector. The results are also presented in Fig. 4 (upper surface). Here, our theory reproduces also the main features observed in the experimental data: an abrupt increase of the sound attenuation near the saturation field $H_s$, and damping with increasing $T'$. All these findings demonstrate the important role the fermionic magnetic excitations play in the vicinity of the QCPs in DTN.
We also tried to reproduce the observed experimental results using the scaling-like procedure, proposed in Ref. . In the framework of that approach we can use $(\chi^{x,y}_{\pi})^{(1)}$, which seems more accurate than the use of $(\chi^{z}_{\pi})^{(1)}$ only. However, the agreement between the theory and experiment was worse than for our effective free-fermion model.
Generally speaking, one could as well use a bosonic, say the Holstein-Primakoff, representation of spin operators [@HP] for any temperatures in DTN. However, to describe the behavior of spins for $T > T_N(H)$ one has to take into account all interactions between these bosons (because the interactions are of the same magnitude as the energy of the free bosons), which is impossible so far. We do not know any other theory, bosonic or fermionic, which can describe the behavior of the magnetization, sound velocity, and attenuation in DTN better than the theory presented here. The situation, e.g., in weakly coupled spin ladders, is very different from the one in DTN, because in our case one cannot consider any of the spin-spin interactions as weak. Also, the use of hard-core bosons for the description of the behavior of DTN for $T > T_N(H)$ cannot help because from the viewpoint of their collective behavior they can be regarded as fermions (i.e., only one fermion, or hard-core boson can be in one state). For $T< T_N(H)$ in DTN, we definitely cannot use hard-core bosons (e.g., for hard-core bosons BEC is impossible, however, see Ref.\[\] for DTN). The advantage of our fermionic description of DTN for $T >T_N(H)$ (and the mentioned bosonic description for $T < T_N(H)$), compared to the use of only a bosonic description of low-energy spin excitations, is that in our approach both fermions and bosons are basically [*non-interacting*]{}. Hence, they have all features of standard fermions and bosons. For strongly interacting bosons, which is the case for hard-core bosons or the Holstein-Primakoff representation for $T > T_N(H)$, one cannot, strictly speaking, use directly the bosonic character of these excitations. We finally want to note, that in our calculations we never used the symmetry of the wave function, the other difference between fermions and bosons.
In summary, our magnetic and magneto-acoustic studies of the quantum spin-chain magnet NiCl$_2$-4SC(NH$_2$)$_2$ show that the behavior of the observed properties at $T > T_N(H)$ can be well described by an effective 1D [*fermionic*]{} model of low-lying spin excitations. This fact, together with previous results showing the bosonic 3D behavior of the magnetic excitations in DTN for $T
< T_N(H)$, [@Zapf] suggests the presence of a crossover from a fermionic to a bosonic character of the magnetic excitations close to the quantum critical points. The fermionic and bosonic nature of the magnetic excitations is related to the short-range correlations in the spin chains and to the long-range three-dimensional order, respectively.
We thank S. A. Zvyagin for stimulating discussions. A.A.Z. acknowledges the support from the Ukrainian Fundamental Research State Fund (F25.4/13).
[99]{}
See, e.g., A. A. Zvyagin [*Finite Size Effects in Correlated Electron Models: Exact Results*]{}, Imperial College Press, London, 2005. See, e.g., L. P. Regnault [*et al.*]{}, Phys. Rev. B [**53**]{}, 5579 (1996). S. E. Sebastian [*et al.*]{}, Nature [**441**]{}, 617 (2006). T. Radu [*et al.*]{}, Phys. Rev. Lett. [**95**]{}, 127202 (2005). Ch. Rüegg [*et al.*]{}, Nature [**423**]{}, 62 (2003). Y. Maeda, C. Hotta, and M. Oshikawa, Phys. Rev. Lett. [**99**]{}, 057205 (2007). V. S. Zapf [*et al.*]{}, Phys. Rev. Lett. [**96**]{}, 077204 (2006). S. A. Zvyagin [*et al.*]{}, Phys. Rev. Lett. [**98**]{}, 047205 (2007). A. Paduan-Filho [*et al.*]{}, Phys. Rev. B [**69**]{}, 020405(R) (2004). See, e.g., B. Lüthi, [*Physical Acoustics in the Solid State*]{}, Springer, Berlin, 2005. V. S. Zapf [*et al.*]{}, Phys. Rev. B [**77**]{}, 020404(R) (2008) M. Tachiki and S. Maekawa, Progr. Theor. Phys. [**51**]{}, 1 (1974). Z. Honda, K. Katsumata, Y.Nishiyama, and I. Harada, Phys. Rev. B [**63**]{}, 064420 (2001). T. Holstein, and H. Primakoff, Phys. Rev. [**58**]{} 1098 (1940).
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abstract: 'Models in which the space-time metric $g_{\mu\nu}$ is not symmetric, $\ie$ $g_{\mu\nu} \ne g_{\nu\mu}$ may make predictions in scattering experiments, for example in a future $e^+e^-$ linear collider, similar to those from noncommutative field theory. We compute the differential cross sections for pair annihilation, Bhabha and M$\o$ller scattering and find that both nonsymmetric gravity theory(NGT) and noncommutative field theory predict a similar dependence of the differential cross section on the azimuthal angle in agreement with all known data, however in NGT Lorentz violation need not be as severe. Astrophysical and cosmological tests may prove very useful in distinguishing these two theories.'
bibliography:
- 'all.bib'
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[**Nonsymmetric Gravity and Noncommutative Signals\
**]{}
**N. Kersting$^{a}$, and Y.L. Ma$^{a}$**
[*$^{a}$ Theoretical Physics Group, Department of Physics Tsinghua University, Beijing P.R.C. 100084* ]{}
Introduction {#sec:intro}
============
If Nature violates isotropy of space or commutivity among coordinate displacements, the basic assumptions at the root of most of physics come into question and it becomes imperative to parametrize this violation, devising tests to put limits on the parameters. Two such parameterizations in this direction involve the theories of noncommuting space coordinates[@Hinchliffe:2002km] and nonsymmetric gravity[@Moffat:1995fc]. Particle physics experiments at future hadronic or linear colliders may provide good testing grounds for these theories.
In nonsymmetric gravity theory (NGT) the metric of space-time is taken to be nonsymmetric, $\ie$ $g_{\mu\nu} \ne g_{\nu\mu}$. In particular, we may write $$\label{g-components}
g_{\mu\nu} = g_{(\mu\nu)} + g_{[\mu\nu]}$$ decomposing $g$ into its symmetric and antisymmetric pieces. The contravariant tensor $g^{\mu\nu}$ is defined as usual: $$g^{\mu\nu}g_{\mu\rho} = \delta^\nu_\rho$$ One can now go on to define a Lagrangian density as in general relativity, ${\cal L} = \sqrt{-g} R$, where $g\equiv det(g_{\mu\nu})$ and $R$ is the Ricci scalar, and derive field equations for $g_{(\mu\nu)}$ and $g_{[\mu\nu]}$. More details on this reformulation of general relativiy can be found in [@Moffat:1995fc]. There has been extensive work analyzing the effects of $g_{[\mu\nu]}$ for black hole solutions of the field equations, galaxy dynamics, stellar stability, and other phenomena of cosmological and astrophysical relevance [@Moffat:1997cc; @Moffat:1995pi; @Moffat:1996dq] where $g_{(\mu\nu)}$ and $g_{[\mu\nu]}$ may be of comparable size.
In the context of particle physics however, we may start with the assumption that the curvature of space in the region of interest is small: $$\label{g-defn}
g_{\mu\nu} \approx \eta_{\mu\nu} + h_{(\mu\nu)} + a_{[\mu\nu]}$$ where $\eta$ is the usual Minkowski metric and both $h$ and $a$ satisfy $a_{\mu\nu}, h_{\mu\nu} \ll 1 ~\forall~ \mu,\nu$. We further assume that these fields’ dynamics are negligable in the region of interest and we may treat them as background fields. The effects of the symmetric tensor $h$ on particle physics in this limit has been studied elsewhere (see for example [@h-study; @Gusev:1998rp; @DiPiazza:2003zp]). We would like to focus our attention here on the effects of the antisymmetric piece $a$.
We therefore take $h=0$ and $a_{\mu\nu} = {\cal O}(\epsilon) \ll 1 ~\forall~ \mu,\nu$. To simplify computations, we take the form of $a$ to be $$\label{a-form}
a_{\mu\nu} =
\left( \begin{array}{cccc}
0 & \epsilon & \epsilon & \epsilon\\
-\epsilon & 0 & \epsilon & \epsilon\\
-\epsilon & -\epsilon & 0 & \epsilon\\
-\epsilon & -\epsilon & -\epsilon & 0\\
\end{array} \right)$$ In this scenario Eqn(\[g-defn\]) states that the NGT is a perturbation of the ordinary flat-space theory in the small parameter $\epsilon$. This parameter may depend on space-time, as one would expect from the metric theory of General Relativity(GR), and from this point on we keep this dependance implicit in all equations: $\epsilon \equiv \epsilon(x)$, with the understanding that odd powers of $\epsilon$ appearing in physical quantities may average to zero over sufficiently large regions of space or time.
The theory of noncommutating space-time coordinates has already received much attention in the literature(see [@Hinchliffe:2002km] for an extensive review). Here we briefly recall its key features.
Noncommutative space-time is a deformation of ordinary space-time in which the space-time coordinates $x_\mu$, representable by Hermitian operators $\widehat{x}_\mu$, do not commute: $$\label{nceqn}
[\widehat{x}_\mu,\widehat{x}_\nu]=i \theta_{\mu \nu}$$ Here $\theta_{\mu \nu}$ is the deformation parameter: ordinary space-time is obtained in the $\theta_{\mu \nu} \to 0$ limit. By convention it is a real tensor antisymmetric under $\mu \leftrightarrow \nu$. To obtain a noncommuting version of a particular field theory, one need only replace ordinary products between fields with the so-called “star product” defined as : $$\label{star}
(f \star g)(x) \equiv e^{i \theta_{\mu \nu} \partial_{\mu}^{y} \partial_{\nu}^{z}}
f(y) g(z) \mid_{y=z=x}$$ In particular, one can transform the Standard Model into a noncommutative Standard Model (ncSM). Noncommuting coordinates are found to follow naturally in the context of string theory, where $\theta_{\mu \nu}$ is related to a background electric field. The direction of this field explicitly breaks Lorentz invariance, strongly constraining the size of $\theta_{\mu \nu}$. Phenomenological constraints on the ncSM, ranging from Hydrogen spectra, $e^+e^-$ scattering, and various CP-violating quantities[@Hinchliffe:2002km] imply that the dimensionful parameters $\theta_{\mu\nu}$ should not exceed $1~(TeV)^{-2}$.
In this paper we demonstrate that some signals of NGT at a high center-of-mass collider bear resemblance to those of the ncSM. We present some calculations in NGT of simple QED processes in Section \[sec:predict\], showing that differential scattering cross sections have an oscillatory dependance on the azimuthal angle $\varphi$ similar to that in the ncSM. We also explicitly show agreement of NGT with known data from various electron scattering experiments. In Section \[sec:constraints\] we collect some results in a purely general relativistic formulation, investigating classical constraints on the theory from Newton’s Laws and cosmological considerations. Section \[sec:concl\] contains our conclusions.
Predictions in Simple Processes {#sec:predict}
===============================
Pair Annihilation: $e^+e^- \to \gamma\gamma$
--------------------------------------------
Pair annihilation occurs to lowest order in $\alpha_{em}$ through the two tree-level diagrams shown in Figure \[ann-fig\].
As in the SM, the spin-averaged squared amplitude is written $$\begin{aligned}
\label{pair-cross}
{1\over4}\sum_{spins}|{\cal
M}|^2&=&{e^4\over4}g_{\mu\rho}g_{\nu\sigma}\cdot
tr\bigg\{(p\hspace{-0.17cm}\slash_1+m)\bigg[\frac{\gamma^\mu
k\hspace{-0.17cm}\slash_1\gamma^\nu-2\gamma^\mu p_1^\nu}{2p_1\cdot
k_1}+\frac{\gamma^\nu
k\hspace{-0.17cm}\slash_2\gamma^\mu-2\gamma^\nu
p_1^\mu}{2p_1\cdot k_2}\bigg]\nonumber\\
&&\cdot(p\hspace{-0.17cm}\slash_1+m)\bigg[\frac{\gamma^\sigma
k\hspace{-0.17cm}\slash_1\gamma^\rho-2\gamma^\rho p_1^\sigma}{2p_1\cdot
k_1}+\frac{\gamma^\rho
k\hspace{-0.17cm}\slash_2\gamma^\sigma-2\gamma^\sigma
p_1^\rho}{2p_1\cdot k_2}\bigg]\bigg\}\nonumber\\\end{aligned}$$ with $m$ the electron mass. Recall the metric tensors are however not as in the SM, but rather $g_{\mu\nu} = \eta_{\mu\nu} + a_{\mu\nu}$ with $a_{\mu\nu}$ defined as in Eqn \[a-form\]. We defer the full calculation to the Appendix and simply state our result for the differential cross section: $$\label{pair-cross}
\frac{d\sigma}{d\Omega}_{pSM} + \frac{d\sigma}{d\Omega}_{p1}
+ \frac{d\sigma}{d\Omega}_{p2}$$ where $$\begin{aligned}
\frac{d\sigma}{d\Omega}_{pSM} & = & \frac{\alpha^2}{ s}
\bigg[\frac{1 + \cos^2\theta}{\sin^2\theta}\bigg]\nonumber\\
&&\nonumber\\
\frac{d\sigma}{d\Omega}_{p1} &= &\epsilon\frac{\alpha^2}{s}
\bigg[\frac{2\cos\theta}{\sin^2\theta}+\frac{(\sin\varphi+\cos\varphi)}{\sin\theta}\bigg]\nonumber\\
&&\nonumber\\
\frac{d\sigma}{d\Omega}_{p2} &=& 2\epsilon^2 \frac{\alpha^2}{s}
\bigg[\frac{\cos^2\theta-3}{\sin^2\theta}+\frac{(1+\cos\theta)\sin2\theta}{\sin^4\theta}(\sin\varphi-\cos\varphi)-\frac{\sin2\theta}{\sin^2\theta}\cos\varphi\bigg] \nonumber\\\end{aligned}$$ Here we are working in the high-energy limit ($ m_e \approx 0$) and the usual angles $\theta$ and $\varphi$ parameterize photon direction in the center of mass frame. We have written the ${\cal O}(\epsilon)$ and ${\cal O}(\epsilon^2)$ contributions separately because we will later see in Section \[sec:constraints\] that constraints on Lorentz violation strongly favor the scenario where $\epsilon$ (but not necessarily $\epsilon^2$) average to zero over small distances. In this case the prediction of NGT for the differential cross section is $\frac{d\sigma}{d\Omega}_{pSM} + \frac{d\sigma}{d\Omega}_{p2}$.
Recently the OPAL Collaboration[@Abbiendi:2003wv] has analyzed pair annihilation data at a center of mass energy near $200~GeV$. In Figure \[opal-t-fig\] we show their data for the differential cross section $d\sigma /d \cos\theta$ and our prediction $\frac{d\sigma}{d\Omega}_{pSM} + \frac{d\sigma}{d\Omega}_{p2}$. In this case, $\epsilon$ may be as large as $\approx 0.14$ without deviating more than $\sim~1~\sigma$ with the data.
Using this value of $\epsilon$ we predict the azimuthal distribution in Figure \[opal-p-fig\], again showing the OPAL data alongside the prediction from the SM (in this case a flat line). As one can see from the figure, the data are consistent with both the SM and NGT for this value of $\epsilon$.
The prediction from noncommutative models similarly consists of a negative correction to $d\sigma/d\cos\theta$ and an oscillatory $d\sigma/d\varphi$ [@Hewett:2000zp]. In this case the OPAL data are consistent with $\theta_{\mu\nu}< (141~ GeV)^{-2}$, to which we refer the reader to the original OPAL report for the details.
Bhabha Scattering: $e^+e^- \to e^+e^-$
--------------------------------------
As seen in the previous section, the $d\sigma/d\cos\theta$ distribution for pair annihiliation served to constrain $\epsilon$ more severely than the azimuthal distribution $d\sigma/d\varphi$. Our aim in this and the following section is to see whether other simple scattering processes can add to this constraint.
The Bhabha scattering amplitude receives contributions from diagrams involving both the photon and the Z boson as shown in Fig \[bha-diag\].
However, it will be simplest to consider this process in the energy range $m_e \ll \sqrt{s} \ll m_Z$ where we may ignore the $Z$ diagrams and the electron mass. In this limit the differential cross section is approximately (see Appendix for details) $$\label{bha-cross}
\frac{d\sigma}{d\Omega}_{bSM} + \frac{d\sigma}{d\Omega}_{b1}
+ \frac{d\sigma}{d\Omega}_{b2}$$ where $$\begin{aligned}
\frac{d\sigma}{d\Omega}_{bSM} & = & \frac{\alpha^2}{2 s}
\bigg[\frac{1 + \cos^4\theta/2}{\sin^4\theta/2}
+ \frac{1}{2}(1 + \cos^2\theta) - 2 \frac{\cos^4\theta/2}{\sin^2\theta/2}
\bigg]\\
&&\\
\frac{d\sigma}{d\Omega}_{b1} &= &\epsilon \frac{\alpha^2}{ s}
\left[ \frac{\cos^2\theta + 6 \cos\theta - 1}{\sin^2\theta/2}
+ \sin\theta (\sin\varphi + \cos\varphi)\right]\\
&&\\
\frac{d\sigma}{d\Omega}_{b2} &=& \epsilon^2 \frac{\alpha^2}{2 s}
\bigg[ \sin 2\theta (\sin\varphi + \cos\varphi) - \sin^2 \theta \\
&&\\
&& - 2 \frac{ \cos^2 \frac{\theta}{2}}{\sin^4 \frac{\theta}{2}}
(\sin \theta (\sin\varphi + \cos\varphi) + \cos \theta + 1) \\
&&\\
&& -4 \frac{ \cos^2 \frac{\theta}{2}}{\sin^2 \frac{\theta}{2}}
( \sin\theta \cos\varphi - \cos\theta + 1) \bigg]\\\end{aligned}$$ Again we see the characteristic oscillatory dependence on the azimuthal angle, similar to the prediction from noncommutative theories[@Hewett:2000zp]. We can compare the prediction in Eqn \[bha-cross\] (setting $\frac{d\sigma}{d\Omega}_{b1}=0$) with a measurement by the PLUTO Collaboration[@Berger:1980rq] performed at $\sqrt{s}=9.4~GeV$ (see Figure \[bhabha-t-fig\]). We conclude that for the case of $\epsilon = 0.14$ there is no conflict with the data.
M$\o$ller Scattering: $e^-e^- \to e^-e^-$ {#subsec:moll}
-----------------------------------------
Finally we consider the constraints from M$\o$ller scattering. Now the differential cross section is (see Appendix) $$\label{moll-cross}
\frac{d\sigma}{d\Omega}_{mSM} + \frac{d\sigma}{d\Omega}_{m1}
+ \frac{d\sigma}{d\Omega}_{m2}$$ where $$\begin{aligned}
\frac{d\sigma}{d\Omega}_{mSM} &= & \frac{\alpha^2}{s}
\bigg[ 1 + \frac{1}{\sin^4\theta/2} + \frac{1}{\cos^4\theta/2}
\bigg]\\
&&\\
\frac{d\sigma}{d\Omega}_{m1} &= &-2\epsilon\frac{\alpha^2}{s}
\bigg[ \frac{\cos^2\theta + 6 \cos\theta - 1}{\sin^2\theta/2}
+ 4\frac{\sin^2 \theta/2}{\sin\theta}(\sin\varphi + \cos\varphi)\bigg]\\
&&\\
\frac{d\sigma}{d\Omega}_{m2} &=& -8\epsilon^2 \frac{\alpha^2}
{s \sin^4\theta}
\bigg[ \sin^4 \theta + \sin\theta \cos\theta
((3+ \cos^2 \theta)\sin\varphi + (5 - \cos^2 \theta) \cos\varphi\bigg] \\\end{aligned}$$ We note a dependence on $\varphi$ similar to that in the other scattering processes and check the constraint from the $\theta$ distribution: Figure \[moller-t-fig\] shows data taken at the Mark III linear accelerator at SLAC[@Barber:1966]. Again we see that the agreement between theory and experiment is excellent for $\epsilon \approx 0.14$.
Constraints from General Relativity {#sec:constraints}
===================================
We have seen in the previous sections that NGT passes several tests in experiments probing high energies. It is relevant to inquire whether the theory likewise satisfies constraints at energies corresponding to macroscopic or cosmological scales. Thus we leave high energy physics to the side for the moment and concentrate on constraints in the framework of general relativity. Although a fair number of papers address this question we will restrict ourselves to a discussion of only a few of these.
Newton’s $2^{nd}$ Law
---------------------
Starting from the geodesic equation in GR, $$\label{newton}
\frac{d^2 x^\mu}{d \sigma^2}
+ \Gamma^\mu_{\nu\lambda}\frac{d x^\nu}{d \sigma}
\frac{d x^\lambda}{d \sigma} = 0$$ and using the metric chosen in Eqn (\[a-form\]), we obtain for a particle in a conservative potential $U(x,y,z)$ the following equations of motion: $$\label{newton-system}
m\frac{d^2}{dt^2}
\left(
\begin{array}{c}
x \\ y \\ z \\
\end{array}
\right)
=
\frac{1}{1 - \epsilon^4}\left(
\begin{array}{ccc}
-1 + \epsilon^2 & -\epsilon(1- \epsilon^2) & -\epsilon(1+ \epsilon)^2 \\
\epsilon(1- \epsilon^2) & -1+ \epsilon^2 & -\epsilon(1- \epsilon^2) \\
\epsilon(1- \epsilon)^2 & \epsilon(1- \epsilon^2) & -1+ \epsilon^2 \\
\end{array}
\right)
\left(
\begin{array}{c}
U_x \\ U_y \\ U_z \\
\end{array}
\right)$$ where $U_i \equiv dU/dx^i$. Thus motion in a given direction is influenced by the gradient of the potential in an orthogonal direction, violating Newton’s Second Law. That planets in the solar system move on Keplerian orbits to an excellent approximation puts very stringent constraints on such deviations from $F=ma$. Therefore, if $\epsilon$ varies only slowly over solar-system distances, its magnitude must be vanishingly small to match the observed trajectories of the planets. On the other hand, if $\epsilon$ averages to zero over much smaller distances($\eg$ sub-micron) then odd powers of $\epsilon$ may be set to zero in Eqn (\[newton-system\]). The even powers of $\epsilon$ may be removed from off-diagonal entries by a suitable coordinate rotation (see Appendix), reducing Eqn (\[newton-system\]) to the usual diagonal form. This is the justification for setting odd powers of $\epsilon$ to zero in the scattering cross section formulae in Eqns \[pair-cross\], \[bha-cross\], and \[moll-cross\].
Cosmology
---------
Strictly homogeneous and isotropic solutions of the NGT field equations always reduce to the Friedmann-Robertson-Walker(FRW) solutions of GR[@Moffat:1997cc]. Relaxing the homogeneous requirement slightly ($\ie$ turning on $g_{[\mu\nu]}$) leads to the approximate FRW metric $$ds^2 = dt^2 -R^2(t)
\left(h(r)dr^2 + r^2(d\theta^2 + \sin^2 \theta~d\varphi^2)\right)$$ where the field equations determine $h(r)$ as well as $b(r)$ in the following equation involving the Hubble variable $H=\dot{R}/R$: $$\begin{aligned}
H^2(t) + \frac{b(r)}{R^2(t)} &=& \Omega(r,t)H^2(t) \\
\Omega(r,t) &\equiv& \Omega_M(t) + \Omega_S(r,t) \\
\Omega_M(t) &=& \frac{8 \pi G \rho_M(t)}{3 H^2(t)}\\\end{aligned}$$ Here $\Omega_M(t)$ is the usual matter density while $\Omega_S(r,t)$ is the contribution to density from $g_{[\mu\nu]}$. Precise measurements of the curvature and density of the universe can therefore put constraints on the magnitude of $g_{[\mu\nu]}$.
Another way to constrain NGT is by measuring the polarization of light arriving from distant cosmic sources[@Moffat:1997cc]. This can be seen immediately from the electromagnetic action: $$I_em \sim \int d^4 x
\sqrt{-g} g_{\mu\alpha}g_{\nu\beta} F^{\mu\nu}F^{\alpha\beta}$$ upon expanding $g$ as in Eqn(\[g-components\]). One can show that terms like $\partial^iA^i\partial^jA^j, i\ne j$ arise with no counterpart in ordinary electromagnetism. Such terms imply a “mixing” between directions as light propagates, leading to a distance-dependent polarization.
Astrophysics
------------
Stellar collapse is predicted to differ markedly from the standard GR prediction: namely, a collapsing star with mass above the Chandrasekhar limit does not lead to a black hole singularity[@Moffat:1995pi]. The collapse is found to asymptotically reach a compact pseduo-stable state which, like a black hole, emits larges amounts of thermal and gravitational radiation, but no Hawking radiation. For practical purposes therefore it may be difficult to distinguish this object from a standard GR black hole.
At galactic scales, NGT may provide an explanation for the flat behaviour of rotation curves in spiral galaxies alternative to the conventional theory that the galactic halo consists of $90$ percent dark matter. NGT can alter Newtonian gravity at galactic scales[@Moffat:1996dq], predicting rotation curves in agreement with data, without measurably affecting gravity at or below solar system scales.
We stress that these and other cosmological or astrophysical tests of NGT are just as important as ones performed at high energy such as those considered in this paper. While the latter tests bear results similar to those from noncommutative theories, it is the former which can most clearly distinguish between the two since NGT is a gravitational phenomenon whereas noncommutivity in the conventional string theory context is not.
Conclusions {#sec:concl}
===========
We have seen in the preceding analysis that, in electron scattering experiments, the predictions of NGT are similar to those from theories of noncommuting coordinates. Although in both theories the deviation of the differential cross section $d\sigma/d\theta$ from the SM prediction offers the strongest constraint, we suggest that as experimental precision improves the oscillatory behaviour of $d\sigma/d\varphi$ should be the clearest prediction of these theories since the SM background is flat. We conclude that the OPAL data is consistent with both a ncSM with $\theta_{\mu\nu} \sim (141~ GeV)^{-2}$ and the particular NGT presented in this paper for values of $\epsilon \sim~0.14$, implying that the metric of space-time could be up to $2$ percent antisymmetric in the neighborhood of terrestrial experiments. Precision data from PLUTO and MarkIII confirm the latter bound on NGT. However, in contrast to the parameter $\epsilon$, which is dimensionless, the parameter $\theta_{\mu\nu}$ is of mass dimension $-2$ and therefore should cause deviations from the SM which scale with the square of center-of-mass energy. As more data becomes available from high energy collider experiments such as those planned at the LHC or a future $e^+ e^-$ linear collider, noncommutative signals should therefore grow stronger and eventually overtake those of NGT.
From a theoretical perspective, NGT is a much cleaner theory than the ncSM. The latter, aside from some difficulties in the gauge sector, suffers from severe Lorentz violation and a still unsolved problem in ultraviolet divergences[@Hinchliffe:2002km]. Nonetheless, both theories represent interesting perturbations of ordinary space-time. Mixing of space-time coordinates is a common feature of both noncommutative theory and NGT, though the origin of this mixing arises from electromagnetism in the former and gravity in the latter. Cosmological and astrophysical phenomenology should therefore readily distinguish the two theories. Work in this area has thus far been encouraging.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the Department of Physics at Tsinghua University.
Appendix {#appendix .unnumbered}
========
Useful Identities {#useful-identities .unnumbered}
-----------------
The following is a partial list of contractions and trace identities for gamma matrices contracted with $a_{\mu\nu}$.
### contraction identities {#contraction-identities .unnumbered}
$$\begin{aligned}
\gamma_\mu &=& (\eta_{\mu\nu} + a_{\mu\nu})\gamma^\nu \\
\gamma^\mu\gamma_\mu &=& 4 + \gamma^\mu a_{\mu\nu}\gamma^\nu \\
\gamma_\mu\gamma^\mu &=& 4 - \gamma^\mu a_{\mu\nu}\gamma^\nu \\
\gamma^\mu \gamma^\nu \gamma_\mu &=& -2 \gamma^\nu
- \gamma^\nu \gamma^\mu a_{\mu\rho}\gamma^\rho
+ 2 \eta^{\nu\mu} a_{\mu\rho}\gamma^\rho \\
\gamma_\mu\gamma^\nu\gamma^\mu &=& -2 \gamma^\nu
+ \gamma^\mu a_{\mu\rho}\gamma^\rho \gamma^\nu
+ 2 \eta^{\nu\mu} a_{\mu\rho}\gamma^\rho \\
\gamma^\mu \gamma^\nu \gamma^\rho \gamma_\mu &=& 4 \eta^{\nu\rho}
+ \gamma^\nu \gamma^\rho \gamma^\mu a_{\mu\lambda}\gamma^\lambda
+ 2(\gamma^\rho \eta^{\nu\mu} -
\gamma^\nu\eta^{\rho\mu}) a_{\mu\lambda}\gamma^\lambda \\
\gamma_\mu \gamma^\nu \gamma^\rho \gamma^\mu &=&
4 \eta^{\nu\rho} -
\gamma^\mu a_{\mu\lambda}\gamma^\lambda \gamma^\nu \gamma^\rho
+2( \eta^{\rho\mu} a_{\mu\lambda} \gamma^\lambda \gamma^\nu -
\eta^{\nu\mu} a_{\mu\lambda} \gamma^\lambda \gamma^\rho)\\
\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma_\mu &=&
-2 \gamma^\sigma \gamma^\rho \gamma^\nu -
\gamma^\nu \gamma^\rho \gamma^\sigma
\gamma^\mu a_{\mu\lambda}\gamma^\lambda
+ 2( \gamma^\nu\gamma^\rho \eta^{\sigma\mu} -
\gamma^\nu\gamma^\sigma\eta^{\rho\mu}
+ \gamma^\rho\gamma^\sigma
\eta^{\nu\mu})a_{\mu\lambda}\gamma^\lambda
\nonumber\\\end{aligned}$$
### trace identities {#trace-identities .unnumbered}
$$\begin{aligned}
tr[\gamma^\mu a_{\mu\nu}\gamma^\nu] &=& 0 \\
tr[\gamma^\lambda a_{\lambda\mu}\gamma^\nu] &=&
4 \eta^{\nu\rho}a_{\rho\mu}\\
tr[\gamma^\mu a_{\mu\nu}\gamma^\nu\gamma^\sigma \gamma^\tau] &=&
-8 \eta^{\sigma\mu}a_{\mu\nu} \eta^{\nu\tau} \\
tr[\gamma^\mu a_{\mu\nu}\gamma^\nu
\gamma^\rho\gamma^\sigma \gamma^\tau\gamma^\lambda] &=&
8( -\eta^{\rho\mu}a_{\mu\nu} \eta^{\nu\sigma}\eta^{\tau\lambda}
+ \eta^{\rho\mu}a_{\mu\nu} \eta^{\nu\tau}\eta^{\sigma\lambda}
- \eta^{\rho\mu}a_{\mu\nu} \eta^{\nu\lambda}\eta^{\sigma\tau}
\nonumber\\
&&
- \eta^{\sigma\mu}a_{\mu\nu} \eta^{\nu\tau} \eta^{\rho\lambda}
+ \eta^{\sigma\mu}a_{\mu\nu} \eta^{\nu\lambda} \eta^{\rho\tau}
- \eta^{\tau\mu}a_{\mu\nu} \eta^{\nu\lambda} \eta^{\rho\sigma}
\nonumber\\\end{aligned}$$
Pair Annihilation {#pair-annihilation .unnumbered}
-----------------
To compute the pair annihilation cross section, we may start from the expression for Compton scattering which is related by crossing symmetry ($p_1 \to p, p_2 \to -p^\prime, k_1 \to -k,
k_2 \to k^\prime$): $$\begin{aligned}
i{\cal
M}&=&\bar{u}(p^\prime)(-ie\gamma^\mu)\epsilon^\ast_\mu(k^\prime)\frac{i(p\hspace{-0.17cm}\slash+k\hspace{-0.17cm}\slash+m)}{(p+k)^2-m^2}(-ie\gamma^\nu)\epsilon_\nu(k)u(p)\nonumber\\
&&+\bar{u}(p^\prime)(-ie\gamma^\nu)\epsilon_\nu(k)\frac{i(p\hspace{-0.17cm}\slash-k\hspace{-0.17cm}\slash^\prime+m)}{(p-k^\prime)^2-m^2}(-ie\gamma^\mu)\epsilon^\ast_\mu(k^\prime)u(p)\nonumber\\
&=&-ie^2\epsilon^\ast_\mu(k^\prime)\epsilon_\nu(k)\bar{u}(p^\prime)\bigg[\frac{\gamma^\mu(p\hspace{-0.17cm}\slash+k\hspace{-0.17cm}\slash+m)\gamma^\nu}{(p+k)^2-m^2}+\frac{\gamma^\nu(p\hspace{-0.17cm}\slash-k\hspace{-0.17cm}\slash^\prime+m)\gamma^\mu}{(p-k^\prime)^2-m^2}\bigg]u(p)\end{aligned}$$ From here, the cross section is $$\begin{aligned}
{1\over4}\sum_{spins}|{\cal
M}|^2&=&{e^4\over4}g_{\mu\rho}g_{\nu\sigma}\cdot
tr\bigg\{(p\hspace{-0.17cm}\slash^\prime+m)\bigg[\frac{\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu+2\gamma^\mu p^\nu}{2p\cdot
k}+\frac{\gamma^\nu
k\hspace{-0.17cm}\slash^\prime\gamma^\mu-2\gamma^\nu
p^\mu}{2p\cdot k^\prime}\bigg]\nonumber\\
&&\cdot(p\hspace{-0.17cm}\slash+m)\bigg[\frac{\gamma^\sigma
k\hspace{-0.17cm}\slash\gamma^\rho+2\gamma^\rho p^\sigma}{2p\cdot
k}+\frac{\gamma^\rho
k\hspace{-0.17cm}\slash^\prime\gamma^\sigma-2\gamma^\sigma
p^\rho}{2p\cdot k^\prime}\bigg]\bigg\}\nonumber\\
&\equiv&{e^4\over4}\bigg[\frac{I}{(2p\cdot
k)^2}+\frac{II}{(2p\cdot k)(2p\cdot k^\prime)}+\frac{III}{(2p\cdot
k^\prime )(2p\cdot k)}+\frac{IV}{(2p\cdot k^\prime)^2}\bigg]\end{aligned}$$ where the first trace is $$\begin{aligned}
I&=&tr[(p\hspace{-0.17cm}\slash^\prime+m)(\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu+2\gamma^\mu
p^\nu)(p\hspace{-0.17cm}\slash+m)(\gamma_\nu
k\hspace{-0.17cm}\slash\gamma_\mu+2\gamma_\mu p_\nu)]\nonumber\\\end{aligned}$$ First we consider the ${\cal O}(\epsilon)$ contributions to this amplitude. In the computation we will need to insert the full expression for $g_{\mu\nu}$ only; note that to leading order Dirac propagators are unchanged, $\eg$ $\Sigma_s u(p)\overline{u}_s(p)=
p\hspace{-0.17cm}\slash+m \equiv p^{\mu}\eta_{\mu\nu}\gamma^{\nu}+m$, which is true even after radiative corrections. After much Dirac algebra using the contraction and trace identities above and taking the high energy limit $m=0$, we find that $I$ has no ${\cal O}(\epsilon)$ piece. However the second trace, $$\begin{aligned}
II&=&tr[(p\hspace{-0.17cm}\slash^\prime+m)(\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu+2\gamma^\mu
p^\nu)(p\hspace{-0.17cm}\slash+m)(\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu-2\gamma_\nu p_\mu)]\nonumber\\
&=&tr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu
p\hspace{-0.17cm}\slash\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-2tr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu p\hspace{-0.17cm}\slash\gamma_\nu p_\mu]\nonumber\\
&&+mtr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-2mtr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu\gamma_\nu p_\mu]\nonumber\\
&&+2tr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu p^\nu
p\hspace{-0.17cm}\slash\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-4tr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu
p^\nu p\hspace{-0.17cm}\slash\gamma_\nu
p_\mu]\nonumber\\
&&+2mtr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu p^\nu\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-4mtr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu p^\nu\gamma_\nu p_\mu]\nonumber\\
&&+mtr[\gamma^\mu
k^\prime\hspace{-0.17cm}\slash\gamma^\nu\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-2mtr[\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu p\hspace{-0.17cm}\slash\gamma_\nu p_\mu]\nonumber\\
&&+m^2tr[\gamma^\mu k\hspace{-0.17cm}\slash\gamma^\nu\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-2m^2tr[\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu\gamma_\nu
p_\mu]\nonumber\\
&&+2mtr[\gamma^\mu p^\nu p\hspace{-0.17cm}\slash\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-4mtr[\gamma^\mu p^\nu
p\hspace{-0.17cm}\slash\gamma_\nu p_\mu]\nonumber\\
&&+2m^2tr[\gamma^\mu p^\nu\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-4m^2tr[\gamma^\mu p^\nu\gamma_\nu p_\mu]\nonumber\\
&=&tr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu
p\hspace{-0.17cm}\slash\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-2tr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu
k\hspace{-0.17cm}\slash\gamma^\nu p\hspace{-0.17cm}\slash\gamma_\nu p_\mu]\nonumber\\
&&+2tr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu p^\nu
p\hspace{-0.17cm}\slash\gamma_\mu
k\hspace{-0.17cm}\slash^\prime\gamma_\nu]-4tr[p\hspace{-0.17cm}\slash^\prime\gamma^\mu
p^\nu p\hspace{-0.17cm}\slash\gamma_\nu
p_\mu]\nonumber\\\end{aligned}$$ yields $$\begin{aligned}
II&=&\mbox{ordinary theory}\nonumber\\
&&-32\epsilon p^\prime\cdot pk\oslash k^\prime+16\epsilon(p\cdot
kp\oslash p^\prime +p^\prime\cdot pk\oslash p)\end{aligned}$$ where we define $$\begin{aligned}
p\cdot k &\equiv& p_\mu \eta^{\mu\nu} k_\nu \nonumber \\
p\oslash k &\equiv&
p_\mu \eta^{\mu\nu} a_{\nu\rho} \eta^{\rho\sigma} k_\sigma \nonumber \\ \end{aligned}$$ and “ordinary theory” refers to the SM ($\epsilon=0$). The other traces are trivially obtained from the above: $$\begin{aligned}
&&III=II\nonumber\\
&&IV=I(k\rightarrow-k^\prime)\end{aligned}$$ Transforming to the pair annihiliation momenta by crossing symmetry, $$\begin{aligned}
p\rightarrow
p_1,p^\prime\rightarrow-p_2,k\rightarrow-k_1,k^\prime\rightarrow
k_2\end{aligned}$$ we obtain $$\begin{aligned}
{1\over4}\sum_{spins}|{\cal
M}|^2&\equiv&{e^4\over4}\bigg[\frac{I}{(2p_1\cdot
k_1)^2}-\frac{II}{(2p_1\cdot k_1)(2p_1\cdot
k_2)}-\frac{III}{(2p_1\cdot k_2 )(2p_1\cdot
k_1)}+\frac{IV}{(2p_1\cdot k_2)^2}\bigg]\nonumber\\\end{aligned}$$ and $$\begin{aligned}
II&=&III\nonumber\\
&=&\mbox{ordinary theory}\nonumber\\
&&-32\epsilon p_2\cdot p_1k_1\oslash k_2+16\epsilon(p_1\cdot
k_1p_1\oslash p_2 +p_2\cdot p_1k_1\oslash p_1)\end{aligned}$$ $$\begin{aligned}
I&=&\mbox{ordinary theory}\end{aligned}$$ $$\begin{aligned}
IV&=&\mbox{ordinary theory}\end{aligned}$$ Define the kinematical variables $E, \theta, \varphi$ in the center of mass frame: $$\begin{aligned}
\label{kin-vars}
&&p_1=(E,E\hat{z})\nonumber\\
&&p_2=(E,-E\hat{z})\nonumber\\
&&k_1=(E,E\sin\theta\cos\varphi,E\sin\theta\sin\varphi,E\cos\theta)\nonumber\\
&&k_2=(E,-E\sin\theta\cos\varphi,-E\sin\theta\sin\varphi,-E\cos\theta)\end{aligned}$$ It is now straightforward to derive the ${\cal O}(\epsilon)$ differential cross section: $$\begin{aligned}
\frac{d
\sigma}{d\Omega}&=&\mbox{ordinary
theory}\nonumber\\
&&+\frac{\alpha^2\epsilon}{2E^2}\cdot\bigg[\frac{2\cos\theta}{\sin^2\theta}+\frac{(\sin\varphi+\cos\varphi)}{\sin\theta}\bigg]\nonumber\\\end{aligned}$$ Now for ${\cal O}(\epsilon^2)$: The first trace eventually reduces to $$\begin{aligned}
I&=&32[p\cdot k p^\prime\otimes k+p^\prime\cdot kp\otimes
k-p^\prime\cdot pk\otimes k]\nonumber\\
&&-32p^\prime\oslash kp\oslash k -16tr(a^2)p^\prime\cdot kp\cdot k\end{aligned}$$ where we introduce the notation $\otimes$: $$\begin{aligned}
k\otimes p&=&k_\mu
\eta^{\mu\nu} a_{\nu\rho}\eta^{\rho\sigma}a_{\sigma\tau}
\eta^{\tau\lambda}p_\lambda =p\otimes k \nonumber\\
tr(a^2) &=& \eta^{\mu\nu} a_{\nu\rho}
\eta^{\rho\sigma}a_{\sigma\tau}\eta^\tau_{\cdot\mu} \nonumber\\\end{aligned}$$ Similarly, the second trace is $$\begin{aligned}
II&=&16\epsilon^2(k^\prime\otimes kp^\prime\cdot p+p^\prime\otimes
pk^\prime\cdot
k )\nonumber\\\end{aligned}$$ Then the other traces follow easily as before: $$\begin{aligned}
III&=&II=16\epsilon^2(k^\prime\otimes kp^\prime\cdot
p+p^\prime\otimes pk^\prime\cdot k )\end{aligned}$$ $$\begin{aligned}
IV&=&I(k\rightarrow-k^\prime)=32[p\cdot k^\prime p^\prime\otimes
k^\prime+p^\prime\cdot k^\prime p\otimes
k^\prime-p^\prime\cdot pk^\prime\otimes k^\prime]\nonumber\\
&&-32p^\prime\oslash k^\prime p\oslash k^\prime
-16tr(a^2)p^\prime\cdot k^\prime p\cdot k^\prime\end{aligned}$$ From the particular choice of $a$ in Eqn \[a-form\], we have $$\begin{aligned}
a_{\nu\rho}\eta^{\rho\sigma}a_{\sigma\tau}
&=&
\left(
\begin{array}{cccc}
3&2&0&-2\\
2&1&0&-2\\
0&0&1&0\\
-2&-2&0&1
\end{array}
\right)\end{aligned}$$ whence it follows from the kinematical assignments that $$\begin{aligned}
{1\over4}\sum_{spins}|{\cal
M}|^2&=&{\rm ordinary} + {\cal O}(\epsilon) +
4e^4\epsilon^2\bigg[\frac{\cos^2\theta-3}{\sin^2\theta}+\frac{(1+\cos\theta)\sin2\theta}{\sin^4\theta}(\sin\varphi-\cos\varphi)-\frac{\sin2\theta}{\sin^2\theta}\cos\varphi\bigg]\nonumber\\\end{aligned}$$ leading to the expression reported in Eqn \[pair-cross\].
Bhabha Scattering {#bhabha-scattering .unnumbered}
-----------------
The two diagrams contributing (see Figure \[bha-diag\]) interfere destructively, with spin averaged squared amplitude $$\begin{aligned}
{1\over4}\sum_{spins}|{\cal
M}|^2&=&
\frac{e^4}{4}\bigg[\frac{I}{(p+p^\prime)^4}
+ \frac{II}{(k^\prime-p^\prime)^4}
- \frac{III + IV}{(p+p^\prime)^2 (k^\prime-p^\prime)^2}
\bigg] \nonumber\\
I &\equiv& tr[p\hspace{-0.17cm}\slash^\prime \gamma^\mu
p\hspace{-0.17cm}\slash \gamma^\sigma]
tr[ k\hspace{-0.17cm}\slash \gamma^\nu
k\hspace{-0.17cm}\slash^\prime\gamma^\rho]g_{\mu\nu}g_{\rho\sigma}
\nonumber\\
II &\equiv& I(p \leftrightarrow k^\prime) \nonumber\\
III &\equiv& tr[p\hspace{-0.17cm}\slash^\prime \gamma^\mu
p\hspace{-0.17cm}\slash \gamma^\rho
k\hspace{-0.17cm}\slash \gamma^\nu
k\hspace{-0.17cm}\slash^\prime \gamma^\sigma]g_{\mu\nu}g_{\rho\sigma}
\nonumber\\
IV &\equiv& III(p \leftrightarrow k^\prime) \nonumber\\\end{aligned}$$ Then $$\begin{aligned}
I &=& p^\prime_\alpha p_\beta (\eta^{\alpha \mu}\eta^{\beta \sigma}
- \eta^{\alpha \beta}\eta^{\mu \sigma}+
\eta^{\alpha \sigma}\eta^{\mu\beta})\oslash
k_\lambda k^\prime_\tau
(\eta^{\lambda \nu}\eta^{\tau \rho} -
\eta^{\lambda \tau}\eta^{\nu \rho} +
\eta^{\lambda \rho}\eta^{\nu \tau})g_{\mu\nu}g_{\rho\sigma}\nonumber\\
&=& {\rm ordinary theory} + \nonumber\\
&& -2 \bigg[(p \oslash k^\prime)(p^\prime \oslash k)
+ (p^\prime \oslash k^\prime)(p \oslash k) +
(k \cdot k^\prime)(p \otimes p^\prime)\bigg] \nonumber\\\end{aligned}$$ ($\ie$ there is no ${\cal O}(\epsilon)$ piece). By repeated use of the contraction identities noted earlier, one can further verify that $$\begin{aligned}
III+IV &=& {\rm ordinary~theory} +
32 \bigg[ (p \cdot p^\prime)(k \oslash k^\prime) +
(p^\prime \cdot k^\prime)(k \oslash p)
+ (p \cdot k^\prime) (p^\prime \otimes k) -
(k \cdot p^\prime) (k^\prime \otimes p)\bigg]\nonumber\\\end{aligned}$$ Using the kinematical assignments as in Eqn \[kin-vars\], replacing $p_1 \to p, p_2 \to p^\prime, k_1 \to k, k_2 \to k^\prime$, the quoted cross section in the text follows straightforwardly.
M$\o$ller Scattering {#moller-scattering .unnumbered}
--------------------
This can be obtained quickly from Bhabha scattering by substituting $p$ for $k$ and vice versa in the traces ( but not in the denominators). Hence $$\begin{aligned}
{1\over4}\sum_{spins}|{\cal
M}|^2&=&
\frac{e^4}{4}\bigg[\frac{I}{(p+p^\prime)^4}
+ \frac{II}{(k^\prime-p^\prime)^4}
- \frac{III + IV}{(p+p^\prime)^2 (k^\prime-p^\prime)^2}
\bigg] \nonumber\\\end{aligned}$$ where $$\begin{aligned}
I &=& {\rm ordinary~theory} + \nonumber\\
&& -2 \bigg[(k \oslash k^\prime)(p^\prime \oslash p)
+ (p^\prime \oslash k^\prime)(k \oslash p) +
(p \cdot k^\prime)(k \otimes p^\prime)\bigg] \nonumber\\
II &=& I(k \leftrightarrow k^\prime) \nonumber\\
III+IV &=& {\rm ordinary~theory} +
32 \bigg[ (k \cdot p^\prime)(p \oslash k^\prime) +
(p^\prime \cdot k^\prime)(k \oslash p)
+ (k \cdot k^\prime) (p^\prime \otimes p) -
(p \cdot p^\prime) (k^\prime \otimes k)\bigg]\nonumber\\\end{aligned}$$ Making the same kinematical assignments as for Bhabha scattering, the quoted cross section in Section \[subsec:moll\] follows.
Preserving $F=ma$ {#preserving-fma .unnumbered}
-----------------
Starting from Eqn(\[newton-system\]), perform a rotation in the $x-z$ plane: $$\label{rotated}
\frac{d^2}{dt^2}
\left(
\begin{array}{c}
x' \\ y' \\ z' \\
\end{array}
\right)
=
{\bf \Omega}
\left(
\begin{array}{c}
U_x' \\ U_y' \\ U_z' \\
\end{array}
\right)$$ where $${\bf \Omega} =
\frac{1}{1 - \epsilon^4}\left(
\begin{array}{ccc}
\epsilon^2 - 1 - 4\epsilon^2 c_\theta s_\theta &
\frac{1-\epsilon^4 }{1+\epsilon^2}\epsilon (s_\theta- c_\theta) &
\epsilon( 1 + 2\epsilon(c^2_\theta - s^2_\theta) + \epsilon^2) \\
-\frac{1-\epsilon^4 }{1+\epsilon^2}\epsilon (s_\theta- c_\theta) &
-\frac{1-\epsilon^4 }{1+\epsilon^2} &
-\frac{1-\epsilon^4 }{1+\epsilon^2}\epsilon (s_\theta+ c_\theta) \\
-\epsilon( 1 + 2\epsilon(s^2_\theta - c^2_\theta) + \epsilon^2) &
\epsilon\frac{1-\epsilon^4 }{1+\epsilon^2} &
\epsilon^2 - 1 + 4\epsilon^2 c_\theta s_\theta \\
\end{array}
\right)$$ Neglecting odd powers of $\epsilon$, this can clearly be brought to diagonal form by setting $ c_\theta = s_\theta$, $\ie$ rotating by 45 degrees.
|
---
abstract: 'A major challenge of semantic parsing is the vocabulary mismatch problem between natural language and target ontology. In this paper, we propose a sentence rewriting based semantic parsing method, which can effectively resolve the mismatch problem by rewriting a sentence into a new form which has the same structure with its target logical form. Specifically, we propose two sentence-rewriting methods for two common types of mismatch: a dictionary-based method for 1-N mismatch and a template-based method for N-1 mismatch. We evaluate our sentence rewriting based semantic parser on the benchmark semantic parsing dataset – WEBQUESTIONS. Experimental results show that our system outperforms the base system with a 3.4% gain in F1, and generates logical forms more accurately and parses sentences more robustly.'
author:
- |
Bo Chen Le Sun Xianpei Han Bo An\
State Key Laboratory of Computer Sciences\
Institute of Software, Chinese Academy of Sciences, China.\
[{chenbo, sunle, xianpei, anbo}@iscas.ac.cn]{}\
bibliography:
- 'acl2016.bib'
title: Sentence Rewriting for Semantic Parsing
---
Introduction
============
Semantic parsing is the task of mapping natural language sentences into logical forms which can be executed on a knowledge base [@zelle:aaai96; @DBLP:conf/uai/ZettlemoyerC05; @kate-mooney:2006:COLACL; @wong-mooney:2007:ACLMain; @lu-EtAl:2008:EMNLP; @kwiatkowksi-EtAl:2010:EMNLP]. Figure 1 shows an example of semantic parsing. Semantic parsing is a fundamental technique of natural language understanding, and has been used in many applications, such as question answering [@liang-jordan-klein:2011:ACL-HLT2011; @he-EtAl:2014:EMNLP20142; @DBLP:conf/aaai/ZhangHL016] and information extraction [@krishnamurthy-mitchell:2012:EMNLP-CoNLL; @choi-kwiatkowski-zettlemoyer:2015:ACL-IJCNLP; @parikh-poon-toutanova:2015:NAACL-HLT].
![An example of semantic parsing.[]{data-label="example-figure"}](f1.pdf){width="50.00000%"}
Semantic parsing, however, is a challenging task. Due to the variety of natural language expressions, the same meaning can be expressed using different sentences. Furthermore, because logical forms depend on the vocabulary of target-ontology, a sentence will be parsed into different logical forms when using different ontologies. For example, in below the two sentences $s_1$ and $s_2$ express the same meaning, and they both can be parsed into the two different logical forms $lf_1$ and $lf_2$ using different ontologies.
-------- -------------------------------------------------
$s_1$ *What is the population of Berlin?*
$s_2$ *How many people live in Berlin?*
$lf_1$ `\lambdax.population(Berlin,x)`
$lf_2$ `count(\lambdax.person(x)\wedgelive(x,Berlin))`
-------- -------------------------------------------------
Based on the above observations, one major challenge of semantic parsing is the structural mismatch between a natural language sentence and its target logical form, which are mainly raised by the vocabulary mismatch between natural language and ontologies. Intuitively, if a sentence has the same structure with its target logical form, it is easy to get the correct parse, e.g., a semantic parser can easily parse $s_1$ into $lf_1$ and $s_2$ into $lf_2$. On the contrary, it is difficult to parse a sentence into its logic form when they have different structures, e.g., $s_1 \rightarrow lf_2$ or $s_2 \rightarrow lf_1$.
To resolve the vocabulary mismatch problem, this paper proposes a sentence rewriting approach for semantic parsing, which can rewrite a sentence into a form which will have the same structure with its target logical form. Table 1 gives an example of our rewriting-based semantic parsing method. In this example, instead of parsing the sentence “*What is the name of Sonia Gandhi’s daughter?*” into its structurally different logical form `childOf.S.G.\wedgegender.female` directly, our method will first rewrite the sentence into the form “*What is the name of Sonia Gandhi’s female child?*”, which has the same structure with its logical form, then our method will get the logical form by parsing this new form. In this way, the semantic parser can get the correct parse more easily. For example, the parse obtained through traditional method will result in the wrong answer “*Rahul Gandhi*”, because it cannot identify the vocabulary mismatch between “*daughter*” and `child\wedgefemale`. By contrast, by rewriting “*daughter*” into “*female child*”, our method can resolve this vocabulary mismatch.
\(a) An example using traditional method\
[|p[205pt]{}|]{}\
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\(b) An example using our method\
[|p[205pt]{}|]{}\
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Specifically, we identify two common types of vocabulary mismatch in semantic parsing:
1. 1-N mismatch: a simple word may correspond to a compound formula. For example, the word “*daughter*” may correspond to the compound formula `child\wedgefemale`.
2. N-1 mismatch: a logical constant may correspond to a complicated natural language expression, e.g., the formula `population` can be expressed using many phrases such as “*how many people*” and “*live in*”.
To resolve the above two vocabulary mismatch problems, this paper proposes two sentence rewriting algorithms: One is a dictionary-based sentence rewriting algorithm, which can resolve the 1-N mismatch problem by rewriting a word using its explanation in a dictionary. The other is a template-based sentence rewriting algorithm, which can resolve the N-1 mismatch problem by rewriting complicated expressions using paraphrase template pairs.
Given the generated rewritings of a sentence, we propose a ranking function to jointly choose the optimal rewriting and the correct logical form, by taking both the rewriting features and the semantic parsing features into consideration.
We conduct experiments on the benchmark WEBQUESTIONS dataset [@berant-EtAl:2013:EMNLP]. Experimental results show that our method can effectively resolve the vocabulary mismatch problem and achieve accurate and robust performance.
The rest of this paper is organized as follows. Section 2 reviews related work. Section 3 describes our sentence rewriting method for semantic parsing. Section 4 presents the scoring function which can jointly ranks rewritings and logical forms. Section 5 discusses experimental results. Section 6 concludes this paper.
Related Work
============
Semantic parsing has attracted considerable research attention in recent years. Generally, semantic parsing methods can be categorized into synchronous context free grammars (SCFG) based methods [@wong-mooney:2007:ACLMain; @TACL654; @li-EtAl:2015:EMNLP3], syntactic structure based methods [@ge-mooney:2009:ACLIJCNLP; @reddy_largescale_2014; @reddy_transforming_2016], combinatory categorical grammars (CCG) based methods [@zettlemoyer-collins:2007:EMNLP-CoNLL2007; @kwiatkowksi-EtAl:2010:EMNLP; @kwiatkowski-EtAl:2011:EMNLP; @krishnamurthy-mitchell:2014:P14-1; @wang-kwiatkowski-zettlemoyer:2014:EMNLP2014; @artzi-lee-zettlemoyer:2015:EMNLP], and dependency-based compositional semantics (DCS) based methods [@liang-jordan-klein:2011:ACL-HLT2011; @berant-EtAl:2013:EMNLP; @berant-liang:2014:P14-1; @TACL646; @pasupat2015compositional; @wang-berant-liang:2015:ACL-IJCNLP].
One major challenge of semantic parsing is how to scale to open-domain situation like Freebase and Web. A possible solution is to learn lexicons from large amount of web text and a knowledge base using a distant supervised method [@krishnamurthy-mitchell:2012:EMNLP-CoNLL; @cai-yates:2013:ACL2013; @berant-EtAl:2013:EMNLP]. Another challenge is how to alleviate the burden of annotation. A possible solution is to employ distant-supervised techniques [@clarke-EtAl:2010:CONLL; @liang-jordan-klein:2011:ACL-HLT2011; @cai-yates:2013:*SEM; @artzi-zettlemoyer:2013:TACL], or unsupervised techniques [@poon-domingos:2009:EMNLP; @goldwasser-EtAl:2011:ACL-HLT2011; @poon:2013:ACL2013].
There were also several approaches focused on the mismatch problem. Kwiatkowski et al. addressed the ontology mismatch problem (i.e., two ontologies using different vocabularies) by first parsing a sentence into a domain-independent underspecified logical form, and then using an ontology matching model to transform this underspecified logical form to the target ontology. However, their method is still hard to deal with the 1-N and the N-1 mismatch problems between natural language and target ontologies. Berant and Liang addressed the structure mismatch problem between natural language and ontology by generating a set of canonical utterances for each candidate logical form, and then using a paraphrasing model to rerank the candidate logical forms. Their method addresses mismatch problem in the reranking stage, cannot resolve the mismatch problem when constructing candidate logical forms. Compared with these two methods, we approach the mismatch problem in the parsing stage, which can greatly reduce the difficulty of constructing the correct logical form, through rewriting sentences into the forms which will be structurally consistent with their target logic forms.
Sentence rewriting (or paraphrase generation) is the task of generating new sentences that have the same meaning as the original one. Sentence rewriting has been used in many different tasks, e.g., used in statistical machine translation to resolve the word order mismatch problem [@collins-koehn-kucerova:2005:ACL; @he-EtAl:2015:EMNLP]. To our best knowledge, this paper is the first work to apply sentence rewriting for vocabulary mismatch problem in semantic parsing.
Sentence Rewriting for Semantic Parsing
=======================================
As discussed before, the vocabulary mismatch between natural language and target ontology is a big challenge in semantic parsing. In this section, we describe our sentence rewriting algorithm for solving the mismatch problem. Specifically, we solve the 1-N mismatch problem by dictionary-based rewriting and solve the N-1 mismatch problem by template-based rewriting. The details are as follows.
Dictionary-based Rewriting
--------------------------
In the 1-N mismatch case, a word will correspond to a compound formula, e.g., the target logical form of the word “*daughter*” is `child\wedgefemale` (Table 2 has more examples).
To resolve the 1-N mismatch problem, we rewrite the original word (“*daughter*”) into an expression (“*female child*”) which will have the same structure with its target logical form (`child\wedgefemale`). In this paper, we rewrite words using their explanations in a dictionary. This is because each word in a dictionary will be defined by a detailed explanation using simple words, which often will have the same structure with its target formula. Table 2 shows how the vocabulary mismatch between a word and its logical form can be resolved using its dictionary explanation. For instance, the word “*daughter*” is explained as “*female child*” in Wiktionary, which has the same structure as `child\wedgefemale`.
-- -- --
-- -- --
: \[wik-table\]Several examples of words, their logical forms and their explanations in Wiktionary.
In most cases, only common nouns will result in the 1-N mismatch problem. Therefore, in order to control the size of rewritings, this paper only rewrite the common nouns in a sentence by replacing them with their dictionary explanations. Because a sentence usually will not contain too many common nouns, the size of candidate rewritings is thus controllable. Given the generated rewritings of a sentence, we propose a sentence selection model to choose the best rewriting using multiple features (See details in Section 4).
Table 3 shows an example of the dictionary-based rewriting. In Table 3, the example sentence $s$ contains two common nouns (“*name*” and “*daughter*”), therefore we will generate three rewritings $r_1$, $r_2$ and $r_3$. Among these rewritings, the candidate rewriting $r_2$ is what we expected, as it has the same structure with the target logical form and doesn’t bring extra noise (i.e., replacing “*name*” with its explanation “*reputation*”).
[|p[205pt]{}|]{}\
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For the dictionary used in rewriting, this paper uses Wiktionary. Specifically, given a word, we use its “Translations” part in the Wiktionary as its explanation. Because most of the 1-N mismatch are caused by common nouns, we only collect the explanations of common nouns. Furthermore, for polysomic words which have several explanations, we only use their most common explanations. Besides, we ignore explanations whose length are longer than 5.
Template-based Rewriting
------------------------
In the N-1 mismatch case, a complicated natural language expression will be mapped to a single logical constant. For example, considering the following mapping from the natural language sentence $s$ to its logical form $lf$ based on Freebase ontology:
----------------------------------------
$s$: *How many people live in Berlin?*
$lf$: `\lambdax.population(Berlin,x)`
----------------------------------------
where the three words: “*how many*” (`count`), “*people*” (`people`) and “*live in*” (`live`) will map to the predicate `population` together. Table 4 shows more N-1 examples.
-- --
-- --
: \[mismatch-table\] Several N-1 mismatch examples.
To resolve the N-1 mismatch problem, we propose a template rewriting algorithm, which can rewrite a complicated expression into its simpler form. Specifically, we rewrite sentences based on a set of paraphrase template pairs $P = \{(t_{i1},t_{i2}) | i=1,2,...,n\}$, where each template $t$ is a sentence with an argument slot \$y, and $t_{i1}$ and $t_{i2}$ are paraphrases. In this paper, we only consider single-slot templates. Table 5 shows several paraphrase template pairs.
-- --
-- --
: \[paraphrase-template-table\] Several examples of paraphrase template pairs.
Given the template pair database and a sentence, our template-based rewriting algorithm works as follows:
1. Firstly, we generate a set of candidate templates $ST = \{st_1, st_2,...,st_n\}$ of the sentence by replacing each named entity within it by “\$y”. For example, we will generate template “*How many people live in \$y*” from the sentence “*How many people live in Berlin*”.
2. Secondly, using the paraphrase template pair database, we retrieve all possible rewriting template pairs $(t_1,t_2)$ with $t_1 \in ST$, e.g., we can retrieve template pair (“*How many people live there in \$y*”, “*What is the population of \$y*” for $t_2$) using the above $ST$.
3. Finally, we get the rewritings by replacing the argument slot “\$y” in template $t_2$ with the corresponding named entity. For example, we get a new candidate sentence “*What is the population of Berlin*” by replacing “\$y” in $t_2$ with Berlin. In this way we can get the rewriting we expected, since this rewriting will match its target logical form `population(Berlin)`.
To control the size and measure the quality of rewritings using a specific template pair, we also define several features and the similarity between template pairs (See Section 4 for details).
[|p[195pt]{}|]{}\
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To build the paraphrase template pair database, we employ the method described in Fader et al. to automatically collect paraphrase template pairs. Specifically, we use the WikiAnswers paraphrase corpus [@fader-zettlemoyer-etzioni:2013:ACL2013], which contains 23 million question-clusters, and all questions in the same cluster express the same meaning. Table 6 shows two paraphrase clusters from the WikiAnswers corpus. To build paraphrase template pairs, we first replace the shared noun words in each cluster with the placeholder “\$y”, then each two templates in a cluster will form a paraphrase template pair. To filter out noisy template pairs, we only retain salient paraphrase template pairs whose co-occurrence count is larger than 3.
Sentence Rewriting based Semantic Parsing
=========================================
In this section we describe our semantic rewriting based semantic parsing system. Figure 2 presents the framework of our system. Given a sentence, we first rewrite it into a set of new sentences, then we generate candidate logical forms for each new sentence using a base semantic parser, finally we score all logical forms using a scoring function and output the best logical form as the final result. In following, we first introduce the used base semantic parser, then we describe the proposed scoring function.
Base Semantic Parser
--------------------
In this paper, we produce logical forms for each sentence rewritings using an agenda-based semantic parser [@TACL646], which is based on the lambda-DCS proposed by Liang . For parsing, we use the lexicons and the grammars released by Berant et al. , where lexicons are used to trigger unary and binary predicates, and grammars are used to conduct logical forms. The only difference is that we also use the composition rule to make the parser can handle complicated questions involving two binary predicates, e.g., `child.obama\wedgegender.female`.
![The framework of our sentence rewriting based semantic parsing.[]{data-label="framework-figure"}](f2.pdf){width="50.00000%"}
For model learning and sentence parsing, the base semantic parser learned a scoring function by modeling the policy as a log-linear distribution over (partial) agenda derivations Q:
$$p_{\theta}(a|s)=\frac{\exp\{\phi(a)^T\theta)\}}{\sum_{a'\in A}\exp\{\phi(a')^T\theta)\}}$$
The policy parameters are updated as follows:
$$\theta\leftarrow\theta+\eta R(h_{target})\sum\nolimits_{t=1}^T\delta(h_{target})$$
$$\label{wave kinematic}
\begin{split}
\delta_t(h) &= \nabla_{\theta}\log p_{\theta}(a_t|s_t) \\
&=\phi(a_t)-E_{p_{\theta}(a_t'|s_t)}[\phi(a_t')]
\end{split}$$
The reward function $R(h)$ measures the compatibility of the resulting derivation, and $\eta$ is the learning rate which is set using the AdaGrad algorithm [@Duchi:2011:ASM:1953048.2021068]. The target history $h_{target}$ is generated from the root derivation $d^*$ with highest reward out of the $K$ (beam size) root derivations, using local reweighting and history compression.
Scoring Function
----------------
To select the best semantic parse, we propose a scoring function which can take both sentence rewriting features and semantic parsing features into consideration. Given a sentence $x$, a generated rewriting $x'$ and the derivation $d$ of $x'$, we score them using follow function:
$$\label{wave kinematic}
\begin{split}
score(x,x',d) &=\theta\cdot\phi(x,x',d) \\
&= \theta_1\cdot\phi(x,x')+\theta_2\cdot\phi(x',d)
\end{split}$$
This scoring function is decomposed into two parts: one for sentence rewriting – $\theta_1\cdot\phi(x,x')$ and the other for semantic parsing – $\theta_2\cdot\phi(x',d)$. Following Berant and Liang , we update the parameters $\theta_2$ of semantic parsing features as the same as (2). Similarly, the parameters $\theta_1$ of sentence rewriting features are updated as follows:
$$\theta_1\leftarrow\theta_1+\eta R(h_{target}^*)\delta(x,x'^*)$$
$$\label{wave kinematic}
\begin{split}
\delta(x,x'^*)&=\nabla \log p_{\theta_1}(x'^*|x) \\
&=\phi(x,x'^*)-E_{p_{\theta_1}(x'|x)}[\phi(x,x')]
\end{split}$$
where the learning rate $\eta$ is set using the same algorithm in Formula (2).
Parameter Learning Algorithm
----------------------------
To estimate the parameters $\theta_1$ and $\theta_2$, our learning algorithm uses a set of question-answer pairs $(x_i,y_i)$. Following Berant and Liang , our updates for $\theta_1$ and $\theta_2$ do not maximize reward nor the log-likelihood. However, the reward provides a way to modulate the magnitude of the updates. Specifically, after each update, our model results in making the derivation, which has the highest reward, to get a bigger score. Table 7 presents our learning algorithm.
[p[205pt]{}]{}\
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\
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Features
--------
As described in Section 4.3, our model uses two kinds of features. One for the semantic parsing module – which are simply the same features described in Berant and Liang . One for the sentence rewriting module –these features are defined over the original sentence, the generated sentence rewritings and the final derivations:
[**Features for dictionary-based rewriting**]{}. Given a sentence $s_0$, when the new sentence $s_1$ is generated by replacing a word to its explanation $w\to ex$, we will generate four features: The first feature indicates the word replaced. The second feature indicates the replacement $w\to ex$ we used. The final two features are the POS tags of the left word and the right word of $w$ in $s_0$.
[**Features for template-based rewriting**]{}. Given a sentence $s_0$, when the new sentence $s_1$ is generated through a template based rewriting $t_1\to t_2$, we generate four features: The first feature indicates the template pair ($t_1$, $t_2$) we used. The second feature is the similarity between the sentence $s_0$ and the template $t_1$, which is calculated using the word overlap between $s_0$ and $t_1$. The third feature is the compatibility of the template pair, which is the pointwise mutual information (PMI) between $t_1$ and $t_2$ in the WikiAnswers corpus. The final feature is triggered when the target logical form only contains an atomic formula (or predicate), and this feature indicates the mapping from template $t_2$ to the predicate $p$.
Experiments
===========
In this section, we assess our method and compare it with other methods.
Experimental Settings
---------------------
[**Dataset**]{}: We evaluate all systems on the benchmark WEBQUESTIONS dataset [@berant-EtAl:2013:EMNLP], which contains 5,810 question-answer pairs. All questions are collected by crawling the Google Suggest API, and their answers are obtained using Amazon Mechanical Turk. This dataset covers several popular topics and its questions are commonly asked on the web. According to Yao , 85% of questions can be answered by predicting a single binary relation. In our experiments, we use the standard train-test split [@berant-EtAl:2013:EMNLP], i.e., 3,778 questions (65%) for training and 2,032 questions (35%) for testing, and divide the training set into 3 random 80%-20% splits for development.
Furthermore, to verify the effectiveness of our method on solving the vocabulary mismatch problem, we manually select 50 mismatch test examples from the WEBQUESTIONS dataset, where all sentences have different structure with their target logical forms, e.g., “*Who is keyshia cole dad?*” and “*What countries have german as the official language?*”.
[**System Settings**]{}: In our experiments, we use the Freebase Search API for entity lookup. We load Freebase using Virtuoso, and execute logical forms by converting them to SPARQL and querying using Virtuoso. We learn the parameters of our system by making three passes over the training dataset, with the beam size $K=200$, the dictionary rewriting size $K_D=100$, and the template rewriting size $K_T=100$.
[**Baselines**]{}: We compare our method with several traditional systems, including semantic parsing based systems [@berant-EtAl:2013:EMNLP; @berant-liang:2014:P14-1; @TACL646; @yih-EtAl:2015:ACL-IJCNLP], information extraction based systems [@yao-vandurme:2014:P14-1; @yao:2015:demos], machine translation based systems [@bao-EtAl:2014:P14-1], embedding based systems [@bordes-chopra-weston:2014:EMNLP2014; @yang-EtAl:2014:EMNLP2014], and QA based system [@DBLP:conf/cikm/BastH15].
[**Evaluation**]{}: Following previous work [@berant-EtAl:2013:EMNLP], we evaluate different systems using the fraction of correctly answered questions. Because golden answers may have multiple values, we use the average F1 score as the main evaluation metric.
Experimental Results
--------------------
Table 8 provides the performance of all base-lines and our method. We can see that:
1. Our method achieved competitive performance: Our system outperforms all baselines and get the best F1-measure of 53.1 on WEBQUESTIONS dataset.
2. Sentence rewriting is a promising technique for semantic parsing: By employing sentence rewriting, our system gains a 3.4% F1 improvement over the base system we used [@TACL646].
3. Compared to all baselines, our system gets the highest precision. This result indicates that our parser can generate more-accurate logical forms by sentence rewriting. Our system also achieves the third highest recall, which is a competitive performance. Interestingly, both the two systems with the highest recall [@DBLP:conf/cikm/BastH15; @yih-EtAl:2015:ACL-IJCNLP] rely on extra-techniques such as entity linking and relation matching.
------------------------------------------------ -- -- --
**System & **Prec. & **Rec. & **F1 (avg)\
Berant et al., 2013 & 48.0 & 41.3 & 35.7\
Yao and Van-Durme, 2014 & 51.7 & 45.8 & 33.0\
Berant and Liang, 2014 & 40.5 & 46.6 & 39.9\
Bao et al., 2014 & – & – & 37.5\
Bordes et al., 2014a & – & – & 39.2\
Yang et al., 2014 & – & – & 41.3\
Bast and Haussmann, 2015 & 49.8 & 60.4 & 49.4\
Yao, 2015 & 52.6 & 54.5 & 44.3\
Berant and Liang, 2015 & 50.5 & 55.7 & 49.7\
Yih et al., 2015 & 52.8 & **60.7** & 52.5\
Our approach & **53.7** & 60.0 & **53.1**\
********
------------------------------------------------ -- -- --
: \[result-table\]The results of our system and recently published systems. The results of other systems are from either original papers or the standard evaluation web.
[**The effectiveness on mismatch problem**]{}. To analyze the commonness of mismatch problem in semantic parsing, we randomly sample 500 questions from the training data and do manually analysis, we found that 12.2% out of the sampled questions have mismatch problems: 3.8% out of them have 1-N mismatch problem and 8.4% out of them have N-1 mismatch problem.
To verify the effectiveness of our method on solving the mismatch problem, we conduct experiments on the 50 mismatch test examples and Table 9 shows the performance. We can see that our system can effectively resolve the mismatch between natural language and target ontology: compared to the base system, our system achieves a significant 54.5% F1 im-provement.
-- -- -- --
-- -- -- --
: \[result-50-table\]The results on the 50 mismatch test dataset.
When scaling a semantic parser to open-domain situation or web situation, the mismatch problem will be more common as the ontology and language complexity increases [@kwiatkowski-EtAl:2013:EMNLP]. Therefore we believe the sentence rewriting method proposed in this paper is an important technique for the scalability of semantic parser.
[**The effect of different rewriting algorithms**]{}. To analyze the contribution of different rewriting methods, we perform experiments using different sentence rewriting methods and the results are presented in Table 10. We can see that:
-- -- -- --
-- -- -- --
: \[result-2032-table\]The results of the base system and our systems on the 2032 test questions.
1. Both sentence rewriting methods improved the parsing performance, they resulted in 1.8% and 3.2% F1 improvements respectively[^1].
2. Compared with the dictionary-based rewriting method, the template-based rewriting method can achieve higher performance improvement. We believe this is because N-1 mismatch problem is more common in the WEBQUESTIONS dataset.
3. The two rewriting methods are good complementary of each other. The semantic parser can achieve a higher performance improvement when using these two rewriting methods together.
[**The effect on improving robustness**]{}. We found that the template-based rewriting method can greatly improve the robustness of the base semantic parser. Specially, the template-based method can rewrite similar sentences into a uniform template, and the (template, predicate) feature can provide additional information to reduce the uncertainty during parsing. For example, using only the uncertain alignments from the words “*people*” and “*speak*” to the two predicates `official_language` and `language_spoken`, the base parser will parse the sentence “*What does jamaican people speak?*” into the incorrect logical form `official_language.jamaican` in our experiments, rather than into the correct form `language_spoken.jamaican` (See the final example in Table 11). By exploiting the alignment from the template “*what language does \$y people speak*” to the predicate , our system can parse the above sentence correctly.
-- --
-- --
: \[examples-table\]Examples which our system generates more accurate logical form than the base semantic parser. **O** is the original sentence; **R** is the generated sentence from sentence rewriting (with the highest score for the model, including rewriting part and parsing part); **LF** is the target logical form.
[**The effect on OOV problem**]{}. We found that the sentence rewriting method can also provide extra profit for solving the OOV problem. Traditionally, if a sentence contains a word which is not covered by the lexicon, it will cannot be correctly parsed. However, with the help of sentence rewriting, we may rewrite the OOV words into the words which are covered by our lexicons. For example, in Table 11 the 3rd question “*What are some of the traditions of islam?*” cannot be correctly parsed as the lexicons don’t cover the word “*tradition*”. Through sentence rewriting, we can generate a new sentence “*What is of the religion of islam?*”, where all words are covered by the lexicons, in this way the sentence can be correctly parsed.
Error Analysis
--------------
To better understand our system, we conduct error analysis on the parse results. Specifically, we randomly choose 100 questions which are not correctly answered by our system. We found that the errors are mainly raised by following four reasons (See Table 12 for detail):
-- -- --
-- -- --
: \[error-table\]The main reasons of parsing errors, the ratio and an example for each reason are also provided.
The first reason is the label issue. The main label issue is incompleteness, i.e., the answers of a question may not be labeled completely. For example, for the question “*Who does nolan ryan play for?*”, our system returns 4 correct teams but the golden answer only contain 2 teams. One another label issue is the error labels. For example, the gold answer of the question “*What state is barack obama from?*” is labeled as “*Illinois*”, however, the correct answer is “*Hawaii*”.
The second reason is the n-ary predicate problem ($n>2$). Currently, it is hard for a parser to conduct the correct logical form of n-ary predicates. For example, the question “*What year did the seahawks win the superbowl?*” describes an n-ary championship event, which gives the championship and the champion of the event, and expects the season. We believe that more research attentions should be given on complicated cases, such as the n-ary predicates parsing.
The third reason is temporal clause. For example, the question “*Who did nasri play for before arsenal?*” contains a temporal clause “*before*”. We found temporal clause is complicated and makes it strenuous for the parser to understand the sentence.
The fourth reason is superlative case, which is a hard problem in semantic parsing. For example, to answer “*What was the name of henry viii first wife?*”, we should choose the first one from a list ordering by time. Unfortunately, it is difficult for the current parser to decide what to be ordered and how to order.
There are also many other miscellaneous error cases, such as spelling error in the question, e.g., “*capitol*” for “*capital*”, “*mary*” for “*marry*”.
Conclusions
===========
In this paper, we present a novel semantic parsing method, which can effectively deal with the mismatch between natural language and target ontology using sentence rewriting. We resolve two common types of mismatch (i) one word in natural language sentence vs one compound formula in target ontology (1-N), (ii) one complicated expression in natural language sentence vs one formula in target ontology (N-1). Then we present two sentence rewriting methods, dictionary-based method for 1-N mismatch and template-based method for N-1 mismatch. The resulting system significantly outperforms the base system on the WEBQUESTIONS dataset.
Currently, our approach only leverages simple sentence rewriting methods. In future work, we will explore more advanced sentence rewriting methods. Furthermore, we also want to employ sentence rewriting techniques for other challenges in semantic parsing, such as the spontaneous, unedited natural language input, etc.
[^1]: Our base system yields a slight drop in accuracy compared to the original system (Berant and Liang, 2015), as we parallelize the learning algorithm, and the order of the data for updating the parameter is different to theirs.
|
---
author:
- 'M. Weżgowiec'
- 'M. Ehle'
- 'R. Beck'
bibliography:
- 'myreferences.bib'
date: 'Received date/Accepted date'
title: 'Hot gas and magnetic arms of : indications for reconnection heating? [^1]'
---
[The grand-design face-on spiral galaxy NGC6946 is remarkable because of its high star formation activity, the massive northern spiral arm, and the magnetic arms, which are observed in polarized radio synchrotron emission and are located between the optical arms and possibly are magnetic reconnection regions. ]{} [ We used electron densities and temperatures in star-forming (active) and less active regions and compared them to findings from the analysis of the radio data to study the energy budget of NGC6946. The hot gas above the magnetic arms between the optical arms might suggest gas heating by reconnection. We also study the population of point sources in NGC6946, including the origin of the puzzling ultra-luminous emission complex MF16. ]{} [ X-ray observations of NGC6946 performed with were used to study the emission from X-ray point sources and diffuse hot gas, including the magnetic arms and the halo. Spectral fitting of the diffuse X-ray emission allowed us to derive temperatures of the hot gas. With assumptions about the emission volume, this allowed us to estimate gas densities, masses, and cooling times. ]{} [ To explain the X-ray emission from the spiral arms of NGC6946 two-temperature plasma models are needed to account for the disk and halo emission. The interarm regions show only one thermal component. We observe that the temperature of the hot gas in and above the magnetic arm regions increases slightly when compared to the average temperatures in the areas in and above the spiral arms. For the southwestern part of the disk, which is depolarized in the radio range by Faraday rotation, we find more efficient mixing of disk and halo gas. ]{} [ We propose magnetic reconnection in the magnetic arm regions of NGC6946 as the possible cause of the additional heating of the gas and ordering of the magnetic fields. In the southwestern part of the galactic disk we observed indications of a possible faster outflow of the hot gas. A very hot gas within the MF16 nebula possibly suggests shock heating by a supernova explosion. ]{}
Introduction
============
NGC6946 (see Table \[astrdat\]) is a Scd spiral galaxy seen face-on, which is listed in Arp’s atlas of peculiar galaxies because of its massive northern spiral arm. Its large optical diameter and low inclination make NGC6946 one of the most prominent grand-design spiral galaxies in the sky. It is known to host a bright starburst nucleus [see e.g. @telesco]. Detections of CO radio emission lines [@nieten; @walsh] gave further evidence for the high star-forming activity of NGC6946 (especially in the northern spiral arm), proved that molecular clouds containing warm and dense gas are distributed throughout the inner disk, and showed that the total molecular gas mass is exceptionally high for a spiral galaxy. The distributions of the emission from NGC6946 in various spectral ranges were analysed with wavelet functions [see @frick].
-------------------- -------------------------------------- -- -- -- --
Morphological type SABc
Inclination 18
Diameter D$_{25}$ 115
R.A.$_{2000}$ 20$^{\rm h}$34$^{\rm m}$53$^{\rm s}$
Dec$_{2000}$ +600913
Distance 7Mpc
-------------------- -------------------------------------- -- -- -- --
: \[astrdat\]Basic astronomical properties of NGC6946
Previous radio continuum findings
---------------------------------
NGC6946 has been thoroughly studied in the radio regime in total and polarized emission [@beck91; @ehleN6946]. Radio polarization observations at 18cm and 20cm wavelengths revealed excess Faraday rotation and strong depolarization in the SW quadrant of NGC6946 that is probably due to a large-scale magnetic field along the line of sight, oriented approximately perpendicular to the disk plane [@beck91]. Analysis of polarization data at four frequencies also suggested strong vertical fields extending far above the disk [@beck07]. Such field lines should enable an outflow of hot gas into a halo. As the SW quadrant of NGC6946 is a region of relatively low star-formation activity, this phenomenon resembles a [**]{} on the Sun.
The average energy density of the warm ionized medium in the interstellar medium (ISM) of the inner disk of NGC6946 was found to be lower by a factor of about 10 than the energy densities of the magnetic field and that of the cosmic rays, resulting in the conclusion that the magnetic field dominates thermal processes in the disk and halos of galaxies [@beck04]. A significant fraction of the diffuse ISM must be unstable, giving rise to gas outflows.
Radio polarization data at $\lambda6$ cm led to the discovery of a new phenomenon: highly aligned magnetic fields that are concentrated in two main spiral features, located almost precisely [**]{} the optical spiral arms of NGC6946 [@beckhoernes their Fig. 1]. No enhanced densities of molecular, neutral, or warm ionized gas have been detected at the positions of these magnetic arms. However, radio observations revealed significant Faraday rotation in these regions so that some ionized gas must be present. @frick analysed the magnetic and optical spiral arms in NGC6946 using 1D wavelet transformations and found that each magnetic arm is similar to the [**]{} optical arm and hence can be regarded as a phase-shifted image.
Rudimentary magnetic arms were also found in other spiral galaxies [@beck15], but NGC6946 still is the most prominent example.
The magnetic arms contradict density-wave models, which predict enhanced ordered magnetic fields at the inner edges of the arms. Several mechanisms were proposed to explain the magnetic arms. For example, the continuous injection and amplification of turbulent fields by supernova shock fronts may suppress the mean-field dynamo in the material arms [@moss13; @moss15]. Alternatively, the introduction of a relaxation time of the magnetic response in the dynamo equation may lead to a phase shift between the material and magnetic spiral arms [@chamandy13a; @chamandy13b]. The mean-field dynamo in the material arms might also be suppressed by outflows driven by star formation [@chamandy15].
We propose here that the strongly polarized radio emission from the magnetic arms may also suggest the existence of reconnection regions where cosmic rays are accelerated. The gas heated by the same process should be detectable in the X-ray domain [@lesch; @hanasz]. By comparing the properties of the hot gas in the magnetic and spiral arms, we may be able to trace a possible additional heating of the gas that would be caused by the reconnection processes.
Earlier X-ray observations
--------------------------
Despite the rather high $N_{\rm H}$-value of about $\rm 2\times10^{21}$ cm$^{-2}$ (see Tab. \[obsred\]), NGC6946 was detected by [*ROSAT*]{}’s All Sky survey with a count rate of 0.1 cts s$^{-1}$. Analysing the 36 ks PSPC pointed observation, @schlegel94a [@schlegel94b; @schlegel94c] apart from SN 1980K additionally reported emission from nine point-like X-ray sources and diffuse emission from NGC 6946. The brightest source (identified at that time with a very luminous supernova remnant MF16) has a count rate of 0.07 cts s$^{-1}$ corresponding to a flux of $8.3\times 10^{-13}$ in the energy range 0.5-2 keV, three sources are fainter by about a factor of 10, and the rest (fainter by a factor of 40) are at the detection threshold.
@holt studied discrete X-ray sources in NGC6946 using a 60 ks ACIS observation and found the source population dominated by high-mass X-ray binaries. Their survey was complete down to approximately $10^{37}$ . However, in contrast to previous results, the ultra-luminous MF16 complex was found to be deficient in line emission expected from an interaction with a dense surrounding medium. Its spectrum lacks pronounced spectral lines and can be fit with a variety of models that are all associated with unusually high luminosities, leaving the origin of the MF16 related X-ray emission unknown. @schlegel03 used the same observations and the luminosity function derived by @holt to distinguish point sources from the diffuse emission. These authors estimated that as much as 10% of the total soft X-ray emission could be due to a hot diffuse component.
We checked the archive and found additional ACIS-S pointings: one on-axis pointing aiming at SN2002HH (30 ks, IAU Circ. 8024 and Roberts & Colbert 2003), and three ($30$ ks each) off-axis pointings centred on SN2004et. The combination of these observations was used by severeal authors [@soria08; @fridriksson08; @kajava09; @liu11] to investigate individual sources (e.g. supernovae, ultra-luminous sources) in NGC6946 and their spectral and temporal variability. Recent observations (20 ks, PI Kochanek) were also aimed at studying point-source populations, which is beyond the scope of this paper.
Immediate objectives
--------------------
This paper focuses on a detailed analysis of the extended emission from the hot gas of NGC6946 with the use of the data acquired by the XMM-Newton X-ray telescope [@jansen]. The parameters of the hot gas acquired from the spectral analysis of selected regions of the galaxy are compared with the properties of the radio emission, especially with its polarized component that traces the structure of the magnetic field of the galaxy.
In the following section (Sect. \[obsred\]) details of the data reduction and analysis are presented. Section \[results\] presents the distribution of the X-ray emission from both diffuse gas and point-source populations, and we also describe the spectra we obtained. In Sect. \[disc\] we thoroughly discuss the results, including correlations and comparisons with the polarized radio emission from NGC6946. We also provide new insight into the nature of the ultra-luminous source MF16. We conclude in Sect. \[cons\].
Observations and data reduction {#obsred}
===============================
---------------------------- ------------ -- -- -- --
Obs ID 0200670301
0500730101
0500730201
0691570101
column density $N_{\rm H}$ 1.84
MOS filter medium
MOS obs. mode FF
pn filter medium
pn obs. mode FF
Total/clean pn time \[ks\] 13.1/8.3
28.4/20.2
33.3/29.7
114.3/98.2
---------------------------- ------------ -- -- -- --
: \[xdat\]Characteristics of the XMM-Newton X-ray observations of NGC6946
NGC6946 has been observed 11 times between 2003 and 2006 with the XMM-Newton telescope [@jansen], but the observations always suffered from heavy high flaring radiation. The only observation that yielded any clean data was made on 13 June 2004 (ObsID 0200670301). Consequently, the galaxy was observed again, on 2 and 8 November 2007 (ObsIDs 0500730201 and 0500730101, respectively, see Table \[xdat\]). The data were still affected by high flaring radiation, but this time it was possible to obtain many good-quality data. The effect of frequent flaring radiation on the observations was caused by visibility constraints that required NGC6946 to be observed at the end of an XMM-Newton revolution. In recent years the orbit has evolved and NGC6946 can be much better observed. Since early 2012 it has become possible to observe this galaxy for almost a full orbit (144ks). The most recent observations, performed between 21 and 23 of December 2012 and aimed at the ultra-luminous source NGC6946 X-1 (ObsID 0691570101), provided a long exposure that was relatively free of high flaring radiation; this resulted in 98ks of good data.
The data were processed using the SAS 13.0.0 package [@sas] with standard reduction procedures. Following the routine of tasks $epchain$ and $emchain$, event lists for two EPIC-MOS cameras [@turner] and the EPIC-pn camera [@strueder] were obtained. Next, the event lists were carefully filtered for periods of intense radiation of high-energy background by creating light curves of high-energy emission. These light curves were used to produce good time interval (GTI) tables, which mark times of low count rates of high-energy emission. Such tables (time ranges) were then used to remove the remaining data when high count rates were observed. The resulting lists were checked for the residual existence of soft proton flare contamination, which could influence the faint extended emission. To do that, we used a script[^2] that performs calculations developed by @spcheck. We found that only the shortest observation (ObsID 0200670301) is contaminated very slightly by soft proton radiation. To ensure the best-quality data (crucial to analyse diffuse emission), we only used events with FLAG=0 and PATTERN$\leq$4 (EPIC-pn) or FLAG=0 and PATTERN$\leq$12 (EPIC-MOS) in the following data processing.
The filtered event lists were used to produce images, background images, exposure maps (without vignetting correction), masked for an acceptable detector area using the images script[^3], modified by the authors to allow adaptive smoothing. All images and maps were produced (with exposure correction) in four energy bands of 0.2 - 1 keV, 1 - 2 keV, 2 - 4.5 keV, and 4.5 - 12 keV. The images were then combined into final EPIC images and adaptively smoothed with a maximum smoothing scale of 30$\arcsec$ FWHM. The rms values were obtained by averaging the emission over a large source-free area in the final map.
Another set of images was also constructed after excluding all point sources found within the D$_{25}$ diameter of NGC6946 from the event lists (see below for details on point source exclusion). This was done with the help of a routine used to create re-filled blank sky background maps - [*ghostholes\_ind*]{}[^4]. In this way we obtained a map of diffuse emission where all regions of excluded point sources are filled with emission close to extracted regions by sampling adjacent events and randomising spatial coordinates[^5]. Although this method is used to handle background maps, we obtained good results when we applied it to real source data. Section \[diffuse\] presents images of soft (0.2-1keV) and medium (1-2keV) emission, together with a corresponding hardness ratio map, defined as $$HR=\frac{{\rm med}-{\rm soft}}{{\rm med}+{\rm soft}},$$ for images with and without detected point sources.
Next, the spectral analysis was performed. To create spectra we only used the event list from the EPIC-pn camera because it offers the highest sensitivity in the soft energy band. Only the emission above 0.3keV was analysed because the internal noise of the pn camera is too high below this limit[^6]. Although this is not crucial when combined with MOS cameras to produce images, it is important to exclude the softest emission below 0.3keV to obtain reliable good-quality spectra.
Unsmoothed images for all bands were used to search for point sources with the standard SAS [*edetect\_chain*]{} procedure. Regions found to include a possible point source were marked. The area was individually chosen for each source to ensure that we excluded all pixels brighter than the surrounding background. These areas were then used to construct spectral regions for which spectra were acquired.
The non-default way of excluding the detected point sources helped to keep more diffuse emission in the final spectra. However, expecting some contribution from the PSF wings, we added a power-law component to our model fits to account for any residual emission. A power-law component was also needed to account for unresolved point sources. The background spectra were obtained using blank sky event lists [see @carter]. These blank sky event lists were filtered using the same procedures as for the source event lists. For each spectrum we produced response matrices and effective area files. For the latter, detector maps needed for extended emission analysis were also created. The spectra were binned, which resulted in a better signal-to-noise ratio. To obtain a reasonable number of bins at the same time, we chose to have 25 total counts per energy bin. The spectra were fitted using XSPEC 11 [@xspec].
Since observations 0500730201 and 0500730101 have identical pointings and position angles, we merged the cleaned event lists using the SAS task [*merge*]{}. The spectra extracted from this merged lists give the same model-fitting results as separate spectra fitted simultaneously, but the errors are better constrained, therefore we used the former spectra in our final analysis. It was not possible to also merge-in the shortest observation (ObsID 0200670301) because it has different parameters (pointing and position angle), and simultaneous fitting with the larger data set showed that both spectra are systematically offset. This resulted in a poorer model fit. We assume that this might arise because the observation was significantly affected by high flaring radiation and because the filtered ”clean” data still show residual contamination that might influence the spectral fitting. Although we used this observation for image production, we therefore excluded the pn data from our spectral analysis. For the same inconsistency reasons (different pointing and position angle), we used the longest observation 0691570101 separately when performing the spectral analysis. This approach resulted in two corresponding spectra for each of the studied regions. Each pair of spectra was merged (as well as their corresponding background spectra) using the SAS task $epicspeccombine$. Although for multiple spectra a most commonly advised routine is a simultaneous fit rather than a fit to a combined spectrum, we note that for spectra with a very different sensitivity (certainly in our case), a combination of spectra leads to a much better handle on the background and consequently a better fit. This is because background subtraction only takes place for the merged spectrum after the source and background spectra are combined (contrary to a simultaneous fit, where each spectrum is background subtracted before the fit).
For the overlays we also used the XMM-Newton Optical Monitor data acquired during the same observations and produced an image in the UVM2 filter using the standard SAS [*omchain*]{} procedure.
Results
=======
Distribution of the X-ray emission {#dist}
----------------------------------
### Diffuse emission {#diffuse}
NGC6946 shows soft extended X-ray emission corresponding to the entire star-forming disk (Fig. \[6946xsoft\]), with the brightest emission closely following the star-forming regions. Although the southern part of the star-forming disk is less pronounced than the northern one, no asymmetries of the emission from the hot gas are visible. In contrast, the X-ray emission seems to extend farther out beyond star-forming regions in the southern part of the galaxy. An area of diffuse emission around the galactic centre forms a structure resembling a very small bar that crosses the central core and is aligned with the H$\alpha$ emission.
The hot gas disk visible in the 1-2keV energy band (Fig. \[6946xmedium\]) is extended in a similar way as the emission in the softer energy band (Fig. \[6946xsoft\]). This may suggest large amounts of very hot gas in the galactic disk and/or halo. To further investigate the contribution from the hottest gas to the X-ray emission from NGC6946, we produced a hardness ratio map using both distributions (Fig. \[6946hr\]).
As mentioned before, the two distribution are similar on average because the values in most parts of the HR map are close to 0. Nevertheless, north-east and south-west of the centre, distinct areas of softer emission are clearly visible. This corresponds well to the orientation of the bright star-forming regions visible in the H$\alpha$ map (Fig. \[6946xsoft\]). However, the softest emission is produced in the south-western part of the disk, where the production of young massive stars is diminished, as seen in the UV map (Fig. \[6946xmedium\]).
### Point sources {#pointdist}
Figure \[6946points\] shows all detected point sources within the D$_{25}$ disk of NGC6946. For all sources we performed a spectral analysis. For weak sources the hardness ratios (HRs) were derived (see Table \[6946sources\]). We used the same energy bands as in @pietsch04: (0.2-0.5)keV, (0.5-1.0)keV, (1.0-2.0)keV, (2.0-4.5)keV, and (4.5-12)keV as bands 1 to 5. Consequently, the hardness ratios are calculated as $HR_i=B_{i+1}-B_i/B_{i+1}+B_i$ for $i$ = 1 to 4, where $B_i$ is the count rate in band $i$, as defined above. For sources with more than 500 net counts in the total energy band (0.2-12keV), spectra were extracted and fitted with models (see Sect. \[sourcespec\]).
------------ ------------------------------------- --------------------------- ------------ -------------- --------------- --------------- -------------- -------------- --------------- -------------- --------------- --------------- ---------------
ID $\textstyle\alpha_{2000}$ $\textstyle\delta_{2000}$ R B$_1$ B$_2$ B$_3$ B$_4$ B$_5$ C$_t$ HR$_1$ HR$_2$ HR$_3$ HR$_4$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)
1 20$^{\rm h}$34$^{\rm m}$214 +6010152 17 5 55 17 34 29 140 0.83 -0.53 0.33 -0.08
2 20$^{\rm h}$34$^{\rm m}$249 +6010356 15 7 87 102 87 86 369 0.85 0.08 -0.08 -0.01
[**3**]{} [**20$^{\rm h}$34$^{\rm m}$259**]{} [**+6009070**]{} [**19**]{} [**52**]{} [**349**]{} [**577**]{} [**322**]{} [**68**]{} [**1368**]{} [**0.74**]{} [**0.25**]{} [**-0.28**]{} [**-0.65**]{}
4 20$^{\rm h}$34$^{\rm m}$266 +6011139 13 22 17 9 0 0 48 -0.13 -0.31 -1 n/a
[**5**]{} [**20$^{\rm h}$34$^{\rm m}$344**]{} [**+6010312**]{} [**16**]{} [**18**]{} [**325**]{} [**262**]{} [**45**]{} [**2**]{} [**652**]{} [**0.9**]{} [**-0.11**]{} [**-0.71**]{} [**-0.91**]{}
[**6**]{} [**20$^{\rm h}$34$^{\rm m}$365**]{} [**+6009305**]{} [**21**]{} [**160**]{} [**723**]{} [**520**]{} [**79**]{} [**42**]{} [**1524**]{} [**0.64**]{} [**-0.16**]{} [**-0.74**]{} [**-0.31**]{}
7 20$^{\rm h}$34$^{\rm m}$378 +6004331 20 13 78 163 108 65 427 0.71 0.35 -0.2 -0.25
8 20$^{\rm h}$34$^{\rm m}$401 +6008233 14 0 111 102 65 28 306 1 -0.04 -0.22 -0.4
9 20$^{\rm h}$34$^{\rm m}$448 +6007180 15 31 192 94 19 19 355 0.72 -0.34 -0.66 0
10 20$^{\rm h}$34$^{\rm m}$466 +6013251 15 71 52 62 69 69 323 -0.15 0.09 0.05 0
[**11**]{} [**20$^{\rm h}$34$^{\rm m}$480**]{} [**+6009322**]{} [**16**]{} [**97**]{} [**583**]{} [**690**]{} [**252**]{} [**80**]{} [**1702**]{} [**0.71**]{} [**0.08**]{} [**-0.46**]{} [**-0.52**]{}
[**12**]{} [**20$^{\rm h}$34$^{\rm m}$488**]{} [**+6008101**]{} [**17**]{} [**42**]{} [**312**]{} [**441**]{} [**203**]{} [**79**]{} [**1077**]{} [**0.76**]{} [**0.17**]{} [**-0.37**]{} [**-0.44**]{}
13 20$^{\rm h}$34$^{\rm m}$490 +6012170 18 6 96 190 101 47 440 0.89 0.33 -0.31 -0.36
[**14**]{} [**20$^{\rm h}$34$^{\rm m}$491**]{} [**+6005538**]{} [**19**]{} [**106**]{} [**599**]{} [**194**]{} [**21**]{} [**9**]{} [**929**]{} [**0.7**]{} [**-0.51**]{} [**-0.8**]{} [**-0.39**]{}
15 20$^{\rm h}$34$^{\rm m}$494 +6013419 11 0 21 24 30 22 97 1 0.07 0.11 -0.15
[**16**]{} [**20$^{\rm h}$34$^{\rm m}$504**]{} [**+6010170**]{} [**18**]{} [**79**]{} [**608**]{} [**470**]{} [**177**]{} [**70**]{} [**1404**]{} [**0.77**]{} [**-0.13**]{} [**-0.45**]{} [**-0.43**]{}
[**17**]{} [**20$^{\rm h}$34$^{\rm m}$514**]{} [**+6010437**]{} [**17**]{} [**123**]{} [**871**]{} [**1468**]{} [**827**]{} [**201**]{} [**3490**]{} [**0.75**]{} [**0.26**]{} [**-0.28**]{} [**-0.61**]{}
18 20$^{\rm h}$34$^{\rm m}$518 +6012486 11 2 31 45 7 7 92 0.88 0.18 -0.73 0
[**19**]{} [**20$^{\rm h}$34$^{\rm m}$524**]{} [**+6009106**]{} [**23**]{} [**269**]{} [**2545**]{} [**4542**]{} [**2061**]{} [**499**]{} [**9916**]{} [**0.81**]{} [**0.28**]{} [**-0.38**]{} [**-0.61**]{}
[**20**]{} [**20$^{\rm h}$34$^{\rm m}$535**]{} [**+6010548**]{} [**16**]{} [**66**]{} [**411**]{} [**465**]{} [**233**]{} [**59**]{} [**1234**]{} [**0.72**]{} [**0.06**]{} [**-0.33**]{} [**-0.6**]{}
[**21**]{} [**20$^{\rm h}$34$^{\rm m}$567**]{} [**+6008326**]{} [**20**]{} [**84**]{} [**651**]{} [**809**]{} [**529**]{} [**279**]{} [**2352**]{} [**0.77**]{} [**0.11**]{} [**-0.21**]{} [**-0.31**]{}
22 20$^{\rm h}$34$^{\rm m}$571 +6005326 17 6 68 82 18 38 212 0.84 0.09 -0.64 0.36
[**23**]{} [**20$^{\rm h}$34$^{\rm m}$578**]{} [**+6009480**]{} [**14**]{} [**33**]{} [**446**]{} [**1004**]{} [**475**]{} [**58**]{} [**2016**]{} [**0.86**]{} [**0.38**]{} [**-0.36**]{} [**-0.78**]{}
[**24**]{} [**20$^{\rm h}$35$^{\rm m}$002**]{} [**+6009080**]{} [**24**]{} [**812**]{} [**4413**]{} [**3982**]{} [**688**]{} [**99**]{} [**9994**]{} [**0.69**]{} [**-0.05**]{} [**-0.71**]{} [**-0.75**]{}
25 20$^{\rm h}$35$^{\rm m}$007 +6012508 15 0 34 53 31 7 125 1 0.22 -0.26 -0.63
[**26**]{} [**20$^{\rm h}$35$^{\rm m}$008**]{} [**+6011299**]{} [**38**]{} [**3684**]{} [**16674**]{} [**17157**]{} [**5547**]{} [**1588**]{} [**44650**]{} [**0.64**]{} [**0.01**]{} [**-0.51**]{} [**-0.55**]{}
[**27**]{} [**20$^{\rm h}$35$^{\rm m}$012**]{} [**+6010094**]{} [**18**]{} [**87**]{} [**614**]{} [**1017**]{} [**638**]{} [**263**]{} [**2619**]{} [**0.75**]{} [**0.25**]{} [**-0.23**]{} [**-0.42**]{}
28 20$^{\rm h}$35$^{\rm m}$054 +6006214 16 7 50 160 115 63 395 0.75 0.52 -0.16 -0.29
29 20$^{\rm h}$35$^{\rm m}$115 +6008578 15 33 194 133 13 2 375 0.71 -0.19 -0.82 -0.73
30 20$^{\rm h}$35$^{\rm m}$127 +6007314 16 0 55 83 53 9 200 1 0.2 -0.22 -0.71
31 20$^{\rm h}$35$^{\rm m}$140 +6005210 16 15 43 74 55 34 221 0.49 0.26 -0.15 -0.24
[**32**]{} [**20$^{\rm h}$35$^{\rm m}$187**]{} [**+6010555**]{} [**18**]{} [**32**]{} [**183**]{} [**767**]{} [**445**]{} [**114**]{} [**1541**]{} [**0.7**]{} [**0.61**]{} [**-0.27**]{} [**-0.59**]{}
33 20$^{\rm h}$35$^{\rm m}$255 +6009543 18 9 221 79 7 2 318 0.92 -0.47 -0.84 -0.6
34 20$^{\rm h}$35$^{\rm m}$281 +6009205 14 3 221 79 7 0 310 0.97 -0.47 -0.84 -1
35 20$^{\rm h}$35$^{\rm m}$324 +6010179 19 18 68 50 48 11 195 0.57 -0.15 -0.02 -0.63
------------ ------------------------------------- --------------------------- ------------ -------------- --------------- --------------- -------------- -------------- --------------- -------------- --------------- --------------- ---------------
The left panel of Fig. \[6946points\] suggests that some of the sources might not originate from within the galaxy and are instead more distant background objects. Moreover, the centre of the most extended region of source 26 does not seem to coincide with the visible H$\alpha$ clump, but more precise astrometry of the Chandra observation associated this source with the galaxy [e.g. @kajava09]. This source is the brightest point source – the ultra-luminous complex MF16 that is assumed to be a supernova remnant [e.g. @matonick]. It is not a pure point source, as it is considerably larger than the point spread function of the XMM-Newton EPIC cameras (of $\simeq$ 12).
Another interesting source is source 17, overlapping in Fig. \[6946points\] with sources 16 and 20. This source was not visible in previous observations (where sources 16 and 20 were detected), but only in the most recent sensitive observations.
Spectral analysis of the X-ray emission {#spectra}
---------------------------------------
For spectra of the hot gas we used a model that attributed one or two thermal plasmas and/or a contribution from unresolved point-like sources. Thermal plasma is represented in this work by a [*mekal*]{} model, which is a model of an emission spectrum from hot diffuse gas based on the model calculations of Mewe and Kaastra [@mewe; @kaastra]. In all models we fixed the metallicity to solar. This was determined by a test spectral fitting of the spectra, which showed that in a wide range of metallicities (0.1 - 1.5 solar), no significant change in gas temperatures or power-law photon indices could be observed. Slight deviations of these parameters were still within the errors provided by the models that used solar metallicity. This suggests that all differences that are found in gas temperatures when fitting spectra do not result from a possible combined effect of abundance gradients and enforcing solar metallicities in the models. It also shows that an additional power-law component in the models of diffuse emission does account for emission from unresolved point-like sources and is not introduced to mimic variable gas abundance because excluding the power-law component from the model and again varying metallicities resulted in unphysical values of the parameters. Consequently, the quality of the model fits remained very low (with reduced $\chi^2 > 2$).
A contribution from unresolved point-like sources is fitted with a simple power law. For some models an additional component to account for the internal absorption needed to be used. For all models we also used a fixed foreground (galactic) absorption. Tables 4 and 8 present an overview of the various spectral models, including the obtained parameters. The errors provided for the model parameters are always 1-$\sigma$ errors. The derived X-ray fluxes and luminosities are shown in Tables \[6946xfs\] and \[6946xfr\].
The same model components were used for the spectral analysis of the brightest point sources (see Sect. \[pointdist\]). Since we did not aim to perform a variability analysis of point sources, we used all data that were suitable for spectral analysis (for the selection and merging of separate data sets, see Sect. \[obsred\]), similarly as for regions of diffuse emission.
The regions of the diffuse X-ray emission from which the spectra were extracted are presented in Fig. \[6946xregs\]. Extraction areas for the brightest point-sources are the same as found by the source-detection analysis (cf. Fig. \[6946points\]). All plots of the modelled spectra together with the fitted models are presented in Figs. \[6946mod1\] and \[6946mod2\].
### Brightest sources {#sourcespec}
For almost all sources (except for source 27) for which spectral fitting was performed, we also used the thermal plasma component (see Table \[6946xtabs\]) to account for diffuse emission projected onto each of the sources. The amount of this emission was often significant as a result of the sensitive observations of a gas-rich galaxy with a relatively large point spread function of the EPIC-pn camera that did not allow clearly separating the studied sources. Many of the spectra also required an additional model component to account for the high internal absorption.
For one source (source 5) a model with a power-law component did not give good results. The spectrum was fitted best when using two thermal components. It is possible that it is simply a hot gas clump that was classified by the source detection routine to be a point source. Alternatively, the lowest count number of all fitted spectra might also be the reason for difficulties in finding a proper fit. The hot component fitted in this model reaches values of almost 2keV, which is fairly unexpected for the hot gas in the galactic disk.
For source 17, which was undetected in earlier observations, a complex model had to be fitted. Its power-law component is poorly constrained, however. The two sources near this position that were detected previously, sources 16 and 20, were not detected in the recent sensitive observation because the brightness of the diffuse emission was higher throughout, and also because the bright source 17 has wings. For consistency of the results, however, we used all observations to extract spectra for all three sources. Nevertheless, the complexity of the model for source 17 might be caused by admixtures from the two adjacent sources.
The spectrum of the core region, source 19, where the densest and the hottest gas might be expected, could be fitted equally well with either one or two thermal components. We present both model fits. The model with two components yielded 0.32$^{+0.02}_{-0.03}$keV and 0.70$^{+0.07}_{-0.06}$keV, while a single temperature component showed a temperature of 0.62$\pm$0.04keV. These values and the derived fluxes for this region (see Table \[6946xfs\]) suggest that most of the emission originates from the hotter component. The main difference between both model fits, however, is a significant variation of the internal absorption (see Table \[6946xtabs\]).
By fitting a model (consisting of one power-law and two thermal components) to the spectrum of MF16 (source 26), we obtained a temperature of 1.04$\pm$0.03keV for the component. Still, most of the flux from MF16 originates from the power-law component that is associated with the central source. To further investigate the spectrum of this source, we followed @kajava09 and added a multicolour disk component to our model. We obtained a similar value for the temperature of the inner disk.
---- ------------------------------------ ------------------------ ------------------------ ------------------------ ------------------------ -------------------- --------
ID Model Internal kT$_1$ kT$_2$ Photon $\chi_{\rm red}^2$ Net
type nH \[keV\] \[keV\] Index counts
3 wabs(mekal+wabs\*power law) 4.19$^{+2.07}_{-1.76}$ 0.23$^{+0.04}_{-0.05}$ – 2.62$\pm$0.23 0.94 1368
5 wabs(mekal+mekal) – 0.54$\pm$0.09 1.93$^{+0.65}_{-0.32}$ – 0.80 652
6 wabs(mekal+wabs\*power law) 1.86$^{+0.47}_{-0.44}$ 0.60$^{+0.11}_{-0.35}$ – 3.91$^{+0.61}_{-0.41}$ 1.04 1524
11 wabs(mekal+wabs\*power law) 1.78$^{+1.66}_{-0.54}$ 0.38$^{+0.16}_{-0.08}$ – 2.48$^{+0.48}_{-0.21}$ 0.99 1702
12 wabs(mekal+power law) – 0.58$^{+0.21}_{-0.22}$ – 1.78$\pm$0.12 0.91 1077
14 wabs(mekal+power law) – 0.46$^{+0.06}_{-0.08}$ – 3.12$^{+0.56}_{-0.91}$ 1.12 929
16 wabs(mekal+power law) – 0.38$^{+0.21}_{-0.06}$ – 2.03$^{+0.15}_{-0.13}$ 1.15 1404
17 wabs(mekal+mekal+wabs\*power law) 5.29$^{+2.05}_{-1.33}$ 0.18$\pm$0.02 0.66$\pm$0.06 2.31$^{+0.83}_{-1.04}$ 0.91 3490
19 wabs(mekal+wabs(mekal+power law)) 6.87$^{+1.02}_{-1.21}$ 0.32$^{+0.02}_{-0.03}$ 0.70$^{+0.07}_{-0.06}$ 2.55$^{+0.07}_{-0.03}$ 1.02 9916
19 wabs(mekal+wabs\*power law) 2.89$^{+0.31}_{-0.33}$ 0.62$\pm$0.04 – 2.36$^{+0.03}_{-0.07}$ 1.03 9916
20 wabs(mekal+power law) – 0.62$^{+0.08}_{-0.12}$ – 1.65$^{+0.12}_{-0.11}$ 0.90 1234
21 wabs(mekal+power law) – 0.45$^{+0.08}_{-0.06}$ – 1.52$^{+0.08}_{-0.07}$ 1.12 2352
23 wabs(mekal+wabs\*power law) 3.29$^{+0.81}_{-1.05}$ 0.30$^{+0.21}_{-0.07}$ – 2.52$^{+0.22}_{-0.20}$ 1.10 2016
24 wabs(mekal+wabs\*power law) 2.16$^{+0.17}_{-0.21}$ 0.78$\pm$0.05 – 3.80$\pm$0.15 1.23 9994
26 wabs(mekal+mekal+power law) – 0.40$^{+0.09}_{-0.03}$ 1.04$\pm$0.03 2.26$\pm$0.03 1.19 44650
26 wabs(mekal+mekal+power law+diskbb) – 0.66$\pm$0.06 1.34$^{+0.32}_{-0.22}$ 1.98$\pm$0.08 1.11 44650
27 wabs\*power law – – – 1.68$\pm$0.05 1.05 2619
32 wabs(mekal+wabs\*power law) 8.83$^{+2.15}_{-1.70}$ 0.25$^{+0.06}_{-0.05}$ – 2.82$^{+0.26}_{-0.17}$ 0.80 1541
---- ------------------------------------ ------------------------ ------------------------ ------------------------ ------------------------ -------------------- --------
ID mekal cold mekal hot power law total luminosity
---- ---------------------- ------------------------ ------------------------- ------------------------- ------------------------
3 1.5$^{+0.6}_{-0.8}$ – 18.8$^{+6.3}_{-4.3}$ 20.3$^{+6.8}_{-5.1}$ 1.10$^{+0.37}_{-0.25}$
5 1.0$^{+0.1}_{-0.2}$ 1.1$^{+0.3}_{-0.2}$ – 2.1$\pm$0.4 –
6 2.0$^{+0.7}_{-1.1}$ – 41.1$^{+31.3}_{-14.0}$ 43.1$^{+32.0}_{-15.2}$ 2.41$^{+1.84}_{-0.82}$
11 0.8$^{+0.9}_{-0.4}$ – 9.7$^{+4.9}_{-2.5}$ 10.5$^{+5.8}_{-2.9}$ 0.57$^{+0.29}_{-0.15}$
12 0.3$\pm$0.2 – 5.3$^{+1.3}_{-1.2}$ 5.6$^{+1.5}_{-1.4}$ 0.31$^{+0.08}_{-0.07}$
14 2.0$^{+0.8}_{-0.4}$ – 1.3$^{+1.0}_{-0.6}$ 3.3$^{+1.1}_{-0.7}$ 0.08$^{+0.06}_{-0.04}$
16 1.2$^{+0.5}_{-0.4}$ – 4.3$^{+1.0}_{-0.8}$ 5.5$^{+1.6}_{-1.3}$ 0.25$^{+0.06}_{-0.05}$
17 1.6$^{+0.6}_{-0.5}$ 1.6$\pm$0.6 25.7$^{+5.2}_{-5.8}$ 28.9$^{+6.3}_{-6.9}$ 1.51$^{+0.30}_{-0.34}$
19 4.9$^{+0.6}_{-0.8}$ 26.0$^{+15.6}_{-11.2}$ 69.6$^{+6.5}_{-2.9}$ 100.5$^{+22.8}_{-14.9}$ 4.08$^{+0.38}_{-0.17}$
19 3.6$\pm$0.6 – 58.6$^{+5.7}_{-2.4}$ 62.2$^{+6.3}_{-3.0}$ 3.44$^{+0.33}_{-0.14}$
20 0.8$\pm$0.2 – 5.6$^{+1.6}_{-1.4}$ 6.4$^{+1.7}_{-1.5}$ 0.33$^{+0.09}_{-0.08}$
21 0.9$\pm$0.3 – 11.4$^{+2.1}_{-1.9}$ 12.4$^{+2.3}_{-2.2}$ 0.67$^{+0.12}_{-0.11}$
23 0.6$^{+0.7}_{-0.4}$ – 17.7$^{+5.3}_{-3.3}$ 18.3$^{+5.9}_{-3.8}$ 1.04$^{+0.31}_{-0.19}$
24 4.5$\pm$0.9 – 108.0$^{+16.7}_{-14.9}$ 112.6$^{+17.6}_{-15.9}$ 6.33$^{+0.98}_{-0.87}$
26 10.5$^{+2.5}_{-2.0}$ 14.5$^{+1.7}_{-1.6}$ 128.3$^{+4.2}_{-4.4}$ 153.3$^{+8.4}_{-8.0}$ 7.52$^{+0.25}_{-0.26}$
26 10.2$\pm$2.4 9.4$^{+2.3}_{-1.4}$ 104.1$^{+14.4}_{-16.9}$ 156.2$^{+32.1}_{-21.2}$ 8.03$^{+2.05}_{-1.76}$
27 – – 14.8$^{+1.4}_{-1.3}$ 14.8$^{+1.4}_{-1.3}$ 0.87$^\pm$0.08
32 0.7$\pm$0.3 – 36.3$^{+9.3}_{-6.0}$ 37.0$^{+9.6}_{-6.2}$ 2.13$^{+0.55}_{-0.35}$
### Regions of diffuse emission {#regspec}
To analyse emission of the hot gas from NGC6946, we used H$\alpha$ and UV images to choose areas that correspond to the star-forming regions. The radio morphology of the galaxy was also taken into account because the most prominent polarized features were found [**]{} the gaseous spiral arms of NGC6946 (see Fig. \[6946radio\]). All selected regions are presented in Fig. \[6946xregs\]. A brief description of all regions is presented in Table \[names\]. For transparency, region letters are used throughout this paper. In the process of extracting the spectra, the emission from point sources (Fig. \[6946points\]) was excluded.
-------- ---------------------------------
Region Region
letter description
A south-western arm
B north-western interarm
C south-western interarm
D central region w/o UV emission
E south-eastern arm
F eastern interarm
G northeastern arm
H western arms
I south-eastern interarm
J central region with UV emission
-------- ---------------------------------
To investigate the temperature of the hot gas in selected regions of NGC6946, we fitted a single thermal plasma model, adding a power-law component to account for undetected point sources and/or residual emission from excluded sources. Only regions C and F did not require this additional power-law component. This could simply be caused by a lower number of net counts in their spectra, resulting in a low signal-to-noise ratio, hence lower accuracy of the fitting, so that the basic models are equally good in fitting the data. On the other hand, both spectra show very little emission above 1keV, which suggests that the harder emission from point sources contributes hardly anything.
In Table \[singles\] we present the results of single thermal plasma model fitting. For regions of the spiral galactic arms (regions A, E, G, and H), very steep photon indices of the power-law component are visible, which is unexpected for typical galactic X-ray point sources. This might be due to enhanced emission in the soft-to-medium energy band (around and above 1keV), however, which can also suggest that a second thermal component is necessary to account for the significant emission from the hot gas in the galactic disk. To check this possibility, we fitted a model consisting of two thermal components for the areas of the spiral arms and kept the previously introduced power-law component. Since both parameters and residuals of the new fits were much more physical, that is, showed values in expected ranges, we used them as the final fits for the subsequent analysis.
To ensure that a single thermal plasma model is the best-fit model for the remaining regions (central areas and interarm regions), we also fitted a model with two thermal components to their corresponding spectra. In all cases we obtained significantly flatter (lower) photon indices of the power-law component, which suggests that some of the harder emission from the unresolved point sources was fitted with the new second thermal component. Furthermore, most of the fitted parameters were poorly constrained. For some of them it was impossible to get any constraints. The final models for all regions are presented in Table \[6946xtabr\].
For regions of the spiral arms for which a model with two thermal components was used, we assumed that the cooler component ($\sim$ 0.3keV) of the hot gas is associated with the galactic halo and the hotter component ($\sim$ 0.7keV) with the emission from the disk, as is observed in edge-on galaxies [e.g. @tuellmann06].
For the remaining regions, that is, for the central areas and the interarm regions, we assumed that the single thermal component can be described as a mixture of gas components from the disk and the halo above. The parameters of these components seem to be similar to the level, which does not allow clearly separating them at a given sensitivity level. For the interarm regions this can be easily explained by the lack of star-forming regions in the disk, which causes the gas to be relatively uniform throughout the entire volume. Nevertheless, we cannot use this argument for the central regions of the galaxy, where the disk emission is certainly significant. A reasonable explanation for this case is a high star-forming activity of NGC6946 and the consequent internal absorption that is highest in the central densest part of the galactic disk. This absorption may cause some of the disk emission from the central regions I and J to remain hidden, and the rest, together with the halo gas, mimics a homogenous medium.
-------- ------------------------ ------------------------ ----------
Region kT Photon Reduced
\[keV\] index $\chi^2$
A 0.53$^{+0.05}_{-0.08}$ 2.55$^{+0.26}_{-0.17}$ 1.21
B 0.60$^{+0.06}_{-0.08}$ 1.84$\pm$0.22 1.13
C 0.55$^{+0.10}_{-0.11}$ – 1.15
D 0.49$^{+0.07}_{-0.14}$ 1.78$^{+0.13}_{-0.08}$ 0.99
E 0.53$^{+0.04}_{-0.12}$ 2.31$^{+0.20}_{-0.23}$ 1.17
F 0.53$\pm$0.06 – 1.05
G 0.50$^{+0.02}_{-0.03}$ 2.54$^{+0.05}_{-0.11}$ 1.35
H 0.53$^{+0.08}_{-0.03}$ 2.68$^{+0.45}_{-0.40}$ 1.24
I 0.50$^{+0.14}_{-0.13}$ 1.86$^{+0.29}_{-0.25}$ 1.07
J 0.42$^{+0.09}_{-0.05}$ 1.69$^{+0.19}_{-0.16}$ 1.01
-------- ------------------------ ------------------------ ----------
-------- ----------------------------- ------------------------ ------------------------ ------------------------ -------------------- --------
Region Model kT$_1$ kT$_2$ Photon $\chi_{\rm red}^2$ Net
type \[keV\] \[keV\] Index counts
A wabs(mekal+mekal+power law) 0.28$^{+0.03}_{-0.04}$ 0.73$^{+0.07}_{-0.06}$ 1.82$^{+0.36}_{-0.44}$ 0.96 3512
B wabs(mekal+power law) 0.60$^{+0.06}_{-0.08}$ – 1.84$\pm$0.22 1.13 2516
C wabs\*mekal 0.55$^{+0.10}_{-0.11}$ – – 1.15 385
D wabs(mekal+power law) 0.49$^{+0.07}_{-0.14}$ – 1.78$^{+0.13}_{-0.08}$ 0.99 1581
E wabs(mekal+mekal+power law) 0.30$^{+0.05}_{-0.03}$ 0.80$^{+0.09}_{-0.14}$ 1.72$^{+0.26}_{-0.21}$ 0.99 2552
F wabs\*mekal 0.53$\pm$0.06 – – 1.05 896
G wabs(mekal+mekal+power law) 0.27$\pm$0.02 0.78$\pm$0.06 1.93$^{+0.17}_{-0.19}$ 1.03 8179
H wabs(mekal+mekal+power law) 0.28$^{+0.04}_{-0.03}$ 0.68$^{+0.07}_{-0.05}$ 1.59$^{+0.70}_{-0.88}$ 1.09 5008
I wabs(mekal+power law) 0.50$^{+0.14}_{-0.13}$ – 1.86$^{+0.29}_{-0.25}$ 1.07 1211
J wabs(mekal+power law) 0.42$^{+0.09}_{-0.05}$ – 1.69$^{+0.19}_{-0.16}$ 1.01 2349
-------- ----------------------------- ------------------------ ------------------------ ------------------------ -------------------- --------
Region B shows almost no H$\alpha$ or UV emission, while region J includes the central parts of the galaxy. Both regions differ significantly in the value of the fitted temperature of the hot gas. It is much hotter (0.60$^{+0.06}_{-0.08}$keV) for the quiet region B than for region J with its clear galactic-disk (nuclear) emission (0.42$^{+0.09}_{-0.05}$keV). The remaining single thermal component model fits show that the temperature of the hot gas is between 0.5keV and 0.55keV. Halo components of the model fits to the regions of spiral arms show a very constant temperature of around 0.28keV. For the disk components, however, differences are observed, as the south-western and western part of the disk (spiral arms A and H) have a significantly lower temperature than the eastern and north-eastern part (spiral arms E and G).
For all regions except for region H (western spiral arm) most of the X-ray flux is produced by the power-law component that is attributed to unresolved and/or residual emission from point sources. For the exceptional region H we note, however, that the value of the power-law flux is only poorly constrained. Therefore, its value might be much higher, which would lead to a higher contribution to the total flux. A high contribution from the power-law components of the fits might arise because NGC6946 is a starburst galaxy with a large population of X-ray point sources (Fig. \[6946points\]). The method of source extraction used in this paper, described in Sect. \[obsred\], might certainly cause this effect. However, as argued before, this ensured that as little as possible of the diffuse emission has been lost by point-source extraction.
For the regions of the spiral arms, more flux comes from the halo components according to the fits, which agrees with a physical picture where halo emission is produced in a much larger volume density of the underlying star formation. These fractions are equal only for the western spiral arm (region H); we discuss this in Sect. \[gasparams\] in more detail.
Region mekal cold mekal hot power law total
-------- ---------------------------- ---------------------------- ----------------------------- -----------------------
A 3.5$^{+0.8}_{-0.9}$ (0.32) 2.5$^{+0.8}_{-0.6}$ (0.23) 4.8$^{+5.1}_{-2.0}$ (0.45) 10.8$^{+6.7}_{-3.5}$
B 2.4$^{+0.3}_{-0.5}$ (0.22) – 8.5$^{+4.0}_{-2.5}$ (0.78) 10.9$^{+4.4}_{-2.9}$
C 1.1$^{+0.1}_{-0.2}$ (1.00) – – 1.1$^{+0.1}_{-0.2}$
D 1.4$^{+0.2}_{-0.3}$ (0.25) – 4.2$^{+1.0}_{-1.1}$ (0.75) 5.6$^{+1.2}_{-1.4}$
E 2.6$\pm$0.9 (0.26) 1.6$^{+0.7}_{-0.6}$ (0.16) 5.7$^{+3.5}_{-2.2}$ (0.58) 9.8$^{+5.0}_{-3.74}$
F 2.1$^{+0.1}_{-0.2}$ (1.00) – – 2.1$^{+0.1}_{-0.2}$
G 9.3$^{+1.5}_{-1.4}$ (0.31) 5.2$^{+1.0}_{-1.3}$ (0.18) 15.3$^{+5.4}_{-3.3}$ (0.51) 29.7$^{+7.8}_{-6.0}$
H 6.6$^{+1.9}_{-1.7}$ (0.43) 6.4$^{+1.5}_{-1.8}$ (0.42) 2.3$^{+12.8}_{-1.9}$ (0.15) 15.4$^{+16.0}_{-5.5}$
I 1.2$\pm$0.2 (0.30) – 2.8$^{+1.7}_{-1.1}$ (0.70) 4.0$^{+1.8}_{-1.3}$
J 2.0$\pm$0.5 (0.21) – 7.5$^{+2.6}_{-1.8}$ (0.79) 9.5$^{+3.1}_{-2.3}$
Discussion {#disc}
==========
Point sources {#pts}
-------------
Of the 35 point sources found in the galactic disk of NGC6946, for 19 of them we were able to calculate only the hardness ratios. All values are presented in Table \[6946sources\]. For the remaining sources we fitted models to the acquired spectra (see Sect. \[sourcespec\]). As mentioned above, all sources except for source 27 required an additional thermal component in the model. This was most likely due to the characteristics of the observations with the XMM-Newton telescope, that is, a relatively low resolution with a high sensitivity to diffuse X-ray emission. As a result, all spectra of the studied point sources included significant information from the diffuse gas present across the entire disk of NGC6946.
For several sources, an additional component to account for internal absorption was needed to obtain a good fit (see Table \[6946xtabs\]). For three sources, 19, 23, and 32, this was crucial. For source 19, which is the core region of NGC6946 and therefore not just a point source, this additional absorption is easy to explain because we observe the densest parts of the galaxy. The two other sources, however, are located in areas relatively free of star formation (see Fig. \[6946points\], left). Furthermore, none of the extended regions used for the spectral analysis of the diffuse emission coming from the hot gas needed an additional internal absorption component to be described by the model. Therefore, the argument of (relatively) low-resolution observations of a gas- and dust-rich galaxy cannot be used here. Instead, for each source that required an internal absorption component in the model, we propose that it is an X-ray binary (or an intermediate-mass black hole, IMBH) surrounded by a dust torus.
To calculate luminosities of the spectrally analysed point sources, we only used the fluxes of the power-law component to exclude the contribution from the galactic hot gas in the modelled spectrum (Table \[6946xfs\]). Half of the sources show luminosities higher than 10$^{39}$ergs$^{-1}$. This value is the most widely used observational definition of an ultra-luminous source (ULX); it is often associated with an accreting IMBH that forms in the core collapse of young dense stellar clusters [e.g. @miller]. Since the abundance of ULXs is often linked to recent star formation activity [e.g. @berghea13], a significant number of ULXs in NGC6946 agrees well with the vivid star formation of this galaxy.
One of these sources is the nebula MF16 (region 26) mentioned above. Although some of its thermal emission (from both model components) can be associated with the galactic hot gas, as mentioned above, a significant contribution from the hotter component needs a different explanation. That the gas temperature exceeds 1keV (Table \[6946xtabs\]), which is much higher than the temperature of the disk gas, may provide further evidence that MF16 might be indeed a supernova remnant, as previously claimed by @matonick, and the very high temperature of the hot gas may result from shock heating of the medium surrounding the explosion region. This contradicts the findings of @berghea, who claimed that no signs of shock heating are present.
Extended emission {#diff}
-----------------
### Parameters of the hot gas {#gasparams}
From our spectral model fits we were able to derive more parameters of the hot gas, including electron densities $n_e$, masses $M$, thermal energies $\epsilon_{th}$, and cooling times $\tau$. To perform our calculations we used the model of thermal cooling and ionization equilibrium of @Nulsen, where $L_X=1.11\cdot \Lambda(T)\,n^2_e\,V\,\eta$, $\eta$ is an unknown filling factor and $\Lambda(T)$ is a cooling coefficient of the order of $10^{-22}\,{\rm erg}\,{\rm cm}^3\,s^{-1}$ for temperatures of a few millions K [@Raymond]. However, the main difficulty in calculating the physical parameters of the hot gas component are the assumptions about the emitting volume $V$.
It seems straightforward to assume that we only see soft X-ray emission from the visible side of the disk (and therefore the halo), with all emission from the other side being absorbed by the neutral hydrogen in the galactic disk. For the visible part of the halo emission, we assumed a cylindrical volume above the disk of NGC6946, extending out to half of the $D_{25}$ diameter of the galaxy (10kpc). In this geometry, the halo around NGC6946 would be roughly spherical. This approach seems to be justified for a starbust galaxy that has significantly extended halos of the X-ray emission. For the disk emission we assumed a disk thickness of 1kpc. For the interarm regions, for which a single thermal model was fitted, we used volumes of 10kpc times the area of a region, as the model describes the mixed emission from both the disk and the halo (see Sect. \[regspec\]).
Tables \[halogas\] and \[diskgas\] present the derived parameters of the hot gas in the areas of diffuse emission in NGC6946.
To verify the obtained values, a comparison with earlier Chandra observations of NGC6946 by @schlegel03 would be an important step. Because of the low sensitivity to diffuse emission of these observations, the analysis was unfortunately made only for the entire galactic disk and no detailed study is available. Still, their temperatures of 0.25$\pm$0.03keV and 0.70$\pm$0.10keV for two thermal components agree very well with our values for the halo and the disk gas temperatures, respectively. The electron density of $\sim$0.012 ($\times\,\eta^{-0.5}$)cm$^{-3}$ derived by @schlegel03 for the 1kpc thick disk also matches our results well if we consider a ten times larger volume than used for our calculations. A similar study of six other nearby face-on late-type spiral galaxies was reported by @owen, who obtained temperatures of two thermal model fits of 0.2-0.3keV and 0.6-0.7keV and the derived electron densities of the gas of the order of a few $10^{-3}$ ($\times\,\eta^{-0.5}$)cm$^{-3}$, depending on the level of the star-forming activity.
### Hot gas components {#components}
As mentioned before, for the regions of the spiral arms we needed a two-temperature model to account for the emission from the hot gas residing in both galactic disk and the surrounding halo. Although the temperatures of the hot gas are almost identical for the halo components and similar for the disk components (where two pairs could be distinguished - A with H and E with G - see Table \[6946xtabr\]), the derived parameters show significant differences (see Table \[diskgas\]). As expected, for the north-eastern spiral arm with the brightest H$\alpha$ and UV emission, marked as region G, we obtained the highest values of number density and energy density of all spiral arms.
-------- ------------------------- ------------------------------ ----------------------------- ------------------------------------- ---------------------------
Region (n$_{halo}\eta^{-0.5}$) (M$^{halo}_{gas}\eta^{0.5}$) (E$^{halo}_{th}\eta^{0.5}$) ($\epsilon^{halo}_{th}\eta^{-0.5}$) ($\tau^{halo}\eta^{0.5}$)
\[10$^{-3}$cm$^{-3}$\] \[10$^6$M$_\odot$\] \[10$^{54}$erg\] \[10$^{-12}$ergcm$^{-3}$\] \[Myr\]
B 0.84$\pm$0.07 5.09$^{+0.37}_{-0.47}$ 8.74$^{+1.58}_{-1.87}$ 1.22$^{+0.22}_{-0.26}$ 1969$^{+98}_{-13}$
C 0.72$^{+0.07}_{-0.06}$ 2.66$^{+0.26}_{-0.24}$ 4.20$^{+1.24}_{-1.14}$ 0.96$^{+0.28}_{-0.26}$ 2063$^{+388}_{-277}$
D 2.03$^{+0.15}_{-0.12}$ 1.22$^{+0.09}_{-0.07}$ 1.71$^{+0.39}_{-0.56}$ 2.40$^{+0.54}_{-0.79}$ 662$^{+48}_{-96}$
F 1.22$\pm$0.06 3.08$^{+0.15}_{-0.16}$ 4.67$^{+0.79}_{-0.74}$ 1.56$^{+0.26}_{-0.25}$ 1202$^{+139}_{-84}$
I 1.22$^{+0.11}_{-0.12}$ 1.86$^{+0.16}_{-0.18}$ 2.66$^{+1.05}_{-0.88}$ 1.47$^{+0.58}_{-0.49}$ 1107$^{+325}_{-146}$
J 2.63$^{+0.22}_{-0.28}$ 1.48$^{+0.12}_{-0.16}$ 1.78$^{+0.56}_{-0.38}$ 2.66$^{+0.84}_{-0.57}$ 480$^{+25}_{-24}$
-------- ------------------------- ------------------------------ ----------------------------- ------------------------------------- ---------------------------
------------ ------------------------- ------------------------------ ----------------------------- ------------------------------------- ---------------------------
Region (n$_{disk}\eta^{-0.5}$) (M$^{disk}_{gas}\eta^{0.5}$) (E$^{disk}_{th}\eta^{0.5}$) ($\epsilon^{disk}_{th}\eta^{-0.5}$) ($\tau^{halo}\eta^{0.5}$)
\[10$^{-3}$cm$^{-3}$\] \[10$^6$M$_\odot$\] \[10$^{54}$erg\] \[10$^{-12}$ergcm$^{-3}$\] \[Myr\]
A$_{disk}$ 4.13$^{+0.52}_{-0.40}$ 1.14$^{+0.15}_{-0.11}$ 2.39$^{+0.56}_{-0.41}$ 7.25$^{+1.69}_{-1.24}$ 517$^{+47}_{-34}$
A$_{halo}$ 1.69$^{+0.15}_{-0.19}$ 4.68$^{+0.42}_{-0.54}$ 3.75$^{+0.78}_{-0.91}$ 1.14$^{+0.24}_{-0.28}$ 579$^{+12}_{-10}$
E$_{disk}$ 2.92$^{+0.47}_{-0.49}$ 1.05$^{+0.17}_{-0.18}$ 2.41$^{+0.70}_{-0.76}$ 5.62$^{+1.63}_{-1.77}$ 815$^{+79}_{-83}$
E$_{halo}$ 1.26$^{+0.16}_{-0.24}$ 4.53$^{+0.56}_{-0.87}$ 3.89$^{+1.21}_{-1.06}$ 0.91$^{+0.28}_{-0.25}$ 809$^{+90}_{-22}$
G$_{disk}$ 4.63$^{+0.31}_{-0.55}$ 2.15$^{+0.14}_{-0.26}$ 4.79$^{+0.72}_{-0.90}$ 8.70$^{+1.30}_{-1.62}$ 498$^{+42}_{-18}$
G$_{halo}$ 2.15$\pm$0.01 9.96$\pm$0.07 7.70$^{+0.63}_{-0.62}$ 1.40$\pm$0.11 448$^{+37}_{-31}$
H$_{disk}$ 3.65$^{+0.37}_{-0.49}$ 3.19$^{+0.32}_{-0.42}$ 6.21$^{+1.33}_{-1.22}$ 5.95$^{+1.28}_{-1.17}$ 524$^{+63}_{-8}$
H$_{halo}$ 1.31$^{+0.14}_{-0.16}$ 11.44$^{+1.22}_{-1.43}$ 9.18$^{+1.34}_{-2.01}$ 0.88$^{+0.13}_{-0.19}$ 752$^{+39}_{-83}$
------------ ------------------------- ------------------------------ ----------------------------- ------------------------------------- ---------------------------
### Magnetic fields in NGC6946 {#bfields}
To analyse the magnetic field parameters we used the same regions as for the spectral analysis. For each region both total intensity and polarized intensity fluxes at a radio wavelength of 6.2cm with a beam of 15$\arcsec$ were obtained. Then, using the energy equipartition formula provided by @beckra, we calculated the strengths of both the total and ordered magnetic fields. The calculations were made assuming a synchrotron spectral index of 1.0, an inclination of the galactic disk of 30$\degr$, and a proton-to-electron ratio of 100. For the emitting volume a disk of 1kpc thickness was assumed. The main uncertainties are introduced by the last two parameters. They may vary by a factor of 2. With the assumed spectral index this amounts to an error of $\sim$30% for the strength of the magnetic field and $\sim$60% for its energy density. We note here, however, that such errors are systematic, which means that we may in fact expect the relative uncertainties between the points to be smaller. Table \[magparams\] summarizes our results and also provides values for the energy densities of the magnetic field.
-------- -------------- ------------- -------------- ---------------------------- -------------
Region S$_{synch}$ p$_{synch}$ B$_{tot}$ $\epsilon_B$ B$_{ord}$
\[mJy/beam\] \[%\] \[$\mu$G\] \[10$^{-12}$ergcm$^{-3}$\] \[$\mu$G\]
A 0.67 7.7 17.2$\pm$5.2 11.8$\pm$7.1 4.9$\pm$1.5
B 0.36 22.6 14.4$\pm$4.3 8.3$\pm$5.0 7.2$\pm$2.2
C 0.31 26.3 13.8$\pm$4.1 7.6$\pm$4.6 7.4$\pm$2.2
D 1.74 9.3 21.8$\pm$6.5 18.9$\pm$11.3 6.8$\pm$2.0
E 0.42 12.7 15.2$\pm$4.6 9.1$\pm$5.5 5.6$\pm$1.7
F 0.39 16.5 14.8$\pm$4.4 8.8$\pm$5.3 6.2$\pm$1.9
G 0.54 6.5 16.3$\pm$4.9 10.6$\pm$6.4 4.2$\pm$1.3
H 0.39 10.8 14.9$\pm$4.5 8.9$\pm$5.3 5.0$\pm$1.5
I 0.35 20.5 14.3$\pm$4.3 8.2$\pm$4.9 6.8$\pm$2.0
J 1.40 14.6 20.4$\pm$6.1 16.6$\pm$10.0 8.1$\pm$2.4
-------- -------------- ------------- -------------- ---------------------------- -------------
Apart from the central regions of the galaxy (regions D and J), the strength of the total magnetic field is roughly constant across the disk. A slight increase can be observed in the most prominent spiral arms (regions A and G). Consequently, these regions show higher energy densities of the magnetic field, with the maximum near the galactic core. The strengths of the ordered magnetic field are also similar in all parts of the disk, with higher values in the areas of the magnetic arms and the central region of the galaxy. As suggested by Fig. \[6946radio\], the areas of the magnetic arms show a much higher degree of polarization than the other regions.
### Hot gas and magnetic fields of spiral and magnetic arms {#spirals}
In addition to the grand-design structure of its gaseous spiral arms, NGC6946 presents a distinct spiral structure of the magnetic fields which resembles magnetic arms that are phase-shifted with regard to the gaseous ones. These magnetic arms coincide well with the interarm regions. Since the spiral arm and interarm regions vary significantly in terms of the ISM structure, we investigated the emission from the hot gas in both areas to obtain more clues about the interplay of the magnetic field and the hot plasma. Because a model with two thermal components was used to analyse the emission from the spiral arms (accounting for the disk and halo components), to compare it with the emission from the hot gas in and above the interarm (magnetic arm) areas, we needed to calculate averages of the values obtained from the two-temperature fits.
We added gas masses and thermal energies for the appropriate regions. Next, we calculated number and energy densities, taking into account the volumes assumed for the disk and halo component emission (the area of a given region times 1 or 10kpc, respectively). Our results are presented in Table \[averagearms\]. We compared the averaged values of number and energy densities for all spiral arm regions with respective values for the interarm regions. We also calculated the ratios of the thermal energy densities to number densities, hence obtaining an average energy per particle, which is independent of the unknown volume-filling factor ($E_p = \epsilon/n$).
-------- ------------------------ ------------------------- ------------------------- ------------------------------
Region ($n\,\eta^{-0.5}$) (M$_{gas}\eta^{0.5}$) (E$_{th}\eta^{0.5}$) ($\epsilon_{th}\eta^{-0.5}$)
\[10$^{-3}$cm$^{-3}$\] \[10$^6$M$_\odot$\] \[10$^{54}$erg\] \[10$^{-12}$ergcm$^{-3}$\]
A 1.91$^{+0.19}_{-0.21}$ 5.82$^{+0.57}_{-0.66}$ 6.14$^{+1.34}_{-1.32}$ 1.70$^{+0.37}_{-0.38}$
E 1.41$^{+0.18}_{-0.27}$ 5.58$^{+0.72}_{-1.05}$ 6.30$^{+1.91}_{-1.82}$ 1.34$^{+0.40}_{-0.39}$
G 2.38$^{+0.03}_{-0.06}$ 12.11$^{+0.21}_{-0.33}$ 12.49$^{+1.35}_{-1.51}$ 2.06$^{+0.22}_{-0.25}$
H 1.52$^{+0.16}_{-0.19}$ 14.63$^{+1.54}_{-1.85}$ 15.39$^{+2.65}_{-3.22}$ 1.35$^{+0.23}_{-0.29}$
-------- ------------------------ ------------------------- ------------------------- ------------------------------
We present our results in Table \[particles\]. The interarm regions show higher values of an energy per particle than for the spiral arm regions. This is consistent with the single-temperature fits (Fig. \[singles\]), which show slightly higher temperatures for the regions of the magnetic arms. This means that there may be an additional effect that provides thermal energy to the interarm regions. Since this additional heating might also be due to magnetic reconnection, we analysed the magnetic field properties of the areas of the spiral arms and the interarm regions.
Spiral arm $E_p$ $\epsilon_{B}$ Magnetic arm $E_p$ $\epsilon_{B}$
------------ ------------------------ ---------------- -------------- ------------------------ ----------------
A 0.89$^{+0.10}_{-0.11}$ 11.8$\pm$7.1 B 1.45$^{+0.14}_{-0.15}$ 8.3$\pm$5.0
E 0.95$^{+0.14}_{-0.12}$ 9.1$\pm$5.5 C 1.33$^{+0.23}_{-0.27}$ 7.6$\pm$4.6
G 0.87$^{+0.08}_{-0.09}$ 10.6$\pm$6.4 F 1.28$^{+0.15}_{-0.16}$ 8.8$\pm$5.3
H 0.88$^{+0.06}_{-0.08}$ 8.9$\pm$5.3 I 1.20$^{+0.34}_{-0.31}$ 8.2$\pm$4.9
### Heating of the gas by magnetic reconnection? {#reconn}
Because for the interarm regions we have the information about the hot gas coming from both the disk and the above halo (one-temperature fit to the spectra), a direct comparison of the energy densities of disk hot gas and magnetic fields is possible for the regions of the spiral arms alone. Nevertheless, since we are interested in the global energy budget of the galaxy, we need information about both thermal and magnetic energy densities for the disk and the halo. Although halo magnetic fields surely exist (as observed in edge-on spiral galaxies), we do not have any direct information on their structure and strength in the case of NGC6946. The observed radio emission is, however, an integration along the line of sight, that is, we see contributions from both the galaxy disk and halo. Since the majority of the cosmic rays originates in the underlying disk, an assumption for the emitting volume (the disk) seems to be justified. As the vertical scale height of the halo magnetic fields of $\sim6-7$kpc [e.g. @beck15] is similar to the assumed size of the hot gas halo (10kpc), we do not expect a significant change of the magnetic energy density in the halo, especially when an uncertainty of its calculation (60%) is considered. It is therefore justified to compare the obtained magnetic energy densities with those of the hot gas in and above the interarm regions.
Still, for all areas of the disk of NGC6946 we see much higher energy densities of the magnetic fields than those of the disk component of the hot gas (Tables \[diskgas\] and \[magparams\]). Interestingly, for the region of the most prominent spiral arm (region G), conditions closest to equilibrium are observed, with the energy density of the magnetic field only 22% higher than that of the hot gas in the galactic disk. For the remaining spiral arms this difference is as high as $50-63\%$. If we compare the magnetic field energy densities to those of the halo (Tables \[halogas\] and \[diskgas\]) or averaged values for the hot gas in and above the spiral arms (disk+halo, Table \[averagearms\]), a distinct dominance of magnetic fields by a factor of a few is visible. This suggests that only in the areas of high star-forming activity it is possible that the thermal energy density of the gas is similar to that of the local magnetic fields.
To investigate the interplay between the thermal gas and the magnetic fields in greater detail, we compared the averaged (i.e. from both disk and halo components) thermal energies per particle with the magnetic field energy densities for the spiral and magnetic arms. We calculated the average values for the gaseous spiral arms and the magnetic arms, which resulted in an energy density of the magnetic field of about 10.1$\times\,10^{-12}$ergcm$^{-3}$ for the spiral arms and 8.2$\times\,10^{-12}$ergcm$^{-3}$ for the interarm regions and for energies per particle of 0.90$\times$10$^{-9}$erg and 1.32$\times$10$^{-9}$erg, respectively. Our results are presented in Table \[particles\]. An anti-correlation between the magnetic field strength (and its energy) and the thermal energy of the gas is visible (Fig. \[epemag\]); with a slight decrease of the energy density of the magnetic field (by 23%), the energy per particle increases significantly (by 68%).
A possible explanation is that in regions that simultaneously show higher thermal energies of the gas and lower energies of the magnetic fields, some of the energy of the magnetic field might have been converted into thermal energy by magnetic reconnection. Fast reconnection should be possible in most astrophysical plasmas [@hanasz], with a heating rate proportional to the Alfvén speed [@lesch; @lazarian]. As the gas density in the interarm regions is lower while the total magnetic field is almost as strong as in the spiral arms, the Alfvén speed is higher, and hence the heating rate is higher in interarm regions. Indeed, we do observe such an additional heating of the gas in the magnetic arm regions (see Tables \[singles\] and \[6946xtabr\]).
In general, such a slight increase in temperature in the interarm regions could be easily explained by longer cooling times that are due to the lower density of the gas. However, if we compare interarm regions B and C with F and I, we note that in magnetic arms B and C higher temperatures, lower number and energy densities than in magnetic arms F and I were obtained. Surprisingly, the difference in cooling times reaches almost a factor of 2. Still, for all four interarm regions practically the same magnetic field strengths and energy densities were observed. The difference is visible, however, when ordered fields are considered – in both regions B and C we see a much higher regularity of the field (i.e. the ratio of the strengths of ordered and total magnetic field). This trend is visible in all magnetic arms when compared to the spiral arms – the regularity of the magnetic field increases with the energy per particle (Fig. \[epregul\]).
The above findings allow constructing a picture of both turbulence and reconnection acting in a galactic disk. Although in the spiral arms reconnection effects are expected to be more efficient (stronger field tangling), their action might be difficult to see because both the heating and field (dis)ordering is dominated by turbulence. In the interarm regions, however, where the magnetic field is highly ordered, reconnection heating may dominate turbulence heating, and the increase in temperature due to reconnection heating might be noticeable. This is what we observe, especially in the magnetic arms B and C, which have slightly higher temperatures than most star-forming spiral arms. Although reconnection is a very local process, acting at distances of a few pc or less, if it is equally efficient throughout the entire magnetic arm, it might contribute to the field ordering, an effect that would last longer because of the weaker turbulence. Again, this high field ordering is most distinctly seen in magnetic arms B and C.
The region of the western spiral arm (region H) is interesting. While for the other spiral arm regions we observe a significant contribution from the halo component (see Table \[6946xfr\]), region H shows almost equal contributions from the halo and disk components. Furthermore, the difference of temperatures of the two components is the lowest in this region, owing mainly to the lowest temperature of all spiral arm disk components. This may suggest that for the western spiral arm mixing of the disk and the halo gas is the most efficient. Since the southern part of region H is in the area of strong Faraday depolarization, as reported by @beck91 [@beck07], this might be a piece of evidence for vertical magnetic fields and enhanced outflow speed in this area of the galactic disk. In particular the region of the south-western spiral arm (region A), which also contributes to the depolarization area, shows a similar trend for the interplay of the halo and disk gas components, although at a lower level.
Summary and conclusions {#cons}
=======================
The detailed analysis of the X-ray emission from the hot gas in NGC6946, together with earlier radio continuum studies, can be summarized as follows:
- A spectral analysis of the point sources revealed a significant number of ULXs, which agrees with the enhanced star formation of the galaxy.
- The galaxy presents significant emission from the hot gas across its entire star-forming disk. Intensity enhancements are found in the regions of the spiral arms, also in the harder energy band. A significant amount of the very soft emission is found in the region of high Faraday depolarization.
- The radio-polarized emission structure is reflected in the spectral properties of the hot gas - areas of magnetic arms visible in the interarm regions are well described with a single thermal plasma model, which shows that the temperature of the hot gas is slightly higher than in the spiral arm regions.
- An increase in temperature of the hot gas in the magnetic arm regions could be described as additional heating due to magnetic reconnection.
- A possible conversion of magnetic field energy into thermal energy of the hot gas in the interarm regions is suggested by the lower energy density and strength of the magnetic field and the higher thermal energy per particle, when compared to the areas of the spiral arms.
- In the conditions of low turbulence in the magnetic arm regions, reconnection, acting mostly on tangled fields, might also contribute to the field ordering, as suggested by both the highest temperatures of the hot gas and the highest degree of polarization in magnetic arms B and C.
- We found signatures of a very hot gas in the area of the ultra-luminous source MF16, which may suggest shock heating of the gas by a supernova explosion.
We thank Wolfgang Pietsch and Stefania Carpano for their collaboration on the original XMM-Newton observing proposals that form the base of this paper. Special thanks go to Harald Lesch and Alex Lazarian, who improved our understanding of the reconnection theory. We also thank Stefanie Komossa for useful comments on an earlier version of the paper, and the anonymous referee for a helpful report.
[^1]: Based on observations obtained with [*XMM-Newton*]{}, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA
[^2]: http://xmm2.esac.esa.int/external/xmm\_sw\_cal/\
background/epic\_scripts.shtml\#flare
[^3]: http://xmm.esac.esa.int/external/xmm\_science/\
gallery/utils/images.shtml
[^4]: ftp://xmm.esac.esa.int/pub/ccf/constituents/extras/\
background/epic/blank\_sky/scripts
[^5]: http://xmm2.esac.esa.int/external/xmm\_sw\_cal/\
background/blank\_sky.shtml\#BGsoft
[^6]: http://xmm.esac.esa.int/external/xmm\_user\_support/\
documentation/uhb
|
---
author:
- 'Henri C. G. de Cagny[^1]'
- 'Bart E. Vos$^*$'
- 'Mahsa Vahabi$^*$'
- 'Nicholas A. Kurniawan'
- Masao Doi
- 'Gijsje H. Koenderink [^2]'
- 'Fred C. MacKintosh [^3]'
- 'Daniel Bonn [^4]'
bibliography:
- 'References.bib'
title: Porosity governs normal stresses in polymer gels
---
**When sheared, most elastic solids such as metals, rubbers and polymer hydrogels dilate in the direction perpendicular to the shear plane. This well-known behaviour known as the Poynting effect is characterized by a positive normal stress [@Poynting1909]. Surprisingly, biopolymer gels made of fibrous proteins such as fibrin and collagen, as well as many biological tissues exhibit the opposite effect, contracting under shear and displaying a *negative* normal stress [@Janmey2007; @horgan2015]. Here we show that this anomalous behaviour originates from the open network structure of biopolymer gels, which facilitates interstitial fluid flow during shear. Using fibrin networks with a controllable pore size as a model system, we show that the normal stress response to an applied shear is positive at short times, but decreases to negative values with a characteristic time scale set by pore size. Using a two-fluid model, we develop a quantitative theory that unifies the opposite behaviours encountered in synthetic and biopolymer gels. Synthetic polymer gels are impermeable to solvent flow and thus effectively incompressible at typical experimental time scales, whereas biopolymer gels are effectively compressible. Our findings suggest a new route to tailor elastic instabilities such as the die-swell effect that often hamper processing of polymer materials and furthermore show that poroelastic effects play a much more important role in the mechanical properties of cells and tissues than previously anticipated.**
When subjected to a shear stress, materials either shrink (shear contraction) or expand (shear dilatancy). As shown by Poynting more than a century ago [@Poynting1909], simple elastic solids exhibit shear dilatancy. Similar behaviour has since been observed in more complex viscoelastic systems, such as granular materials, rubbers and polymer glasses [@Reynolds1885; @Larson1998]. In case of polymer materials, shear dilatancy is usually described by the classical Mooney-Rivlin model [@Mooney1940; @Rivlin1948], which predicts a normal stress difference $N_1 \sim G\gamma^2$, where $\gamma$ is the shear strain and $G$ the network shear modulus. In Fig. 1a this behaviour is illustrated for polyacrylamide [(PAAm)]{} hydrogels of varying stiffness subjected to an oscillatory shear deformation. Surprisingly, biopolymer networks have been reported to exhibit the opposite response, contracting when sheared [@Janmey2007; @Kang2009; @horgan2015]. This behaviour is clearly illustrated in Fig. 1b, which shows that aqueous gels of the blood clotting protein fibrin develop a negative normal stress under shear. The magnitude of the normal stress again increases quadratically with strain, but it reaches values comparable to the shear modulus at significantly lower shear strain ($\gamma\simeq 1/10$) than for polyacrylamide ($\gamma\simeq1$). The origin for the remarkable difference in the sign and magnitude of the normal stress between synthetic hydrogels and biopolymer gels is still unknown.
![Normal stress difference $N_1=\frac{2F}{\pi R^2}$, where $F$ is the normal force (thrust) reported by the rheometer and $R$ is the sample radius, as a function of the amplitude of the applied oscillatory shear strain for **(a)** PAAm [@SI] prepared with various ratios of monomer-to-cross-linker concentrations. The line indicates a quadratic dependence of $N_1\sim\gamma^2$, as expected from the Mooney-Rivlin model [@Mooney1940; @Rivlin1948]. In **(b)**, [$\frac{2F}{\pi R^2}$]{} is shown for fibrin gels polymerized at 22C at various fibrinogen concentrations (in mg/mL). The line indicates a [$\sim\gamma^2$ dependence,]{} but with negative sign.](Figure1.pdf){width="50.00000%"}
[Here, we aim to understand the mechanistic basis of the fundamentally different response of synthetic and biopolymer gels and to develop a minimal model that can capture the behaviour of both types of gels. In either case, the normal stress response is fundamentally nonlinear, since its sign cannot reverse when the shear train $\gamma$ is reversed. Thus, to lowest order, normal stress is expected to vary as $\gamma^2$, even while the shear stress remains linear in $\gamma$. Although both gels in Fig. 1 are hydrogels containing over $90\%$ interstitial water, structurally there is a profound difference in the pore size. While polyacrylamide gels have a pore size of order 10 nanometers [@tombs1965], fibrin networks have pore sizes that can be in the micrometer range [@Okada1985; @Pieters2012; @Lang2013]. Fluid permeability can therefore play an important role in the mechanical response.]{} For hydrogels with a small pore size, we expect a strong viscous coupling between the network and the solvent, which will suppress motion of the network relative to the solvent and effectively render the gel as a whole incompressible. By contrast, biopolymer gels can expel interstitial fluid to relax pressure gradients on long enough time scales, allowing the network to contract upon shearing [@heussinger2007nonaffine; @Conti2009; @Kang2009; @Tighe2013].
![Fluorescence confocal microscopy images of fibrin networks whose pore size is tuned by polymerizing under different conditions: at 22C **(a)** and 27C **(b)**. The scale bars are $10~\mu$m. Protein content is (8 mg/mL) in both samples. **(c)**Normal stress $\sigma_N$, given by the apparent normal stress difference $\frac{2F}{\pi R^2}$ obtained from the rheometer thrust $F$, for four fibrin networks differing in pore size as a function of time after the application of a constant shear stress at $t=0$. The stress relaxation curves are fitted to an exponential decay derived from the two-fluid model in [@SI] (black lines). **(d)** Schematic representation of the two-fluid model showing an inward, radial contraction of the network (black) relative to the solvent (blue) upon shearing.](Figure2.pdf){width="70.00000%"}
[To study the role of porosity, we choose fibrin networks as a model system, the pore size of which can be tuned]{} from nanometers to micrometers by simply changing the temperature, ionic strength and pH during self-assembly [@Nunes1995]. This is demonstrated in Figs. 2a and b, which show fluorescence microscopy images of two fibrin gels that are assembled at the same monomer concentration of 8 mg/mL but at different temperatures. Using quantitative measurements of the fiber mass-length ratio by light scattering, we calculate average mesh sizes of $0.36\ \mu$m and $0.29\ \mu$m for these networks (see [@SI] for further details).
To test the influence of pore size on the normal stress response, we subject each network to a constant shear stress and we monitor the normal stress as a function of time. Intriguingly, we find that in each case, the normal stress relaxes from an initially positive or close-to-zero value to a negative steady-state value with a rate that strongly varies with pore size. The characteristic relaxation time, $\tau$, increases from just a few seconds to $\sim$100 s as the pore size [of fibrin]{} decreases from $0.36\ \mu$m to $0.08\ \mu$m. [For PAAm gels with a pore size $\sim10$ nm [@tombs1965], the relaxation time grows to over 15 hours (Fig. 2c).]{} These observations support our hypothesis that the sign of the normal stress is controlled by the time scale for solvent flow through the network. The data suggest that the normal stress is positive as long as the polymer network and the fluid remain viscously coupled, and switches sign to become negative when the fluid can move relative to the network. Importantly, the timescale separating these behaviours is unrelated to the timescales apparent in the linear viscoelastic response (see Fig. 2c).
To quantitatively model the effects of network poroelasticity in the shear rheology of polymer networks, we start from the two-fluid model [@Brochard1977; @Milner1993; @Gittes1997; @Levine2000] that describes a polymer gel as a biphasic system comprised of a linear elastic network immersed in a viscous and incompressible liquid [@SI]. The two components are coupled by a force per unit volume, $\Gamma\left(\dot{\vec{u}}-\vec v\right)$, acting on the liquid and opposite to the force on the network. This dissipative force arises from the relative motion of the solvent, which moves with velocity $\vec v$ and the network, with velocity $\dot{\vec{u}}$. For a network with pore size $\xi$ and a fluid with viscosity $\eta$, $\Gamma \sim \eta/\xi^2$, [since the Stokes drag force on a network strand of size $\sim\xi$ moving with relative velocity $\sim v$ is $\sim\eta\xi\Delta v$ and this acts on a volume $\sim\xi^3$ [@SI]. Given the small polymer volume fraction $\phi$ of most hydrogels and biopolymer networks ($\phi\sim10^{-3}$ for fibrin gels), the radial velocity component of the incompressible fluid effectively vanishes and the only radial motion is due to the network.]{} This radial motion, $\dot u_r$, generates a radial pressure gradient in the solvent given by: \_rP=u\_r=-/r-(K/r\^2)u\_r,\[2Fluidshort\] for a cone-plate geometry. The net force on the network has two distinct elastic contributions. The first contribution comes from the hoop stress $\tilde{\sigma}$, which tends to drive the network radially inward (Fig. 2d) [@SI]: hoop stresses generated by shearing tend to drive radial contraction of the network and expulsion of the solvent, [much as a twisted sponge expels water by contracting radially.]{} [By symmetry, $\tilde\sigma\sim\gamma^2$ to lowest order, as noted above, although Eq. is linear in $u_r$.]{} The second contribution to the net force on the network comes from a restoring force that balances the radial contraction on long time scales (i.e., as $\dot u_r\rightarrow0$). This restoring force originates from the gradient in the elastic shear stress $\sim G\nabla_z u_r$ that results from the axial ($z$) variation of $u$ (see Fig. 2d). For a cone-plate rheometer with small gap size $d$ and small cone angle $\alpha$, the restoring force $\sim G/d^2u_r $. Thus, since $d=\tan(\alpha) r$, $K\sim G/\tan(\alpha)^2$ in Eq. . We thus predict a characteristic relaxation time $\tau\sim \eta d^2/G\xi^2$. Indeed, we experimentally observe a rapid decrease of the relaxation time with increasing pore size, consistent with the predicted scaling (see Fig. S1 in [@SI]).
We can consider two opposite limits of Eq. . [In the limit of small pore size and $\Gamma\rightarrow\infty$, the radial displacement $u_r\rightarrow0$ (with finite $\Gamma\dot u_r$) and Eq. reduces to $\nabla_rP=-\tilde\sigma/r$.]{} Shearing will thus increase the pressure toward the axis of the rheometer, which results in a positive contribution to the normal force. Dense hydrogels will therefore effectively behave as incompressible materials for which the normal force $F$ is related to the normal stress difference $\sigma_{xx}-\sigma_{zz}$ by $N_1=2F/\pi R^2$, where $R$ is the sample radius [@Venerus2007]. $N_1$ is positive for a rubber-like material, consistent with measurements on polyacrylamide gels [@Rivlin1948]. In the opposite limit of networks with a large pore size, the pressure difference can relax by water efflux and in steady state the two terms on the right hand side of Eq. cancel.
A key prediction of the two-fluid model is that the of the normal stress measured in a rheology experiment should depend on the experimental time scale relative to the characteristic relaxation time, $\tau\sim \eta d^2/G\xi^2$. To quantitatively test this prediction, we subject the fibrin gels to an oscillatory shear stress with frequencies between 0.001 Hz and 5 Hz, allowing us to conveniently probe a range of time scales from 0.2 to 1000 s in a single experiment. We measure the normal stress response after the system has reached steady state (Fig. S2 in [@SI]).
When we plot the time-dependent normal stress (Fig. 3d-f) as a function of shear stress, we obtain the Lissajous curves shown in Fig. 3a-c. Strikingly, the Lissajous curves completely change with changing frequency. For oscillation periods longer than $\tau$ (Fig. 3a), the normal stress decreases with increasing shear stress, demonstrating contractile behaviour under shear. By contrast, for oscillation periods shorter than $\tau$, the normal stress increases with increasing shear stress, demonstrating extensile behaviour (Fig. 3c). The transition occurs at an intermediate frequency that is of order $1/\tau$ (Fig. 3b). This experiment unambiguously shows that the normal stress response of a polymer gel is governed by fluid flow, which is suppressed at higher frequencies. The normal stress response is therefore controlled by the network pore size, and is furthermore dependent on the shear modulus $G$ and the gap size $d$ between the cone and the plate (see Fig. S3 in [@SI]).
The two-fluid model allows us to perform an even more rigorous test of the mechanism governing the normal stress response of polymer gels, since we can calculate the time-dependence of the normal stress and compare it to experiments. For symmetry reasons, the normal-stress components $\sigma_{xx}$ and $\sigma_{zz}$ are expected to have a leading $\gamma^2$ dependence on strain. Since the shear stress $\sigma_{xz}\simeq G\gamma$, we define $\sigma_{xx}\equiv A_xG\gamma^2\quad\mbox{and}\quad\sigma_{zz}\equiv A_zG\gamma^2$. For an oscillatory strain $\gamma(t)=\gamma_0\sin(\omega t)$, the steady state solution of the time-dependent Eq. is [@SI] $$N_1^{\mbox{\scriptsize (app)}} =-2A_zG\gamma(t)^2+\tilde AG\gamma_0^2({\mathcal A}\cos (2 \omega t) +{\mathcal B}\sin (2\omega t)),\label{Napp}
$$ where $${\mathcal A}=-\frac{1}{8 \omega\tau}
\Big[2 \tan ^{-1}\left(1+2 \sqrt{\omega\tau}\right) +2 \tan ^{-1}\left(1-2 \sqrt{\omega\tau}\right)-\pi+4 \omega\tau\Big],\label{calA}$$ =(1+4\^2\^2),\[calB\] with $\tilde A=A_x-A_z$ and $\tau=\Gamma R^2/K \sim\eta d^2/G\xi^2$. In the limit where $\omega\tau\gg1$, Eq. reduces to the well-known Mooney-Rivlin expression for incompressible materials, $N_1=G\gamma^2$. In the opposite limit where $\omega\tau\ll1$, Eq. instead reduces to $N_1^{\mbox{\scriptsize (app)}}=-2A_zG\gamma^2$.
Based on prior measurements on a range of biopolymer gels in the $\omega\tau\ll1$ limit [@Janmey2007; @Kang2009; @Storm2005] as well as models of fibrous networks [@MacKintosh1995; @Storm2005; @heussinger2007nonaffine; @Conti2009], we anticipate $A_{z}\sim 1/\gamma_c$, where $\gamma_c$ is the onset strain for nonlinear elasticity, which is typically $\sim 1/10$. Thus, in the limit of low frequencies, not only is $N_1^{\mbox{\rm\scriptsize (app)}}$ negative, but its magnitude can actually be much larger than $\sigma_{xy}\gamma$.
To test these predictions experimentally, we fit the oscillatory normal-stress data shown in Fig. 3 to Eq. (2). The only fit parameters are $A_z$ and $\tilde{A}$, since the shear modulus $G$ is measured independently from the shear stress at small strain and the relaxation time $\tau$ is measured independently from the normal stress relaxation upon applying a constant shear stress ($\tau = 12.5$ s, Fig. 2c). We observe excellent agreement between the data (symbols) and the model (solid lines) over the entire range of oscillation frequencies (Fig. 3), with fitting parameters that are insensitive to frequency (Fig. S1 in [@SI]).
Our observations reveal that poroelastic effects involving interstitial fluid flow play an unexpectedly important role in the shear rheology of polymer gels. Poroelastic effects in porous media such as fluid-imbibed polymer gels are usually considered to affect only volume-changing deformations such as compression and extension [@Biot1956; @Casares2015; @Oosten2016]. Our experiments and theory demonstrate that the shear response of polymer gels is highly sensitive to fluid flow and network compressibility, in spite of the volume-conserving nature of simple shear deformations. Depending on the timescale of deformation and the hydrodynamic coupling of the polymer network with the surrounding solvent, polymer gels behave as either incompressible materials with a positive normal stress or compressible materials with a negative normal stress.
We demonstrated that the normal stress response of both synthetic and biopolymer gels is quantitatively captured by a minimal model that takes into account the biphasic nature of hydrogels. This model can explain why synthetic hydrogels exhibit shear-dilation, while biopolymer gels have been reported to exhibit shear-contraction. This suggests a new route to tailor the sign and magnitude of the normal stresses for polymer materials by tuning the pore size, solvent viscosity, and nonlinear shear elasticity. This could prove valuable in the context of materials science, since normal stresses can cause elastic instabilities that severely complicate processing [@Larson1998]. Finally, our findings highlight the important role of poroelastic effects in tissue and extracellular matrix mechanics, where normal stresses can become a dominant stress component, even for small strains of order 10% [@Kang2009]. Related poroelastic effects in intracellular networks have previously been shown to govern the rheology of cells. However, the much smaller cellular dimensions $d\simeq 1~\mu$m, can be expected to limit the corresponding poroelastic relaxation time to of order 1 s, even for the smaller mesh sizes of order 10 nm [@Moeendarbary2013], which renders cells effectively compressible on timescales $\gtrsim$ 1 s.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).
**Supplemental Materials\
**
Materials and Methods {#materials-and-methods .unnumbered}
=====================
Chemicals were purchased from Sigma Aldrich (Zwijndrecht, the Netherlands). Human plasma fibrinogen and thrombin were purchased from Enzyme Research Laboratories (Swansea, United Kingdom). Fibrinogen stock solution was diluted to 8 mg/mL in assembly buffer (150 mM $\textrm{NaCl}$, 20 mM HEPES and 5 mM $\textrm{CaCl}_2$) at a pH of 7.4. Dense networks (fine clots) with an average pore size of 0.08 $\mu$m were obtained in fine-clot assembly buffer (400 mM $\textrm{NaCl}$ and 50 mM Tris-HCl) at a pH of 8.5 [@Bale1985]. The mixtures were prewarmed to the desired polymerization temperature and 0.5 U/mL thrombin was added to initiate network formation. The fibrin gels were allowed to polymerize *in situ* for at least 12 hours (22C and 27C samples) or 4 hours (37C samples) between the prewarmed cone and plate geometry of an Anton Paar rheometer (Physics MCR 302, Graz). Polyacrylamide gels were polymerized by preparing a mixture of polyacrylamide and N,N’-methylenebis(acrylamide)(Bis) followed by dilution to the desired final concentration. Polymerization was initiated by adding ammonium persulfate (0.5 $\mu$g/mL) and tetramethylethylenediamine (1 $\mu$L/mL) and allowed to proceed in situ between the rheometer plates at 20 C. , and a series of stainless steel cone-plate and plate-plate geometries in additional experiments reported in the Supplementary Information. Solvent evaporation was prevented by adding a layer of mineral oil (Sigma Aldrich, M3516) to cover the liquid-air interface. We checked that this procedure did not influence the normal stress response. The composition of the polyacrylamide gels displayed in Figure 1a of the main text are the following:
Gel Total polymer mass fraction (%) Acrylamide-MBAA mass ratio
----- --------------------------------- ----------------------------
A 4 149:1
B 5 199:1
C 4 99:1
D 6 299:1
E 3 39:1
We measured the linear elastic shear modulus $G$ of the gels by measuring the stress response to a small oscillatory shear strain with an amplitude of $0.1\%$ and frequency of 1 Hz. The normal stress response to an applied shear was obtained by applying Large-Amplitude Oscillatory Shear (LAOS) at a range of frequencies ($0.001$ Hz to $7$ Hz) at a shear stress amplitude of 800 Pa. We measured the time-resolved strain and normal stress response using an oscilloscope coupled to the analogue outputs of the rheometer. The characteristic normal-stress relaxation time was obtained by applying a constant shear stress (400 Pa for the 22 C and the 27 C gels, 550 Pa for the 37 C gel, 500 Pa for the fine clot gel and a strain of 150% for PAAm).
To characterize the pore size of the fibrin gels, we performed light scattering measurements using a spectrophotometer (Lambda35 UV/VIS Perkin Elmer, Waltham, MA, USA). Gels were polymerized in quartz cuvettes and absorbance spectra were taken over a wavelength range from 450 to 900 nm. The mass-length ratio of the fibers was obtained by fitting the spectra to a scattering model that assumes a random network of rigid cylindrical fibers [@Yeromonahos2010; @Piechocka2016]. The average mesh size $\xi\sim\sqrt{1/\rho}$ follows from the fiber length density $\rho = \frac{\mu}{c_p}$, where $\mu$ is the fiber mass-length ratio and $c_p$ the fibrinogen concentration. To validate the light scattering measurements, we also performed image analysis of confocal fluorescence microscopy images of fibrin networks doped with 5 mole% AlexaFluor 488-conjugated fibrinogen (Life Technologies, Eugene, Oregon, USA). Images were obtained on a Nikon Eclipse TI confocal system with a 100x oil immersion objective (NA=1.40). Image analysis is discussed in the Supplementary Information section ’Pore size analysis’.
Two-fluid model and relaxation of hoop stress {#two-fluid-model-and-relaxation-of-hoop-stress .unnumbered}
=============================================
Thus, to lowest order, a quadratic dependence on shear strain $\gamma$ is expected. We define this force (per unit volume) to be f\_r=--AG\^2,\[hoop\] where the coefficient $\tilde A$ is dimensionless. In an incompressible medium, in which net radial motion is not possible, this radial force must be balanced by a pressure that builds toward the axis of the rheometer. In the presence of a free surface, as opposed to a rheometer plate, this gives rise to rod-climbing behavior. In the case of a rheometer, the pressure results in a positive, upward thrust $F$.
For a polymer gel consisting of both network and solvent, radial motion of the network is possible, which can lead to a relaxation of the pressure contribution to the measured thrust. To model this relaxation, we use the two-fluid model [@Brochard1977; @Milner1993; @Gittes1997; @Levine2000], in which the network displacement $u$ and solvent velocity $v$ satisfy the coupled equations. The equation for the net force per unit volume acting on the fluid in the non-inertial limit is 0=\^2v-P-(v-),\[2fluidViscous\] while the corresponding (non-inertial) equation for the force on the network is 0=G\^2u+(G+)(u)+(v-),\[2fluidElastic\] where $\eta$ is the solvent viscosity, $G$ is the shear modulus and $\lambda$ is a Lamé coefficient that is typically of order $G$. These equations are coupled by a term representing the force on the fluid (and opposite to the force on the network) due to the relative motion of the two components. The coupling constant $\Gamma$ is expected to be of order $\eta/\xi^2$ for a network mesh size $\xi$. This can be estimated in a free-draining approximation by considering the drag force on a network strand of length $\sim\xi$ moving with velocity $\dot{\vec u}$ with respect to a stationary solvent. Apart from a weak logarithmic correction, this drag force is approximately $4\pi\xi\eta\dot u$. This is the force per mesh volume $\sim\xi^3$, meaning that the force per unit volume is of order $\eta\dot u/\xi^2$. This is the net force per unit volume on the fluid given by the last term on the right-hand side of with $\vec v=0$.
If the volume fraction of the network is small, as it is for most biopolymer gels ($\sim10^{-3}$), then we can safely assume that only the network moves radially in response to strain-induced hoop stresses $\tilde\sigma$, while the solvent remains stationary since it is incompressible. In this case, the radial component of reduces to $\nabla_r P=\Gamma\dot{u_r}$. Again, the radial motion of the network at low volume fraction is well approximated by G u\_r.\[urcomp1\] Here, we have assumed a cone-plate geometry with gap $d\ll r$. Corrections to , from both $\nabla^2\vec u$ and $(G+\lambda)\vec\nabla\cdot(\vec\nabla\cdot\vec u)$ terms are smaller by of order $(d/r)^2$. Together with the boundary conditions that $u_r=0$ at both $z=0$ and $z=d$, we find a characteristic relaxation time \~\[tau\] for the radial motion with axial profile $u_r\propto\sin(\pi z/d)$. The simplified linear equation of motion is -(\^2 G/d\^2)u\_r.\[urcomp2\]
What is still missing from this analysis is the additional force in acting on the network due to hoop stresses. Being fundamentally nonlinear, this is not captured by , so we add this to the results of the (linear) two-fluid model to obtain the following phenomenological equation of motion: \_rP=u\_r=-K-AG\^2.\[2Fluid\] Here, $K\simeq\pi^2G/\tan\left(\alpha\right)^2$ and we have used the fact that $d=\tan\left(\alpha \right)r$.
Incompressible limit {#incompressible-limit .unnumbered}
--------------------
First, let’s consider the case of an incompressible medium, as one has for the limit $\Gamma\rightarrow\infty$, where $u\rightarrow0$ and \_rP=-. This equation can be integrated to give P(R)-P(r)=-(R/r), where $P(R)$ is the pressure at the sample boundary, i.e., atmospheric pressure $P_0$. The excess pressure, P=P(r)-P\_0 can be integrated to give a positive (upward) contribution to the thrust $F$ 2\_0\^R r(R/r)dr=. Adding this to the direct contribution -R\^2\_[zz]{} from $\sigma_{zz}$, we find that =N\_1=\_[xx]{}-\_[zz]{},\[N1-incomp\] implying that, =\_[xx]{}+\_[zz]{}AG\^2, where $\tilde A=\left(A_x+A_z\right)$, \_[xx]{}A\_xG\^2\_[zz]{}A\_zG\^2. We expect $A_{z}\sim 1/\gamma_0$, based on the prior low-frequency model [@Janmey2007; @Kang2009]. Since $\sigma_{xx}$ usually is of order but larger than $\sigma_{zz}$, we expect both $A_{x,z}\sim 1/\gamma_0$, which is typically of order 10 for biopolymer networks. Moreover, as defined both stress components are strictly positive (tensile) and we expect that $A_x>A_z$, for $N_1>0$ in the incompressible limit.
Compressible limit {#compressible-limit .unnumbered}
------------------
In the limit of long times $t\gg\tau$ and low frequencies $\omega\tau\ll 1$ in Eq. (\[2Fluid\]), the pressure vanishes, and the apparent $N_1$ measured is that of Refs. [@Janmey2007; @Kang2009] =N\_1\^=-2\_[zz]{}=-2A\_zG\^2.\[N1-comp\] This describes the long-time value in Fig. 2c of the main text, to which the normal stress relaxes. For intermediate times/frequencies, we solve Eq. (\[2Fluid\]), with $\gamma(t)=\gamma_0\sin(\omega t)$, to find both steady-state (ss) and transient (tr) solutions, where the general $u_r(t)=u^{\mbox{\scriptsize (ss)}}(t)+u^{\mbox{\scriptsize (tr)}}(t)$. We find u\^(t) =- and u\^(t)=e\^[-t]{}. The transient is found by choosing a homogeneous solution of to give $u_r(t)=u^{\mbox{\scriptsize (ss)}}(t)+u^{\mbox{\scriptsize (tr)}}(t)=0$ at $t=0$. The transient is most relevant to the case where its characteristic relaxation time $\tau$ is large compared with the period of oscillation $\sim1/\omega$. Thus, we approximate u\^(t)e\^[-t]{}, from which we find \_rP=u\_r-e\^[-t]{}.\[last\] As noted before, there is no single relaxation time in this system, since $\tau\sim\eta d^2/(G\xi^2)$ depends on the gap $d$. This can be seen in , where the relaxation is $r$-dependent. This can be integrated to find the transient contribution to $N_1^{\mbox{\scriptsize app}}$ given by N\_1\^ AG\_0\_0\^2 (+e\^[-t/]{}) AG\_0\_0\^2,\[taufit\] where the latter approximation is valid to within less than 2% until the transient has decayed to less than 2% of its initial value.
For the steady-state, N\_1\^=-2A\_zG\^2+AG\_0\^2 ([A]{}(2 t)+[B]{}(2t)),\[N1app\] where =- \[calA\] and =(1+4\^2\^2).\[calB\] For an incompressible system, is recovered for $u_r(t)=u^{\mbox{\scriptsize (ss)}}(t)+u^{\mbox{\scriptsize (tr)}}(t)$ as $\tau\rightarrow\infty$. In the limit of low frequency, this reduces to the fully compressible limit of .
Application of the model to the experimental results {#application-of-the-model-to-the-experimental-results .unnumbered}
----------------------------------------------------
The experiments are done on fibrin samples with different mesh sizes, obtained by varying the temperature, ionic conditions and pH. To explain the experimental results using the two-fluid model, we fit the steady state normal stress data to Eq. (\[N1app\]). The free parameters in this equation are $A_z$ and $\tilde{A}$. In this formula $\omega$ is the frequency of oscillatory shear stress or strain. The other parameters $G$, $\gamma_0$ and $\tau$ can all be directly obtained experimentally. The amplitude of the shear strain, $\gamma_0$ is evaluated by fitting a sinusoidal function to the recorded shear strain data. The linear shear modulus $G$ is obtained by fitting a linear stress-strain relation to the stress-strain curves. In all cases, the relaxation time $\tau$ is obtained by fitting the normal stress relaxation curves versus time to Eq. (\[taufit\]), For fibrin gels polymerized at $27^\circ$ C, we find an average relaxation time $\tau=12.5$s . We apply frequencies in the range 0.001 Hz to 1 Hz. In Fig. \[fig:fitparam\], the fit parameters from the two-fluid model are plotted versus frequency. In Fig. 3 of the main text, we have chosen three frequencies (0.001, 0.01 and 1 Hz) out of this frequency range, to show how the normal stress response changes from extensile to contractile behavior with changing frequency.
![ Fit parameters ($A_z$ and $\tilde{A}$) for the fibrin sample polymerized at $27^{\circ}$C versus frequency.[]{data-label="fig:fitparam"}](SIfig1a.pdf "fig:"){width="40.00000%"} ![ Fit parameters ($A_z$ and $\tilde{A}$) for the fibrin sample polymerized at $27^{\circ}$C versus frequency.[]{data-label="fig:fitparam"}](SIfig1b.pdf "fig:"){width="40.00000%"}
Transient regime during oscillatory shear measurements {#transient-regime-during-oscillatory-shear-measurements .unnumbered}
======================================================
To measure the transition time in a fibrin gel polymerized under fine clot conditions (pore size $0.08\ \mu$m), we applied an oscillatory shear stress and measured the normal force as a function of time. Fig. \[Fn\_Transient\] shows the transient regime, where the normal stress relaxes from a positive to a negative value over a characteristic time scale $\tau$. This relaxation is similar ($\tau = 100$ s) to what is observed for experiments where a constant shear stress is applied (as shown in the main text in Figure 2).
Influence of rheometer shear cell geometry on the normal stress signal {#influence-of-rheometer-shear-cell-geometry-on-the-normal-stress-signal .unnumbered}
======================================================================
In this section, we show normal stress measurements obtained by shearing fibrin gels polymerized at 27 C with different geometries. The dimensional analysis explained in the section ’Two-fluid model and relaxation of hoop stress’ predicts that the characteristic time constant of the gel should scale as $\tau\sim \frac{\eta d^2}{G\xi^2}$ (Eq. (\[tau\])), where $d$ is a characteristic length scale of the problem. By changing the measurement geometries, we thus expect the normal force signal at a given frequency to change.
We measure for different frequencies the phase shift $\phi$ between the normal force signal, $N(t)=A\cos(2.(2\pi\nu) t + \phi)$, and the squared shear strain, $\gamma(t)^{2}=(\gamma_{0}\cos((2\pi\nu).t))^{2}=\frac{\gamma_{0}^{2}}{2}(1+\cos(2.(2\pi\nu).t)$. We choose to compare $N(t)$ with $\gamma(t)^{2}$ because both quantities have the same frequency ($2\nu$), and because of the analogy with the Mooney-Rivlin model prediction, $N=G\gamma^{2}$. The phase shift as a function of frequency measured for several plate-plate (PP) geometries and cone-plate (CP) geometries is shown in Fig. \[SIgeometries\]. We observe a marked dependence on gap size and cone angle.
As shown in Fig. \[SIgeometries\]b, we can scale out these differences by rescaling the frequency axis by the characteristic relaxation time $\tau\sim\frac{\eta d^2}{(G\xi^2)}$, where $\eta$ is the viscosity of the interstitial fluid ($10^{-3}$ Pa s), $G$ is the storage modulus of the gel (which varies slightly with each experiment), $d$ is the gap of the rheometer (chosen at the edges for the cone-plate geometries) and $\xi$ is the mesh size of the gel (0.29 $\mu$m at 27C). We observe a collapse of all the curves (Fig. \[SIgeometries\]b), showing that the gap size is the relevant length scale that governs the time-dependence of the normal force, in accordance with predictions from the dimensional analysis.
Pore size analysis {#pore-size-analysis .unnumbered}
==================
The pore size of the fibrin gels was determined using turbidimetry, and verified by confocal microscopy. Turbidimetry data was analyzed based on a model for light scattering from isotropic networks of rigid rods [@Piechocka2016; @Yeromonahos2010] in order to obtain the average mass-length ratio of the fibrin fibers. The fiber density in terms of the total length per volume, $\rho$, was calculated from the mass-length ratio and the fibrinogen mass concentration; we then obtained the mesh size $\xi\sim\sqrt{1/\rho}$.
An independent way to obtain the gel’s pore size is through analysis of confocal images [@Munster2013]. Analysis was done on 2 independently polymerized samples. Planes were analyzed separately. A FFT bandpass filter was applied to each confocal plane, filtering out features smaller than 2 pixels and larger than 30 pixels. As fibers were typically 5 pixels in diameter, this filtering step preserves their structure. Various thresholding techniques were applied: the built-in Matlab thresholding function (imgbw), Kapur’s thresholding method (also known as the Maximum Entropy method) [@Kapur1985] and Otsu’s thresholding method [@Otsu1979]. An overview of images with the FFT filter and thresholding applied is shown in Figure \[Thresholding\]. The pore sizes obtained using analysis of turbidity spectra and image analysis are shown in Table \[Pore\_size\_table\]. Although absolute values differ, the ratios between the 22C and the 27C show a consistent picture where the pore size of the fibrin network decreases upon an increase in the polymerization temperature by a factor of approximately 1.2. A histogram of the resulting pore size distributions is shown in Figure \[Pore\_size\].
22C, pore size ($\mu$m) 27C, pore size ($\mu$m) Ratio
--------------------------- ------------------------- ------------------------- -------
BP filter, Matlab (imgbw) 0.66 (0.00) 0.54 (0.03) 1.21
BP filter, Max. entropy 0.68 (0.01) 0.60 (0.03) 1.12
BP filter, Otsu 1.19 (0.10) 0.92 (0.12) 1.29
Photospectrometry 0.36 (0.01) 0.29 (0.01) 1.24
: Pore sizes in $\mu m$ of 8 mg/ml fibrin gels polymerized at 22C and 27C, obtained using bubble analysis of confocal images with various thresholding methods, and turbidimetry. The number between brackets is the standard deviation between measurements.[]{data-label="Pore_size_table"}
[^1]: authors contributed equally
[^2]: Email: [email protected]
[^3]: Email: [email protected]
[^4]: Email: [email protected]
|
---
abstract: 'We introduce and investigate different definitions of effective amenability, in terms of computability of Følner sets, Reiter functions, and Følner functions. As a consequence, we prove that recursively presented amenable groups have subrecursive Følner function, answering a question of Gromov; for the same class of groups we prove that solvability of the Equality Problem on a generic set (generic EP) is equivalent to solvability of the Word Problem on the whole group (WP), thus providing the first examples of finitely presented groups with unsolvable generic EP. In particular, we prove that for finitely presented groups, solvability of generic WP doesn’t imply solvability of generic EP.'
address: 'Università degli Studi Niccolò Cusano - Via Don Carlo Gnocchi, 3 00166 Roma, Italia'
author:
- Matteo Cavaleri
title: Følner functions and the generic Word Problem for finitely generated amenable groups
---
Introduction
============
In this paper we define and study some effective versions of amenability for finitely generated groups, in terms of computability of Følner sets, computability of Reiter functions and subrecursivity of Følner functions.
Let $\Gamma$ be a group generated by a finite subset $X$. Given $n\in\mathbb N$, we say (cf. [@VER]) that a non-empty finite subset $\Omega \subset \Gamma$ is an *$n$-Følner set* (with respect to $X$) if $$\label{folner}\frac{|\Omega \setminus x \Omega|}{|\Omega|}\leq n^{-1}, \;\;\;\forall x\in X.$$ We denote by $\mathfrak F\o l_{\Gamma,X}(n)$ the set of all $n$-Følner sets of $\Gamma$ with respect to $X$. Moreover, we say that a sequence $(\Omega_n)_{n\in \mathbb N}$ of subsets of $\Gamma$ is a *Følner sequence* if for every $n\in \mathbb N$, $\Omega_n\in \mathfrak F\o l_{\Gamma,X}(n).$ A related important notion is the Følner function $F_{\Gamma,X}$, introduced by Vershik [@VER], that measures the cardinality of the smallest Følner sets: $$F_{\Gamma,X}(n):=\min\{|\Omega|:\;\; \Omega\in \mathfrak F\o l_{\Gamma,X}(n)\},$$ with the convention that $\min \emptyset := \infty$. It is well known that the existence of a Følner sequence and the asymptotic behaviour of the function $F_{\Gamma,X}$ does not depend on the choice of $X$: we say that $\Gamma$ is *amenable* if it admits a Følner sequence (and therefore $F_{\Gamma,X}(n)<\infty,\;\forall n\in \mathbb N$).
A function $f\colon \mathbb N\to \mathbb N$ is said to be *recursive* if there exists an algorithm (Turing machine) that:\
(i) stops for every input $n$;\
(ii) *computes $f$*, that is, gives $f(n)$ as an output.\
A function is *subrecursive* if it admits a recursive upper bound. We refer to [@Mal] for general computability theory.
Vershik himself was interested in algorithmic behaviour of Følner functions, conjecturing the existence of arbitrarily fast growing Følner functions. This was confirmed by Erschler [@A2], who provided examples of finitely generated groups with Følner function growing faster than any given function, even non-subrecursive. In particular, the Følner sets of those groups are missing any algorithmic description. Analogous results were recovered in [@G; @OO]. We finally mention that, most recently, Brieussel and Zheng [@Zheng Cor 4.7] have shown that any non-decreasing function is asymptotically equivalent to the Følner function of some finitely generated group.
However the behaviour for finitely presented groups remained open:
[[@G p.578, Gromov]]{} “(d) Is there an universal bound on the asymptotic growth of the Følner functions of finitely presented amenable groups by a recursive (primitively recursive?) function? (Maybe there is such a bound in every given recursive class of presentations?). Or, at another extreme, are there finitely presented amenable groups with so fast growing Følner function, such that their amenability is unprovable in Arithmetic? (An enticing possibility would be this situation for the Thompson group).”
The above-mentioned possibility about Thompson group was studied in [@Moo]: if the Thompson group $F$ is amenable then its Følner function grows faster than any iterated exponential. For recursively presented groups, in [@A1] Erschler showed that the asymptotics of the Følner function of the $k$-iterated wreath-product of $\mathbb Z$ is the $k$-th tetration of $n$.
One of our main results (Section \[reisec\]) is the following partial answer to the aforementioned question of Gromov:
\[A\] The Følner function of a recursively presented amenable group is subrecursive. Moreover, every recursively enumerable class of recursive amenable presentations admits a uniform recursive upper bound for the asymptotic growth of the corresponding Følner functions.
The first sentence follows from Theorem \[imp\], the second from Corollary \[c3\].
The main tool used in the proof of the above theorem is the construction of a uniform algorithm $\widehat{\mathfrak{K}}$, described in Theorem \[imp\], that for any $n\in \mathbb N$ and any recursive presentation, provides, if it exists, a function on the associated free group whose pushforward on the group is $n$-invariant (an equivalent notion for amenability, see Section \[prel\]). Let us fix some notation.
With any finite set $X$ of generators of $\Gamma$, we associate a set $\mathsf X$ and a bijection $\varphi\colon \mathsf X\to X$. We denote by $\mathbb F_{\mathsf X}$ the free group generated by $\mathsf X$, and by $\pi_\Gamma\colon\mathbb F_{\mathsf X}\rightarrow\Gamma$ the unique epimorphism extending $\varphi$. The group $\Gamma$ has *solvable Word Problem* (WP) if there exists an algorithm that for every $\omega \in \mathbb F_{\mathsf X}$ as an input, stops and establishes whether or not $\omega$ represents the identity in $\Gamma$ (i.e. $\pi_\Gamma(\omega)=1_\Gamma$). This is equivalent to saying that $\ker \pi_\Gamma\subset \mathbb F_{\mathsf X} $ is *recursive*. We also say that $\Gamma$ is recursively (resp. finitely) presentable if there exists $R\subset \mathbb F_{\mathsf X}$ recursive (resp. finite), such that the normal closure $R^{\mathbb F_{\mathsf X}}=\ker \pi_\Gamma$. Dehn in [@Dehn] first formulated the Word Problem, several years before the study about computability started. Only in the 1950s [@Boone; @Novikov] examples of finitely presented groups with unsolvable WP appeared.
From a practical point of view, often in computer science it is not important the behaviour of an algorithm for the totality of the inputs, because it is possible that it is strongly influenced by a small, negligible, subset of inputs. Sometimes it is more interesting to study the *average* or the behaviour for *most* of the inputs. This concept was developed even in group theory [@G1; @G2; @G3; @AO]: we refer to [@KMSS] for an extensive discussion on the subject. In particular, Kapovich, Miasnikov, Schupp and Shpilrain formally defined the concept of *generic computability* and *generic-case complexity*, especially focusing on algorithmic problems for finitely generated groups. We now present the *generic Equality Problem*.
Following [@Afinite], we say that the Equality Problem (EP) is solvable on a subset $S\subset \mathbb F_{\mathsf X}$ if there exists an algorithm with input $(\omega_1, \omega_2) \in \mathbb F_{\mathsf X}\times \mathbb F_{\mathsf X}$, such that whenever $(\omega_1, \omega_2) \in S\times S$ the algorithm stops, establishing whether $\pi_\Gamma(\omega_1)=\pi_\Gamma(\omega_2)$ or not. Notice that when $S$ is a subgroup, EP is equivalent to the Word Problem for $S$.
Denoting by $B_n$ the ball of radius $n$ in $\mathbb F_{\mathsf X}$, a subset $S\subset \mathbb F_{\mathsf X}$ is called *generic* if $$\label{ggg} \lim_{n\to\infty} \frac{|S\cap B_n|}{|B_n|}=1 ;$$ a subset is *negligible* if its complement is generic.
The group $\Gamma$ has *solvable generic EP* if there exist a finite set of generators $X$ and a generic subset $S\subset \mathbb F_{\mathsf X}$ such that the EP is solvable on $S$.
The dependence on the choice of the generating set $X\subset \Gamma$ in the above definition is, to our knowledge, presently unknown. Passing from classical computability problems to their generic version fails, in general, to preserve independence of the choice of the generating set. However, we believe that, in the present setting, this is not the case.
The transition to genericity makes solvable some classical unsolvable problems; the literature in this direction is very rich, starting from [@KMSS] to [@JS; @DJS; @KS; @KKS]. But, not less important, especially for cryptography, is to produce examples [@Afinite; @GMO; @MR; @JS] of problems generically hard or even generically undecidable. Up to now there were no examples of finitely presented groups with unsolvable generic WP or unsolvable generic EP. Here we provide examples of the latter by proving a sort of “stability" for the Word Problem in recursively presented amenable groups:
\[B\] In the class of recursively presented amenable groups: $$\mbox{ solvable WP } \Longleftrightarrow \mbox{solvable generic EP}$$
Section \[GEP\] is devoted to proving this theorem.
To prove this, we use a variation of the algorithm $\widehat{\mathfrak K}$, and the following: in a recursively presented group computability, for every $n$, of a one-to-one preimage of an $n$-Følner set, gives solvability of WP (Theorem \[WP\]). Thus, more generally, solvability of EP on a set containing a preimage of a Følner sequence implies solvability of the WP.
As a byproduct, the following provides a solution to [@Afinite Problem 1.5, b] (we denote by $G(M)$ the Kharlampovich groups, see [@H; @KMS]):
The finitely presented groups $G(M)$ have unsolvable generic Equality Problem.
Indeed, the groups $G(M)$ are finitely presented, solvable and therefore amenable, and have unsolvable Word Problem ([@H; @KMS]).
Note that in [@KMSS] (linear) solvability of the generic Word Problem for solvable groups is proved. Thus even if the Equality Problem is the natural generalization of the Word Problem, however the generic EP is different from the generic WP.
Let $\mathcal C_A$ denote the class of recursively presented amenable groups and consider the following subclasses: $\mathcal C_{WP}$ (with solvable WP), $\mathcal C_{CF}$ (with computable Følner sets), $\mathcal C_{CFI}$ (with computable Følner sets by one-to-one preimages), $\mathcal C_{CR}$ (with computable Reiter functions), $\mathcal C_{SF}$ (with subrecursive Følner function) (see next section for the definitions).\
The following theorem summarizes the current understanding about the relations among these several notions of effective amenability.
\[C\] $$\mathcal C_{CFI}=\mathcal C_{WP}\subsetneq \mathcal C_{CF} \subset \mathcal C_{SF}=\mathcal C_{CR}=\mathcal C_A$$
The first equality is Theorem \[WP\], the other equalities follow from Theorem \[imp\], the remaining relations were already proved in [@CAV; @Pre].
Whether or not the inclusion $\mathcal C_{CF} \subset \mathcal C_{SF}$ is strict is an open question.
The paper is organized as follows.
- We introduce notation and the definitions of *computable Følner sets* and *computable Reiter functions*. We present some basic properties, fundamental for all the next sections.
- We prove that every amenable recursively presented group has computable Reiter functions and therefore has subrecursive Følner function, equivalently, $\mathcal C_{SF}=\mathcal C_{CR}=\mathcal C_A$ (Theorem \[imp\]). We analyze the existence of uniform recursive upper bounds for the Følner functions of recursively presented amenable groups (Corollary \[c1\], \[c2\], \[c3\]). Theorem \[imp\] and its proof are fundamental for Section \[GEP\].
- In the class of amenable recursively presented groups, we characterize those groups with solvable WP as the groups with computable Følner sets by one-to-one preimages, equivalently, we show $\mathcal C_{CFI}=\mathcal C_{WP}$ (Theorem \[WP\]). Moreover, in Corollary \[ovvio\], we easily show that, in case of solvability of WP, all definitions of effective amenability are equivalent. Theorem \[WP\] is fundamental for Section \[GEP\].
- We prove that a recursively presented amenable group with solvable generic EP has solvable WP (Theorem \[uno\]). The proof uses the algorithm described in the proof of Theorem \[imp\] and the characterization of the WP given by Theorem \[WP\].
- Questions and final remarks.
Acknowledgement {#acknowledgement .unnumbered}
---------------
This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS - UEFISCDI, project number PN-II-RU-TE-2014-4-0669. I thank Tullio Ceccherini-Silberstein for the long and precious discussions, and the anonymous referees: a first one who suggested me to investigate the notion of computability of Reiter functions and to strengthen the notion of computability of Følner sets (this suggestion turned very precious for my further development in computability theory), and a second one for the careful reading and the precious advices on the organization of the presentation of the paper.
Preliminaries {#prel}
=============
Throughout this paper, $\Gamma$ is a group generated by a finite set $X$. We fix a set $\mathsf X$ and a bijection $\varphi\colon \mathsf X\to X$, and denote by $\pi_\Gamma\colon\mathbb F_{\mathsf X}\rightarrow\Gamma$ the unique epimorphism extending $\varphi$, where $\mathbb F_{\mathsf X}$ is the free group based on $\mathsf X$. For $x\in X$ we set $\mathsf x:=\varphi^{-1}(x)\in \mathsf X$: we believe that this use of different fonts, avoiding possible ambiguities, considerably simplifies notation. Given an element $\omega$ in the free group $\mathbb F_{\mathsf X}$ we denote by $|\omega|$ the natural word length of $\omega$ with respect to $\mathsf X \cup \mathsf X^{-1}$; we denote by $B_n:=\{\omega\in\mathbb F_{\mathsf X}:\; |\omega|\leq n\}$ the ball of radius $n$ and by ${S_n:= B_n\setminus B_{n-1}\subset\mathbb F_{\mathsf X},}$ the sphere of radius $n$. For a natural number $k$, we denote by ${[k]:=\{1,2,\ldots,k\}}$, and recall that $\mathfrak F\o l_{\Gamma,X}(n)$ is the family of $n$-Følner sets of $\Gamma$ with respect to $X$. The function $\chi_A$ is the characteristic function of the subset $A$ (both for $A\subset \Gamma$ or $A\subset \mathbb F_{\mathsf X}$).
A summable non-zero function $h\colon \Gamma \to \mathbb R^+$, $\|h\|_{1,\Gamma}:=\sum_{g\in\Gamma} |h(g)|<\infty $, is *$n$-invariant* with respect to $X$ if for all $x\in X$ $$\label{nin}\frac{\|h-_x\!h\|_{1,\Gamma}}{\|h\|_{1,\Gamma}}\leq n^{-1};$$ where $_xh \colon \Gamma \to \mathbb R^+$ is the function defined by $_xh(g):=h(x^{-1}g)$.\
We denote by $\mathfrak Reit_{\Gamma,X}(n)$ (from the *Reiter condition* for amenability [@rei]) the set of all summable non-zero functions from $\Gamma$ to $\mathbb R^+$ that are $n$-invariant with respect to $X$.
\[RF\] The following facts are well known and/or easy to prove (see [@tullio; @Coo])
- $\Omega \in \mathfrak F\o l_{\Gamma,X}(n) \implies \Omega g \in \mathfrak F\o l_{\Gamma,X}(n), \; \forall g\in \Gamma;$
- $\Omega \in \mathfrak F\o l_{\Gamma,X}(n) \implies \frac{|\Omega\setminus x^{-1} \Omega|}{|\Omega|}\leq \frac{1}{n},\; \forall x\in X;$
- $\Omega\in \mathfrak F\o l_{\Gamma,X}(n) \Leftrightarrow \frac{|\Omega\cap x \Omega|}{|\Omega|}\geq 1-\frac{1}{n},\; \forall x\in X;$
- $\Omega\in \mathfrak F\o l_{\Gamma,X}(2n)\: \Leftrightarrow \chi_\Omega\in \mathfrak Reit_{\Gamma,X}(n),$\
since $\frac{\|\chi_\Omega-_x\!\chi_\Omega\|_{1,\Gamma}}{\|\chi_\Omega\|_{1,\Gamma}}=
\frac{\|\chi_\Omega-\!\chi_{x\Omega}\|_{1,\Gamma}}{\|\chi_\Omega\|_{1,\Gamma}}=2 \frac{|\Omega\setminus x \Omega|}{|\Omega|};$
- $h\in \mathfrak Reit_{\Gamma,X}(n) \implies \exists \Omega\subset Supp(h):=\{ g\in \Gamma:\; h(g)\neq 0\},\; \Omega\in \mathfrak F\o l_{\Gamma,X}(n)$,\
precisely, by the so-called layer cake decomposition, or Namioka’s trick, there exists ${\epsilon\in \mathbb R^+}$ such that $\{g\in \Gamma:\; h(g)>\epsilon\}\in \mathfrak F\o l_{\Gamma,X}(n);$
Thus $\Gamma$ is amenable if and only if $\mathfrak Reit_{\Gamma,X}(n)\neq \emptyset$ for every $n\in\mathbb N$ or, equivalently, ${\mathfrak F\o l_{\Gamma,X}(n)\neq \emptyset}$ for every $n\in\mathbb N$. In order to define a notion of effective amenability for $\Gamma$ we require the existence of an algorithm computing, in some sense, either Følner sets or Reiter functions. Since in general $\Gamma$ has unsolvable Word Problem we “lift" the output to $\mathbb F_{\mathsf X}$. The following notion was introduced and studied in [@CAV; @Pre]:
$\Gamma$ has *computable Følner sets* if there exists an algorithm with:\
INPUT: $n\in \mathbb N$\
OUTPUT: $F \subset \mathbb F_X$ finite, such that $\pi_{\Gamma}(F)\in\mathfrak F\o l_{\Gamma,X}(n)$.
The computability of Følner sets does not depend on the choice of the finite set of generators and, in particular, for finitely presented groups, if we change a given finite presentation we can algorithmically update the algorithm.\
The following is the analogue definition for the Reiter condition:
\[rei\] $\Gamma$ has *computable Reiter functions* with respect to $X$ if there exists an algorithm with\
INPUT: $n\in \mathbb N$\
OUTPUT: $f\colon \mathbb F_{\mathsf X}\to \mathbb Q^+$, finitely supported, such that ${\pi_{\Gamma}}_*(f)\in \mathfrak Reit_{\Gamma,X}(n)$,\
where ${\pi_{\Gamma}}_*(f)\colon \Gamma \to \mathbb Q^+$ is the *pushforward* of $f$, defined by ${\pi_{\Gamma}}_*(f)(g):=\sum_{\nu\in \pi_{\Gamma}^{-1}(g)}f(\nu)$.
Consider the commutative diagram of group epimorphisms:
$$\begin{tikzcd}
G_1 \arrow{r}{\pi_1} \arrow[swap]{dr}{\pi_3} & G_2 \arrow{d}{\pi_2} \\ & G_3
\end{tikzcd}$$ and $f \colon G_1 \to \mathbb R.$ Then the following holds:
- ${\pi_2}_*({\pi_1}_*(f))={\pi_3}_*(f)$ and if $f$ is finitely supported then ${\pi_1}_*(f)\colon G_2\to \mathbb R$ is finitely supported;\
as a consequence, the definition of computability of Reiter functions does not depend on the choice of the finite set of generators;
- ${\pi_1}_*(_gf)=\,_{\pi_1(g)}\!\, {\pi_1}_*(f),\; \forall g\in G_1$;
- $\|f\|_{1,G_1}\geq \|{\pi_1}_*(f)\|_{1,G_2}\geq \|{\pi_3}_*(f)\|_{1,G_3}$, and, if $f$ is positive, equalities hold;
- ${\pi_1}_*(f) \in \mathfrak Reit_{G_2}(n)\implies {\pi_3}_*(f)\in \mathfrak Reit_{G_3}(n)$,\
thus computability of Reiter functions passes to quotients.
Recursive bounds for Følner functions {#reisec}
=====================================
\[imp\] Suppose that $\Gamma$ is recursively presentable. Then the following are equivalent:
- $\Gamma$ is amenable;
- $\Gamma$ has subrecursive Følner function;
- there exists an algorithm with\
INPUT: $n\in \mathbb N$\
OUTPUT: $F \subset \mathbb F_{\mathsf X}$ finite, such that $\pi_{\Gamma}(F)$ contains an $n$-Følner set;
- $\Gamma$ has computable Reiter functions.
It is clear that $(iii)\implies (ii)\implies (i)$;\
$(iv)\implies (iii)$\
For every $n\in \mathbb N$ the output of the algorithm in Definition \[rei\] is a function $f\colon \mathbb F_{\mathsf X}\to \mathbb Q^+$ with finite support, say $F \subset \mathbb F_{\mathsf X}$. Let $h:={\pi_{\Gamma}}_*(f)$ be the pushforward of $f$, so that $h\in \mathfrak Reit_{\Gamma,X}(n)$. Then, as mentioned in Remark \[RF\], there exists $\epsilon\in\mathbb R^+$ such that $\Omega_\epsilon:=\{g\in \Gamma: h(g)>\epsilon\}\in \mathfrak F\o l_{\Gamma,X}(n)$. We complete by observing that $\Omega_\epsilon\subset \pi_\Gamma(F)$.\
$(i)\implies (iv)$\
The first step is to write, fixing $n\in\mathbb N$, a subroutine $\mathfrak K(n)$ that, taken a function $f\colon \mathbb F_{\mathsf X}\to \mathbb Q^+$ with finite support $F\subset \mathbb F_{\mathsf X}$, stops if ${\pi_{\Gamma}}_*(f)\in \mathfrak Reit_{\Gamma,X}(n)$. In fact, even if we cannot compute the pushforward (because we have no assumptions on WP), we can estimate the $n$-invariance after the following arguments.
With every partition $\mathcal Q$ of the finite support $F$ we associate the positive rational numbers $$M^{\mathsf x}_{\mathcal Q}(f):=\frac{\sum_{V\in \mathcal Q}|\sum_{\nu\in V}(f(\nu)-f(\mathsf x^{-1}\nu))|}{\sum_{\nu\in F}f(\nu)},\; \mathsf x\in \mathsf X.$$ Denoting by $\mathcal P$ the canonical partition of $F$ associated with $\pi_\Gamma$ ($\forall \nu_1,\nu_2\in F$ there exists $V\in \mathcal P$ such that $\nu_1,\nu_2\in V$ if and only if $\pi_\Gamma(\nu_1)=\pi_\Gamma(\nu_2))$, we have
$$\label{est}
\frac{\|{\pi_{\Gamma}}_*(f)-_x\!\!{\pi_{\Gamma}}_*(f) \|_{1,\Gamma}}{\|{\pi_{\Gamma}}_*(f)\|_{1,\Gamma}}=M^{\mathsf x}_{\mathcal P}(f),\; \forall \mathsf x\in \mathsf X.$$
By the triangle inequality, for any two partitions $\mathcal Q$ and $\mathcal Q'$ of $F$ if ${\mathcal Q \leq \mathcal Q'}$ then $M^{\mathsf x}_{\mathcal Q}(f)\geq M^{\mathsf x}_{\mathcal Q'}(f)$. In particular for any partition $\mathcal P'$ of $F$ such that $\mathcal P'\leq \mathcal P$, or equivalently, such that ${\nu_1,\nu_2\in V\in \mathcal P'}\implies \pi_\Gamma(\nu_1)=\pi_\Gamma(\nu_2)$, using equation we have
$$\label{est1}
\frac{\|{\pi_{\Gamma}}_*(f)-_x{\pi_{\Gamma}}_*(f) \|_{1,\Gamma}}{\|{\pi_{\Gamma}}_*(f)\|_{1,\Gamma}}\leq M^{\mathsf x}_{\mathcal P'}(f),\; \forall \mathsf x\in \mathsf X.$$
So we define $\mathfrak K(n)$ as follows: with input $f$, it sets $\mathcal P_0:= \{\{f\}: f\in F\}$, the finest partition of $F$. As $\Gamma$ is recursively presented, there is a recursive enumeration $\eta_1, \eta_2,\ldots$ of the words in $\ker \pi_\Gamma$. When $\mathfrak K(n)$ reads $\eta_m$, for every pair of distinct $V_1, V_2\in \mathcal P_{m-1}$ such that $\eta_m\in V_1 V_2^{-1}$, it merges $V_1$ and $V_2$, defining a new partition $\mathcal P_{m}$; then it computes $M^{\mathsf x}_{\mathcal P_m}(f)$ and, if $M^{\mathsf x}_{\mathcal P_m}(f)\leq n^{-1}$ for every $\mathsf x\in \mathsf X$, it stops, if not, it goes to the next trivial word $\eta_{m+1}$.
By construction $\mathcal P_m\leq \mathcal P$ and the inequality holds (with $\mathcal P'=\mathcal P_m$); thus, when $\mathfrak K(n)$ stops, $M^{\mathsf x}_{\mathcal P_m}(f)\leq n^{-1}$ for every $\mathsf x\in \mathsf X$, and therefore ${\pi_{\Gamma}}_*(f)$ is $n$-invariant. Conversely, if ${\pi_{\Gamma}}_*(f)$ is $n$-invariant, at latest when $\mathcal P_m=\mathcal P$ we have $M^{\mathsf x}_{\mathcal P_m}(f)\leq n^{-1}$, for any $\mathsf x\in\mathsf X$, by equality .
Now, using hypothesis (i), for every $n\in \mathbb N$ there exists a non-empty finite subset $F\in \mathbb F_{\mathsf X}$ such that $\pi_{\Gamma}(F)\in \mathfrak F\o l_{\Gamma,X}(2n)$ and $|F|=|\pi_\Gamma(F)|$: the pushforward of the characteristic function $\chi_F$ of $F$ is the characteristic function $\chi_{\pi_\Gamma(F)}\in \mathfrak Reit_{\Gamma,X}(n)$, by Remark \[RF\]. We list all finite subsets of $\mathbb F_{\mathsf X}$: $F_1,F_2,\ldots$ (they are countably many) and we simultaneously run $\mathfrak K(n)$ on $\chi_{F_1}$, $\chi_{F_2} \ldots$ until one of the subroutines stops, providing a function with $n$-invariant pushforward (the sought Reiter funtion).
In general, the algorithm $\mathfrak K(n)$ may stop also with a function $\chi_F$ whose pushforward is not a characteristic function in $\Gamma$. This obstruction to reach $n$-Følner sets cannot be avoided because if we could change $\mathfrak K(n)$ in order to stop only when ${\pi_{\Gamma}}_*(\chi_F)$ is characteristic, this would imply that $\Gamma$ has solvable Word Problem (this is a consequence of Theorem \[WP\] that we will see in the next section). This is in general impossible, even for finitely presented groups with subrecursive Følner function.
The question –whether we can obtain computability of Følner sets (i.e. of a preimage not necessarily 1-1) with a similar algorithm– remains open: actually, we can estimate better and better $|\pi_\Gamma(F)\setminus x\pi_\Gamma(F)|$ from above listing the elements in $\ker \pi_\Gamma$, but in this case the denominator $| \pi_\Gamma(F)| $ is not computable and, at least for a general set, it is impossible to estimate from below its cardinality without solvability of the Word Problem. The same issue appears for stability of computability of Følner sets under quotients, see [@Pre].
Consider an enumeration $(P_i)_{i\in\mathbb N}$ of all finitely generated recursive presentations, $P_i=\{\mathsf X_i| \mathsf R_i\}$, $\Gamma_i:=\mathbb F_{\mathsf X_i}\slash \mathsf R_i^{\mathbb F_{\mathsf X_i}}$. Clearly, we can extend $\mathfrak K$ to the universal algorithm $\widehat{\mathfrak K}$, that taking as an input $n$ and a presentation $P_i$, runs as $\mathfrak K(n)$ on $\mathbb F_{\mathsf X_i}$, using only the recursive set of relations $\mathsf R_i$, and stops if the group $\Gamma_i$ admits $n$-Følner sets with respect to $X_i$.
Recall (cf. [@Mal]) that a *partially recursive, $k$-place function* is a function $\mathcal U\colon D_{\mathcal U}\to \mathbb N$, where $D_{\mathcal U}\subset\mathbb N^k$, such that there exists an algorithm that for every input $(n_1,n_2,\ldots,n_k)\in D_{\mathcal U}$ stops and gives $\mathcal U(n_1,n_2,\ldots,n_k)$ as an output.
\[c1\] There exists a $2$-place partial recursive function $\mathcal U$ such that $$F_{\Gamma_i,\mathsf X_i}(n)\leq \mathcal U(i,n)$$ on the domain $\{(i,n)\in\mathbb N^2:\;F_{\Gamma_i,\mathsf X_i}(n)< \infty \}.$
\[c2\] For every $n\in \mathbb N$ fixed, the set of finitely generated recursive presentations of groups admitting $n$-Følner sets is recursively enumerable.
For every $n\in \mathbb N$ fixed the property of admitting $n$-Følner sets is a presentation property, not a group property.
\[c3\] For every recursively enumerable class $\mathcal C$ of finitely generated recursive presentations of amenable groups there exists a recursive function $U_{\mathcal C}$ such that for every $P_i\in \mathcal C$: $$F_{\Gamma_i,\mathsf X_i}\leq U_{\mathcal C} \;\; \mbox{eventually}.$$
More generally, suppose that $(f_i)_{i\in\mathbb N}$ is a recursively enumerable set of recursive functions $f_i\colon \mathbb N\to \mathbb N$. Then the function $U\colon \mathbb N\to \mathbb N$, defined as $$U(n):= \max_{i\leq n} f_i(n)$$ is recursive and eventually dominates $f_i$, for every $i\in\mathbb N$.
This concludes the proof of Theorem \[A\] in the Introduction.
Amenability and the Word Problem {#swp}
================================
\[WP\] The following are equivalent:
- $\Gamma$ is amenable with solvable Word Problem;
- $\Gamma$ is recursively presentable and there exists an algorithm with\
INPUT: $n\in \mathbb N$\
OUTPUT: $F \subset \mathbb F_{\mathsf X}\,$ finite, such that $\pi_{\Gamma}(F)\in \mathfrak F\o l_{\Gamma,X}(n)$ and $|F|=|\pi_\Gamma(F)|.$
${}$\
$(i)\implies (ii)$\
Suppose $\Gamma$ is amenable with solvable Word Problem. Then, by the latter property, for any given finite subset $F \subset \mathbb F_X$ we can algorithmically check if $\pi_{\Gamma}(F)\in \mathfrak F\o l_{\Gamma,X}(n)$ and $|F|=|\pi_\Gamma(F)|.$ Fixing an enumeration of the finite subsets of $\mathbb F_X$, we check these conditions until we find a suitable $F$, whose existence is guaranteed by amenability of $\Gamma$.\
Finally, solvability of the Word Problem ensures existence of a recursive set $R:=\ker \pi_\Gamma$ of defining relations of $\Gamma$.\
$(ii)\implies (i)$\
It is clear that $(ii)$ implies amenability of $\Gamma$. It remains to show that $\Gamma$ has solvable Word Problem. By virtue of Remark \[RF\], we have that $\mathfrak F\o l_{\Gamma,X\cup X^{-1}}(n)=\mathfrak F\o l_{\Gamma,X}(n)$. Moreover, solvability of the Word Problem does not depend on the choice of the generating set. We can therefore assume, without loss of generality, that $X=X^{-1}$. For a given $\omega\in \mathbb F_{\mathsf X}$, we denote by $n:=\max\{|\omega|,3\}$ and compute a finite subset $F$ of $\mathbb F_{\mathsf X}$ such that $\pi_{\Gamma}(F)\in \mathfrak F\o l_{\Gamma,X}(n^2)$ and $|F|=|\pi_\Gamma(F)|=:k$. We write $F=:\{f_1,f_2,\ldots, f_k\}$ and ${\mathsf X=:\{ \mathsf x_1, \mathsf x_2,\ldots, \mathsf x_d\}}$. We are going to algorithmically construct $d$ permutations $\sigma_1,\ldots,\sigma_d\in Sym(k)$ that are “approximations" for the left action of $x_1,\ldots x_d$ on $\pi_\Gamma(F)$, interpreting $[k]$ as a copy of $\pi_\Gamma(F)$. We have no assumptions on the Word Problem but the group $\Gamma$ is recursively presented, thus $\ker \pi_\Gamma$ is recursively enumerable: in order to obtain the sought permutations we list the trivial words $\eta_1,\eta_2,\ldots, \eta_t,\ldots$ and then, for every $\ell\in [d]$, we construct, in a way that we will describe soon, a sequence of approximations $$\Sigma^0_\ell\subset \Sigma^1_\ell\subset \cdots \subset \Sigma^t_\ell \subset\cdots,$$ where $\Sigma^t_\ell $, for $t=0,1,\ldots$, is not yet a permutation of $[k]$ but just a subset of $[k]^2$, with the following property: $$\label{leftaction}
(i,j)\in \Sigma^t_\ell \implies \pi_\Gamma(\mathsf x_\ell f_i)=\pi_\Gamma(f_j).$$ We start by setting $\Sigma^0_\ell=\emptyset$ for $\ell=1,\ldots d$. So, for $t=0$, property trivially holds.\
As we list the elements of $\ker \pi_\Gamma$, we update the $\Sigma^t_\ell$’s in this way: we read $\eta_t \in \ker \pi_\Gamma$, for each $(i,j)\in[k]^2$ and each $\ell\in[d]$ such that $\mathsf x_\ell f_i f_j^{-1}= \eta_t$ in $\mathbb F_{\mathsf X}$, we set $\Sigma^t_\ell=\Sigma^{t-1}_\ell\cup \{(i,j)\}$. In this way, property is maintained for every $t$.\
We stop when we meet $\hat t$ such that $\min_\ell |\Sigma^{\hat t}_\ell|>(1-\frac{1}{n^2})k$. We then simply write $\Sigma_\ell$ instead of $\Sigma^{\hat t}_\ell$.\
Indeed, since $\pi_{\Gamma}(F)\in \mathfrak F\o l_{\Gamma,X}(n^2)$, by Remark \[RF\], we have that
$$\label{dannata}
\frac{|\{(i,j):\,\mathsf x_\ell f_i f_j^{-1}\in \ker \pi_\Gamma\}|}{k}\geq
\frac{|\pi_{\Gamma}(F)\cap x_\ell \pi_{\Gamma}( F)|}{|\pi_{\Gamma}(F)|}>1-\frac{1}{n^2}.$$
This guarantees that our procedure will stop. Injectivity of $\pi_\Gamma$ on $F$ guarantees that if $(i,j),(i',j')\in \Sigma_\ell$ are distinct then $i\neq i'$ and $j\neq j'$. Then for $\ell=1,\ldots,d$ we can algorithmically choose $\sigma_\ell\in Sym(k)$, a permutation of $[k]$ such that $(i,j)\in \Sigma_\ell \implies \sigma_\ell(i)=j.$
The permutations $\sigma_1,\ldots,\sigma_d$ have the following property $$\label{dogma}
\ell_H(\omega(\sigma_1,\ldots,\sigma_d))
\begin{cases}
\leq \frac{1}{n}, \; \mbox{ if }\omega \in B_n\cap \ker \pi_\Gamma \\
\geq 1 - \frac{1}{n},\;\mbox{ if } \omega \in B_n \setminus \ker \pi_\Gamma
\end{cases}$$ where for $\sigma\in Sym(k)$ the positive real number $\ell_H(\sigma):=\frac{|\{i\in[k]:\;\sigma(i)\neq i\}|}{k}$ is the *normalized Hamming length* of $\sigma$.
Suppose $\omega=\mathsf x_{l_n}\ldots\mathsf x_{l_2}\mathsf x_{l_1}$, where $l_z\in [d]$, $z=1,2,\ldots,n$. We define the subset $$I_{\omega}:=\{i_0\in [k]: \,
\exists i_1, i_2,
\ldots, i_n \in [k] : \,(i_{t-1},i_{t})
\in \Sigma_{l_z},
\forall z\in [n] \}.$$
Informally, $ I_{\omega}$ is the set of $i \in [k]$ for which we can compute $\omega(\sigma_1,\ldots,\sigma_d)(i)=\sigma_{l_n}\ldots\sigma_{l_2}\sigma_{l_1}(i)$ only looking at $\Sigma_1,\ldots,\Sigma_d$. In particular, by property of the $\Sigma_\ell$’s, we have: $$\label{parola}
i\in I_\omega \implies \pi_\Gamma(\omega f_i)=\pi_\Gamma(f_{\omega(\sigma_1,\ldots,\sigma_d)(i)}).$$ Setting $N_{\ell}:=\{i\in[k]: (i,j)\notin \Sigma_\ell \; \forall j\in [k] \}$, we can also write $I_\omega=\{i_0\in [k]: \, \sigma_{l_{n'}}\ldots\sigma_{l_2}\sigma_{l_1}(i_0)\notin N_{l_{n'}},\, \forall n'\in[n] \}.$\
In order to estimate the cardinality of $I_\omega$, we define $\phi \colon [k]\setminus I_\omega \hookrightarrow N_{l_n}\sqcup\ldots \sqcup N_{l_2}\sqcup N_{l_1}$,\
$\phi(i):=( n',i')$ where $n'$ is the smallest number in $[n]$ such that $\sigma_{l_{n'}}\ldots\sigma_{l_2}\sigma_{l_1}(i)\in N_{l_{n'}}$, and $i':=\sigma_{l_{n'}}\ldots\sigma_{l_2}\sigma_{l_1}(i)$. By construction of $\Sigma_\ell$, $|N_\ell|\leq \frac{k}{n^2}$, combining with the fact that the map $\phi$ is injective, we have $$\label{acc}\begin{split}
&|[k]\setminus I_\omega|\leq \sum^n_{z=1} |N_{l_z}^{s_z}|\leq \frac{k}{n},\\
&|I_\omega|\geq (1-\frac{1}{n})k.\end{split}$$
Suppose $\omega\in \ker \pi_\Gamma$. Then, for $i\in I_\omega$, property implies $\pi_\Gamma(f_{\omega(\sigma_1,\ldots,\sigma_d)(i)})=\pi_\Gamma(f_{i})$. By injectivity of $\pi_\Gamma$ on $F$, $i$ is a fixed point of $\omega(\sigma_1,\ldots,\sigma_d)$; by virtue of estimate , we the have $\ell_H(\omega(\sigma_1,\ldots,\sigma_d))\leq \frac{|[k]\setminus I_\omega|}{k}\leq \frac{1}{n}.$
If $\omega\notin \ker \pi_\Gamma$, then again by property we have that, for $i\in I_\omega$, $\pi_\Gamma(f_{\omega(\sigma_1,\ldots,\sigma_d)(i)})\neq\pi_\Gamma(f_{i})$. This means that $I_\omega$ contains only non-fixed points and therefore, by virtue of estimate ,\
$\ell_H(\omega(\sigma_1,\ldots,\sigma_d))\geq \frac{|I_\omega|}{k}\geq 1- \frac{1}{n}.$ This ends the proof of the claim.
We are now in position to complete the proof of the theorem. Since the number $\ell_H(\omega(\sigma_1,\ldots,\sigma_d))$ is computable, by property we can algorithmically determine whether $\omega$ belongs to $\ker \pi_\Gamma$ or not. Thus $\Gamma$ has solvable Word Problem (in the terminology of [@CAV] we actually proved that $\Gamma$ has *computable sofic approximations*, see Theorem 3.3.1 in [@CAV]).
In combination with Theorem \[WP\] and the results in [@Pre], this proves Theorem \[C\] in the Introduction.
\[ovvio\] Suppose that $\Gamma$ has solvable Word Problem. Then the following are equivalent:
- $\Gamma$ is amenable;
- there exists an algorithm with\
INPUT: $n\in \mathbb N$\
OUTPUT: $F \subset \mathbb F_X$ finite, such that $\pi_{\Gamma}(F)\in \mathfrak F\o l_{\Gamma,X}(n)$ and $|F|=|\pi_\Gamma(F)|;$
- $\Gamma$ has computable Følner sets;
- $\Gamma$ has computable Reiter functions;
- $\Gamma$ has subrecursive Følner function.
By virtue of Theorem \[WP\] we have $(i)\implies (ii)$. It is obvious that $(ii)\implies (iii)\implies (i)$ and that $(ii)\implies (v)\implies (i)$; by Remark \[RF\] we have $(iv)\implies (i)$.\
Finally $(ii)\implies(iv)$ because if $F \subset \mathbb F_X$ is finite, such that $\pi_{\Gamma}(F)\in \mathfrak F\o l_{\Gamma,X}(2n)$ and ${|F|=|\pi_\Gamma(F)|}$ then the pushforward of the characteristic function $\chi_F$ of $F$ is the characteristic function $\chi_{\pi_\Gamma(F)}$ of $\pi_\Gamma(F)$: this is $n$-invariant by Remark \[RF\].
Generic EP {#GEP}
==========
This section is devoted to proving the following theorem (cf. Theorem \[B\] in the Introduction).
\[uno\] Suppose that $\Gamma$ is amenable and recursively presentable. Then the following are equivalent:
- $\Gamma$ has solvable Word Problem;
- $\Gamma$ has solvable generic Equality Problem.
As stated in the Introduction, the Kharlampovich groups $G(M)$ are finitely presented, solvable and therefore amenable, and have unsolvable Word Problem (see [@H; @KMS]). Therefore, by the previous theorem, they have unsolvable generic Equality Problem, thus providing a solution to [@Afinite Problem 1.5, b].
In order to prove Theorem \[uno\], we need some preliminary results.
\[elenco\] Suppose $\Gamma$ has solvable Equality Problem on $S$, where $S\subset \mathbb F_{\mathsf X}$. Then there exists a family $\mathcal A$ of finite subsets of $\mathbb F_{\mathsf X}$, with the following properties:
1. $\mathcal A$ is recursively enumerable;
2. ${\pi_\Gamma}_{|_A}$ is injective $\forall A\in \mathcal A$;
3. $\forall S'\subset S, \mbox{ $S'$ finite, }\; \exists A\in \mathcal A$ such that $ \pi_\Gamma(A)= \pi_\Gamma(S')$.
Let $\mathfrak A$ be the associated algorithm for the solvability of the Equality Problem. Recall that $\mathfrak A$ (at least) stops on $S\times S$. We can easily define an algorithm $\mathfrak A'$ with input $B$, any finite subset of $\mathbb F_{\mathsf X}$, that checks if any two words in $B$ represent the same elements in $\Gamma$, that is, it checks if ${\pi_\Gamma}_{|_B}$ is injective. Clearly $\mathfrak A'$ stops at least for every finite $B\subset S$. Thus we enumerate all finite subsets of $\mathbb F_{\mathsf X}$: $B_1, B_2,\ldots$ and we simultaneously (diagonally) run $\mathfrak A'$ on these sets, and give as an output only those subsets $B$ for which the two following conditions are met: $\mathfrak A'$ stops and $\mathfrak A'$ has checked that ${\pi_\Gamma}_{|_B}$ is injective. Let $\mathcal A$ be the set of these outputs. Propeties (1-[$\mathcal A$]{}) and (2-[$\mathcal A$]{}) hold by construction of $\mathcal A$. For any finite $S'\subset S$, for each element of $\pi_\Gamma(S')$ we choose only one representative word in $S'$, obtaining a subset $A\subset S'\subset S$ such that $\pi_\Gamma(A)= \pi_\Gamma(S')$ and ${\pi_\Gamma}_{|_A}$ is injective. Then $A\in \mathcal A$ and the property (3-[$\mathcal A$]{}) is proved.
\[genefol\] Suppose that $S$ is a generic subset of $\mathbb F_{\mathsf X}$. Then for every finite subset $F\subset \mathbb F_{\mathsf X}$ there exists $y\in \mathbb F_{\mathsf X}$ such that $Fy\subset S$.
Since for every finite set $F$ there exists $k\in \mathbb N$ such that $F\subset B_k$, without loss of generality we may reduce to the case $F=B_k$. We denote by $N:=S^c$, the complement of $S$; so that, being $S$ generic, $N$ is *negligible*, that is $\frac{|N\cap B_n|}{|B_n|}\to 0$. We want to prove that there exists $y\in \mathbb F_{\mathsf X}$ such that $N\cap B_k y=\emptyset$. Recall that $S_n:=B_n\setminus B_{n-1}$ is the $n$-sphere in $\mathbb F_{\mathsf X}$.
For every $m\in \mathbb N$ we have: $$B_{m+2k}\supset \bigsqcup_{\omega\in S_m}
B_k a_{\omega} \omega,$$ where, for every $\omega\in S_m$, the word $a_{\omega}$ is a suitable element of $S_k$ such that $|a_{\omega} \omega|=m+k.$ Let’s check the disjointness of the union. For all distinct $ \omega,\, \omega'\in S_m$, since $|\omega \omega'^{-1}|\geq 2$ we have $|a_{\omega}
\omega
\omega'^{-1}
a_{\omega'}^{-1}|\geq 2k+2.$ By the triangular inequality, $B_k a_{\omega}\omega$ and $B_k a_{\omega'}\omega'$ are disjoint.
Suppose, by contradiction, that $N\cap B_k y\neq \emptyset$ for every $y$, then we have $$\label{quasi}\frac{|B_n\cap N|}{|B_n|}\geq
\frac{|S_{n-2k}|}{|B_n|}
\rightarrow
\frac{2|\mathsf X|-2}{(2|\mathsf X|-1)^{2k+1}}.$$ If $|\mathsf X|\geq 2$, this is impossible since the set $N$ is negligible.\
If $|\mathsf X|=1$, we notice that in $B_n$ there are approximately $\frac{n}{k}$ disjoint copies of $B_k$ and the limit in equals $\frac{1}{k}$, providing again a contradiction.
There are other notions of genericity: for instance, one may replace the balls $B_n$ by the spheres $S_n=B_n\setminus B_{n-1}$ in Equation . It follows from Cesaro’s theorem that any $(S_n)$-generic set is also $(B_n)$-generic. As a consequence, Lemma \[genefol\] remains true if we suppose that $S$ is $(S_n)$-generic. Moreover, upper Banach genericity is strictly weaker than genericity: fixing $\mathsf x\in \mathsf X$, for any function $f\colon \mathbb N \to \mathbb N$ the subset $T_f:=\bigcup_{n\in\mathbb N} B_n \mathsf x^{f(n)}$ clearly contains an increasing sequence of translated balls, but the asymptotic behavior of the ratio $\frac{|T_f\cap B_n|}{|B_n|}$ can be arbitrary (it depends on the growth of $f$).
\[GF\] Suppose that $\Gamma$ is amenable and $S\subset \mathbb F_{\mathsf X}$ is generic. Then $\pi_\Gamma(S)$ contains a Følner sequence: $\forall n\in\mathbb N\; \exists\, \Omega_n\in \mathfrak F\o l_{\Gamma,X}(n) \mbox{ such that } \Omega_n\subset \pi_\Gamma(S).$
Since $\Gamma$ is amenable, for every $n\in\mathbb N$ there exists a finite subset $F_n\subset \mathbb F_{\mathsf X}$ such that ${\pi_\Gamma(F_n)\in \mathfrak F\o l_{\Gamma,X}(n)}$. Since $S$ is generic, then by virtue of Lemma \[genefol\] there exists ${y_n\in \mathbb F_{\mathsf X}}$ such that $F_n y_n\subset S$; by Remark \[RF\], the set $\Omega_n:=\pi_\Gamma(F_n y_n)\in \mathfrak F\o l_{\Gamma,X}(n)$.
${}$\
$(i)\implies (ii)$ is true in general.\
$(ii)\implies (i)$\
By virtue of Theorem \[WP\], it is enough to show the existence of a finite generating set $Y$ and an algorithm with:\
INPUT: $n\in \mathbb N$\
OUTPUT: $F \subset \mathbb F_{\mathsf Y}$ finite, such that $\pi_{\Gamma}(F)\in \mathfrak F\o l_{\Gamma,Y}(n)$ and $|F|=|\pi_\Gamma(F)|.$\
Since $\Gamma$ has solvable generic Equality Problem, there exists a set of generators, say $Y$, and a generic subset $S\subset \mathbb F_{\mathsf Y}$ with solvable EP.\
Let $\mathcal A$ be the family given by Lemma \[elenco\]. By property (1-${\mathcal A}$) we have a recursive enumeration of $\mathcal A$: $E_1,E_2,\ldots$ . Thanks to property (3-[$\mathcal A$]{}), the family $\pi_\Gamma(\mathcal A):=\{\pi_\Gamma(E_1), \pi_\Gamma(E_2),\ldots\}$ contains $\{\pi_\Gamma (S'):\; S'\subset S,\; S'\mbox{ finite}\}$ and, by Lemma \[GF\], for every $n \in \mathbb N$ we have $$\pi_\Gamma(\mathcal A) \cap \mathfrak F\o l_{\Gamma,Y}(n)\neq \emptyset.$$ The property (2-[$\mathcal A$]{}) ensures that ${\pi_\Gamma}_*(\chi_{E_i})=\chi_{\pi_{\Gamma}(E_i)}$, and therefore by Remark \[RF\], for all $n\in\mathbb N$ $$\{{\pi_\Gamma}_*(\chi_{E_1}), {\pi_\Gamma}_*(\chi_{E_2}),\ldots\}\cap \mathfrak Reit_{\Gamma,Y}(n) \neq\emptyset.$$ We now are in position to define the sought algorithm:\
for every $n\in \mathbb N$ we run the algorithm $\mathfrak K(n)$ used in the proof of Theorem \[imp\], simultaneously on the functions $\chi_{E_1}, \chi_{E_2},\ldots$ until one of the subroutines stops, providing a function $\chi$ such that ${{\pi_\Gamma}_*(\chi)\in \mathfrak Reit_{\Gamma,Y}(n)}$. Again, by the property (2-[$\mathcal A$]{}), the pushforward ${\pi_\Gamma}_*(\chi)$ is still a characteristic function and then by Remark \[RF\], the output $F:=Supp(\chi)$ (i.e. $\chi=\chi_F$) satisfies the required conditions.
Questions and final remarks {#QR}
===========================
The existence of a recursive universal bound for recursively (resp. finitely) presented amenable groups can be related to the arithmetic hierarchy of the property of being amenable. But there is no hope to establish, using our algorithm, if the bound is primitively recursive, since the stopping time depends on the bound itself.
Is the class of recursively (finitely) presented amenable groups recursively enumerable?
For solvable groups the question is open (see [@miller]), even if in this case a universal bound for Følner functions of groups of this class is known [@SC]. In [@gri] there are some questions and remarks about decidability of amenability and bounds for Følner function in some subclasses of groups.
The Kharlampovich groups $G(M)$ have:
- unsolvable Word Problem [@H];
- solvable generic Word Problem [@KMSS];
- unsolvable strongly generic Word Problem [@GMO];
- unsolvable generic Equality Problem (Corollary in the Introduction);
- computable Følner sets [@CAV; @Pre].
Here a subset $ S \subset \mathbb F_{\mathsf X}$ is *strongly generic* if $\frac{|S\cap B_n|}{|B_n|}\to 1$ exponentially fast, and a strongly generic problem is solvable if it is solvable on a strongly generic set (for some generating set).\
As an easy consequence, we deduce that solvability of generic WP does not imply solvability of generic EP.
Does solvability of the strongly generic WP imply solvability of the (strongly) generic EP?
An answer to this question would make clearer the relation between Theorem \[B\] and the following.
Let $G$ be a finitely presented amenable group with unsolvable word problem. Then for any choice of generators $W\to G$ the word problem in $G$ is not solvable on any exponentially generic subset of $W$.
We can also measure genericity for the Equality Problem in $\mathbb F_{\mathsf X}\times \mathbb F_{\mathsf X}$ with a general subset, not necessarily of type $S\times S$, that is $$T\subset \mathbb F_{\mathsf X}\times \mathbb F_{\mathsf X} \mbox{ is $(B_n\times B_n)$-\emph{generic} if } \frac{|T\cap (B_n\times B_n)|}{|B_n\times B_n|}\to 1.$$ With this weaker notion of genericity for the EP it is not clear if we can reach the analogous thesis of Theorem \[B\].
Finally, the last question that we asked in [@Pre]: –Does subrecursivity of the Følner function imply computability of Følner sets?– can be replaced, in view of Theorem \[C\], by the following.
Does there exist a recursively presented amenable group that has not computable Følner sets?
|
---
abstract: 'Parsing articulated objects, humans and animals, into semantic parts (body, head and arms, ) from natural images is a challenging and fundamental problem for computer vision. A big difficulty is the large variability of scale and location for objects and their corresponding parts. Even limited mistakes in estimating scale and location will degrade the parsing output and cause errors in boundary details. To tackle these difficulties, we propose a “Hierarchical Auto-Zoom Net” (HAZN) for object part parsing which adapts to the local scales of objects and parts. HAZN is a sequence of two “Auto-Zoom Nets" (AZNs), each employing fully convolutional networks that perform two tasks: (1) predict the locations and scales of object instances (the first AZN) or their parts (the second AZN); (2) estimate the part scores for predicted object instance or part regions. Our model can adaptively “zoom” (resize) predicted image regions into their proper scales to refine the parsing. We conduct extensive experiments over the PASCAL part datasets on humans, horses, and cows. For humans, our approach significantly outperforms the state-of-the-arts by $5\%$ mIOU and is especially better at segmenting small instances and small parts. We obtain similar improvements for parsing cows and horses over alternative methods. In summary, our strategy of first zooming into objects and then zooming into parts is very effective. It also enables us to process different regions of the image at different scales adaptively so that, for example, we do not need to waste computational resources scaling the entire image.'
author:
- 'Fangting Xia, Peng Wang, Liang-Chieh Chen & Alan L. Yuille'
bibliography:
- 'eccv2016.bib'
title: 'Zoom Better to See Clearer: Human and Object Parsing with Hierarchical Auto-Zoom Net'
---
16SubNumber[0000]{}
Introduction
============
When people look at natural images, they often first locate regions that contain objects, and then perform the more detailed task of object parsing, decomposing each object instance into its semantic parts. Object parsing, of humans and horses, is important for estimating their poses and understanding their semantic interactions with others and with the environment.
In computer vision, object parsing plays a key role for real understanding of objects in images and helps for many visual tasks, e.g., segmentation [@eslami2012generative; @wang2015joint], pose estimation [@dong2014towards], and fine-grained recognition [@zhang2014part]. It also has many industrial applications such as robotics and image descriptions for the blind.
There has been a growing literature on the related task of object semantic segmentation due to the availability of evaluation benchmarks such as PASCAL VOC [@everingham2014pascal] and MS-COCO [@lin2014microsoft]. There has been work on human parsing, segmenting humans into their semantic parts, but this has mainly studied under constrained conditions which pre-suppose known scale, fairly accurate localization, clear appearances, and/or relatively simple poses [@bo2011shape; @zhu2011max; @eslami2012generative; @yamaguchi2012parsing; @dong2014towards; @LiuCVPR15]. There are few works done on parsing animals, like cows and horses, and these often had similar restriction, e.g., roughly known size and location [@wang2014semantic; @wang2015joint].
In this paper we address the task of parsing objects, such as humans and horses, in “the wild" where there are large variations in scale, location, occlusion, and pose. This motivates us to work with PASCAL images [@everingham2014pascal] because these were chosen for studying multiple visual tasks, do not suffer from dataset design bias [@li2014secrets], and include large variations of objects, particularly of scale. Hence parsing humans in PASCAL is considerably more difficult than in datasets, such as Fashionista [@yamaguchi2012parsing], that were constructed solely to evaluate human parsing.
Recently, deep learning methods have led to big improvements on object parsing [@hariharan2014hypercolumns; @wang2015joint]. These improvements are due to fully convolutional nets (FCNs) [@long2014fully] and the availability of object part annotations on large-scale datasets, e.g. PASCAL [@chen2014detect]. Although these methods worked well, they can make mistakes on small or large scale objects and, in particular, they have no mechanism to adapt to the size of the object.
{width="1.0\linewidth"}
\[fig:intuition\]
In this paper, we present a hierarchical method for object parsing which performs scale estimation and object parsing jointly and is able to adapt its scale to objects and parts. It is partly motivated by the proposal-free end-to-end detection strategies [@huang2015densebox; @ren2015faster; @DBLP:journals/corr/RedmonDGF15; @DBLP:journals/corr/LiangWSYLY15]. To get some intuition for our approach observe, in Fig. \[fig:intuition\](a), that the scale and location of a target object, and of its corresponding parts, can be estimated accurately from the field-of-view (FOV) window by applying a deep net. We call our approach “Hierarchical Auto-Zoom Net" (HAZN) which parses the objects at three levels of granularity, namely image-level, object-level, and part-level, gradually giving clearer and better parsing results, see Fig. \[fig:intuition\](b). The HAZN sequentially combines two “Auto-Zoom Nets" (AZNs), each of which predicts the locations and scales for objects (the first AZN) or parts (the second AZN), properly zooms (resizes) the predicted image regions, and refines the object parsing result for those image regions (see Fig \[fig:framework\]). The HAZN uses three fully convolutional neural networks (FCNs) [@long2014fully] that share the same structure. The first FCN acts directly on the image to estimate a finite set of possible locations and sizes of objects (e.g., bounding boxes) with confidence scores, together with a part score map of the image. The part score map (in the bottom left of Fig. \[fig:intuition\](b)) is similar to that proposed by previous deep-learned methods. The object bounding boxes are scaled to a fixed size by zooming in or zooming out (as applicable) and the image and part score maps within the boxes are also scaled (e.g.,by bilinear interpolation for zooming in or downsampling for zooming out). Then the second FCN is applied to the scaled object bounding boxes to make proposals (bounding boxes) for the positions and sizes of the parts, with confidence values, and to re-estimate the part scores within the object bounding boxes. This yields improved part scores, see the bottom-middle score map of Fig. \[fig:intuition\](b). We then apply the third FCN to the scaled part bounding boxes to produce new estimates of the part scores and to combine all of them (for different object and part bounding boxes) to output final part scores, see the bottom right of Fig. \[fig:intuition\](b), which are our parse of the object. This strategy is modified slightly so that, for example, we scale humans differently depending on whether we have detected a complete human or only the upper part of a human, which can be determined automatically from the part score map.
{width="1.0\linewidth"}
\[fig:framework\]
We now briefly comment on the advantages of our approach for dealing with scale and how it differs from more traditional methods. Previous methods mainly select a fixed set of scales in advance and then perform fusion on the outputs of a deep net at different layers. Computational requirements mean that the number of scales must be small and it is impractical (due to memory limitations) to use very fine scales. Our approach is considerably more flexible because we adaptively estimate scales at different regions in the image which allows us to search over a large range of scales. In particular, we can use very fine scales because we will probably only need to do this within small image regions. For example, our largest zooming ratio is 2.5 (at part level) on PASCAL while that number is 1.5 if we have to zoom the whole image. This is a big advantage when trying to detect small parts, such as the tail of a cow, as is shown by the experiments. In short, the adaptiveness of our approach and the way it combines scale estimation with parsing give novel computational advantages.
We illustrate our approach by extensive experiments for parsing humans on the PASCAL-Person-Part dataset [@chen2014detect] and for parsing animals on a horse-cow dataset [@wang2014semantic]. These datasets are challenging because they have large variations in scale, pose, and location of the objects. Our zooming approach outperforms previous state of the art methods by a large margin. We are particulary good at detecting small object parts.
Background
==========
The study of human part parsing has been largely restricted to constrained environments, where a human instance in an image is well localized and has a relatively simple pose like standing or walking [@bo2011shape; @zhu2011max; @eslami2012generative; @yamaguchi2012parsing; @dong2014towards; @LiuCVPR15; @DBLP:journals/corr/XiaZWY15]. These works, though useful for parsing a well cropped human instance from simple commercial product images, are limited when applied to parsing human instances in the wild, since humans in real-world images are often in various poses, scales, and may be occluded or highly deformed. The high flexibility of poses, scales and occlusion patterns is difficult to handle by shape-based and appearance-based models with hand-crafted features or bottom-up segments.
Over the past few years, with the powerful deep convolutional neural networks (DCNNs) [@lecun1998gradient] and big data, researchers have made significant performance improvement for semantic object segmentation in the wild [@chen2014semantic; @dai2015boxsup; @liu2015semantic; @noh2015learning; @papandreou2015weakly; @wang2015towards; @tsogkas2015semantic], showing that DCNNs can also be applied to segment object parts in the wild. These deep segmentation models work on the whole image, regarding each semantic part as a class label. But this strategy suffers from the large scale variation of objects and parts, and many details can be easily missed. [@hariharan2014hypercolumns] proposed to sequentially perform object detection, object segmentation and part segmentation, in which the object is first localized by a RCNN [@girshick2014rich], then the object (in the form of a bounding box) is segmented by a fully convolutional network (FCN) [@long2014fully] to produce an object mask, and finally part segmentation is performed by partitioning the mask. The process has two potential drawbacks: (1) it is complex to train all components of the model; (2) the error from object masks, local confusion and inaccurate edges, propagates to the part segments. Our model follows this general coarse-to-fine strategy, but is more unified (with all three FCNs employing the same structure) and more importantly, we do not make premature decisions. In order to better discover object details and use object-level context, [@wang2015joint] employed a two-stream FCN to jointly infer object and part segmentations for animals, where the part stream was performed to discover part-level details and the object stream was performed to find object-level context. Although this work discovers object-level context to help part parsing, it only uses a single-scale network for both object and part score prediction, where small-scale objects might be missed at the beginning and the scale variation of parts still remains unsolved.
Many studies in computer vision has addressed the scale problem to improve recognition or segmentation. These include exploiting multiple cues [@DBLP:journals/ijcv/HoiemEH08], hierarchical region grouping [@arbelaez2011contour; @florack1996gaussian], and applying general or salient object proposals combined with iterative localization [@DBLP:journals/pami/AlexeDF12; @Yukun_CVPR15; @DBLP:conf/cvpr/WangWZFZL12]. However, most of these works either adopted low-level features or only considered constrained scene layouts, making it hard to handle wild scene variations and difficult to unify with DCNNs. Some recent works try to handle the scale issue within a DCNN structure. They commonly use multi-scale features from intermediate layers, and perform late fusion on them [@long2014fully; @hariharan2014hypercolumns; @chen2014semantic] in order to achieve scale invariance. Most recently, [@chen2015attention] proposed a scale attention model, which learns pixel-wise weights for merging the outputs from three fixed scales. These approaches, though developed on powerful DCNNs, are all limited by the number of scales they can select and the possibility that the scales they select may not cover a proper one. Our model avoids the scale selection error by directly regressing the bounding boxes for objects/parts and zooming the regions into proper scales. In addition, this mechanism allows us to explore a broader range of scales, contributing a lot to the discovery of missing object instances and the accuracy of part boundaries.
The Model
=========
As shown in Fig. \[fig:framework\], our Hierarchical Auto-Zoom model (HAZN) has three levels of granularity for tackling scale variation in object parsing, image-level, object-level, and part-level. At each level, a fully convolutional neural network (FCN) is used to perform scale/location estimation and part parsing simultaneously. The three levels of FCNs are all built on the same network structure, a modified FCN proposed by [@chen2014semantic], namely DeepLab-LargeFOV. This network structure is one of the most effective FCNs in segmentation, so we also treat it as our baseline for final performance comparison.
To handle scale variation in objects and parts, the HAZN concatenates two Auto-Zoom Nets (AZNs), namely object-scale AZN and part-scale AZN, into a unified network. The object-scale AZN refines the image-level part score map with object bounding box proposals while the part-scale AZN further refines the object-level part score map with part bounding box proposals. Each AZN employs an auto-zoom process: first estimates the region of interest (ROI), then properly resizes the predicted regions, and finally refine the part scores within the resized regions.
Object-scale Auto-Zoom Net (AZN)
--------------------------------
\[subsec:AZN\] For the task of object part parsing, we are provided with $n$ training examples $\{{{\mathbf I}}_i, {{\mathbf L}}_i\}_{i=1}^{n}$, where ${{\mathbf I}}$ is the given image and ${{\mathbf L}}$ is the supervision information that provides discrete semantic labels of interest. Our target is to learn the posterior distribution $P(l_j | {{\mathbf I}},j)$ for each pixel j of an image ${{\mathbf I}}$. This distribution is approximated by our object-scale AZN, as shown in Fig. \[fig:prob\_model\].
![Object-scale Auto-Zoom model from a probabilistic view, which predicts ROI region $N(k)$ at object-scale, and then refines part scores based on the properly zoomed region $N(k)$. Details are in Sec. \[subsec:AZN\].](fig/prob_model_eccv_peng.pdf){width="0.8\linewidth"}
\[fig:prob\_model\]
We first use the image-level FCN (see Fig. \[fig:framework\]) to produce the image-level part score map $P_{\iota_1}(l_j | {{\mathbf I}},j)$, which gives comparable performance to our baseline method (DeepLab-LargeFOV). This is a normal ***part parsing network*** that uses the original image as input and outputs the pixel-wise part score map. Our object-scale AZN aims to refine this part score map with consideration of object instance scales. To do so, we add a second component to the image-level FCN, performing regression to estimate the size and location of an object bounding box (or ROI) for each pixel, together with a confidence map indicating the likelihood that the box is an object. This component is called a ***scale estimation network*** (**SEN**), which shares the first few layers with the part parsing network in the image-level FCN. In math, the SEN corresponds to a probabilistic model $P(b_j | {{\mathbf I}}, j)$, where $b_j$ is the estimated bounding box for pixel $j$, and $P(b_j|...)$ is the confidence score of $b_j$.
After getting $\{b_j |\forall j\in{{\mathbf I}}\}$, we threshold the confidence map and perform non-maximum suppresion to yield a finite set of object ROIs (typically 5-10 per image, with some overlap): $\{b_k | k \in {{\mathbf I}}\}$. Each $b_k$ is associated with a confidence score $P(b_k)$. As shown in Fig. \[fig:framework\], a **region zooming** operation is then performed on each $b_k$, resizing $b_k$ to a standard-sized ROI $N(k)$. Specifically, this zooming operation computes a zooming ratio $f(b_k, L^{b_k}_p)$ for bounding box $b_k$, based on the bounding box $b_k$ and the computed image-level part labels $L^{b_k}_p$ within the bounding box, and then enlarges or shrinks the image within the bounding box by the zooming ratio. We will discuss $f()$ in detail in Sec. \[sec:exp\].
Now we have a set of zoomed ROI proposals $\{N(k)|k\in{{\mathbf I}}\}$, each $N(k)$ associated with score $P(b_k)$. We learn another probabilistic model $P(l_j | N(k), {{\mathbf I}}, j)$, which re-estimates the part label for each pixel $j$ within the zoomed ROI $N(k)$. This probabilistic model corresponds to the part parsing network in the object-level FCN (see Fig. \[fig:framework\]), which takes as input the zoomed object bounding boxes and outputs the part scores within those object bounding boxes.
The new part scores for the zoomed ROIs need to be merged to produce the object-level part score map for the whole image. Since there may be multiple ROIs that cover a pixel $j$, we define the neighbouring region set for pixel j as ${{\mathcal Q}}(j)=\{N(k) | j \in N(k), k \in {{\mathbf I}}\}$. Under this definition of ${{\mathcal Q}}(j)$, the **score merging** process can be expressed as Equ. \[eqn:AZN\], which essentially computes the weighted sum of part scores for pixel $j$, from the zoomed ROIs that cover $j$. For a pixel that is not covered by any zoomed ROI, we simply use its image-level part score as the current part score. Formally, the object-level part score $P_{\iota_2}(l_j|{{\mathbf I}}, j)$, is computed as, $$\begin{aligned}
\vspace{-1.6\baselineskip}
P_{\iota_2}(l_j|{{\mathbf I}}, j) &= \sum\nolimits_{N(k) \in {{\mathcal Q}}(j)} P(l_j | N(k), {{\mathbf I}}, j) P(N(k) | {{\mathbf I}}, j); \nonumber \\
P(N(k) | {{\mathbf I}}, j) &= P(b_k) / \textstyle{\sum_{k: N(k) \in {{\mathcal Q}}(j)} P(b_k)}
\label{eqn:AZN}\end{aligned}$$
Hierarchical Auto-Zoom Net (HAZN)
---------------------------------
\[subsec:HAZN\] The scale of object parts can also vary considerably even if the scale of the object is fixed. This leads to a hierarchical strategy with multiple stages, called the Hierarchical Auto-Zoom Net (HAZN), which applies AZNs to images to find objects and then on objects to find parts, followed by a refinement stage. As shown in Fig. \[fig:framework\], we add the part-scale AZN to the end of the object-scale AZN.
We add a second component to the object-level FCN, the SEN network, to estimate the size and location of part bounding boxes, together with confidence maps for every pixel within each zoomed object ROI. Again the confidence map is thresholded, and non-maximal suppresion is applied, to yield a finite set of part ROIs (typically 5-30 per image, with some overlap). Each part ROI is zoomed to a fixed size. Then, we re-estimate the part scores within each zoomed part ROI using the part parsing network in the part-level FCN. The part parsing network is the only component of the part-level FCN, which takes the zoomed part ROI and the zoomed object-level part scores (within the part ROI) as inputs. After getting the part scores within each zoomed Part ROI, the score merging process is the same as in the object-scale AZN.
We can easily extend our HAZN to include more AZNs at finer scale levels if we focus on smaller object parts such as human eyes. If the HAZN contains $n$ AZNs, there are $n+1$ FCNs needed to be trained. The $\iota$-th FCN learns $P_\iota(l_j | N(j)_{\iota-1}, ...)$ to refine the part scores based on the scale/location estimation results $P_{\iota-1}(N(j)_{\iota-1} | ...)$ and the part parsing results $P_{\iota-1}(l_j | N(j)_{\iota-1}, ...)$ from the previous level $\iota-1$. At the same time, the $\iota$-th FCN also learns $P_\iota(N(j)_l | ...)$ to estimate the region of interest (ROI) for the next level $\iota+1$.
Training and Testing Phases for Object-scale AZN {#subsec:train_test}
------------------------------------------------
In this section, we introduce the specific networks to learn our probalistic models. Specifically, we use a modern version of FCN, **DeepLab-LargeFOV** [@chen2014semantic], as our basic network structure. DeepLab-LargeFOV is a stronger variant of DeepLab [@chen2014semantic], which takes the raw image as input, and outputs dense feature maps. DeepLab-LargeFOV modifies the filter weights at the $fc_6$ layer so that its field-of-view is larger. Due to the space limits, we refer readers to the original paper for details.
![Ground truth regression target for training the scale estimation network (SEN) in the image-level FCN. Details in Sec. \[subsec:train\_test\].](fig/model_eccv.pdf){width="0.8\linewidth"}
\[fig:train\_sen\]
#### Training the SEN.
The scale estimation network (SEN) aims to regress the region of interest (ROI) for each pixel $j$ in the form of a bounding box, $b_j$. Here we borrow the idea presented in the DenseBox [@huang2015densebox] to do scale estimation, since it is simple and performing well enough for our task. In detail, at object level, the ROI of pixel $j$ corresponds to the object instance box that pixel $j$ belongs to. For training the SEN, two output label maps are needed as visualized in Fig. \[fig:train\_sen\]. The first one is the bounding box regression map ${{\mathbf L}}_{b}$, which is a four-channel output for each pixel $j$ to represent its ROI $b_j$: ${{\mathbf l}}_{bj}=\{dx_j, dy_j, w_j, h_j\}$. Here $(dx_j, dy_j)$ is the relative position from pixel $j$ to the center of $b_j$; $h_j$ and $w_j$ are the height and width of $b_j$. We then re-scale the outputs by dividing them with $400$. The other target output map is a binary confidence seed map ${{\mathbf L}}_{c}$, in which ${{\mathbf l}}_{cj}\in\{0,1\}$ is the ROI selection indicator at pixel $j$. It indicates the preferred pixels for us to use for ROI prediction, which helps the algorithm prevent many false positives. In practice, we choose the central pixels of each object instance as the confidence seeds, which tend to predict the object bounding boxes more accurately than those pixels at the boundary of an object instance region.
Given the ground-truth label map of object part parsing, we can easily derive the training examples for the SEN: ${{\mathcal H}} = \{{{\mathbf I}}_i, {{\mathbf L}}_{bi}, {{\mathbf L}}_{ci}\}_{i=1}^{n}$, where $n$ is the number of training instances. We minimize the negative log likelihood to learn the weights ${{\mathbf W}}$ for the SEN, and the loss $l_{SEN}$ is defined in Equ. \[eqn:loss\_sen\]. [$$\begin{aligned}
l_{SEN}({{\mathcal H}}|{{\mathbf W}}) &= \frac{1}{n}\sum\nolimits_{i}(l_{b}({{\mathbf I}}_i, {{\mathbf L}}_{bi} | {{\mathbf W}}) + \lambda l_{c}({{\mathbf I}}_i, {{\mathbf L}}_{ci} | {{\mathbf W}})); \nonumber\\
l_{c}({{\mathbf I}}, {{\mathbf L}}_{c}|{{\mathbf W}}) &= -\beta\sum\limits_{j:l_{cj}=1} \log P(l_{cj}^*=1|{{\mathbf I}}, {{\mathbf W}}) -(1-\beta)\sum\limits_{j:l_{cj}=0}\log P(l_{cj}^*=0|{{\mathbf I}}, {{\mathbf W}}); \nonumber \\
l_{b}({{\mathbf I}}, {{\mathbf L}}_{b}|{{\mathbf W}}) &= \frac{1}{|{{\mathbf L}}_{cj}^+|}\sum\nolimits_{j:l_{cj}=1}\|{{\mathbf l}}_{bj}-{{\mathbf l}}_{bj}^*\|^2
\label{eqn:loss_sen}\end{aligned}$$ ]{} For the confidence seeds, we employ the balanced cross entropy loss, where $l_{cj}^*$ and $l_{cj}$ are the predicted value and ground truth value respectively. The probability is from a sigmoid function performing on the activation of the last layer of the CNN at pixel $j$. $\beta$ is defined as the proportion of pixels with $l_{cj}=0$ in the image, which is used to balance the positive and negative instances. The loss for bounding box regression is the Euclidean distance over the confidence seed points, and $|{{\mathbf L}}_{cj}^+|$ is the number of pixels with $l_{cj}=1$.
#### Testing the SEN.
For testing, the SEN outputs both the confidence score map $P(l_{cj}^*=1|{{\mathbf I}}, {{\mathbf W}})$ and a four-dimensional bounding box ${{\mathbf l}}_{bj}^*$ for each pixel $j$. We regard a pixel $j$ with confidence score higher than $0.5$ to be reliable and output its bounding box $b_j = {{\mathbf l}}_{bj}^*$, associated with confidence score $P(b_j) = P(l_{cj}^*=1|{{\mathbf I}}, {{\mathbf W}})$. We perform non-maximum suppression based on the confidence scores, yielding several bounding boxes $\{{{\mathbf b}}_j | j=1,2,...\}$ as candidate ROIs with confidence scores $P({{\mathbf b}}_j)$. Each candidate ROI $b_j$ is then properly zoomed, becoming $N(j)$.
#### Training the part parsing.
The training of the part parsing network is standard. For the object-level FCN, the part parsing network is trained based on all the the zoomed image regions (ROIs), with the ground-truth part label maps ${{\mathcal H_p}} = \{{{\mathbf L}}_{pi}\}_{i=1}^{n}$ within the zoomed ROIs. For the image-level FCN, the part parsing network is trained based on the original training images, and has the same structure as the scale estimation network (SEN) in the FCN. Therefore, we merge the part parsing network with the SEN, yielding the image-level FCN with loss defined in Equ. \[eqn:AZN\_loss\]. Here, $l_{p}({{\mathbf I}}, {{\mathbf L}}_{p})$ is the commonly used multinomial logistic regression loss for classification. $$\begin{aligned}
l_{AZN}({{\mathcal H}},{{\mathcal H_p}}|{{\mathbf W}}) =\frac{1}{n}\sum\nolimits_il_{p}({{\mathbf I}}_i, {{\mathbf L}}_{pi}) + l_{SEN}({{\mathcal H}}|{{\mathbf W}});
\label{eqn:AZN_loss}\end{aligned}$$
#### Testing the part parsing.
For testing the object-scale AZN, we first run the image-level FCN, yielding part score maps at the image level and bounding boxes for the object level. Then we zoom onto the bounding boxes and parse these regions based on the object-level FCN model, yielding part score maps at the object level. By merging the part score maps from the two levels, we get better parsing results for the whole image.
Experiments {#sec:exp}
===========
Implementation Details {#subsec:impDetails}
----------------------
#### Selection of confidence seeds.
To train the scale estimation network (SEN), we need to select confidence seeds for object instances or parts. For human instances, we use the human instance masks from the PASCAL-Person-Part Dataset [@chen2014detect] and select the central $7\times7$ pixels within each instance mask as the confidence seeds. To get the confidence seeds for human parts, we first compute connected part segments from the groundtruth part label map, and then also select the central $7\times7$ pixels within each part segment. We present the details of our approach for humans because the extension to horses and cows is straightforward.
#### Zooming ratio of ROIs.
The SEN networks in the FCNs provide a set of human/part bounding boxes (ROIs), $\{b_j | j\in {{\mathbf I}}\}$, which are then zoomed to a proper human/part scale. The zooming ratio of $b_j$, $f(b_j, L^{b_j}_p)$, is decided based on the size of $b_j$ and the previously computed part label map $L^{b_j}_p$ within $b_j$. We use slightly different strategies to compute the zooming ratio at the human and part levels. For the part level, we simply resize the bounding box to a fixed size, $f_p(b_j) = s_t / max(w_j, h_j)$, where $s_t = 255$ is the target size. Here $w_j$ and $h_j$ are the width and height of $b_j$. For the human level, we need to consider the frequently occurred truncation case when only the upper half of a human instance is visible. In practice, we use the image-level part label map $L^{b_j}_p$ within the box, and check the existence of legs to decide whether the full body is visible. If the full body is visible, we use the same strategy as parts. Otherwise, we change the target size $s_t$ to $140$, yielding relative smaller region than the full body visible case. We select the target size based on a validation set. Finally, we limit all zooming ratio $f_p(b_j)$ within the range $[0.4, 2.5]$ for both human and part bounding boxes to avoid artifacts from up or down sampling of images.
Experimental Protocol
---------------------
#### Dataset.
We conduct experiments on humans part parsing using the PASCAL-Person-Part dataset annotated by [@chen2014detect] which is a subset from the PASCAL VOC 2010 dataset. The dataset contains detailed part annotations for every person, e.g., head, torso, . We merge the annotations into six clases: Head, Torso, Upper/Lower Arms and Upper/Lower Legs (plus one background class). We only use those images containing humans for training (1716 images in the training set) and testing (1817 images in the validation set), the same as [@chen2015attention]. Note that parsing humans in PASCAL is challenging because it has larger variations in scale and pose than other human parsing datasets. In addtion, we also perform parsing experiments on the horse-cow dataset [@wang2014semantic], which contains animal instances in a rough bounding box. In this dataset, we keep the same experimental setting with [@wang2015joint].
#### Training.
We train the FCNs using stochastic gradient descent with mini-batches. Each mini-batch contains 30 images. The initial learning rate is 0.001 (0.01 for the final classifier layer) and is decreased by a factor of 0.1 after every 2000 iterations. We set the momentum to be 0.9 and the weight decay to be 0.0005. The initialization model is a modified VGG-16 network pre-trained on ImageNet. Fine-tuning our network on all the reported experiments takes about 30 hours on a NVIDIA Tesla K40 GPU. After training, the average inference time for one PASCAL image is 1.3 s/image.
#### Evaluation metric.
The object parsing results is evaluated in terms of mean IOU (mIOU). It is computed as the pixel intersection-over-union (IOU) averaged across classes [@everingham2014pascal], which is also adopted recently to evaluate parts [@wang2015joint; @chen2015attention]. We also evaluate the part parsing performance each object instance in terms of $AP^r_{part}$ as defined in [@hariharan2014hypercolumns].
#### Network architecture.
We use DeepLab-LargeFOV [@chen2014semantic] as building blocks for the FCNs in our Hierarchical Auto-Zoom Net (HAZN). Recall that our HAZN consists of three FCNs working at different levels of granularity: image level, object level, and part level. At each level, HAZN outputs part parsing scores, and estimats locations and scales for the next level of granularity (objects or parts).
Experimental Results on Parsing Humans in the Wild {#subsec:human_parsing}
--------------------------------------------------
#### Comparison with state-of-the-arts.
As shown in [Tab \[table:seg\_eval\]]{}, we compare our full model (HAZN) with four baselines. The first one is DeepLab-LargeFOV [@chen2014semantic]. The second one is DeepLab-LargeFOV-CRF, which adds a post-processing step to DeepLab-LargeFOV by means of a fully-connected Conditional Random Field (CRF) [@KrahenbuhlK11]. CRFs are commonly used as postprocessing for object semantic segmentation to refine boundaries [@chen2014semantic]. The third one is Multi-Scale Averaging, which feeds the DeepLab-LargeFOV model with images resized to three fixed scales (0.5, 1.0 and 1.5) and then takes the average of the three part score maps to produce the final parsing result. The fourth one is Multi-Scale Attention [@chen2015attention], a most recent work which uses a scale attention model to handle the scale variations in object parsing.
Our HAZN obtains the performance of 57.5%, which is 5.8% better than DeepLab-LargeFOV, and 4.5% better than DeepLab-LargeFOV-CRF. Our model significantly improves the segmentation accuracy in all parts. Note we do not use any CRF for post processing. The CRF, though proven effective in refining boundaries in object segmentation, is not strong enough at recovering details of human parts as well as correcting the errors made by the DeepLab-LargeFOV.
The third baseline (Multi-Scale Averaging) enumerates multi-scale features which is commonly used to handle the scale variations, yet its performance is poorer than ours, indicating the effectiveness of our Auto-Zoom framework.
Our overall mIOU is 1.15% better than the fourth baseline (Multi-Scale Attention), but we are much better in terms of detailed parts like upper legs (around 3% improvement). In addition, we further analyze the scale-invariant ability in Tab. \[table:seg\_eval\_scalewise\], which both methods aim to improve. We can see that our model surpasses Multi-Scale Attention in all instance sizes especially at size XS (9.5%) and size S (5.5%).
#### Importance of object and part scale.
As shown in [Tab \[table:seg\_eval\]]{}, we study the effect of the two scales in our HAZN. In practice, we remove either the object-scale AZN or the part-scale AZN from the full HAZN model, yielding two sub-models: (1) **HAZN (no object scale)**, which only handles the scale variation at part level. (2)**HAZN (no part scale)**, which only handles the scale variation at object instance level.
Compared with our full model, removing the object-scale AZN causes 2.8% mIOU degradation while removing the part-scale AZN results in 1% mIOU degradation. We can see that the object-scale AZN, which handles the scale variation at object instance level, contributes a lot to our final parsing performance. For the part-scale AZN, it further improves the parsing by refining the detailed part predictions, around 3% improvement of lower arms as shown in [Tab \[table:seg\_eval\]]{}, yielding visually more satisfactory results. This demonstrates the effectiveness of the two scales in our HAZN model.
\[table:seg\_eval\_scalewise\]
#### Part parsing accuracy w.r.t. size of human instance.
Since we handle human with various sizes, it is important to check how our model performs with respect to the change of human size in images. In our experiments, we categorize all the ground truth human instances into four different sizes according to the bounding box area of each instance $s_b$ (the square root of the bounding box area). Then we compute the mean IOU (within the bounding box) for each of these four scales.
The four sizes are defined as follows: (1) Size XS: $s_b \in [0,80]$, where the human instance is extremely small in the image; (2) Size S: $s_b \in [80,140]$; (3) Size M: $s_b \in [140,220]$; (4) Size L: $s_b \in [220,520]$, which usually corresponds to truncated human instances where the human’s head or torso covers the majority of the image. The results are given in [Tab \[table:seg\_eval\_scalewise\]]{}. The baseline DeepLab-LargeFOV performs badly at size XS or S (usually only the head or the torso can be detected by the baseline), while our HAZN (full model) improves over it significantly by 14.6% for size XS and 10.8% for size S. This shows that HAZN is particularly good for small objects, where the parsing is difficult to obtain. For instances in size M and L, our model also significantly improve the baselines by around 5%. In general, by using HAZN, we achieve much better scale invariant property to object size than a generally used FCN type of model. We also list the results for the other three baselines for reference.
In addition, it is also important to jointly perform the two scale AZNs in a sequence. To show this, we additionally list the results from our model without object/part scale AZN in the $5_{th}$ and the $6_{th}$ row respectively. By jumping over object scale (HAZN no object scale), the performance becomes significantly worse at size XS, since the model can barely detect the object parts at the image-level when the object is too small. However, if we remove part scale (HAZN no part scale), the performance also dropped in all sizes. This is because using part scale AZN can recover the part details much better than only using object scale. Our HAZN (full model), which sequentially leverage the benefits from both the object scale and part scale, yielding the best performance overall.
#### Instance-wise part parsing accuracy.
We evaluate our part parsing results each human instance in terms of $AP^r_{part}$ as defined in [@hariharan2014hypercolumns]. The segment IOU threshold is set to 0.5. A human instance segment is correct only when it overlaps enough with a groundtruth instance segment. To compute the intersection of two segments, we only consider the pixels whose part labels are also right.
To generate instance segmentation (which is not our major task), we follow a similar strategy to [@hariharan2014hypercolumns] by first generating object detection box and then doing instance segmentation. Specifically, we use fast R-CNN to produce a set of object bounding box proposals, and each box is associated with a confidence score. Then within each bounding box, we use FCN to predict a coarse object instance mask, and use the coarse instance mask to retrieve corresponding part segments from our final HAZN part label map. Last, we use the retrieved part segments to compose a new instance mask where we keep the boundary of part segments. In the instance overlapping cases, we follow the boundary from the predicted instance mask.
We first directly compare with the number reported by [@hariharan2014hypercolumns], on the whole validation set of PASCAL 2010. Our full HAZN achieves **43.08%** in $AP^r_{part}$, **14%** higher than [@hariharan2014hypercolumns]. We also compare with two state-of-the-art baselines (DeepLab-LargeFOV [@chen2014semantic] and Multi-Scale Attention [@chen2015attention]) on the PASCAL-Person-Part dataset. For both baselines, we applied the same strategy to generate instances but with different part parsing results. As shown in Tab.\[table:apr\_part\], our model is 12% points higher than DeepLab-LargeFOV and 6% points higher than Multi-Scale Attention, in terms of $AP^r_{part}$.
\[table:apr\_part\]
#### Qualitative results
We visually show several example results from the PASCAL-Person-Part dataset in [Fig \[fig:seg\_image\]]{}. The baseline DeepLab-LargeFOV-CRF produces several errors due to lack of object and part scale information, background confusion ($1_{st}$ row), human part confusion ($3_{rd}$ row), or important part missing ($4_{th}$ row), , yielding non-satisfactory part parsing results. Our HAZN (no part scale), which only contains object-scale AZN, already successfully relieves the confusions for large scale human instances while recovers the parts for small scale human instances. By further introducing part scale, the part details and boundaries are recovered even better, which are more visually satisfactory.
More visual examples are provided in Fig. \[fig:human\_res\], comparing with more baselines. It can be seen that our full model (HAZN) gives much more satisfied part parsing results than the state-of-the-art baselines. Specifically, for small-scale human instances (the 1, 2, 5 rows of the figure), our HAZN recovers human parts like lower arms and lower legs and gives more accurate part boundaries; for medium-scale or large-scale human instances (the 3, 4, 9, 10 rows of the figure), our model relieves the local confusion with other parts or with the background.
\[t\] ![Qualitative comparison on the PASCAL-Person-Part dataset. We compare with DeepLab-LargeFOV-CRF [@chen2014semantic] and HAZN (no part scale). Our proposed HAZN models (the $3_{rd}$ and $4_{th}$ columns) attain better visual parsing results, especially for small scale human instances and small parts such as legs and arms.](./fig/results_v3.pdf "fig:"){width="1.00\linewidth"}
\[fig:seg\_image\]
#### Failure cases.
Our typical failure modes are shown in [Fig \[fig:seg\_image\_fail\]]{}. Compared with the baseline DeepLab-LargeFOV-CRF, our models give more reasonable parsing results with less local confusion, but they still suffer from heavy occlusion and unusual poses.
{width="1.0\linewidth"}
\[fig:seg\_image\_fail\]
![More qualitative comparison on PASCAL-Person-Part. The baselines are explained in Sec. \[subsec:human\_parsing\].[]{data-label="fig:human_res"}](supp_fig/supplementary_results.pdf){width="1.05\linewidth"}
Experiments on the Horse-Cow Dataset {#sec:horses}
------------------------------------
To show the generality of our method to instance-wise object part parsing, we also applied our method to horse instances and cow instances presented in [@wang2014semantic]. All the testing procedures are the same as those described above for humans.
We copy the baseline numbers from [@wang2015joint], and give the evaluation results in Tab. \[table:part\_bbox\_eval\]. It shows that our baseline models from the DeepLab-LargeFOV [@chen2014semantic] already achieve competative results with the state-of-the-arts, while our HAZN provides a big improvement for horses and cows. The improvement over the state-of-the-art method [@wang2015joint] is roughly 5% mIOU. It is most noticeable for small parts, e.g. the improvement for detecting horse/cow head and cow tails is more than 10$\%$. This shows that our auto-zoom strategy can be effectively generalized to other objects for part parsing.
We also provide qualitative evaluations in Fig. \[fig:horse\_cow\_res\], comparing our full model with three state-of-the-art baselines. The three baselines are explained in Sec. \[subsec:human\_parsing\]. We can observe that using our model, small parts such as legs and tails have been effectively recovered, and the boundary accuracy of all parts has been improved.
![Qualitative comparison on the Horse-Cow Dataset. The baselines are explained in Sec. \[subsec:human\_parsing\].[]{data-label="fig:horse_cow_res"}](supp_fig/horse_cow_visual_results.pdf){width="1.00\linewidth"}
Conclusions
===========
In this paper, we propose the “Hierarachical Auto-Zoom Net” (HAZN) to parse objects in the wild, yielding per-pixel segmentation of the object parts. It adaptably estimates the scales of objects, and their parts, by a two-stage process of Auto-Zoom Nets. We show that on the challenging PASCAL dataset, HAZN performs significantly better (by 5% mIOU), compared to state of the art methods, when applied to humans, horses, and cows. Unlike standard methods which process the image at a fixed range of scales, HAZN’s strategy of searching for objects and then for parts enables it, for example, to zoom in to small image regions and enlarge them to scales which would be prohibitively expensive (in terms of memory) if applied to the entire image (as fixed scale methods would require).
In the future, we would love to extend our HAZN to parse more detailed parts, such as human hand and human eyes. Also, the idea of our AZN can be applied to other tasks like pose estimation in the wild, to make further progress.
|
---
abstract: 'In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. A 3-Lie algebra enjoys a product structure if and only if it is the direct sum (as vector spaces) of two subalgebras. We find that there are four types special integrability conditions, and each of them gives rise to a special decomposition of the original 3-Lie algebra. They are also related to ${\mathcal{O}}$-operators, Rota-Baxter operators and matched pairs of 3-Lie algebras. Parallelly, we introduce the notion of a complex structure on a 3-Lie algebra and there are also four types special integrability conditions. Finally, we add compatibility conditions between a complex structure and a product structure, between a symplectic structure and a paracomplex structure, between a symplectic structure and a complex structure, to introduce the notions of a complex product structure, a para-Kähler structure and a pseudo-Kähler structure on a 3-Lie algebra. We use 3-pre-Lie algebras to construct these structures. Furthermore, a Levi-Civita product is introduced associated to a pseudo-Riemannian 3-Lie algebra and deeply studied.'
author:
- Yunhe Sheng and Rong Tang
title: ' [Symplectic, product and complex structures on 3-Lie algebras ]{} '
---
Introduction
============
A symplectic structure on a Lie algebra ${\mathfrak g}$ is a nondegenerate 2-cocycle $\omega\in\wedge^2{\mathfrak g}^*$. The underlying structure of a symplectic Lie algebra is a quadratic pre-Lie algebra [@Chu]. An almost product structure on a Lie algebra ${\mathfrak g}$ is a linear map $E$ satisfying $E^2={\rm{Id}}$. If in addition, $E$ also satisfies the following integrability condition $$[Ex,Ey]=E([Ex,y]+[x,Ey]-E[x,y]),\quad \forall x,y\in{\mathfrak g},$$ then $E$ is called a product structure. The above integrability condition is called the Nijenhuis condition. An equivalent characterization of a product structure is that ${\mathfrak g}$ is the direct sum (as vector spaces) of two subalgebras. An almost complex structure on a Lie algebra ${\mathfrak g}$ is a linear map $J$ satisfying $J^2=-{\rm{Id}}$. A complex structure on a Lie algebra is an almost complex structure that satisfies the Nijenhuis condition. Adding compatibility conditions between a complex structure and a product structure, between a symplectic structure and a paracomplex structure, between a symplectic structure and a complex structure, one obtains a complex product structure, a paraKähler structure and a pseudo-Kähler structure respectively. These structures play important roles in algebra, geometry and mathematical physics, and are widely studied. See [@Alek; @Andrada0; @ABD; @ABDO; @AS; @Baibialgebra; @Bai-2; @Banayadi0; @Benayadi; @Calvaruso0; @Calvaruso; @Poon1; @Poon2; @Li; @Salamon] for more details.
Generalizations of Lie algebras to higher arities, including 3-Lie algebras and more generally, $n$-Lie algebras [@Filippov; @Kasymov; @Tcohomology], have attracted attention from several fields of mathematics and physics. It is the algebraic structure corresponding to Nambu mechanics [@Gautheron; @N; @T]. In particular, the study of 3-Lie algebras plays an important role in string theory. In [@Basu], Basu and Harvey suggested to replace the Lie algebra appearing in the Nahm equation by a 3-Lie algebra for the lifted Nahm equations. Furthermore, in the context of Bagger-Lambert-Gustavsson model of multiple M2-branes, Bagger-Lambert managed to construct, using a ternary bracket, an $N=2$ supersymmetric version of the worldvolume theory of the M-theory membrane, see [@BL0]. An extensive literatures are related to this pioneering work, see [@BL3; @BL2; @HHM; @P]. See the review article [@review] for more details. In particular, metric 3-algebras were deeply studied in the seminal works [@DFM; @DFMR; @DFMR2]. In [@Liu-Sheng-Bai-Chen], the authors introduced the notion a Nijenhuis operator on an $n$-Lie algebra, which generates a trivial deformation.
The purpose of this paper is to study symplectic structures, product structure and complex structures on 3-Lie algebras and these combined structures. In the case of Lie algebras, pre-Lie algebras play important roles in these studies. It is believable that 3-pre-Lie algebras will play important roles in the corresponding studies. Thus, first we introduce the notion of a representation of a 3-pre-Lie algebra and construct the associated semidirect product 3-pre-Lie algebra. Several important properties of representations of 3-pre-Lie algebras are studied.
Note that the notion of a symplectic structure on a 3-Lie algebra was introduced in [@BGS] and it is shown that the underlying structure of a symplectic 3-Lie algebra is a quadratic 3-pre-Lie algebra. We introduce the notion of a phase space of a 3-Lie algebra ${\mathfrak g}$, which is a symplectic 3-Lie algebra ${\mathfrak g}\oplus {\mathfrak g}^*$ satisfying some conditions, and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. We also introduce the notion of a Manin triple of 3-pre-Lie algebras and show that there is a one-to-one correspondence between Manin triples of 3-pre-Lie algebras and phase spaces of 3-Lie algebras.
An almost product structure on a 3-Lie algebra ${\mathfrak g}$ is defined to be a linear map $E:{\mathfrak g}\longrightarrow{\mathfrak g}$ satisfying $E^2={\rm{Id}}$. It is challengeable to add an integrability condition on an almost product structure to obtain a product structure on a 3-Lie algebra. We note that the Nijenhuis condition (see ) given in [@Liu-Sheng-Bai-Chen] is the correct integrability condition. Let us explain this issue. Denote by ${\mathfrak g}_{\pm}$ the eigenspaces corresponding to eigenvalues $\pm1$ of an almost product structure $E$. Then it is obvious that ${\mathfrak g}={\mathfrak g}_+\oplus {\mathfrak g}_-$ as vector spaces. The Nijenhuis condition ensures that both ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are subalgebras. This is what “integrability” means. Moreover, we find that there are four types special integrability conditions, which are called strict product structure, abelian product structure, strong abelian product structure and perfect product structure respectively, each of them gives rise to a special decomposition of the original 3-Lie algebra. See the following table for a precise description:
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product $E[x,y,z]_{\mathfrak g}=[Ex,y,z]_{\mathfrak g}+[x,Ey,z]_{\mathfrak g}+[x,y,Ez]_{\mathfrak g}$ ${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-$
structure $-E([Ex,Ey,z]_{\mathfrak g}+[Ex,y,Ez]_{\mathfrak g}+[x,Ey,Ez]_{\mathfrak g})$ $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_+$
$+[Ex,Ey,Ez]_{\mathfrak g}$ $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_-$
strict product $E[x,y,z]_{\mathfrak g}=[Ex,y,z]_{\mathfrak g}$ $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}=0$
structure $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}=0$
abelian product $[x,y,z]_{\mathfrak g}=-[x,Ey,Ez]_{\mathfrak g}-[Ex,y,Ez]_{\mathfrak g}-[Ex,Ey,z]_{\mathfrak g}$ $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_+]_{\mathfrak g}=0$
structure $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_-]_{\mathfrak g}=0$
$[x,y,z]_{\mathfrak g}=E[Ex,y,z]_{\mathfrak g}+E[x,Ey,z]_{\mathfrak g}+E[x,y,Ez]_{\mathfrak g}$ $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_+]_{\mathfrak g}=0$
strong abelian $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_-]_{\mathfrak g}=0$
product structure [${\mathcal{O}}$-operators]{} $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_+$
[Rota-Baxter operators]{} $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_-$
perfect product $E[x,y,z]_{\mathfrak g}=[Ex,Ey,Ez]_{\mathfrak g}$ $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_-$
[structure]{} [involutive automorphisms]{} $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_+$
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It is surprised that a strong abelian product structure is also an ${\mathcal{O}}$-operator on a 3-Lie algebra associated to the adjoint representation. Since an ${\mathcal{O}}$-operator on a 3-Lie algebra associated to the adjoint representation is also a Rota-Baxter operator [@RB3Lie; @PBG], it turns out that involutive Rota-Baxter operator can also serve as an integrability condition. This is totally different from the case of Lie algebras. Furthermore, by the definition of a perfect product structure, an involutive automorphism of a 3-Lie algebra can also serve as an integrability condition. This is also a new phenomenon. Note that the decomposition that a perfect product structure gives is exactly the condition required in the definition of a matched pair of 3-Lie algebras [@BGS]. Thus, this kind of product structure will be frequently used in our studies.
An almost complex structure on a 3-Lie algebra ${\mathfrak g}$ is defined to be a linear map $J:{\mathfrak g}\longrightarrow{\mathfrak g}$ satisfying $J^2=-{\rm{Id}}$. With the above motivation, we define a complex structure on a 3-Lie algebra ${\mathfrak g}$ to be an almost complex structure satisfying the Nijenhuis condition. Then ${\mathfrak g}_i$ and ${\mathfrak g}_{-i}$, which are eigenspaces of eigenvalues $\pm i$ of a complex linear map $J_{\mathbb C}$ (the complexification of $J$) are subalgebras of the 3-Lie algebra ${\mathfrak g}_{\mathbb C}$, the complexification of ${\mathfrak g}$. Parallel to the case of product structures, there are also four types special integrability conditions, and each of them gives rise to a special decomposition of ${\mathfrak g}_{\mathbb C}$:
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complex $J[x,y,z]_{\mathfrak g}=[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}$ ${\mathfrak g}_{\mathbb C}={\mathfrak g}_i\oplus{\mathfrak g}_{-i}$
structure $+J([Jx,Jy,z]_{\mathfrak g}+[Jx,y,Jz]_{\mathfrak g}+[x,Jy,Jz]_{\mathfrak g})$ $[{\mathfrak g}_i,{\mathfrak g}_i,{\mathfrak g}_i]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak g}_i$
$-[Jx,Jy,Jz]_{\mathfrak g}$ $[{\mathfrak g}_{-i},{\mathfrak g}_{-i},{\mathfrak g}_{-i}]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak g}_{-i}$
strict complex $J[x,y,z]_{\mathfrak g}=[Jx,y,z]_{\mathfrak g}$ $[{\mathfrak g}_i,{\mathfrak g}_i,{\mathfrak g}_{-i}]_{{\mathfrak g}_{\mathbb C}}=0$
structure $[{\mathfrak g}_{-i},{\mathfrak g}_{-i},{\mathfrak g}_i]_{{\mathfrak g}_{\mathbb C}}=0$
abelian complex $[x,y,z]_{\mathfrak g}=[x,Jy,Jz]_{\mathfrak g}+[Jx,y,Jz]_{\mathfrak g}+[Jx,Jy,z]_{\mathfrak g}$ $[{\mathfrak g}_i,{\mathfrak g}_i,{\mathfrak g}_i]_{{\mathfrak g}_{\mathbb C}}=0$
structure $[{\mathfrak g}_{-i},{\mathfrak g}_{-i},{\mathfrak g}_{-i}]_{{\mathfrak g}_{\mathbb C}}=0$
$[x,y,z]_{\mathfrak g}=-J([Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g})$ $[{\mathfrak g}_i,{\mathfrak g}_i,{\mathfrak g}_i]_{{\mathfrak g}_{\mathbb C}}=0$
strong abelian $[{\mathfrak g}_{-i},{\mathfrak g}_{-i},{\mathfrak g}_{-i}]_{{\mathfrak g}_{\mathbb C}}=0$
complex structure [${\mathcal{O}}$-operators]{} $[{\mathfrak g}_i,{\mathfrak g}_i,{\mathfrak g}_{-i}]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak g}_i$
[Rota-Baxter operators]{} $[{\mathfrak g}_{-i},{\mathfrak g}_{-i},{\mathfrak g}_i]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak g}_{-i}$
perfect complex $J[x,y,z]_{\mathfrak g}=-[Jx,Jy,Jz]_{\mathfrak g}$ $[{\mathfrak g}_i,{\mathfrak g}_i,{\mathfrak g}_{-i}]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak g}_{-i}$
[structure]{} [anti-involutive automorphisms]{} $[{\mathfrak g}_{-i},{\mathfrak g}_{-i},{\mathfrak g}_i]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak g}_i$
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Then we add a compatibility condition between a complex structure and a product structure on a 3-Lie algebra to define a complex product structure on a 3-Lie algebra. We give an equivalent characterization of a complex product structure on a 3-Lie algebra ${\mathfrak g}$ using the decomposition of ${\mathfrak g}$. We add a compatibility condition between a symplectic structure and a paracomplex structure on a 3-Lie algebra to define a paraKähler structure on a 3-Lie algebra. An equivalent characterization of a paraKähler structure on a 3-Lie algebra ${\mathfrak g}$ is also given using the decomposition of ${\mathfrak g}$. Associated to a paraKähler structure on a 3-Lie algebra, there is also a pseudo-Riemannian structure. We introduce the notion of a Livi-Civita product associated to a pseudo-Riemannian 3-Lie algebra, and give its precise formulas. Finally, we add a compatibility condition between a symplectic structure and a complex structure on a 3-Lie algebra to define a pseudo-Kähler structure on a 3-Lie algebra. The relation between a paraKähler structure and a pseudo-Kähler structure on a 3-Lie algebra is investigated. We construct complex product structures, paraKähler structures and pseudo-Kähler structures in terms of 3-pre-Lie algebras. We also give examples of symplectic structures, product structures, complex structures, complex product structure, paraKähler structures and pseudo-Kähler structures on the $4$-dimensional Euclidean $3$-Lie algebra $A_{4}$ given in [@BL0].
The paper is organized as follows. In Section 2, we recall Nijenhuis operators on 3-Lie algebras and 3-pre-Lie algebras. In Section 3, we study representations of 3-pre-Lie algebras. In Section 4, we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. We also introduce the notion of a Manin triple of 3-pre-Lie algebras and study its relation with phase spaces of 3-Lie algebras. In Section 5, we introduce the notion of a product structure on a 3-Lie algebra and give four special integrability conditions. In Section 6, we introduce the notion of a complex structure on a 3-Lie algebra and give four special integrability conditions. In Section 7, we introduce the notion of a complex product structure on a 3-Lie algebra and give its equivalent characterization. In Section 8, we introduce the notion of a paraKähler structure on a 3-Lie algebra and give its equivalent characterization. Moreover, we give a detailed study on the associated Levi-Civita product. In Section 9, we introduce the notion of a pseudo-Kähler structure on a 3-Lie algebra and study the relation with a paraKähler structure.
In this paper, we work over the real field $\mathbb R$ and the complex field $\mathbb C$ and all the vector spaces are finite-dimensional.
[**Acknowledgement:**]{} We give our warmest thanks to Chengming Bai for very useful comments and discussions. This research is supported by NSFC (11471139) and NSF of Jilin Province (20170101050JC).
Preliminaries
=============
In this section, first we recall the notion of a Nijenhuis operator on a 3-Lie algebra, which will be frequently used as the integrability condition in our later studies. Then we recall the notion of a 3-pre-Lie algebra, which is the main tool to construct examples of symplectic, product and complex structures on 3-Lie algebras.
\[defi of n-LA\] A [**$3$-Lie algebra**]{} is a vector space ${\mathfrak g}$ together with a trilinear skew-symmetric bracket $[\cdot,\cdot,\cdot]_{\mathfrak g}:\wedge^3{\mathfrak g}\longrightarrow{\mathfrak g}$ such that the following [**fundamental**]{} identity holds: $$\begin{aligned}
\label{FI}
[x,y,[z,w,v]_{\mathfrak g}]_{\mathfrak g}=[[x,y,z]_{\mathfrak g},w,v]_{\mathfrak g}+[z,[x,y,w]_{\mathfrak g},v]_{\mathfrak g}+[z,w,[x,y,v]_{\mathfrak g}]_{\mathfrak g},\quad \forall x,y,z,w,v\in{\mathfrak g}.\end{aligned}$$
For $x,y\in{\mathfrak g}$, define ${\mathrm{ad}}:\wedge^{2}{\mathfrak g}\longrightarrow{\mathfrak {gl}}({\mathfrak g})$ by $${\mathrm{ad}}_{x,y}z=[x,y,z]_{\mathfrak g}, \quad\forall z\in {\mathfrak g}.$$ Then is equivalent to that ${\mathrm{ad}}_{x,y}$ is a derivation, i.e. $${\mathrm{ad}}_{x,y}[z,w,v]_{\mathfrak g}=[{\mathrm{ad}}_{x,y}z,w,v]_{\mathfrak g}+[z,{\mathrm{ad}}_{x,y}w,v]_{\mathfrak g}+[z,w,{\mathrm{ad}}_{x,y}v]_{\mathfrak g},\quad \forall x,y\in{\mathfrak g}.$$
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra, and $N:{\mathfrak g}\longrightarrow{\mathfrak g}$ a linear map. Define a $3$-ary bracket $[\cdot,\cdot,\cdot]_N^1:\wedge^3{\mathfrak g}\longrightarrow{\mathfrak g}$ by $$\label{eq:bracket(1)}
[x,y,z]_N^{1}=[Nx,y,z]_{\mathfrak g}+[x,Ny,z]_{\mathfrak g}+[x,y,Nz]_{\mathfrak g}-N[x,y,z]_{\mathfrak g}.$$ Then we define $3$-ary bracket $[\cdot,\cdot,\cdot]_N^2:\wedge^3{\mathfrak g}\longrightarrow{\mathfrak g}$ by $$\label{eq:bracket (j)}
[x,y,z]_N^{2}=[Nx,Ny,z]_{\mathfrak g}+[x,Ny,Nz]_{\mathfrak g}+[Nx,y,Nz]_{\mathfrak g}-N[x,y,z]_N^{1}.$$
\[defi:Nijenhuis\][([@Liu-Sheng-Bai-Chen])]{} Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra. A linear map $N:{\mathfrak g}\longrightarrow{\mathfrak g}$ is called a [**Nijenhuis operator**]{} if the following [**Nijenhuis condition**]{} is satisfied: $$\label{eq:Nijenhuis(n)}
[Nx,Ny,Nz]_{\mathfrak g}=N[x,y,z]_N^{2},\quad\forall x,y,z\in {\mathfrak g}.$$
More precisely, a linear map $N:{\mathfrak g}\longrightarrow{\mathfrak g}$ of a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is a Nijenhuis operator if and only if $$\begin{aligned}
\nonumber[Nx,Ny,Nz]_{\mathfrak g}&=&N[Nx,Ny,z]_{\mathfrak g}+N[x,Ny,Nz]_{\mathfrak g}+N[Nx,y,Nz]_{\mathfrak g}\\\nonumber
&&-N^2[Nx,y,z]_{\mathfrak g}-N^2[x,Ny,z]_{\mathfrak g}-N^2[x,y,Nz]_{\mathfrak g}\\
\label{eq:Nejenhuiscon}&&+N^3[x,y,z]_{\mathfrak g}.\end{aligned}$$
[([@Kasymov])]{}\[defi:usualrep\] A [**representation**]{} of a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ on a vector space $V$ is a linear map $\rho:\wedge^2{\mathfrak g}\longrightarrow {\mathfrak {gl}}( V),$ such that for all $x_1,x_2,x_3,x_4\in{\mathfrak g},$ there holds: $$\begin{aligned}
&\rho([x_1,x_2,x_3]_{\mathfrak g}, x_4) +\rho(x_3,[x_1,x_2, x_4]_{\mathfrak g}) =[\rho(x_1,x_2),\rho(x_3,x_4)];\\
&\rho([x_1,x_2,x_3]_{\mathfrak g},x_4)=\rho(x_1,x_2)\circ\rho(x_3,x_4)+\rho(x_2,x_3)\circ\rho(x_1,x_4)+\rho(x_3,x_1)\circ\rho(x_2,x_4).\end{aligned}$$
[Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra. The linear map ${\mathrm{ad}}:\wedge^{2}{\mathfrak g}\longrightarrow{\mathfrak {gl}}({\mathfrak g})$ defines a representation of the $3$-Lie algebra ${\mathfrak g}$ on itself, which we call the [**adjoint representation**]{} of ${\mathfrak g}$. ]{}
Let $A$ be a vector space. For a linear map $\phi:A\otimes A\lon{\mathfrak {gl}}(V)$, we define a linear map $\phi^*: A\otimes A\lon{\mathfrak {gl}}(V^*)$ by $$\begin{aligned}
\langle \phi^*(x,y)\alpha,v\rangle=-\langle\alpha, \phi(x,y)v\rangle,\,\,\,\,\forall \alpha\in V^*,x,y\in{\mathfrak g},v\in V.\end{aligned}$$
[([@BGS])]{}\[dual-rep-3-Lie\] Let $(V,\rho)$ be a representation of a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then $(V^*,\rho^*)$ is a representation of the $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$, which is called the [**dual representation**]{}.
\[lem:semidirectp\] Let ${\mathfrak g}$ be a $3$-Lie algebra, $V$ a vector space and $\rho:
\wedge^2{\mathfrak g}\rightarrow {\mathfrak {gl}}(V)$ a skew-symmetric linear map. Then $(V;\rho)$ is a representation of ${\mathfrak g}$ if and only if there is a $3$-Lie algebra structure (called the [**semidirect product**]{}) on the direct sum of vector spaces ${\mathfrak g}\oplus V$, defined by $$\label{eq:sum}
[x_1+v_1,x_2+v_2,x_3+v_3]_{\rho}=[x_1,x_2,x_3]_{\mathfrak g}+\rho(x_1,x_2)v_3+\rho(x_2,x_3)v_1+\rho(x_3,x_1)v_2,$$ for all $x_i\in {\mathfrak g}, v_i\in V, 1\leq i\leq 3$. We denote this semidirect product $3$-Lie algebra by ${\mathfrak g}\ltimes_\rho V.$
Let $A$ be a vector space with a linear map $\{\cdot,\cdot,\cdot\}:\otimes^3 A{\,\rightarrow\,}A$. The pair $(A,\{\cdot,\cdot,\cdot\})$ is called a $3$-[**pre-Lie algebra**]{} if the following identities hold: $$\begin{aligned}
\{x,y,z\} &=&-\{y,x,z\}\\
\nonumber\{x_1,x_2,\{x_3,x_4,x_5\}\} &=&\{[x_1,x_2,x_3]_C,x_4,x_5\}+\{x_3,[x_1,x_2,x_4]_C,x_5\}\\
&&+\{x_3,x_4,\{x_1,x_2,x_5\}\}\\
\nonumber \{[x_1,x_2,x_3]_C,x_4,x_5\} &=&\{x_1,x_2,\{x_3,x_4,x_5\}\}+\{x_2,x_3,\{x_1,x_4,x_5\}\}\\
&&+\{x_3,x_1,\{x_2,x_4,x_5\}\},\end{aligned}$$ where $x,y,z,x_i\in A,1\le i\le 5$ and $[\cdot,\cdot,\cdot]_C$ is defined by $$\begin{aligned}
[x,y,z]_C\triangleq \{x,y,z\}+\{y,z,x\}+\{z,x,y\},\,\,\,\,\forall x,y,z\in A.\end{aligned}$$
[([@BGS Proposition 3.21])]{}\[3-pre-Lie\] Let $(A,\{\cdot,\cdot,\cdot\})$ be a $3$-pre-Lie algebra. Then $(A,[\cdot,\cdot,\cdot]_C)$ is a $3$-Lie algebra, which is called the sub-adjacent $3$-Lie algebra of $A$, and denoted by $A^c$. $(A,\{\cdot,\cdot,\cdot\})$ is called the compatible $3$-pre-Lie algebra structure on the $3$-Lie algebra $A^c$.
Define the left multiplication $L:\wedge^2 A\longrightarrow{\mathfrak {gl}}(A)$ by $L(x,y)z=\{x,y,z\}$ for all $x,y,z\in A$. Then $(A,L)$ is a representation of the $3$-Lie algebra $A^c$. Moreover, we define the right multiplication $R:\otimes^2 A\lon{\mathfrak {gl}}(A)$ by $R(x,y)z=\{z,x,y\}$. If there is a $3$-pre-Lie algebra structure on its dual space $A^*$, we denote the left multiplication and right multiplication by ${\mathcal{L}}$ and ${\mathcal{R}}$ respectively.
[([@BGS Definition 3.16])]{}\[3-Lie-O-operator\] Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra and $(V,\rho)$ a representation. A linear operator $T:V\lon{\mathfrak g}$ is called an [**${\mathcal{O}}$-operator**]{} associated to $(V,\rho)$ if $T$ satisfies: $$\begin{aligned}
[Tu,Tv,Tw]_{\mathfrak g}=T(\rho(Tu,Tv)w+\rho(Tv,Tw)u+\rho(Tw,Tu)v),\,\,\,\,\forall u,v,w\in V.\end{aligned}$$
[([@BGS Proposition 3.27])]{}\[3-Lie-compatible-3-pre-Lie\] Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra. Then there is a compatible $3$-pre-Lie algebra if and only if there exists an invertible ${\mathcal{O}}$-operator $T:V\lon{\mathfrak g}$ associated to a representation $(V,\rho)$. Furthermore, the compatible $3$-pre-Lie structure on ${\mathfrak g}$ is given by $$\begin{aligned}
\{x,y,z\}=T\rho(x,y)T^{-1}(z),\,\,\,\,\forall x,y,z\in {\mathfrak g}.\end{aligned}$$
Representations of 3-pre-Lie algebras
=====================================
In this section, we introduce the notion of a representation of a 3-pre-Lie algebra, construct the corresponding semidirect product 3-pre-Lie algebra and give the dual representation.
\[defi:rep3-pre-Lie\] A [**representation**]{} of a $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$ on a vector space $V$ consists of a pair $(\rho,\mu)$, where $\rho:\wedge^2 A\rightarrow {\mathfrak {gl}}(V)$ is a representation of the $3$-Lie algebra $A^c$ on $V$ and $\mu:\otimes^2 A\rightarrow {\mathfrak {gl}}(V)$ is a linear map such that for all $x_1,x_2,x_3,x_4\in A$, the following equalities hold: $$\begin{aligned}
\nonumber \rho(x_1,x_2)\mu(x_3,x_4) &=&\mu(x_3,x_4)\rho(x_1,x_2)-\mu(x_3,x_4)\mu(x_2,x_1)\\
\label{rep1}&&+\mu(x_3,x_4)\mu(x_1,x_2)+\mu([x_1,x_2,x_3]_C,x_4)+\mu(x_3,\{x_1,x_2,x_4\}),\\
\label{rep2} \mu([x_1,x_2,x_3]_C,x_4)&=&\rho(x_1,x_2)\mu(x_3,x_4)+\rho(x_2,x_3)\mu(x_1,x_4)+\rho(x_3,x_1)\mu(x_2,x_4),\\
\nonumber \mu(x_1,\{x_2,x_3,x_4\}) &=&\mu(x_3,x_4)\mu(x_1,x_2)+\mu(x_3,x_4)\rho(x_1,x_2)\\
\nonumber &&-\mu(x_3,x_4)\mu(x_2,x_1)-\mu(x_2,x_4)\mu(x_1,x_3)\\
\label{rep3} &&-\mu(x_2,x_4)\rho(x_1,x_3)+\mu(x_2,x_4)\mu(x_3,x_1)+\rho(x_2,x_3)\mu(x_1,x_4),\\
\nonumber \mu(x_3,x_4)\rho(x_1,x_2) &=&\mu(x_3,x_4)\mu(x_2,x_1)-\mu(x_3,x_4)\mu(x_1,x_2)\\
\label{rep4} &&+\rho(x_1,x_2)\mu(x_3,x_4)-\mu(x_2,\{x_1,x_3,x_4\})+\mu(x_1,\{x_2,x_3,x_4\}).\end{aligned}$$
Let $(A,\{\cdot,\cdot,\cdot\})$ be a $3$-pre-Lie algebra and $\rho$ a representation of the sub-adjacent $3$-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$ on the vector space $V$. Then $(\rho,0)$ is a representation of the $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$ on the vector space $V$. It is obvious that $(L,R)$ is a representation of a $3$-pre-Lie algebra on itself, which is called the [**regular representation**]{}.
Let $(V,\rho,\mu)$ be a representation of a $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$. Define a trilinear bracket operation $\{\cdot,\cdot,\cdot\}_{\rho,\mu}:\otimes^3(A\oplus V){\,\rightarrow\,}A\oplus V$ by $$\begin{aligned}
\label{semidirect-3-pre-Lie-bracket}
\{x_1+v_1,x_2+v_2,x_3+v_3\}_{\rho,\mu}\triangleq\{x_1,x_2,x_3\}+\rho(x_1,x_2)v_3+\mu(x_2,x_3)v_1-\mu(x_1,x_3)v_2.\end{aligned}$$ By straightforward computations, we have
\[semidirect-3-pre-Lie\] With the above notation, $(A\oplus V,\{\cdot,\cdot,\cdot\}_{\rho,\mu})$ is a $3$-pre-Lie algebra.
This 3-pre-Lie algebra is called the [**semidirect product**]{} of the $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$ and $(V,\rho,\mu)$, and denoted by $A\ltimes_{\rho,\mu}V$.
Let $V$ be a vector space. Define the switching operator $\tau:\otimes^2 V\longrightarrow \otimes^2 V$ by $$\begin{aligned}
\tau(T)=x_2\otimes x_1,\quad \forall T=x_1\otimes x_2\in\otimes^2 V.\end{aligned}$$
\[pro:representa\] Let $(\rho,\mu)$ be a representation of a $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$ on a vector space $V$. Then $\rho-\mu\tau+\mu$ is a representation of the sub-adjacent $3$-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$ on the vector space $V$.
[[**Proof.**]{} ]{}By Theorem \[semidirect-3-pre-Lie\], we have the semidirect product 3-pre-Lie algebra $A\ltimes_{\rho,\mu}V$. Consider its sub-adjacent 3-Lie algebra structure $[\cdot,\cdot,\cdot]_C$, we have $$\begin{aligned}
\nonumber[x_1+v_1,x_2+v_2,x_3+v_3]{_C}&=&\{x_1+v_1,x_2+v_2,x_3+v_3\}_{\rho,\mu}+\{x_2+v_2,x_3+v_3,x_1+v_1\}_{\rho,\mu}\\
\nonumber &&\{x_3+v_3,x_1+v_1,x_2+v_2\}_{\rho,\mu}\\
\nonumber &=&\{x_1,x_2,x_3\}+\rho(x_1,x_2)v_3+\mu(x_2,x_3)v_1-\mu(x_1,x_3)v_2\\
\nonumber&&+\{x_2,x_3,x_1\}+\rho(x_2,x_3)v_1+\mu(x_3,x_1)v_2-\mu(x_2,x_1)v_3\\
\nonumber&&+\{x_3,x_1,x_2\}+\rho(x_3,x_1)v_2+\mu(x_1,x_2)v_3-\mu(x_3,x_2)v_1\\
\nonumber &=&[x_1,x_2,x_3]_C+((\rho-\mu\tau+\mu)(x_1,x_2))v_3\\
\label{eq:samesubadj} && +((\rho-\mu\tau+\mu)(x_2,x_3))v_1+((\rho-\mu\tau+\mu)(x_3,x_1))v_2.\end{aligned}$$ By Lemma \[lem:semidirectp\], $\rho-\mu\tau+\mu$ is a representation of the sub-adjacent $3$-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$ on the vector space $V$. The proof is finished.
If $(\rho,\mu)=(L,R)$ is the regular representation of a 3-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$, then $\rho-\mu\tau+\mu={\mathrm{ad}}$ is the adjoint representation of the sub-adjacent 3-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$.
\[sub-adjacent-3-Lie\] Let $(\rho,\mu)$ be a representation of a $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$ on a vector space $V$. Then the semidirect product $3$-pre-Lie algebras $A\ltimes_{\rho,\mu}V$ and $A\ltimes_{\rho-\mu\tau+\mu,0}V$ given by the representations $(\rho,\mu)$ and $(\rho-\mu\tau+\mu,0)$ respectively have the same sub-adjacent $3$-Lie algebra $A^c\ltimes_{\rho-\mu\tau+\mu}V$ given by , which is the semidirect product of the $3$-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$ and its representation $(V,\rho-\mu\tau+\mu)$.
Let $(\rho,\mu)$ be a representation of a $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$ on a vector space $V$. Then $(\rho^*-\mu^*\tau+\mu^*,-\mu^*)$ is a representation of the $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$ on the vector space $V^*$, which is called the [**dual representation**]{} of the representation $(V,\rho,\mu)$.
[[**Proof.**]{} ]{}By Proposition \[pro:representa\], $\rho-\mu\tau+\mu$ is a representation of the sub-adjacent $3$-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$ on the vector space $V$. By Lemma \[dual-rep-3-Lie\], $\rho^*-\mu^*\tau+\mu^*$ is a representation of the sub-adjacent $3$-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$ on the dual vector space $V^*$. It is straightforward to deduce that other conditions in Definition \[defi:rep3-pre-Lie\] also hold. We leave details to readers.
Let $(V,\rho,\mu)$ be a representation of a $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$. Then the semidirect product $3$-pre-Lie algebras $A\ltimes_{\rho^*,0}V^*$ and $A\ltimes_{\rho^*-\mu^*\tau+\mu^*,-\mu^*}V^*$ given by the representations $(\rho^*,0)$ and $(\rho^*-\mu^*\tau+\mu^*,-\mu^*)$ respectively have the same sub-adjacent $3$-Lie algebra $A^c\ltimes_{\rho^*}V^*$, which is the semidirect product of the $3$-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$ and its representation $(V^*,\rho^*)$.
If $(\rho,\mu)=(L,R)$ is the regular representation of a 3-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$, then $(\rho^*-\mu^*\tau+\mu^*,-\mu^*)=({\mathrm{ad}}^*,-R^*)$ and the corresponding semidirect product 3-Lie algebra is $A^c\ltimes_{L^*}A^*$, which is the key object when we construct phase spaces of 3-Lie algebras in the next section.
Symplectic structures and phase spaces of $3$-Lie algebras
==========================================================
In this section, we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Moreover, we introduce the notion of a Manin triple of 3-pre-Lie algebras and show that there is a one-to-one correspondence between Manin triples of 3-pre-Lie algebras and perfect phase spaces of 3-Lie algebras.
[([@BGS])]{} A [**symplectic structure**]{} on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is a nondegenerate skew-symmetric bilinear form $\omega\in\wedge^2{\mathfrak g}^*$ satisfying the following equality: $$\begin{aligned}
\label{symplectic-structure}
\omega([x,y,z]_{\mathfrak g},w)-\omega([y,z,w]_{\mathfrak g},x)+\omega([z,w,x]_{\mathfrak g},y)-\omega([w,x,y]_{\mathfrak g},z)=0,\quad\forall x,y,z,w\in{\mathfrak g}.\end{aligned}$$
\[ex:A4symplectic\][Consider the $4$-dimensional Euclidean $3$-Lie algebra $A_{4}$ given in [@BL0]. The underlying vector space is $\mathbb R^4$. Relative to an orthogonal basis $\{e_1,e_2,e_3,e_4\}$, the $3$-Lie bracket is given by $$[e_1,e_2,e_3]=e_4, \quad [e_2,e_3,e_4]=e_1,\quad
[e_1,e_3,e_4]=e_2,\quad[e_1,e_2,e_4]=e_3.$$ Then it is straightforward to see that any nondegenerate skew-symmetric bilinear form is a symplectic structure on $A_4$. In particular, $$\begin{aligned}
\omega_1=e_3^*\wedge e^*_1+e_4^*\wedge e_2^*,\quad \omega_2=e_2^*\wedge e^*_1+e_4^*\wedge e_3^*,\quad \omega_3=e_2^*\wedge e^*_1+e_3^*\wedge
e_4^*,\\ \omega_4=e_1^*\wedge e^*_2+e_4^*\wedge e_3^*,\quad\omega_5=e_1^*\wedge e^*_2+e_3^*\wedge e_4^*,\quad\omega_6=e_1^*\wedge e_3^*+e_2^*\wedge e_4^*\end{aligned}$$ are symplectic structures on $A_4$, where $\{e_1^*,e_2^*,e_3^*,e_4^*\}$ are the dual basis. ]{}
[([@BGS])]{}\[3-pre-Lie-under-3-Lie\] Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g},\omega)$ be a symplectic $3$-Lie algebra. Then there exists a compatible $3$-pre-Lie algebra structure $\{\cdot,\cdot,\cdot\}$ on ${\mathfrak g}$ given by $$\label{3-pre-Lie-omega}
\omega(\{x,y,z\},w)=-\omega(z,[x,y,w]_{\mathfrak g}),\quad \forall x,y,z,w\in {\mathfrak g}.$$
A [**quadratic 3-pre-Lie algebra**]{} is a 3-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$ equipped with a nondegenerate skew-symmetric bilinear form $\omega\in\wedge^2A^*$ such that the following invariant condition holds: $$\label{eq:quadratic}
\omega(\{x,y,z\},w)=-\omega(z,[x,y,w]_C),\quad \forall x,y,z,w\in A.$$ Proposition \[3-pre-Lie-under-3-Lie\] tells us that quadratic 3-pre-Lie algebras are the underlying structures of symplectic 3-Lie algebras.
Let $V$ be a vector space and $V^*={\mathrm{Hom}}(V,\mathbb R)$ its dual space. Then there is a natural nondegenerate skew-symmetric bilinear form $\omega$ on $T^*V=V\oplus V^*$ given by: $$\begin{aligned}
\label{phase-space}
\omega(x+\alpha,y+\beta)=\langle \alpha,y\rangle-\langle \beta,x\rangle,\,\,\,\,\forall x,y\in V,\alpha,\beta\in V^*.\end{aligned}$$
Let $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$ be a $3$-Lie algebra and ${\mathfrak h}^*$ its dual space.
- If there is a $3$-Lie algebra structure $[\cdot,\cdot,\cdot]$ on the direct sum vector space $T^*{\mathfrak h}={\mathfrak h}\oplus{\mathfrak h}^*$ such that $({\mathfrak h}\oplus{\mathfrak h}^*,[\cdot,\cdot,\cdot],\omega)$ is a symplectic $3$-Lie algebra, where $\omega$ given by , and $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$ and $({\mathfrak h}^*,[\cdot,\cdot,\cdot]|_{{\mathfrak h}^*})$ are $3$-Lie subalgebras of $ ({\mathfrak h}\oplus{\mathfrak h}^*,[\cdot,\cdot,\cdot])$, then the symplectic $3$-Lie algebra $({\mathfrak h}\oplus{\mathfrak h}^*,[\cdot,\cdot,\cdot],\omega)$ is called a [**phase space**]{} of the $3$-Lie algebra $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$.
- A phase space $({\mathfrak h}\oplus{\mathfrak h}^*,[\cdot,\cdot,\cdot],\omega)$ is called [**perfect**]{} if the following conditions are satisfied: $$\label{eq:conperfectPS}
[x,y,\alpha]\in{\mathfrak h}^*,\quad [\alpha,\beta,x]\in{\mathfrak h},\quad \forall x,y\in{\mathfrak h}, \alpha,\beta\in{\mathfrak h}^*.$$
3-pre-Lie algebras play important role in the study of phase spaces of 3-Lie algebras.
\[3-pre-Lie-phase-space\] A $3$-Lie algebra has a phase space if and only if it is sub-adjacent to a $3$-pre-Lie algebra.
[[**Proof.**]{} ]{}Let $(A,\{\cdot,\cdot,\cdot\})$ be a $3$-pre-Lie algebra. By Proposition \[3-pre-Lie\], the left multiplication $L$ is a representation of the sub-adjacent $3$-Lie algebra $A^c$ on $A$. By Lemma \[dual-rep-3-Lie\], $L^*$ is a representation of the sub-adjacent $3$-Lie algebra $A^c$ on $A^*$. Thus, we have the semidirect product 3-Lie algebra $A^c\ltimes_{L^*}A^*=(A^c\oplus A^*,[\cdot,\cdot,\cdot]_{L^*})$. Then $(A^c\ltimes_{L^*}A^*,\omega)$ is a symplectic $3$-Lie algebra, which is a phase space of the sub-adjacent $3$-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$. In fact, for all $x_1,x_2,x_3,x_4\in A$ and $\alpha_1,\alpha_2,\alpha_3,\alpha_4\in A^*$, we have $$\begin{aligned}
&&\omega([x_1+\alpha_1,x_2+\alpha_2,x_3+\alpha_3]_{L^*},x_4+\alpha_4)\\&=&\omega([x_1,x_2,x_3]_C+L^*(x_1,x_2)\alpha_3+L^*(x_2,x_3)\alpha_1+L^*(x_3,x_1)\alpha_2,x_4+\alpha_4)\\
&=&\langle L^*(x_1,x_2)\alpha_3+L^*(x_2,x_3)\alpha_1+L^*(x_3,x_1)\alpha_2,x_4\rangle-\langle \alpha_4,[x_1,x_2,x_3]_C\rangle\\
&=&-\langle \alpha_3,\{x_1,x_2,x_4\}\rangle-\langle \alpha_1,\{x_2,x_3,x_4\}\rangle-\langle \alpha_2,\{x_3,x_1,x_4\}\rangle\\
&&-\langle \alpha_4,\{x_1,x_2,x_3\}\rangle-\langle \alpha_4,\{x_2,x_3,x_1\}\rangle-\langle \alpha_4,\{x_3,x_1,x_2\}\rangle.\end{aligned}$$ Similarly, we have $$\begin{aligned}
&&\omega([x_2+\alpha_2,x_3+\alpha_3,x_4+\alpha_4]_{L^*},x_1+\alpha_1)\\&=&-\langle \alpha_4,\{x_2,x_3,x_1\}\rangle-\langle \alpha_2,\{x_3,x_4,x_1\}\rangle
-\langle \alpha_3,\{x_4,x_2,x_1\}\rangle\\
&&-\langle \alpha_1,\{x_2,x_3,x_4\}\rangle-\langle \alpha_1,\{x_3,x_4,x_2\}\rangle-\langle \alpha_1,\{x_4,x_2,x_3\}\rangle,\\
&&\omega([x_3+\alpha_3,x_4+\alpha_4,x_1+\alpha_1]_{L^*},x_2+\alpha_2)\\&=&-\langle \alpha_1,\{x_3,x_4,x_2\}\rangle-\langle \alpha_3,\{x_4,x_1,x_2\}\rangle
-\langle \alpha_4,\{x_1,x_3,x_2\}\rangle\\
&&-\langle \alpha_2,\{x_3,x_4,x_1\}\rangle-\langle \alpha_2,\{x_4,x_1,x_3\}\rangle-\langle \alpha_2,\{x_1,x_3,x_4\}\rangle,\\
&&\omega([x_4+\alpha_4,x_1+\alpha_1,x_2+\alpha_2]_{L^*},x_3+\alpha_3)\\&=&-\langle \alpha_2,\{x_4,x_1,x_3\}\rangle-\langle \alpha_4,\{x_1,x_2,x_3\}\rangle
-\langle \alpha_1,\{x_2,x_4,x_3\}\rangle\\
&&-\langle \alpha_3,\{x_4,x_1,x_2\}\rangle-\langle \alpha_3,\{x_1,x_2,x_4\}\rangle-\langle \alpha_3,\{x_2,x_4,x_1\}\rangle.\end{aligned}$$ Since $\{x_1,x_2,x_3\}=-\{x_2,x_1,x_3\}$, we deduce that $\omega$ is a symplectic structure on the semidirect product 3-Lie algebra $A^c\ltimes_{L^*}A^*$. Moreover, $(A^c,[\cdot,\cdot,\cdot]_C)$ is a subalgebra of $A^c\ltimes_{L^*}A^*$ and $A^*$ is an abelian subalgebra of $A^c\ltimes_{L^*}A^*$. Thus, the symplectic 3-Lie algebra $(A^c\ltimes_{L^*}A^*,\omega)$ is a phase space of the sub-adjacent $3$-Lie algebra $(A^c,[\cdot,\cdot,\cdot]_C)$.
Conversely, let $(T^*{\mathfrak h}={\mathfrak h}\oplus{\mathfrak h}^*,[\cdot,\cdot,\cdot],\omega)$ be a phase space of a $3$-Lie algebra $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$. By Proposition \[3-pre-Lie-under-3-Lie\], there exists a compatible $3$-pre-Lie algebra structure $\{\cdot,\cdot,\cdot\}$ on $T^*{\mathfrak h}$ given by . Since $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$ is a subalgebra of $({\mathfrak h}\oplus{\mathfrak h}^*,[\cdot,\cdot,\cdot])$, we have $$\begin{aligned}
\omega(\{x,y,z\},w)=-\omega(z,[x,y,w])=-\omega(z,[x,y,w]_{{\mathfrak h}})=0,\quad\forall x,y,z,w\in{\mathfrak h}.\end{aligned}$$ Thus, $\{x,y,z\}\in{\mathfrak h}$, which implies that $({\mathfrak h},\{\cdot,\cdot,\cdot\}|_{\mathfrak h})$ is a subalgebra of the $3$-pre-Lie algebra $(T^*{\mathfrak h},\{\cdot,\cdot,\cdot\})$. Its sub-adjacent 3-Lie algebra $({\mathfrak h}^c,[\cdot,\cdot,\cdot]_C)$ is exactly the original $3$-Lie algebra $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$.
\[3-pre-Lie-sub\] Let $(T^*{\mathfrak h}={\mathfrak h}\oplus{\mathfrak h}^*,[\cdot,\cdot,\cdot],\omega)$ be a phase space of a $3$-Lie algebra $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$ and $({\mathfrak h}\oplus {\mathfrak h}^*,\{\cdot,\cdot,\cdot\})$ the associated $3$-pre-Lie algebra. Then both $({\mathfrak h},\{\cdot,\cdot,\cdot\}|_{\mathfrak h})$ and $({\mathfrak h}^*,\{\cdot,\cdot,\cdot\}|_{{\mathfrak h}^*})$ are subalgebras of the $3$-pre-Lie algebra $({\mathfrak h}\oplus {\mathfrak h}^*,\{\cdot,\cdot,\cdot\})$.
If $({\mathfrak h}\oplus{\mathfrak h}^*,[\cdot,\cdot,\cdot],\omega)$ is a phase space of a $3$-Lie algebra $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$ such that the $3$-Lie algebra $({\mathfrak h}\oplus{\mathfrak h}^*,[\cdot,\cdot,\cdot])$ is a semidirect product ${\mathfrak h}\ltimes_{\rho^*}{\mathfrak h}^*$, where $\rho$ is a representation of $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$ on ${\mathfrak h}$ and $\rho^*$ is its dual representation, then $$\{x,y,z\}\triangleq \rho(x,y)z,\quad \forall x,y,z\in{\mathfrak h},$$ defines a $3$-pre-Lie algebra structure on ${\mathfrak h}$.
[[**Proof.**]{} ]{}For all $x,y,z\in{\mathfrak h}$ and $\alpha\in{\mathfrak h}^*$, we have $$\begin{aligned}
\langle \alpha,\{x,y,z\}\rangle&=&-\omega(\{x,y,z\},\alpha)=\omega(z,[x,y,\alpha]_{{\mathfrak g}\oplus{\mathfrak g}^*})=\omega(z,\rho^*(x,y)\alpha)=-\langle \rho^*(x,y)\alpha,z\rangle\\
&=&\langle \alpha,\rho(x,y)z\rangle.\end{aligned}$$ Therefore, $\{x,y,z\}=\rho(x,y)z$.
[Let $(A,\{\cdot,\cdot,\cdot\}_A)$ be a $3$-pre-Lie algebra. Since there is a semidirect product $3$-pre-Lie algebra structure $(A\ltimes_{L^*,0}A^*,\{\cdot,\cdot,\cdot\}_{L^*,0})$ on the phase space $T^*A^c=A^c\ltimes_{L^*}A^{*}$, one can construct a new phase space $T^*A^c\ltimes_{L^*}(T^*A^c)^*$. This process can be continued indefinitely. Hence, there exist a series of phase spaces $\{A_{(n)}\}_{n\ge2}:$ $$A_{(1)}=A^c,\,\,\,\,A_{(2)}=T^*A_{(1)}=A^c\ltimes_{L^*}A^{*},\cdots,\,\,\,\,A_{(n)}=T^*A_{(n-1)},\cdots.$$ $A_{(n)}~~(n\ge2)$ is called the symplectic double of $A_{(n-1)}.$ ]{}
At the end of this section, we introduce the notion of a Manin triple of 3-pre-Lie algebras.
A [**Manin triple of $3$-pre-Lie algebras**]{} is a triple $({\mathcal{A}};A,A')$, where
- $({\mathcal{A}},\{\cdot,\cdot,\cdot\},\omega)$ is a quadratic $3$-pre-Lie algebra;
- both $A$ and $A'$ are isotropic subalgebras of $({\mathcal{A}},\{\cdot,\cdot,\cdot\})$;
- ${\mathcal{A}}=A\oplus A'$ as vector spaces;
- for all $x,y\in A$ and $\alpha,\beta\in A'$, there holds: $$\label{eq:conMT}
\{x,y,\alpha\}\in A',\quad \{\alpha, x,y\}\in A',\quad \{\alpha,\beta, x\}\in A,\quad \{ x,\alpha,\beta\}\in A.$$
In a Manin triple of $3$-pre-Lie algebras, since the skewsymmetric bilinear form $\omega$ is nondegenerate, $A'$ can be identified with $A^*$ via $$\langle \alpha,x\rangle\triangleq \omega(\alpha,x),\quad\forall x\in A, \alpha\in A'.$$ Thus, ${\mathcal{A}}$ is isomorphic to $A\oplus A^*$ naturally and the bilinear form $\omega$ is exactly given by . By the invariant condition , we can obtain the precise form of the 3-pre-Lie structure $\{\cdot,\cdot,\cdot\}$ on $A\oplus A^*$.
\[pro:stuctureMP3preLie\] Let $(A\oplus A^*;A,A^*)$ be a Manin triple of $3$-pre-Lie algebras, where the nondegenerate skewsymmetric bilinear form $\omega$ on the $3$-pre-Lie algebra is given by . Then we have $$\begin{aligned}
\label{eq:m1}\{x,y,\alpha\}&=&(L^*-R^*\tau+R^*)(x,y)\alpha,\\
\label{eq:m2}\{\alpha,x,y\}&=&-R^*(x,y)\alpha,\\
\label{eq:m3}\{\alpha,\beta,x\}&=&({\mathcal{L}}^*-{\mathcal{R}}^*\tau+{\mathcal{R}}^*)(\alpha,\beta)x,\\
\label{eq:m4}\{x,\alpha,\beta\}&=&-{\mathcal{R}}^*(\alpha,\beta)x.
\end{aligned}$$
[[**Proof.**]{} ]{}For all $x,y,z\in A,\alpha\in A^*$, we have $$\begin{aligned}
\langle\{x,y,\alpha\},z\rangle&=&\omega(\{x,y,\alpha\},z)=-\omega(\alpha,[x,y,z]_C)\\
&=&-\omega(\alpha,\{x,y,z\}+\{y,z,x\}+\{z,x,y\})\\
&=&-\omega(\alpha,L(x,y)z-R(y,x)z+R(x,y)z)\\
&=&-\langle\alpha,L(x,y)z-R(y,x)z+R(x,y)z\rangle\\
&=&\langle(L^*-R^*\tau+R^*)(x,y)\alpha,z\rangle,\end{aligned}$$ which implies that holds. We have $$\begin{aligned}
\langle\{\alpha,x,y\},z\rangle&=&\omega(\{\alpha,x,y\},z)=-\omega(y,[\alpha,x,z]_C)=\omega(y,[z,x,\alpha]_C)=- \omega(\{z,x,y\},\alpha)\\
&=&\langle\alpha,R(x,y)z\rangle=-\langle R^*(x,y)\alpha,z\rangle,\end{aligned}$$ which implies that holds. Similarly, we can deduce that and hold.
\[thm:MT-ps\] There is a one-to-one correspondence between Manin triples of $3$-pre-Lie algebras and perfect phase spaces of $3$-Lie algebras. More precisely, if $(A\oplus A^*;A,A^*)$ is a Manin triple of $3$-pre-Lie algebras, then $(A\oplus A^*,[\cdot,\cdot,\cdot]_C,\omega)$ is a symplectic $3$-Lie algebra, where $\omega$ is given by . Conversely, if $({\mathfrak h}\oplus {\mathfrak h}^*,[\cdot,\cdot,\cdot],\omega)$ is a perfect phase space of a $3$-Lie algebra $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$, then $({\mathfrak h}\oplus {\mathfrak h}^*;{\mathfrak h},{\mathfrak h}^*)$ is a Manin triple of $3$-pre-Lie algebras, where the $3$-pre-Lie algebra structure on ${\mathfrak h}\oplus {\mathfrak h}^*$ is given by .
[[**Proof.**]{} ]{}Let $(A\oplus A^*;A,A^*)$ be a Manin triple of $3$-pre-Lie algebras. Denote by $\{\cdot,\cdot,\cdot\}_A$ and $\{\cdot,\cdot,\cdot\}_{A^*}$ the 3-pre-Lie algebra structure on $A$ and $A^*$ respectively, and denote by $[\cdot,\cdot,\cdot]_A$ and $[\cdot,\cdot,\cdot]_{A^*}$ the corresponding sub-adjacent 3-Lie algebra structure on $A$ and $A^*$ respectively. By Proposition \[pro:stuctureMP3preLie\], it is straightforward to deduce that the corresponding 3-Lie algebra structure $[\cdot,\cdot,\cdot]_C$ on $A\oplus A^*$ is given by $$\begin{aligned}
\nonumber [x+\alpha,y+\beta,z+\gamma]_C&=&[x,y,z]_A+{\mathcal{L}}^*(\alpha,\beta)z+{\mathcal{L}}^*(\beta,\gamma)x+{\mathcal{L}}^*(\gamma,\alpha)y\\
\label{eq:MP3Lie} &&+[\alpha,\beta,\gamma]_{A^*}+L^*(x,y)\gamma+L^*(y,z)\alpha+L^*(z,x)\beta.\end{aligned}$$ For all $x_1,x_2,x_3,x_4\in A$ and $\alpha_1,\alpha_2,\alpha_3,\alpha_4\in A^*$, we have $$\begin{aligned}
&&\omega([x_1+\alpha_1,x_2+\alpha_2,x_3+\alpha_3]_C,x_4+\alpha_4)\\&=&\omega([x_1,x_2,x_3]_A+{\mathcal{L}}^*(\alpha_1,\alpha_2)x_3+{\mathcal{L}}^*(\alpha_2,\alpha_3)x_1+{\mathcal{L}}^*(\alpha_3,\alpha_1)x_2\\
&&+[\alpha_1,\alpha_2,\alpha_3]_{A^*}+L^*(x_1,x_2)\alpha_3+L^*(x_2,x_3)\alpha_1+L^*(x_3,x_1)\alpha_2,x_4+\alpha_4)\\
&=&\langle [\alpha_1,\alpha_2,\alpha_3]_{A^*}+L^*(x_1,x_2)\alpha_3+L^*(x_2,x_3)\alpha_1+L^*(x_3,x_1)\alpha_2,x_4\rangle\\
&&-\langle \alpha_4,[x_1,x_2,x_3]_A+{\mathcal{L}}^*(\alpha_1,\alpha_2)x_3+{\mathcal{L}}^*(\alpha_2,\alpha_3)x_1+{\mathcal{L}}^*(\alpha_3,\alpha_1)x_2\rangle\\
&=&\langle [\alpha_1,\alpha_2,\alpha_3]_{A^*},x_4\rangle-\langle \alpha_3,\{x_1,x_2,x_4\}_A\rangle-\langle \alpha_1,\{x_2,x_3,x_4\}_A\rangle-\langle \alpha_2,\{x_3,x_1,x_4\}_A\rangle\\
&&-\langle \alpha_4,[x_1,x_2,x_3]_A\rangle+\langle\{\alpha_1,\alpha_2,\alpha_4\}_{A^*},x_3\rangle+\langle\{\alpha_2,\alpha_3,\alpha_4\}_{A^*},x_1\rangle
+\langle\{\alpha_3,\alpha_1,\alpha_4\}_{A^*},x_2\rangle.\end{aligned}$$ Similarly, we have $$\begin{aligned}
&&\omega([x_2+\alpha_2,x_3+\alpha_3,x_4+\alpha_4],x_1+\alpha_1)\\&=&\langle [\alpha_2,\alpha_3,\alpha_4]_C,x_1\rangle-\langle \alpha_4,\{x_2,x_3,x_1\}_A\rangle-\langle \alpha_2,\{x_3,x_4,x_1\}_A\rangle-\langle \alpha_3,\{x_4,x_2,x_1\}_A\rangle\\
&&-\langle \alpha_1,[x_2,x_3,x_4]_C\rangle+\langle\{\alpha_2,\alpha_3,\alpha_1\}_{A^*},x_4\rangle+\langle\{\alpha_3,\alpha_4,\alpha_1\}_{A^*},x_2\rangle
+\langle\{\alpha_4,\alpha_2,\alpha_1\}_{A^*},x_3\rangle,\\
&&\omega([x_3+\alpha_3,x_4+\alpha_4,x_1+\alpha_1],x_2+\alpha_2)\\&=&\langle [\alpha_3,\alpha_4,\alpha_1]_C,x_2\rangle-\langle \alpha_1,\{x_3,x_4,x_2\}_A\rangle-\langle \alpha_3,\{x_4,x_1,x_2\}_A\rangle-\langle \alpha_4,\{x_1,x_3,x_2\}_A\rangle\\
&&-\langle \alpha_2,[x_3,x_4,x_1]_C\rangle+\langle\{\alpha_3,\alpha_4,\alpha_2\}_{A^*},x_1\rangle+\langle\{\alpha_4,\alpha_1,\alpha_2\}_{A^*},x_3\rangle
+\langle\{\alpha_1,\alpha_3,\alpha_2\}_{A^*},x_4\rangle,\\
&&\omega([x_4+\alpha_4,x_1+\alpha_1,x_2+\alpha_2],x_3+\alpha_3)\\&=&\langle [\alpha_4,\alpha_1,\alpha_2]_C,x_3\rangle-\langle \alpha_2,\{x_4,x_1,x_3\}_A\rangle-\langle \alpha_4,\{x_1,x_2,x_3\}_A\rangle-\langle \alpha_1,\{x_2,x_4,x_3\}_A\rangle\\
&&-\langle \alpha_3,[x_4,x_1,x_2]_C\rangle+\langle\{\alpha_4,\alpha_1,\alpha_3\}_{A^*},x_2\rangle+\langle\{\alpha_1,\alpha_2,\alpha_3\}_{A^*},x_4\rangle
+\langle\{\alpha_2,\alpha_4,\alpha_3\}_{A^*},x_1\rangle.\end{aligned}$$ By $\{x_1,x_2,x_3\}_A=-\{x_2,x_1,x_3\}_A$ and $\{\alpha_1,\alpha_2,\alpha_3\}_{A^*}=-\{\alpha_2,\alpha_1,\alpha_3\}_{A^*}$, we deduce that $\omega$ is a symplectic structure on the 3-Lie algebra $(A\oplus A^*,[\cdot,\cdot,\cdot]_C)$. Therefore, it is a phase space.
Conversely, let $({\mathfrak h}\oplus {\mathfrak h}^*,[\cdot,\cdot,\cdot],\omega)$ be a phase space of the $3$-Lie algebra $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h})$. By Proposition \[3-pre-Lie-under-3-Lie\], there exists a $3$-pre-Lie algebra structure $ \{\cdot,\cdot,\cdot\}$ on $ {\mathfrak h}\oplus {\mathfrak h}^*$ given by such that $({\mathfrak h}\oplus {\mathfrak h}^*,\{\cdot,\cdot,\cdot\},\omega)$ is a quadratic 3-pre-Lie algebra. By Corollary \[3-pre-Lie-sub\], $({\mathfrak h},\{\cdot,\cdot,\cdot\}|_{{\mathfrak h}})$ and $({\mathfrak h}^*,\{\cdot,\cdot,\cdot\}|_{{\mathfrak h}^*})$ are $3$-pre-Lie subalgebras of $({\mathfrak h}\oplus {\mathfrak h}^*,\{\cdot,\cdot,\cdot\})$. It is obvious that both ${\mathfrak h}$ and ${\mathfrak h}^*$ are isotropic. Thus, we only need to show that holds. By , for all $x_1,x_2\in{\mathfrak h}$ and $\alpha_1,\alpha_2\in{\mathfrak h}^*$, we have $$\begin{aligned}
\omega(\{x_1,x_2,\alpha_1\},\alpha_2)=-\omega(\alpha_1,[x_1,x_2,\alpha_2]_C)=0,\end{aligned}$$ which implies that $\{x_1,x_2,\alpha_1\}\in{\mathfrak h}^*$. Similarly, we can show that the other conditions in also hold. The proof is finished.
The notions of a matched pair of $3$-Lie algebras and a Manin triple of $3$-Lie algebras were introduced in [@BGS]. By , we obtain that $(A^c,{A^*}^c;L^*,{\mathcal{L}}^*)$ is a matched pair of $3$-Lie algebras and the phase space is exactly the double of this matched pair. However, one should note that a Manin triple of $3$-pre-Lie algebras does not give rise to a Manin triple of $3$-Lie algebras.
For pre-Lie algebras, there are equivalent description between Manin triples of pre-Lie algebras, matched pairs of pre-Lie algebras associated to the dual representations of the regular representations and pre-Lie bialgebras [@Baibialgebra]. Here we only study Manin triples of $3$-pre-Lie algebras, which are closely related to phase spaces of $3$-Lie algebras and para-Kähler $3$-Lie algebras that studied in Section 8. We postpone the study of matched pairs of $3$-pre-Lie algebras and $3$-pre-Lie bialgebras in the future.
Product structures on $3$-Lie algebras
======================================
In this section, we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. We find four special integrability conditions, each of them gives a special decomposition of the original 3-Lie algebra. At the end of this section, we introduce the notion of a (perfect) paracomplex structure on a $3$-Lie algebra and give examples.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra. An [**almost product structure**]{} on the $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is a linear endomorphism $E:{\mathfrak g}\lon{\mathfrak g}$ satisfying $E^2={\rm{Id}}$ $(E\not=\pm{\rm{Id}})$. An almost product structure is called a [**product**]{} structure if the following integrability condition is satisfied: $$\begin{aligned}
\nonumber\label{product-structure}
E[x,y,z]_{\mathfrak g}&=&[Ex,Ey,Ez]_{\mathfrak g}+[Ex,y,z]_{\mathfrak g}+[x,Ey,z]_{\mathfrak g}+[x,y,Ez]_{\mathfrak g}\\
&&-E[Ex,Ey,z]_{\mathfrak g}-E[x,Ey,Ez]_{\mathfrak g}-E[Ex,y,Ez]_{\mathfrak g}.\end{aligned}$$
One can understand a product structure on a $3$-Lie algebra as a Nijenhuis operator $E$ on a $3$-Lie algebra satisfying $E^2={\rm{Id}}.$
\[product-structure-subalgebra\] Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra. Then $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ has a product structure if and only if ${\mathfrak g}$ admits a decomposition: $$\begin{aligned}
{\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-,\end{aligned}$$ where ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are subalgebras of ${\mathfrak g}$.
[[**Proof.**]{} ]{}Let $E$ be a product structure on ${\mathfrak g}$. By $E^2={\rm{Id}}$ , we have ${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-$, where ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are the eigenspaces of ${\mathfrak g}$ associated to the eigenvalues $\pm1$. For all $x_1,x_2,x_3\in{\mathfrak g}_+$, we have $$\begin{aligned}
E[x_1,x_2,x_3]_{\mathfrak g}&=&[Ex_1,Ex_2,Ex_3]_{\mathfrak g}+[Ex_1,x_2,x_3]_{\mathfrak g}+[x_1,Ex_2,x_3]_{\mathfrak g}+[x_1,x_2,Ex_3]_{\mathfrak g}\\
&&-E[Ex_1,Ex_2,x_3]_{\mathfrak g}-E[x_1,Ex_2,Ex_3]_{\mathfrak g}-E[Ex_1,x_2,Ex_3]_{\mathfrak g}\\
&=&4[x_1,x_2,x_3]_{\mathfrak g}-3E[x_1,x_2,x_3]_{\mathfrak g}.\end{aligned}$$ Thus, we have $[x_1,x_2,x_3]_{\mathfrak g}\in{\mathfrak g}_{+}$, which implies that ${\mathfrak g}_+$ is a subalgebra. Similarly, we can show that ${\mathfrak g}_-$ is a subalgebra.
Conversely, we define a linear endomorphism $E:{\mathfrak g}\lon{\mathfrak g}$ by $$\begin{aligned}
\label{eq:productE}
E(x+\alpha)=x-\alpha,\,\,\,\,\forall x\in{\mathfrak g}_+,\alpha\in{\mathfrak g}_-.\end{aligned}$$ Obviously we have $E^2={\rm{Id}}$. Since ${\mathfrak g}_+$ is a subalgebra of ${\mathfrak g}$, for all $x_1,x_2,x_3\in{\mathfrak g}_+$, we have $$\begin{aligned}
&&[Ex_1,Ex_2,Ex_3]_{\mathfrak g}+[Ex_1,x_2,x_3]_{\mathfrak g}+[x_1,Ex_2,x_3]_{\mathfrak g}+[x_1,x_2,Ex_3]_{\mathfrak g}\\
&&-E[Ex_1,Ex_2,x_3]_{\mathfrak g}-E[x_1,Ex_2,Ex_3]_{\mathfrak g}-E[Ex_1,x_2,Ex_3]_{\mathfrak g}\\
&=&4[x_1,x_2,x_3]_{\mathfrak g}-3E[x_1,x_2,x_3]_{\mathfrak g}=[x_1,x_2,x_3]_{\mathfrak g}\\
&=&E[x_1,x_2,x_3]_{\mathfrak g},\end{aligned}$$ which implies that holds for all $x_1,x_2,x_3\in{\mathfrak g}_+$. Similarly, we can show that holds for all $x,y,z\in{\mathfrak g}$. Therefore, $E$ is a product structure on ${\mathfrak g}$.
Let $E$ be an almost product structure on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. If $E$ satisfies the following equation $$\begin{aligned}
\label{abel-product-0}
E[x,y,z]_{\mathfrak g}=[Ex,y,z]_{\mathfrak g},\end{aligned}$$ then $E$ is a product structure on ${\mathfrak g}$ such that $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}=0$ and $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}=0,$ i.e. ${\mathfrak g}$ is the $3$-Lie algebra direct sum of ${\mathfrak g}_+$ and ${\mathfrak g}_-.$
[[**Proof.**]{} ]{}By and $E^2={\rm{Id}}$, we have $$\begin{aligned}
&&[Ex,Ey,Ez]_{\mathfrak g}+[Ex,y,z]_{\mathfrak g}+[x,Ey,z]_{\mathfrak g}+[x,y,Ez]_{\mathfrak g}\\
&&-E[Ex,Ey,z]_{\mathfrak g}-E[x,Ey,Ez]_{\mathfrak g}-E[Ex,y,Ez]_{\mathfrak g}\\
&=&[Ex,Ey,Ez]_{\mathfrak g}+E[x,y,z]_{\mathfrak g}+[x,Ey,z]_{\mathfrak g}+[x,y,Ez]_{\mathfrak g}\\
&&-[E^2x,Ey,z]_{\mathfrak g}-[Ex,Ey,Ez]_{\mathfrak g}-[E^2x,y,Ez]_{\mathfrak g}\\
&=&E[x,y,z]_{\mathfrak g}.\end{aligned}$$ Thus, $E$ is a product structure on ${\mathfrak g}$. For all $x_1,x_2\in{\mathfrak g}_+,\alpha_1\in{\mathfrak g}_-$, on one hand we have $$\begin{aligned}
E[\alpha_1,x_1,x_2]_{\mathfrak g}=[E\alpha_1,x_1,x_2]_{\mathfrak g}=-[\alpha_1,x_1,x_2]_{\mathfrak g}.\end{aligned}$$ On the other hand, we have $$\begin{aligned}
E[\alpha_1,x_1,x_2]_{\mathfrak g}=E[x_1,x_2,\alpha_1]_{\mathfrak g}=[Ex_1,x_2,\alpha_1]_{\mathfrak g}=[x_1,x_2,\alpha_1]_{\mathfrak g}.\end{aligned}$$ Thus, we obtain $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}=0$. Similarly, we have $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}=0$. The proof is finished.
[**(Integrability condition I)**]{} An almost product structure $E$ on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is called a [**strict product structure**]{} if holds.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra. Then $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ has a strict product structure if and only if ${\mathfrak g}$ admits a decomposition: $${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-,$$ where ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are subalgebras of ${\mathfrak g}$ such that $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}=0$ and $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}=0.$
[[**Proof.**]{} ]{}We leave the details to readers.
Let $E$ be an almost product structure on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. If $E$ satisfies the following equation $$\begin{aligned}
\label{abel-product}
[x,y,z]_{\mathfrak g}=-[x,Ey,Ez]_{\mathfrak g}-[Ex,y,Ez]_{\mathfrak g}-[Ex,Ey,z]_{\mathfrak g},\end{aligned}$$ then $E$ is a product structure on ${\mathfrak g}$.
[[**Proof.**]{} ]{}By and $E^2={\rm{Id}}$, we have $$\begin{aligned}
&&[Ex,Ey,Ez]_{\mathfrak g}+[Ex,y,z]_{\mathfrak g}+[x,Ey,z]_{\mathfrak g}+[x,y,Ez]_{\mathfrak g}\\
&&-E[Ex,Ey,z]_{\mathfrak g}-E[x,Ey,Ez]_{\mathfrak g}-E[Ex,y,Ez]_{\mathfrak g}\\
&=&-[Ex,E^2y,E^2z]_{\mathfrak g}-[E^2x,Ey,E^2z]_{\mathfrak g}-[E^2x,E^2y,Ez]_{\mathfrak g}\\
&&+[Ex,y,z]_{\mathfrak g}+[x,Ey,z]_{\mathfrak g}+[x,y,Ez]_{\mathfrak g}+E[x,y,z]_{\mathfrak g}\\
&=&E[x,y,z]_{\mathfrak g}.\end{aligned}$$ Thus, $E$ is a product structure on ${\mathfrak g}$.
[**(Integrability condition II)**]{} An almost product structure $E$ on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is called an [**abelian product structure**]{} if holds.
\[abelian-product-structure\] Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra. Then $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ has an abelian product structure if and only if ${\mathfrak g}$ admits a decomposition: $${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-,$$ where ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are abelian subalgebras of ${\mathfrak g}$.
[[**Proof.**]{} ]{}Let $E$ be an abelian product structure on ${\mathfrak g}$. For all $x_1,x_2,x_3\in{\mathfrak g}_{+}$, we have $$\begin{aligned}
[x_1,x_2,x_3]_{\mathfrak g}&=& -[Ex_1,Ex_2,x_3]_{\mathfrak g}-[x_1,Ex_2,Ex_3]_{\mathfrak g}-[Ex_1,x_2,Ex_3]_{\mathfrak g}\\
&=&-3[x_1,x_2,x_3]_{\mathfrak g},\end{aligned}$$ which implies that $[x_1,x_2,x_3]_{\mathfrak g}=0$. Similarly, for all $\alpha_1,\alpha_2,\alpha_3\in{\mathfrak g}_{-}$, we also have $[\alpha_1,\alpha_2,\alpha_3]_{\mathfrak g}=0$. Thus, both ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are abelian subalgebras.
Conversely, define a linear endomorphism $E:{\mathfrak g}\lon{\mathfrak g}$ by . Then it is straightforward to deduce that $E$ is an abelian product structure on ${\mathfrak g}$.
Let $E$ be an almost product structure on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. If $E$ satisfies the following equation $$\begin{aligned}
\label{product-integrability}
[x,y,z]_{\mathfrak g}&=&E[Ex,y,z]_{\mathfrak g}+E[x,Ey,z]_{\mathfrak g}+E[x,y,Ez]_{\mathfrak g},\end{aligned}$$ then $E$ is an abelian product structure on ${\mathfrak g}$ such that $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_+$ and $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_-.$
[[**Proof.**]{} ]{}By and $E^2={\rm{Id}}$, we have $$\begin{aligned}
&&[Ex,Ey,Ez]_{\mathfrak g}+[Ex,y,z]_{\mathfrak g}+[x,Ey,z]_{\mathfrak g}+[x,y,Ez]_{\mathfrak g}\\
&&-E[Ex,Ey,z]_{\mathfrak g}-E[x,Ey,Ez]_{\mathfrak g}-E[Ex,y,Ez]_{\mathfrak g}\\
&=&E[x,Ey,Ez]_{\mathfrak g}+E[Ex,y,Ez]_{\mathfrak g}+E[Ex,Ey,z]_{\mathfrak g}+E[x,y,z]_{\mathfrak g}\\
&&-E[Ex,Ey,z]_{\mathfrak g}-E[x,Ey,Ez]_{\mathfrak g}-E[Ex,y,Ez]_{\mathfrak g}\\
&=&E[x,y,z]_{\mathfrak g}.\end{aligned}$$ Thus, we obtain that $E$ is a product structure on ${\mathfrak g}$. For all $x_1,x_2,x_3\in{\mathfrak g}_+$, by , we have $$\begin{aligned}
[x_1,x_2,x_3]_{\mathfrak g}&=&E[Ex_1,x_2,x_3]_{\mathfrak g}+E[x_1,Ex_2,x_3]_{\mathfrak g}+E[x_1,x_2,Ex_3]_{\mathfrak g}\\
&=&3E[x_1,x_2,x_3]_{\mathfrak g}=3[x_1,x_2,x_3]_{\mathfrak g}.\end{aligned}$$ Thus, we obtain $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_+]_{\mathfrak g}=0$. Similarly, we have $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_-]_{\mathfrak g}=0$. By Corollary \[abelian-product-structure\], $E$ is an abelian product structure on ${\mathfrak g}$. Moreover, for all $x_1,x_2\in{\mathfrak g}_+,\alpha_1\in{\mathfrak g}_-$, we have $$\begin{aligned}
[x_1,x_2,\alpha_1]_{\mathfrak g}&=&E[Ex_1,x_2,\alpha_1]_{\mathfrak g}+E[x_1,Ex_2,\alpha_1]_{\mathfrak g}+E[x_1,x_2,E\alpha_1]_{\mathfrak g}\\
&=&E[x_1,x_2,\alpha_1]_{\mathfrak g},\end{aligned}$$ which implies that $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_+$. Similarly, we have $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_-$.
[**(Integrability condition III)**]{} An almost product structure $E$ on a $3$-Lie algebra $ ({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is called a [**strong abelian product structure**]{} if holds.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra. Then $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ has a strong abelian product structure if and only if ${\mathfrak g}$ admits a decomposition: $${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-,$$ where ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are abelian subalgebras of ${\mathfrak g}$ such that $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_+$ and $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_-.$
Let $E$ be a strong abelian product structure on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then we can define $\nu_+:{\mathfrak g}_+\longrightarrow {\mathrm{Hom}}(\wedge^2{\mathfrak g}_-,{\mathfrak g}_-)$ and $\nu_-:{\mathfrak g}_-\longrightarrow {\mathrm{Hom}}(\wedge^2{\mathfrak g}_+,{\mathfrak g}_+)$ by $$\nu_+(x)(\alpha,\beta)=[\alpha,\beta,x]_{\mathfrak g},\quad \nu_-(\alpha)(x,y)=[x,y,\alpha]_{\mathfrak g},\quad\forall x,y\in{\mathfrak g}_+, \alpha,\beta\in{\mathfrak g}_-.$$ It turns out $\nu_+$ and $\nu_-$ are generalized representations of abelian $3$-Lie algebras ${\mathfrak g}_+$ and ${\mathfrak g}_-$ on ${\mathfrak g}_-$ and ${\mathfrak g}_+$ respectively. See [@Liu-Makhlouf-Sheng] for more details about generalized representations of $3$-Lie algebras.
More surprisingly, a strong abelian product structure is an ${\mathcal{O}}$-operator as well as a Rota-Baxter operator [@RB3Lie; @PBG]. Thus, some ${\mathcal{O}}$-operators and Rota-Baxter operators on 3-Lie algebras can serve as integrability conditions.
Let $E$ be an almost product structure on a $3$-Lie algebra $ ({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then $E$ is a strong abelian structure on ${\mathfrak g}$ if and only if $E$ is an ${\mathcal{O}}$-operator associated to the adjoint representation $({\mathfrak g},{\mathrm{ad}})$. Furthermore, there exists a compatible $3$-pre-Lie algebra $({\mathfrak g},\{\cdot,\cdot,\cdot\})$ on the $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$, here the $3$-pre-Lie algebra structure on ${\mathfrak g}$ is given by $$\begin{aligned}
\{x,y,z\}=E[x,y,Ez]_{\mathfrak g},\,\,\,\,\forall x,y,z\in{\mathfrak g}.\end{aligned}$$
[[**Proof.**]{} ]{}By , for all $x,y,z\in{\mathfrak g}$ we have $$\begin{aligned}
[Ex,Ey,Ez]_{\mathfrak g}&=&E[E^2x,Ey,Ez]_{\mathfrak g}+E[Ex,E^2y,Ez]_{\mathfrak g}+E[Ex,Ey,E^2z]_{\mathfrak g}\\
&=&E({\mathrm{ad}}_{Ex,Ey}z+{\mathrm{ad}}_{Ey,Ez}x+{\mathrm{ad}}_{Ez,Ex}y).\end{aligned}$$ Thus, $E$ is an ${\mathcal{O}}$-operator associated to the adjoint representation $({\mathfrak g},{\mathrm{ad}})$.
Conversely, if for all $x,y,z\in{\mathfrak g}$, we have $$\begin{aligned}
[Ex,Ey,Ez]_{\mathfrak g}&=&E({\mathrm{ad}}_{Ex,Ey}z+{\mathrm{ad}}_{Ey,Ez}x+{\mathrm{ad}}_{Ez,Ex}y)\\
&=&E([Ex,Ey,z]_{\mathfrak g}+[x,Ey,Ez]_{\mathfrak g}+[Ex,y,Ez]_{\mathfrak g}),\end{aligned}$$ then $
[x,y,z]_{\mathfrak g}=E[x,y,Ez]_{\mathfrak g}+E[Ex,y,z]_{\mathfrak g}+E[x,Ey,z]_{\mathfrak g}$ by $E^{-1}=E$. Thus, $E$ is a strong abelian structure on ${\mathfrak g}$.
Furthermore, by $E^{-1}=E$ and Proposition \[3-Lie-compatible-3-pre-Lie\], there exists a compatible $3$-pre-Lie algebra on ${\mathfrak g}$ given by $
\{x,y,z\}=E{\mathrm{ad}}_{x,y}E^{-1}(z)=E[x,y,Ez]_{\mathfrak g}.
$ The proof is finished.
There is a new phenomenon that an involutive automorphism of a 3-Lie algebra also serves as an integrability condition.
Let $E$ be an almost product structure on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. If $E$ satisfies the following equation $$\begin{aligned}
\label{product-integrability-1}
E[x,y,z]_{\mathfrak g}=[Ex,Ey,Ez]_{\mathfrak g},\end{aligned}$$ then $E$ is a product structure on ${\mathfrak g}$ such that $$\label{eq:coherenceconPP}
[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_-,\quad [{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_+.$$
[[**Proof.**]{} ]{}By and $E^2={\rm{Id}}$, we have $$\begin{aligned}
&&[Ex,Ey,Ez]_{\mathfrak g}+[Ex,y,z]_{\mathfrak g}+[x,Ey,z]_{\mathfrak g}+[x,y,Ez]_{\mathfrak g}\\
&&-E[Ex,Ey,z]_{\mathfrak g}-E[x,Ey,Ez]_{\mathfrak g}-E[Ex,y,Ez]_{\mathfrak g}\\
&=&E[x,y,z]_{\mathfrak g}+[Ex,y,z]_{\mathfrak g}+[x,Ey,z]_{\mathfrak g}+[x,y,Ez]_{\mathfrak g}\\
&&-[E^2x,E^2y,Ez]_{\mathfrak g}-[Ex,E^2y,E^2z]_{\mathfrak g}-[E^2x,Ey,E^2z]_{\mathfrak g}\\
&=&E[x,y,z]_{\mathfrak g}.\end{aligned}$$ Thus, $E$ is a product structure on ${\mathfrak g}$. Moreover, for all $x_1,x_2\in{\mathfrak g}_+,\alpha_1\in{\mathfrak g}_-$, we have $$\begin{aligned}
E[x_1,x_2,\alpha_1]_{\mathfrak g}=[Ex_1,Ex_2,E\alpha_1]_{\mathfrak g}=-[x_1,x_2,\alpha_1]_{\mathfrak g},\end{aligned}$$ which implies that $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_-$. Similarly, we have $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_+$.
[**(Integrability condition IV)**]{} An almost product structure $E$ on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is called a [**perfect product structure**]{} if holds.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a $3$-Lie algebra. Then $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ has a perfect product structure if and only if ${\mathfrak g}$ admits a decomposition: $${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-,$$ where ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are subalgebras of ${\mathfrak g}$ such that $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_-$ and $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_+.$
[[**Proof.**]{} ]{}We leave the details to readers.
A strict product structure on a $3$-Lie algebra is a perfect product structure.
Let $E$ be a product structure on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. By Theorem \[product-structure-subalgebra\], ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are subalgebras. However, the brackets of mixed terms are very complicated. But a perfect product structure $E$ on $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ ensures $[{\mathfrak g}_+,{\mathfrak g}_+,{\mathfrak g}_-]_{\mathfrak g}\subset{\mathfrak g}_-$ and $[{\mathfrak g}_-,{\mathfrak g}_-,{\mathfrak g}_+]_{\mathfrak g}\subset{\mathfrak g}_+$. Note that this is exactly the condition required in the definition of a matched pair of $3$-Lie algebras [@BGS]. Thus, $E$ is a perfect product structure if and only if $({\mathfrak g}_+,{\mathfrak g}_-)$ is a matched pair of $3$-Lie algebras. This type of product structures are very important in our later studies.
- A [**paracomplex structure**]{} on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is a product structure $E$ on ${\mathfrak g}$ such that the eigenspaces of ${\mathfrak g}$ associated to the eigenvalues $\pm1$ have the same dimension, i.e. $\dim({\mathfrak g}_+)=\dim({\mathfrak g}_-)$.
- A [**perfect paracomplex structure**]{} on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is a perfect product structure $E$ on ${\mathfrak g}$ such that the eigenspaces of ${\mathfrak g}$ associated to the eigenvalues $\pm1$ have the same dimension, i.e. $\dim({\mathfrak g}_+)=\dim({\mathfrak g}_-)$.
\[paracomplex-3-pre-Lie\] Let $(A,\{\cdot,\cdot,\cdot\})$ be a $3$-pre-Lie algebra. Then, on the semidirect product $3$-Lie algebra $ A^c\ltimes_{L^*}A^*$, there is a perfect paracomplex structure $E:A^c\ltimes_{L^*}A^*{\,\rightarrow\,}A^c\ltimes_{L^*}A^*$ given by $$\begin{aligned}
\label{eq:defiE}
E(x+\alpha)=x-\alpha,\,\,\,\,\forall x\in A^c, \alpha\in A^*.\end{aligned}$$
[[**Proof.**]{} ]{}It is obvious that $E^2={\rm{Id}}$. Moreover, we have $(A^c\ltimes_{L^*}A^*)_+=A$, $(A^c\ltimes_{L^*}A^*)_-=A^*$ and they are two subalgebras of the semidirect product $3$-Lie algebra $A^c\ltimes_{L^*}A^*$. By Theorem \[product-structure-subalgebra\], $E$ is a product structure on $A^c\ltimes_{L^*}A^*$. Since $A$ and $A^*$ have the same dimension, $E$ is a paracomplex structure on $A^c\ltimes_{L^*}A^*$. It is obvious that $E$ is perfect.
At the end of this section, we give some examples of product structures.
[ There is a unique non-trivial $3$-dimensional $3$-Lie algebra. It has a basis $\{e_1,e_2,e_3\}$ with respect to which the non-zero product is given by $$[e_1,e_2,e_3]=e_1.$$ Then $E=\left(\begin{array}{ccc}1&0&0\\
0&1&0\\
0&0&-1\end{array}\right)$ and $E=\left(\begin{array}{ccc}1&0&0\\
0&-1&0\\
0&0&1\end{array}\right)$ are strong abelian product structures and $E=\left(\begin{array}{ccc}-1&0&0\\
0&1&0\\
0&0&1\end{array}\right)$ is a perfect product structure. ]{}
\[ex:A4product\][Consider the $4$-dimensional Euclidean $3$-Lie algebra $A_4$ given in Example \[ex:A4symplectic\]. Then $$\begin{aligned}
E_1=\left(\begin{array}{cccc}1&0&0&0\\
0&1&0&0\\
0&0&-1&0\\
0&0&0&-1\end{array}\right),~
E_2=\left(\begin{array}{cccc}1&0&0&0\\
0&-1&0&0\\
0&0&1&0\\
0&0&0&-1\end{array}\right),~
E_3=\left(\begin{array}{cccc}1&0&0&0\\
0&-1&0&0\\
0&0&-1&0\\
0&0&0&1\end{array}\right),\\
E_4=\left(\begin{array}{cccc}-1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&-1\end{array}\right),~
E_5=\left(\begin{array}{cccc}-1&0&0&0\\
0&1&0&0\\
0&0&-1&0\\
0&0&0&1\end{array}\right),
E_6=\left(\begin{array}{cccc}-1&0&0&0\\
0&-1&0&0\\
0&0&1&0\\
0&0&0&1\end{array}\right)\end{aligned}$$are perfect and abelian product structures. ]{}
Complex structures on $3$-Lie algebras
======================================
In this section, we introduce the notion of a complex structure on a real 3-Lie algebra using the Nijenhuis condition as the integrability condition. Parallel to the case of product structures, we also find four special integrability conditions.
\[complex\] Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a real $3$-Lie algebra. An [**almost complex structure**]{} on ${\mathfrak g}$ is a linear endomorphism $J:{\mathfrak g}\lon{\mathfrak g}$ satisfying $J^2=-{\rm{Id}}$. An almost complex structure is called a [**complex**]{} structure if the following integrability condition is satisfied: $$\begin{aligned}
\nonumber\label{complex-structure}
J[x,y,z]_{\mathfrak g}&=&-[Jx,Jy,Jz]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}\\
&&+J[Jx,Jy,z]_{\mathfrak g}+J[x,Jy,Jz]_{\mathfrak g}+J[Jx,y,Jz]_{\mathfrak g}.\end{aligned}$$
One can understand a complex structure on a $3$-Lie algebra as a Nijenhuis operator $J$ on a $3$-Lie algebra satisfying $J^2=-{\rm{Id}}.$
One can also use definition \[complex\] to define the notion of a complex structure on a complex $3$-Lie algebra, considering $J$ to be $\mathbb C$-linear. However, this is not very interesting since for a complex $3$-Lie algebra, there is a one-to-one correspondence between such $\mathbb C$-linear complex structures and product structures (see Proposition \[equivalent\]).
Consider ${\mathfrak g}_{\mathbb C}={\mathfrak g}\otimes_{\mathbb R} \mathbb C\cong\{x+iy|x,y\in{\mathfrak g}\}$, the complexification of the real Lie algebra ${\mathfrak g}$, which turns out to be a complex $3$-Lie algebra by extending the $3$-Lie bracket on ${\mathfrak g}$ complex trilinearly, and we denote it by $({\mathfrak g}_{\mathbb C},[\cdot,\cdot,\cdot]_{{\mathfrak g}_{\mathbb C}})$. We have an equivalent description of the integrability condition given in Definition \[complex\]. We denote by $\sigma$ the conjugation in ${\mathfrak g}_{\mathbb C}$ with respect to the real form ${\mathfrak g}$, that is, $\sigma(x+iy)=x-iy,\,\,x,y\in{\mathfrak g}$. Then, $\sigma$ is a complex antilinear, involutive automorphism of the complex vector space ${\mathfrak g}_{\mathbb C}$.
\[complex-structure-subalgebra\] Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a real $3$-Lie algebra. Then ${\mathfrak g}$ has a complex structure if and only if ${\mathfrak g}_{\mathbb C}$ admits a decomposition: $$\begin{aligned}
{\mathfrak g}_{\mathbb C}={\mathfrak q}\oplus{\mathfrak p},\end{aligned}$$ where ${\mathfrak q}$ and ${\mathfrak p}=\sigma({\mathfrak q})$ are complex subalgebras of ${\mathfrak g}_{\mathbb C}$.
[[**Proof.**]{} ]{}We extend the complex structure $J$ complex linearly, which is denoted by $J_{\mathbb C}$, i.e. $J_{\mathbb C}:{\mathfrak g}_{\mathbb C}\longrightarrow {\mathfrak g}_{\mathbb C}$ is defined by $$\label{eq:JC}
J_{\mathbb C}(x+iy)=Jx+iJy,\quad \forall x,y\in{\mathfrak g}.$$ Then $J_{\mathbb C}$ is a complex linear endomorphism on ${\mathfrak g}_{\mathbb C}$ satisfying $J_{\mathbb C}^2=-{\rm{Id}}$ and the integrability condition on ${\mathfrak g}_{\mathbb C}$. Denote by ${\mathfrak g}_{\pm i}$ the corresponding eigenspaces of ${\mathfrak g}_{\mathbb C}$ associated to the eigenvalues $\pm i$ and there holds: $$\begin{aligned}
{\mathfrak g}_{\mathbb C}={\mathfrak g}_{i}\oplus{\mathfrak g}_{-i}.\end{aligned}$$ It is straightforward to see that ${\mathfrak g}_{i}=\{x-iJx|x\in{\mathfrak g}\}$ and ${\mathfrak g}_{-i}=\{x+iJx|x\in{\mathfrak g}\}$. Therefore, we have ${\mathfrak g}_{-i}=\sigma({\mathfrak g}_{i})$.
For all $X,Y,Z\in{\mathfrak g}_{i}$, we have $$\begin{aligned}
J_{\mathbb C}[X,Y,Z]_{{\mathfrak g}_{\mathbb C}}&=&-[J_{\mathbb C}X,J_{\mathbb C}Y,J_{\mathbb C}Z]_{{\mathfrak g}_{\mathbb C}}+[J_{\mathbb C}X,Y,Z]_{{\mathfrak g}_{\mathbb C}}+[X,J_{\mathbb C}Y,Z]_{{\mathfrak g}_{\mathbb C}}+[X,Y,J_{\mathbb C}Z]_{{\mathfrak g}_{\mathbb C}}\\
&&+J_{\mathbb C}[J_{\mathbb C}X,J_{\mathbb C}Y,Z]_{{\mathfrak g}_{\mathbb C}}+J_{\mathbb C}[X,J_{\mathbb C}Y,J_{\mathbb C}Z]_{{\mathfrak g}_{\mathbb C}}+J_{\mathbb C}[J_{\mathbb C}X,Y,J_{\mathbb C}Z]_{{\mathfrak g}_{\mathbb C}}\\
&=&4i[X,Y,Z]_{{\mathfrak g}_{\mathbb C}}-3J_{\mathbb C}[X,Y,Z]_{{\mathfrak g}_{\mathbb C}}.\end{aligned}$$ Thus, we have $[X,Y,Z]_{{\mathfrak g}_{\mathbb C}}\in{\mathfrak g}_{i}$, which implies that ${\mathfrak g}_i$ is a subalgebra. Similarly, we can show that ${\mathfrak g}_{-i}$ is also a subalgebra. Conversely, we define a complex linear endomorphism $J_{\mathbb C}:{\mathfrak g}_{\mathbb C}\lon{\mathfrak g}_{\mathbb C}$ by $$\begin{aligned}
\label{defi-complex-structure}
J_{\mathbb C}(X+\sigma(Y))=iX-i\sigma(Y),\,\,\,\,\forall X,Y\in{\mathfrak q}.\end{aligned}$$ Since $\sigma$ is a complex antilinear, involutive automorphism of ${\mathfrak g}_{\mathbb C}$, we have $$\begin{aligned}
J_{\mathbb C}^2(X+\sigma(Y))=J_{\mathbb C}(iX-i\sigma(Y))=J_{\mathbb C}(iX+\sigma(iY))=i(iX)-i\sigma(iY)=-X-\sigma(Y),\end{aligned}$$ i.e. $J_{\mathbb C}^2=-{\rm{Id}}$. Since ${\mathfrak q}$ is a subalgebra of ${\mathfrak g}_{\mathbb C}$, for all $X,Y,Z\in{\mathfrak q}$, we have $$\begin{aligned}
&&-[J_{\mathbb C}X,J_{\mathbb C}Y,J_{\mathbb C}Z]_{{\mathfrak g}_{\mathbb C}}+[J_{\mathbb C}X,Y,Z]_{{\mathfrak g}_{\mathbb C}}+[X,J_{\mathbb C}Y,Z]_{{\mathfrak g}_{\mathbb C}}+[X,Y,J_{\mathbb C}Z]_{{\mathfrak g}_{\mathbb C}}\\
&&+J_{\mathbb C}[J_{\mathbb C}X,J_{\mathbb C}Y,Z]_{{\mathfrak g}_{\mathbb C}}+J_{\mathbb C}[X,J_{\mathbb C}Y,J_{\mathbb C}Z]_{{\mathfrak g}_{\mathbb C}}+J_{\mathbb C}[J_{\mathbb C}X,Y,J_{\mathbb C}Z]_{{\mathfrak g}_{\mathbb C}}\\
&=&4i[X,Y,Z]_{{\mathfrak g}_{\mathbb C}}-3J_{\mathbb C}[X,Y,Z]_{{\mathfrak g}_{\mathbb C}}=i[X,Y,Z]_{{\mathfrak g}_{\mathbb C}}\\
&=&J_{\mathbb C}[X,Y,Z]_{{\mathfrak g}_{\mathbb C}},\end{aligned}$$ which implies that $J_{\mathbb C}$ satisfies for all $X,Y,Z\in{\mathfrak q}$. Similarly, we can show that $J_{\mathbb C}$ satisfies for all ${\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}}\in{\mathfrak g}_{\mathbb C}$. Since ${\mathfrak g}_{\mathbb C}={\mathfrak q}\oplus{\mathfrak p}$, we can write ${\mathcal{X}}\in{\mathfrak g}_{\mathbb C}$ as ${\mathcal{X}}=X+\sigma(Y),$ for some $X,Y\in{\mathfrak q}$. Since $\sigma$ is a complex antilinear, involutive automorphism of ${\mathfrak g}_{\mathbb C}$, we have $$\begin{aligned}
(J_{\mathbb C}\circ\sigma)(X+\sigma(Y))=J_{\mathbb C}(Y+\sigma(X))=iY-i\sigma(X)=\sigma(iX-i\sigma(Y))=(\sigma\circ J_{\mathbb C})(X+\sigma(Y)),\end{aligned}$$ which implies that $J_{\mathbb C}\circ\sigma=\sigma\circ J_{\mathbb C}$. Moreover, since $\sigma({\mathcal{X}})={\mathcal{X}}$ is equivalent to ${\mathcal{X}}\in{\mathfrak g}$, we deduce that the set of fixed points of $\sigma$ is the real vector space ${\mathfrak g}$. By $J_{\mathbb C}\circ\sigma=\sigma\circ J_{\mathbb C}$, there is a well-defined $J\in{\mathfrak {gl}}({\mathfrak g})$ given by $$J\triangleq J_{\mathbb C}|_{{\mathfrak g}}.$$ Follows from that $J_{\mathbb C}$ satisfies and $J_{\mathbb C}^2=-{\rm{Id}}$ on ${\mathfrak g}_{\mathbb C}$, $J$ is a complex structure on ${\mathfrak g}$.
Let $J$ be an almost complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. If $J$ satisfies $$\begin{aligned}
\label{adapt}
J[x,y,z]_{\mathfrak g}=[Jx,y,z]_{\mathfrak g},\,\,\,\,\forall x,y,z\in{\mathfrak g},\end{aligned}$$ then $J$ is a complex structure on $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$.
[[**Proof.**]{} ]{}By and $J^2=-{\rm{Id}}$, we have $$\begin{aligned}
&&-[Jx,Jy,Jz]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}\\
&&+J[Jx,Jy,z]_{\mathfrak g}+J[x,Jy,Jz]_{\mathfrak g}+J[Jx,y,Jz]_{\mathfrak g}\\
&=&-[Jx,Jy,Jz]_{\mathfrak g}+J[x,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}\\
&&+[J^2x,Jy,z]_{\mathfrak g}+[Jx,Jy,Jz]_{\mathfrak g}+[J^2x,y,Jz]_{\mathfrak g}\\
&=&J[x,y,z]_{\mathfrak g}.\end{aligned}$$ Thus, we obtain that $J$ is a complex structure on ${\mathfrak g}$.
[**(Integrability condition I)**]{} An almost complex structure $J$ on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is called a [**strict complex structure**]{} if holds.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a real $3$-Lie algebra. Then there is a strict complex structure on $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ if and only if ${\mathfrak g}_{\mathbb C}$ admits a decomposition: $$\begin{aligned}
{\mathfrak g}_{\mathbb C}={\mathfrak q}\oplus{\mathfrak p},\end{aligned}$$ where ${\mathfrak q}$ and ${\mathfrak p}=\sigma({\mathfrak q})$ are complex subalgebras of ${\mathfrak g}_{\mathbb C}$ such that $[{\mathfrak q},{\mathfrak q},{\mathfrak p}]_{{\mathfrak g}_{\mathbb C}}=0$ and $[{\mathfrak p},{\mathfrak p},{\mathfrak q}]_{{\mathfrak g}_{\mathbb C}}=0,$ i.e. ${\mathfrak g}_{\mathbb C}$ is a $3$-Lie algebra direct sum of ${\mathfrak q}$ and ${\mathfrak p}$.
[[**Proof.**]{} ]{}Let $J$ be a strict complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then, $J_{\mathbb C}$ is a strict complex structure on the complex $3$-Lie algebra $({\mathfrak g}_{\mathbb C},[\cdot,\cdot,\cdot]_{{\mathfrak g}_{\mathbb C}})$. For all $X,Y\in{\mathfrak g}_{i}$ and $\sigma(Z)\in{\mathfrak g}_{-i}$, on one hand we have $$\begin{aligned}
J_{\mathbb C}[X,Y,\sigma(Z)]_{{\mathfrak g}_{\mathbb C}}=[J_{\mathbb C}X,Y,\sigma(Z)]_{{\mathfrak g}_{\mathbb C}}=i[X,Y,\sigma(Z)]_{{\mathfrak g}_{\mathbb C}}.\end{aligned}$$ On the other hand, we have $$\begin{aligned}
J_{\mathbb C}[X,Y,\sigma(Z)]_{{\mathfrak g}_{\mathbb C}}=J_{\mathbb C}[\sigma(Z),X,Y]_{{\mathfrak g}_{\mathbb C}}=[J_{\mathbb C}\sigma(Z),X,Y]_{{\mathfrak g}_{\mathbb C}}=-i[\sigma(Z),X,Y]_{{\mathfrak g}_{\mathbb C}}.\end{aligned}$$ Thus, we obtain $[{\mathfrak g}_i,{\mathfrak g}_i,{\mathfrak g}_{-i}]_{{\mathfrak g}_{\mathbb C}}=0$. Similarly, we can show $[{\mathfrak g}_{-i},{\mathfrak g}_{-i},{\mathfrak g}_{i}]_{{\mathfrak g}_{\mathbb C}}=0$.
Conversely, define a complex linear endomorphism $J_{\mathbb C}:{\mathfrak g}_{\mathbb C}\lon{\mathfrak g}_{\mathbb C}$ by . Then it is straightforward to deduce that $J_{\mathbb C}^2=-{\rm{Id}}$. Since ${\mathfrak q}$ is a subalgebra of ${\mathfrak g}_{\mathbb C}$, for all $X,Y,Z\in{\mathfrak q}$, we have $$\begin{aligned}
J_{\mathbb C}[X,Y,Z]_{{\mathfrak g}_{\mathbb C}}=i[X,Y,Z]_{{\mathfrak g}_{\mathbb C}}=[J_{\mathbb C}X,Y,Z]_{{\mathfrak g}_{\mathbb C}},\end{aligned}$$ which implies that $J_{\mathbb C}$ satisfies for all $X,Y,Z\in{\mathfrak q}$. Similarly, we can show that $J_{\mathbb C}$ satisfies for all ${\mathcal{X}},{\mathcal{Y}},{\mathcal{Z}}\in{\mathfrak g}_{\mathbb C}$. By the proof of Theorem \[complex-structure-subalgebra\], we obtain that $J\triangleq J_{\mathbb C}|_{{\mathfrak g}}$ is a strict complex structure on the real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. The proof is finished.
Let $J$ be an almost complex structure on a real 3-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. We can define a complex vector space structure on the real vector space ${\mathfrak g}$ by $$\begin{aligned}
\label{complex-space}
(a+bi)x\triangleq ax+bJx,\,\,\,\forall a,b\in\mathbb R,x\in{\mathfrak g}.\end{aligned}$$ Define two maps $\varphi:{\mathfrak g}\lon{\mathfrak g}_i$ and $\psi:{\mathfrak g}\lon{\mathfrak g}_{-i}$ as following: $$\begin{aligned}
\varphi(x)&=&\frac{1}{2}(x-iJx),\\
\psi(x) &=&\frac{1}{2}(x+iJx).\end{aligned}$$ It is straightforward to deduce that $\varphi$ is complex linear isomorphism and $\psi=\sigma\circ\varphi$ is a complex antilinear isomorphism between complex vector spaces.
Let $J$ be a strict complex structure on a real 3-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then with the complex vector space structure defined above, $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is a complex $3$-Lie algebra. In fact, the fact that the $3$-Lie bracket is complex trilinear follows from $$\begin{aligned}
[(a+bi)x,y,z]_{\mathfrak g}&=&[ax+bJx,y,z]_{\mathfrak g}=a[x,y,z]_{\mathfrak g}+b[Jx,y,z]_{\mathfrak g}\\
&=&a[x,y,z]_{\mathfrak g}+bJ[x,y,z]_{\mathfrak g}=(a+bi)[x,y,z]_{\mathfrak g}\end{aligned}$$ using and .
Let $J$ be a complex structure on ${\mathfrak g}$. Define a new bracket $[\cdot,\cdot,\cdot]_J:\wedge^3{\mathfrak g}\lon{\mathfrak g}$ by $$\begin{aligned}
\label{J-bracket}
[x,y,z]_J\triangleq \frac{1}{4}([x,y,z]_{\mathfrak g}-[x,Jy,Jz]_{\mathfrak g}-[Jx,y,Jz]_{\mathfrak g}-[Jx,Jy,z]_{\mathfrak g}),\,\,\,\,\forall x,y,z\in{\mathfrak g}.\end{aligned}$$
\[subalgebra-iso\] Let $J$ be a complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ is a real $3$-Lie algebra. Moreover, $J$ is a strict complex structure on $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ and the corresponding complex $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ is isomorphic to the complex $3$-Lie algebra ${\mathfrak g}_{i}$.
[[**Proof.**]{} ]{}One can show that $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ is a real $3$-Lie algebra directly. Here we use a different approach to prove this result. By , for all $x,y,z\in{\mathfrak g}$, we have $$\begin{aligned}
\nonumber[\varphi(x),\varphi(y),\varphi(z)]_{{\mathfrak g}_{\mathbb C}}&=&\frac{1}{8}[x-iJx,y-iJy,z-iJz]_{{\mathfrak g}_{\mathbb C}}\\
\nonumber &=&\frac{1}{8}([x,y,z]_{\mathfrak g}-[x,Jy,Jz]_{\mathfrak g}-[Jx,y,Jz]_{\mathfrak g}-[Jx,Jy,z]_{\mathfrak g})\\
\nonumber&&-\frac{1}{8}i([x,y,Jz]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}-[Jx,Jy,Jz]_{\mathfrak g})\\
\nonumber &=&\frac{1}{8}([x,y,z]_{\mathfrak g}-[x,Jy,Jz]_{\mathfrak g}-[Jx,y,Jz]_{\mathfrak g}-[Jx,Jy,z]_{\mathfrak g})\\
\nonumber &&-\frac{1}{8}iJ([x,y,z]_{\mathfrak g}-[x,Jy,Jz]_{\mathfrak g}-[Jx,y,Jz]_{\mathfrak g}-[Jx,Jy,z]_{\mathfrak g})\\
\label{eq:Jiso}&=&\varphi[x,y,z]_J.\end{aligned}$$ Thus, we have $[x,y,z]_J=\varphi^{-1}[\varphi(x),\varphi(y),\varphi(z)]_{{\mathfrak g}_{\mathbb C}}$. Since $J$ is a complex structure, ${\mathfrak g}_i$ is a $3$-Lie subalgebra. Therefore, $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ is a real $3$-Lie algebra.
By , for all $x,y,z\in{\mathfrak g}$, we have $$\begin{aligned}
J[x,y,z]_J&=&\frac{1}{4}J([x,y,z]_{\mathfrak g}-[x,Jy,Jz]_{\mathfrak g}-[Jx,y,Jz]_{\mathfrak g}-[Jx,Jy,z]_{\mathfrak g})\\
&=&\frac{1}{4}(-[Jx,Jy,Jz]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g})\\
&=&[Jx,y,z]_J,\end{aligned}$$ which implies that $J$ is a strict complex structure on $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$. By , $\varphi$ is a complex $3$-Lie algebra isomorphism. The proof is finished.
Let $J$ be a complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then $J$ is a strict complex structure on $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ if and only if $[\cdot,\cdot,\cdot]_J=[\cdot,\cdot,\cdot]_{\mathfrak g}.$
[[**Proof.**]{} ]{}If $J$ is a strict complex structure on $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$, by $J[x,y,z]_{\mathfrak g}=[Jx,y,z]_{\mathfrak g}$, we have $$\begin{aligned}
[x,y,z]_J=\frac{1}{4}([x,y,z]_{\mathfrak g}-[x,Jy,Jz]_{\mathfrak g}-[Jx,y,Jz]_{\mathfrak g}-[Jx,Jy,z]_{\mathfrak g})=[x,y,z]_{\mathfrak g}.\end{aligned}$$ Conversely, if $[\cdot,\cdot,\cdot]_J=[\cdot,\cdot,\cdot]_{\mathfrak g}$, we have $$-3[x,y,z]_{\mathfrak g}=[x,Jy,Jz]_{\mathfrak g}+[Jx,y,Jz]_{\mathfrak g}+[Jx,Jy,z]_{\mathfrak g}.$$ Then by the integrability condition of $J$, we obtain $$\begin{aligned}
4J[x,y,z]_J&=&-[Jx,Jy,Jz]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}\\
&=&3[Jx,y,z]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}\\
&=&4[Jx,y,z]_{\mathfrak g},\end{aligned}$$ which implies that $J[x,y,z]_{\mathfrak g}=[Jx,y,z]_{\mathfrak g}$. The proof is finished.
Let $J$ be an almost complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. If $J$ satisfies the following equation $$\begin{aligned}
\label{abel-complex}
[x,y,z]_{\mathfrak g}=[x,Jy,Jz]_{\mathfrak g}+[Jx,y,Jz]_{\mathfrak g}+[Jx,Jy,z]_{\mathfrak g},\end{aligned}$$ Then, $J$ is a complex structure on ${\mathfrak g}$.
[[**Proof.**]{} ]{}By and $J^2=-{\rm{Id}}$, we have $$\begin{aligned}
&&-[Jx,Jy,Jz]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}\\
&&+J[Jx,Jy,z]_{\mathfrak g}+J[x,Jy,Jz]_{\mathfrak g}+J[Jx,y,Jz]_{\mathfrak g}\\
&=&-[Jx,J^2y,J^2z]_{\mathfrak g}-[J^2x,Jy,J^2z]_{\mathfrak g}-[J^2x,J^2y,Jz]_{\mathfrak g}\\
&&+[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}+J[x,y,z]_{\mathfrak g}\\
&=&J[x,y,z]_{\mathfrak g}.\end{aligned}$$ Thus, we obtain that $J$ is a complex structure on ${\mathfrak g}$.
[**(Integrability condition II)**]{} An almost complex structure $J$ on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is called an [**abelian complex structure**]{} if holds.
Let $J$ be an abelian complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ is an abelian $3$-Lie algebra.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a real $3$-Lie algebra. Then ${\mathfrak g}$ has an abelian complex structure if and only if ${\mathfrak g}_{\mathbb C}$ admits a decomposition: $${\mathfrak g}_{\mathbb C}={\mathfrak q}\oplus{\mathfrak p},$$ where ${\mathfrak q}$ and ${\mathfrak p}=\sigma({\mathfrak q})$ are complex abelian subalgebras of ${\mathfrak g}_{\mathbb C}$.
[[**Proof.**]{} ]{}Let $J$ be an abelian complex structure on ${\mathfrak g}$. By Proposition \[subalgebra-iso\], we obtain that $\varphi$ is a complex $3$-Lie algebra isomorphism from $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ to $({\mathfrak g}_{i},[\cdot,\cdot,\cdot]_{{\mathfrak g}_{\mathbb C}})$. Since $J$ is abelian, $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ is an abelian $3$-Lie algebra. Therefore, ${\mathfrak q}={\mathfrak g}_{i}$ is an abelian subalgebra of ${\mathfrak g}_{\mathbb C}$. Since ${\mathfrak p}={\mathfrak g}_{-i}=\sigma({\mathfrak g}_{i})$, for all $x_1+iy_1,x_2+iy_2,x_3+iy_3\in{\mathfrak g}_i$, we have $$\begin{aligned}
&&[\sigma(x_1+iy_1),\sigma(x_2+iy_2),\sigma(x_3+iy_3)]_{{\mathfrak g}_{\mathbb C}}\\&=&[x_1-iy_1,x_2-iy_2,x_3-iy_3]_{{\mathfrak g}_{\mathbb C}}\\
&=&([x_1,x_2,x_3]_{\mathfrak g}-[x_1,y_2,y_3]_{\mathfrak g}-[y_1,x_2,y_3]_{\mathfrak g}-[y_1,y_2,x_3]_{\mathfrak g})\\
&&-i([x_1,x_2,y_3]_{\mathfrak g}+[x_1,y_2,x_3]_{\mathfrak g}+[y_1,x_2,x_3]_{\mathfrak g}-[y_1,y_2,y_3]_{\mathfrak g})\\
&=&\sigma[x_1+iy_1,x_2+iy_2,x_3+iy_3]_{{\mathfrak g}_{\mathbb C}}\\
&=&0.\end{aligned}$$ Thus, ${\mathfrak p}$ is an abelian subalgebra of ${\mathfrak g}_{\mathbb C}$.
Conversely, by Theorem \[complex-structure-subalgebra\], there is a complex structure $J$ on ${\mathfrak g}$. Moreover, by Proposition \[subalgebra-iso\], we have a complex $3$-Lie algebra isomorphism $\varphi$ from $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ to $({\mathfrak q},[\cdot,\cdot,\cdot]_{{\mathfrak g}_{\mathbb C}})$. Thus, $({\mathfrak g},[\cdot,\cdot,\cdot]_J)$ is an abelian $3$-Lie algebra. By the definition of $[\cdot,\cdot,\cdot]_J$, we obtain that $J$ is an abelian complex structure on ${\mathfrak g}$. The proof is finished.
Let $J$ be an almost complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. If $J$ satisfies the following equation $$\begin{aligned}
\label{complex-integrability}
[x,y,z]_{\mathfrak g}&=&-J[Jx,y,z]_{\mathfrak g}-J[x,Jy,z]_{\mathfrak g}-J[x,y,Jz]_{\mathfrak g},\end{aligned}$$ then $J$ is a complex structure on ${\mathfrak g}$.
[[**Proof.**]{} ]{}By and $J^2=-{\rm{Id}}$, we have $$\begin{aligned}
&&-[Jx,Jy,Jz]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}\\
&&+J[Jx,Jy,z]_{\mathfrak g}+J[x,Jy,Jz]_{\mathfrak g}+J[Jx,y,Jz]_{\mathfrak g}\\
&=&J[J^2x,Jy,Jz]_{\mathfrak g}+J[Jx,J^2y,Jz]_{\mathfrak g}+J[Jx,Jy,J^2z]_{\mathfrak g}+J[x,y,z]_{\mathfrak g}\\
&&+J[Jx,Jy,z]_{\mathfrak g}+J[x,Jy,Jz]_{\mathfrak g}+J[Jx,y,Jz]_{\mathfrak g}\\
&=&J[x,y,z]_{\mathfrak g}.\end{aligned}$$ Thus, $J$ is a complex structure on ${\mathfrak g}$.
[**(Integrability condition III)**]{} An almost complex structure $J$ on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is called a [**strong abelian complex structure**]{} if holds.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a real $3$-Lie algebra. Then ${\mathfrak g}$ has a strong abelian complex structure if and only if ${\mathfrak g}_{\mathbb C}$ admits a decomposition: $${\mathfrak g}_{\mathbb C}={\mathfrak q}\oplus{\mathfrak p},$$ where ${\mathfrak q}$ and ${\mathfrak p}=\sigma({\mathfrak q})$ are abelian complex subalgebras of ${\mathfrak g}_{\mathbb C}$ such that $[{\mathfrak q},{\mathfrak q},{\mathfrak p}]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak q}$ and $[{\mathfrak p},{\mathfrak p},{\mathfrak q}]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak p}.$
Parallel to the case of strong abelian product structures on a 3-Lie algebra, strong abelian complex structures on a $3$-Lie algebra are also ${\mathcal{O}}$-operators associated to the adjoint representation.
Let $J$ be an almost complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then $J$ is a strong abelian complex structure on a $3$-Lie algebra $ ({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ if and only if $-J$ is an ${\mathcal{O}}$-operator on $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ associated to the adjoint representation $({\mathfrak g},{\mathrm{ad}})$. Furthermore, there exists a compatible $3$-pre-Lie algebra $({\mathfrak g},\{\cdot,\cdot,\cdot\})$ on the $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$, here the $3$-pre-Lie algebra structure on ${\mathfrak g}$ is given by $$\begin{aligned}
\{x,y,z\}=-J[x,y,Jz]_{\mathfrak g},\,\,\,\,\forall x,y,z\in{\mathfrak g}.\end{aligned}$$
[[**Proof.**]{} ]{}By , for all $x,y,z\in{\mathfrak g}$ we have $$\begin{aligned}
[-Jx,-Jy,-Jz]_{\mathfrak g}&=&J[J^2x,Jy,Jz]_{\mathfrak g}+J[Jx,J^2y,Jz]_{\mathfrak g}+J[Jx,Jy,J^2z]_{\mathfrak g}\\
&=&-J({\mathrm{ad}}_{-Jx,-Jy}z+{\mathrm{ad}}_{-Jy,-Jz}x+{\mathrm{ad}}_{-Jz,-Jx}y).\end{aligned}$$ Thus, $-J$ is an ${\mathcal{O}}$-operator associated to the adjoint representation $({\mathfrak g},{\mathrm{ad}})$.
Conversely, if for all $x,y,z\in{\mathfrak g}$, we have $$\begin{aligned}
[-Jx,-Jy,-Jz]_{\mathfrak g}&=&-J({\mathrm{ad}}_{-Jx,-Jy}z+{\mathrm{ad}}_{-Jy,-Jz}x+{\mathrm{ad}}_{-Jz,-Jx}y)\\
&=&-J([-Jx,-Jy,z]_{\mathfrak g}+[x,-Jy,-Jz]_{\mathfrak g}+[-Jx,y,-Jz]_{\mathfrak g}),\end{aligned}$$ then we obtain $[x,y,z]_{\mathfrak g}=-J[x,y,Jz]_{\mathfrak g}-J[Jx,y,z]_{\mathfrak g}-J[x,Jy,z]_{\mathfrak g}$ by $({-J})^{-1}=J$.
Furthermore, by $(-J)^{-1}=J$ and Proposition \[3-Lie-compatible-3-pre-Lie\], there exists a compatible $3$-pre-Lie algebra on ${\mathfrak g}$ given by $
\{x,y,z\}=-J{\mathrm{ad}}_{x,y}({-J}^{-1}(z))=-J[x,y,Jz]_{\mathfrak g}.
$ The proof is finished.
Let $J$ be an almost complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. If $J$ satisfies the following equation $$\begin{aligned}
\label{complex-integrability-1}
J[x,y,z]_{\mathfrak g}=-[Jx,Jy,Jz]_{\mathfrak g},\end{aligned}$$ then $J$ is a complex structure on ${\mathfrak g}$.
[[**Proof.**]{} ]{}By and $J^2={\rm{Id}}$, we have $$\begin{aligned}
&&-[Jx,Jy,Jz]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}\\
&&+J[Jx,Jy,z]_{\mathfrak g}+J[x,Jy,Jz]_{\mathfrak g}+J[Jx,y,Jz]_{\mathfrak g}\\
&=&J[x,y,z]_{\mathfrak g}+[Jx,y,z]_{\mathfrak g}+[x,Jy,z]_{\mathfrak g}+[x,y,Jz]_{\mathfrak g}\\
&&-[J^2x,J^2y,Jz]_{\mathfrak g}-[Jx,J^2y,J^2z]_{\mathfrak g}-[J^2x,Jy,J^2z]_{\mathfrak g}\\
&=&J[x,y,z]_{\mathfrak g}.\end{aligned}$$ Thus, $J$ is a complex structure on ${\mathfrak g}$.
[**(Integrability condition IV)**]{} An almost complex structure $J$ on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ is called a [**perfect complex structure**]{} if holds.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a real $3$-Lie algebra. Then ${\mathfrak g}$ has a perfect complex structure if and only if ${\mathfrak g}_{\mathbb C}$ admits a decomposition: $${\mathfrak g}_{\mathbb C}={\mathfrak q}\oplus{\mathfrak p},$$ where ${\mathfrak q}$ and ${\mathfrak p}=\sigma({\mathfrak q})$ are complex subalgebras of ${\mathfrak g}_{\mathbb C}$ such that $[{\mathfrak q},{\mathfrak q},{\mathfrak p}]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak p}$ and $[{\mathfrak p},{\mathfrak p},{\mathfrak q}]_{{\mathfrak g}_{\mathbb C}}\subset{\mathfrak q}.$
Let $J$ be a strict complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then $J$ is a perfect complex structure on ${\mathfrak g}$.
\[ex:A4complex\]
Consider the $4$-dimensional Euclidean $3$-Lie algebra $A_4$ given in Example \[ex:A4symplectic\]. Then $$\begin{aligned}
J_1=\left(\begin{array}{cccc}0&0&-1&0\\
0&0&0&-1\\
1&0&0&0\\
0&1&0&0\end{array}\right),~
J_2=\left(\begin{array}{cccc}0&-1&0&0\\
1&0&0&0\\
0&0&0&-1\\
0&0&1&0\end{array}\right),~
J_3=\left(\begin{array}{cccc}0&-1&0&0\\
1&0&0&0\\
0&0&0&1\\
0&0&-1&0\end{array}\right),~\\
J_4=\left(\begin{array}{cccc} 0&1&0&0\\
-1&0&0&0\\
0&0&0&-1\\
0&0&1&0\end{array}\right),~
J_5=\left(\begin{array}{cccc} 0&1&0&0\\
-1&0&0&0\\
0&0&0&1\\
0&0&-1&0\end{array}\right),~
J_6=\left(\begin{array}{cccc} 0&0&1&0\\
0&0&0&1\\
-1&0&0&0\\
0&-1&0&0\end{array}\right)\end{aligned}$$are abelian complex structures. Moreover, $J_1,J_6$ are strong abelian complex structures and $J_2,J_3,J_4,J_5$ are perfect complex structures.
Complex product structures on $3$-Lie algebras
==============================================
In this section, we add a compatibility condition between a complex structure and a product structure on a 3-Lie algebra to introduce the notion of a complex product structure. We construct complex product structures using 3-pre-Lie algebras. First we illustrate the relation between a complex structure and a product structure on a complex $3$-Lie algebra.
\[equivalent\] Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a complex $3$-Lie algebra. Then $E$ is a product structure on ${\mathfrak g}$ if and only if $J=iE$ is a complex structure on ${\mathfrak g}$.
[[**Proof.**]{} ]{}Let $E$ be a product structure on ${\mathfrak g}$. We have $J^2=i^2E^2=-{\rm{Id}}.$ Thus, $J$ is an almost complex structure on ${\mathfrak g}$. Since $E$ satisfies the integrability condition , we have $$\begin{aligned}
J[x,y,z]_{\mathfrak g}&=&iE[x,y,z]_{\mathfrak g}\\
&=&-[iEx,iEy,iEz]_{\mathfrak g}+[iEx,y,z]_{\mathfrak g}+[x,iEy,z]_{\mathfrak g}+[x,y,iEz]_{\mathfrak g}\\
&&+iE[iEx,iEy,z]_{\mathfrak g}+iE[x,iEy,iEz]_{\mathfrak g}+iE[iEx,y,iEz]_{\mathfrak g}.\end{aligned}$$ Thus, $J$ is a complex structure on the complex $3$-Lie algebra ${\mathfrak g}$.
The converse part can be proved similarly and we omit details.
\[complex-to-special-paracomplex\] Let $J$ be a complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then, $-iJ_{\mathbb C}$ is a paracomplex structure on the complex $3$-Lie algebra $({\mathfrak g}_{\mathbb C},[\cdot,\cdot,\cdot]_{{\mathfrak g}_{\mathbb C}})$, where $J_{\mathbb C}$ is defined by .
[[**Proof.**]{} ]{}By Theorem \[complex-structure-subalgebra\], ${\mathfrak g}_{\mathbb C}={\mathfrak g}_{i}\oplus{\mathfrak g}_{-i}$ and ${\mathfrak g}_{-i}=\sigma({\mathfrak g}_{i})$, where ${\mathfrak g}_{i}$ and ${\mathfrak g}_{-i}$ are subalgebras of ${\mathfrak g}_{\mathbb C}$. It is obvious that $\dim({\mathfrak g}_i)=\dim({\mathfrak g}_{-i})$. By Proposition \[product-structure-subalgebra\], there is a paracomplex structure on ${\mathfrak g}_{\mathbb C}$. On the other hand, it is obvious that $J_{\mathbb C}$ is a complex structure on ${\mathfrak g}_{\mathbb C}$. By Proposition \[equivalent\], $-iJ_{\mathbb C}$ a product structure on the complex $3$-Lie algebra $({\mathfrak g}_{\mathbb C},[\cdot,\cdot,\cdot]_{{\mathfrak g}_{\mathbb C}})$. It is straightforward to see that ${\mathfrak g}_i$ and ${\mathfrak g}_{-i}$ are eigenspaces of $-iJ_{\mathbb C}$ corresponding to $+1$ and $-1$. Thus, $-iJ_{\mathbb C}$ is a paracomplex structure.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a real $3$-Lie algebra. A [**complex product**]{} structure on the $3$-Lie algebra ${\mathfrak g}$ is a pair $\{J,E\}$ of a complex structure $J$ and a product structure $E$ satisfying $$\label{eq:compro}
J\circ E=-E\circ J.$$ If $E$ is perfect, we call $\{J,E\}$ a [**perfect complex product**]{} structure on ${\mathfrak g}.$
Let $\{J,E\}$ be a complex product structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. For all $x\in{\mathfrak g}_+$, by , we have $E(Jx)=-Jx$, which implies that $J({\mathfrak g}_+)\subset{\mathfrak g}_-$. Analogously, we obtain $J({\mathfrak g}_-)\subset{\mathfrak g}_+$. Thus, we get $J({\mathfrak g}_-)={\mathfrak g}_+$ and $J({\mathfrak g}_+)={\mathfrak g}_-$. Therefore, $\dim({\mathfrak g}_+)=\dim({\mathfrak g}_-)$ and $E$ is a paracomplex structure on ${\mathfrak g}$.
Let $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ be a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then the following statements are equivalent:
- ${\mathfrak g}$ has a complex product structure;
- ${\mathfrak g}$ has a complex structure $J$ and can be decomposed as ${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-$, where ${\mathfrak g}_+,{\mathfrak g}_-$ are $3$-Lie subalgebras of ${\mathfrak g}$ and ${\mathfrak g}_-=J{\mathfrak g}_+$.
[[**Proof.**]{} ]{}Let $\{J,E\}$ be a complex product structure and let ${\mathfrak g}_{\pm}$ denote the eigenspaces corresponding to the eigenvalues $\pm1$ of $E$. By Theorem \[product-structure-subalgebra\], both ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are $3$-Lie subalgebras of ${\mathfrak g}$ and $J\circ E=-E\circ J$ implies ${\mathfrak g}_-=J{\mathfrak g}_+.$
Conversely, we can define a linear map $E:{\mathfrak g}\lon{\mathfrak g}$ by $$\begin{aligned}
E(x+\alpha)=x-\alpha,\,\,\,\,\forall x\in{\mathfrak g}_+,\alpha\in{\mathfrak g}_-.\end{aligned}$$ By Theorem \[product-structure-subalgebra\], $E$ is a product structure on ${\mathfrak g}$. By ${\mathfrak g}_-=J{\mathfrak g}_+$ and $J^2=-{\rm{Id}}$, we have $$\begin{aligned}
E(J(x+\alpha))=E(J(x)+J(\alpha))=-J(x)+J(\alpha)=-J(E(x+\alpha)).\end{aligned}$$ Thus, $\{J,E\}$ is a complex product structure on ${\mathfrak g}$. The proof is finished.
\[ex:A4cp\][ Consider the product structures and the complex structures on the $4$-dimensional Euclidean $3$-Lie algebra $A_4$ given in Example \[ex:A4product\] and Example \[ex:A4complex\] respectively. Then $\{J_i,E_i\}$ for $i=1,2,3,4,5,6$ are complex product structures on $A_4$. ]{}
We give a characterization of a perfect complex product structure on a 3-Lie algebra.
\[3-pre-Lie-complex-product\] Let $E$ be a perfect paracomplex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. Then there is a perfect complex product structure $\{J,E\}$ on ${\mathfrak g}$ if and only if there exists a linear isomorphism $\phi:{\mathfrak g}_+\lon{\mathfrak g}_-$ satisfying the following equation $$\begin{aligned}
\label{complex-perfect-product}
\nonumber\phi[x,y,z]_{\mathfrak g}&=&-[\phi(x),\phi(y),\phi(z)]_{\mathfrak g}+[\phi(x),y,z]_{\mathfrak g}+[x,\phi(y),z]_{\mathfrak g}+[x,y,\phi(z)]_{\mathfrak g}\\
&&+\phi[\phi(x),\phi(y),z]_{\mathfrak g}+\phi[x,\phi(y),\phi(z)]_{\mathfrak g}+\phi[\phi(x),y,\phi(z)]_{\mathfrak g},\quad \forall x,y,z\in{\mathfrak g}_+.\end{aligned}$$
[[**Proof.**]{} ]{}Let $\{J,E\}$ be a perfect complex product structure on ${\mathfrak g}$. Define a linear isomorphism $\phi:{\mathfrak g}_+\lon{\mathfrak g}_-$ by $\phi\triangleq J|_{{\mathfrak g}_+}:{\mathfrak g}_+\lon{\mathfrak g}_-$. By the compatibility condition that the complex structure $J$ satisfies and the coherence condition that a perfect product structure $E$ satisfies, we deduce that holds.
Conversely, we define an endomorphism $J$ of ${\mathfrak g}$ by $$\begin{aligned}
\label{complex-perfect-product-structure}
J(x+\alpha)=-\phi^{-1}(\alpha)+\phi(x),\,\,\,\,\forall x\in{\mathfrak g}_+,\alpha\in{\mathfrak g}_-.\end{aligned}$$ It is obvious that $J$ is an almost complex structure on ${\mathfrak g}$ and $J\circ E=-E\circ J$. For all $\alpha,\beta,\gamma\in{\mathfrak g}_-$, let $x,y,z\in{\mathfrak g}_+$ such that $\phi(x)=\alpha,\phi(y)=\beta$ and $\phi(z)=\gamma$. By and , we have $$\begin{aligned}
&&-[J\alpha,J\beta,J\gamma]_{\mathfrak g}+[J\alpha,\beta,\gamma]_{\mathfrak g}+[\alpha,J\beta,\gamma]_{\mathfrak g}+[\alpha,\beta,J\gamma]_{\mathfrak g}\\
&&+J[J\alpha,J\beta,\gamma]_{\mathfrak g}+J[\alpha,J\beta,J\gamma]_{\mathfrak g}+J[J\alpha,\beta,J\gamma]_{\mathfrak g}\\
&=&[x,y,z]_{\mathfrak g}-[x,\phi(y),\phi(z)]_{\mathfrak g}-[\phi(x),y,\phi(z)]_{\mathfrak g}-[\phi(x),\phi(y),z]_{\mathfrak g}\\
&&-\phi^{-1}[x,y,\phi(z)]_{\mathfrak g}-\phi^{-1}[\phi(x),y,z]_{\mathfrak g}-\phi^{-1}[x,\phi(y),z]_{\mathfrak g}\\
&=&-\phi^{-1}[\phi(x),\phi(y),\phi(z)]_{\mathfrak g}\\
&=&J[\alpha,\beta,\gamma]_{\mathfrak g},\end{aligned}$$ which implies that holds for all $\alpha,\beta,\gamma\in{\mathfrak g}_-$. Similarly, we can deduce that holds for all the other cases. Thus, $J$ is a complex structure and $\{J,E\}$ is a perfect complex product structure on the 3-Lie algebra ${\mathfrak g}$.
At the end of this section, we construct perfect complex product structure using 3-pre-Lie algebras.
A nondegenerate symmetric bilinear form ${\mathcal{B}}\in A^*\otimes A^*$ on a real $3$-pre-Lie algebra $(A,\{\cdot,\cdot,\cdot\})$ is called [**invariant**]{} if $$\begin{aligned}
\label{3-pre-Lie-symmetric-bilinear}
{\mathcal{B}}(\{x,y,z\},w)=-{\mathcal{B}}(z,\{x,y,w\}),\,\,\,\,\forall x,y,z,w\in A.\end{aligned}$$ Then ${\mathcal{B}}$ induces a linear isomorphism ${\mathcal{B}}^{\sharp}:A{\,\rightarrow\,}A^*$ by $$\begin{aligned}
\langle{\mathcal{B}}^{\sharp}(x),y\rangle={\mathcal{B}}(x,y),\,\,\,\,\forall x,y\in A.\end{aligned}$$
\[pro:compro\] Let $(A,\{\cdot,\cdot,\cdot\})$ be a real $3$-pre-Lie algebra with a nondegenerate symmetric bilinear from ${\mathcal{B}}$. Then there is a perfect complex product structure $\{J,E\}$ on the semidirect product $3$-Lie algebra $ A^c\ltimes_{L^*}A^*$, where $E$ is given by and the complex structure $J$ is given as follows: $$\begin{aligned}
\label{3-pre-Lie-complex}
J(x+\alpha)=-{{\mathcal{B}}^{\sharp}}^{-1}(\alpha)+{\mathcal{B}}^{\sharp}(x),\,\,\,\,\forall x\in A,\alpha\in A^*.\end{aligned}$$
[[**Proof.**]{} ]{}By Proposition \[paracomplex-3-pre-Lie\], $E$ is a perfect product structure on $A^c\ltimes_{L^*}A^*$. For all $x,y,z\in A^c$, we have $$\begin{aligned}
&&-[{\mathcal{B}}^{\sharp}(x),{\mathcal{B}}^{\sharp}(y),{\mathcal{B}}^{\sharp}(z)]_{L^*}+[{\mathcal{B}}^{\sharp}(x),y,z]_{L^*}+[x,{\mathcal{B}}^{\sharp}(y),z]_{L^*}+[x,y,{\mathcal{B}}^{\sharp}(z)]_{L^*}\\
&&+{\mathcal{B}}^{\sharp}[{\mathcal{B}}^{\sharp}(x),{\mathcal{B}}^{\sharp}(y),z]_{L^*}+{\mathcal{B}}^{\sharp}[x,{\mathcal{B}}^{\sharp}(y),{\mathcal{B}}^{\sharp}(z)]_{L^*}+{\mathcal{B}}^{\sharp}[{\mathcal{B}}^{\sharp}(x),y,{\mathcal{B}}^{\sharp}(z)]_{L^*}\\
&=&[{\mathcal{B}}^{\sharp}(x),y,z]_{L^*}+[x,{\mathcal{B}}^{\sharp}(y),z]_{L^*}+[x,y,{\mathcal{B}}^{\sharp}(z)]_{L^*}\\
&=&L^*(x,y){\mathcal{B}}^{\sharp}(z)+L^*(y,z){\mathcal{B}}^{\sharp}(x)+L^*(z,x){\mathcal{B}}^{\sharp}(y).\end{aligned}$$ By , we have $$\begin{aligned}
\langle {\mathcal{B}}^{\sharp}[x,y,z]_C,w\rangle&=&\langle {\mathcal{B}}^{\sharp}\{x,y,z\},w\rangle+\langle {\mathcal{B}}^{\sharp}\{y,z,x\},w\rangle+\langle {\mathcal{B}}^{\sharp}\{z,x,y\},w\rangle\\
&=&{\mathcal{B}}(\{x,y,z\},w)+{\mathcal{B}}(\{y,z,x\},w)+{\mathcal{B}}(\{z,x,y\},w)\\
&=&-{\mathcal{B}}(z,\{x,y,w\})-{\mathcal{B}}(x,\{y,z,w\})-{\mathcal{B}}(y,\{z,x,w\})\\
&=&-\langle{\mathcal{B}}^{\sharp}(z),\{x,y,w\}\rangle-\langle{\mathcal{B}}^{\sharp}(x),\{y,z,w\}\rangle
-\langle{\mathcal{B}}^{\sharp}(y),\{z,x,w\}\rangle\\
&=&\langle L^*(x,y){\mathcal{B}}^{\sharp}(z),w\rangle+\langle L^*(y,z){\mathcal{B}}^{\sharp}(x),w\rangle+\langle L^*(z,x){\mathcal{B}}^{\sharp}(y),w\rangle,\end{aligned}$$ which implies that $${\mathcal{B}}^{\sharp}[x,y,z]_C=L^*(x,y){\mathcal{B}}^{\sharp}(z)+L^*(y,z){\mathcal{B}}^{\sharp}(x)+L^*(z,x){\mathcal{B}}^{\sharp}(y).$$ Thus, we have $$\begin{aligned}
{\mathcal{B}}^{\sharp}[x,y,z]_C&=&-[{\mathcal{B}}^{\sharp}(x),{\mathcal{B}}^{\sharp}(y),{\mathcal{B}}^{\sharp}(z)]_{L^*}+[{\mathcal{B}}^{\sharp}(x),y,z]_{L^*}+[x,{\mathcal{B}}^{\sharp}(y),z]_{L^*}
+[x,y,{\mathcal{B}}^{\sharp}(z)]_{L^*}\\
&&+{\mathcal{B}}^{\sharp}[{\mathcal{B}}^{\sharp}(x),{\mathcal{B}}^{\sharp}(y),z]_{L^*}+{\mathcal{B}}^{\sharp}[x,{\mathcal{B}}^{\sharp}(y),{\mathcal{B}}^{\sharp}(z)]_{L^*}+{\mathcal{B}}^{\sharp}[{\mathcal{B}}^{\sharp}(x),y,{\mathcal{B}}^{\sharp}(z)]_{L^*}.\end{aligned}$$ By Proposition \[3-pre-Lie-complex-product\], we obtain that $\{J,E\}$ is a perfect complex product structure on $ A^c\ltimes_{L^*}A^*.$
Let $(A,\{\cdot,\cdot,\cdot\})$ be a real $3$-pre-Lie algebra. On the real $3$-Lie algebra ${\mathrm{aff}}(A)=A^c\ltimes_L A$, we consider two endomorphisms $J$ and $E$ given by $$\begin{aligned}
J(x,y)=(-y,x),\,\,\,\,E(x,y)=(x,-y),\,\,\,\,\forall x,y\in A.\end{aligned}$$
With the above notations, $\{J,E\}$ is a perfect complex product structure on the $3$-Lie algebra ${\mathrm{aff}}(A)$.
[[**Proof.**]{} ]{}It is obvious that $E$ is a perfect product structure on ${\mathrm{aff}}(A)$. Moreover, we have $J^2=-{\rm{Id}}$ and $J\circ E=-E\circ J$. Obviously ${\mathrm{aff}}(A)_+=\{(x,0)|x\in A\}, {\mathrm{aff}}(A)_-=\{(0,y)|y\in A\}$. Define $\phi:{\mathrm{aff}}(A)_+\lon{\mathrm{aff}}(A)_-$ by $\phi\triangleq J|_{{\mathrm{aff}}(A)_+}:{\mathrm{aff}}(A)_+\lon{\mathrm{aff}}(A)_-$. More precisely, $\phi(x,0)=(0,x)$. Then for all $(x,0),(y,0),(z,0)\in{\mathrm{aff}}(A)_+$, we have $$\begin{aligned}
&&-[\phi(x,0),\phi(y,0),\phi(z,0)]_L+[\phi(x,0),(y,0),(z,0)]_L+[(x,0),\phi(y,0),(z,0)]_L\\
&&+[(x,0),(y,0),\phi(z,0)]_L
+\phi[\phi(x,0),\phi(y,0),(z,0)]_L+\phi[(x,0),\phi(y,0),\phi(z,0)]_L\\
&&+\phi[\phi(x,0),(y,0),\phi(z,0)]_L\\
&=&[\phi(x,0),(y,0),(z,0)]_L+[(x,0),\phi(y,0),(z,0)]_L+[(x,0),(y,0),\phi(z,0)]_L\\
&=&(0,\{y,z,x\})+(0,\{z,x,y\})+(0,\{x,y,z\})\\
&=&\phi[(x,0),(y,0),(z,0)]_L.\end{aligned}$$ By Proposition \[3-pre-Lie-complex-product\], $\{J,E\}$ is a perfect complex product structure on the $3$-Lie algebra ${\mathrm{aff}}(A)$.
Para-Kähler structures on $3$-Lie algebras
==========================================
In this section, we add a compatibility condition between a symplectic structure and a paracomplex structure on a 3-Lie algebra to introduce the notion of a para-Kähler structure on a $3$-Lie algebra. A para-Kähler structure gives rise to a pseudo-Riemannian structure. We introduce the notion of a Livi-Civita product associated to a pseudo-Riemannian 3-Lie algebra and give its precise formulas using the decomposition of the original 3-Lie algebra.
Let $\omega$ be a symplectic structure and $E$ a paracomplex structure on a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. The triple $({\mathfrak g},\omega,E)$ is called a [**para-Kähler**]{} $3$-Lie algebra if the following equality holds: $$\label{eq:pk}
\omega(Ex,Ey)=-\omega(x,y),\quad \forall x,y\in{\mathfrak g}.$$ If $E$ is perfect, we call $({\mathfrak g},\omega,E)$ a [**perfect para-Kähler**]{} $3$-Lie algebra.
Let $(A,\{\cdot,\cdot,\cdot\})$ be a $3$-pre-Lie algebra. Then $(A^c\ltimes_{L^*}A^*,\omega,E)$ is a perfect para-Kähler $3$-Lie algebra, where $\omega$ is given by and $E$ is defined by .
[[**Proof.**]{} ]{}By Theorem \[3-pre-Lie-phase-space\], $(A^c\ltimes_{L^*}A^*,\omega)$ is a symplectic 3-Lie algebra. By Proposition \[paracomplex-3-pre-Lie\], $E$ is a perfect paracomplex structure on the phase space $T^*A^c$. For all $x_1,x_2\in A,\alpha_1,\alpha_2\in A^*$, we have $$\begin{aligned}
\omega(E(x_1+\alpha_1),E(x_2+\alpha_2))&=&\omega(x_1-\alpha_1,x_2-\alpha_2)=\langle -\alpha_1, x_2\rangle-\langle -\alpha_2, x_1\rangle\\
&=&-\omega(x_1+\alpha_1,x_2+\alpha_2).\end{aligned}$$ Therefore, $(T^*A^c=A^c\ltimes_{L^*}A^*,\omega,E)$ is a perfect paraKähler 3-Lie algebra.
Similar as the case of para-Kähler Lie algebras, we have the following equivalent description of a para-Kähler $3$-Lie algebra.
Let $({\mathfrak g},\omega)$ be a symplectic $3$-Lie algebra. Then there exists a paracomplex structure $E$ on the $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ such that $({\mathfrak g},\omega,E)$ is a para-Kähler $3$-Lie algebra if and only if there exist two isotropic $3$-Lie subalgebras ${\mathfrak g}_+$ and ${\mathfrak g}_-$ such that ${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-$ as the direct sum of vector spaces.
[[**Proof.**]{} ]{}Let $({\mathfrak g},\omega,E)$ be a para-Kähler $3$-Lie algebra. Since $E$ is a paracomplex structure on ${\mathfrak g}$, we have $
{\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-,
$ where ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are $3$-Lie subalgebras of ${\mathfrak g}$. For all $x_1,x_2\in{\mathfrak g}_+$, by , we have $$\begin{aligned}
\omega(Ex_1,Ex_2)=\omega(x_1,x_2)=-\omega(x_1,x_2),\end{aligned}$$ which implies that $\omega({\mathfrak g}_+,{\mathfrak g}_+)=0$. Thus, ${\mathfrak g}_+$ is isotropic. Similarly, ${\mathfrak g}_-$ is also isotropic.
Conversely, since ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are subalgebras, ${\mathfrak g}={\mathfrak g}_+\oplus {\mathfrak g}_-$ as vector spaces, there is a product structure $E$ on ${\mathfrak g}$ defined by . Moreover, since ${\mathfrak g}={\mathfrak g}_+\oplus {\mathfrak g}_-$ as vector spaces and both ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are isotropic, we obtain that dim ${\mathfrak g}_+$=dim ${\mathfrak g}_-$. Thus, $E$ is a paracomplex structure on ${\mathfrak g}$. For all $x_1,x_2\in{\mathfrak g}_+,\alpha_1,\alpha_2\in{\mathfrak g}_-$, since ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are isotropic, we have $$\begin{aligned}
\omega(E(x_1+\alpha_1),E(x_2+\alpha_2))&=&\omega(x_1-\alpha_1,x_2-\alpha_2)=-\omega(x_1,\alpha_2)-\omega(\alpha_1,x_2)\\
&=&-\omega(x_1+\alpha_1,x_2+\alpha_2).\end{aligned}$$ Thus, $({\mathfrak g},\omega,E)$ is a para-Kähler $3$-Lie algebra. The proof is finished.
\[ex:A4pK\][ Consider the symplectic structures and the perfect paracomplex structures on the $4$-dimensional Euclidean $3$-Lie algebra $A_4$ given in Example \[ex:A4symplectic\] and Example \[ex:A4product\] respectively. Then $\{\omega_i,E_i\}$ for $i=1,2,3,4,5,6$ are perfect para-Kähler structures on $A_4$. ]{}
\[ex:standardpK\][Let $({\mathfrak h},[\cdot,\cdot,\cdot]_{\mathfrak h}$ be a 3-Lie algebra and $({\mathfrak h}\oplus {\mathfrak h}^*,\omega)$ its (perfect) phase space, where $\omega$ is given by . Then $E:{\mathfrak h}\oplus {\mathfrak h}^*\longrightarrow{\mathfrak h}\oplus {\mathfrak h}^*$ defined by $$\label{eq:Ephasespace}
E(x+\alpha)=x-\alpha,\quad \forall x\in{\mathfrak h},\alpha\in{\mathfrak h}^*,$$ is a (perfect) paracomplex structure and $({\mathfrak h}\oplus {\mathfrak h}^*,\omega,E)$ is a (perfect) para-Kähler $3$-Lie algebra. ]{}
Let $({\mathfrak g},\omega,E)$ be a para-Kähler $3$-Lie algebra. Then it is obvious that ${\mathfrak g}_-$ is isomorphic to ${\mathfrak g}_+^*$ via the symplectic structure $\omega$. Moreover, it is straightforward to deduce that
\[pro:standardpK\] Any para-Kähler $3$-Lie algebra is isomorphic to the para-Kähler $3$-Lie algebra associated to a phase space of a $3$-Lie algebra.
In the sequel, we study the Levi-Civita product associated to a perfect para-Kähler 3-Lie algebra.
A [**pseudo-Riemannian $3$-Lie algebra**]{} is a $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$ endowed with a nondegenerate symmetric bilinear form $S$. The associated Levi-Civita product is the product on ${\mathfrak g}$, $\nabla:\otimes^3{\mathfrak g}\longrightarrow{\mathfrak g}$ with $(x,y,z)\longmapsto \nabla_{x,y}z$, given by the following formula: $$\begin{aligned}
\label{Levi-Civita product}
3S(\nabla_{x,y}z,w)=S([x,y,z]_{\mathfrak g},w)-2S([x,y,w]_{\mathfrak g},z)+S([y,z,w]_{\mathfrak g},x)+S([z,x,w]_{\mathfrak g},y).\end{aligned}$$
Let $({\mathfrak g},S)$ be a pseudo-Riemannian $3$-Lie algebra. Then the Levi-Civita product $\{\cdot,\cdot,\cdot\}$ satisfies the following equations: $$\begin{aligned}
\nabla_{x,y}z&=&-\nabla_{y,x}z,\\
\nabla_{x,y}z+\nabla_{y,z}x+\nabla_{z,x}y&=&[x,y,z]_{\mathfrak g}.\end{aligned}$$
[[**Proof.**]{} ]{}For all $w\in{\mathfrak g}$, it is obvious that $$\begin{aligned}
3S(\nabla_{y,x}z,w)&=&S([y,x,z]_{\mathfrak g},w)-2S([y,x,w]_{\mathfrak g},z)+S([x,z,w]_{\mathfrak g},y)+S([z,y,w]_{\mathfrak g},x)\\
&=&-3S(\nabla_{x,y}z,w).\end{aligned}$$ By the nondegeneracy of $S$, we obtain $\nabla_{x,y}z=-\nabla_{y,x}z.$
For all $x,y,z,w\in{\mathfrak g}$, we have $$\begin{aligned}
3S(\nabla_{x,y}z,w)&=&S([x,y,z]_{\mathfrak g},w)-2S([x,y,w]_{\mathfrak g},z)+S([y,z,w]_{\mathfrak g},x)+S([z,x,w]_{\mathfrak g},y),\\
3S(\nabla_{y,z}x,w)&=&S([y,z,x]_{\mathfrak g},w)-2S([y,z,w]_{\mathfrak g},x)+S([z,x,w]_{\mathfrak g},y)+S([x,y,w]_{\mathfrak g},z),\\
3S(\nabla_{z,x}y,w)&=&S([z,x,y]_{\mathfrak g},w)-2S([z,x,w]_{\mathfrak g},y)+S([x,y,w]_{\mathfrak g},z)+S([y,z,w]_{\mathfrak g},x).\end{aligned}$$ Add up the three equations, we have $$3S(\nabla_{x,y}z+\nabla_{y,z}x+\nabla_{z,x}y,w)=3S([x,y,z]_{\mathfrak g},w),$$ which implies that $\nabla_{x,y}z+\nabla_{y,z}x+\nabla_{z,x}y=[x,y,z]_{\mathfrak g}.$ The proof is finished.
Let $({\mathfrak g},\omega,E)$ be a perfect para-Kähler $3$-Lie algebra. Define a bilinear form $S$ on ${\mathfrak g}$ by $$\begin{aligned}
S(x,y)\triangleq \omega(x,Ey),\,\,\,\,\forall x,y\in{\mathfrak g}.\end{aligned}$$
With the above notations, $({\mathfrak g},S)$ is a pseudo-Riemannian $3$-Lie algebra. Moreover, the associated Levi-Civita product $\nabla$ and the perfect paracomplex structure $E$ satisfy the following compatibility condition: $$E\nabla _{x,y}z=\nabla_{Ex,Ey}Ez.$$
[[**Proof.**]{} ]{}Since $\omega$ is skewsymmetric and $\omega(Ex,Ey)=-\omega(x,y)$, we have $$\begin{aligned}
S(y,x)=\omega(y,Ex)=-\omega(Ey,E^2x)=-\omega(Ey,x)=\omega(x,Ey)=S(x,y),\end{aligned}$$ which implies that $S$ is symmetric. Moreover, since $\omega$ is nondegenerate and $E^2={\rm{Id}}$, it is obvious that $S$ is nondegenerate. Thus, $S$ is a pseudo-Riemannian metric on the 3-Lie algebra ${\mathfrak g}$. Moreover, we have $$\begin{aligned}
&&3S(\nabla_{Ex,Ey}Ez,w)\\&=&S([Ex,Ey,Ez]_{\mathfrak g},w)-2S([Ex,Ey,w]_{\mathfrak g},Ez)+S([Ey,Ez,w]_{\mathfrak g},Ex)+S([Ez,Ex,w]_{\mathfrak g},Ey)\\
&=& S(E[x,y,z]_{\mathfrak g},w)-2S(E[x,y,Ew]_{\mathfrak g},Ez)+S(E[y,z,Ew]_{\mathfrak g},Ex)+S(E[z,x,Ew]_{\mathfrak g},Ey)\\
&=&-( S([x,y,z]_{\mathfrak g},Ew)-2S([x,y,Ew]_{\mathfrak g},z)+S([y,z,Ew]_{\mathfrak g},x)+S([z,x,Ew]_{\mathfrak g},y))\\
&=&-3S(\nabla_{x,y}z,Ew)\\
&=&3S(E\nabla_{x,y}z,w).\end{aligned}$$ Thus, we have $ E\nabla _{x,y}z=\nabla_{Ex,Ey}Ez.$
The following two propositions clarifies the relationship between the Levi-Civita product and the 3-pre-Lie multiplication on a para-Kähler $3$-Lie algebra.
\[Levi-Civita-3-pre-Lie\] Let $({\mathfrak g},\omega,E)$ be a para-Kähler $3$-Lie algebra and $\nabla$ the associated Levi-Civita product. Then for all $x_1,x_2,x_3\in{\mathfrak g}_+$ and $\alpha_1,\alpha_2,\alpha_3\in{\mathfrak g}_-$, we have $$\nabla_{x_1,x_2}x_3=\{x_1,x_2,x_3\},\quad \nabla_{\alpha_1,\alpha_2}\alpha_3=\{\alpha_1,\alpha_2,\alpha_3\}.$$
[[**Proof.**]{} ]{}Since $({\mathfrak g},\omega,E)$ is a para-Kähler $3$-Lie algebra, $3$-Lie subalgebras ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are isotropic and ${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-$ as vector spaces. For all $x_1,x_2,x_3,x_4\in{\mathfrak g}_+$, we have $$\begin{aligned}
&&3\omega(\nabla_{x_1,x_2}x_3,x_4)\\&=&3S(\nabla_{x_1,x_2}x_3,Ex_4)=3S(\nabla_{x_1,x_2}x_3,x_4)\\
&=&S([x_1,x_2,x_3]_{\mathfrak g},x_4)-2S([x_1,x_2,x_4]_{\mathfrak g},x_3)+S([x_2,x_3,x_4]_{\mathfrak g},x_1)+S([x_3,x_1,x_4]_{\mathfrak g},x_2)\\
&=&\omega([x_1,x_2,x_3]_{\mathfrak g},x_4)-2\omega([x_1,x_2,x_4]_{\mathfrak g},x_3)+\omega([x_2,x_3,x_4]_{\mathfrak g},x_1)+\omega([x_3,x_1,x_4]_{\mathfrak g},x_2)\\
&=&0.\end{aligned}$$ By $({\mathfrak g}_+)^{\perp}={\mathfrak g}_+$, we obtain $\nabla_{x_1,x_2}x_3\in{\mathfrak g}_+$. Similarly, for all $\alpha_1,\alpha_2,\alpha_3\in{\mathfrak g}_-$, $\nabla_{\alpha_1,\alpha_2}\alpha_3\in{\mathfrak g}_-$. Furthermore, for all $x_1,x_2,x_3\in{\mathfrak g}_+,$ and $\alpha\in{\mathfrak g}_-$, we have $$\begin{aligned}
&&3\omega(\nabla_{x_1,x_2}x_3,\alpha)\\&=&3S(\nabla_{x_1,x_2}x_3,E\alpha)=-3S(\nabla_{x_1,x_2}x_3,\alpha)\\
&=&-S([x_1,x_2,x_3]_{\mathfrak g},\alpha)+2S([x_1,x_2,\alpha]_{\mathfrak g},x_3)-S([x_2,x_3,\alpha]_{\mathfrak g},x_1)-S([x_3,x_1,\alpha]_{\mathfrak g},x_2)\\
&=&\omega([x_1,x_2,x_3]_{\mathfrak g},\alpha)+2\omega([x_1,x_2,\alpha]_{\mathfrak g},x_3)-\omega([x_2,x_3,\alpha]_{\mathfrak g},x_1)-\omega([x_3,x_1,\alpha]_{\mathfrak g},x_2)\\
&=&\omega([\alpha,x_1,x_2]_{\mathfrak g},x_3)+2\omega([x_1,x_2,\alpha]_{\mathfrak g},x_3)\\
&=&-3\omega(x_3,[x_1,x_2,\alpha]_{\mathfrak g})\\
&=&3\omega(\{x_1,x_2,x_3\},\alpha).\end{aligned}$$ Thus, $\nabla_{x_1,x_2}x_3=\{x_1,x_2,x_3\}.$ Similarly, we have $\nabla_{\alpha_1,\alpha_2}\alpha_3=\{\alpha_1,\alpha_2,\alpha_3\}$. The proof is finished.
Let $({\mathfrak g},\omega,E)$ be a perfect para-Kähler $3$-Lie algebra and $\nabla$ the associated Levi-Civita product. Then for all $x_1,x_2\in{\mathfrak g}_+$ and $\alpha_1,\alpha_2\in{\mathfrak g}_-$, we have $$\begin{aligned}
\label{eq:conn1}\nabla_{x_1,x_2}\alpha_1&=&\{x_1,x_2,\alpha_1\}+\frac{2}{3}(\{x_2,\alpha_1,x_1\}+\{\alpha_1,x_1,x_2\}),\\
\label{eq:conn2}\nabla_{\alpha_1,x_1}x_2&=&-\frac{1}{3}\{\alpha_1,x_1,x_2\}+\frac{2}{3}\{x_2,\alpha_1,x_1\},\\
\label{eq:conn3}\nabla_{\alpha_1,\alpha_2}x_1&=&\{\alpha_1,\alpha_2,x_1\}+\frac{2}{3}(\{\alpha_2,x_1,\alpha_1\}+\{x_1,\alpha_1,\alpha_2\}),\\
\label{eq:conn4}\nabla_{x_1,\alpha_1}\alpha_2&=&-\frac{1}{3}\{x_1,\alpha_1,\alpha_2\}+\frac{2}{3}\{\alpha_2,x_1,\alpha_1\}.\end{aligned}$$
[[**Proof.**]{} ]{}Since $({\mathfrak g},\omega,E)$ is a perfect para-Kähler $3$-Lie algebra, $3$-Lie subalgebras ${\mathfrak g}_+$ and ${\mathfrak g}_-$ are isotropic and ${\mathfrak g}={\mathfrak g}_+\oplus{\mathfrak g}_-$ as vector spaces. Thus, we have $S({\mathfrak g}_+,{\mathfrak g}_+)=S({\mathfrak g}_-,{\mathfrak g}_-)=0.$ For all $x_1,x_2\in{\mathfrak g}_+$ and $\alpha_1,\alpha_2\in{\mathfrak g}_-$, we have $$\begin{aligned}
&&3S(\nabla_{x_1,x_2}\alpha_1,\alpha_2)\\
&=&S([x_1,x_2,\alpha_1]_{\mathfrak g},\alpha_2)-2S([x_1,x_2,\alpha_2]_{\mathfrak g},\alpha_1)
+S([x_2,\alpha_1,\alpha_2]_{\mathfrak g},x_1)+S([\alpha_1,x_1,\alpha_2]_{\mathfrak g},x_2)=0.\end{aligned}$$ Since $S$ is nondegenerate, we have $\nabla_{x_1,x_2}\alpha_1\in{\mathfrak g}_-.$ Moreover, For all $x_1,x_2,x_3\in{\mathfrak g}_+$ and $\alpha_1\in{\mathfrak g}_-$, we have $$\begin{aligned}
&&3\omega(\nabla_{x_1,x_2}\alpha_1,x_3)\\&=&3S(\nabla_{x_1,x_2}\alpha_1,Ex_3)=3S(\nabla_{x_1,x_2}\alpha_1,x_3)\\
&=&S([x_1,x_2,\alpha_1]_{\mathfrak g},x_3)-2S([x_1,x_2,x_3]_{\mathfrak g},\alpha_1)
+S([x_2,\alpha_1,x_3]_{\mathfrak g},x_1)+S([\alpha_1,x_1,x_3]_{\mathfrak g},x_2)\\
&=&\omega([x_1,x_2,\alpha_1]_{\mathfrak g},Ex_3)-2\omega([x_1,x_2,x_3]_{\mathfrak g},E\alpha_1)
+\omega([x_2,\alpha_1,x_3]_{\mathfrak g},Ex_1)+\omega([\alpha_1,x_1,x_3]_{\mathfrak g},Ex_2)\\
&=&\omega([x_1,x_2,\alpha_1]_{\mathfrak g},x_3)+2\omega([x_1,x_2,x_3]_{\mathfrak g},\alpha_1)
+\omega([x_2,\alpha_1,x_3]_{\mathfrak g},x_1)+\omega([\alpha_1,x_1,x_3]_{\mathfrak g},x_2)\\ &=&\omega([x_1,x_2,\alpha_1]_{\mathfrak g},x_3)+2\omega(\{x_1,x_2,\alpha_1\},x_3)
+\omega(\{x_2,\alpha_1,x_1\},x_3)+\omega(\{\alpha_1,x_1,x_2\},x_3).\end{aligned}$$ Thus, we obtain $$\begin{aligned}
\nabla_{x_1,x_2}\alpha_1&=&\{x_1,x_2,\alpha_1\}+\frac{2}{3}(\{x_2,\alpha_1,x_1\}+\{\alpha_1,x_1,x_2\}),\end{aligned}$$ which implies that holds.
For all $x_1,x_2\in{\mathfrak g}_+$ and $\alpha_1,\alpha_2\in{\mathfrak g}_-$, we have $$\begin{aligned}
&&3S(\nabla_{\alpha_1,x_1}x_2,\alpha_2)\\
&=&S([\alpha_1,x_1,x_2]_{\mathfrak g},\alpha_2)-2S([\alpha_1,x_1,\alpha_2]_{\mathfrak g},x_2)
+S([x_1,x_2,\alpha_2]_{\mathfrak g},\alpha_1)+S([x_2,\alpha_1,\alpha_2]_{\mathfrak g},x_1)=0.\end{aligned}$$ Since $S$ is nondegenerate, we have $\nabla_{\alpha_1,x_1}x_2\in{\mathfrak g}_-.$ Moreover, For all $x_1,x_2,x_3\in{\mathfrak g}_+$ and $\alpha_1\in{\mathfrak g}_-$, we have $$\begin{aligned}
&&3\omega(\nabla_{\alpha_1,x_1}x_2,x_3)\\&=&3S(\nabla_{\alpha_1,x_1}x_2,Ex_3)=3S(\nabla_{\alpha_1,x_1}x_2,x_3)\\
&=&S([\alpha_1,x_1,x_2]_{\mathfrak g},x_3)-2S([\alpha_1,x_1,x_3]_{\mathfrak g},x_2)
+S([x_1,x_2,x_3]_{\mathfrak g},\alpha_1)+S([x_2,\alpha_1,x_3]_{\mathfrak g},x_1)\\ &=&\omega([\alpha_1,x_1,x_2]_{\mathfrak g},x_3)-2\omega([\alpha_1,x_1,x_3]_{\mathfrak g},x_2)
-\omega([x_1,x_2,x_3]_{\mathfrak g},\alpha_1)+\omega([x_2,\alpha_1,x_3]_{\mathfrak g},x_1)\\
&=&\omega([\alpha_1,x_1,x_2]_{\mathfrak g},x_3)-2\omega(\{\alpha_1,x_1,x_2\},x_3)
-\omega(\{x_1,x_2,\alpha_1\},x_3)+\omega(\{x_2,\alpha_1,x_1\},x_3).\end{aligned}$$ Thus, we obtain $$\begin{aligned}
\nabla_{\alpha_1,x_1}x_2&=&-\frac{1}{3}\{\alpha_1,x_1,x_2\}+\frac{2}{3}\{x_2,\alpha_1,x_1\},\end{aligned}$$ which implies that holds.
and can be proved similarly. We omit details. The proof is finished.
Under the isomorphism given in Proposition \[pro:standardpK\] and the correspondence given in Theorem \[thm:MT-ps\], using the formulas provided in Proposition \[pro:stuctureMP3preLie\], we get
For the perfect para-Kähler $3$-Lie algebra $({\mathfrak h}\oplus {\mathfrak h}^*,\omega,E)$ given in Example \[ex:standardpK\], for all $x_1,x_2\in{\mathfrak h}$ and $\alpha_1,\alpha_2\in{\mathfrak h}^*$, we have $$\begin{aligned}
\label{eq:conn11}\nabla_{x_1,x_2}\alpha_1&=&(L^*(x_1,x_2)-\frac{1}{3}R^*(x_2,x_1)+\frac{1}{3}R^*(x_1,x_2))\alpha_1,\\
\label{eq:conn22}\nabla_{\alpha_1,x_1}x_2&=&(\frac{1}{3}R^*(x_1,x_2)+\frac{2}{3}R^*(x_2,x_1))\alpha_1,\\
\label{eq:conn33}\nabla_{\alpha_1,\alpha_2}x_1&=&({\mathcal{L}}^*(\alpha_1,\alpha_2)-\frac{1}{3}{\mathcal{R}}^*(\alpha_2,\alpha_1)+\frac{1}{3}{\mathcal{R}}^*(\alpha_1,\alpha_2))x_1,\\
\label{eq:conn44}\nabla_{x_1,\alpha_1}\alpha_2&=&(\frac{1}{3}{\mathcal{R}}^*(\alpha_1,\alpha_2)+\frac{2}{3}{\mathcal{R}}^*(\alpha_2,\alpha_1))x_1.\end{aligned}$$
Pseudo-Kähler structures on $3$-Lie algebras
============================================
In this section, we add a compatibility condition between a symplectic structure and a complex structure on a 3-Lie algebra to introduce the notion of a pseudo-Kähler structure on a $3$-Lie algebra. The relation between para-Kähler structures and pseudo-Kähler structures on a 3-Lie algebra is investigated.
Let $\omega$ be a symplectic structure and $J$ a complex structure on a real $3$-Lie algebra $({\mathfrak g},[\cdot,\cdot,\cdot]_{\mathfrak g})$. The triple $({\mathfrak g},\omega,J)$ is called a real [**pseudo-Kähler**]{} $3$-Lie algebra if $$\label{eq:pK}
\omega(Jx,Jy)=\omega(x,y),\quad \forall x,y\in{\mathfrak g}.$$
\[ex:A4sK\][ Consider the symplectic structures and the complex structures on the $4$-dimensional Euclidean $3$-Lie algebra $A_4$ given in Example \[ex:A4symplectic\] and Example \[ex:A4complex\] respectively. Then $\{\omega_i,J_i\}$ for $i=1,2,3,4,5,6$ are pseudo-Kähler structures on $A_4$. ]{}
Let $({\mathfrak g},\omega,J)$ be a real pseudo-Kähler $3$-Lie algebra. Define a bilinear form $S$ on ${\mathfrak g}$ by $$\begin{aligned}
S(x,y)\triangleq \omega(x,Jy),\,\,\,\,\forall x,y\in{\mathfrak g}.\end{aligned}$$ Then $({\mathfrak g},S)$ is a pseudo-Riemannian $3$-Lie algebra.
[[**Proof.**]{} ]{}By , we have $$\begin{aligned}
S(y,x)=\omega(y,Jx)=\omega(Jy,J^2x)=-\omega(Jy,x)=\omega(x,Jy)=S(x,y),\end{aligned}$$ which implies that $S$ is symmetric. Moreover, since $\omega$ is nondegenerate and $J^2=-{\rm{Id}}$, it is obvious that $S$ is nondegenerate. Thus, $S$ is a pseudo-Riemannian metric on the 3-Lie algebra ${\mathfrak g}$.
Let $({\mathfrak g},\omega,J)$ be a real pseudo-Kähler $3$-Lie algebra. If the associated pseudo-Riemannian metric is positive definite, we call $({\mathfrak g},\omega,J)$ a real [**Kähler**]{} $3$-Lie algebra.
Let $({\mathfrak g},\omega,E)$ be a complex para-Kähler $3$-Lie algebra. Then $({\mathfrak g}_{\mathbb R},\omega_{\mathbb R},J)$ is a real pseudo-Kähler $3$-Lie algebra, where ${\mathfrak g}_{\mathbb R}$ is the underlying real $3$-Lie algebra, $J=iE$ and $\omega_{\mathbb R}={\mathrm{Re}}(\omega)$ is the real part of $\omega.$
[[**Proof.**]{} ]{}By Proposition \[equivalent\], $J=iE$ is a complex structure on the complex $3$-Lie algebra ${\mathfrak g}$. Thus, $J$ is also a complex structure on the real $3$-Lie algebra ${\mathfrak g}_{\mathbb R}$. It is obvious that $\omega_{\mathbb R}$ is skew-symmetric. If for all $x\in{\mathfrak g}$, $\omega_{\mathbb R}(x,y)=0$. Then we have $$\begin{aligned}
\omega(x,y)=\omega_{\mathbb R}(x,y)+i\omega_{\mathbb R}(-ix,y)=0.\end{aligned}$$ By the nondegeneracy of $\omega$, we obtain $y=0$. Thus, $\omega_{\mathbb R}$ is nondegenerate. Therefore, $\omega_{\mathbb R}$ is a symplectic structure on the real $3$-Lie algebra ${\mathfrak g}_{\mathbb R}$. By $\omega(Ex,Ey)=-\omega(x,y)$, we have $$\begin{aligned}
\omega_{\mathbb R}(Jx,Jy)={\mathrm{Re}}(\omega(iEx,iEy))={\mathrm{Re}}(-\omega(Ex,Ey))={\mathrm{Re}}(\omega(x,y))=\omega_{\mathbb R}(x,y).\end{aligned}$$ Thus, $({\mathfrak g}_{\mathbb R},iE,\omega_{\mathbb R})$ is a real pseudo-Kähler $3$-Lie algebra.
Conversely, we have
Let $({\mathfrak g},\omega,J)$ be a real pseudo-Kähler $3$-Lie algebra. Then $({\mathfrak g}_{\mathbb C},\omega_{\mathbb C},E)$ is a complex para-Kähler $3$-Lie algebra, where ${\mathfrak g}_{\mathbb C}={\mathfrak g}\otimes_{\mathbb R}\mathbb C$ is the complexification of ${\mathfrak g}$, $E=-iJ_{\mathbb C}$ and $\omega_{\mathbb C}$ is the complexification of $\omega$, more precisely, $$\begin{aligned}
\label{complex-omega}
\omega_{\mathbb C}(x_1+iy_1,x_2+iy_2)=\omega(x_1,x_2)-\omega(y_1,y_2)+i\omega(x_1,y_2)+i\omega(y_1,x_2), \quad\forall x_1,x_2,y_1,y_2\in{\mathfrak g}.\end{aligned}$$
[[**Proof.**]{} ]{}By Corollary \[complex-to-special-paracomplex\], $E=-iJ_{\mathbb C}$ is a paracomplex structure on the complex $3$-Lie algebra ${\mathfrak g}_{\mathbb C}$. It is obvious that $\omega_{\mathbb C}$ is skew-symmetric and nondegenerate. Moreover, since $\omega$ is a symplectic structure on ${\mathfrak g}$, we deduce that $\omega_{\mathbb C}$ is a symplectic structure on ${\mathfrak g}_{\mathbb C}.$ Finally, by $\omega(Jx,Jy)=\omega(x,y)$, we have $$\begin{aligned}
\omega_{\mathbb C}(E(x_1+iy_1),E(x_2+iy_2))&=&\omega_{\mathbb C}(Jy_1-iJx_1,Jy_2-iJx_2)\\
&=&\omega(Jy_1,Jy_2)-\omega(Jx_1,Jx_2)-i\omega(Jx_1,Jy_2)-i\omega(Jy_1,Jx_2)\\
&=&\omega(y_1,y_2)-\omega(x_1,x_2)-i\omega(x_1,y_2)-i\omega(y_1,x_2)\\
&=&-\omega_{\mathbb C}(x_1+iy_1,x_2+iy_2).\end{aligned}$$ Therefore, $({\mathfrak g}_{\mathbb C},\omega_{\mathbb C},-iJ_{\mathbb C})$ is a complex para-Kähler $3$-Lie algebra.
At the end of this section, we construct a Kähler $3$-Lie algebra using a $3$-pre-Lie algebra with a symmetric and positive definite invariant bilinear form.
Let $(A,\{\cdot,\cdot,\cdot\})$ be a real $3$-pre-Lie algebra with a symmetric and positive definite invariant bilinear form ${\mathcal{B}}$. Then $(A^c\ltimes_{L^*}A^*,\omega,-J)$ is a real Kähler $3$-Lie algebra, where $J$ is given by and $\omega$ is given by .
[[**Proof.**]{} ]{}By Theorem \[3-pre-Lie-phase-space\] and Proposition \[pro:compro\], $\omega$ is a symplectic structure and $J$ is a perfect complex structure on the semidirect product 3-Lie algebra $( A^c\ltimes_{L^*}A^*,[\cdot,\cdot,\cdot]_{L^*})$. Obviously, $-J$ is also a perfect complex structure on $A^c\ltimes_{L^*}A^*$. Let $\{e_1,\cdots,e_n\}$ be a basis of $A$ such that ${\mathcal{B}}(e_i,e_j)=\delta_{ij}$ and $e_1^*,\cdots,e_n^*$ be the dual basis of $A^*$. Then for all $i,j,k,l$, we have $$\begin{aligned}
\omega(e_i+e_j^*,e_k+e_l^*)&=&\delta_{jk}-\delta_{li},\\
\omega(-J(e_i+e_j^*),-J(e_k+e_l^*))&=&\omega(e_j-e_i^*,e_l-e_k^*)=-\delta_{il}+\delta_{kj},\end{aligned}$$ which implies that $\omega(-J(x+\alpha),-J(y+\beta))=\omega(x+\alpha,y+\beta)$ for all $x,y\in A$ and $\alpha,\beta\in A^*$. Therefore, $(A^c\ltimes_{L^*}A^*,\omega,-J)$ is a pseudo-Kähler $3$-Lie algebra. Finally, Let $x=\sum_{i=1}^{n}\lambda_ie_i\in A,\alpha=\sum_{i=1}^{n}\mu_ie_i^*\in A^*$ such that $x+\alpha\not=0.$ We have $$\begin{aligned}
S(x+\alpha,x+\alpha)&=&\omega(x+\alpha,-J(x+\alpha))\\
&=&\omega\big(\sum_{i=1}^{n}\lambda_ie_i+\sum_{i=1}^{n}\mu_ie_i^*,\sum_{i=1}^{n}\mu_ie_i-\sum_{i=1}^{n}\lambda_ie_i^*)\big)\\
&=&\sum_{i=1}^{n}\mu_i^2+\sum_{i=1}^{n}\lambda_i^2>0.\end{aligned}$$ Thus, $S$ is positive definite. Therefore, $\{A^c\ltimes_{L^*}A^*,\omega,-J\}$ is a real Kähler $3$-Lie algebra.
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Department of Mathematics, Jilin University, Changchun 130012, China
Email: [email protected], [email protected]
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abstract: 'We study global behavior of small solutions of the Gross-Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schrödinger equation, this implies asymptotic stability of all plane wave solutions for such disturbances. We also prove that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it. The key ingredients are: (1) some quadratic transforms of the solutions, which effectively linearize the nonlinear energy space, (2) a bilinear Fourier multiplier estimate, which allows irregular denominators due to a degenerate non-resonance property of the quadratic interactions, and (3) geometric investigation of the degeneracy in the Fourier space to minimize its influence.'
author:
- 'Stephen Gustafson,Kenji Nakanishi,Tai-Peng Tsai'
title: |
Scattering theory for\
the Gross-Pitaevskii equation\
in three dimensions
---
Introduction
============
We continue the study [@vac; @vac2] of the global dispersive nature of solutions for the Gross-Pitaevskii equation (GP) [$$\begin{split} \label{GP}
{&}i\psi_t + {\Delta}\psi = (|\psi|^2-1)\psi,\quad \psi:{\mathbb{R}}^{1+3}\to{\mathbb{C}}\end{split}$$]{} with the boundary condition $\lim_{|x|\to{\infty}}\psi=1$. The equation itself is equivalent to the defocusing nonlinear Schrödinger equation (NLS) by putting ${\varphi}=e^{-it}\psi$ [$$\begin{split} \label{NLS}
{&}i{\varphi}_t + {\Delta}{\varphi}= |{\varphi}|^2{\varphi},
\end{split}$$]{} but the nonzero boundary condition has nontrivial and often remarkable effects on the space-time global behavior of the solutions both in dispersive and non-dispersive regimes, on which there has been extensive studies [@BGS; @BOS; @BS1; @BSm; @Ch; @CJ1; @CJ2; @Ga; @Ge; @Gr2; @Gr3; @vac; @vac2; @J2; @LaSch; @LX; @OS; @Sp]. This boundary condition, or more generally $\lim_{|x|\to{\infty}}|\psi|=1$, is natural in some physical contexts such as superfluids and nonlinear optics, or generally in the hydrodynamic interpretation of NLS, where $|\psi|^2$ is the fluid density and the zeros correspond to vortices. (For more physical backgrounds, see [@BS2; @FS; @JR; @JPR; @SS] and the references therein.)
Hence it is not surprising that our GP equation, after some transformations , is very similar to a version of the Boussinesq equation [$$\begin{split} \label{Bsq}
u_{tt} - 2{\Delta}u + {\Delta}^2 u = -{\Delta}(u^2),
\end{split}$$]{} which is a model for water waves and other fluid dynamics. More precisely, this equation can be regarded as a simplified version of our transformed equations (See Remark \[transBsq\] for the details). Our analysis will reveal that the linear part, which is exactly the same for and our GP, has better dispersive properties than both the low frequency limit (the wave equation) and the hight frequency limit (the Schrödinger equation) for this type of bilinear interaction.
We recall some known facts about GP . For any solution $\psi=1+u$, we have conservation of renormalized energy: [$$\begin{split}
E_1(\psi) {&}:= \int_{{\mathbb{R}}^3} |{\nabla}\psi|^2 + \frac{(|\psi|^2-1)^2}{2} dx
= \int_{{\mathbb{R}}^3} |{\nabla}u|^2 + \frac{(2{\mathop{\mathrm{Re}}}u+|u|^2)^2}{2} dx,
\end{split}$$]{} provided it is initially finite, and then the solution $\psi$ is unique and global [@Ge]. As a remarkable feature of GP, there is a family of traveling wave solutions [@BS1; @Ch] of the form $\psi(t,x)=v_c(x-ct)$ with finite energy for $0<|c|<\sqrt{2}$, where $\sqrt{2}$ is the sound speed. They are quite different from the solitary waves of the focusing NLS in their slow (algebraic) decay at the spatial infinity [@Gr3] and bounded (subsonic) range of speeds [@Gr2].
Recently it is proved [@BGS] that there is a lower bound on the energy of all possible traveling waves for in three dimensions: [$$\begin{split} \label{def E0}
{\mathcal{E}}_0 := \inf\{E_1(\psi) \mid \text{$\psi(t,x)=v(x-ct)$ solves \eqref{GP} for some $c$}\}>0.
\end{split}$$]{} If one believes that there is no more stable structure supported by , it is natural to expect that the regime below the threshold ${\mathcal{E}}_0$ is dominated by dispersion, as was conjectured in [@BS2]. Understanding the effect of dispersion in the nonlinearity is an essential step towards investigating asymptotic stability of traveling waves. However the problem is not quite easy, because we have quadratic interactions of the perturbation $u=\psi-1$, without any decaying factor, due to the nonzero constant background: [$$\begin{split} \label{eq u0}
iu_t + {\Delta}u - 2{\mathop{\mathrm{Re}}}u &= u^2+2|u|^2+|u|^2u.
\end{split}$$]{} They are much stronger, for dispersive waves, than cubic interactions and quadratic ones with decaying factors, which typically arise in the stability analysis of solitary waves for NLS. In fact, if one considers the NLS with general quadratic terms as a model equation, [$$\begin{split}
iu_t + {\Delta}u = {\lambda}_+ u^2 + {\lambda}_0 |u|^2 + {\lambda}_- {\overline}{u}^2,
\end{split}$$]{} then the asymptotic behavior for $t\to{\infty}$ of solutions from small localized initial data is known [@HN; @HMN] only if ${\lambda}_0=0$ in three dimensions. From the technical view point, the quadratic power in three dimensions corresponds to the so-called Strauss exponent, because the $L^p$ decay estimate [$$\begin{split}
\|e^{it{\Delta}}{\varphi}\|_{L^3({\mathbb{R}}^3)} {{\ \lesssim \ }}|t|^{-1/2}\|{\varphi}\|_{L^{3/2}({\mathbb{R}}^3)},
\end{split}$$]{} implies that $L^3$ is mapped back to the dual $L^{3/2}$ by the quadratic nonlinearity, with the critical decay order $|t|^{-1}$ for integrability. Thus one cannot get closed nonlinear estimates just from the $L^p$ decay, unless one starts with given final states ([@vac2] gives such a result for GP). Hence we have to take account of the oscillatory property of the quadratic terms. Then the term $|u|^2$ is worse than the others, because its phase by the linear approximation is stationary both in space and time at zero frequency ${\xi}=0$.
Actually, we cannot neglect the first order term $2{\mathop{\mathrm{Re}}}u$ for large time behavior, so we should not expect too strong an analogy with the quadratic NLS. By the diagonalizing transform [$$\begin{split}
u=u_1+iu_2 \mapsto v=u_1+iUu_2, {\quad}U=\sqrt{-{\Delta}(2-{\Delta})^{-1}},
\end{split}$$]{} we get the equation for $v$ with a self-adjoint operator $H=\sqrt{-{\Delta}(2-{\Delta})}$ : [$$\begin{split}
iv_t - H v = U(3u_1^2+u_2^2+|u|^2u_1) + i(2u_1u_2+|u|^2u_2).
\end{split}$$]{} The equation for $U^{-1}v$ is[^1] similar to the quadratic wave equation around ${\xi}\to 0$ [$$\begin{split} \label{NLW}
u_{tt} - {\Delta}u = |{\nabla}u|^2,
\end{split}$$]{} if we pick up the first nonlinearity $U(u_1^2)$ only. This equation is known [@John] to blow up from arbitrarily small initial data in $C_0^{\infty}$. Note that the other terms $U(u_2^2)+2i(u_1u_2)$ are even worse, enhancing the low frequency through $u_2=U^{-1}{\mathop{\mathrm{Im}}}v$. The wave equation behaves worse in the quadratic interactions than the Schrödinger (and the Klein-Gordon) equation because, beside the slower $L^p$ decay of $e^{it\sqrt{-{\Delta}}}$, the waves propagating in parallel directions are strongly resonant in the bilinear forms, unless they have the null structure (special coefficients killing exactly the parallel interactions).
Despite of all these observations, we can prove that the solutions for GP with small localized initial energy disperse linearly at time infinity. Below we denote ${{{\langle}x {\rangle}}}=\sqrt{2+|x|^2}$ and ${{{\langle}{\nabla}{\rangle}}}=\sqrt{2-{\Delta}}$. $H^1$ is the standard Sobolev space with the norm $\|v\|_{H^1}=\|{{{{\langle}{\nabla}{\rangle}}}}v\|_{L^2}$, and ${{{\langle}x {\rangle}}}^{-1}H^1$ is the weighted space with the norm $\|{{{\langle}x {\rangle}}}v\|_{H^1}$.
\[thm:init\] There exists ${\delta}>0$ such that: For any $u(0)\in H^1({\mathbb{R}}^3)$ satisfying [$$\begin{split} \label{init u}
\int_{{\mathbb{R}}^3} {{{\langle}x {\rangle}}}^2(|{\mathop{\mathrm{Re}}}u(0,x)|^2 + |{\nabla}u(0,x)|^2) dx < {\delta}^2,
\end{split}$$]{} we have a unique global solution $\psi=1+u$ of such that $v := {\mathop{\mathrm{Re}}}u + i U{\mathop{\mathrm{Im}}}u$ satisfies $e^{itH}v\in C({\mathbb{R}};{{{\langle}x {\rangle}}}^{-1}H^1({\mathbb{R}}^3))$ and for some $v_+\in{{{\langle}x {\rangle}}}^{-1}H^1({\mathbb{R}}^3)$, [$$\begin{split} \label{scatt}
{&}\left\|v(t) - e^{-itH}v_+\right\|_{H^1} \le O(t^{-1/2}),
{\quad}\left\|{{{\langle}x {\rangle}}}\left\{e^{itH}v(t) - v_+\right\}\right\|_{H^1} \to 0,
\end{split}$$]{} as $t\to{\infty}$. Moreover, we have $E_1(\psi)=\|{{{{\langle}{\nabla}{\rangle}}}}v_+\|_{L^2}^2$, and the correspondence $v(0)\mapsto v_+$ defines a bi-Lipschitz map between $0$-neighborhoods of ${{{\langle}x {\rangle}}}^{-1}H^1({\mathbb{R}}^3)$.
Actually we have some decay $t^{-{\varepsilon}}$ also for the weighted norm in , but we will not try to specify it, since it is quite small. For pointwise decay, we can derive from the weighted energy estimate together with the $L^p$ estimate on $e^{-itH}$ that [$$\begin{split} \label{pt dec u}
\|u_1(t)\|_{L^{\infty}({\mathbb{R}}^3)} \le O(t^{-1}),
{\quad}\|u_2(t)\|_{L^{\infty}({\mathbb{R}}^3)} \le O(t^{-9/10}).
\end{split}$$]{}
In view of the scaling property, the optimal (weakest) weight should be ${{{\langle}x {\rangle}}}^{-1/2}$ instead of ${{{\langle}x {\rangle}}}^{-1}$. The latter is however more convenient to estimate in the Fourier space, where it turns into one derivative. We can see that all the traveling waves with finite energy have finite, from their asymptotic behavior [@Gr3] as $|x|\to{\infty}$. For that purpose, we should not use a weight stronger than ${{{\langle}x {\rangle}}}^{-3/2}$, hence ${{{\langle}x {\rangle}}}^{-1}$ is the unique integer choice.
By using fractional derivatives in the Fourier space, we could reduce the weight slightly, but probably not to the optimal rate $1/2$, since our argument needs to sacrifice part of the decay in some bilinear estimates with Fourier singularities.
Those singularities arise because the linearized operator $e^{-itH}$ behaves like the wave equation around ${\xi}\to 0$, where the parallel interactions become stronger. Since we know that solutions of nonlinear wave equations such as blow up, we must use the difference between $H$ and $\sqrt{-2{\Delta}}$ in estimating the bilinear terms, which gives us some degenerate non-resonance property. The singular multipliers appear when we integrate on the phase in the Fourier regions for such interactions. In order to treat them, we will derive an estimate for bilinear Fourier multipliers (Lemma \[sbil\]), which allows much less regular multipliers than the standard Coifman-Meyer type estimates. It will also be crucial to investigate carefully the geometric structure of the Fourier regions for those singularities, in order to exploit their smallness in volume.
On the other hand, we also have to take care of those bad quadratic terms which contain $u_2=U^{-1}{\mathop{\mathrm{Im}}}v$. The remedy was already given in [@vac; @vac2], by the quadratic transform [$$\begin{split} \label{def M}
u \mapsto M(u) := v + {{{{\langle}{\nabla}{\rangle}}}}^{-2}|u|^2,
\end{split}$$]{} which effectively removes those terms from the equation. We will introduce a new quadratic transform, which behaves better in terms of regularity for high-high interactions. It turns out that the wave-like behavior of $e^{-itH}$ becomes an advantage when estimating the quadratic parts of those transforms, because it gives more decay for ${\xi}\to 0$ than $e^{it{\Delta}}$. Indeed that is crucial to get the scattering in the same topology ${{{\langle}x {\rangle}}}^{-1}H^1$ as the initial data space.
Extending the above result to the energy space is a supercritical problem from the scaling view point, and so it seems beyond our current technology (it is critical in four dimensions, where we have the small energy scattering [@vac]). However, we can still solve the final state problem without uniqueness. Since the energy space is essentially nonlinear for $u$, we need the explicit linearizing map $M(u)$ to state our scattering result.
\[thm:fin\] For any $z_+\in H^1({\mathbb{R}}^3)$, there exists a global solution $\psi=1+u$ of satisfying $E_1(\psi)=\|{{{{\langle}{\nabla}{\rangle}}}}z_+\|_{L^2({\mathbb{R}}^3)}^2$ and as $t\to{\infty}$ [$$\begin{split} \label{scat z}
{&}\|M(u) - e^{-itH}z_+\|_{H^1({\mathbb{R}}^3)} \to 0.
\end{split}$$]{}
The proof depends on a compactness argument, so we get almost no information about the initial data set for the scattering. However, the above initial data result Theorem \[thm:init\] implies that it contains at least those data satisfying .
In , we can replace $M(u)$ with $v$ if the norm is also replaced with $\dot H^1$, but the quadratic part is generally not even bounded in $L^2$ (see Remark \[L2 unbd\] for the details). Hence the nonlinear effect is not really vanishing for large time in the energy sense, but instead the map $M$ essentially linearizes the (renormalized) energy space [$$\begin{split}
{&}F_1 := \{f \in \dot H^1({\mathbb{R}}^3) \mid 2{\mathop{\mathrm{Re}}}f +|f|^2 \in L^2({\mathbb{R}}^3)\},
\end{split}$$]{} where $\dot H^1({\mathbb{R}}^3)=\{f\in L^6({\mathbb{R}}^3)\mid {\nabla}f\in L^2({\mathbb{R}}^3)\}$ is the homogeneous Sobolev space. $F_1$ was introduced in [@Ge] with the distance ${\delta}(f,g)$ defined by[^2] [$$\begin{split}
{\delta}(f,g)^2 = \|{\nabla}(f-g)\|_{L^2}^2 + \frac{1}{2}\||f|^2+2f_1-|g|^2-2g_1\|_{L^2}^2.
\end{split}$$]{} More precisely, the energy space in three dimensions was characterized as [$$\begin{split}
\{e^{i{\theta}}(1+f)\mid {\theta}\in{\mathbb{R}},\ f \in F_1\}.
\end{split}$$]{} As for the mapping property of $M$ on the energy space, we have the following. Let for any $L>0$ and ${\kappa}>0$, [$$\begin{split}
{&}F_1(L,{\kappa}) := \{f\in F_1\mid E_1(1+f)\le L^2,\ \|f\|_{L^6}\le {\kappa}\},
{\\ &}H^1(L,{\kappa}) := \{f\in H^1\mid \|f\|_{H^1}\le L,\ \|f\|_{L^6}\le {\kappa}\}.
\end{split}$$]{}
\[thm:map\] (1) For any $f,g\in F_1$, we have [$$\begin{split} \label{energy mapping}
{\delta}(f,g)^2 = \|{{{{\langle}{\nabla}{\rangle}}}}(M(f)-M(g))\|_{L^2}^2 + \frac{1}{2}\|U(|f|^2-|g|^2)\|_{L^2}^2,
\end{split}$$]{} hence the map $u\mapsto M(u)$ is Lipschitz continuous from $F_1$ to $H^1$.
\(2) For any $L>0$, there exist ${\kappa}>0$, ${\kappa}'>2{\kappa}$, and a unique map $R:H^1(L,{\kappa})\to F_1(2L,{\kappa}')$ satisfying $M\circ R=id$. Moreover we have [$$\begin{split}
{\delta}(R(f),R(g)) \le 2\|{{{{\langle}{\nabla}{\rangle}}}}(f-g)\|_{L^2},
\end{split}$$]{} namely the inverse is also Lipschitz in this domain.
Remark that the smallness is required only in $L^6$, not in the energy, and the $L^6$ norm decays if the solution scatters. Hence we can apply the invertible maps to scattering solutions and their asymptotic profiles for sufficiently large time. Thus we can rewrite the scattering statement as [$$\begin{split}
{\delta}(u(t),R(e^{-itH}z_+))\to 0 {\quad}(t\to{\infty}).
\end{split}$$]{}
We can also translate the above results to [*asymptotically plane wave solutions*]{} for the cubic NLS . First, it is easy to see that for any $a\ge 0$ and any $d\in{\mathbb{N}}$, every weak solution ${\varphi}\in L^3_{loc}({\mathbb{R}}^{1+d})$ satisfying $|{\varphi}|=a$ on ${\mathbb{R}}^{1+d}$ is written in the form [$$\begin{split} \label{plane}
{\varphi}(t,x) = a e^{-i(a^2+|b|^2)t+ibx+ic},
\end{split}$$]{} for some $b\in{\mathbb{R}}^d$ and $c\in{\mathbb{R}}$. The above solutions for can be regarded as perturbations of the plane wave ${\varphi}=e^{-it}(1+u)$, and perturbations of the other plane waves are generated by the invariance of NLS under the scaling and Galilean transforms. More explicitly, for any solution $\psi=1+u$ of , for any $a>0$, $b \in {\mathbb{R}}^3$ and $c\in{\mathbb{R}}$, we have a solution ${\varphi}$ for of the form [$$\begin{split} \label{asy plane}
{\varphi}(t,x) = ae^{-i(a^2+|b|^2)t+ibx+ic}(1+u(a^2t,a(x-2bt))).
\end{split}$$]{} In particular, Theorem \[thm:init\] implies the asymptotic stability of all plane waves in the weighted energy norm , as well as in $L^{\infty}$ , for perturbations $u(0)$ satisfying . Similarly, Theorem \[thm:fin\] implies that there exist plenty of asymptotically plane wave solutions with finite (renormalized) energy, at least one for each asymptotic profile from $H^1$, $a>0$, $b\in{\mathbb{R}}^3$ and $c\in{\mathbb{R}}$.
Notice that the situation is quite different for the standard boundary condition $a=0$. Indeed the invariant scaling ${\varphi}\mapsto {\lambda}{\varphi}({\lambda}^2 t,{\lambda}x)$ also changes the energy [$$\begin{split}
E_0({\varphi}) = \int_{{\mathbb{R}}^3} |{\nabla}{\varphi}|^2 + \frac{|{\varphi}|^4}{2} dx,
\end{split}$$]{} homogeneously, so the smallness of the energy $E_0({\varphi})$ has no impact on the scattering, and the transform $M$ is reduced to the identity. Moreover, it is easy to observe that the free propagator $e^{it{\Delta}}$ is continuous on the energy space $\dot H^1\cap L^4$, but not globally bounded. Hence some free solutions from this space cannot be asymptotic to any nonlinear solution in the full energy. Once we restrict the solutions to $L^2({\mathbb{R}}^3)$, then the scattering problem was completely solved by [@GV], extending the earlier result [@LS] in a smaller space. But without $L^2$ finiteness, it is not even clear whether we can have a result similar to Theorem \[thm:fin\] in $\dot H^1$, or we should instead modify the set of asymptotic profiles.
We conclude the introduction by reinforcing the conjecture in [@BS2] into a precise statement of nonlinear scattering. Let ${\mathcal{E}}_0>0$ be as in .
\[conj\] For any global solution $\psi\in C({\mathbb{R}};1+F_1)$ of satisfying $E_1(\psi)<{\mathcal{E}}_0$, there is a unique $z_+\in H^1({\mathbb{R}}^3)$ satisfying $E_1(\psi)=\|{{{{\langle}{\nabla}{\rangle}}}}z_+\|_{L^2}^2$ and [$$\begin{split}
\|M(u(t))-e^{-itH}z_+\|_{H^1({\mathbb{R}}^3)} \to 0 {\quad}(t\to{\infty}).
\end{split}$$]{} Moreover, the map $u(0)\mapsto z_+$ is a homeomorphism between the open balls of radius ${\mathcal{E}}_0^{1/2}$ around $0$ in $F_1$ and $H^1$.
The rest of this paper is organized as follows. Sections 2–3 are preparatory. Section 4 deals with the final data problem. Sections 5–13 are devoted to the initial data problem. In the appendix, we give a correction to our previous paper [@vac2].
In more detail: in Section 2, we collect some notation and basic estimates used throughout the paper. In Section 3, we compute the normal forms including the previous and the new ones. In Section 4, we prove the final data result Theorem \[thm:fin\] and also Theorem \[thm:map\]. In Section 5, we start with the initial data problem by setting up the main function space $X$ and deriving its decay properties. In Section 6, we derive the weighted estimates for the normal form transform. Section 7 deals with the initial data. In Section 8, we estimate the Strichartz norms without weight. In Section 9, we treat the easier parts of the weighted estimate, namely those without the phase derivative, and also the quartic terms. In Section 10, we treat the main terms, namely the bilinear forms with the phase derivative, using a bilinear multiplier estimate allowing singularities, and assuming some bounds on the singular bilinear multipliers. Those bounds are proved in Section 11. The cubic terms with the phase derivative are estimated in Section 12. Finally we prove Theorem \[thm:init\] in Section 13.
Preliminaries
=============
In this section we establish some notation and basic setting for our problem. First, for function spaces, we denote the Lebesgue, the Lorentz, the Sobolev and the Besov spaces by $L^p$, $L^{p,q}$, $H^{s,p}$ and $B^s_{p,q}$ respectively, for $1\le p,q\le {\infty}$ and $s\in{\mathbb{R}}$. The homogeneous versions of the latter two are denoted by $\dot H^{s,p}$ and $\dot B^s_{p,q}$. We do not use the spaces with $0\le p<1$, but instead we use the following convention [$$\begin{split}
L^{1/p} = L^p,{\quad}L^{1/p,1/q} = L^{p,q},{\quad}H^{s,1/p} = H^{s,p},
\end{split}$$]{} for convenience of explicit computation of the Hölder exponents. For any Banach space $B$, we denote the same space with the weak topology by ${{\text{w-}B}}$.
We denote the Fourier transform on ${\mathbb{R}}^d$ by [$$\begin{split}
{&}{\mathcal{F}}{\varphi}= {\widetilde}{\varphi}({\xi}) := \int_{{\mathbb{R}}^d} {\varphi}(x)e^{-ix{\xi}} dx,
{\\ &}{\mathcal{F}}_x^{\xi}[f(x,y)] = ({\mathcal{F}}_x^{\xi}f)({\xi},y):= \int_{{\mathbb{R}}^d} f(x,y)e^{-ix{\xi}} dx,
\end{split}$$]{} and the Fourier multiplier of any function ${\varphi}$ by [$$\begin{split}
{&}{\varphi}(-i{\nabla})f := {\mathcal{F}}^{-1}[{\varphi}({\xi}){\widetilde}f({\xi})],
{\\ &}{\varphi}(-i{\nabla})_x f(x,y) := ({\mathcal{F}}_x^{\xi})^{-1}[{\varphi}({\xi}){\mathcal{F}}_x^{\xi}[f(x,y)]].
\end{split}$$]{}
Next we introduce a Littlewood-Paley decomposition (homogeneous version) in the standard way. Fix a cut-off function $\chi\in C_0^{\infty}({\mathbb{R}})$ satisfying $\chi(x)=1$ for $|x|\le 1$ and $\chi(x)=0$ for $|x|\ge 2$. For each $k\in 2^{\mathbb{Z}}$, we define [$$\begin{split}
\chi^k(x) := \chi(|x|/k) - \chi(2|x|/k),
\end{split}$$]{} so that we have $\chi^k\in C_0^{\infty}({\mathbb{R}}^d)$, [$$\begin{split}
{\operatorname{supp}}\chi^k \subset\{k/2<|x|<2k\}, {\quad}\sum_{k\in 2^{\mathbb{Z}}} \chi^k(x) = 1{\quad}(x\not=0).
\end{split}$$]{} Hence the Littlewood-Paley decomposition is given by [$$\begin{split} \label{LP decop}
f = \sum_{k\in 2^{\mathbb{Z}}} \chi^k({\nabla}) f.
\end{split}$$]{} We denote the decomposition into the lower and higher frequency by [$$\begin{split} \label{freq split}
f_{<k} := \sum_{j<k} \chi^j({\nabla}) f,
{\quad}f_{\ge k} := \sum_{j\ge k} \chi^j({\nabla}) f.
\end{split}$$]{}
For any function $B({\xi}_1,\dots,{\xi}_N)$ on $({\mathbb{R}}^d)^N$, we associate the $N$-multilinear operator $B[f_1,\dots f_N]$ defined by [$$\begin{split}
{&}{\mathcal{F}}_x^{\xi}B[f_1,\dots f_N]
{}= \int_{{\xi}={\xi}_1+\cdots+{\xi}_N} B({\xi}_1\dots{\xi}_N){\widetilde}f_1({\xi}_1)\cdots{\widetilde}{f_N}({\xi}_N) d{\xi}_2\cdots d{\xi}_N,
\end{split}$$]{} which is called a multilinear Fourier multiplier with symbol $B$, thus we identify the symbol and the operator. Whenever we write a symbol in the variables $({\xi},{\xi}_1,\dots{\xi}_N)$, it should be understood as a funciton of $({\xi}_1,\dots{\xi}_N)$ by substitution ${\xi}={\xi}_1+\dots+{\xi}_N$. Hence for example, a product of single multipliers can be written as [$$\begin{split}
B_0({\xi})B_1({\xi}_1)B_2({\xi}_2)[f,g] = B_0(-i{\nabla})\left\{(B_1(-i{\nabla})f)\cdot(B_2(-i{\nabla})g)\right\}.
\end{split}$$]{}
In addition, we introduce the following convention for bilinear multipliers. For any bilinear symbol $B({\xi}_1,{\xi}_2)$, we assign the variables ${\eta}$ and ${\zeta}$ such that ${\eta}={\xi}_1$ and ${\zeta}={\xi}_2$, but regarding $({\xi},{\eta})$ and $({\xi},{\zeta})$ respectively as the independent variables. Hence the partial derivatives of the symbol $B$ in each coordinates are given by [$$\begin{split}
{&}({{\nabla}_{\xi}^{({\eta})}}B,{\nabla}_{\eta}B) = ({\nabla}_{{\xi}_2} B({\eta},{\xi}-{\eta}), ({\nabla}_{{\xi}_1}-{\nabla}_{{\xi}_2})B({\eta},{\xi}-{\eta})),
{\\ &}({{\nabla}_{\xi}^{({\zeta})}}B,{\nabla}_{\zeta}B) = ({\nabla}_{{\xi}_1} B({\xi}-{\zeta},{\zeta}), ({\nabla}_{{\xi}_2}-{\nabla}_{{\xi}_1})B({\xi}-{\zeta},{\zeta})).
\end{split}$$]{}
For any number or vector $a$, we denote [$$\begin{split} \label{def UHetc}
{&}{{{{\langle}a {\rangle}}}}:=\sqrt{2+|a|^2},{\quad}{\widehat}{a}:=\frac{a}{|a|},
{\quad}U(a):=\frac{|a|}{{{{{\langle}a {\rangle}}}}},{\quad}H(a):=|a|{{{{\langle}a {\rangle}}}},
\end{split}$$]{} which will mostly appear in the Fourier spaces, in particular $U= \sqrt{-{\Delta}/(2-{\Delta})}$ and $H= \sqrt{-{\Delta}(2-{\Delta})}$. For any complex-valued function $f$, we often denote the complex conjugate by [$$\begin{split}
f^+ := f, {\quad}f^-:={\overline}{f},
\end{split}$$]{} to treat them in a symmetric way. Also we denote [$$\begin{split} \label{def J}
J f = e^{-itH}xe^{itH}f, {\quad}{\check}f = {\mathcal{F}}e^{itH} f,{\quad}{{{\check}{f}}^\pm} = {\mathcal{F}}e^{\pm itH} f^\pm.
\end{split}$$]{} Remark that these operations are time dependent.
Finally, we collect a few basic estimates, which will be used throughout the paper.
Let $2\le p\le{\infty}$, $0\le{\theta}\le 1$, $s\in{\mathbb{R}}$, and ${\sigma}=1/2-1/p$. Then we have [$$\begin{split} \label{Bp dec H}
\|e^{-itH}{\varphi}\|_{\dot B^s_{p,2}} {{\ \lesssim \ }}|t|^{-(d-{\theta}){\sigma}}\|U^{(d-2+3{\theta}){\sigma}}{{{\langle}{\nabla}{\rangle}}}^{2{\theta}{\sigma}}{\varphi}\|_{\dot B^s_{p',2}},
\end{split}$$]{} where $p'=p/(p-1)$ is the Hölder conjugate. For $2\le p<{\infty}$, we have also [$$\begin{split} \label{Lp dec H}
\|e^{-itH}{\varphi}\|_{L^{p,2}} {{\ \lesssim \ }}|t|^{-(d-{\theta}){\sigma}}\|U^{(d-2+3{\theta}){\sigma}}{{{\langle}{\nabla}{\rangle}}}^{2{\theta}{\sigma}}{\varphi}\|_{L^{p',2}}.
\end{split}$$]{}
The case ${\theta}=0$ is similar to the Schrödinger equation, but we gain $U^{(d-2){\sigma}}$ around ${\xi}=0$. The case ${\theta}=1$ with $|{\xi}|<1$ is the same as the wave equation, where we have worse decay for $t\to{\infty}$, but better decay for ${\xi}\to 0$, than the case ${\theta}=0$. This reflects the fact that the group velocity for $e^{-itH}$ is bounded from below $|{\nabla}H({\xi})|>\sqrt{2}$. For $|{\xi}|{{\ \gtrsim \ }}1$, the estimate with ${\theta}>0$ follows from interpolation between ${\theta}=0$ and the Sobolev embedding, hence it is just the same as for Schrödinger.
The estimate in the Besov space is proved by the standard stationary phase argument after the Littlewood-Paley decomposition, and its interpolation with $L^2$ conservation. See [@vac] for the details in the case ${\theta}=0$. Then we can replace Besov with $L^p$ spaces by the embedding [$$\begin{split}
\dot B^0_{p,2} \subset L^p, {\quad}\dot B^0_{p',2} \supset L^{p'},
\end{split}$$]{} for $2\le p<{\infty}$. The Lorentz version follows from real interpolation.
From the above decay estimate, we get the Strichartz estimate (cf. [@vac]), [$$\begin{split} \label{Strz}
\|e^{-itH}f\|_{L^{\infty}H^1 \cap U^{1/6}L^2 B^1_{6,2}} {{\ \lesssim \ }}\|f\|_{U^{-1/6}L^2 B^1_{6/5,2} + L^1 H^1},
\end{split}$$]{} where we gain $U^{1/6}$ for the endpoint norm.
For bilinear Fourier multipliers, we will use the Coifman-Meyer bilinear multiplier estimate [@CM] (see also [@GK]): if [$$\begin{split} \label{CMcond}
|{\partial}_{{\xi}_1}^{\alpha}{\partial}_{{\xi}_2}^{\beta}B({\xi}_1,{\xi}_2)| {{\ \lesssim \ }}(|{\xi}_1|+|{\xi}_2|)^{-|{\alpha}|-|{\beta}|},
\end{split}$$]{} for up to sufficiently large ${\alpha},{\beta}$, then for any $p_j\in(1,{\infty})$ satisfying $1/p_0=1/p_1+1/p_2$, we have [$$\begin{split} \label{bilCM}
\|B[f,g]\|_{L^{p_0}({\mathbb{R}}^d)} {{\ \lesssim \ }}\|f\|_{L^{p_1}({\mathbb{R}}^d)} \|g\|_{L^{p_2}({\mathbb{R}}^d)}.
\end{split}$$]{} As was shown in [@GK], one cannot generally replace the right hand side of by $|{\xi}_1|^{-|{\alpha}|}|{\xi}_2|^{-|{\beta}|}$, but we can reduce our regular multipliers to the above case. However the above estimate is not applicable when the multipliers are really singular due to divisors coming from the phase integrations in the main estimates for Theorem \[thm:init\]. In this case, we will use another bilinear estimate (Lemma \[sbil\]), which allows general multipliers with singularities. As for regular multipliers, the following specific form will be often sufficient:
Let $d\in{\mathbb{N}}$ and $k\in{\mathbb{N}}$. Then we have [$$\begin{split} \label{bil nonsing}
{&}\sup_{0\le a\le 1}\left\|\frac{{{{{\langle}{\xi}_1 {\rangle}}}}^{2k(1-a)}{{{{\langle}{\xi}_2 {\rangle}}}}^{2ka}}{{{{{\langle}({\xi}_1,{\xi}_2) {\rangle}}}}^{2k}}[f,g]\right\|_{L^{p_0}_x({\mathbb{R}}^d)}
{{\ \lesssim \ }}\|f\|_{L^{p_1}_x({\mathbb{R}}^d)} \|g\|_{L^{p_2}_x({\mathbb{R}}^d)},
\end{split}$$]{} for any $p_0,p_1,p_2\in (1,{\infty})$ satisfying $1/p_0=1/p_1+1/p_2$, where $({\xi}_1,{\xi}_2)$ denotes the $2d$ dimensional vector and so [$$\begin{split}
{{{{\langle}({\xi}_1,{\xi}_2) {\rangle}}}} = \sqrt{2+|{\xi}_1|^2+|{\xi}_2|^2}.
\end{split}$$]{}
Denote the above operators by $B_a$, which is analytic on the strip $0\le{\mathop{\mathrm{Re}}}a\le 1$ as bilinear operators $(H^d)^2\to L^2$. When $a=0$ or $a=1$, $B_a$ satisfies , hence the estimate follows from Coifman-Meyer. Since the Fourier multiplier ${{{\langle}t {\rangle}}}^{-1-d}{{{\langle}{\xi}{\rangle}}}^{it}$ is bounded on $L^p$ uniformly for $t\in{\mathbb{R}}$, we get [$$\begin{split}
\sup_{t\in{\mathbb{R}}} {{{\langle}t {\rangle}}}^{1+d}(\|B_{it}[f,g]\|_{L^{p_0}} + \|B_{1+it}[f,g]\|_{L^{p_0}}) {{\ \lesssim \ }}\|f\|_{L^{p_1}} \|g\|_{L^{p_2}},
\end{split}$$]{} which is extended to the whole strip $0\le{\mathop{\mathrm{Re}}}a\le 1$ by the three lines theorem for analytic functions.
Normal forms {#s:normal}
============
The idea of using normal forms for asymptotic behavior of nonlinear dispersive equations goes back to [@Sh] in the case of nonlinear Klein-Gordon equations. For the NLS, quadratic terms have in general resonance sets with one codimension (except for ${\overline}{u}^2$), which give rise to singularities in the normal forms. Here the resonance set is given in the spatial frequency where the time oscillation of the nonlinearity coincides with the linear evolution. We call it temporal resonance.
In [@HMN] for the scattering of NLS in three dimensions, such singularity was avoided by applying the normal form to remove only some of the quadratic terms, leaving the others with better decay properties. Here the decay for the remainders comes from the fact that the oscillation phase of the nonlinearity relative to the linear evolution is not stationary at those spatial frequencies. We call it spatial resonance. When we simply say “non-resonant", it means that it is [*not simultaneously*]{} temporally and spatially resonant.
Our transforms follow the same idea as [@HMN]. However, the actual forms and computations are totally different from the NLS, due to the lower order term $2{\mathop{\mathrm{Re}}}u$ and the special structure of the nonlinearity. In addition, we can not completely avoid singularities because of some degeneracy in the non-resonance property, for which we need another idea (a bilinear estimate for irregular multipliers).
In the previous paper [@vac2], we introduced the normal form $M(u)$ as defined in . As is indicated in Theorem \[thm:map\], this transform is quite natural in view of the nonlinear energy, and will be essential for the proof of Theorem \[thm:fin\]. However, it does not fit the weighted estimate for Theorem \[thm:init\] very well, where it suffers from a derivative loss. To avoid assuming higher regularity, we introduce another normal form which is more suited to the weighted estimate. For that purpose, we start with general bilinear transforms.
First we rewrite the equation for $u=\psi-1=u_1+iu_2$ in terms of the real variables: [$$\begin{split} \label{eq u}
{&}\dot u_1 = -{\Delta}u_2 + 2 u_1u_2 + |u|^2 u_2,
{\\ &}\dot u_2 = -(2-{\Delta})u_1 - 3u_1^2 - u_2^2 - |u|^2u_1.
\end{split}$$]{} We consider a nonlinear transform given in the form[^3] [$$\begin{split}
w := u+B_1'[u_1,u_1]+B_2'[u_2,u_2]
\end{split}$$]{} where $B_j'$ are real-valued symmetric bilinear Fourier multipliers, which we will specify later. From the above equation for $u$, we derive [$$\begin{split}
{&}(i{\partial}_t+{\Delta}-2{\mathop{\mathrm{Re}}})B_1'[u_1,u_1]
{\\ &}= -(2-{\Delta})B_1'[u_1,u_1] + 2iB_1'[u_1,-{\Delta}u_2+2u_1u_2 + |u|^2u_2],
{\\ &}(i{\partial}_t+{\Delta}-2{\mathop{\mathrm{Re}}})B_2'[u_2,u_2]
{\\ &}= -(2-{\Delta})B_2'[u_2,u_2] + 2iB_2'[u_2,({\Delta}-2)u_1-3u_1^2 -u_2^2 -|u|^2u_1],
\end{split}$$]{} and hence we get the equation for $w$: [$$\begin{split}
{}(i{\partial}_t + {\Delta}- 2{\mathop{\mathrm{Re}}})w
{&}= B'_3[u_1,u_1] + B'_4[u_2,u_2]
{}+ iB'_5[u_1,u_2]
{\\ &}+ |u|^2u_1
{}+ iC'_3[u_1,u_1,u_2] + iC'_4[u_2,u_2,u_2]
{}+ iQ_1(u),
\end{split}$$]{} where $B_j'$ ($j=3,4,5$) are bilinear multipliers defined by, [$$\begin{split}
{&}B'_3 = 3-{{{{\langle}{\xi}{\rangle}}}}^2B_1',
{\quad}B'_4 = 1-{{{{\langle}{\xi}{\rangle}}}}^2B_2',
{\quad}B'_5 = 2+2|{\xi}_2|^2B_1'-2{{{{\langle}{\xi}_1 {\rangle}}}}^2B_2',
\end{split}$$]{} $C_j'$ ($j=3,4$) are cubic multipliers defined by [$$\begin{split} \label{def C'}
{&}C'_3({\xi}_1,{\xi}_2,{\xi}_3) = 1 + 4B_1'({\xi}_1,{\xi}_2+{\xi}_3) - 6B_2'({\xi}_1+{\xi}_2,{\xi}_3),
{\\ &}C'_4({\xi}_1,{\xi}_2,{\xi}_3) = 1 - 2 B_2'({\xi}_1+{\xi}_2,{\xi}_3),
\end{split}$$]{} and $Q_1(u)= 2 B_1'[u_1,|u|^2u_2]- 2B_2'[u_2,|u|^2u_1]$ gathers the quartic terms.
Now we choose our $B_1'$ and $B_2'$. Recall the diagonalized equation for $v=u_1 + i U u_2$: [$$\begin{split} \label{eq v}
iv_t - Hv = {\mathcal{N}_v}(u) := U(3u_1^2+u_2^2+|u|^2u_1) + i(2u_1u_2+|u|^2u_2),
\end{split}$$]{} where we observe that we have $U^{-1}$ in each $u_2$ compared with $v$, and also on the imaginary part relative to the real part. To counteract this effect, we want [$$\begin{split}
{&}B'_4 = O(|{\xi}_1||{\xi}_2|), {\quad}B'_5 = O(|{\xi}||{\xi}_2|), {\quad}C_4'=O(|{\xi}_1|+|{\xi}_2|+|{\xi}_3|).
\end{split}$$]{} It is easy to see that $B_4'=B_5'=0$ is impossible due to the symmetry of $B_1'$ and $B_2'$. The simplest choice for $B_4'=0$ is given by [$$\begin{split} \label{old def B12}
{&}B_1'=B_2'={{{{\langle}{\xi}{\rangle}}}}^{-2},
{\quad}B_4' = 0, {\quad}B_5' = 4{{{{\langle}{\xi}{\rangle}}}}^{-2}{\xi}{\xi}_2,
{\quad}C_3'=C_4'=U^2,
\end{split}$$]{} where ${\xi}{\xi}_2={\xi}\cdot{\xi}_2$ denotes the inner product in ${\mathbb{R}}^3$. This leads to $M(u)$ in , and is the most natural from the energy viewpoint . Moreover, the cubic part has a subtle but remarkable non-resonance property, which will be revealed in Section \[s:fin\] for the proof of Theorem \[thm:fin\].
On the other hand, the simplest choice for $B_5'=0$ is given by [$$\begin{split} \label{def B12}
{&}-B_1' = B_2' = {{{{\langle}({\xi}_1,{\xi}_2) {\rangle}}}}^{-2} = \frac{1}{2+|{\xi}_1|^2+|{\xi}_2|^2},
{\\ &}B'_4 = \frac{-2{\xi}_1{\xi}_2}{2+|{\xi}_1|^2+|{\xi}_2|^2},
{\quad}B'_5 = 0,
{\quad}C'_4 = \frac{|{\xi}_1+{\xi}_2|^2+|{\xi}_3|^2}{2+|{\xi}_1+{\xi}_2|^2+|{\xi}_3|^2},
\end{split}$$]{} where we omit $C_3'$ since it is just complicated without any important structure. This choice is not directly linked to the energy, and the cancellation in the cubic part is weaker. Its advantage is that we do not lose a derivative for ${{{\langle}{\xi}{\rangle}}}\ll|{\xi}_1|\sim|{\xi}_2|$ as in $B_5'$ of . Such a derivative loss for high-high interactions can be saved by increasing the regularity. However, in the weighted estimate for bilinear terms, we will encounter another derivative loss by integration on the phase, and therefore we would need $H^2_x$ to close the estimates with the choice .
The latter derivative loss can be observed in the Schrödinger case as well: look at the weighted estimate for the typical bilinear term $|e^{-it{\Delta}}{\varphi}|^2$ in the Fourier space [$$\begin{split}
{\mathcal{F}}_x^{\xi}xe^{it{\Delta}}|e^{-it{\Delta}}{\varphi}|^2 = i{\nabla}_{\xi}\int e^{it{\Omega}} {\overline}{{\mathcal{F}}{\varphi}}(-{\eta}){\mathcal{F}}{\varphi}({\xi}-{\eta}) d{\eta},
\end{split}$$]{} where ${\Omega}:= |{\xi}|^2 - |{\eta}|^2 + |{\xi}-{\eta}|^2$. Partial integration on $e^{it{\Omega}}$ gives a factor of the form [$$\begin{split}
\frac{|{{\nabla}_{\xi}^{({\eta})}}{\Omega}|}{|{\nabla}_{\eta}{\Omega}|} = \frac{|2{\xi}-{\eta}|}{|{\xi}|},
\end{split}$$]{} which causes a derivative loss in the region $|{\xi}|\ll|{\eta}|\sim|{\eta}-{\xi}|$. However, those two normal forms eventually lead to the same asymptotic behavior, in the case of weighted energy scattering. See Section \[ss:other normal\].
We remark that there are other possible choices. For example, if we want decay in $B_3'$ as well, we may choose [$$\begin{split}
{&}B_1'={{{{\langle}{\xi}{\rangle}}}}^{-2}(3-4{{{{\langle}{\xi}{\rangle}}}}^{-2}{\xi}_1{\xi}_2),
{\quad}B_2' = {{{{\langle}{\xi}{\rangle}}}}^{-2}(1-4{{{{\langle}{\xi}{\rangle}}}}^{-2}{\xi}_1{\xi}_2),
{\\ &}B_3'=B_4'=4{{{{\langle}{\xi}{\rangle}}}}^{-2}{\xi}_1{\xi}_2,
{\\ &}B_5'=4{{{{\langle}{\xi}{\rangle}}}}^{-4}(4{\xi}_0{\xi}_2+4|{\xi}_0|^2|{\xi}_2|^2+2({\xi}_1{\xi}_2)[{\xi}_0({\xi}_0+4{\xi}_1-4{\xi}_2)],
\end{split}$$]{} but it suffers from derivative loss in all $B_j'$, so we will not use it. In the rest of the paper, those notations $B_j'$, $C_j'$ and $Q_1$ are reserved for the second choice .
Now we fix the notation for our transforms corresponding to and respectively by[^4] [$$\begin{split} \label{w to z}
{&}{z}:= M(u) = v + {{{{\langle}{\xi}{\rangle}}}}^{-2}|u|^2,
{\\ &}{Z}:= v + b(u):= v-{{{{\langle}({\xi}_1,{\xi}_2) {\rangle}}}}^{-2}[u_1,u_1] + {{{{\langle}({\xi}_1,{\xi}_2) {\rangle}}}}^{-2}[u_2,u_2].
\end{split}$$]{} Then we get coupled equations for $(v,{z})$ and $(v,{Z})$ in the form [$$\begin{split} \label{eq uzo}
{&}v = {z}- {{{{\langle}{\xi}{\rangle}}}}^{-2}|u|^2, {\quad}i\dot {z}- H{z}= {\mathcal{N}_{z}}(u),
\end{split}$$]{} [$$\begin{split} \label{eq uzn}
{&}v = {Z}- b(u), {\quad}i\dot {Z}- H{Z}= {\mathcal{N}_{Z}}(v).
\end{split}$$]{} The nonlinear terms are given by [$$\begin{split} \label{def NO}
{\mathcal{N}_{z}}(u) := U\left\{2u_1^2+|u|^2u_1\right\}
{}- i{{{{\langle}{\nabla}{\rangle}}}}^{-2}{\nabla}\cdot\left\{4u_1{\nabla}u_2+{\nabla}(|u|^2u_2)\right\},
\end{split}$$]{} [$$\begin{split} \label{def NN}
{\mathcal{N}_{Z}}(v) {&}:= B_3[v_1,v_1] + B_4[v_2,v_2]
{}+ C_1[v_1,v_1,v_1] +C_2[v_2,v_2,v_1]
{\\ &\quad}+ i C_3[v_1,v_1,v_2] +i C_4[v_2,v_2,v_2]
{}+ iQ_1(u),
\end{split}$$]{} with the bilinear multipliers $B_j$ defined by using $B_j'$ in the case , [$$\begin{split}
{&}B_3 = U({\xi})B'_3 = \frac{-2U({\xi})(4+4|{\xi}_1|^2+4|{\xi}_2|^2-{\xi}_1{\xi}_2)}{2+|{\xi}_1|^2+|{\xi}_2|^2},
{\\ &}B_4 = U({\xi})U({\xi}_1)^{-1}U({\xi}_2)^{-1}B_4' = \frac{-2U({\xi}){{{{\langle}{\xi}_1 {\rangle}}}}{{{{\langle}{\xi}_2 {\rangle}}}}{\widehat}{{\xi}_1}{\widehat}{{\xi}_2}}{2+|{\xi}_1|^2+|{\xi}_2|^2},
\end{split}$$]{} and the cubic multipliers $C_j$ defined by using $C_j'$ in , [$$\begin{split}
{&}C_1 = U({\xi}),
{\quad}C_2 = U({\xi})U({\xi}_1)^{-1}U({\xi}_2)^{-1},
{\\ &}C_3 = U({\xi}_3)^{-1}C'_3,
{\quad}C_4 = U({\xi}_1)^{-1}U({\xi}_2)^{-1}U({\xi}_3)^{-1}C'_4,
\end{split}$$]{} and the quartic term $Q_1$ is given by [$$\begin{split}
Q_1(u) = -2{{{{\langle}({\xi}_1,{\xi}_2) {\rangle}}}}^{-2}[u_1,|u|^2u_2] - 2{{{{\langle}({\xi}_1,{\xi}_2) {\rangle}}}}^{-2}[u_2,|u|^2u_1].
\end{split}$$]{} We have the following bounds: [$$\begin{split} \label{bound on B}
{&}|B_3|+|B_4| {{\ \lesssim \ }}U({\xi}),
{\\ &}|C_j| {{\ \lesssim \ }}U({\xi}_1)^{-1}U({\xi}_2)^{-1} + U({\xi}_2)^{-1}U({\xi}_3)^{-1} +
U({\xi}_3)^{-1}U({\xi}_1)^{-1}.
\end{split}$$]{} More precisely, they are written as linear combinations of products of those terms on the r.h.s. of and regular multipliers as in , together with the Riesz operator.
\[transBsq\] By the transform $u\mapsto v:=u-iH^{-1}\dot u$, the Boussinesq equation can be rewritten as [$$\begin{split}
iv_t + H v = U(v_1^2),
\end{split}$$]{} hence it can be regarded as a simplified version of our equation for $(v,{z})$. In fact, our proof for Theorem \[thm:init\] directly applies to this equation, while Theorem \[thm:fin\] and Theorem \[thm:map\] rely much more on the nonlinear energy structure, which is different for . For the (physically relevant) two dimensional case, refer to [@vac2] for a final data result, which is also applicable to this equation.
Final data problem {#s:fin}
==================
In this section, we consider the final state problem in the energy space, proving Theorems \[thm:fin\] and \[thm:map\]. Our strategy for Theorem \[thm:fin\] is essentially the same as [@asshort] for NLS, using compactness to get weak convergence, and energy conservation to make it strong. The normal form transform $M(u)$ is essential both for the uniform bound and for the equicontinuity of approximating sequence of solutions. In addition, we will use a hidden non-resonance property in the cubic terms of to get the equicontinuity. It is noteworthy that the argument in [@asshort] does not work for NLS without $L^2$ finiteness, whereas for GP we are using only the energy conservation. We start with the proof of Theorem \[thm:map\], since the result will be used in the proof of Theorem \[thm:fin\].
The identity is straightforward. We construct $R$ by solving the following equation for $g$ with any given $f\in H^1(L,{\kappa})$: [$$\begin{split}
g_1=R_f(g_1) := f_1-{{{{\langle}{\nabla}{\rangle}}}}^{-2}\left\{g_1^2+(U^{-1}f_2)^2\right\}, {\quad}g_2=U^{-1}f_2.
\end{split}$$]{} We use the following product estimate for general $u,v$: [$$\begin{split} \label{prod in Rf}
\|{{{{\langle}{\nabla}{\rangle}}}}^{-2}uv\|_{\dot H^1} {&}{{\ \lesssim \ }}\|U(uv)\|_{L^2} \sim \|{\nabla}(uv)\|_{H^{-1}}
{\\ &}{{\ \lesssim \ }}\|v\|_{\dot H^1}\|{\nabla}u\|_{H^{-1,3}} + \|u\|_{L^{\infty}+L^6}\|{\nabla}v\|_{L^2}
{\\ &}{{\ \lesssim \ }}\|v\|_{\dot H^1}\left\{\|Uu\|_{L^3} + \|u\|_{L^6}^{1/2}\|Uu\|_{L^6}^{1/2} \right\}
{\\ &}{{\ \lesssim \ }}\|v\|_{\dot H^1}\|u\|_{\dot H^1}^{1/2}\|Uu\|_{L^6}^{1/2}.
\end{split}$$]{} Applying it to $R_f(g_1)$, we get [$$\begin{split}
{&}\|{{{{\langle}{\nabla}{\rangle}}}}^{-2}g_2^2\|_{\dot H^1}
{{\ \lesssim \ }}\|Ug_2^2\|_{L^2} {{\ \lesssim \ }}\|f\|_{L^6}^{1/2}\|f\|_{H^1}^{3/2},
{\\ &}\|R_f(u)-R_f(v)\|_{\dot H^1}
{{\ \lesssim \ }}\|u+v\|_{\dot H^1}^{1/2}\|u+v\|_{L^6}^{1/2}\|u-v\|_{\dot H^1},
\end{split}$$]{} and in addition, [$$\begin{split}
{&}\|{{{{\langle}{\nabla}{\rangle}}}}^{-2}g_1^2\|_{\dot H^1} {{\ \lesssim \ }}\|Ug_1^2\|_{L^2} {{\ \lesssim \ }}\|g_1{\nabla}g_1\|_{L^{3/2}}
{{\ \lesssim \ }}\|g_1\|_{L^6}\|g_1\|_{\dot H^1},
{\\ &}\|{{{{\langle}{\nabla}{\rangle}}}}^{-2}g_1^2\|_{L^6} {{\ \lesssim \ }}\|g_1^2\|_{L^3} = \|g_1\|_{L^6}^2.
\end{split}$$]{} Hence the map $g\mapsto R_f(g_1)+ig_2$ is a contraction on the set [$$\begin{split}
\{g\in F_1(2L,C({\kappa}+L^{3/2}{\kappa}^{1/2})) \mid g_2=U^{-1}f_2\},
\end{split}$$]{} for the $\dot H^1$ norm and some $C>0$, if ${\kappa}>0$ is sufficiently small. Hence it has a unique fixed point $g_1=R_f(g_1)$, by which we can define $R(f)=g_1+iU^{-1}f_2$. Then we have $M\circ R=id$ and the uniqueness of $R$ follows from that for the fixed point. For the continuity of $R$, let $f,g\in H^1(L,{\kappa})$ and $u=R(f)$, $v=R(g)$. Then we have by , [$$\begin{split}
\|U(|u|^2-|v|^2)\|_{L^2}
{&}{{\ \lesssim \ }}\|u+v\|_{\dot H^1}^{1/2}\|u+v\|_{L^6}^{1/2}\|u-v\|_{\dot H^1}
\ll \|u-v\|_{\dot H^1},
\end{split}$$]{} if ${\kappa}$ is small. Moreover, [$$\begin{split}
\|u-v\|_{\dot H^1}^2 = \|u_1-v_1\|_{\dot H^1}^2 + \|f_2-g_2\|_{H^1}^2.
\end{split}$$]{} For the real component, we have [$$\begin{split}
{&}\|u_1-v_1\|_{\dot H^1}
=\|R_f(u_1)-R_g(v_1)\|_{\dot H^1}
{\\ &}\le \|f_1-g_1\|_{\dot H^1} + \|{{{{\langle}{\nabla}{\rangle}}}}^{-2}(u_1^2-v_1^2)\|_{\dot H^1}
+ \|{{{{\langle}{\nabla}{\rangle}}}}^{-2}(U^{-1}f_2)^2-(U^{-1}g_2)^2\|_{\dot H^1}.
\end{split}$$]{} where the second term is bounded by [$$\begin{split}
\|u_1+v_1\|_{L^6} \|u_1-v_1\|_{\dot H^1} \ll \|u_1-v_1\|_{\dot H^1},
\end{split}$$]{} and the third term is bounded by using , [$$\begin{split}
\|f_2-g_2\|_{H^1}\|f_2+g_2\|_{H^1}^{1/2}\|f_2+g_2\|_{L^6}^{1/2}
\ll \|f_2-g_2\|_{H^1}.
\end{split}$$]{} Hence for any ${\varepsilon}>0$, we can get by choosing ${\kappa}>0$ small enough, [$$\begin{split}
\|u_1-v_1\|_{\dot H^1}
\le \|f_1-g_1\|_{H^1} + {\varepsilon}\|f_2-g_2\|_{H^1},
\end{split}$$]{} and the desired Lipschitz condition follows.
First we rewrite the equation for ${z}=M(u)$, using the renormalized charge density $q(u):=|\psi|^2-1=2u_1+|u|^2$: [$$\begin{split} \label{eqz}
{&}i{z}_t - H {z}= {\mathcal{N}_{z}}^1(u)+{\mathcal{N}_{z}}^2(u),
{\\ &\quad}{\mathcal{N}_{z}}^1 := \frac{U}{2}(q^2 - |u|^4 - 2u_1^3)
{}- \frac{i{\nabla}}{{{{{\langle}{\nabla}{\rangle}}}}^2}\cdot\left\{2q{\nabla}u_2+u_1(2u_2{\nabla}u_1-u_1{\nabla}u_2)\right\},
{\\ &\quad}{\mathcal{N}_{z}}^2 := -U(u_2^2u_1) - i{{{{\langle}{\nabla}{\rangle}}}}^{-2}{\nabla}\cdot(u_2^2{\nabla}u_2).
\end{split}$$]{} Let $L=2\|{{{{\langle}{\nabla}{\rangle}}}}z_+\|_{L^2}+1$ and $z^0:=e^{-itH}z_+$. Since $\|z^0(t)\|_{L^6_x}\to 0$ as $t\to{\infty}$, for large $T>0$ we have $z^0(T)\in H^1(L,{\kappa})$. Let $\psi^T$ be the global solution of satisfying [$$\begin{split}
\psi^T = 1 + u^T,{\quad}u^T(T) = R(z^0(T)),
\end{split}$$]{} and let ${z}^T=M(u^T)$. Then we have ${z}^T(T)=z^0(T)$ and the above equation with $({z},u)=({z}^T,u^T)$ and $q=q^T=q(u^T)$. Moreover we have [$$\begin{split}
\|{{{{\langle}{\nabla}{\rangle}}}}{z}^T\|_{L^2} \le E_1(\psi^T)^{1/2} = {\delta}(R(z^0(T)),0)\le 2\|{{{{\langle}{\nabla}{\rangle}}}}z_+\|_{L^2},
\end{split}$$]{} so ${z}^T$, $u^T$ and $q^T$ are uniformly bounded in $H^1$, $F_1$ and $L^2$ respectively. By the Sobolev embedding, $u^T$ is bounded in $L^6_x$. Combined with the $L^2$ bound on $q^T$ and the fact that $(L^3+L^6)\cap L^2\subset L^3$, it implies also that $u_1^T$ and $q^T$ are bounded in $L^3_x$.
Moreover, $u^T$ and ${z}^T$ are equicontinuous in $C({\mathbb{R}};{\mathcal{S}}'({\mathbb{R}}^3))$ by their equations. Hence for some sequence $T\to{\infty}$, $u^T$ and ${z}^T$ converge in $C({\mathbb{R}};{{\text{w-}\dot H^1}})$ and $C({\mathbb{R}};{{\text{w-}H^1}})$ respectively. We denote their limits by $u^{\infty}$ and $z^{\infty}$. Then $q^T$ also converges to $q^{\infty}=q(u^{\infty})$ in $C({\mathbb{R}};L^2_{loc})$ and $C({\mathbb{R}};{{\text{w-}L^2}})$. Also it is easy to check that $v^{\infty}:=u_1^{\infty}+iUu_2^{\infty}$ and $z^{\infty}$ satisfy the equation .
To get weak convergence of $e^{itH}{z}^T$ globally in time, we need its equicontinuity at $t={\infty}$, i.e., for any test function ${\varphi}\in{\mathcal{S}}({\mathbb{R}}^3)$ we have [$$\begin{split}
\int_{t_1}^{t_2} {\big\langle {\varphi}\big| e^{itH}{\mathcal{N}_{z}}(u^T(t)) \big\rangle_x} dt \to 0
\end{split}$$]{} as $t_2>t_1\to{\infty}$ uniformly in $T$. Those terms in ${\mathcal{N}_{z}}^1(u)$ are bounded respectively by (here we omit $T$), [$$\begin{split}
{&}\|q^2\|_{L^{12/11}} {{\ \lesssim \ }}\|q\|_{L^2\cap L^3}^2,
{\\ &}\|U|u|^4\|_{L^{12/11}} {{\ \lesssim \ }}\|u^3{\nabla}u\|_{L^1} {{\ \lesssim \ }}\|u\|_{L^6}^3\|{\nabla}u\|_{L^2},
{\\ &}\|U(u_1^3)\|_{L^{12/11}} {{\ \lesssim \ }}\|u_1^2{\nabla}u_1\|_{L^1} {{\ \lesssim \ }}\|u_1\|_{L^3\cap L^6}^2\|{\nabla}u\|_{L^2}.
{\\ &}\|q{\nabla}u_2\|_{L^{12/11}} {{\ \lesssim \ }}\|q\|_{L^2\cap L^3}\|{\nabla}u\|_{L^2},
{\\ &}\|u_1u{\nabla}u\|_{L^{12/11}} {{\ \lesssim \ }}\|u_1\|_{L^3\cap L^6}\|u\|_{L^6}\|{\nabla}u\|_{L^2},
\end{split}$$]{} hence by using the $L^{12}$ decay of $e^{itH}$ , [$$\begin{split}
\int_{t_1}^{t_2} |{\big\langle {\varphi}\big| e^{itH}{\mathcal{N}_{z}}^1(u^T(t)) \big\rangle_x}| dt {{\ \lesssim \ }}\int_{t_1}^{t_2} t^{-5/4} dt {{\ \lesssim \ }}t_1^{-1/4} \to 0.
\end{split}$$]{} The remaining term ${\mathcal{N}_{z}}^2$ is bounded only in $L^{6/5}_x$, which gives by the $L^p$ decay the critical decay rate $1/t$, and so we cannot get compactness in this way. More precisely, when we decompose them into dyadic frequencies by [$$\begin{split}
{\mathcal{N}_{z}}^2 = -\sum_{j,k\in 2^{\mathbb{Z}}} U\left\{\chi({\nabla})^j(u_2^2) \chi({\nabla})^k u_1\right\} + i\frac{{\nabla}}{{{{{\langle}{\nabla}{\rangle}}}}^2}\cdot\left\{\chi({\nabla})^j(u_2^2)\chi({\nabla})^k{\nabla}u_2\right\},
\end{split}$$]{} we have trouble only when $j\ll\min(1,k)$, since otherwise $U$ or ${\nabla}$ outside gives the factor $j$ and so those terms are bounded in $L^{12/11}_x$ as above. In order to bound the remaining part, we use a non-resonance property due to its special structure: [$$\begin{split} \label{decop NO1}
{\mathcal{N}_{z}}^2 = -U(u_2^2{\overline}{v}) - i{{{{\langle}{\nabla}{\rangle}}}}^{-2}\left\{H({\xi})U({\xi}_2)-{\xi}{\xi}_2\right\}[u_2^2,u_2],
\end{split}$$]{} where the second bilinear multiplier has the small factor ${\xi}_1$, since it equals [$$\begin{split}
{&}{{{{\langle}{\xi}{\rangle}}}}^{-2}\left\{|{\xi}|({{{{\langle}{\xi}{\rangle}}}}U({\xi}_2)-|{\xi}|)+|{\xi}|^2-{\xi}{\xi}_2\right\}
{\\ &}= \left\{ \frac{|{\xi}||{\xi}_2|}{({{{{\langle}{\xi}{\rangle}}}}+{{{{\langle}{\xi}_2 {\rangle}}}}){{{{\langle}{\xi}_2 {\rangle}}}}}
- \frac{|{\xi}|}{|{\xi}|+|{\xi}_2|} \right\}\frac{({\xi}_1+2{\xi}_2)\cdot{\xi}_1}{{{{{\langle}{\xi}{\rangle}}}}^2} + \frac{{\xi}{\xi}_1}{{{{{\langle}{\xi}{\rangle}}}}^2},
\end{split}$$]{} and the multiplier inside the big braces is bounded by the Coifman-Meyer estimate if restricted to $|{\xi}_1|\ll|{\xi}_2|\sim|{\xi}|$. So this term is bounded in $L^{12/11}_x$.
For the first term in , we integrate on $e^{it(H({\xi})+H({\xi}_2))}$ [$$\begin{split} \label{IBP for fin}
{&}\int_{t_1}^{t_2} e^{it(H({\xi})+H({\xi}_2))}U[u_2^2,{\overline}{e^{itH}v}] dt
{\\ &}= [-ie^{itH}{\mathcal{B}}[u_2^2,{\overline}{v}]]_{t_1}^{t_2}
+ \int_{t_1}^{t_2} ie^{itH}{\mathcal{B}}[2u_2\dot u_2,{\overline}{v}] - e^{itH}{\mathcal{B}}[u_2^2,{\overline}{{\mathcal{N}_v}(u)}] dt,
\end{split}$$]{} with the bilinear multiplier ${\mathcal{B}}$ defined by [$$\begin{split}
{\mathcal{B}}= (H({\xi})+H({\xi}_2))^{-1}U({\xi}) = {{{{\langle}{\xi}{\rangle}}}}^{-2}\left\{1+H({\xi}_2)H({\xi})^{-1}\right\}^{-1},
\end{split}$$]{} where the multiplier in the braces is bounded by Coifman-Meyer if $|{\xi}_1|\ll|{\xi}_2|\sim|{\xi}|$. From the equation , we have [$$\begin{split}
{&}\dot u_2 = -q + {\Delta}u_1 - q u_1 \in L^{\infty}_t H^{-1}_x,
{\\ &}{\mathcal{N}_v}(u) = U(|u|^2+q u_1) + iqu_2 \in L^{\infty}_t(L^{3/2}\cap L^2)_x,
\end{split}$$]{} so $u_2\dot u_2\in \dot H^1\times H^{-1}$ is bounded in $H^{-1,3/2}\cap H^{-2}$. Hence restricted to $|{\xi}_1|\ll\min(1,|{\xi}_2|)$ is bounded in $t_1^{-1/2}L^3_x + t_1^{-1/4} L^{12}_x$.
Thus we conclude that ${{\acute}{z}}^T:=e^{itH}{z}^T$ is equicontinuous in $C([-{\infty},{\infty}];{\mathcal{S}}'({\mathbb{R}}^3))$, and so by choosing appropriate sequence $T\to{\infty}$, we get [$$\begin{split}
{{\acute}{z}}^T \to {{\acute}{z}}^{\infty}{\text{ in }C([-{\infty},{\infty}];{{\text{w-}H^1}})},
\end{split}$$]{} together with ${{\acute}{z}}^{\infty}({\infty})=z_+$ from the initial condition ${z}^T(T)=e^{-iTH}z_+$. By the lower semi-continuity for $T\to{\infty}$ (along the sequence), we have [$$\begin{split}
E_1(\psi^{\infty}) {&}= \|{{{{\langle}{\nabla}{\rangle}}}}{z}^{\infty}\|_{L^2}^2 + \frac{1}{2}\|U|u^{\infty}|^2\|_{L^2}^2
{}\le \liminf_{T\to{\infty}} E_1(\psi^T(T))
= \|{{{{\langle}{\nabla}{\rangle}}}}z_+\|_{L^2}^2,
\end{split}$$]{} and for $t\to{\infty}$, [$$\begin{split}
\|{{{{\langle}{\nabla}{\rangle}}}}z_+\|_{L^2}^2 \le \liminf_{t\to{\infty}} E_1(\psi^{\infty}(t)) = E_1(\psi^{\infty}).
\end{split}$$]{} Hence both inequalities must be equality, which implies that the convergence is strong in both cases. Hence $e^{itH}{z}^{\infty}(t)\to z_+$ strongly in $H^1_x$ as $t\to{\infty}$.
\[L2 unbd\] We can show that for some asymptotic profiles $z_+\in H^1$, the above constructed solution has $v_1$ unbounded in $L^2_x$ for large $t$ ($v_2$ is bounded in $L^2_x$ by the energy). So we really need the nonlinear map $M$ to have the scattering statement in $H^1$. The proof goes by contradiction. If $v_1=u_1$ is uniformly bounded in $L^2$, by the $L^2$ bound of $q$, $|u|^2$ is also bounded. Then it is easy to see that ${\mathcal{N}_{z}}(u)$ is bounded in $UL^p_x$ for $1\le p\le 6/5$. Thus we have by the $L^p$ decay with $1/p:=1/6-{\varepsilon}$ for ${\varepsilon}>0$ small, [$$\begin{split}
\|e^{i(t-T)H}{z}(t) - {z}(T)\|_{U^{1+1/3+{\varepsilon}}L^p_x}
{&}= \|\int_T^t e^{isH}{\mathcal{N}_{z}}(u(s))ds\|_{U^{4/3+{\varepsilon}}L^p_x}
{\\ &}{{\ \lesssim \ }}\int_T^t s^{-1-3{\varepsilon}}\|{\mathcal{N}_{z}}(u(s))\|_{UL^{5/6+{\varepsilon}}_x} ds {{\ \lesssim \ }}T^{-3{\varepsilon}},
\end{split}$$]{} and so by letting $t\to{\infty}$, we get [$$\begin{split}
e^{-iTH}z_+-{z}(T) \in U^{4/3+{\varepsilon}}L^p_x.
\end{split}$$]{} Since $U^{-1}z_2(T)=u_2(T)\in L^{\infty}_TL^4_x\subset L^{\infty}_T\dot H^{-1/2-{\varepsilon}/3,p}$ for $T$ large, we have [$$\begin{split}
({\mathop{\mathrm{Im}}}e^{-iTH}z_+)_{<1} = (Uu_2(T) + U^{4/3+{\varepsilon}}L^{\infty}_TL^p_x)_{<1}
\subset U^{4/3}L^{\infty}_TL^p_x,
\end{split}$$]{} which is clearly not implied by the assumption $z_+\in H^1$. For example, let [$$\begin{split}
z_+ = \sum_{k\in 4^{-{\mathbb{N}}}}ie^{iH/k}k^{3/2+3{\varepsilon}}f(k x),
\end{split}$$]{} for some $f\in{\mathcal{S}}$ satisfying ${\operatorname{supp}}{\widetilde}f\subset\{1<|{\xi}|<2\}$. Then $z_+\in H^1$ but [$$\begin{split}
\|U^{-4/3}e^{-iH/k}z_+\|_{\dot B^0_{p,{\infty}}} {{\ \gtrsim \ }}k^{-4/3+3/2+3{\varepsilon}-3/p}\|f\|_{L^p_x} \to {\infty}{\quad}(1/k\to{\infty}).
\end{split}$$]{}
However it is not clear whether $\|v_1(t)\|_{L^2_x}$ is growing, or infinite for all time ($v(t)\in L^2_x$ persists in time by the equation for $v$). We expect that both cases do happen, depending on the asymptotic data.
Initial data problem
====================
The strategy for Theorem \[thm:init\] largely follows that in [@HMN] for the NLS: we will derive an a priori bound on the weighted energy norm of the solution after removing the free propagator, namely $e^{itH}v(t)$. Then the a priori bound gives space and time decay properties of the solution by getting back the propagator, and this decay property is used in turn to bound the Duhamel integral for the a priori estimate. However, the linear decay property is not sufficient to bound the quadratic terms, so that we have to integrate by parts on the phase to gain more integrability and decay from the oscillations. At this point we meet the special difficulty of our equation, that is the singularity due to degeneration of the nonresonance property around ${\xi}\to 0$. To treat it, we will derive a bilinear estimate for singular Fourier multipliers, and investigate the shape of the degeneration in a similar way as in our previous paper [@vac2] for the two dimensional case.
Now we set up our function spaces. We will derive iterative estimates for both $v$ and ${Z}$ of in the following space-time norms. Recall the notation for $J$. [$$\begin{split} \label{main est}
{&}\|{Z}(t)\|_{X(t)} := \|{Z}(t)\|_{H^1_x} + \|J{Z}(t)\|_{H^1_x},
{\quad}\|{Z}\|_X := \sup_t \|{Z}(t)\|_{X(t)},
{\\ &}\|{Z}\|_S := \|{Z}\|_{L^{\infty}_t H^1_x} + \|U^{-1/6}{Z}\|_{L^2_t H^{1,6}_x}.
\end{split}$$]{} The last norm is finite for $e^{-itH}{\varphi}$ if ${\varphi}\in H^1({\mathbb{R}}^3)$, by the Strichartz estimate . The key ingredient is the $H^1$ bound on $J{Z}$, which can be written in the Fourier space [$$\begin{split}
\|J{Z}\|_{H^1_x} \sim \|{{{\langle}{\xi}{\rangle}}} {\nabla}_{\xi}{\check}{Z}\|_{L^2_{\xi}}.
\end{split}$$]{} More precisely, we are going to prove [$$\begin{split}
\left\|\int_T^t e^{-i(t-s)H}{\mathcal{N}_{Z}}(s) ds\right\|_{X(T,{\infty})} {{\ \lesssim \ }}{{{\langle}T {\rangle}}}^{-{\varepsilon}},
\end{split}$$]{} for some small ${\varepsilon}>0$. This estimate implies the desired scattering for ${Z}$ in ${{{\langle}x {\rangle}}}^{-1}H^1$. The scattering for $v$ is the same, because we will prove that the difference $b(u)$ is vanishing faster in the same space $X$.
Decay property by the weighted norm
-----------------------------------
We derive a few decay properties as $t\to{\infty}$ and ${\xi}\to 0$ of the $X$ space . First, the commutator relations $[{\nabla}_j,J_k] = {\delta}_{j,k}$ and $[{{{{\langle}{\nabla}{\rangle}}}},J]=-{{{{\langle}{\nabla}{\rangle}}}}^{-1}{\nabla}$ imply [$$\begin{split}
\|J {\nabla}v(t)\|_{L^2_x} + \|J{{{{\langle}{\nabla}{\rangle}}}}v(t)\|_{L^2_x} {{\ \lesssim \ }}\|Jv(t)\|_{H^1_x} + \|v(t)\|_{L^2_x}.
\end{split}$$]{} Since $1/|x|\in L^{d,{\infty}}({\mathbb{R}}^d)$, we have by the Sobolev and the Hölder, [$$\begin{split}
\|v(t)\|_{\dot H^{-1}_x} {&}\sim \|e^{itH}v(t)\|_{\dot H^{-1}_x}
{\\ &}{{\ \lesssim \ }}\|e^{itH}v(t)\|_{L^{6/5,2}_x} {{\ \lesssim \ }}\|xe^{itH}v(t)\|_{L^2_x}
\sim \|Jv(t)\|_{L^2_x}.
\end{split}$$]{} Thus we obtain [$$\begin{split} \label{v L2 bounds}
\|U^{-2}v\|_{L^6_x}
{&}{{\ \lesssim \ }}\|v\|_{\dot H^{-1}} + \|v\|_{H^1_x} \sim \|U^{-1}v\|_{H^1_x}
{}{{\ \lesssim \ }}\|v(t)\|_{X(t)}.
\end{split}$$]{}
Choosing ${\theta}=0$ and $p=6$ in the $L^p$ decay estimate , we have [$$\begin{split}
\|{{{\langle}{\nabla}{\rangle}}}U^{-1/3}v(t)\|_{L^6_x}
{&}{{\ \lesssim \ }}t^{-1}\|{{{\langle}{\nabla}{\rangle}}}e^{itH}v(t)\|_{L^{6/5,2}_x}
{\\ &}{{\ \lesssim \ }}t^{-1}\|J{{{{\langle}{\nabla}{\rangle}}}}v(t)\|_{L^2_x}
{{\ \lesssim \ }}t^{-1}\|v(t)\|_{X(t)}.
\end{split}$$]{} Combining with the Sobolev bound , we get [$$\begin{split} \label{v decay}
{&}\||{\nabla}|^{-2+5{\theta}/3} v_{<1}(t)\|_{L^6_x} {{\ \lesssim \ }}\min(1,t^{-{\theta}})\|v(t)\|_{X(t)},
{\\ &}\||{\nabla}|^{{\theta}} v_{\ge 1}(t)\|_{L^6_x} {{\ \lesssim \ }}\min(t^{-{\theta}},t^{-1})\|v(t)\|_{X(t)},
\end{split}$$]{} for $0\le{\theta}\le 1$, where $v_{<1}$ and $v_{\ge 1}$ denote the smooth separation of frequency . In particular, we have [$$\begin{split} \label{U-1vL6}
{&}\|U^{-1}v(t)\|_{L^6_x} {{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-3/5}\|v(t)\|_{X(t)},
{\quad}\|U^{-1}v\|_{L^2_t H^{1,6}_x} {{\ \lesssim \ }}\|v\|_{S\cap X}.
\end{split}$$]{} Also we obtain the Strichartz bound on $u=v_1+iU^{-1}v_2$: [$$\begin{split} \label{Stz u}
{&}\|u\|_{L^{\infty}H^1} {{\ \lesssim \ }}\|U^{-1}v\|_{L^{\infty}H^1} {{\ \lesssim \ }}\|v\|_X,
{\\ &}\|u\|_{L^2 H^{1,6}}
{{\ \lesssim \ }}\|v_{\ge 1}\|_{L^2 H^{1,6}} + \|U^{-1}v_{<1}\|_{L^2 L^6} {{\ \lesssim \ }}\|v\|_{X\cap S}.
\end{split}$$]{}
For the estimate on $b(u)$, it is better to use the wave-type decay. Choosing $p=4$ and ${\theta}=2/3$ in , and using complex interpolation, we have [$$\begin{split} \label{U-1vL4}
\|{{{\langle}{\nabla}{\rangle}}}^{2/3}U^{-1}v(t)\|_{L^4_x}
{&}{{\ \lesssim \ }}t^{-7/12}\|U^{-1/4}{{{\langle}{\nabla}{\rangle}}}e^{itH}v(t)\|_{L^{4/3,2}_x}
{\\ &}{{\ \lesssim \ }}t^{-7/12}\|{{{\langle}{\nabla}{\rangle}}}e^{itH}v(t)\|_{L^{6/5,2}_x}^{3/4}\|U^{-1}{{{\langle}{\nabla}{\rangle}}}e^{itH}v(t)\|_{L^2_x}^{1/4}
{\\ &}{{\ \lesssim \ }}t^{-7/12}\|v(t)\|_{X(t)}.
\end{split}$$]{}
Estimates on the normal form
============================
In this section, we derive decay estimates on $b(u)$ (defined in ), and the invertibility of the mapping $v\mapsto {Z}$. First from the $L^p$ decay property of $v$ and the bilinear estimate , we have [$$\begin{split} \label{b bound}
{}\|b(u)\|_{H^{2,p}_x}
{&}{{\ \lesssim \ }}\|U^{-1}v\|_{L^{p_1}_x} \|U^{-1}v\|_{L^{p_2}_x} {{\ \lesssim \ }}\|U^{-1}v\|_{L^2_x}^{2-{\theta}} \|U^{-1}v\|_{L^6_x}^{{\theta}}
{\\ &}{{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-3{\theta}/5}\|v(t)\|_{X(t)}^2,
\end{split}$$]{} for $0<{\theta}\le 2$, where [$$\begin{split}
\frac{1}{p} = 1 - \frac{{\theta}}{3} = \frac{1}{p_1}+\frac{1}{p_2},
{\quad}2\le p_1,p_2\le 6.
\end{split}$$]{} For the weighted norm, we compute $Jb(u)$ in the Fourier space, using the notation in . It is given by a linear combination of terms of the form [$$\begin{split}
{&}e^{-itH({\xi})}{\nabla}_{\xi}\int_{{\xi}={\eta}+{\zeta}} e^{itH({\xi})\mp H({\eta})\mp H({\zeta})}B({\eta},{\zeta}){{{\check}{v}}^\pm}({\eta}) {{{\check}{v}}^\pm}({\zeta}) d{\eta}{\\ &}= e^{-itH({\xi})}\int e^{it{\Omega}}\Bigl[({{\nabla}_{\xi}^{({\eta})}}B + it{{\nabla}_{\xi}^{({\eta})}}{\Omega}\cdot B) {{{\check}{v}}^\pm}({\eta}) {{{\check}{v}}^\pm}({\zeta}) + B {{{\check}{v}}^\pm}({\eta}) {\nabla}{{{\check}{v}}^\pm}({\zeta})\Bigr] d{\eta}{\\ &}= {\mathcal{F}}\left[({{\nabla}_{\xi}^{({\eta})}}B+it{{\nabla}_{\xi}^{({\eta})}}{\Omega}\cdot B)[v^\pm,v^\pm] + B[v^\pm,(Jv)^\pm]\right],
\end{split}$$]{} with $B={{{{\langle}({\xi}_1,{\xi}_2) {\rangle}}}}^{-2}$ or ${{{{\langle}({\xi}_1,{\xi}_2) {\rangle}}}}^{-2}U({\xi}_1)^{-1}U({\xi}_2)^{-1}$, and [$$\begin{split}
{&}{\Omega}:= H({\xi})\mp H({\eta})\mp H({\zeta}), {\quad}{{\nabla}_{\xi}^{({\eta})}}{\Omega}={\nabla}H({\xi})\mp{\nabla}H({\zeta}).
\end{split}$$]{} By the bilinear estimate and the $L^p$ decay property , and , we have [$$\begin{split} \label{est Jb}
{&}\|{{\nabla}_{\xi}^{({\eta})}}B[v^\pm,v^\pm]\|_{H^1_x}
{{\ \lesssim \ }}\|U^{-1}v\|_{L^3_x} \|U^{-2}v\|_{L^6_x} {{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-3/10}\|v\|_X^2,
{\\ &}\|t{{\nabla}_{\xi}^{({\eta})}}{\Omega}\cdot B[v^\pm,v^\pm]\|_{H^1_x}
{{\ \lesssim \ }}\|t^{1/2}U^{-1}v\|_{L^4_x}^2
{{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-1/6}\|v\|_X^2,
{\\ &}\|B[v^\pm,(Jv)^\pm]\|_{H^1_x}
{{\ \lesssim \ }}\|U^{-1}v\|_{L^3_x} \|U^{-1}Jv\|_{L^6_x}
{{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-3/10}\|v\|_X^2.
\end{split}$$]{} Thus we obtain [$$\begin{split} \label{dec bX}
{&}\|b(u)(t)\|_{H^1_x} {{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-9/10}\|v\|_{X(t)}^2,
{\quad}\|Jb(u)(t)\|_{H^1_x} {{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-1/6}\|v\|_{X(t)}^2,
{\\ &}\|b(u)\|_{S(T_1,T_2)} {{\ \lesssim \ }}{{{\langle}T_1 {\rangle}}}^{-7/10}\|v\|_{X(T_1,T_2)}^2,
\end{split}$$]{} for any $t\ge 0$ and any $T_2\ge T_1\ge 0$. Hence by the Banach fixed point theorem, the map $v\mapsto v+b(u)$ is bi-Lipschitz in small balls around $0$ of $X(t)$ uniformly for all $t\in{\mathbb{R}}$, and also globally in $X\cap S$.
Other normal forms {#ss:other normal}
------------------
It is a natural question if we get the same asymptotics for other normal forms, in particular, for ${z}$ defined in . It is easy to see that the above argument works to get [$$\begin{split}
\|{{{{\langle}{\nabla}{\rangle}}}}^{-2}|u|^2\|_{X(t)} {{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-1/6}\|v\|_X^2.
\end{split}$$]{} For this, we need to modify the second inequality in for the interaction with $|{\xi}|+1\ll|{\xi}_1|\sim|{\xi}_2|$ because we lose regularity. Using the Schrödinger type decay estimate in that region, we get nevertheless [$$\begin{split}
\|t{{\nabla}_{\xi}^{({\eta})}}{\Omega}\cdot{{{{\langle}{\xi}{\rangle}}}}^{-2}[u^\pm,u^\pm]\|_{H^1_x}
{&}{{\ \lesssim \ }}\|t^{1/2}U^{-1}v\|_{L^4_x}^2 + \|tv_{>1}\|_{H^{1,6}_x}\|v_{>1}\|_{L^3_x}
{\\ &}{{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-1/6}\|v\|_{X}^2.
\end{split}$$]{} Therefore we have [$$\begin{split}
\|{z}(t)-v(t)\|_{X(t)} {{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-1/6}\|v\|_X^2,
\end{split}$$]{} in other words, ${z}$, $v$ and ${Z}$ have the same asymptotics in the strong topology of $X$, as long as $v$ is in $X$.
Initial condition
=================
Here we check that the smallness condition on $u(0)$ is equivalent to smallness of $v(0)$ in $X(0)$.
By Sobolev and Hölder in Lorentz spaces, implies [$$\begin{split}
\|u(0)\|_{L^2_x} {{\ \lesssim \ }}\|{\nabla}u(0)\|_{L^{6/5,2}_x}
{{\ \lesssim \ }}\|x{\nabla}u(0)\|_{L^2_x} {{\ \lesssim \ }}{\delta}.
\end{split}$$]{} Hence we have [$$\begin{split}
\|u(0)\|_{H^1_x} + \|xu_1(0)\|_{L^2_x} + \|x{\nabla}u_1(0)\|_{L^2_x}
+ \|x{\nabla}u_2(0)\|_{L^2_x} {{\ \lesssim \ }}{\delta}.
\end{split}$$]{} Moreover, by using the commutator $[x,U]={\mathcal{F}}^{-1}[i{\nabla}_{\xi},U({\xi})]{\mathcal{F}}= i{\nabla}_{\xi}U$, [$$\begin{split}
\|{{{\langle}{\nabla}{\rangle}}}x U u_2\|_{L^2_x}
{&}\le \|{{{\langle}{\nabla}{\rangle}}}U x u_2\|_{L^2_x} + \|{{{\langle}{\nabla}{\rangle}}}({\nabla}_{\xi}U)u_2\|_{L^2_x}
{\\ &}{{\ \lesssim \ }}\|{\nabla}xu_2\|_{L^2_x} + \|u_2\|_{L^2_x},
\end{split}$$]{} hence we get $\|{{{\langle}x {\rangle}}}v(0)\|_{H^1_x} {{\ \lesssim \ }}{\delta}$.
On the other hand, since [$$\begin{split}
[J_j,{\nabla}_k U^{-1}] = -{\delta}_{j,k}U^{-1} + {\nabla}_{x_k} i {\nabla}_{{\xi}_j} U^{-1},
\end{split}$$]{} we have [$$\begin{split} \label{com Ju}
\|J{\nabla}u\|_{L^2_x}+\|{\nabla}Ju\|_{L^2_x}
{&}{{\ \lesssim \ }}\|U^{-1}{\nabla}Jv\|_{L^2_x} + \|[J,{\nabla}U^{-1}]v\|_{L^2_x} + \|u\|_{L^2_x}
{\\ &}{{\ \lesssim \ }}\|Jv\|_{H^1_x} + \|U^{-1}v\|_{L^2_x} {{\ \lesssim \ }}\|v(t)\|_{X(t)}.
\end{split}$$]{} In particular, by letting $t=0$, [$$\begin{split}
\|{{{{\langle}x {\rangle}}}}u_1(0)\|_{L^2_x} + \|{{{{\langle}x {\rangle}}}}{\nabla}u(0)\|_{L^2_x}
{{\ \lesssim \ }}\|{{{{\langle}x {\rangle}}}}v(0)\|_{H^1_x}.
\end{split}$$]{}
Estimates without weight {#ss:est w/o w}
========================
In this section, we estimate ${Z}$ in the Strichartz norm $L^{\infty}_t H^1_x \cap U^{1/6}L^2_t H^{1,6}_x$, by putting the nonlinear terms ${\mathcal{N}_{Z}}(v)$ in the dual Strichartz space, as in .
The bilinear terms are bounded by using : [$$\begin{split} \label{Stz on B}
{&}\|B_j[v^\pm,v^\pm]\|_{L^{4/3}_t H^{1,3/2}_x}
{{\ \lesssim \ }}\|v\|_{L^{\infty}H^1} \|v\|_{L^{4/3} L^6} {{\ \lesssim \ }}\|v\|_{X\cap S}^2,
\end{split}$$]{} for $j=3,4$, where the last factor is bounded by the decay estimate . On interval $t\in(T,{\infty})$ for $T>1$, we can estimate instead [$$\begin{split}
{}\|B_j[v^\pm,v^\pm]\|_{L^1_t H^1_x}
{&}{{\ \lesssim \ }}\|v\|_{L^4_{t>T} H^{1,3}} \|v\|_{L^{4/3}_{t>T} L^6}
{\\ &}{{\ \lesssim \ }}T^{-1/2}\|v\|_{L^{\infty}H^1}^{1/2} \|tv\|_{L^{\infty}_{t>T} H^{1,6}_x}^{3/2}
{}{{\ \lesssim \ }}T^{-1/2} \|v\|_X^2.
\end{split}$$]{}
The cubic terms $C_j$ with $j=1,2,3,4$ are similarly bounded by [$$\begin{split} \label{Stz on C}
\|C_j[v^\pm,v^\pm,v^\pm]\|_{L^2_t H^{1,6/5}_x}
{&}{{\ \lesssim \ }}\|U^{-1}v\|_{L^{\infty}H^1} \|U^{-1}v\|_{L^4 L^6}^2
{}{{\ \lesssim \ }}\|v\|_{X}^3,
\end{split}$$]{} by using and , together with the regular bilinear estimate . For $t>T$ we gain at least $T^{2(-3/5+1/4)}=T^{-7/10}$ from the $L^6_x$ decay of $U^{-1}v$. The quartic term $Q_1$ is estimated by using , [$$\begin{split}
{&}\|Q_1(u)\|_{L^{4/3} H^{1,3/2}}
{}{{\ \lesssim \ }}\|u\|_{L^4 H^{1,3}}^3 \|u\|_{L^{\infty}L^6}
{{\ \lesssim \ }}\|v\|_{X\cap S}^4.
\end{split}$$]{} For $t>T$ we gain at least $T^{-3/5}$ from the $L^6_x$ decay of $u=v_1+iU^{-1}v_2$.
In conclusion, we have [$$\begin{split}
{}\left\|\int_T^t e^{-i(t-s)H}{\mathcal{N}_{Z}}ds\right\|_{S(T,{\infty})}
{&}{{\ \lesssim \ }}\|{\mathcal{N}_{Z}}\|_{L^2_{t>T} H^{1,6/5} + L^1_{t>T} H^1} {\\ &}{{\ \lesssim \ }}T^{-1/2}(\|v\|_{X\cap S}^2+\|v\|_{X\cap S}^4),
\end{split}$$]{} so once we get a uniform bound on $\|v\|_{X\cap S}$, the scattering of ${Z}$ and $v$ in $H^1$ follows: [$$\begin{split} \label{scat z H1}
\exists! v_+\in H^1,{\quad}\|{Z}(t)-e^{-itH}v_+\|_{H^1} + \|v(t)-e^{-itH}v_+\|_{H^1} {{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-1/2},
\end{split}$$]{} where we used .
Estimates without phase derivative
==================================
Now we proceed to the $L^{\infty}_t H^1_x$ bound on $J{Z}$. First we rewrite the equation for ${Z}$ by replacing $v$ with ${Z}$ in the bilinear terms: [$$\begin{split}
{&}{\mathcal{N}_{Z}}(v)=B_3[{Z}_1,{Z}_1] + B_4[{Z}_2,{Z}_2]
+ \sum_{j=1}^5 C_j + Q_1(u) + Q_2(u) =: {\mathcal{N}_{Z}}'(v,{Z}),
{\\ &}C_5(v,v,{Z}):=-2B_3[b(u),{Z}_1], {\quad}Q_2(u):=B_3[b(u),b(u)],
\end{split}$$]{} because we are going to integrate on the phase in time (in some Fourier region), where the difference of oscillation between ${Z}$ and $v$ will become essential. In the higher order terms, it is negligible thanks to the time decay of $b(u)$.
Applying $J$ to ${\mathcal{N}_{Z}}'$, we get bilinear terms in the Fourier space like [$$\begin{split}
\int_0^t\int {{\nabla}_{\xi}^{({\eta})}}\left\{e^{is{\Omega}}B_j({\eta},{\xi}-{\eta}){{{\check}{{Z}}}^\pm}(s,{\eta}) {{{\check}{{Z}}}^\pm}(s,{\xi}-{\eta})\right\} d{\eta}ds,
\end{split}$$]{} with $j=3,4$ and the phase ${\Omega}:= H({\xi})\mp H({\eta})\mp H({\xi}-{\eta})$. If the derivative ${{\nabla}_{\xi}^{({\eta})}}$ lands on $B_j$, then since ${{\nabla}_{\xi}^{({\eta})}}B_j$ is a bounded multiplier, its contribution is estimated as above. If ${{\nabla}_{\xi}^{({\eta})}}$ lands on ${{{\check}{{Z}}}^\pm}$, we get terms of the form in the physical space [$$\begin{split}
\int_0^t e^{isH} B_j[{Z}^\pm,(J {Z})^\pm] ds,
\end{split}$$]{} whose contribution in $H^1_x$ is bounded by using the Strichartz together with the regular bilinear estimate , [$$\begin{split}
\|B_j[{Z}^\pm,(J {Z})^\pm]\|_{L^{4/3}_t H^{1,3/2}_x}
{&}{{\ \lesssim \ }}\|{Z}\|_{L^{4/3} H^{1,6}} \|J {Z}\|_{L^{\infty}H^1}
{}{{\ \lesssim \ }}\|{Z}\|_{X\cap S}^2,
\end{split}$$]{} where the $L^{4/3} H^{1,6}$ norm was bounded by the Strichartz $S$ for small $t$ and by the decay for large $t$. Hence if we restrict on $t\in(T,{\infty})$, then we get additional factor $T^{-1/4}$.
Similarly, we have cubic terms like [$$\begin{split} \label{C phaseD}
{\nabla}_{\xi}\int_0^t\iint_{{\xi}={\xi}_1+{\xi}_2+{\xi}_3} e^{is{\Omega}}C_j({\xi}_1,{\xi}_2,{\xi}_3){{{\check}{v}}^\pm}(s,{\xi}_1){{{\check}{v}}^\pm}(s,{\xi}_2){{{\check}{v}}^\pm}(s,{\xi}_3) d{\xi}_1 d{\xi}_2 ds,
\end{split}$$]{} for $1\le j\le 4$, where ${\Omega}=H({\xi})\mp H({\xi}_1)\mp H({\xi}_2)\mp H({\xi}_3)$. If ${\nabla}_{\xi}$ hits $C_j$, its contribution is bounded in the same way as before. If ${\nabla}_{\xi}$ hits ${{{\check}{v}}^\pm}$, then its contribution is estimated by using the Strichartz and the bilinear estimate : [$$\begin{split} \label{JC}
{}\|C_j[v^\pm,v^\pm,(J v)^\pm]\|_{L^2_t H^{1,6/5}_x}
{&}{{\ \lesssim \ }}\|J v\|_{L^{\infty}H^1} \|U^{-1}v\|_{L^2 H^{1,6}} \|U^{-1}v\|_{L^{\infty}L^6}
{\\ &\quad}+ \|U^{-1}J v\|_{L^{\infty}L^6} \|U^{-1}v\|_{L^{\infty}H^1} \|v\|_{L^2 H^{1,6}}
{\\ &}{{\ \lesssim \ }}\|v\|_{X\cap S}^3,
\end{split}$$]{} where if the derivative in the $H^{1,6/5}_x$ norm lands on $Jv$ (with large frequency), it is dominated by the first term on the right, otherwise we use the second term. Refer to for the $L^2 H^{1,6}$ norm on $U^{-1}v$. For large $t>T$, we gain at least $T^{-1/2}$.
For $C_5$, we have just to replace the last ${{{\check}{v}}^\pm}$ with ${{{\check}{{Z}}}^\pm}$ in , hence the final bound in is replaced by $\|v\|_{X\cap S}^2\|{Z}\|_{X\cap S}$.
The quartic terms $Q_j(u)$ with $j=1,2$ are regular. So we write in the physical space [$$\begin{split}
J\int_0^t e^{-i(t-s)H}Q_j(u)ds
=\int_0^t e^{-i(t-s)H}(x-s{\nabla}H({\xi}))Q_j(u) ds.
\end{split}$$]{} Then by Strichartz, their contribution in $L^{\infty}H^1$ is bounded by [$$\begin{split} \label{JN1,4}
\|x Q_j(u)\|_{L^{2} H^{1,6/5}}
+ \|t Q_j(u)\|_{L^2 H^{2,6/5}}.
\end{split}$$]{} For the first term we need to estimate $xu$. Since $Ju=xu-t({\nabla}_{\xi}H) u$, we have by using , [$$\begin{split}
{&}\|xu\|_{L^6_x+tL^2_x}
{{\ \lesssim \ }}\|Ju\|_{\dot H^1_x} + \|({\nabla}_{\xi}H) u\|_{L^2_x}
{{\ \lesssim \ }}\|v(t)\|_{X(t)}.
\end{split}$$]{} Then we have [$$\begin{split}
\|xQ_j(u)\|_{L^2 H^{1,6/5}}
{&}{{\ \lesssim \ }}\|u\|_{L^{\infty}L^6} \|u\|_{L^2 H^{1,6}} \|u\|_{L^{\infty}L^3} \|u\|_{L^{\infty}L^6}
{\\ &\quad}+ \|xu\|_{L^{\infty}(L^6+tL^2)} \|{{{\langle}t {\rangle}}}^{1/2}u\|_{L^{\infty}L^6}^2 \|u\|_{L^{\infty}L^3\cap L^2 L^{\infty}}
{\\ &}{{\ \lesssim \ }}\|v\|_{X\cap S}^4,
\end{split}$$]{} and [$$\begin{split}
\|tQ_j(u)\|_{L^2 H^{2,6/5}}
{{\ \lesssim \ }}\|t^{1/2}u\|_{L^{\infty}L^6}^2 \|u\|_{L^2 L^6} \|u\|_{L^{\infty}L^3}
{{\ \lesssim \ }}\|v\|_{X\cap S}^4.
\end{split}$$]{} For $t>T$, we gain at least $T^{-1/4}$ from the decay of $u$ in $L^{\infty}L^6\cap L^{\infty}L^3\cap L^2 L^{\infty}$ by .
Thus it remains only to estimate the terms with the derivative ${\nabla}_{\xi}$ landing on the phase $e^{is{\Omega}}$ in $B_j$ and $C_j$.
Estimates with phase derivative in bilinear terms
=================================================
This and the next sections make the heart of this paper. If the derivative lands on the phase in the bilinear terms, then the pointwise decay as above is at best $t^{-1}L^3_x$, which is far from sufficient. So we should exploit the non-resonance property, through integration by parts in ${\eta}$ and $s$. We decompose $B_j$ with $j=3,4$ smoothly into two parts: [$$\begin{split} \label{ST reson decop}
B_j = B_j^X + B_j^T,
\end{split}$$]{} where $B_j^X$ is supported in $({\xi},{\eta})$ where the interaction is spatially non-resonant, and $B_j^T$ in the temporally non-resonant region. Here non-resonance means simply that either ${\eta}$ or $s$ derivative of $e^{is{\Omega}}$ does not vanish.
We integrate the phase in ${\eta}$ for $B_j^X$ and in $s$ for $B_j^T$. The strict intersection of the spatially resonant and temporally resonant regions is only at ${\xi}=0$, which can be compensated by the decay at ${\xi}=0$ of the bilinear forms. This is the reason why the above decomposition is possible.
The same type of argument has been used in [@vac2] for the 2D final data problem. The main difference from there is that now we need $L^p$-type bilinear estimates to close our argument for the initial data problem, and that we are not free to integrate by parts because the entry functions are the unknown solutions, whereas in the final data problem they were the given asymptotic profile.
After the integration by parts, we get multipliers whose [*derivatives*]{} have much stronger singularity than allowed in the standard $L^p$ multiplier estimate, even if it were linear. The singularity is due to the behavior of $H({\xi})$ around ${\xi}\to 0$, which is close to the wave equation and thus enhancing resonance between the parallel interactions. Roughly speaking, the singularity increases for each derivative twice as fast as for the standard multipliers, even with the best decomposition of $B^X+B^T$.
To overcome this difficulty, we take advantage of the room in decay in the 3D case to reduce differentiation of the symbol, employing a bilinear estimate with loss in the Hölder exponent. It turns out that there is a narrow balance between the singularity and the loss in $L^p$, such that we can close all the estimates.
Specifically, we use only $1/2+{\varepsilon}$ or $1+{\varepsilon}$ derivatives for small frequencies, while the standard $L^p$ estimates need at least $3/2+{\varepsilon}$ derivatives of the symbol. Moreover, it will be important for us to exploit the smallness of region where the singularity is the strongest.
The plan of these two sections is as follows. First we prove the bilinear estimate allowing some singularities. Secondly we carry out the estimates on all the bilinear terms, assuming some bounds on the bilinear Fourier multipliers with divisors. Finally in the next section, we prove those bounds for each multiplier, using geometric properties of the resonance sets.
Singular bilinear multiplier with Strichartz
--------------------------------------------
We introduce mixed (semi-) norms ${\mathcal{L}}^p \dot B^s_{q,r}$ for the symbols by [$$\begin{split} \label{def LB}
\|f({\xi},{\eta})\|_{{\mathcal{L}}^p \dot B^s_{q,r}}
:= \|j^s \chi^j({\nabla})_{\eta}f({\xi},{\eta})\|_{\ell_j^r L^p_{\xi}L^q_{\eta}(2^{\mathbb{Z}}\times{\mathbb{R}}^d\times{\mathbb{R}}^d)},
\end{split}$$]{} where $\chi^j$ is as in .
\[sbil\] Let $0\le s\le d/2$, and $(p,q)$ be any dual Strichartz exponent except for the endpoint, namely [$$\begin{split}
1\le p<2,{\quad}1< q\le 2,{\quad}\frac{2}{p} + \frac{d}{q} = 2 + \frac{d}{2}.
\end{split}$$]{} Let $(p_1,q_1)$ and $(p_2,q_2)$ satisfy [$$\begin{split}
{&}\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p},
{\quad}\frac{1}{q_1}+\frac{1}{q_2}=\frac{1}{q} + \frac{1}{q(s)},
{\quad}\frac{1}{q(s)}:=\frac{1}{2}-\frac{s}{d},
{\\ &}p\le p_1,p_2\le{\infty}, {\quad}q\le q_1,q_2\le{\infty}.
\end{split}$$]{} Then for any bilinear Fourier multiplier $B$ we have [$$\begin{split}
{&}\left\|\int e^{itH}B[u(t),v(t)]dt\right\|_{L^2_x}
{}{{\ \lesssim \ }}\|B\|_{{\mathcal{L}}^{\infty}_{{\xi}}\dot B^s_{2,1,{\eta}}+{\mathcal{L}}^{\infty}_{{\xi}}\dot B^s_{2,1,{\zeta}}}
\|u\|_{L^{p_1}_tL^{q_1}_x}\|v\|_{L^{p_2}_tL^{q_2}_x}.
\end{split}$$]{} where the first norm of $B$ is in the $({\xi},{\eta})$ coordinates and the second in $({\xi},{\zeta})=({\xi},{\xi}-{\eta})$.
$q(s)$ is the Sobolev exponent for the embedding $\dot B^s_{2,1}\subset L^{q(s)}$, and $1/q(s)$ gives the precise loss compared with the Hölder inequality.
The above estimate is a sort of composition of the Strichartz and the bilinear Fourier multiplier. Indeed, by choosing $u(t,x)=f_n(t){\varphi}(x)$ and $v(t,x)=\psi(x)$ with $\|f_n\|_{L^1}\le 1$ and $f_n\to {\delta}\in {\mathcal{S}}'({\mathbb{R}})$, we can deduce
\[cor:sbil\] Let $0\le s\le d/2$, and let $(q_1,q_2)$ satisfy [$$\begin{split}
{&}\frac{1}{q_1}+\frac{1}{q_2}=\frac{1}{2} + \frac{1}{q(s)},
{\quad}2\le q_1,q_2\le q(s),
{\quad}\frac{1}{q(s)}:=\frac{1}{2}-\frac{s}{d}.
\end{split}$$]{} Then for any bilinear Fourier multiplier $B$ we have [$$\begin{split}
\|B[{\varphi},\psi]\|_{L^2_x}
{{\ \lesssim \ }}\|B\|_{{\mathcal{L}}^{\infty}_{{\xi}}\dot B^s_{2,1,{\eta}}+{\mathcal{L}}^{\infty}_{{\xi}}\dot B^s_{2,1,{\zeta}}}\|{\varphi}\|_{L^{q_1}_x}\|\psi\|_{L^{q_2}_x}.
\end{split}$$]{}
Practically, the above norm on $B$ can be estimated by the interpolation [$$\begin{split} \label{interpol for B}
{&}\|B\|_{{\mathcal{L}}^{\infty}\dot B^s_{2,1}}
{{\ \lesssim \ }}\|B\|_{L^{\infty}\dot H^{s_0}}^{1-{\theta}} \|B\|_{L^{\infty}\dot H^{s_1}}^{{\theta}},
\end{split}$$]{} for $s=(1-{\theta})s_0 + {\theta}s_1$, with $s_0\not=s_1$ and ${\theta}\in (0,1)$. For intermediate exponents, we can instead apply multilinear real interpolation to the above estimates, replacing ${\mathcal{L}}^{\infty}\dot B^s_{2,1}$ by $L^{\infty}\dot B^s_{2,{\infty}}$ for free. But this does not work in the boundary cases, and indeed the estimates would become false, for example when $s=0$ or $s=d/2$, which will be used later. We abbreviate those norms by [$$\begin{split} \label{def HB}
{&}[H^s] := L^{\infty}_{\xi}\dot H^s_{\eta}+ L^{\infty}_{\xi}\dot H^s_{\zeta},
{\quad}[B^s] := {\mathcal{L}}^{\infty}_{\xi}\dot B^s_{2,1,{\eta}} + {\mathcal{L}}^{\infty}_{\xi}\dot B^s_{2,1,{\zeta}}.
\end{split}$$]{}
The proof given below essentially contains one for the corollary, but it seems difficult to decouple the lemma into the Strichartz and a time independent estimate. On the other hand, the corollary just barely fails to be sufficient for our application. In fact, more use of the Strichartz simply causes more loss in decay, compared with the weighted decay estimate. That is why we do not pursue the endpoint Strichartz, which corresponds to the worst case. Nevertheless, we prefer the flexibility in the space-time Hölder exponents provided by Strichartz.
By symmetry between ${\eta}$ and ${\zeta}$, it suffices to prove for the norm in the $({\xi},{\eta})$ coordinates. In the case $s=0$, we get the above just by Plancherel. For $s>0$, we want to integrate by parts in the Fourier space. Since the Fourier transform of $u$ and $v$ themselves are not smooth, we will use the cubic decomposition in $x$. The Fourier transform of the bilinear operator is given by [$$\begin{split}
X := \int B f({\xi},{\eta}) d{\eta},
{\quad}f({\xi},{\eta}) {&}:= \int e^{itH({\xi})} {\widetilde}u(t,{\eta}){\widetilde}v(t,{\xi}-{\eta}) dt
{\\ &}= {\mathcal{F}}_x^{\xi}{\mathcal{F}}_y^{\eta}\int e^{itH_x}[u(t,x+y)v(t,x)] dt.
\end{split}$$]{} First we dyadically decompose $B$ in the frequency for ${\eta}$: [$$\begin{split}
{&}X = \sum_{n\in 2^{\mathbb{Z}}} X_n,
{\quad}X_n := \int B_n f d{\eta},
{\quad}B_n := \chi^n({\nabla})_{\eta}B({\eta},{\xi}-{\eta}).
\end{split}$$]{} By the definition of our norm on $B$, it suffices to bound each $X_n$ uniformly. Next, we decompose $u(t,x)$ and $v(t,x)$ into cubes of size $n$ in $x$. Fix $c\in C^{\infty}_0({\mathbb{R}}^d)$ such that for all $x\in{\mathbb{R}}^d$ we have [$$\begin{split}
\sum_{j\in {\mathbb{Z}}^d} c(x-j) = 1.
\end{split}$$]{} Then for any ${\varphi}\in{\mathcal{S}}'({\mathbb{R}}^d)$ we have the following decomposition [$$\begin{split}
{&}{\widetilde}{\varphi}({\xi}) = \sum_{j\in{\mathbb{Z}}^d} e^{-inj{\xi}}{\widehat}{\varphi}_j^n({\xi}),
{\quad}{\varphi}_j^n(x):=c(x/n-j){\varphi}(x),
{\\ &}{\widehat}{\varphi}_j^n({\xi}):=e^{inj{\xi}}{\widetilde}{\varphi}_j^n({\xi})
= {\mathcal{F}}_x^{\xi}[c(x/n){\varphi}(x+nj)],
\end{split}$$]{} where ${\varphi}_j^n$ is supported on $\{|x/n-j|{{\ \lesssim \ }}1\}$. Applying this decomposition to both $u$ and $v$, we get [$$\begin{split}
{&}X_n = \sum_{j,k\in{\mathbb{Z}}^d} \int B_n f_{j,k}^n({\xi},{\eta}) d{\eta},
{\\ &}f_{j,k}^n := \int e^{itH({\xi})} e^{-inj{\eta}-ink({\xi}-{\eta})}{\widehat}u_j^n(t,{\eta}){\widehat}v_k^n(t,{\xi}-{\eta}) dt.
\end{split}$$]{} If $j\not=k$, then we choose $a\in\{1,\dots,d\}$ such that $|(j-k)_a|\sim|j-k|$ and integrate the phase $e^{-in(j-k){\eta}}$ in the variable ${\eta}_a$ in $K$ times for arbitrary $K\in{\mathbb{N}}$. Then we get [$$\begin{split}
\int B_n f_{j,k}^n d{\eta}= \sum_{{\alpha}+{\beta}+{\gamma}=K} {&}[in(j-k)_a]^{-K} \iint e^{itH({\xi})} e^{-inj{\eta}-ink({\xi}-{\eta})}
{\\ &\qquad\times}{\partial}_{{\eta}_a}^{\alpha}B_n \,{\partial}_a^{\beta}{\widehat}u_j^n(t,{\eta})\,(-{\partial}_a)^{\gamma}{\widehat}v_k^n(t,{\xi}-{\eta}) dt d{\eta}.
\end{split}$$]{} Thus denoting [$$\begin{split} \label{def P**}
{\varphi}_{a,j}^{{\gamma},n} := {\mathcal{F}}^{-1}_{\xi}e^{-inj{\xi}} {\partial}_{a}^{\gamma}{\widehat}{\varphi}_j^n({\xi}) = n^{\gamma}[-i(x/n-j)_a]^{\gamma}c(x/n-j){\varphi}(x),
\end{split}$$]{} we have [$$\begin{split}
{&}X_n = \sum_{a=0}^d \sum_{{\alpha}=0}^K \int [{\partial}_{{\eta}_a}^{\alpha}B_n] G_{a,{\alpha}}^n({\xi},{\eta}) d{\eta},
{\\ &}G_{a,{\alpha}}^n({\xi},{\eta}) := {\mathcal{F}}_x^{\xi}{\mathcal{F}}_y^{\eta}\int e^{itH_x} F_{a,{\alpha}}^n(t,x,y) dt,
{\\ &}F_{0,0}^n := \sum_{j\in{\mathbb{Z}}^d} u_j(t,x+y) v_j(t,x),{\quad}F_{0,{\alpha}}^n := 0 {\quad}({\alpha}>0),
{\\ &}F_{a,{\alpha}}^n(x,y) := \sum_{j,k} \sum_{{\beta}+{\gamma}=K-{\alpha}} \frac{(-1)^{\gamma}}{[in(j-k)_a]^K} u_{a,j}^{{\beta},n} (t,x+y) v_{a,k}^{{\gamma},n}(t,x),
\end{split}$$]{} where in the last term, the summation for $(j,k)$ is constrained to the set $\{(j-k)_a\sim|j-k|\}$, which should be chosen to partition $\{j\not=k\}$ by $a=1,\dots,d$.
Applying Hölder in $({\xi},{\eta})$, we have [$$\begin{split}
\|X_n\|_{L^2_{\xi}}
{{\ \lesssim \ }}\sum_{a=0}^d \sum_{{\alpha}=0}^K \|{\partial}_{{\eta}_a}^{\alpha}B_n\|_{L^{\infty}_{\xi}L^2_{\eta}}
\|G_{a,{\alpha}}^n\|_{L^2_{{\xi},{\eta}}.}
\end{split}$$]{} To estimate the $L^2_{{\xi},{\eta}}$ norm, we recall the standard proof of the (non-endpoint) Strichartz estimate from the decay of $e^{itH}$. After using Plancherel in $({\xi},{\eta})$, we get [$$\begin{split}
\|G_{a,{\alpha}}^n\|_{L^2_{{\xi},{\eta}}}^2
= \iiint {\big\langle e^{i(t-t')H_x}F_{a,{\alpha}}^n(t,x,y) \big| F_{a,{\alpha}}^n(t',x,y) \big\rangle_x} dy dt dt',
\end{split}$$]{} and the $L^2_x$ inner product is bounded by using the $L^p$ decay of $e^{itH}$, [$$\begin{split}
{{\ \lesssim \ }}|t-t'|^{-{\sigma}} \|F_{a,{\alpha}}^n(t,x,y)\|_{L^q_x} \|F_{a,{\alpha}}^n(t',x,y)\|_{L^q_x},
\end{split}$$]{} where $1\le q\le 2$ and ${\sigma}= d(1/q-1/2)$. Applying the Schwarz inequality to $dy$, and Hardy-Littlewood-Sobolev to $dt'dt$, we get [$$\begin{split}
\|G_{a,{\alpha}}^n\|_{L^2_{{\xi},{\eta}}}^2
{{\ \lesssim \ }}\|F_{a,{\alpha}}^n(t,x,y)\|^2_{L^p_t L^2_y L^q_x},
\end{split}$$]{} for any dual Strichartz exponent $(p,q)$, except for the endpoint, which we do not consider for simplicity. By the support property of $u_{a,j}^{{\beta},n}$ and $v_{a,k}^{{\gamma},n}$ , we have, noting that $|y|{{\ \gtrsim \ }}n$ when ${\alpha}>0$, [$$\begin{split}
\|F_{a,{\alpha}}^n(t,x,y)\|_{L^p_t L^2_y L^q_x}
{&}{{\ \lesssim \ }}n^{-{\alpha}}\|\min(1,|y/n|^{-K})u(t,x+y)v(t,x)\|_{L^p_t L^2_y L^q_x}.
\end{split}$$]{} On the other hand, by the Fourier support property of $B_n$, we have [$$\begin{split}
\|{\partial}_{{\eta}_a}^{\alpha}B_n\|_{L^{\infty}_{\xi}L^2_{\eta}}
{{\ \lesssim \ }}n^{{\alpha}-s}\|B_n\|_{L^{\infty}_{\xi}\dot H^s_{\eta}}.
\end{split}$$]{} Putting them together, we obtain [$$\begin{split}
\|X_n\|_{L^2_{\xi}}
{{\ \lesssim \ }}\|B_n\|_{L^{\infty}_{\xi}\dot H^s_{\eta}} n^{-s}\|{{{\langle}y/n {\rangle}}}^{-K}u(x+y)v(x)\|_{L^p_t L^2_y L^q_x}
\end{split}$$]{} for any $K\in{\mathbb{N}}$. By using Young’s inequality, we have for sufficiently large $K$, [$$\begin{split}
\|{{{\langle}y/n {\rangle}}}^{-K}{\varphi}(x+y)\psi(x)\|_{L^2_yL^q_x}^q
{&}=\|{{{\langle}y/n {\rangle}}}^{-Kq}(|{\varphi}|^q*|\psi|^q)(y)\|_{L^{2/q}_y}
{\\ &}{{\ \lesssim \ }}\|{{{\langle}y/n {\rangle}}}^{-Kq}\|_{L^{d/(sq)}_y} \||{\varphi}|^q\|_{L^{q_1/q}} \||\psi|^q\|_{L^{q_2/q}}
{\\ &}\sim n^{sq}\|{\varphi}\|_{L^{q_1}}^q \|\psi\|_{L^{q_2}}^q.
\end{split}$$]{} Applying this to the above at each $t$, then using Hölder in $t$, and summing for all $n\in 2^{\mathbb{Z}}$, we get the desired estimate.
Spatial integration of phase
----------------------------
First we consider the spatially non-resonant part $B_j^X$. More precisely, we first decompose $B_j$ dyadically by using , [$$\begin{split} \label{decop BB}
B_j^{a,b,c} := \chi^a({\xi})\chi^b({\eta})\chi^c({\zeta}) B_j = B_j^{a,b,c,X} + B_j^{a,b,c,T},
\end{split}$$]{} such that $|{\xi}|\sim a$, $|{\eta}|\sim b$ and $|{\zeta}|\sim c$ in the support of $B_j^{a,b,c}$. The sum over $a,b,c\in 2^{\mathbb{Z}}$ can be restricted to [$$\begin{split} \label{dyprod}
a {{\ \lesssim \ }}b \sim c,{\quad}b{{\ \lesssim \ }}c\sim a, {\quad}c{{\ \lesssim \ }}a\sim b.
\end{split}$$]{} The smooth decomposition into $B_j^{a,b,c,X}$ and $B_j^{a,b,c,T}$ will be given in Section \[s:multest\]. For each $B_j^{a,b,c,X}$, we integrate the phase in ${\eta}$, by using the identity [$$\begin{split}
e^{is{\Omega}} = \frac{{\nabla}_{\eta}{\Omega}}{is|{\nabla}_{\eta}{\Omega}|^2}\cdot{\nabla}_{\eta}e^{is{\Omega}}.
\end{split}$$]{} Then we get terms like [$$\begin{split} \label{phase1}
{&}{\mathcal{F}}^{-1}\int_0^t ds \int e^{is{\Omega}} \left\{{\mathcal{B}}_1\cdot{\nabla}_{\eta}[{{{\check}{{Z}}}^\pm}({\eta}) {{{\check}{{Z}}}^\pm}({\xi}-{\eta})]
+ {\mathcal{B}}_2 {{{\check}{{Z}}}^\pm}({\eta}) {{{\check}{{Z}}}^\pm}({\xi}-{\eta}) \right\}d{\eta}{\\ &}= \int_0^t e^{isH}\left\{{\mathcal{B}}_1[(J{Z})^\pm,{Z}^\pm] - {\mathcal{B}}_1[{Z}^\pm, (J{Z})^\pm] + {\mathcal{B}}_2[{Z}^\pm,{Z}^\pm]\right\} ds,
\end{split}$$]{} where ${\Omega}=H({\xi})\mp H({\eta})\mp H({\xi}-{\eta})$ and ${\mathcal{B}}_k={\mathcal{B}}_{k,j}^{a,b,c}$ is defined by [$$\begin{split}
{&}{\mathcal{B}}_{1,j}^{a,b,c} := \frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}\cdot{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}B_j^{a,b,c,X},
{\quad}{\mathcal{B}}_{2,j}^{a,b,c} := {\nabla}_{\eta}\cdot\frac{{\nabla}_{\eta}{\Omega}\cdot {{\nabla}_{\xi}^{({\eta})}}{\Omega}\cdot B_j^{a,b,c,X}}{|{\nabla}_{\eta}{\Omega}|^2}.
\end{split}$$]{} Denoting [$$\begin{split}
{&}M:=\max(a,b,c),{\quad}m:=\min(a,b,c),{\quad}l:=\min(b,c),
\end{split}$$]{} we assume that if $M\ll 1$ then [$$\begin{split} \label{B^X bound1}
{&}\|{\mathcal{B}}_1^{a,b,c}\|_{[H^{1+{\varepsilon}}]} {{\ \lesssim \ }}l^{1/2-2{\varepsilon}},
{\quad}\|{\mathcal{B}}_2^{a,b,c}\|_{[H^{1+{\varepsilon}}]} {{\ \lesssim \ }}l^{1/2-2{\varepsilon}}M^{-1},
\end{split}$$]{} for ${\varepsilon}>0$ small, and if $M{{\ \gtrsim \ }}1$ then [$$\begin{split} \label{B^X bound2}
{&}\|{\mathcal{B}}_1^{a,b,c}\|_{[H^{3/2+{\varepsilon}}]} {{\ \lesssim \ }}l^{1-{\varepsilon}}{{{\langle}a {\rangle}}}^{-1}+m^{-{\varepsilon}},
{\quad}\|{\mathcal{B}}_2^{a,b,c}\|_{[H^{3/2+{\varepsilon}}]} {{\ \lesssim \ }}l^{-{\varepsilon}}{{{\langle}a {\rangle}}}^{-1} + m^{-{\varepsilon}},
\end{split}$$]{} for $|{\varepsilon}|$ small. Note that we have a derivative loss for the first term if ${{{\langle}a {\rangle}}}\ll b\sim c$. This is the reason for our new normal form (cf. the discussion in Section \[s:normal\]). For the definition of those norms, see and . The above estimates will be proved in Section \[s:multest\]. Then applying Lemma \[sbil\] with some fixed $s=1+{\varepsilon}>1$ and $3/2$, we get [$$\begin{split}
{&}\left\|\sum_{a,b,c} \int e^{isH}\{{\mathcal{B}}_1[(J{Z})^\pm,{Z}^\pm] \pm {\mathcal{B}}_1[{Z}^\pm,(J{Z})^\pm]\} ds\right\|_{H^1_x}
{\\ &}{{\ \lesssim \ }}\|\sum_{M\ll 1}m^{{\varepsilon}/2}{\mathcal{B}}_1^{a,b,c}\|_{[B^{1+{\varepsilon}}]}
{}\|U^{-{\varepsilon}/2}J{Z}\|_{L^{\infty}_t L^{1/2-{\varepsilon}/6}_x} \|U^{-1/6}{Z}\|_{L^{1-{\varepsilon}/4}_t L^6_x}
{\\ &\quad}+ \|\sum_{M{\gtrsim}1}U(m)^{{\varepsilon}/2}{{{\langle}a {\rangle}}}{{{\langle}b {\rangle}}}^{-1}{{{\langle}c {\rangle}}}^{-1}{\mathcal{B}}_1^{a,b,c}\|_{[B^{3/2}]}
{\\ &\qquad\times}\|U^{-{\varepsilon}/2}{{{\langle}{\nabla}{\rangle}}}J{Z}\|_{L^{\infty}_t L^{1/2-{\varepsilon}/6}_x} \|U^{-1/6}{{{\langle}{\nabla}{\rangle}}}{Z}\|_{L^{3/4+{\varepsilon}/4}_t L^6_x},
\end{split}$$]{} where the sums are for $a,b,c$ satisfying , and we used the convention $L^{1/p}=L^p$. The norms on ${\mathcal{B}}_1^{a,b,c}$ are bounded by the above assumption and interpolation as follows. For the first case $M\ll 1$, we have [$$\begin{split}
\|\sum_{M\ll 1}m^{{\varepsilon}/2}{\mathcal{B}}_1^{a,b,c}\|_{[H^{1+{\varepsilon}'}]}{{\ \lesssim \ }}\sum_{l\ll 1}m^{{\varepsilon}/2}l^{1/2-2{\varepsilon}'} {{\ \lesssim \ }}1,
\end{split}$$]{} for ${\varepsilon}'>0$ small, so we can replace the norm with $[B^{1+{\varepsilon}}]$ by the interpolation for varying ${\varepsilon}'$. For the second case $M{{\ \gtrsim \ }}1$ we have [$$\begin{split}
{&}\|\sum_{M{\gtrsim}1}U(m)^{{\varepsilon}/2}{{{\langle}a {\rangle}}}{{{\langle}b {\rangle}}}^{-1}{{{\langle}c {\rangle}}}^{-1}{\mathcal{B}}_1^{a,b,c}\|_{[H^{3/2+{\varepsilon}'}]}
{}{{\ \lesssim \ }}\sum_{l\in 2^{\mathbb{Z}}} U(m)^{\varepsilon}{{{\langle}l {\rangle}}}^{-{\varepsilon}'-1} {{\ \lesssim \ }}1,
\end{split}$$]{} for $|{\varepsilon}'|$ small. Then we can change the norm to $[B^{3/2}]$ by the interpolation .
The norm on $J{Z}$ is bounded by the Sobolev embedding [$$\begin{split}
\|U^{-{\varepsilon}/2}{{{\langle}{\nabla}{\rangle}}}J{Z}\|_{L^{\infty}_t L^{1/2-{\varepsilon}/6}_x} {{\ \lesssim \ }}\|J{Z}\|_{L^{\infty}_t H^1_x},
\end{split}$$]{} and the norms on ${Z}$ are bounded by splitting into $|t|<1$ and $|t|>1$, [$$\begin{split}
{&}\|U^{-1/6}{Z}\|_{L^{1-{\varepsilon}/4}_t L^6_x} + \|U^{-1/6}{{{\langle}{\nabla}{\rangle}}} {Z}\|_{L^{3/4+{\varepsilon}/4}_t L^6_x}
{\\ &}{{\ \lesssim \ }}\|U^{-1/6}{Z}\|_{L^2_t H^{1,6}_x}
+ \|tU^{-1/3}{Z}\|_{L^{\infty}_t H^{1,6}_x}
{{\ \lesssim \ }}\|{Z}\|_{S\cap X}.
\end{split}$$]{} When restricted for large $t>T$, the above is bounded by $T^{-{\varepsilon}/4}$.
In the same way, we estimate the term with ${\mathcal{B}}_2$ by [$$\begin{split}
{&}\left\|\int_0^t e^{isH}{\mathcal{B}}_2[{Z}^\pm,{Z}^\pm] ds\right\|_{H^1_x}
{\\ &}{{\ \lesssim \ }}\|\sum_{a,b,c}U(l)U(M)^{1/6}{{{\langle}a {\rangle}}}{{{\langle}b {\rangle}}}^{-1}{{{\langle}c {\rangle}}}^{-1}{\mathcal{B}}_2^{a,b,c}\|_{[B^{1+{\varepsilon}}]+[B^{3/2}]}
{\\ &\qquad\times}\|U^{-1}{Z}\|_{L^{\infty}_t H^1_x} \|U^{-1/6}{Z}\|_{(L^{1-{\varepsilon}/2}\cap L^{4/3})_t H^{1,6}_x},
\end{split}$$]{} where the $L^2_x$ norm is applied to the $l$ frequency component, while the $L^6_x$ norm is applied to the other with frequency $\max(b,c)\sim M$. We use $[B^{1+{\varepsilon}}]$ for $M\le 1$ with $L^{1-{\varepsilon}/2}_t L^6_x$, and $[B^{3/2}]$ for $M\ge 1$ with $L^{4/3}_t L^6_x$. Then the norm on ${\mathcal{B}}_2$ is estimated by [$$\begin{split}
{&}\|\sum_{M<1}\cdots{\mathcal{B}}_2^{a,b,c}\|_{[H^{1+{\varepsilon}'}]}
{{\ \lesssim \ }}\sum_{l\le 1} l^{3/2-2{\varepsilon}'}M^{-5/6} {{\ \lesssim \ }}1,
{\\ &}\|\sum_{M\ge 1}\cdots{\mathcal{B}}_2^{a,b,c}\|_{[H^{3/2+{\varepsilon}''}]}
{{\ \lesssim \ }}\sum_{l\in 2^{\mathbb{Z}}} U(l) {{{\langle}l {\rangle}}}^{-{\varepsilon}''-1} {{\ \lesssim \ }}1,
\end{split}$$]{} for ${\varepsilon}'>0$ small and $|{\varepsilon}''|$ small. Then we can replace the norms with $[B^{1+{\varepsilon}}]$ and $[B^{3/2}]$ respectively, by the interpolation .
The norms on ${Z}$ are treated in the same way as for ${\mathcal{B}}_1$. For large $t>T$, we have additional decay factor $T^{-{\varepsilon}/2}$.
Time integration of phase
-------------------------
For the temporally non-resonant part $B_j^{a,b,c,T}$, we integrate the phase in $s$. Then we get the time derivative of the bilinear functions, for which we use the equation again. Thus we get terms like [$$\begin{split} \label{integ in s}
{&}\int_0^t e^{isH}\left\{{\mathcal{B}}_3[{Z}^\pm,{Z}^\pm]+{\mathcal{B}}_3[s{\mathcal{N}_{Z}}^\pm,{Z}^\pm] + {\mathcal{B}}_3[{Z}^\pm,s{\mathcal{N}_{Z}}^\pm]\right\}ds
{\\ &}+ \left[e^{isH}{\mathcal{B}}_3[s{Z}^\pm,{Z}^\pm]\right]_{s=0}^t,
\end{split}$$]{} where we put [$$\begin{split}
{&}{\mathcal{B}}_3 = {\mathcal{B}}_{3,j}^{a,b,c} := \frac{{\nabla}_{\xi}{\Omega}}{{\Omega}}B_j^{a,b,c,T},
\end{split}$$]{} with $j=3,4$. Recall $|{\xi}|\sim a$, $|{\eta}|\sim b$, $|{\zeta}|\sim c$, $M:=\max(a,b,c)$, $m=\min(a,b,c)$ and $l=\min(b,c)$. We assume that [$$\begin{split} \label{B^T bound}
\|{\mathcal{B}}_3^{a,b,c}\|_{[H^{s}]}
{{\ \lesssim \ }}({{{\langle}M {\rangle}}}/M)^{s}l^{3/2-s}{{{\langle}a {\rangle}}}^{-1},
\end{split}$$]{} for $0<s<2$. See with for the definition of norm. This bound will be proved in Section \[s:multest\].
For the first term in , we use Lemma \[sbil\] with some fixed $s=1+{\varepsilon}>1$ for $M\ll 1$ and $s=3/2$ for $M{{\ \gtrsim \ }}1$. Thus we get [$$\begin{split}
{&}\left\|\sum_{a,b,c}\int e^{isH} {\mathcal{B}}_3[{Z}^\pm,{Z}^\pm] ds\right\|_{H^1_x}
{\\ &}{{\ \lesssim \ }}\|\sum_{a,b,c} U(l)U(M)^{1/6}{{{\langle}a {\rangle}}}{{{\langle}b {\rangle}}}^{-1}{{{\langle}c {\rangle}}}^{-1}{\mathcal{B}}_3^{a,b,c}\|_{[B^{1+{\varepsilon}}] + [B^{3/2}]}
{\\ &\qquad\times}\|U^{-1/6}{{{\langle}{\nabla}{\rangle}}}{Z}\|_{(L^{1-{\varepsilon}/2}\cap L^{4/3})_tL^6_x} \|U^{-1}{{{\langle}{\nabla}{\rangle}}}{Z}\|_{L^{\infty}_t L^2_x}.
\end{split}$$]{} For the norm on ${\mathcal{B}}_3$, we have [$$\begin{split}
{&}\|\sum_{M< 1}\cdots{\mathcal{B}}_3^{a,b,c}\|_{[H^{1+{\varepsilon}'}]}
{{\ \lesssim \ }}\sum_{l\le 1}lM^{-1/3-2{\varepsilon}'} {{\ \lesssim \ }}1,
{\\ &}\|\sum_{M\ge 1}\cdots{\mathcal{B}}_3^{a,b,c}\|_{[H^{3/2+{\varepsilon}'}]}
{{\ \lesssim \ }}\sum_{l\in 2^{\mathbb{Z}}}U(l){{{\langle}l {\rangle}}}^{-{\varepsilon}'-2} {{\ \lesssim \ }}1,
\end{split}$$]{} for $|{\varepsilon}'|$ small. Then by the interpolation , we can change the norms to $[B^{1+{\varepsilon}}]$ and $[B^{3/2}]$, respectively. If the time interval is restricted on $(T,{\infty})$, then we get additional decay factor $T^{-{\varepsilon}/2}$ from the $L^6_x$ decay of ${Z}$.
For the time boundary term, we use Corollary \[cor:sbil\] with $s=1$ and $3/2$. Then [$$\begin{split}
{&}\|\sum_{a,b,c}e^{itH}{\mathcal{B}}_3[t{Z}^\pm,{Z}^\pm]\|_{H^1_x}
{\\ &}{{\ \lesssim \ }}\|\sum_{a,b,c}U(l)U(M)^{1/3}{{{\langle}a {\rangle}}}{{{\langle}l {\rangle}}}^{-1/2}{{{\langle}M {\rangle}}}^{-1}{\mathcal{B}}_3^{a,b,c}\|_{[B^{1}]+[B^{3/2}]}
{\\ &\qquad\times}\|U^{-1/3}t{Z}\|_{H^{1,6}_x} \|U^{-1}{Z}\|_{H^1_x},
\end{split}$$]{} where we used $H^1\subset H^{1/2,3}$ for the lower frequency $l$ term when $M{{\ \gtrsim \ }}1$. The norm on ${\mathcal{B}}_3$ is bounded by [$$\begin{split}
{&}\|\sum_{M< 1}\cdots{\mathcal{B}}_3^{a,b,c}\|_{[H^{1+{\varepsilon}'}]}
{{\ \lesssim \ }}\sum_{l \le 1}lM^{{\varepsilon}'-1/6} {{\ \lesssim \ }}1,
{\\ &}\|\sum_{M\ge 1}\cdots{\mathcal{B}}_3^{a,b,c}\|_{[H^{3/2+{\varepsilon}'}]}
{{\ \lesssim \ }}\sum_{l\in 2^{\mathbb{Z}}}U(l){{{\langle}l {\rangle}}}^{-{\varepsilon}'-3/2} {{\ \lesssim \ }}1,
\end{split}$$]{} for $|{\varepsilon}'|$ small, which change into the desired norms by the interpolation. For large $t>T$, we can gain $T^{-{\varepsilon}}$ by replacing $[B^1]$ with $[B^{1+{\varepsilon}}]$, and $H^1_x$ with $H^{1,1/2-{\varepsilon}/3}_x$.
To the remaining terms in , we apply Lemma \[sbil\] with some fixed $s=1/2+{\varepsilon}>1/2$ and $s=3/2$. We obtain [$$\begin{split} \label{B3 aIBP}
{&}\left\|\sum_{a,b,c}\int e^{isH}\{{\mathcal{B}}_3[s{\mathcal{N}_{Z}}^\pm,{Z}^\pm]\pm{\mathcal{B}}_3[{Z}^\pm,s{\mathcal{N}_{Z}}^\pm]\}ds \right\|_{H^1_x}
{\\ &}{{\ \lesssim \ }}\|\sum_{a,b,c}U(l)^{1/2}U(M){{{\langle}a {\rangle}}}{{{\langle}b {\rangle}}}^{-1}{{{\langle}c {\rangle}}}^{-1}{\mathcal{B}}_3^{a,b,c}\|_{[B^{1/2+{\varepsilon}}]+[B^{3/2}]}
{\\ &\qquad\times}\|U^{-1/2}{{{\langle}{\nabla}{\rangle}}} t{\mathcal{N}_{Z}}\|_{(L^{1-{\varepsilon}/2}\cap L^2)_t L^3_x} \|U^{-1}{{{\langle}{\nabla}{\rangle}}}{Z}\|_{L^{\infty}_t L^2_x}.
\end{split}$$]{} For the norm on ${\mathcal{B}}_3$, we estimate [$$\begin{split}
{&}\|\sum_{M<1}\cdots{\mathcal{B}}_3^{a,b,c}\|_{[H^{1/2+{\varepsilon}'}]}
{{\ \lesssim \ }}\sum_{l \le 1} l^{1/2} {{\ \lesssim \ }}1,
{\\ &}\|\sum_{M\ge 1}\cdots{\mathcal{B}}_3^{a,b,c}\|_{[H^{3/2+{\varepsilon}'}]}
{{\ \lesssim \ }}\sum_{l\in 2^{\mathbb{Z}}}U(l) {{{\langle}l {\rangle}}}^{-{\varepsilon}'-2} {{\ \lesssim \ }}1,
\end{split}$$]{} for $|{\varepsilon}'|$ small, which yields the bound on the desired norm via the interpolation.
It remains to estimate the above norm on ${\mathcal{N}_{Z}}$. For the bilinear part of ${\mathcal{N}_{Z}}$, we use , splitting into $|t|<1$ and $|t|>1$, [$$\begin{split}
{&}\|U^{-1/2}B_j[v^\pm,v^\pm]\|_{L^2_t H^{1,3}_x}
{{\ \lesssim \ }}\|v\|_{L^2_t H^{1,6}_x} \|v\|_{L^{\infty}_t L^6_x},
{\\ &}\|t^2U^{-1/2}B_j[v^\pm,v^\pm]\|_{L^{\infty}_t H^{1,3}_x}
{{\ \lesssim \ }}\|tv\|_{L^{\infty}_t H^{1,6}_x}^2,
\end{split}$$]{} for $j=3,4$. The cubic part of ${\mathcal{N}_{Z}}$ is bounded by using [$$\begin{split}
{&}\|tC_j[v^\pm,v^\pm,v^\pm]\|_{(L^1\cap L^2)_t (H^1\cap H^{1,3})_x}
{}{{\ \lesssim \ }}\|tv\|_{L^{\infty}H^{1,6}} \|U^{-1}v\|_{L^{\infty}H^1 \cap L^2 H^{1,6}}^2,
\end{split}$$]{} for $j=1,2,3,4$. Then we apply Sobolev $U^{-1/2}(H^1\cap H^{1,3})_x\subset H^{1,3}_x$ at each $t$. For the quartic term we have [$$\begin{split}
\|Q_1(u)\|_{(H^1\cap H^{1,3})_x}
{{\ \lesssim \ }}\|u\|_{L^6_x}^4 {{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-2}\|v\|_X^4,
\end{split}$$]{} which gives a better bound than the above. Thus we get the desired bound on ${\mathcal{N}_{Z}}$, completing the estimate . If we restrict the time interval into $(T,{\infty})$, then we get additional decay $T^{-{\varepsilon}/2}$, where the worst decay comes from the bilinear part.
Estimates on the bilinear multipliers with divisors {#s:multest}
===================================================
In this section we show that we can decompose the bilinear multipliers $B_j$, $j=3,4$ such that , and hold, for all combinations of ${Z}^\pm$, i.e., $B_j[{Z},{Z}]$, $B_j[{\overline}{{Z}},{Z}]$ and $B_j[{\overline}{{Z}},{\overline}{{Z}}]$. We will use elementary geometry in the Fourier space in a way similar to [@vac2]. Recall our notation in . First we have for $|{\xi}|\ge |{\eta}|$ [$$\begin{split} \label{first bound}
|{\nabla}H({\xi})-{\nabla}H({\eta})| &\sim |H'({\xi})-H'({\eta})| + H'({\eta})|{\widehat}{{\xi}}-{\widehat}{{\eta}}|
{\\ &}\sim \frac{|{\xi}|}{{{{\langle}{\xi}{\rangle}}}}||{\xi}|-|{\eta}|| + {{{\langle}{\eta}{\rangle}}}|{\widehat}{{\xi}}-{\widehat}{{\eta}}|,
\end{split}$$]{} where $H'$ denotes the radial derivative of $H({\xi})$. For the higher derivatives, we have [$$\begin{split}
|{\nabla}^k H({\xi})| {{\ \lesssim \ }}\frac{{{{\langle}{\xi}{\rangle}}}}{|{\xi}|^{k-1}},
\end{split}$$]{} for any $k\in{\mathbb{N}}$, hence by using the Taylor’s formula when $|{\xi}|\sim|{\eta}|$, we obtain [$$\begin{split} \label{higher bound}
{&}|{\nabla}^k H({\xi}) - {\nabla}^k H({\eta})|
{{\ \lesssim \ }}\frac{{{{\langle}{\xi}{\rangle}}}|{\xi}-{\eta}|}{|{\xi}||{\eta}|^{k-1}},
\end{split}$$]{} for $|{\xi}|\ge|{\eta}|$.
Multiplier estimates for ${\overline}{{Z}}{Z}$
----------------------------------------------
In this subsection, we deal with the bilinear terms of the form [$$\begin{split}
e^{itH}B_j[{\overline}{{Z}},{Z}] = {\mathcal{F}}^{-1} \int e^{it{\Omega}} B_j({\eta},{\xi}-{\eta}) {\check}{{Z}}^-({\eta}) {\check}{{Z}}^+({\xi}-{\eta}) d{\eta},
\end{split}$$]{} with $j=3,4$. The phase ${\Omega}$ and its derivatives are given by [$$\begin{split}
{&}{\Omega}:= H({\xi}) + H({\eta}) - H({\xi}-{\eta}),
{\\ &}{{\nabla}_{\xi}^{({\eta})}}{\Omega}= {\nabla}H({\xi}) - {\nabla}H({\xi}-{\eta}),
{\quad}{\nabla}_{\eta}{\Omega}= {\nabla}H({\eta}) + {\nabla}H({\xi}-{\eta}).
\end{split}$$]{} Recall our notation for the dyadic component [$$\begin{split}
{&}|{\xi}|\sim a,{\quad}|{\eta}|\sim b,{\quad}|{\zeta}|\sim c,
{\\ &}M=\max(a,b,c), {\quad}m=\min(a,b,c), {\quad}l=\min(b,c).
\end{split}$$]{} We also denote in this subsection [$$\begin{split}
{&}{\alpha}:= |{\widehat}{{\zeta}}-{\widehat}{{\xi}}|,
{\quad}{\beta}:= |{\widehat}{{\zeta}}+{\widehat}{{\eta}}|,
{\quad}{\eta}^\perp := -{\widehat}{\xi}\times{\widehat}{\xi}\times{\eta}.
\end{split}$$]{} We decompose the $({\xi},{\eta},{\zeta})$ region (${\xi}={\eta}+{\zeta}$) into the following five cases smoothly and exclusively. Namely, we exclude the earlier cases from the later cases.
1. $|{\eta}|\sim|{\xi}|\gg |{\zeta}|$; Temporally non-resonant.
2. ${\alpha}>\sqrt{3}$; Temporally non-resonant.
3. $|{\zeta}|{{\ \gtrsim \ }}1$; Spatially non-resonant.
4. $|{\eta}^\perp|\ll M|{\eta}|$; Temporally non-resonant.
5. Otherwise, spatially non-resonant.
Now we confirm the desired estimates in each case.
### $|{\eta}|\sim|{\xi}| \gg |{\zeta}|$, into $B^T$
We put those dyadic pieces with $\min(b,a)\ge 8c$ into $B_j^T$ in . In this case we need to show . Since ${\Omega}=H({\xi})+H({\eta})-H({\zeta})$ and $m=|{\zeta}|\ll M$, we have [$$\begin{split}
|{\Omega}| = {\Omega}{{\ \gtrsim \ }}H(M) \sim M{{{{\langle}M {\rangle}}}}.
\end{split}$$]{} We choose the $({\xi},{\zeta})$ coordinates to use the smallness of ${\zeta}$ region. Then [$$\begin{split}
{\nabla}_{\zeta}^k{\Omega}= -{\nabla}^k H({\zeta}) + {\nabla}^k H(-{\eta}),
{\quad}{\nabla}_{\zeta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}= -{\nabla}^{1+k} H({\zeta}),
\end{split}$$]{} with trivial bounds [$$\begin{split}
{&}|{\nabla}_{\zeta}{\Omega}| {{\ \lesssim \ }}{{{\langle}M {\rangle}}},
{\quad}|{\nabla}_{\zeta}^2{\Omega}| {{\ \lesssim \ }}{{{\langle}m {\rangle}}}/m,
{\\ &}|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}{{{\langle}M {\rangle}}},
{\quad}|{\nabla}_{\zeta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}{{{\langle}m {\rangle}}}/m^k.
\end{split}$$]{} Thus in this dyadic region we obtain [$$\begin{split}
\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}}{{\Omega}}B_j^{a,b,c}\right\|_{L^{\infty}_{\xi}\dot H^s_{\zeta}}
{{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}}{M{{{\langle}M {\rangle}}}}\frac{m^{3/2}}{m^s}\frac{M}{{{{\langle}M {\rangle}}}} = m^{3/2-s}{{{\langle}M {\rangle}}}^{-1}
\end{split}$$]{} for $j=3,4$ and $s=0,1,2$, which implies by interpolation.
In the remaining cases, we use the coordinates $({\xi},{\eta})$ since $M\sim|{\zeta}|$ and $l\sim|{\eta}|$.
### ${\alpha}>\sqrt{3}$, into $B^T$
More precisely, we cut-off the multipliers by [$$\begin{split} \label{cut al}
\chi_{[{\alpha}]} := \Gamma({\widehat}{\xi}-{\widehat}{\zeta}),
\end{split}$$]{} for a fixed $\Gamma \in C^{\infty}({\mathbb{R}}^3)$ satisfying $\Gamma(x)=1$ for $|x|\ge \sqrt{3}$ and $\Gamma(x)=0$ for $|x|\le 3/2$. Thus we have ${\alpha}>3/2$ in this case, and ${\alpha}<\sqrt{3}$ in the remaining case. By exclusion of the previous case, we have $M\sim|{\zeta}|$ also. Hence we have [$$\begin{split}
|{\nabla}_{\eta}^k \chi_{[{\alpha}]}| {{\ \lesssim \ }}c^{-k} \sim M^{-k},
\end{split}$$]{} for any $k\in{\mathbb{N}}$, which are acceptable errors in all of the following cases.
Now we show in this region. Since ${\alpha}>3/2$, we have $|{\eta}|-|{\zeta}|{{\ \gtrsim \ }}|{\xi}|$. Hence $m\sim|{\xi}|$ and [$$\begin{split}
|{\Omega}|={\Omega}\ge H({\eta})-H({\zeta}) {{\ \gtrsim \ }}{{{\langle}M {\rangle}}}|{\xi}| \sim {{{\langle}M {\rangle}}}m,
\end{split}$$]{} and from , [$$\begin{split}
{&}|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}\frac{M}{{{{\langle}M {\rangle}}}}|{\eta}| + {{{\langle}{\xi}{\rangle}}}{\alpha},
{\quad}|{\nabla}_{\eta}{\Omega}| {{\ \lesssim \ }}\frac{M}{{{{\langle}M {\rangle}}}}|{\xi}| + {{{\langle}{\eta}{\rangle}}}{\beta}.
\end{split}$$]{} Since ${\alpha}>3/2$, we have ${\beta}<1/2$ and so by the sine theorem ${\beta}\sim a{\alpha}/b \sim m/M$. Thus we obtain [$$\begin{split}
{&}|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}\frac{M^2}{{{{\langle}M {\rangle}}}} + {{{\langle}m {\rangle}}} {{\ \lesssim \ }}{{{\langle}M {\rangle}}},
{\quad}|{\nabla}_{\eta}{\Omega}| {{\ \lesssim \ }}\frac{Mm}{{{{\langle}M {\rangle}}}} + \frac{{{{\langle}M {\rangle}}}m}{M} {{\ \lesssim \ }}\frac{|{\Omega}|}{M},
{\\ &}|{\nabla}_{\eta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}| = |{\nabla}^{1+k}H({\zeta})| {{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}}{M^{k}}.
\end{split}$$]{} By , we have also [$$\begin{split}
|{\nabla}_{\eta}^2{\Omega}| {{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}m}{M^2} {{\ \lesssim \ }}\frac{|{\Omega}|}{M^2}.
\end{split}$$]{} Hence we have in this region [$$\begin{split}
\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}}{{\Omega}}\chi_{[{\alpha}]}B_j^{a,b,c}\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}}{{{{\langle}M {\rangle}}}m}\frac{M^{3/2}}{M^s} \frac{m}{{{{\langle}m {\rangle}}}} = \frac{M^{3/2-s}}{{{{\langle}m {\rangle}}}}\sim\frac{l^{3/2-s}}{{{{\langle}a {\rangle}}}},
\end{split}$$]{} for $j=3,4$, and for $s=0,1,2$. This yields by interpolation.
### $|{\zeta}|{{\ \gtrsim \ }}1$, into $B^X$
We put those remaining parts with $c\ge 1$ into $B^X$. By exclusion of the previous cases, we have $M\sim|{\zeta}|{{\ \gtrsim \ }}1$ and ${\alpha}<\sqrt{3}$. We want to show in this region. For the first derivatives, we have from , [$$\begin{split}
{&}|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| \sim ||{\zeta}|-|{\xi}|| + {{{\langle}{\xi}{\rangle}}}{\alpha},
{\quad}|{\nabla}_{\eta}{\Omega}| \sim ||{\zeta}|-|{\eta}|| + {{{\langle}{\eta}{\rangle}}}{\beta}.
\end{split}$$]{} On the other hand, we have [$$\begin{split} \label{cos exp}
{&}|{\xi}|^2 = ||{\zeta}|-|{\eta}||^2 + 2|{\zeta}||{\eta}|(1+{\widehat}{{\zeta}}\cdot{\widehat}{{\eta}})
\sim ||{\zeta}|-|{\eta}||^2 + |{\zeta}||{\eta}|{\beta}^2,
{\\ &}|{\eta}|^2 = ||{\zeta}|-|{\xi}||^2 + 2|{\zeta}||{\xi}|(1-{\widehat}{{\zeta}}\cdot{\widehat}{{\xi}})
\sim ||{\zeta}|-|{\xi}||^2 + |{\zeta}||{\xi}|{\alpha}^2.
\end{split}$$]{} Hence we have [$$\begin{split}
|{{\nabla}_{\xi}^{({\eta})}}{\Omega}|{{\ \lesssim \ }}|{\eta}|, {\quad}|{\nabla}_{\eta}{\Omega}| \sim ||{\zeta}|-|{\eta}||+{{{\langle}{\zeta}{\rangle}}}{\beta}{{\ \gtrsim \ }}|{\xi}|.
\end{split}$$]{} For the higher derivatives, we have from , [$$\begin{split}
{&}|{\nabla}_{\eta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}{{{\langle}{\zeta}{\rangle}}}|{\zeta}|^{-k} \sim M^{1-k},
{\quad}|{\nabla}_{\eta}^k{\Omega}| {{\ \lesssim \ }}|{\xi}||{\eta}|^{1-k} {{\ \lesssim \ }}|{\nabla}_{\eta}{\Omega}||{\eta}|^{1-k}.
\end{split}$$]{} Therefore we have in this region [$$\begin{split}
{&}\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\frac{b^{1+3/2}}{ab^s}U(a)
= l^{5/2-s}{{{\langle}a {\rangle}}}^{-1},
{\\ &}\left\|\frac{{\nabla}_{\eta}{{\nabla}_{\xi}^{({\eta})}}{\Omega}{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
+ \left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}({\nabla}_{\eta}{\Omega})^2{\nabla}_{\eta}^2{\Omega}}{|{\nabla}_{\eta}{\Omega}|^4}\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
{\\ &}+ \left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}{\nabla}_{\eta}[\chi_{[{\alpha}]}^C B_j]\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\frac{b^{3/2}}{ab^s}U(a)
= l^{3/2-s}{{{\langle}a {\rangle}}}^{-1},
\end{split}$$]{} for $j=3,4$ and $s=0,1,2$, where $\chi_{[{\alpha}]}^C=1-\chi_{[{\alpha}]}$ is excluding the previous case. The above estimates imply by interpolation.
### $|{\eta}^\perp|\ll M|{\eta}|$, into $B^T$
More precisely, we cut-off the multipliers by [$$\begin{split} \label{cut perp}
\chi_{[\perp]}:= \chi({{{\langle}M {\rangle}}}|{\eta}\times{\widehat}{{\xi}}|/(100Mb)),
\end{split}$$]{} with $\chi \in C^{\infty}_0({\mathbb{R}})$ satisfying $\chi(\mu)=1$ for $|\mu|\le 1$ and $\chi(\mu)=0$ for $|\mu|\ge 2$. Then we have [$$\begin{split}
|{\nabla}_{\eta}^k \chi_{[\perp]}| {{\ \lesssim \ }}({{{\langle}M {\rangle}}}/Mb)^{k},
\end{split}$$]{} which is supported around $|{\eta}^\perp|\sim Mb/{{{\langle}M {\rangle}}}$ for all $k\ge 1$. We have included ${{{\langle}M {\rangle}}}$ for later reuse, but in this subsection we can ignore it because $M\ll 1$. More precisely, we have by exclusion of the previous cases, [$$\begin{split}
1\gg M\sim |{\zeta}|, {\quad}{\alpha}<\sqrt{3}.
\end{split}$$]{} Now we prove in this region. First if $M\not=|{\zeta}|$, then ${\Omega}\ge H(m){{\ \gtrsim \ }}m$ and [$$\begin{split}
{&}|{\nabla}_{\eta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}\frac{|{\eta}|^{1-k}}{M}, {\quad}|{\nabla}_{\eta}^{k+1}{\Omega}| {{\ \lesssim \ }}\frac{|{\xi}|}{M|{\eta}|^k},
\end{split}$$]{} for $k\ge 0$, by . Hence we have in this region [$$\begin{split}
\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}}{{\Omega}}\chi_{[\perp]}\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\frac{b^{1+3/2}}{mM(Mb)^s}a
\sim l^{3/2}(Ml)^{-s},
\end{split}$$]{} for $j=3,4$ and $s=0,1,2$, where $\chi_{[{\alpha}]}^C$ is due to exclusion of the second case. Note that in this case the smallest divisor $(Mb)^s$ is from the cut-off $\chi_{[\perp]}$. The above estimate implies by interpolation.
Next we consider the main case $M=|{\zeta}|$. Let [$$\begin{split}
{\lambda}:= |{\xi}|+|{\eta}|-|{\zeta}|.
\end{split}$$]{} Then by elementary geometry, we have [$$\begin{split}
{&}|{\eta}^\perp| \sim |{\eta}|({\alpha}+{\beta}),
{\quad}{\lambda}\sim m({\alpha}^2+{\beta}^2) \sim m(|{\eta}^\perp|/|{\eta}|)^2.
\end{split}$$]{} Hence the assumption in this case implies that ${\lambda}\ll M^2m$. Now we use the small but non-zero curvature of $H({\xi})$ in the radial direction around ${\xi}\sim 0$, to get a lower bound on ${\Omega}$ (cf. [@vac2 (4.52)]): [$$\begin{split} \label{est degH''}
{&}-{\Omega}= [H(|{\xi}|+|{\eta}|)-H({\xi})-H({\eta})] + [H({\xi}-{\eta})-H(|{\xi}|+|{\eta}|)],
{\\ &}|H({\xi}-{\eta})-H(|{\xi}|+|{\eta}|)| {{\ \lesssim \ }}{{{\langle}M {\rangle}}}{\lambda},
{\\ &}H(|{\xi}|+|{\eta}|)-H({\xi})-H({\eta}) \sim \frac{|{\xi}||{\eta}|(|{\xi}|+|{\eta}|)}{{{{\langle}{\xi}{\rangle}}}+{{{\langle}{\eta}{\rangle}}}}.
\end{split}$$]{} Thus we obtain [$$\begin{split}
|{\Omega}| \sim M^2 m.
\end{split}$$]{} For the first derivatives we have [$$\begin{split}
{&}|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}M|{\eta}| + {\alpha}\ll |{\eta}|,
{\quad}|{\nabla}_{\eta}{\Omega}| {{\ \lesssim \ }}M|{\xi}| + {\beta}\ll |{\xi}|,
\end{split}$$]{} since [$$\begin{split}
|{\xi}|{\alpha}\sim |{\eta}|{\beta}\sim m({\alpha}+{\beta}) \ll mM \sim |{\xi}||{\eta}|.
\end{split}$$]{} For the higher derivatives we have from , [$$\begin{split} \label{hider1}
|{\nabla}_{\eta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}M^{-k},
{\quad}|{\nabla}_{\eta}^{1+k}{\Omega}| {{\ \lesssim \ }}\frac{|{\xi}|}{M|{\eta}|^k}.
\end{split}$$]{} Using the volume bound $M^2b^3$ as well, we obtain in this case [$$\begin{split}
\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}}{{\Omega}}\chi_{[\perp]}\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\frac{b Mb^{3/2}}{M^2m(Mb)^s}a
\sim l^{3/2}(Ml)^{-s},
\end{split}$$]{} for $j=3,4$ and $s=0,1,2$, which implies by interpolation.
### Otherwise, into $B^X$
By excluding all of the previous cases, we have now [$$\begin{split}
1\gg M\sim|{\zeta}|,{\quad}{\alpha}<\sqrt{3}, {\quad}|{\eta}^\perp|{{\ \gtrsim \ }}M|{\eta}|.
\end{split}$$]{} We want to prove in this region. For the first derivatives, we have from , [$$\begin{split}
|{\nabla}_{\eta}{\Omega}| \sim M||{\zeta}|-|{\eta}||+{\beta},
{\quad}|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| \sim M||{\zeta}|-|{\xi}||+{\alpha}.
\end{split}$$]{} For the angular part, we have by the sine theorem [$$\begin{split}
{\alpha}/{\beta}{{\ \lesssim \ }}|{\eta}|/|{\xi}|,
\end{split}$$]{} since ${\alpha}$ is away from $2$. For the radial part, we have from [$$\begin{split}
M||{\zeta}|-|{\xi}||{{\ \lesssim \ }}M|{\eta}|, {\quad}|{\nabla}_{\eta}{\Omega}| {{\ \gtrsim \ }}M|{\xi}|.
\end{split}$$]{} Thus we obtain [$$\begin{split}
|{{\nabla}_{\xi}^{({\eta})}}{\Omega}|/|{\nabla}_{\eta}{\Omega}| {{\ \lesssim \ }}|{\eta}|/|{\xi}|.
\end{split}$$]{} Using for the higher derivatives, we get [$$\begin{split}
{&}\frac{|{\nabla}_{\eta}^k{\Omega}|}{|{\nabla}_{\eta}{\Omega}|} {{\ \lesssim \ }}\frac{|{\xi}|}{M|{\eta}|^{k-1}{\beta}}
{{\ \lesssim \ }}\frac{1}{|{\eta}|^{k-2}|{\eta}^\perp|},
{\quad}\frac{|{\nabla}_{\eta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}|}{|{\nabla}_{\eta}{\Omega}|} {{\ \lesssim \ }}\frac{1}{M^k{\beta}}
{{\ \lesssim \ }}\frac{|{\eta}|}{M^{k-1}|{\xi}||{\eta}^\perp|},
\end{split}$$]{} where we used [$$\begin{split}
|{\xi}||{\eta}^\perp| {{\ \lesssim \ }}|{\zeta}||{\eta}|{\beta}.
\end{split}$$]{} Now we apply the dyadic decomposition to [$$\begin{split}
|{\eta}^\perp|\sim \mu\in\{k\in 2^{\mathbb{Z}}\mid k\ge Mb\}.
\end{split}$$]{} Getting a factor $\mu|{\eta}|^{1/2}$ from the $L^2_{\eta}$ volume of each dyadic piece, we have [$$\begin{split}
{&}\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}\chi_{[\perp]}^C\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\sum_{\mu \ge Mb} \frac{b\mu b^{1/2}}{a \mu^{s}}a
{{\ \lesssim \ }}l^{3/2}(Ml)^{1-s},
{\\ &}\left\|\frac{{\nabla}_{\eta}{{\nabla}_{\xi}^{({\eta})}}{\Omega}{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}\chi_{[\perp]}^C\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
+ \left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}({\nabla}_{\eta}{\Omega})^2{\nabla}_{\eta}^2{\Omega}}{|{\nabla}_{\eta}{\Omega}|^4}\chi_{[\perp]}^C\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
{\\ &}+ \left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}{\nabla}_{\eta}[\chi_{[\perp]}^C\chi_{[{\alpha}]}^C B_j]\right\|_{\dot H^s_{\eta}} {{\ \lesssim \ }}\sum_{\mu \ge Mb} \frac{b\mu b^{1/2}}{a\mu^{1+s}}a
{{\ \lesssim \ }}l^{3/2}(Ml)^{-s},
\end{split}$$]{} for $j=3,4$ and $1<s\le 2$, by using interpolation for each dyadic piece in $\mu$. The condition $s>1$ is only to have convergence of the sum in $\mu$. As before, $\chi_{[{\alpha}]}^C=1-\chi_{[{\alpha}]}$ and $\chi_{[\perp]}^C=1-\chi_{[\perp]}$ are excluding the previous cases. The above estimates imply .
Multiplier estimates for ${Z}{Z}$ {#ss:zz}
---------------------------------
We consider the bilinear terms of the form [$$\begin{split}
e^{itH}B_j[{Z},{Z}] = {\mathcal{F}}^{-1}\int e^{it{\Omega}}B_j({\eta},{\xi}-{\eta}) {\check}{{Z}}^+({\eta}) {\check}{{Z}}^+({\xi}-{\eta}) d{\eta},
\end{split}$$]{} with $j=3,4$, where the phase is given by [$$\begin{split}
{&}{\Omega}= H({\xi}) - H({\eta}) - H({\xi}-{\eta}),
{\\ &}{{\nabla}_{\xi}^{({\eta})}}{\Omega}= {\nabla}H({\xi}) - {\nabla}H({\xi}-{\eta}),
{\quad}{\nabla}_{\eta}{\Omega}= - {\nabla}H({\eta}) + {\nabla}H({\xi}-{\eta}).
\end{split}$$]{} Recall that $|{\xi}|\sim a$, $|{\eta}|\sim b$, $|{\zeta}|\sim c$, $M=\max(a,b,c)$, $m=\min(a,b,c)$ and $l=\min(b,c)$. By symmetry, we may assume without loss of generality that [$$\begin{split}
b \le c,
\end{split}$$]{} in this subsection. We also denote in this subsection [$$\begin{split}
{&}{\alpha}:= |{\widehat}{{\xi}}-{\widehat}{{\zeta}}|,
{\quad}{\beta}' := |{\widehat}{{\eta}}-{\widehat}{{\zeta}}|,
{\quad}{\gamma}:= |{\widehat}{{\xi}}-{\widehat}{{\eta}}|,
{\quad}{\eta}^\perp := -{\widehat}{\xi}\times{\widehat}{\xi}\times{\eta}.
\end{split}$$]{} We decompose the $({\xi},{\eta},{\zeta})$ region into the following four cases exclusively.
1. $|{\eta}|\sim|{\zeta}|\gg|{\xi}|$; Temporally non-resonant.
2. ${\alpha}>\sqrt{3}$; Temporally non-resonant.
3. $|{\eta}^\perp|\ll M|{\eta}|/{{{\langle}M {\rangle}}}$; Temporally non-resonant.
4. Otherwise, spatially non-resonant.
### $|{\eta}|\sim|{\zeta}|\gg|{\xi}|$, into $B^T$
We put the dyadic pieces with $\min(b,c)\ge 4a$ into $B^T$. Then we have $M\sim |{\zeta}|\sim|{\eta}|\gg|{\xi}|= m$ and [$$\begin{split}
{&}|{\Omega}| \ge H({\eta}) \sim M{{{\langle}M {\rangle}}},
{\quad}|{\nabla}_{\eta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}{{{\langle}M {\rangle}}}M^{-k}, {\quad}|{\nabla}_{\eta}{\Omega}| {{\ \lesssim \ }}{{{\langle}M {\rangle}}}M^{-k},
\end{split}$$]{} such that in this case [$$\begin{split}
\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}}{{\Omega}}B_j^{a,b,c}\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}}{M{{{\langle}M {\rangle}}}}\frac{M^{3/2}}{M^s}\frac{m}{{{{\langle}m {\rangle}}}}
\sim l^{3/2-s}\frac{m}{M}{{{\langle}a {\rangle}}}^{-1},
\end{split}$$]{} for $j=3,4$ and $s=0,1,2$. By interpolation, this is better than .
### ${\alpha}>\sqrt{3}$, into $B^T$
We use the same cut-off $\chi_{[{\alpha}]}$ as defined in , so that we have ${\alpha}>3/2$ in this case, and ${\alpha}<\sqrt{3}$ in the remaining case. Then in this region we have $M=|{\eta}|\sim|{\zeta}|\sim|{\xi}|$, and [$$\begin{split}
|{\Omega}| \ge H({\zeta}) {{\ \gtrsim \ }}M{{{\langle}M {\rangle}}},
\end{split}$$]{} so that we end up with the same bound as in the previous case.
In the remaining cases, ${\alpha}<\sqrt{3}$ and $|{\eta}|{{\ \lesssim \ }}|{\zeta}|$, hence we have [$$\begin{split}
{\alpha}{{\ \lesssim \ }}{\gamma}\sim {\beta}'.
\end{split}$$]{}
### $|{\eta}^\perp|\ll M|{\eta}|/{{{\langle}M {\rangle}}}$, into $B^T$
More precisely, we decompose the multipliers by the cut-off $\chi_{[\perp]}$ defined in . By exclusion of the previous cases, we have now [$$\begin{split}
M\sim|{\xi}|\sim|{\zeta}|{{\ \gtrsim \ }}|{\eta}|\sim m.
\end{split}$$]{} For the higher derivatives, we have from [$$\begin{split} \label{hider2}
{&}|{\nabla}_{\eta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}}{M^k},
{\quad}|{\nabla}_{\eta}^{1+k}{\Omega}| {{\ \lesssim \ }}\frac{{{{\langle}m {\rangle}}}}{m^k}.
\end{split}$$]{} If $|{\zeta}|-|{\xi}|{{\ \gtrsim \ }}|{\eta}|$, then we have [$$\begin{split}
{&}|{\Omega}|\ge H({\zeta})-H({\xi}) {{\ \gtrsim \ }}{{{\langle}M {\rangle}}}m,
{\quad}|{\nabla}_{\xi}{\Omega}| {{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}}{M}m, {\quad}|{\nabla}_{\eta}{\Omega}| {{\ \lesssim \ }}{{{\langle}M {\rangle}}},
\end{split}$$]{} and so [$$\begin{split}
\left\|\frac{{\nabla}_{\xi}{\Omega}}{{\Omega}}\chi_{[\perp]}\chi_{[{\alpha}]}^CB_j\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\frac{m^{3/2}}{M} \left(\frac{{{{\langle}M {\rangle}}}}{Mm}\right)^{s} \frac{M}{{{{\langle}M {\rangle}}}}
= \frac{m^{3/2}}{{{{\langle}M {\rangle}}}}\left(\frac{{{{\langle}M {\rangle}}}}{Mm}\right)^{s},
\end{split}$$]{} for $j=3,4$ and $s=0,1,2$, where $\chi_{[{\alpha}]}^C=1-\chi_{[{\alpha}]}$ eliminates the previous case. By interpolation, it gives .
In the main case $|{\zeta}|-|{\xi}|\ll|{\eta}|$, let ${\lambda}' := |{\zeta}|+|{\eta}|-|{\xi}|$. Since ${\alpha}{{\ \lesssim \ }}{\gamma}$ and both are away from $2$, we have [$$\begin{split}
{\lambda}' \sim m({\alpha}^2+{\gamma}^2) \sim m{\gamma}^2 \sim m(|{\eta}^\perp|/|{\eta}|)^2 \ll \frac{M^2}{{{{\langle}M {\rangle}}}^2}m.
\end{split}$$]{} Then by the same argument as in , we get [$$\begin{split}
|{\Omega}| \sim \frac{M^2}{{{{\langle}M {\rangle}}}}m.
\end{split}$$]{} For the first derivatives, we have from , [$$\begin{split} \label{est first der}
{&}|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| \sim \frac{M}{{{{\langle}M {\rangle}}}}||{\zeta}|-|{\xi}|| + {{{\langle}M {\rangle}}}{\alpha},
{\quad}|{\nabla}_{\eta}{\Omega}| \sim \frac{M}{{{{\langle}M {\rangle}}}}||{\zeta}|-|{\eta}|| + {{{\langle}m {\rangle}}}{\beta}'.
\end{split}$$]{} Combining them with [$$\begin{split}
{\alpha}{{\ \lesssim \ }}\frac{m}{M}{\gamma}, {\quad}{\beta}'\sim{\gamma}\sim\frac{|{\eta}^\perp|}{|{\eta}|} \ll \frac{M}{{{{\langle}M {\rangle}}}},
\end{split}$$]{} we get [$$\begin{split}
|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}m, {\quad}|{\nabla}_{\eta}{\Omega}| {{\ \lesssim \ }}M.
\end{split}$$]{} Using the volume bound $(M/{{{\langle}M {\rangle}}})^2m^3$ as well, we obtain in this region [$$\begin{split}
\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}}{{\Omega}}\chi_{[\perp]}\chi_{[{\alpha}]}^CB_j\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}}{M^2} \frac{Mm^{3/2}}{{{{\langle}M {\rangle}}}}\left(\frac{{{{\langle}M {\rangle}}}}{Mm}\right)^s \frac{M}{{{{\langle}M {\rangle}}}}
= \frac{m^{3/2}}{{{{\langle}M {\rangle}}}}\left(\frac{{{{\langle}M {\rangle}}}}{Mm}\right)^{s},
\end{split}$$]{} for $j=3,4$ and $s=0,1,2$, which gives by interpolation.
### Otherwise, into $B^X$ {#mest zz3}
In the remaining case, we have [$$\begin{split}
{&}M\sim|{\xi}|\sim|{\zeta}|{{\ \gtrsim \ }}|{\eta}|\sim m,
{\quad}|{\eta}^\perp|{{\ \gtrsim \ }}Mm/{{{\langle}M {\rangle}}}.
\end{split}$$]{} Using ${\alpha}{{\ \lesssim \ }}{\gamma}\sim{\beta}'$ also, we have [$$\begin{split}
{&}\frac{{\alpha}}{{\beta}'}\sim\frac{{\alpha}}{{\gamma}}{{\ \lesssim \ }}\frac{|{\eta}|}{|{\zeta}|} \sim \frac{m}{M},
{\quad}\frac{M}{{{{\langle}M {\rangle}}}}{{\ \lesssim \ }}\frac{|{\eta}^\perp|}{|{\eta}|} {{\ \lesssim \ }}{\gamma}\sim{\beta}',
\end{split}$$]{} and in particular, ${\gamma}{{\ \gtrsim \ }}1$ if $M{{\ \gtrsim \ }}1$. In that case, we have from and , [$$\begin{split}
{&}|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| \sim ||{\zeta}|-|{\xi}||+M{\alpha}\sim |{\eta}|,
{\quad}|{\nabla}_{\eta}{\Omega}| {{\ \gtrsim \ }}{{{\langle}m {\rangle}}}.
\end{split}$$]{} If $M{{\ \lesssim \ }}1$, we have [$$\begin{split}
|{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}Mm + \frac{m}{M}{\beta}' \sim \frac{m}{M}|{\nabla}_{\eta}{\Omega}|.
\end{split}$$]{} Thus in both cases we have [$$\begin{split}
\frac{|{{\nabla}_{\xi}^{({\eta})}}{\Omega}|}{|{\nabla}_{\eta}{\Omega}|} {{\ \lesssim \ }}\frac{m{{{\langle}M {\rangle}}}}{M{{{\langle}m {\rangle}}}}.
\end{split}$$]{} Using for the higher derivatives, we get [$$\begin{split}
\frac{|{\nabla}_{\eta}^{1+k}{\Omega}|}{|{\nabla}_{\eta}{\Omega}|} {{\ \lesssim \ }}\frac{1}{m^k{\beta}'}
\sim \frac{1}{|{\eta}|^k{\gamma}} {{\ \lesssim \ }}\frac{1}{m^{k-1}|{\eta}^\perp|},
{\quad}\frac{|{\nabla}_{\eta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}|}{|{\nabla}_{\eta}{\Omega}|}
{{\ \lesssim \ }}\frac{m{{{\langle}M {\rangle}}}}{M^k{{{\langle}m {\rangle}}}|{\eta}^\perp|}.
\end{split}$$]{} Thus in this region we obtain [$$\begin{split}
{&}\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}\chi_{[\perp]}^C\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\sum_{\mu{{\ \gtrsim \ }}\mu_0}\frac{m{{{\langle}M {\rangle}}} \mu m^{1/2}}{M{{{\langle}m {\rangle}}}\mu^{s}}\frac{M}{{{{\langle}M {\rangle}}}}
{{\ \lesssim \ }}\mu_0^{1-s}\frac{m^{3/2}}{{{{\langle}m {\rangle}}}},
{\\ &}\left\|\frac{{\nabla}_{\eta}{{\nabla}_{\xi}^{({\eta})}}{\Omega}{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}\chi_{[\perp]}^C\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
+ \left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}({\nabla}_{\eta}{\Omega})^2{\nabla}_{\eta}^2{\Omega}}{|{\nabla}_{\eta}{\Omega}|^4}\chi_{[\perp]}^C\chi_{[{\alpha}]}^C B_j\right\|_{\dot H^s_{\eta}}
{\\ &}+ \left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}{\nabla}_{\eta}{\Omega}}{|{\nabla}_{\eta}{\Omega}|^2}{\nabla}_{\eta}[\chi_{[\perp]}^C\chi_{[{\alpha}]}^C B_j]\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\sum_{\mu{{\ \gtrsim \ }}\mu_0}\frac{m{{{\langle}M {\rangle}}} \mu m^{1/2}}{M{{{\langle}m {\rangle}}}\mu^{1+s}}\frac{M}{{{{\langle}M {\rangle}}}}
{{\ \lesssim \ }}\mu_0^{-s}\frac{m^{3/2}}{{{{\langle}m {\rangle}}}},
\end{split}$$]{} for $j=3,4$ and $1<s\le 2$ by interpolation for each dyadic piece of $\mu$, where $\mu_0:=Mm/{{{\langle}M {\rangle}}}$, the sum for $\mu$ is on $2^{\mathbb{Z}}$, and $\chi_{[\perp]}^C$ and $\chi_{[{\alpha}]}^C$ are removing the previous regions. The above estimates imply and .
Multiplier estimates for ${\overline}{{Z}}{\overline}{{Z}}$
-----------------------------------------------------------
For the bilinear terms of the form [$$\begin{split}
e^{itH}B_j[{\overline}{{Z}},{\overline}{{Z}}] = \int e^{it{\Omega}}B_j({\eta},{\xi}-{\eta}) {\check}{{Z}}^-({\eta}) {\check}{{Z}}^-({\xi}-{\eta}) d{\eta},
\end{split}$$]{} we need not decompose, since it is always temporally non-resonant. Indeed we have [$$\begin{split}
{&}{\Omega}= H({\xi})+H({\eta})+H({\xi}-{\eta}) {{\ \gtrsim \ }}M{{{\langle}M {\rangle}}}.
\end{split}$$]{} By symmetry, we may assume that $|{\eta}|{{\ \lesssim \ }}|{\zeta}|$. Then [$$\begin{split}
{&}|{{\nabla}_{\xi}^{({\eta})}}{\Omega}|+|{\nabla}_{\eta}{\Omega}| {{\ \lesssim \ }}{{{\langle}M {\rangle}}},
{\quad}|{\nabla}_{\eta}^k{{\nabla}_{\xi}^{({\eta})}}{\Omega}| {{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}}{M^k},
{\quad}|{\nabla}_{\eta}^{1+k}{\Omega}| {{\ \lesssim \ }}\frac{{{{\langle}{\eta}{\rangle}}}}{|{\eta}|^k},
\end{split}$$]{} hence we get [$$\begin{split}
\left\|\frac{{{\nabla}_{\xi}^{({\eta})}}{\Omega}}{{\Omega}}B_j\right\|_{\dot H^s_{\eta}}
{{\ \lesssim \ }}\frac{{{{\langle}M {\rangle}}}b^{3/2}}{M{{{\langle}M {\rangle}}}b^s} U(a)
{{\ \lesssim \ }}l^{3/2-s}{{{\langle}M {\rangle}}}^{-1},
\end{split}$$]{} for $j=3,4$ and $s=0,1,2$, which gives via interpolation.
Estimates with phase derivative in cubic terms
==============================================
In this section we estimate the phase derivatives in the cubic terms $C_j$, $1\le j\le 5$. The higher order terms $Q_j(u)$ have been already estimated in , by using regularity gain. Some parts in the frequency interactions of $C_j$ can be also estimated in a similar way, but if we thereby estimate the whole interactions, we can not avoid losing regularity by the factor $s{\nabla}_{\xi}{\Omega}$.
On the other hand, the cubic terms have in general a large set of interactions which are resonant both in space and time. Hence we cannot always integrate by parts as for the bilinear terms.
Fortunately, it turns out that those two difficulties are disjoint in the Fourier space. Namely, when the derivative loss by ${\nabla}_{\xi}{\Omega}$ becomes really essential, the interaction is not temporally resonant, i.e., $|{\Omega}|$ is even bigger than $|{\nabla}_{\xi}{\Omega}|$. Hence we integrate on the phase in $s$ for those parts, and the other parts are bounded in the form ${\nabla}_{\xi}{\Omega}$. Here we do not need such precise estimates on the resulting multipliers as in the bilinear case, since the divisor appear only when $|{\xi}|\gg 1$.
Before applying $J$ to the cubic terms $C_j$, we apply the frequency decomposition to each $v$ as in : [$$\begin{split}
v = \sum_{k\in 2^{\mathbb{Z}}} v_k, {\quad}v_k := \chi^k({\nabla})v,
\end{split}$$]{} Thus in the Fourier space we get terms like [$$\begin{split} \label{cubic with phase}
\sum_{k_1,k_2,k_3\in 2^{\mathbb{Z}}} \int_0^t s {\nabla}_{\xi}{\Omega}e^{isH} C_j[v^\pm_{k_1},v^\pm_{k_2},v^\pm_{k_3}] ds,
\end{split}$$]{} where [$$\begin{split}
{\Omega}= H({\xi}) \mp H({\xi}_1) \mp H({\xi}_2) \mp H({\xi}_3),{\quad}{\nabla}_{\xi}{\Omega}= {\nabla}H({\xi}) \mp {\nabla}H({\xi}_1),
\end{split}$$]{} and one of $v$ has to be replaced with ${Z}$ if $j=5$. We consider the summand with $k_1\ge\max(k_2,k_3)$, choosing the smaller ${\xi}_2,{\xi}_3$ as integral variables. The other cases are treated in the same way.
If $k_1{{\ \lesssim \ }}1$, then all $|{\nabla}H({\xi}_k)|$ are bounded, hence we can bound it in $L^{\infty}H^1$ by [$$\begin{split}
\|tC_j[v^\pm_{k_1},v^\pm_{k_2},v^\pm_{k_3}]\|_{L^2 L^{6/5}}
{{\ \lesssim \ }}\|tv\|_{L^{\infty}L^6} \|U^{-1}v\|_{L^{\infty}L^2} \|U^{-1}v\|_{L^2 L^6}.
\end{split}$$]{} If $1\ll k_1\sim\max(k_2,k_3)$, then we have $|{\nabla}_{\xi}{\Omega}|{{\ \lesssim \ }}\max(k_2,k_3)$. Hence the $L^{\infty}H^1$ norm is bounded by [$$\begin{split} \label{C hhl}
\|k_1^2 tC_j[v_{k_1}^\pm,v_{k_2}^\pm,v_{k_3}^\pm]\|_{L^2 L^{6/5}}
{{\ \lesssim \ }}\|tv\|_{L^{\infty}H^{1,6}} \|U^{-1}v\|_{L^{\infty}H^1} \|U^{-1}v\|_{L^2 H^{1,6}}.
\end{split}$$]{} In both cases we get $T^{-1/10}$ for $t>T$ from the $L^6_x$ decay of $U^{-1}v$. If $j=5$, we just replace one of $v$ by ${Z}$.
It remains to estimate the terms with $k_1\gg\max(1,k_2,k_3)$. If the sign in the phase is ${\Omega}=H({\xi})-H({\xi}_1)\pm\cdots$, then we have [$$\begin{split}
{&}{\nabla}_{\xi}{\Omega}= F({\xi},{\xi}_2+{\xi}_3)\cdot({\xi}_2+{\xi}_3),
{\quad}F({\xi},{\eta}):=\int_0^1 {\nabla}^2 H({\xi}-{\theta}{\eta})d{\theta},
\end{split}$$]{} getting the factor ${\xi}_2+{\xi}_3$. In the region $|{\xi}|\gg{{{\langle}{\eta}{\rangle}}}$, we have [$$\begin{split}
|{\nabla}_{\xi}^k {\nabla}_{\eta}^l F({\xi},{\eta})| {{\ \lesssim \ }}\int_0^1 |{\nabla}^{2+k+l}H({\xi}-{\theta}{\eta})| d{\theta}{{\ \lesssim \ }}|{\xi}|^{-k-l}.
\end{split}$$]{} Hence the Coifman-Meyer estimate applies to $F$, and so we can bound the Strichartz norm as in the previous case.
Finally, if ${\Omega}=H({\xi})+H({\xi}_1)\pm\cdots$ with $k_1\gg \max(1,k_2,k_3)$, we have [$$\begin{split}
|{\Omega}| {{\ \gtrsim \ }}k_1^2, {\quad}|{\nabla}_{\xi}{\Omega}| {{\ \lesssim \ }}k_1,
\end{split}$$]{} so that we can integrate on $e^{is{\Omega}}$ in $s$. Thus we get terms like [$$\begin{split} \label{IBP C}
\int_0^t \frac{{\nabla}_{\xi}{\Omega}}{{\Omega}} e^{isH}\left\{C[v,v,v]+sC[{\mathcal{N}_v},v,v]\right\} ds
+ \left[\frac{{\nabla}_{\xi}{\Omega}}{{\Omega}}e^{isH}sC[v,v,v]\right]_0^t,
\end{split}$$]{} where we omitted indexes $j$, $k_*$ and $\pm$. Moreover if $j=5$, one of $v$ (or ${\mathcal{N}_v}$) should be replaced with ${Z}$ (resp. ${\mathcal{N}_{Z}}$). Recall that ${\mathcal{N}_v}$ and ${\mathcal{N}_{Z}}$ are the nonlinearities defined in and , respectively. The multipliers have no singularity thanks to the frequency restrictions. Hence using the Coifman-Meyer estimate , we can bound the above term in $L^{\infty}H^1$ by [$$\begin{split}
{&}\|C[v,v,v]\|_{L^2 L^{6/5}} + \|tC[{\mathcal{N}_v},v,v]\|_{L^2 L^{6/5} + L^{4/3} L^{3/2}}
+ \|tC[v,v,v]\|_{L^{\infty}L^2}
{\\ &}{{\ \lesssim \ }}\|v\|_{L^2 L^6}\|U^{-1}v\|_{L^{\infty}L^3}^2
+ (\|{\mathcal{N}_v}\|_{L^2 L^2 + L^{4/3} L^3}+\|v\|_{L^{\infty}L^6})\|t^{1/2}U^{-1}v\|_{L^{\infty}L^6}^2,
\end{split}$$]{} where the norm on ${\mathcal{N}_v}$ is bounded by $L^2 H^{1,6/5}+L^{4/3} H^{1,3/2}$. For ${\mathcal{N}_{Z}}$, it was already estimated in Section \[ss:est w/o w\], and the same estimates and hold for ${\mathcal{N}_v}$ as well. For $t>T$, we gain $T^{-1/5}$ from the $L^6_x$ and $L^3_x$ decay of $U^{-1}v$ (or $U^{-1}{Z}$).
Proof of the initial data result
================================
Now that we have closed estimates in $X$ for $(v,{Z})$ solving the system , we can easily finish the proof for Theorem \[thm:init\]. More precisely, we have obtained the following estimates: For any interval $I=(T_0,T_1)\subset{\mathbb{R}}$, and for any $(v,z)\in C(I;X)$ satisfying and on $I$, we have [$$\begin{split}
{&}\|v(t)-{Z}(t)\|_{X(t)} {{\ \lesssim \ }}{{{\langle}t {\rangle}}}^{-1/6}\|v(t)\|_{X(t)}^2{\quad}(t\in I),
{\\ &}\|v-{Z}\|_{S(I)} {{\ \lesssim \ }}{{{\langle}T_0 {\rangle}}}^{-7/10}\|v\|_{X(I)}^2,
{\\ &}\|{Z}-e^{-i(t-T_0)H}{Z}(T_0)\|_{S(I)} {{\ \lesssim \ }}{{{\langle}T_0 {\rangle}}}^{-1/2}\left\{\|v\|_{(X\cap S)(I)}^2 + \|v\|_{(X\cap S)(I)}^4 \right\},
{\\ &}\|{Z}-e^{-i(t-T_0)H}{Z}(T_0)\|_{X(I)}
{\\ &}{{\ \lesssim \ }}{{{\langle}T_0 {\rangle}}}^{-{\varepsilon}}\left\{\|v\|_{(X\cap S)(I)}^2 + \|v\|_{(X\cap S)(I)}^6
+ \|{Z}\|_{(X\cap S)(I)}^2\right\},
\end{split}$$]{} for some small ${\varepsilon}>0$, where the highest order $6$ comes from the partial integration replacing one of $v$ in $C_j$ by the quartic term $Q_1(u)$.
There are at least two ways to procede: (1) construct global solutions with the desired properties by an iteration argument, or (2) use those estimates as a priori bounds for the global solutions already known in $H^1$.
Since the equations for the iteration sequence are a bit complicated due to the partial integration and the nonlinear transforms, we employ the latter (2) approach. Then we just need to ensure that the global solutions from our initial data stay in $X$ so that the a priori bounds can be rigorously applied. Thanks to the global wellposedness [@BS1] in $\psi\in C({\mathbb{R}};1+H^1({\mathbb{R}}^3))$ for , it suffices to prove that only for a subset of initial data which is dense with respect to the $H^1$ norm for $u$.
It is quite standard to see that for any $u(0)\in H^2\cap{{{\langle}x {\rangle}}}^{-1}H^1$, the unique global solution of stays in the same space: since we already have it in $L^{\infty}H^1$, we just apply the fixed point argument to the following estimates on $(0,T)$ [$$\begin{split}
{&}\|u\|_{L^{\infty}H^2} {{\ \lesssim \ }}\|{\mathcal{N}_u}\|_{L^2 H^{2,6/5}+L^1 H^2}
{{\ \lesssim \ }}(T^{1/2}+T)\|u\|_{L^{\infty}H^2}(1+\|u\|_{L^{\infty}H^1})^2,
{\\ &}\|(x+2it{\nabla})u\|_{H^1} {{\ \lesssim \ }}\|(x+2it{\nabla}){\mathcal{N}_u}\|_{L^2 H^{1,6/5}+L^1 H^1}
{\\ &\quad}{{\ \lesssim \ }}(T^{1/2}+T^2)(\|xu\|_{L^{\infty}H^1}+\|u\|_{L^{\infty}H^2})(1+\|u\|_{L^{\infty}H^1})^2,
\end{split}$$]{} where we denote the nonlinearity in the equation by [$$\begin{split}
{\mathcal{N}_u}:= u^2+2|u|^2+|u|^2u.
\end{split}$$]{} The above estimates follow from Strichartz applied to , and the identity $x+2it{\nabla}=e^{it{\Delta}}xe^{-it{\Delta}}$. Thus we get the solution locally in this space, and then extend it globally by Gronwall.
To those solutions $u\in H^2\cap{{{\langle}x {\rangle}}}^{-1}H^1$ satisfying with $v=u_1+iUu_2$ and ${Z}=v+b(u)$, we can apply the above a priori estimates on any interval, thereby getting global bounds [$$\begin{split}
\|v\|_{X\cap S} + \|{Z}\|_{X\cap S} {{\ \lesssim \ }}\|v(0)\|_{X(0)} {{\ \lesssim \ }}{\delta},
\end{split}$$]{} provided ${\delta}$ is sufficiently small. By a density argument, all solutions satisfying our initial assumption can be obtained as limits of such solutions. Hence the above global bounds are transferred to them, and we get the desired scattering for them, using the identities [$$\begin{split}
{&}\|e^{itH}{Z}(t)-e^{iT_0H}{Z}(T_0)\|_{H^1} = \|{Z}(t)-e^{-i(t-T_0)H}{Z}(T_0)\|_{H^1},
{\\ &}\left\|x\left\{e^{itH}{Z}(t)-e^{iT_0H}{Z}(T_0)\right\}\right\|_{H^1}
= \left\|J\left\{{Z}(t)-e^{-i(t-T_0)H}{Z}(T_0)\right\}\right\|_{H^1}.
\end{split}$$]{}
Denote the map $W:v(0)\mapsto v_+$ thereby obtained (the inverse wave operator) locally around $0$ in ${{{\langle}x {\rangle}}}^{-1}H^1$. The continuity and the invertibility of $W$ are also proved in a standard way. The above estimates are adapted directly to differences of solutions, since after writing the system for $(v,{Z})$, we did not use any genuinely nonlinear structure, but only multi-linear estimates. Let $(v^{(j)},{Z}^{(j)})$ be small solutions of obtained above, and let $f^{(j)}_T=e^{-i(t-T)H}{Z}^{(j)}(T)$, for $j=1,2$ and $T\ge 0$. Then we have [$$\begin{split}
{&}\|{Z}^{(1)}-{Z}^{(2)}-(f^{(1)}_T-f^{(2)}_T)\|_{X\cap S}
{\\ &\quad}{{\ \lesssim \ }}\sum_{j=1}^2(\|v^{(j)}\|_{X\cap S}+\|{Z}^{(j)}\|_{X\cap S})
{}(\|v^{(1)}-v^{(2)}\|_{X\cap S}+\|{Z}^{(1)}-{Z}^{(2)}\|_{X\cap S}),
\end{split}$$]{} and also [$$\begin{split}
\|v^{(1)}-v^{(2)}-({Z}^{(1)}-{Z}^{(2)})\|_{X\cap S}
{{\ \lesssim \ }}\sum_{j=1}^2\|v^{(j)}\|_{X\cap S}\|v^{(1)}-v^{(2)}\|_{X\cap S}.
\end{split}$$]{} Hence by using the smallness assumption, we obtain [$$\begin{split}
{&}\|{Z}^{(1)}-{Z}^{(2)}\|_{X\cap S} \sim \|v^{(1)}-v^{(2)}\|_{X\cap S} \sim \|f^{(1)}_T-f^{(2)}_T\|_{X\cap S}
{\\ &}\sim \|{Z}^{(1)}(T)-{Z}^{(2)}(T)\|_{X(T)} \sim \|v^{(1)}(T)-v^{(2)}(T)\|_{X(T)}.
\end{split}$$]{} Since $W(v^{(j)}(0))=\lim_{t\to{\infty}}e^{itH}{Z}^{(j)}(t)$ in ${{{\langle}x {\rangle}}}^{-1}H^1$, the bi-Lipschitz property of $W$ follows from the above equivalence.
Concerning the invertibility, or construction of the wave operator, we can argue as for Theorem \[thm:fin\], namely consider an approximating sequence of solutions $(v^T,{Z}^T)$ with ${Z}^T(T)=e^{-iTH}v_+$ for any given small $v_+\in{{{\langle}x {\rangle}}}^{-1}H^1$, and let $T\to{\infty}$. Then the above a priori bound implies weak convergence of some subsequence in $X\cap S$, whose limit has the desired scattering property.
Correction for the 2D asymptotic profile in [@vac2]
===================================================
The asymptotic profile given in [@vac2] for the two dimensional case was incorrect. The equation for $z:=U^{-1}[u_1+(2-{\Delta})^{-1}|u|^2]+iu_2$ was given by[^5] [$$\begin{split}
i\dot z - Hz &= 2u_1^2 + 4i{\nabla}(2-{\Delta})^{-1}U^{-1}{\nabla}(u_1{\nabla}u_2) + O(u^3)
{\\ &}= 2[U(z+{\overline}{z})/2]^2 + iH^{-1}{\nabla}[U(z+{\overline}{z})\cdot{\nabla}(z-{\overline}{z})] + O(u^3),
\end{split}$$]{} so the correction term $z'$ coming from $z{\overline}{z}$ should be given by [$$\begin{split}
z' := i\int_{\infty}^t e^{iH(s-t)}\left\{|Uz^0|^2 + 2i{\mathop{\mathrm{Im}}}H^{-1}{\nabla}(U{\overline}{z^0}\cdot{\nabla}z^0)\right\}ds,
\end{split}$$]{} where $z^0:=e^{-itH}{\varphi}$ is the given free solution. The second term was missing in [@vac2]. Nevertheless the proof was correct because it was carried out assuming that all the bilinear terms with $z^0{\overline}{z^0}$ had been subtracted, and those correction terms did not really affect the estimates. Both terms can give non-$L^2$ contribution around $|{\xi}|\sim 0$.
[10]{} F. Bethuel, P. Gravejat and J. C. Saut, [*Travelling waves for the Gross-Pitaevskii equation II.*]{} Preprint, arXiv:0711.2408. F. Bethuel, G. Orlandi, D. Smets, [*Vortex rings for the Gross-Pitaevskii equation*]{}. J. Euro. Math. Soc. [**6**]{} (2004), no. 1, 17–94. F. Bethuel and J. C. Saut, [*Travelling waves for the Gross-Pitaevskii equation. I*]{}. Ann. Inst. H. Poincaré Phys. Théor. [**70**]{} (1999), no. 2, 147–238. F. Bethuel and J. C. Saut, [*Vortices and sound waves for the Gross-Pitaevskii equation*]{}. Nonlinear PDE’s in Condensed Matter and Reactive Flows, 339–354, NATO Sci. Ser. C Math. Phys. Sci., [bf 569]{} Kluwer Acad. Publ., Dordrecht, 2002. F. Bethuel and D. Smets, [*A remark on the Cauchy problem for the 2D Gross-Pitaevskii equation with nonzero degree at infinity.*]{} Differential Integral Equations [**20**]{} (2007), no. 3, 325–338. D. Chiron, [*Travelling waves for the Gross-Pitaevskii equation in dimension larger than two*]{}. Nonlinear Anal. [**58**]{} (2004), no. 1-2, 175–204. R. Coifman and Y. Meyer, [*Commutateurs díntegrales singulieres et operateurs multilineaires.*]{} Ann. Inst. Fourier (Grenoble) [**28**]{} (1978), no. 3, xi, 177–202. J. E. Colliander and R. L. Jerrard, [*Vortex dynamics for the Ginzburg-Landau-Schrödinger equation*]{}. Internat. Math. Res. Notices [**1998**]{}, no. 7, 333–358. J. E. Colliander and R. L. Jerrard, [*Ginzburg-Landau vortices: weak stability and Schrödinger equation dynamics*]{}. J. Anal. Math. [**77**]{} (1999), 129–205. A.L. Fetter and A.A. Svidzinsky, [*Vortices in a trapped dilute Bose-Einstein condensate.*]{} J. Phys. Condens. Matter [**13**]{} (2001) R135–R194. C. Gallo, [*Schrödinger group on Zhidkov spaces.*]{} Adv. Differential Equations [**9**]{} (2004), no. 5-6, 509–538. P. Gérard, [*The Cauchy problem for the Gross-Pitaevskii equation.*]{} Ann. Inst. H. Poincaré Anal. Non Linéaire [**23**]{} (2006), no. 5, 765–779. J. Ginibre and G. Velo, [*Scattering theory in the energy space for a class of nonlinear Schrödinger equations*]{}. J. Math. Pures Appl. (9) [**64**]{} (1985), no. 4, 363–401. L. Grafakos and N. J. Kalton, [*The Marcinkiewicz multiplier condition for bilinear operators.*]{} Studia Math. [**146**]{} (2001), no. 2, 115–156. P. Gravejat, [*A non-existence result for supersonic travelling waves in the Gross-Pitaevskii equation*]{}. Comm. Math. Phys. [**243**]{} (2003), no. 1, 93–103. P. Gravejat, [*Asymptotics for the travelling waves in the Gross-Pitaevskii equation*]{}. Asymptot. Anal. [**45**]{} (2005) 227–299. S. Gustafson, K. Nakanishi and T.-P. Tsai, [*Scattering theory for the Gross-Pitaevskii equation.*]{} Math. Res. Lett. [**13**]{} (2006), no. 2, 273–285. S. Gustafson, K. Nakanishi and T.-P. Tsai, [*Global dispersive solutions for the Gross-Pitaevskii equation in two and three dimensions.*]{} Ann. Henri Poincaré. [**8**]{} (2007), 1303–1331. N. Hayashi, T. Mizumachi and P. I. Naumkin, [*Time decay of small solutions to quadratic nonlinear Schrödinger equations in 3D*]{}. Differential Integral Equations [**16**]{} (2003), no. 2, 159–179. N. Hayashi and P. I. Naumkin, [*On the quadratic nonlinear Schrödinger equation in three space dimensions.*]{} Internat. Math. Res. Notices [**2000**]{}, no. 3, 115–132. R.L. Jerrard, [*Vortex filament dynamics for Gross-Pitaevsky type equations.*]{} Ann. Sc. Norm. Super. Pisa Cl. Sci. [**1**]{} (2002) no. 4, 733-768. F. John, [*Blow-up for quasilinear wave equations in three space dimensions.*]{} Comm. Pure Appl. Math. [**34**]{} (1981), no. 1, 29–51. C. A. Jones, P. H. Roberts, [*Motions in a Bose condensate: IV. Axisymmetric solitary waves*]{}, J. Phys. A: Math. Gen. [**15**]{} (1982) 2599–2619. C. A. Jones, S. J. Putterman and P. H. Roberts, [*Motions in a Bose condensate: V. Stability of solitary wave solutions of non-linear Schrödinger equations in two and three dimensions*]{}, J. Phys. A: Math. Gen. [**19**]{} (1986) 2991–3011. O. Lange and B. J. Schroers, [*Unstable manifolds and Schrödinger dynamics of Ginzburg-Landau vortices*]{}. Nonlinearity [**15**]{} (2002), no. 5, 1471–1488. F. H. Lin and J. X. Xin, [*On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation*]{}. Comm. Math. Phys. [**200**]{} (1999), no. 2, 249–274. J. E. Lin and W. A. Strauss, [*Decay and scattering of solutions of a nonlinear Schrödinger equation.*]{} J. Funct. Anal. [**30**]{} (1978), no. 2, 245–263. K. Nakanishi, [*Asymptotically-free solutions for the short-range nonlinear Schrödinger equation.*]{} SIAM J. Math. Anal. [**32**]{} (2001), no. 6, 1265–1271. Y. N. Ovchinnikov and I. M. Sigal, [*Long-time behaviour of Ginzburg-Landau vortices*]{}. Nonlinearity [**11**]{} (1998), no. 5, 1295–1309. J. Shatah, [*Normal forms and quadratic nonlinear Klein-Gordon equations.*]{} Comm. Pure Appl. Math. [**38**]{} (1985), no. 5, 685–696. D. Spirn, [*Vortex motion law for the Schrödinger-Ginzburg-Landau equations*]{}. SIAM J. Math. Anal. [**34**]{} (2003), no. 6, 1435–1476. C. Sulem and P. -L. Sulem, [*The nonlinear Schrödinger equation. Self-focusing and wave collapse.*]{} Applied Mathematical Sciences, [**139**]{}. Springer-Verlag, New York, 1999. xvi+350 pp.
[Stephen Gustafson]{}, [email protected]\
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
[Kenji Nakanishi]{}, [email protected]\
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
[Tai-Peng Tsai]{}, [email protected]\
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
[^1]: Here we naturally put $U^{-1}$ on $v$ to map the linear energy space $H^1({\mathbb{R}}^3)$ for $v$ onto that for the wave equation, $\dot H^1({\mathbb{R}}^3)$.
[^2]: The distance in [@Ge] is slightly different but equivalent to the above.
[^3]: We will introduce $B_j$ and $C_j$ afterward by transforming $B_j'$ and $C_j'$.
[^4]: The pair $(v,z)$ in the previous paper [@vac2] is equal to $U^{-1}(v,{z})$ in this paper.
[^5]: This $z$ in [@vac2] is equal to $U^{-1}{z}$ in this paper.
|
---
abstract: 'The propagation of nonlinear waves in a lattice of repelling particles is studied theoretically and experimentally. A simple experimental setup is proposed, consisting of an array of coupled magnetic dipoles. By driving harmonically the lattice at one boundary, we excite propagating waves and demonstrate different regimes of mode conversion into higher harmonics, strongly influenced by dispersion and discreteness. The phenomenon of acoustic dilatation of the chain is also predicted and discussed. The results are compared with the theoretical predictions of $\alpha$-FPU equation, describing a chain of masses connected by nonlinear quadratic springs and numerical simulations. The results can be extrapolated to other systems described by this equation.'
author:
- 'A. Mehrem'
- 'N. Jiménez'
- 'L. J. Salmerón-Contreras'
- 'X. García-Andrés'
- 'L. M. García-Raffi'
- 'R. Picó'
- 'V. J. Sánchez-Morcillo'
title: Nonlinear dispersive waves in repulsive lattices
---
Introduction
=============
Repulsive interactions among particles are known to form ordered states of matter. One example is Coulomb interaction, which is on the basis of the solid state physics [@Kittel]. In a crystal, atoms and ions are organized in ordered lattices by means of repulsive forces acting among them. Such non-contact forces provide also the coupling between neighbour atoms, what allows the propagation of perturbations in the form of phonons, or elementary excitations of the lattice. This picture is not restricted to the atomic scale. At a higher scale, the interaction of charged particles other from atoms and ions has shown the formation of crystal lattices. A remarkable case is ionic crystals in a trap [@raizen1992ionic]. Such crystals, which are considered a particular form of condensed matter, are formed by charged particles, e.g. atomic ions, confined by external electromagnetic potentials (Paul or other traps), and interacting by means of the Coulomb repulsion. Crystallization requires low temperature that is achieved by laser cooling techniques. Different crystallization patterns have been observed by tuning the shape and strengths of the traps. Crystals of trapped ions have been subject of great attention as a possible configuration to perform quantum computation [@porras2008mesoscopic]. Crystallization of a gas of confined electrons, known as Wigner crystals, have been also predicted and observed [@Matveev10; @Deshpande08].
Waves in such crystals show strong dispersion at wavelengths comparable with the lattice periodicity. The linear (infinitesimal amplitude) dispersion relation, and some nonlinear characteristics of wave propagation have been experimentally determined in an electrically charged, micrometer-sized dust particles immersed in the sheath of a parallel plate rf discharge in helium in a rf plasma [@homann1997determination], where the waves are excited by transferring the momentum from a laser to the first particle in the chain.
In other type of plasma crystals, linear wave mixing and harmonic generation of compressional waves has been theoretically [@avinash2003nonlinear] and experimentally [@nosenko2004nonlinear] demonstrated. Also, nonlinear standing waves have been discussed in a two-dimensional system of charged particles [@denardo1988theory]. Here the generation of second and third harmonics was predicted on the long-wavelength (non- or weakly dispersive) limit.
Some experiments with analogue models of repulsive lattices have been done using magnets as interacting particles, with the aim of demonstrating the generation and propagation of localized perturbations (discrete breathers and solitons). For example, in the seminal work of Russel [@russell1997moving] a chain of magnetic pendulums (very similar to the setup presented in this paper) was used to simulate at the macroscopic level some natural layered silicate crystals, such as muscovite mica. More recently, in [@moleron2014solitary], the authors proposed another configuration of a chain of repelling magnets, for the study of solitary waves, similar to the highly discrete kinks studied theoretically in Coulomb chains including realistic interatomic and substrate potentials [@archilla2015ultradiscrete].
We finally note that repulsive potentials are not restricted to those of electric or magnetic nature. A celebrated case is the granular chain of spherical particles interacting via Hertz potentials. Many studies have been done in this system, theoretical and experimental, on the propagation of the solitary waves. Recently, several nonlinear effects related to the propagation of intense harmonic waves in such granular lattices has been described in [@sanchez2013second], with special attention to the dispersive regime.
In this work, we investigate experimentally and numerically the propagation of nonlinear and dispersive waves in harmonically driven repulsive lattices with on-site potentials. In particular, we study the harmonic generation of monochromatic waves travelling in an array of coupled magnetic dipoles, comparing the observations with the predictions from the $\alpha$-FPU equation and numerical results including an on-site potential. Two main results are reported: first, the experimental observation of the generation of second harmonic in highly dispersive nonlinear lattices and, second, the saturation in the generation of the evanescent zero frequency mode in lattices with on-site potential. The paper is organized as follows: In Sec. \[sec:theory\], the theoretical model, the equation of motion of a lattice of particles interacting by inverse power-law forces, is presented. The weakly nonlinear limit is considered, where the model approaches to the celebrated $\alpha$-FPU equation. The linear dispersion relation and the analytical solutions for propagating and evanescent nonlinear periodic waves, are given. In Sec. \[sec:magnets\], the theory is particularized for the case of an array of coupled magnetic pendula, and it is presented the experimental setup based on a lattice of magnetic pendula rotating by means of a magnetic bearing system that guarantees low friction. In Sec. \[sec:results\] we discuss the experimental results, concerning the generation of harmonics and a static displacement (dilatation) mode. Finally, the conclusions of the study are given in Sec. \[sec:conclusions\].
Theoretical model {#sec:theory}
=================
Equation of motion
------------------
![Scheme of the lattice of non-linearly coupled oscillators.[]{data-label="schem2"}](figure1.eps){width="0.90\columnwidth"}
We consider an infinite chain of identical particles with mass $M$ aligned along the $x$-axis, interacting with their nearest neighbours via repulsive potentials, $V_\mathrm{int}$. In the absence of perturbations, every mass has a fixed equilibrium position, with the interparticle distance given by $a$, as shown in Figure $\ref{schem2}$. Since the forces are repulsive, note that for a finite chain this is only possible if there is an external potential $V_\mathrm{ext}$ that keeps the particles confined. This effect can be provided by a periodic on-site potential, or a force keeping the boundary particles at fixed positions. The equation of motion can be written as $$M \ddot{u}_n=V'_\mathrm{int}\left(u_{n+1}-u_{n}\right)
-V'_\mathrm{int}\left(u_{n}-u_{n-1}\right) +V'_\mathrm{ext},
\label{eq:eqmotion}$$ where $u_n$ stands for the displacement of the $n$-th particle measured with respect to its equilibrium position, $M$ is the mass of the particle, $V$ are the potentials and $V'$ their derivatives with respect to the spatial coordinate, i.e., the forces. For small displacements, the interaction forces, $V'_\mathrm{int}$, can be considered linear with respect to the distance between the particles, $r$, i.e., $V'(r)=\kappa r$ where $\kappa$ is a constant, then Eq. (\[eq:eqmotion\]) represents a system of coupled harmonic oscillators. For higher amplitude displacements, the linear approximation of the interaction force cannot be assumed in most real systems and nonlinearity must be considered. Chains of nonlinearly coupled oscillators have been extensively studied in the past for different types of anharmonic interaction potentials. Some relevant cases are the $\alpha$-FPU lattice, where $V'(r)=\kappa_1 r+\kappa_2 r^2$ (quadratic interaction), the $\beta$-FPU lattice where $V'(r)=\kappa_1 r+\kappa_3 r^3$ (cubic interaction), the Toda lattice, with $V'(r)=\exp({-r})-1$ or the granular lattice, with $V'(r)=\kappa r^{3/2}$. Here we consider the case of forces that decrease with an inverse power law of the distance, $V'(r)=\beta r^{-\alpha}$, typical of interatomic interactions, e.g. as the Coulomb repulsive interaction. For such a force, the equation of motion results in $$\label{eq:eqmotion2}
M \ddot{u}_n= \frac{\beta}{(a-u_{n+1}+u_{n})^\alpha}-\frac{\beta}{(a-u_{n}+u_{n-1})^\alpha} +V'_\mathrm{ext}.$$
The exponent $\alpha$ can take different values depending on the particular system: $\alpha=2$ for electrically charged particles, e.g. in ion Coulomb crystals [@raizen1992ionic] or dusty plasma crystals [@nosenko2004nonlinear], $\alpha=4$ for distant magnetic dipoles [@russell1997moving], or any other non-integer power [@moleron2014solitary].
In general, Eq. (\[eq:eqmotion2\]) do not possess analytical solutions. Approximate analytical solutions can be obtained in the small amplitude limit, i.e., assuming that the particle displacement $|u_n|$ is small compared to the lattice constant $a$. Under this assumption, the forces can be expanded in Taylor series and Eq.(\[eq:eqmotion2\]) can be reduced, neglecting cubic and higher order terms, to an equation in the normalized form $$\begin{aligned}
\label{eq:FPU}
\ddot{u}_n=&\frac{1}{4}\left(u_{n-1}-2u_{n}+u_{n+1}\right)- \nonumber \\
&\frac{\varepsilon}{8}\left(u_{n-1}-2u_{n}+u_{n+1}\right)\left(u_{n-1}-u_{n+1}\right) + \Omega_0^2 u_n,\end{aligned}$$ where the normalization $u_n=u_n/a$ has been introduced, dots indicate now derivative with respect to a dimensionless time $\tau=\omega_m t$, where $\omega_m=\sqrt{4 \alpha \beta/M a^{\alpha+1}}$ is the maximum frequency of propagating waves (upper cutoff frequency of the dispersion relation), $\varepsilon=(1+\alpha)u_0$ is the nonlinearity coefficient and $\Omega_0=\omega_0/\omega_m$ is the on-site potential characteristic frequency. The on-site restoring force $V'_\mathrm{ext}$ is in general nonlinear. However, for small displacements, as considered here, it may be represented by a term $V'_\mathrm{ext}=M \Omega_0^2 {u_n}{a}$, where $\Omega_0$ is related with the frequency of oscillation of the particle in the external potential. The particular form of this term for the proposed experimental setup will be discussed later. If the on-site potential term is neglected (no external forces acting on the chain), Eq. (\[eq:FPU\]) reduces to the celebrated $\alpha$-FPU equation. It has been considered as an approximate description of many different physical systems, and has played a central role in the study of solitons and chaos [@GavallottiFPU].
Dispersion Relation
-------------------
Some important features of the propagation of waves in a lattice can be understood by analyzing its dispersion relation. For infinitesimal amplitude waves, it can be obtained analytically by neglecting the nonlinear terms in the equation of motion, and solving for a harmonic discrete solution in the form $u_n = \exp {i(\Omega t-k n)}$, where $\Omega=\omega/\omega_m$ is the normalized wave frequency and $k$ is the wavenumber. By replacing this solution in the linearized Eq. (\[eq:FPU\]), we obtain the well-known dispersion relation for a monoatomic lattice, that in normalized form reads $$\label{disp2}
\Omega= \sqrt{\sin^2\left(\frac{k}{2}\right)+\Omega_0^2}.$$ On one hand, there is an upper cutoff frequency at which the transition from propagative to evanescent solutions is produced, i.e., $\mathrm{|Im(k)|>0}$, and it is given in this normalization by $\Omega=\sqrt{1+\Omega_0^2}$. On the other hand, the effect of the on-site potential is to create a low frequency bandgap in the dispersion relation, i. e., $\Omega_0$ represents the lower cut-off frequency. In the absence of external confining potential, $\Omega_0 \rightarrow 0$, the dispersion relation reduces to $\Omega= \left| \sin\left({ka}/{2}\right) \right|$. In this case the upper cutoff normalized frequency is $\Omega=1$.
Although the dispersion relation has been derived assuming infinitesimal amplitude (linear) waves, it describes also the propagation of other modes as the higher harmonics of a fundamental harmonic wave (FW), when these are generated by weakly nonlinear processes, as described in the following sections.
Analytical solutions
--------------------
One known effect of the quadratic nonlinearity is the generation of second and higher harmonics of an input signal. This is the basic effect, for example, of nonlinear acoustic waves propagating in homogeneous, non-dispersive media [@naugolnykh1998nonlinear; @hamilton1998nonlinear], where the amplitude of the harmonics depends on the nonlinearity of the medium, the excitation signal and the propagated distance (the excitation amplitude in the chain $u_0$, the frequency $\Omega$ and its position $n$).
In general, the generation of harmonics is strongly dependent of the dispersion of the system, as occurs in the discrete lattice described by Eq. (\[eq:FPU\]). To study the process of harmonic generation, an analytical solution can be obtained by perturbative techniques, such as the successive approximations method. We follow this approach, by assuming that the nonlinear parameter $\varepsilon$ is small (which implies displacements much smaller than the interparticle separation), and expressing the displacement as a power series in terms of $\varepsilon$, in the form $u_n=u_n^{(0)}+\varepsilon u_n^{(1)}+\varepsilon^2 u_n^{(2)}+\ldots$. After substituting the expansion into Eq. (\[eq:FPU\]), and collecting terms at each order in $\varepsilon$, we obtain a hierarchy of linear equations that can be recursively solved. This has been done in Ref. [@sanchez2013second] to study nonlinear waves in a granular chain, formed by spherical particles in contact interacting by Hertz potentials, and the result is readily extendible to chain of particles interacting by inverse power laws of arbitrary exponent, which results in a particular value of the nonlinearity coefficient. The equation of motion is always given by Eq. (\[eq:FPU\]), the value of $\varepsilon$ being dependent on the exponent $\alpha$. In case of granular chain, it was shown that $\varepsilon=u_0/2$. In this work a chain with quasi-dipolar interaction, $\alpha=4$, gives $\varepsilon=5u_0$, i.e. the nonlinear effects are one order of magnitude higher.
{width="99.00000%"}
Up to second order of accuracy in $\varepsilon$, the displacement field can be expressed as (the details of the derivation can be found in Ref. [@sanchez2013second]) $$\begin{aligned}
\label{eq:Analytic}
u_n=&\varepsilon \Omega^2 n + \nonumber\\
&\frac{1}{2}\left[1+\frac{1}{4} i \varepsilon^2 C_\Omega \sin\left(\frac{\Delta k}{2}n\right) e^{i\frac{\Delta k}{2}n}\right] e^{i\theta_n}+\nonumber\\
&\frac{\varepsilon}{4} \cot\left(\frac{k}{2}\right) \sin\left(\frac{\Delta k}{2}n\right) e^{i\frac{\Delta k}{2}n} e^{2i\theta_n} + \mathrm{c.c.},\end{aligned}$$ where $\theta_n=\Omega t-kna$, $n$ is the oscillator number corresponding to the discrete propagation coordinate, and $$\label{eq:Di1}
C_\Omega=1-\frac{\sin[k(2\Omega)/2]}{ \sin[\Delta k(\Omega)/2]},$$ where $\Delta k=2 k(\Omega)-k(2 \Omega)$ is the wavenumber mismatch between the forced, $2k(\Omega)$, and free, $k(2\Omega)$, contributions to the second harmonic.
The solution given by Eq. (\[eq:Analytic\]) describes wave propagation in the system when the frequency of the second harmonic belongs to the dispersion relation, which is the case for driving frequencies $\Omega< 1/2$. For higher driving frequencies, the second harmonic frequency is outside the propagation band (becoming an evanescent mode) and the solution takes the form $$\begin{aligned}
\label{eq:Analytic2}
u_n = &\varepsilon \Omega^2 n + \nonumber \\
&\frac{1}{2}\left[1+\frac{1}{8} \varepsilon^2 C_\Omega \left(1-e^{-k{''}n}e^{i k{'}n}\right)\right] e^{i\theta_n}+ \nonumber \\
&\frac{\varepsilon}{8} \cot \left(\frac{k}{2}\right) \left(1-e^{-k{''}n}e^{i k{'}n}\right) e^{2i\theta_n} + \mathrm{c.c.},\end{aligned}$$ where $k{''}=2 \cosh^{-1}(2\Omega)$ and $k{'}=2 k(\Omega)-1$ and the mismatch take now the form $\Delta k=k{'}+i k{''}$.
The previous analytical solutions (\[eq:Analytic\]-\[eq:Analytic2\]) predict a number of distinctive features in the nonlinear dynamics of the system, depending on the frequency regime. In the case of the second harmonic belonging to the propagation band, Eq. (\[eq:Analytic\]), dispersion causes a beating in the amplitudes of the different harmonics, since two components of the second harmonic with different wavenumbers propagate asynchronously. Both, the fundamental wave and its second harmonic oscillate out of phase in space: the displacement of the fundamental is maximum where the second harmonic vanishes, which occurs at positions satisfying the following condition: $n=2 \pi/\Delta k$. This process repeats periodically in space as energy is transferred between the two waves as they propagate. The half distance of the spatial beating period corresponds to the coherence length $l_c$: $$\label{coherence}
l_c =\frac{\pi}{\Delta k},$$ and it physically corresponds to the position where the free and forced waves are exactly in phase, i.e., the location of the maximum of first spatial beat.
When the second harmonic frequency lies beyond the cut-off frequency, the free wave is evanescent. There still exists however a forced wave, driven by the first harmonic at any point in the chain. Due to this continuous forcing, the amplitudes of the fundamental and its second harmonic do not oscillate, reaching the amplitude of the second harmonic a constant value after a short transient of growth. This implies propagation of the second harmonic even in the forbidden region. We note that similar results about the behaviour of harmonics have been obtained for nonlinear acoustic waves propagating in a 1D periodic medium or superlattice [@jimenez2016nonlinear]. Finally, we note that the theory predicts the existence of a zero-frequency mode, $u_n=\varepsilon \Omega^2 n$, which represents an static deformation of the lattice, i.e., a constant dilatation. This effect will be studied in detail in Sect. IV.
The lattice of magnetic dipoles {#sec:magnets}
===============================
Forces acting on a magnet
-------------------------
Consider two magnetic dipoles, with magnetic moments $\vec{m}_1$ and $\vec{m}_2$. The force between them is given by the exact relation [@Griffiths07] $$\label{eq:force1}
\vec{F}_{1,2}=
\frac{\mu_0}{4\pi} \vec{\nabla} \cdot \left[ \frac{\vec{m}_1 \cdot \vec{m}_2}{r^3} - 3\frac{\left(\vec{m}_1 \cdot \vec{r} \right)\left( \vec{m}_2 \cdot \vec{r}\right)} {r^5}\right],$$ where $\vec{r}$ is the vector joining the centres of the dipoles. This relation implies that, in general, the force depends on the angle between the dipoles. In the particular case when the dipole moments are equal in magnitude, parallel to each other, and perpendicular to $\vec{r}$ (dipoles in the same plane), the force takes the simpler form $$\label{eq:force2}
\vec{F}_{1,2}=\frac{3\mu_0}{4\pi}\frac{m^2}{r^4} \hat{x},$$ where $m=|\vec{m}_1|=|\vec{m}_2|$, $\mu_0$ is the permeability of the medium and $\hat{x}$ is an unitary vector in the direction of the axis that connect the centres of the magnets. Eq. (\[eq:force2\]) gives the force at equilibrium position ($r=a$) at a magnetic dipole ($n=1$) of the chain produced by is neighbour ($n=2$) in the chain. A opposite force is produced on the oscillator $n=2$.
In the case of the perturbed chain of magnets with nearest neighbour interactions, the distance between centres is a dynamic variable. Assuming small displacements of the magnets, i.e., the angles between the dipole moments are small, we can use Eq. (\[eq:force2\]) with $r = a-u_n+u_{n+1}$ to describe the interaction between two neighbour oscillators $$\label{eq:force3}
\vec{F}_{n,n+1}=\frac{3\mu_0 m^2}{4\pi} \frac{1}{\left(a-u_n+u_{n+1}\right)^4}.$$ Comparing with the equation of motion of the chain, given by Eq. (\[eq:eqmotion2\]), we identify the parameters $$\beta=(3/4\pi)\mu_0 m^2,\quad \alpha=4.$$ This small angle Eq. (\[eq:force3\]) for the forces is a crude approximation and exact expressions can be found in Ref. [@russell1997moving]. However, since our aim is to obtain simple analytical expressions based on the FPU equation, Eq. (\[eq:FPU\]), we will keep this degree of accuracy. The validity of this approximation to describe our setup will be tested in the next sections comparing with the experimental results and numerical simulations.
The above expressions for the forces between magnetic dipoles are valid for loop currents or magnets of negligible dimensions. Expressions for finite size magnets can be found in the literature [@Camacho13] and are in general lengthy and cumbersome. Gilbert’s model of magnetic field of magnets used here results in approximate but simple expressions for the forces [@Griffiths07]. For cylindrical magnets of length $h$, with their magnetic moments parallel, and their axis perpendicular to the line joining the centres, the force between adjacent magnets can be expressed as $$\label{eq:force4}
\vec{F}_{1,2}=\frac{\mu_0 m^2}{2\pi h^2} \left( \frac{1}{r^2}- \frac{r}{\left(r^2+h^2\right)^{3/2}} \right) \hat{x},$$ where the magnetic moment is $m={\cal{M}} h \pi R^2$, $\cal{M}$ is the magnetization and $R$ the radius of a the cylindrical magnet. In the limit $h\ll r$, Eq. (\[eq:force4\]) reduces to Eq. (\[eq:force2\]), i.e., magnets with small dimensions compared to their separation interact via dipolar forces, i.e., $\alpha=4$. In the opposite limit $h\gg r$, parallel magnets close to each other, the interaction law approaches to a Coulomb-type force, i.e., $\alpha=2$. In general, the interaction law of magnets can be approximated by an inverse-law with any given exponent that ranges between monopole and dipole cases.
Experimental setup
------------------
A chain of coupled magnets was built in order to test the theoretical predictions. The experimental setup is shown in Fig. \[setup\]. The chain was composed by 53 identical cylindrical neodymium magnets (Webcraft GmbH, DE, magnet type N45), with mass $M=2$ g, arranged in a one-dimensional periodic lattice. The radius and height of the magnets were $R=2.5$ mm and $h=14$ mm, respectively, and its magnetization was ${\cal{M}} = 1.07~10^6$ A/m. The magnets were oriented with the closest poles being those of the same polarity, therefore the produced forces were repulsive.
To achieve the necessary stability of the chain, the magnets were attached to a rigid bar which allows them to oscillate around a T-shaped support, being each magnet actually a pendulum (see Fig. \[setup\]). The length of the vertical bars was $L=100$ mm, and the distance between supports (and therefore the distance between magnets at equilibrium) was $a=20$ mm. The bearing of the T-shaped support was specially designed to minimize the effects of friction and giving stability to the system. This was achieved by using an additional ring-shaped magnets which keep the oscillators quasi-levitating on air, with just one contact point, as shown in the inset of Fig. (\[setup\]).
The effect of the pendulums is to introduce an additional external force to the dynamics of the chain, corresponding to the term $V'_\mathrm{ext}$ in Eq. (\[eq:eqmotion2\]). If $\theta_n$ is the angle formed by a magnet with respect to its vertical equilibrium position, the restoring force due to gravity is $F_z=M g \sin\theta_n$. For small angles $\theta_n$, and using the notation of Eq. (\[eq:eqmotion2\]), the force per mass can be approximated as $V'_\mathrm{ext}\simeq \Omega_0^2 u_n$, with $\Omega_0=\sqrt{g/L}/\omega_m$.
All magnets oscillate freely except the outermost boundary magnets. The last magnet is fixed, and the first one is attached to the excitation system. The driving system consists of an electrodynamic sub-woofer (Fostex-L363) connected to an audio amplifier (Europower EPS2500) and excited by an arbitrary function generator (Tektronix AFG-2021). The first magnet is attached to the loudspeaker’s diaphragm, thus, being it forced with a sinusoidal motion for different values of frequencies and amplitudes.
The motion of the chain is recorded by using a GoPro-Hero3 camera. The camera is placed at a proper distance from the chain in order to track the motion of a certain number of magnets. In this work, the first $18$ magnets were recorded simultaneously. Then, each pendulum was optically tracked using image post-processing techniques. Image calibration was employed here to correct the lens aberration using the image processing toolbox in Matlab^^, allowing the measurement of the displacement waveforms $u_n$. We considered the travelling wave regime, ignoring the reflected wave by time windowing the recorded video. The measurement in a finite time window guarantees no reflections from the $n=N$ boundary. Due to the quasi-instantaneous temporal duration of the impulse response of the system, after some temporal cycles of measurement the system reaches the stationary. Therefore, the transient measurement is equivalent to the response of an infinite chain and the finite size effects of the chain do not influence the experiments.
The duration of each record was about $3.5$ s, the camera resolution was set to $960$p with a frame rate of 100 frames per second, i.e., leading to a sampling frequency of 100 Hz. Using the measured waveforms, the amplitude of each harmonic was estimated as usual using the Fourier transform.
Experimental results {#sec:results}
====================
Dispersion relation
-------------------
{width="13cm"}
To obtain the dispersion relation experimentally, the first magnet was excited with short duration impulse with low amplitude excitation in order to ensure that the excited waves are described by linear theory. The generated travelling pulse was recorded at two consecutive magnets, i.e., $n$ and $n+1$. The real part of the wavenumber was calculated by estimating the phase difference between them and the imaginary part of the wavenumber was calculated estimating the attenuation, as $$\begin{aligned}
\mathrm{Re}(k) &= \frac{\omega}{\mathrm{Re}(c_p)} = \frac{\arg [U_{n+1}(\omega)/U_n(\omega)]}{a}, \\
\mathrm{Im}(k) &= \frac{\omega}{\mathrm{Im}(c_p)} = \frac{\log\left|U_{n+1}(\omega)/U_n(\omega)\right|}{a}, \end{aligned}$$ where $U_n(\omega)$ is the Fourier transform of the measured displacement of the $n$-th magnet and $c_p$ is the phase velocity. A set of 10 measurements at the oscillator $n=3$ were used to compute the mean value of the phase speed. Figures \[Disp\_Mono\] (a-b) show the real and imaginary part of the wavenumber respectively, where the experimental results and the dispersion relation of Eq. (\[disp2\]) were evaluated at frequencies with step of $\Delta f= 0.66$ Hz. The small magnitude of the experimental errors in the propagating band indicates good repetitiveness of the measurements. The experimental lower frequency cut-off was $f_0=1.68$ Hz, which agrees with the theoretical value $f_0=(1/2\pi) \sqrt{g/L} = 1.48$ Hz, ($f_0=1.56$ Hz if we consider the rigid-body pendulum taking into account the momentum of inertia of the steel rod). The measured upper cut-off frequency was $f_m=17.7$ Hz. This value was used to fit Eq. (\[eq:force4\]) to an inverse power law, using the theoretical prediction $f_m=(1/2\pi)\sqrt{4 \alpha \beta/M a^{\alpha+1}} = 17.6$ we obtained an inverse power law with exponent $\alpha=3.6$ (quasi-dipolar interaction), which is in agreement with the ratio between the height and the separation distance between of the magnets given by Eq. (\[eq:force4\]). Both, upper and lower values of the dispersion relation obtained experimentally can slightly change with the amplitude of the input excitation $u_0$, which is in fact a signature of nonlinear dispersion caused by the finite amplitude of the wave. Note that for higher amplitudes the pulsed excitation used in this experiments leads to the generation of KdV-like compression solitons [@moleron2014solitary]. However, as long as the condition $u_0\ll a$ is fulfilled, the chain propagates linear modes and the dispersion relation can be obtained.
One remarkable result is the low damping of the system, given by the smallness of the imaginary part of the wavenumber in the propagating band.
The complex dispersion relation obtained by numerical integration of Eq. (\[eq:eqmotion2\]) adding a damping term $\gamma \partial u_n / \partial t$ to the equation of motion is shown in Figs \[Disp\_Mono\] (a-b). The damping coefficient, $\gamma$, was fitted to the experiments and corresponds to 0.52 dB/m (note the chain is 1 m long). The damping terms produces a force that opposes the pendulum movement. It is worth noting here that, in the propagating band, the total drag force is roughly twice the viscous drag force estimated for a cylinder of the size of a single magnet oscillating in air [@brouwers1985]: the magnetic bearing system itself produces only small damping. The effect of the small losses is to smooth the limits of the band gap, as it is also observed in other highly dispersive systems, e.g., as in Acoustics [@jimenez2017], and to produce a small attenuation in the propagating band. The damping term is used in the numerical simulations in the following sections.
{width="13cm"}
Harmonic generation
--------------------
By driving the first magnet with a sinusoidal motion, $u_1=u_0 \sin \omega t$, harmonic waves are excited and they propagate along the chain. On one hand, the amplitude $u_0$ was $u_0 = 2.4$ mm. On the other hand, according to the dispersion relation shown in Fig. \[Disp\_Mono\], the driving frequency, $\Omega$, can be chosen among to three different regimes regarding the propagation of the second harmonic: (a) weakly dispersive, (b) strongly dispersive, and (c) evanescent.
The first case (a) is obtained when the frequency of the fundamental wave lies in the lower part of the pass band and, the generated second harmonic is also in the pass band, in the region of weak dispersion. Thus, in this regime the motion equations of the lattice can be approximated by a continuum whose dynamics follows the Boussinesq equation [@sanchez2013second] and the wave roughly propagates without dispersion. In this low frequency regime, the lower harmonics propagate with nearly the same phase velocity. The amplitude of the second harmonic increases roughly linearly with distance while the first harmonic amplitude decreases due to the energy transfer from the fundamental component to the higher harmonics. This case is shown in Fig. \[fig-harmonics\] (a), where a fundamental wave with frequency $f=5$ Hz ($\Omega=0.27$) generates a second harmonic whose frequency $2f=10$ Hz lies on the weakly-dispersive region of the propagative band ($\Omega=0.54$). We note that in this regime, third harmonic is also generated, as shown Fig. \[fig-harmonics\] (b), although it is not predicted by the perturbative analytical solution due to its second-order accuracy.
Secondly, the case (b), corresponding to strongly dispersive second harmonic, is shown in Fig. \[fig-harmonics\] (b). Here, the driving frequency approaches the half of the passband frequency. The second harmonic lies in the highly dispersive part of the band, but still in a propagative region (slightly below the cutoff frequency $f_m$). As observed in the previous case, the amplitude of the second harmonic increases with distance, but now at a particular distance given by the coherence length, $l_c$, it decreases. Both, the fundamental wave and its second harmonic present spatial oscillations, i.e., spatial beatings. Figures \[fig-harmonics\] (c-d) illustrate this case for a fundamental wave with frequency $f=8.8$ Hz, i.e. a second harmonic with frequency $\Omega=0.96$. The experimental value of the coherence length was $l_c \approx 4.5a$, which is in agreement with the theoretical given by Eq. (\[coherence\]).
Finally, the case (c) corresponds to the second harmonic lying within the band-gap, as shown in Fig. \[fig-harmonics\] (e-f) for a excitation frequency of $f=10.1$ Hz ($\Omega=0.55$). In this case, the second harmonic is evanescent and its amplitude does not change with distance. One would expect the absence of the SH field (since it is an evanescent mode) but a finite amplitude is observed in agreement with theory and numerical simulations. The second harmonic component is generated locally as it is “pumped" by the fundamental wave. Its amplitude value remains constant all along the chain, being its amplitude dependent on the driving amplitude and on the properties of the medium (the non-linearity and the magnitude of the dispersion).
The experimental results shown in Fig. \[fig-harmonics\] are in good agreement with the analytical predictions of the asymptotic theory (solid lines), and also with the numerical simulation of Eq. (\[eq:FPU\]). However, small discrepancies can be observed between the theory and the experiments, as well as between the theory and the simulations. The value of the nonlinear coefficient used in the experiments was $\varepsilon=(1+\alpha)u_0 = 0.55$. Thus the small disagreements between the theory and the experiments and simulations are mainly explained due to the non smallness of the nonlinear parameter $\varepsilon$. For small excitation amplitudes, i.e., small $\varepsilon$, the theory and numerical solutions converge to the similar result. However, due to the precision of the motion-tracking acquisition system, it was difficult to accurately measure small amplitude perturbations.
{width="14cm"}
Chain dilatation
----------------
Besides the harmonic generation, the FPU equation also predicts the presence of a static (zero-frequency) mode. It physically represents an incremental shift of the average position of each oscillator, which in turn results in a constant dilatation or expansion of the chain. This term is accounted for by the first term in Eq. (\[eq:Analytic\]). Since the average displacement grows linearly with distance, it can be interpreted as a constant strain produced by the acoustic mode along the lattice.
The phenomenon was originally reported for acoustic waves propagating in a solid described by a nonlinear wave equation [@cantrell1991acoustic], which is actually the continuous (long-wavelength) analogue of Eq. (\[eq:FPU\]). The effect was described there as an acoustic-radiation-induced strain. The physical origin of the expansion of the discrete chain (and also in the continuous solid) is the anharmonicity of the interaction potential, and therefore is a general nonlinear effect. Note radiation forces also appear in other nonlinear systems as acoustic waves in fluids, soft solids or even light (radiation pressure), being the generation of acoustic radiation forces a general mechanism of any wave motion [@sarvazyan2010]. We remark that the phenomenon of acoustic expansion is analogous to the thermal expansion of solids, which also has its physical origin in the lattice anharmonicity. The link between these two effects and its relation with the acoustic nonlinear parameter has been pointed out in Ref. [@Cantrell82].
We have shown in Fig. \[expansion\] the generation of the zero-mode in the particular case of the chain of coupled oscillators and for different excitation frequencies. The experimental results agree with simulations of the full equations of motion including the restoring force. We can see that for all the frequencies the linear increasing of the displacement predicted by the analytical solutions is not observed. Instead, we can observe two regimes, and a transition between them at a particular distance. First, in the region near the boundary (extending up to $n\approx 8$ in our experiment), the displacement grows roughly linearly with distance, as predicted by the theory without restoring force. However, beyond a given distance the growth of the static displacement mode saturates, and the chain attains an unstrained state, with the oscillators moving around positions shifted with respect to their initial values. This behaviour is not predicted by the theory.
The saturation effect can be understood if we recall that the theory was developed assuming that there was not a prescribed equilibrium position for any oscillator in the chain, the chain was assumed semi-infinite, and the only force acting on the masses was the nearest neighbours interaction. However, in the experimental setup an additional restoring force is present, due to gravity. For small perturbations this is equivalent to an on-site potential. Since the magnets are pendula, the maximum shift of a magnet with respect to the equilibrium position is also bounded. Note in Fig. \[expansion\] the oscillators are displaced less than a lattice step. Note also that for a finite value of the on-site potential $\Omega_0$ the zero-th mode is always evanescent. Then, as described previously with the second harmonic in the evanescent case, only the forced contribution to the zero-th order mode is present, leading to a constant value of the zero-th mode.
Finally, Fig. \[expansion2\] shows the dependence of the zero-mode with frequency, measured at $n=3$, $n=5$ and $n=10$. It can be observed that the experimental results agree with the simulations of the full equations of motion, while the simulations of the FPU equation roughly does with the theory (in this case the excitation amplitude was $u_0=4.8$ mm, leading to a value of the nonlinear parameter of $\varepsilon=0.96$). For frequencies below $\Omega\approx0.8$, the amplitude of the zero-mode roughly follows a quadratic dependence with frequency. In addition, the period average displacement of an oscillator corresponds to the position where there exist a balance between the gravity restoring force and the equivalent acoustic-radiation force produced by the nonlinear compressional wave, corresponding to $F_\mathrm{ARF} = \Omega_0^2 \left\langle u_n\right\rangle$, where $\left\langle u_n\right\rangle$ is the amplitude of the zero mode. Thus, the induced acoustic-radiation force in the experimental chain also follows a quadratic dependence with frequency for low frequency waves.
{width="100.00000%"}
Conclusions {#sec:conclusions}
===========
The propagation of nonlinear monochromatic waves in a lattice of particles coupled by repulsive forces following an inverse power-law with distance has been studied theoretically, numerically and experimentally. In the limit of small amplitudes, the system is described by a FPU equation with quadratic nonlinearity, where analytical solutions were generalized for the case of an arbitrary inverse frequency power-law interactions. In particular, it has been developed an experiment consisting in a lattice of coupled magnetic dipoles sinusoidally driven at one boundary, while a magnetic bearing system for the rotation of each pendulum provides low mechanical damping.
In spite of the simplifying assumptions made in the theoretical analysis, the observations agree quite well with the model concerning the generation of the second harmonic, e.g., the characteristic spatial beatings of the second harmonic due to the dispersion of the lattice are observed.
One particular feature of the studied lattice is the existence of a restoring force due to the action of gravity on the pendula. This is roughly equivalent to the introduction of an on-site potential, leading to the generation of a low frequency band gap. In this work, it has been observed for the first time that the generated zero mode is evanescent due to the presence of the on-site potential, therefore, only the forced component of the zero mode propagates through the chain and a saturation of the amplitude of the zero mode is observed. There exist discrepancies between the analytical FPU theory and the experimental measurements of the static dilatation mode. They are caused, mainly, because the developed theory is based on a FPU equation that lacks of the on-site potential that produces the low frequency band gap. Therefore, while the FPU theory predicts a linear monotonic growth of the zero-mode, the presence of the low-frequency band gap makes the zero-frequency mode to be evanescent, and, as a consequence, a saturation of the dilatation of the chain is observed in the experiments and in the numerical simulations. The particular dynamics of the generated zero-mode are discussed in analogy with the radiation force produced by a nonlinear monochromatic travelling wave. This result has an interest beyond the particular studied system, since there exist a number of systems, e.g. as condensed matter or granular crystals, that present similar of dispersion relations, with a low-frequency band gap.
Additionally, the present low-friction experimental setup can be used to explore other effects of nonlinear discrete systems that have been predicted in the literature, e.g., nonlinear localized modes. Under the assumption of small amplitude, these results indicate that the lattice of magnetic dipoles is well described by an $\alpha$-FPU equation, which opens the possibility of extending the results to other systems which are described by the same generic equation. The proposed system can be also viewed as a mechanical analogue of a microscopic crystal of interacting charged particles (atoms or ions) at a macroscopic scale. Despite the limited applicability of this simple one dimensional lattice to describe real crystals, the approach possess however some advantages, as the possibility of varying parameters that are normally fixed, as the strength of the interaction and on-site potentials, or exploring strongly nonlinear regimes which are hardly achievable at atomic scales.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was funded by Spanish Ministerio de Economia e Innovacion (MINECO), grant FIS2015-65998-C2-2-P. AM gratefully acknowledge to Generalitat Valenciana (Santiago Grisolia program). LJSC gratefully acknowledge the support of PAID-01-14 at Universitat Politècnica de València.
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|
---
abstract: |
While the GPGPU paradigm is widely recognized as an effective approach to high performance computing, its adoption in , systems is still in its early stages.
Although GPUs typically show deterministic behaviour in terms of latency in executing computational kernels as soon as data is available in their internal memories, assessment of features of a standard GPGPU system needs careful characterization of all subsystems along data stream path. The networking subsystem results in being the most critical one in terms of absolute value and fluctuations of its response latency.
Our envisioned solution to this issue is [NaNet]{}, an [PCIe]{}Network Interface Card (NIC) design featuring a configurable and extensible set of network channels with direct access through [GPUDirect]{}to NVIDIA Fermi/Kepler GPU memories.
[NaNet]{}design currently supports both standard - [GbE]{}(1000BASE-T) and [10-GbE]{}(10Base-R) - and custom - 34 Gbps [APElink]{}and 2.5 Gbps deterministic latency KM3link - channels, but its modularity allows for a straightforward inclusion of other link technologies.
To avoid host OS intervention on data stream and remove a possible source of jitter, the design includes a network/transport layer offload module with , latency, supporting UDP, KM3link Time Division Multiplexing and [APElink]{}protocols.
After [NaNet]{}architecture description and its latency/bandwidth characterization for all supported links, two real world use cases will be presented: the low level trigger for the RICH detector in NA62 experiment at CERN and the data link for KM3 underwater neutrino telescope.
Results of [NaNet]{}performances in both experiments will be reported and discussed.
author:
- |
A. Lonardo, F. Ameli, R. Ammendola, A. Biagioni, O. Frezza, G. Lamanna, F. Lo Cicero, M. Martinelli, C. Nicolau, P. S. Paolucci, E. Pastorelli, L. Pontisso, D. Rossetti, F. Simeone, F. Simula, M. Sozzi , L. Tosoratto, and P. Vicini\
[^1] [^2] [^3] [^4] [^5] [^6] [^7] [^8] [^9]
title: '[NaNet]{}: a , , Network Interface Card with GPUDirect Features'
---
[NaNet]{}design overview {#sec:nanetone}
========================
is a modular design of a [PCIe]{}[RDMA]{}NIC supporting different network links, namely standard [GbE]{}(1000BASE-T) and [10-GbE]{}(10Base-R), besides custom 34 Gbps [APElink]{} [@APEnetTwepp:2013] and 2.5 Gbps deterministic latency optical KM3link [@Aloisio:2011:NSS]. The design includes a network stack protocol offload engine yielding a very stable communication latency, a feature making [NaNet]{}suitable for use in contexts; [NaNet]{}[GPUDirect]{}[RDMA]{}capability, inherited from the [APEnet+]{}3D torus NIC dedicated to HPC systems [@APEnetChep:2012], extends its into the world of GPGPU heterogeneous computing.
[NaNet]{}design is partitioned into 4 main modules: *I/O Interface*, *Router*, *Network Interface* and *[PCIe]{}Core* (see Fig. \[fig:NaNet\]).
I/O Interface module performs a processing on the data stream: following the OSI Model, the Physical Link Coding stage implements, as the name suggests, the channel physical layer ([*e.g.*]{}1000BASE-T) while the Protocol Manager stage handles, depending on the kind of channel, data/network/transport layers ([*e.g.*]{}Time Division Multiplexing or UDP); the Data Processing stage implements application dependent transformations on data streams ([*e.g.*]{}performing compression/decompression) while the APEnet Protocol Encoder performs protocol adaptation, encapsulating inbound payload data in [APElink]{}packet protocol, used in the inner [NaNet]{}logic, and decapsulating outbound [APElink]{}packets before their payload in output channel transport protocol ([*e.g.*]{}UDP).
The Router module supports a configurable number of ports implementing a full crossbar switch responsible for data routing and dispatch. Number and of the switch ports and the routing algorithm can each be defined by the user to automatically achieve a desired configuration. The Router block dynamically interconnects the ports and comprises a fully connected switch, plus routing and arbitration blocks managing multiple data flows @2.8 GB/s
The *Network Interface* block acts on the trasmitting side by gathering data coming in from the [PCIe]{}port and forwarding them to the Router destination ports while on the receiving side it provides support for [RDMA]{}in communications involving both the host and the GPU (via the dedicated *GPU I/O Accelerator* module). A [Nios II]{}$\mu$controller in included to support configuration and runtime operations.
Finally, the [PCIe]{}Core module is built upon a powerful commercial core from PLDA that sports a simplified but efficient backend interface and multiple DMA engines.
As will be shown in the following, this general architecture has been specialized to be employed in several contexts, and implemented on several devices: [Altera]{}Stratix IV and V FPGA development kit and Terasic [DE5-net]{}board.
![[NaNet]{}architecture schematic.[]{data-label="fig:NaNet"}](NaNet_internals.pdf){width=".49\textwidth"}
: a NIC for the NA62 low level trigger
======================================
The NA62 experiment at CERN [@Lamanna:2011zz] aims at measuring the Branching Ratio of the decay of the charged Kaon into a pion and a pair. The NA62 goal is to collect $\sim100$ events with a signal to background ratio 10:1, using a novel technique with a (75 GeV) unseparated hadron beam decaying in flight. In order to manage the data stream due to a $\sim$ 10 MHz rate of particle decays illuminating the detectors, a set of trigger levels will have to reduce this rate by three orders of magnitude. The entire trigger chain works on the main digitized data stream [@Avanzini:2010zz].
The low-level trigger (L0), implemented in hardware by means of FPGAs on the readout boards, reduces the data stream by a factor 10 to meet the maximum design rate for event readout of 1 MHz. The upper trigger levels (L1 and L2) are on a commodity PC farm for further reconstruction and event building.
In the standard implementation, the FPGAs on the readout boards compute simple trigger primitives on the fly, such as hit multiplicities and rough hit patterns, which are then and sent to a central processor for matching and trigger decision. Thus the maximum latency allowed for the synchronous L0 trigger is related to the maximum data storage time available on the data acquisition boards. For NA62 this value is up to 1 ms, in principle allowing use of more compute demanding implementations at this level, [*i.e.*]{}the GPUs.
As a first example of GPU application in the NA62 trigger system we studied the possibility to reconstruct rings in the RICH. The RICH L0 trigger processor is a synchronous level and the possibility to use the GPU must be verified. In order to test feasibility and performances, as a starting point 5 algorithms for single ring finding in a sparse matrix of 1000 points (centered on the PMs in the RICH spot) with 20 firing PMs (“hits”) on average have been implemented. Results of this study are available in [@Collazuol:2012zz] and show that GPU processing latency is stable and reproducible once data are available in the device internal memory.
In order to fully characterize latency and throughput of the RICH L0 trigger processor (GRL0TP), data communication between the detector readout boards (TEL62) and the L0 trigger processor (L0TP) need to be kept under control. The requisite on bandwidth is 400$\div$700 MB/s, depending on the final choice of the primitives data protocol which in turn depends on the amount of preprocessing actually to be implemented in the TEL62 FPGA. Therefore, in the final system 4$\div$6 [GbE]{}links will be used to extract primitives data from the readout board towards the L0TP.
The NIC was integrated in the GRL0TP prototype, using the “system loopback” setup described in section \[sec:perf\].
implementation {#sec:hw}
--------------
The is a [PCIe]{}Gen2 x8 NIC featuring a standard [GbE]{}interface implemented on [Altera]{}Stratix IV FPGA Development Kit (see Fig. \[fig:board\]). A custom mezzanine mounting 3 QSFP+ connectors, was designed to be optionally mounted on top of the [Altera]{}board and makes able to manage 3 [APElink]{}channels with switching capabilities up to 34 Gbps. [APElink]{}adopts a proprietary data transmission word stuffing protocol; this is pulled for free into .
![on [Altera]{}Stratix IV dev. board EP4SGX230KF40C2 with custom mezzanine card + 3 [APElink]{}channels.[]{data-label="fig:board"}](test_system.jpg){width=".49\textwidth"}
The [GbE]{}transmission system follows the general I/O interface architecture description of Fig. \[fig:NaNet\].
The [Altera]{}Triple Speed Ethernet Megacore (TSE MAC) is the Physical Link Coding, providing complete 10/100/1000 Mbps Ethernet IP modules. The *UDP Offloader* analyzes and interprets the data protocol. It deals with UDP packets payload extraction and provides a wide channel achieving 6.4 Gbps which is a 6 times greater bandwidth than what the standard [GbE]{}requires.
The data coming from TSE MAC are collected by the UDP Offloader through the [Altera]{}Avalon Streaming Interface and redirected into the [NaNet]{}hardware processing data path, avoiding the use of the the FPGA $\mu$controller ([Nios II]{}) from UDP traffic management. The *[NaNet]{}Controller* translates the data packets into [APEnet+]{}encapsulated ones, then hands them over to the Network Interface that takes care of moving them to their GPU memory buffer destination.
Results {#sec:perf}
-------
performances were assessed on a Supermicro SuperServer . The setup comprised a (Tylersburg chipset — Intel 5520) dual socket motherboard, 2 Intel 82576 [GbE]{}ports and [NVIDIA]{}M2070 GPU; sockets were populated with Intel Xeon X5570 @2.93 GHz. The host simulates the RO board by sending UDP packets containing primitives data from the host system [GbE]{}port to the [GbE]{}port hosted by , which in turn streams data directly towards CLOPS in GPU memory that are sequentially consumed by the CUDA kernel implementing the ring reconstruction algorithm. This measurement setup is called “system loopback”. Exploiting the x86 Time Stamp Counter (TSC) register as a common time reference, it was possible in a single process test application to measure latency as time difference between when a received buffer is signalled to the application and the moment before the first UDP packet of a bunch (needed to fill the receive buffer) is sent through the host [GbE]{}port. Communication and kernel processing tasks were serialized in order to perform the measure; This represents a situation: given RDMA capabilities, during normal operation this serialization does not occur and kernel processing seamlessly overlaps with data transfer. Similarly, we closed in a loopback configuration two of the three available [APElink]{}ports and performed the same measurement.
![Latency of [GbE]{}data transfer and of ring reconstruction CUDA kernel processing.[]{data-label="fig:latenza_nanet_fermi"}](nanet_comm_calc_lat_fermi.pdf){width=".49\textwidth"}
![Latency of [APElink]{}data transfer and of ring reconstruction CUDA kernel processing.[]{data-label="fig:latenza_apelink"}](apenet_comm_calc_lat_fermi.pdf){width=".49\textwidth"}
In Fig. \[fig:latenza\_nanet\_fermi\] latencies for varying size buffer transfers in GPU memory using the [GbE]{}link are represented. Besides the smooth behaviour increasing receive buffer sizes, fluctuations are minimal, matching both constraints for and, compatibly with link bandwidth, on data transfers; for a more detailed performance analysis, see [@NanetTwepp:2013].
Bandwidth and latency performances for [APElink]{}channel are in Fig. \[fig:latenza\_apelink\]. It is clear that the system remains within the 1 ms time budget with GPU receive buffer sizes in the $128\div1024$ events range. Although real system physical link and data protocol were used to show the behaviour on , we measured on a reduced bandwidth single [GbE]{}port system that could not match the 10 MEvents/s experiment requirement for the GRL0TP.
To demonstrate the suitability of design for the RICH L0TP, we decided to perform equivalent benchmarks using one of its [APElink]{}ports instead of the [GbE]{}one. Current implementation of [APElink]{}is able to sustain a data flow up to $\sim20$ Gbps. Results for latency of the RICH L0TP are shown in Fig. \[fig:latenza\_apelink\]: a single [APElink]{}data channel between RICH RO and GRL0TP systems roughly matches trigger throughput and latency requirements for receiving buffer size in the $4\div5$ Kevents range.
[NaNet$^3$]{}: the readout and board for the [KM3NeT-IT]{}underwater neutrino telescope
=======================================================================================
[KM3NeT-IT]{}is an underwater experimental apparatus for the detection of high energy neutrinos in the TeV$\div$PeV range based on the [Čerenkov]{}technique. The detector measures the visible [Čerenkov]{}photons induced by charged particles propagating in sea water at speed larger than that of light in the medium, and consists of an array of photomultipliers (PMT).
The charged particle track can be reconstructed measuring the time of arrival of the [Čerenkov]{}photons on the PMTs, whose positions must be known.
The [KM3NeT-IT]{}detection unit is called *tower* and consists of 14 floors vertically spaced 20 meters apart. The floor arms are about 8 m long and support 6 glass spheres called Optical Modules (OM): 2 OMs are located at each floor end and 2 OMs in the middle of the floor; each OM contains one 10 inches PMT and the electronics needed to digitize the PMT signal, format and transmit the data. Each floor hosts also two hydrophones, used to reconstruct in the OM position, and, where needed, oceanographic instrumentation to monitor site conditions relevant for the detector.
All data produced by OMs, hydrophones, and instruments, are collected by an electronic board contained in a vessel at the centre of the floor; this board, called *Floor Control Module* (FCM) manages the communication between the laboratory and the underwater devices, also distributing the timing information and signals. Timing resolution is fundamental in track reconstruction, [*i.e.*]{}pointing accuracy in reconstructing the source position in the sky. An overall time resolution of about 3 ns yields an angular resolution of 0.1 degrees for neutrino energies greater than 1 TeV. Such resolution depends on electronics but also on position measurement of the OMs, which is, in fact, continuously tracked.
The spatial accuracy required should be better than 40 cm.
The [KM3NeT-IT]{}DAQ and data transport architecture {#sec:DAQArc}
----------------------------------------------------
The DAQ architecture is heavily influenced by the need of a common timing distributed all over the system in order to correlate signals from different parts of the apparatus with the required nanosecond resolution. The aim of the data acquisition and transport electronics is to label each signal with a “time stamp”, [*i.e.*]{}the hit arrival time, in order to reconstruct tracks. This need implies that the readout electronics, which is spatially distributed, require common timing and a known delay with respect to a fixed reference. The described constraints hinted to the choice of a synchronous link protocol which embeds clock and data with a deterministic latency; due to the distance between the apparatus and shoreland, the transmission medium is forced to be an optical fiber.
All floor data produced by the OMs, the hydrophones and other devices used to monitor the apparatus status and the environmental conditions, are collected by the Floor Control Module (FCM) board, packed together and transmitted through the optical link. Each floor is independent from the others and is connected by an optical bidirectional virtual connection to the laboratory.
The data stream that a single floor delivers to shore has a rate of $\sim$300 [Mbps]{}, while the communication data rate is much lower, consisting only of data for the apparatus. To preserve optical power budget, the link speed is operated at 800 [Mbps]{}, which, using an 8B10B encoding, accounts for a 640 [Mbps]{}of user payload, well beyond experimental requirement.
Each FCM needs an communication endpoint counterpart. The limited data rate per FCM compared with link technologies led us to designing [NaNet$^3$]{}, an readout board able to manage multiple FCM data channels.
This design represents a [NaNet]{}customization for the [KM3NeT-IT]{}experiment, adding support in its I/O interface for a synchronous link protocol with deterministic latency at physical level and for a Time Division Multiplexing protocol at data level (see Fig. \[fig:NaNet\]).
[NaNet$^3$]{}implementation
---------------------------
The first stage design for [NaNet$^3$]{}was implemented on an evaluation board from Terasic, the [DE5-net]{}board, which is based on [Altera]{}Stratix-V GX FPGA, supports up to 4 SFP+ channels and a [PCIe]{}x8 edge connector.
The first constraint to be satisfied requires having a time delta with nanosecond precision between the wavefronts of three clocks:
- the first clock is an reference one (typically coming from a GPS and redistributed by custom fanout boards) and is used for the optical link transmission from [NaNet$^3$]{}towards the underwater FCM;
- the second clock is recovered from the incoming data stream by a CDR module at the receiving end of the FCM which uses it for sending its data payload from the apparatus back ;
- a third clock is again recovered by [NaNet$^3$]{}while decoding this payload at the end of the loop.
The link established in this way is fully synchronous.
The second fundamental constraint is the deterministic latency that the [Altera]{}Stratix device must enforce — as the FCM does — on both forward and backward path to allow correct time stamping of events on the PMT.
In this way, the [NaNet$^3$]{}board plays the role of a bridge between the 4 FCMs and the FCMServer — [*i.e.*]{}the hosting PC — through the [PCIe]{}bus. Control data en route to the underwater apparatus are correctly sent over the [PCIe]{}bus to the [NaNet$^3$]{}board, which then routes the data to the required optical link. On the opposite direction, both control and hydrophones data plus signals from the boards are extracted from the optical link and on the [PCIe]{}bus towards an application managing all the data. Since the data rate supported by the [PCIe]{}bus is much higher than the data produced by the electronics, we foresee to develop a custom board supporting more than 4 optical links. On the other hand, the [GPUDirect]{}[RDMA]{}features of [NaNet]{}, fully imported in [NaNet$^3$]{}design, will allow us, at a later stage, to build an effective, , platform, in order to investigate improved trigger and data reconstruction algorithms.
At a higher level, two systems handle the data that come from and go to : the Trigger System, which is in charge of analysing the data from PMTs extracting meaningful data from noise, and the Data Manager, which controls the apparatus. The FCMServer communicates with these two systems using standard [10-GbE]{}network links.
The [NaNet$^3$]{}preliminary results
------------------------------------
Preliminary results show that the interoperability between different vendors FPGA devices can be achieved and the timing resolution complies with physics requirements.\
We develop a test setup to explore the fixed latency capabilities of a complete links chain.\
We leverage on the fixed latency native mode of the [Altera]{}transceivers and on the hardware fixed latency implementation for Xilinx device [@Giordano:2011]. The testbed is composed by the [NaNet$^3$]{}board and the FCM board respectively emulating the and boards connected by optical fibers (see Fig. \[fig:testbed\]).\
![[NaNet$^3$]{}testbed.[]{data-label="fig:testbed"}](testbed1.jpg){width=".49\textwidth"}
The external clock has been input to the [NaNet$^3$]{}to clock the transmitting side of the device. A sequence of dummy parallel data are serialised, 8b/10b encoded and transmitted, together with the embedded serial clock, at a data rate of 800 Mbps along the fiber towards the receiver side of the FCM system. The FCM system recovers from the received clock and transmits the received data and recovered clock back to the [NaNet$^3$]{}boards. Lastly, the received side of [NaNet$^3$]{}deserializes data and produces the received clock.\
The way to test the fixed latency features of the SerDes hardware implementation is quite easy taking into account that every time a new initialisation sequence, following an hardware reset or a powerup of the SerDes hardware, has been done, we should be able to measure the same phase shift between transmitted and received clock, equal to the fixed number of serial clock cycles shift used to correctly align the deserialised data stream. Fig. \[fig:Fixed\_Lat\] is a picture taken from scope acquisition in Infinity Persistence showing the results of a preliminary 12 h test where every 10 s a new *reset and align* procedure has been issued. The [NaNet$^3$]{}transmitter parallel clock (the purple signal) maintains exactly the same phase difference with the receiver parallel clock (the yellow signal) and with the FCM recovered clock (the green signal).
![Deterministic latency feature of [NaNet$^3$]{}SerDes: the plot scope shows the phase alignment of the transmitting (purple) and receiving (yellow) parallel clocks after 12 h test of periodic reset and initialisation sequence.[]{data-label="fig:Fixed_Lat"}](Fixed_Latency_Scope.png){width=".49\textwidth"}
Conclusions and future work
===========================
Our [NaNet]{}design proved to be efficient in performing data communication between the NA62 RICH readout system and the GPU-based L0 trigger processor over a single GbE link. Preliminary results of its customization for the data transport system of the [KM3NeT-IT]{}experiment shows that the fundamental requirement of a deterministic latency link can be implemented using [NaNet]{}, paving the way to the use of hybrid trigger and data reconstruction systems. 10 GbE board, currently under development, will allow for a full integration of our architecture in the NA62 experiment and smooth the path to [NaNet]{}usage in other contexts.
[10]{}
R. Ammendola, A. Biagioni, O. Frezza, A. Lonardo, F. L. Cicero, P. S. Paolucci, D. Rossetti, F. Simula, L. Tosoratto, and P. Vicini, [*[APEnet+]{} 34 [Gbps]{} data transmission system and custom transmission logic*]{}, [*Journal of Instrumentation*]{} [**8**]{} (2013), no. 12 C12022.
A. Aloisio, F. Ameli, A. D’Amico, R. Giordano, V. Izzo, and F. Simeone, “The [NEMO]{} experiment data acquisition and timing distribution systems,” in *Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC), 2011 IEEE*, Oct 2011, pp. 147–152.
R. Ammendola, A. Biagioni, O. Frezza, F. L. Cicero, A. Lonardo, P. S. Paolucci, D. Rossetti, F. Simula, L. Tosoratto, and P. Vicini, “[APEnet+]{}: a 3[D]{} [T]{}orus network optimized for [GPU]{}-based [HPC]{} systems,” *Journal of Physics: Conference Series*, vol. 396, no. 4, p. 042059, 2012. \[Online\]. Available: <http://stacks.iop.org/1742-6596/396/i=4/a=042059>
G. Lamanna, “The [NA62]{} experiment at [CERN]{},” *Journal of Physics: Conference Series*, vol. 335, no. 1, p. 012071, 2011. \[Online\]. Available: <http://stacks.iop.org/1742-6596/335/i=1/a=012071>
C. Avanzini *et al.*, “[The trigger and DAQ system for the NA62 experiment]{},” *Nucl. Instrum. Methods Phys. Res., A*, vol. 623, pp. 543–545, 2010.
G. [Collazuol]{}, G. [Lamanna]{}, J. [Pinzino]{}, and M. S. [Sozzi]{}, “[Fast online triggering in high-energy physics experiments using [GPU]{}s]{},” *Nuclear Instruments and Methods in Physics Research A*, vol. 662, pp. 49–54, Jan. 2012.
R. Ammendola, A. Biagioni, O. Frezza, G. Lamanna, A. Lonardo, F. L. Cicero, P. S. Paolucci, F. Pantaleo, D. Rossetti, F. Simula, M. Sozzi, L. Tosoratto, and P. Vicini, “Nanet: a flexible and configurable low-latency nic for real-time trigger systems based on gpus,” *Journal of Instrumentation*, vol. 9, no. 02, p. C02023, 2014. \[Online\]. Available: <http://stacks.iop.org/1748-0221/9/i=02/a=C02023>
R. Giordano and A. Aloisio, “Fixed latency multi-gigabit serial links with [X]{}ilinx [FPGA]{},” *IEEE Transaction On Nuclear Science*, vol. 58, no. 1, pp. 194–201, 2011.
[^1]: INFN Sezione di Roma, Italy.
[^2]: INFN Sezione di Tor Vergata, Italy.
[^3]: INFN Sezione di Pisa, Italy.
[^4]: CERN, Switzerland.
[^5]: Università di Roma Sapienza, Dipartimento di Fisica, Italy.
[^6]: NVIDIA Corporation, U.S.A.
[^7]: A. Lonardo is the corresponding author (.)
[^8]: This work was supported in part by the EU Framework Programme 7 EURETILE project, grant number 247846; R. Ammendola was supported by MIUR (Italy) through the INFN SUMA project; G. Lamanna and M. Sozzi thank the GAP project, partially supported by MIUR under grant RBFR12JF2Z “Futuro in ricerca 2012”.
[^9]: Manuscript received May 22, 2014.
|
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abstract: 'One of the most fundamental problems in network study is community detection. The stochastic block model (SBM) is one widely used model for network data with different estimation methods developed with their community detection consistency results unveiled. However, the SBM is restricted by the strong assumption that all nodes in the same community are stochastically equivalent, which may not be suitable for practical applications. We introduce a pairwise covariates-adjusted stochastic block model (PCABM), a generalization of SBM that incorporates pairwise covariate information. We study the maximum likelihood estimates of the coefficients for the covariates as well as the community assignments. It is shown that both the coefficient estimates of the covariates and the community assignments are consistent under suitable sparsity conditions. Spectral clustering with adjustment (SCWA) is introduced to efficiently solve PCABM. Under certain conditions, we derive the error bound of community estimation under SCWA and show that it is community detection consistent. PCABM compares favorably with the SBM or degree-corrected stochastic block model (DCBM) under a wide range of simulated and real networks when covariate information is accessible.'
address:
- |
Department of Statistics\
Columbia University\
New York, NY 10027\
- |
Department of Biostatistics\
New York University\
New York, NY 10003\
author:
-
-
-
-
bibliography:
- 'reference.bib'
title:
- 'Pairwise Covariates-adjusted Block Model for Community Detection'
- 'SUPPLEMENTARY MATERIAL TO “Pairwise Covariates-adjusted Block Model for Community Detection"'
---
Introduction {#sec:intro}
============
Networks describe the connections among subjects in a population of interest. Its wide applications have attracted researchers from different fields. In social media, people’s behaviors and interests can be unveiled by network analysis (Facebook friends and Twitter followers). In ecology, a food web of predator-prey interactions can provide more information about the habits of individuals and the structure of biocoenosis. There are also wide applications in computer science, biology, physics, and economics [@getoor2005link; @goldenberg2010survey; @newman1963introduction; @graham2014econometric].
Community detection is one of the most studied problems for network data. Communities can be intuitively understood as groups of nodes that are densely connected within groups while sparsely connected between groups. Identifying network communities not only help better understand the structural features of the network but also offers practical benefits. For example, communities in social networks tend to share similar interests, which could provide useful information to build recommendation systems. Most community detection methods fall into two categories: algorithm-based and model-based. For algorithm-based methods [@bickel2009nonparametric; @newman2006modularity; @zhao2011community; @wilson2014testing; @wilson2017community], they come up with an objective function (e.g., modularity) and then optimize it to obtain the community estimation. Model-based methods focus on how the edges are generated from a probabilistic model. Some popular models include stochastic block model [@holland1983stochastic], mixture model [@newman2007mixture], degree-corrected stochastic block model [@karrer2011stochastic], latent space models [@hoff2002latent; @handcock2007model; @hoff2008modeling] and so on. For a systematic review of statistical network models, see [@goldenberg2010survey; @fortunato2010community].
The classical stochastic block model (SBM) assumes that the connection between each pair of nodes only depends on the communities they belong to. For SBM, community detection consistency has been established for various methods, including modularity maximization [@newman2006modularity], profile likelihood [@bickel2009nonparametric], spectral clustering [@rohe2011spectral; @lei2015consistency], maximum likelihood [@choi2012stochastic], variational inference [@bickel2013], penalized local maximum likelihood estimation [@gao2017achieving], among others. In the real world, the connection of nodes may depend on not only community structure but also on nodal covariates. For example, in an ecological network, the predator-prey link between species may depend on their prey types as well as their habits, body sizes, and living environment. Incorporating nodal information into the network model should help us recover a more accurate community structure.
Depending on the relationship between communities and covariates, there are, in general, two classes of models, as shown in Figure \[fig:model\]: *covariates-adjusted* and *covariates-confounding*. $\mathbf{c}$, $Z$ and $A$ respectively stands for latent community label, pairwise covariates, and adjacency matrix. In Figure \[fig:subfig1\], the latent community, and the covariates jointly determine the network structure. One typical example of this model is the friendship network between students. Students become friends for various reasons: they are in the same class; they have the same hobbies; they are of the same ethnic group. Without adjusting those covariates, it is hard to believe $A$ represents any single community membership. We will analyze one such example in detail in Section \[sec:realdata\]. On the other hand, covariates sometimes carry the same community information as the adjacency matrix, which is shown in Figure \[fig:subfig2\]. The name “confounding" comes from graph model [@greenland1999confounding]. The Citation network is a perfect example of this model [@tan2016topic]. When the research topic is treated as the community label for each article, the citation links would largely depend on the research topics of the article pair. Meanwhile, the distribution of the keywords is also likely to be driven by the specific topic the article is about.
[0.35]{}
[0.35]{}
Most researchers modify SBM in the above two ways to incorporate covariates’ information. For the covariates-adjusted model, [@newman2016structure] uses covariates to construct the prior for community label and then generates edges by degree-corrected model; [@yan2016statistical] proposes a directed network model with logistic function, but it does not consider possible community structure. For the covariates-confounding model, [@weng2016community] uses a logistic model as the prior for community labels. [@zhang2016community] proposes a joint community detection criterion, which is an analog of modularity, to incorporate node features. [@ma2017exploration] presents algorithms for two special class of latent space model that incorporates edge covariates. [@wu2017generalized] proposes a generalized linear model with low-rank effects to model network edges, which implies the community structure though not mentioned explicitly. In practice, the true model is usually a compromise between those two. We will use an example to show in Section \[sec:simu\] the connection between those two models.
In this work, we propose a simple yet effective model called *pairwise covariates-adjusted stochastic block model* (PCABM), which extends the SBM by adjusting the probability of connections by the contribution of pairwise covariates. Through this model, we can learn how each covariate affects the connections by looking at its corresponding regression coefficient. In addition, we show the consistency and asymptotic normality for MLE. Besides likelihood methods, we also propose a novel spectral clustering method called *spectral clustering with adjustment* (SCWA). Note that [@binkiewicz2014covariate] also uses a modified version of spectral clustering to incorporate nodal covariates, but it is not based on a specific model. We prove desirable theoretical properties for SCWA applied to PCABM, and show that as a fast algorithm, using it as an initial estimator for the likelihood method usually leads to more accurate community detection than random initialization.
The rest of the paper is organized as follows. In Section \[sec:pcabm\], we introduce the PCABM. We then show the asymptotic properties of the coefficient estimates in Section \[sec:gamma\]. Following that, we introduce two methods for community estimation in Section \[sec:likethm\] and \[sec:scwa\]. Simulations and applications on real networks will be discussed in Section \[sec:simu\]. We conclude the paper with a short discussion in Section \[sec:realdata\]. All proofs are relegated to the Appendix and Supplementary Materials.
Here, we introduce some notations to facilitate the discussion. For a square matrix $M\in \mathbb{R}^{n\times n}$, let $\|M\|$ be the operator norm of $M$, $\|M\|_F=\sqrt{\mbox{trace}(M^TM)}$, $\|M\|_{\infty}=\max_i\sum_{j=1}^n|M_{ij}|$, $\|M\|_0=\#\{(i,j)|M_{ij}\neq0\}$, and $\|M\|_{\max}=\max_{ij}|M_{ij}|$. $\lambda_{\min}(M)$ is the minimum eigenvalue of $M$. For index sets $I,J\subset[n]:=\{1,2,\cdots,n\}$, $M_{I\cdot}$ and $M_{\cdot J}$ are the sub-matrices of $M$ consisting the corresponding rows and columns, respectively. For a vector $\mathbf{x}\in \mathbb{R}^n$, let $\|\mathbf{x}\|=\sqrt{\sum_{i=1}^n x_i^2}$ and $\|\mathbf{x}\|_{\infty}=\max_i |x_i|$. We define the Kronecker power by $\mathbf{x}^{\otimes(k+1)}=\mathbf{x}^{\otimes k}\otimes\mathbf{x}$, where $\otimes$ is the Kronecker product.
For any positive integer $K$, we define $I_K\in\mathbb{R}^{K\times K}$ to be the identity matrix, $J_K\in\mathbb{R}^{K\times K}$ to be all-one matrix, $\mathbf{1}_K$ to be all-one vector. When there is no confusion, we will sometimes omit the subscript $K$. For a vector $\mathbf{x}\in\mathbb{R}^K$, $D(\mathbf{x})\in\mathbb{R}^{K\times K}$ represents the diagonal matrix whose diagonal elements take the value of $\mathbf{x}$. For an event $A$, the indicator function is written as $\mathbbm{1}(A)$. Some Bachmann-Landau notations are clarified as follows. For two real number sequences $x_n$ and $y_n$, we say $x_n=o(y_n)$ if $\lim_{n\to\infty}x_n/y_n=0$, $x_n=O(y_n)$ if $\limsup_{n\to\infty}|x_n/y_n|\leq\infty$.
Pairwise Covariates-Adjusted Stochastic Block Model {#sec:pcabm}
===================================================
We consider a graph with $n$ nodes and $K$ communities, where $K$ is assumed fixed and does not increase with $n$. In this paper, we focus on undirected weighted graphs without self-loops. All edge information is incorporated into a symmetric adjacency matrix $A=[A_{ij}]\in\mathbb{N}^{n\times n}$ with diagonal elements being zero, where $\mathbb{N}$ represents the set of nonnegative integers. The total number of possible edges is denoted by $N_n=n(n-1)/2$. The true node labels $\mathbf{c}=\{c_1,\cdots,c_n\}\in \{1,\cdots,K\}^n$ are drawn independently from a multinomial distribution with parameter vector $\boldsymbol\pi=(\pi_1,\cdots,\pi_K)^T$, where $\sum_{i=1}^K\pi_k=1$ and $\pi_k>0$ for all $k$. The community detection problem is aiming to find a disjoint partition of the node, or equivalently, a set of node labels $\mathbf{e}=\{e_1,\cdots,e_n\}\in \{1,\cdots,K\}^n$ that is close to $\mathbf{c}$, where $e_i\in\{1,\cdots,K\}$ is the label for node $i$.
In classical SBM, we assume $\text{Pr}(A_{ij}=1|\boldsymbol{c}) =B_{c_ic_j}$, where $B=[B_{ab}]\in [0,1]^{K\times K}$ is a symmetric matrix with no identical rows. In practice, the connection between two nodes may depend not only on the communities they belong to, but also on the nodal covariates (e.g., gender, age, religion). To fix idea, assume in addition to $A$, we have observed a pairwise $p$-dimensional vector $\mathbf{z}_{ij}$ between nodes $i$ and $j$. Denote the collection of the pairwise covariates among nodes as $Z=[\mathbf{z}^T_{ij}]\in\mathbb{R}^{n^2\times p}$. Here, we assume $\mathbf{z}_{ij} = \mathbf{z}_{ji}$ and $\mathbf{z}_{ii}=0$.
Now, we are ready to introduce the *Pairwise Covariates-Adjusted Stochastic Block Model* (PCABM). For $i<j$, conditional on the membership vector $\mathbf{c}$ and the pairwise covariate matrix $Z$, $A_{ij}$’s are independent and $$A_{ij}\sim{\rm Poisson}(\lambda_{ij}),\ \lambda_{ij}=B_{c_ic_j}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0},$$ where $\boldsymbol{\gamma}^0$ is the true coefficient vector for the pairwise covariates. In addition to the goal of recovering the community membership vector $\mathbf{c}$, we would also like to get an accurate estimate for $\boldsymbol{\gamma}^0$. The specific term $\exp(\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0)$ is introduced here to adjust the connectivity between nodes $i$ and $j$. Here, as in the vanilla SBM, we assume a sparse setting for $B=\rho_n\bar{B}$, with $\bar{B}$ fixed and $\rho_n\to0$ as $n\to\infty$. Note that due to the contribution of Z, $\varphi_n=n\rho_n$ is no longer the expected degree as in the vanilla SBM [@zhao2012consistency], but it is still useful as a measure of the network sparsity. It is easy to observe that when $\boldsymbol{\gamma}^0=0$, PCABM reduces into the vanilla Poisson SBM.
Under PCABM, the likelihood function is $$\begin{aligned}
\mathcal{L}(\mathbf{e},\boldsymbol{\gamma},B,{\boldsymbol{\pi}}|A,Z)\propto\prod_{i=1}^n\pi_{e_i}\prod_{i<j}B_{e_ie_j}^{A_{ij}}e^{A_{ij}\mathbf{z}_{ij}^T\boldsymbol{\gamma}}\exp\left(-B_{e_ie_j}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}}\right).\end{aligned}$$ Define $$\begin{aligned}
n_k(\mathbf{e})&=\sum_{i=1}^n\mathbbm{1}(e_i=k),\ O_{kl}(\mathbf{e})=\sum_{ij}A_{ij}\mathbbm{1}(e_i=k, e_j=l),\\
E_{kl}(\mathbf{e},\boldsymbol{\gamma})&=\sum_{i\neq j}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}}\mathbbm{1}(e_i=k, e_j=l)=\sum_{(i,j)\in s_{\mathbf{e}}(k,l)}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}},\end{aligned}$$ where $s_{\mathbf{e}}(k,l)=\{(i,j)|e_i=k,e_j=l,i\neq j\}$. Under assignment $\mathbf{e}$, $n_k(\mathbf{e})$ represents the number of nodes in community $k$. For $k\neq l$, $O_{kl}$ is the total number of edges between communities $k$ and $l$; for $k=l$, $O_{kk}$ is twice of the number of edges within community $k$. $E_{kl}$ is the summation of all pair-level factors between communities $k$ and $l$. We can write the log-likelihood function as $$\begin{aligned}
\log\mathcal{L}(\mathbf{e},\boldsymbol{\gamma},B,\boldsymbol{\pi}|A,Z)\propto&\sum_kn_k(\mathbf{e})\log\pi_k+\frac{1}{2}\sum_{kl}O_{kl}(\mathbf{e})\log B_{kl}\\
&-\frac{1}{2}\sum_{kl}B_{kl}E_{kl}(\mathbf{e},\boldsymbol{\gamma})+\sum_{i<j}A_{ij}\mathbf{z}_{ij}^T\boldsymbol{\gamma}.\end{aligned}$$ Given $\mathbf{e}$ and $\boldsymbol{\gamma}$, we derive the MLE $\hat{\pi}_k(\mathbf{e})=\frac{n_k(\mathbf{e})}{n}$ and $\hat{B}_{kl}(\mathbf{e},\boldsymbol{\gamma})=\frac{O_{kl}(\mathbf{e})}{E_{kl}(\mathbf{e},\boldsymbol{\gamma})}$. Plugging $\hat{B}(\mathbf{e},\boldsymbol{\gamma})$ and $\hat{\boldsymbol{\pi}}(\mathbf{e})$ into the original log-likelihood and discarding the constant terms, we have $$\begin{aligned}
\label{eq:log-lik}
\begin{split}
&\log\mathcal{L}(\mathbf{e},\boldsymbol{\gamma},\hat{B},\hat{{\boldsymbol{\pi}}}|A,Z)\\
\propto&\frac{1}{2}\sum_{kl}O_{kl}(\mathbf{e})\log\frac{O_{kl}(\mathbf{e})}{E_{kl}(\mathbf{e},\boldsymbol{\gamma})}+\sum_{i<j}A_{ij}\mathbf{z}_{ij}^T\boldsymbol{\gamma}+\sum_kn_k(\mathbf{e})\log\frac{n_k(\mathbf{e})}{n}.
\end{split}\end{aligned}$$ Out target is to maximize (\[eq:log-lik\]) w.r.t. $\mathbf{e}$ and $\boldsymbol{\gamma}$. We consider a two-step sequential estimation procedure by first studying the estimation of $\boldsymbol{\gamma}^0$ in Section \[sec:gamma\] and then the estimation of $\mathbf{c}$ in Section \[sec:likethm\] (likelihood method) and Section \[sec:scwa\] (spectral method).
Estimation of Coefficients for Pairwise Covariates\[sec:gamma\]
===============================================================
As the first step to maximize the log-likelihood, we consider the estimation of coefficients $\boldsymbol{\gamma}^0$ for pairwise covariates. To this end, we impose the following conditions on $Z$.
\[cond:zbd\] $\{\mathbf{z}_{ij},i<j\}$ are i.i.d. and uniformly bounded, i.e., for $\forall i<j$, $\|\mathbf{z}_{ij}\|_{\infty}\leq\zeta$, where $\zeta>0$ is some constant.
The bounded support condition for $\boldsymbol{z}_{ij}$ is introduced to simplify the proof. It could be relaxed to allow the upper bound to grow slowly with network size $n$.
By Condition \[cond:zbd\], we know that $e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}$ is also uniformly bounded. We define the bound as $\beta_l\leq e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\leq\beta_u$. Thus, the following expectations exist: $\theta(\boldsymbol{\gamma}^0)\equiv\mathbb{E}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\in\mathbb{R}^+$, $\boldsymbol{\mu}(\boldsymbol{\gamma}^0)\equiv\mathbb{E}\mathbf{z}_{ij}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\in\mathbb{R}^p$, and $\Sigma(\boldsymbol{\gamma}^0)\equiv\mathbb{E}\mathbf{z}_{ij}\mathbf{z}_{ij}^Te^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\in\mathbb{R}^{p\times p}$.
To ensure $\boldsymbol{\gamma}^0$ is the unique solution to maximize the likelihood in the population version, we impose the following regularity condition is at the true $\boldsymbol{\gamma}^0$.
\[cond:zpd\] $\Sigma(\boldsymbol{\gamma}^0)-\theta(\boldsymbol{\gamma}^0)^{-1}\boldsymbol{\mu}(\boldsymbol{\gamma}^0)^{\otimes 2}$ is positive definite.
To understand the implication of Condition \[cond:zpd\], consider the function $g(\boldsymbol{\gamma})=\theta(\boldsymbol{\gamma})\Sigma(\boldsymbol{\gamma})-\boldsymbol{\mu}(\boldsymbol{\gamma})^{\otimes2}$. In the special case of SBM where $\boldsymbol{\gamma}^0=\mathbf{0}$, we have $g(\mathbf{0})=\mathbb{E}[\mathbf{z}^{\otimes2}]-\mathbb{E}[\mathbf{z}]^{\otimes2}={\rm cov}(\mathbf{z})$. To avoid multicollinearity, it’s natural for us to require ${\rm cov}(\mathbf{z})$ to be positive definite. For a general PCABM, we require $g(\boldsymbol{\gamma^0})$ to be positive definite at the true value $\boldsymbol{\gamma^0}$.
For a given initial community assignment $\boldsymbol{e}$, denote by $\ell_{\mathbf{e}}$ the log-likelihood terms in (\[eq:log-lik\]) containing $\boldsymbol{\gamma}$, which is $$\ell_{\mathbf{e}}(\boldsymbol{\gamma})\equiv\sum_{i<j}A_{ij}\mathbf{z}_{ij}^T\boldsymbol{\gamma}-\frac{1}{2}\sum_{kl}O_{kl}(\mathbf{e})\log E_{kl}(\mathbf{e},\boldsymbol{\gamma}).$$ We consider the following estimate $\hat{\boldsymbol\gamma}$: $$\begin{aligned}
\label{eq::gamma_hat}
\hat{\boldsymbol{\gamma}}(\boldsymbol{e}) = \arg\max_{\boldsymbol{\gamma}} \ell_{\textbf{e}}(\boldsymbol{\gamma}).\end{aligned}$$
To investigate the asymptotic properties of MLE $\hat{\boldsymbol{\gamma}}(\mathbf{e})$, we impose the following condition of the initial community assignment$\mathbf{e}$.
\[cond:e1\] For any $k\in[K]$, $\kappa_1\leq n_k(\mathbf{e})/n \leq \kappa_2$, where $0<\kappa_1\leq \kappa_2<1$ are some positive constants.
Condition \[cond:e1\] is very mild and can be satisfied by any non-degenerate assignment $\mathbf{e}$ (e.g., a random community assignment with fixed size for each community that is proportional to the total network size). Now, we are ready to present the consistency of $\hat{\boldsymbol{\gamma}}$ with the following theorem.
\[CGM\] Under PCABM, assume Conditions \[cond:zbd\], \[cond:zpd\], and \[cond:e1\] hold. As $n\to\infty$, if $N_n\rho_n\to\infty$, $\hat{\boldsymbol{\gamma}}(\mathbf{e})\xrightarrow{p}\boldsymbol{\gamma}^0.$
This theorem shows that we can get a consistent estimate of $\boldsymbol\gamma^0$ given any community assignment $\mathbf{e}$ under mild conditions. Next, we present the asymptotic normality property for $\hat{\boldsymbol{\gamma}}(\mathbf{e})$.
\[THM:ASY\] Under PCABM, assume Conditions \[cond:zbd\], \[cond:zpd\], and \[cond:e1\] hold. As $n\to\infty$, if $N_n\rho_n\to\infty$, $$\begin{aligned}
\sqrt{N_n\rho_n}(\hat{\boldsymbol{\gamma}}(\mathbf{e})-\boldsymbol{\gamma}^0)\stackrel{d}{\to} \mathcal{N}(\mathbf{0}, \Sigma_\infty^{-1}(\boldsymbol{\gamma}^0)),
\end{aligned}$$ where $\Sigma_\infty(\boldsymbol{\gamma}^0)=\sum_{ab}\bar{B}_{ab}\pi_a\pi_b[\Sigma(\boldsymbol{\gamma}^0)-\theta(\boldsymbol{\gamma}^0)^{-1}\boldsymbol{\mu}(\boldsymbol{\gamma}^0)^{\otimes 2}]$.
Different from [@yan2016statistical], in which the network is dense, the convergence rate is $\sqrt{N_n\rho_n}$ rather than $\sqrt{N_n}$ since the effective number of edges is reduced from $N_n$ to $N_n\rho_n$. The asymptotic covariance matrix $\Sigma_\infty^{-1}(\boldsymbol{\gamma}^0)$ depends on $\theta(\boldsymbol{\gamma}^0)$, $\boldsymbol{\mu}(\boldsymbol{\gamma}^0)$, and $\Sigma(\boldsymbol{\gamma}^0)$, which can be estimated empirically by the plug-in method.
Now, with a consistent estimate of $\boldsymbol{\gamma}^0$, we are ready to study the estimation of $\boldsymbol{c}$. In the next two sections, we will present two different methods for estimating $\boldsymbol{c}$, namely the likelihood-based estimate in Section \[sec:likethm\] and the spectral method in Section \[sec:scwa\].
Likelihood Based Estimate for Community Labels {#sec:likethm}
==============================================
This section presents a likelihood based estimate for community labels by maximizing $\log\mathcal{L}$ regarding $\mathbf{e}$ with $\hat{\boldsymbol{\gamma}}$ from Section \[sec:gamma\]. We demonstrate that under a large class of criteria $Q$, given $\hat{\boldsymbol{\gamma}}(\mathbf{e})$ is consistent, the consistency of $\hat{\mathbf{c}}(\mathbf{e})$ is guaranteed. Here, we write $\hat{\boldsymbol{\gamma}}$ and $\hat{\mathbf{c}}$ as functions of $\mathbf{e}$ to emphasize they depend on the initial label assignments.
Plugging $\hat{\boldsymbol{\gamma}}$ into , the log-likelihood function can be rewritten as $$\ell_{\hat{\boldsymbol{\gamma}}}(\mathbf{e})=\frac{1}{2}\sum_{kl}O_{kl}(\mathbf{e})\log\frac{O_{kl}(\mathbf{e})}{E_{kl}(\mathbf{e},\hat{\boldsymbol{\gamma}})}+\sum_kn_k(\mathbf{e})\log\frac{n_k(\mathbf{e})}{n}.$$ Then, our maximum likelihood estimate for the community label is $$\begin{aligned}
\label{eq::mle-e}
\hat{\mathbf{c}} = \hat{\mathbf{c}} (\hat{\boldsymbol{\gamma}}):=
\arg\max_{\mathbf{e}}\ell_{\hat{\boldsymbol{\gamma}}}(\mathbf{e}). \end{aligned}$$ Instead of directly analyzing $\ell_{\hat{\boldsymbol{\gamma}}}(\mathbf{e})$, similar to [@zhao2012consistency], we first investigate the maximizer of a general class of criteria defined as $$\begin{aligned}
\label{eq:cri}
Q(\mathbf{e},\hat{\boldsymbol{\gamma}}):= F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right),
\end{aligned}$$ where $O(\mathbf{e})=[O_{kl}(\mathbf{e}), k,l\in[K]]$ and $E(\mathbf{e},\hat{\boldsymbol{\gamma}})=[E_{kl}(\mathbf{e},\hat{\boldsymbol{\gamma}}), k,l\in[K]]$. Then, we show our log-likelihood function falls in this class of criteria, implying the consistency of label estimation. We say the criterion $Q$ is *consistent* if the estimated labels, obtained by maximizing the criterion, $\hat{\mathbf{c}}=\arg\max_{\mathbf{e}}Q(\mathbf{e},\hat{\boldsymbol{\gamma}})$ is *consistent*.
Following [@zhao2012consistency], we consider two versions of community detection consistency. Note that the consistency in community detection is understood under any permutation of the labels. To be more precise, let $\mathcal{P}_k$ be the collection of all permutation functions of $[K]$. (1) We say the label estimate $\hat{\mathbf{c}}$ is *weakly consistent* if $\text{Pr}[n^{-1}\min_{\sigma\in \mathcal{P}_K} \sum_{i=1}^n \mathbbm{1}(\sigma(\hat c_i)\neq c_i)<\varepsilon]\to1$ for any $\varepsilon>0$ as $n\to\infty$. (2) We say $\hat{\mathbf{c}}$ is *strongly consistent* if $\text{Pr}[\min_{\sigma\in \mathcal{P}_K} \sum_{i=1}^n \mathbbm{1}(\sigma(\hat c_i)\neq c_i)=0]\to1$, as $n\to\infty$. We will investigate the required sparsity conditions for both versions of consistency, respectively. One key condition of $Q$ for implying consistent community detection is that it reaches the maximum at $\boldsymbol{c}$ under the true parameter $\boldsymbol{\gamma}^0$ in the “population version", which is $F\left(\frac{\mathbb{E}[O(\mathbf{c})]}{2N_n\rho_n},\frac{\mathbb{E}[E(\mathbf{c},\boldsymbol{\gamma}^0)]}{2N_n}\right)$. To further demonstrate what the “population version" is, we introduce some notations. Given a community assignment $\mathbf{e}\in[K]^n$, we define $R(\mathbf{e})\in\mathbb{R}^{K\times K}$ with its elements being $R_{ka}(\mathbf{e})=\frac{1}{n}\sum_{i=1}^n\mathbbm{1}(e_i=k, c_i=a)$. One can view $R$ as the empirical joint distribution of $\mathbf{e}$ and $\mathbf{c}$. Next, we introduce the key condition for the function $F$ in terms of $R$. as follows.
\[cond:max\] $F(R\bar{B}R^T,RJR^T)$ is uniquely [^1] maximized over $\mathcal{R}=\{R:R\geq0,R^T\mathbf{1}=\boldsymbol{\pi}\}$ by $R=D(\boldsymbol{\pi})$.
Besides the common factor $\mathbb{E}[\exp(\mathbf{z}_{ij}^T\boldsymbol{\gamma})^0]$, the first term is $\bar{B}$ weighted by community pairwise proportions, the second term is the normalized pairwise count between two communities. This reduces the criteria to the form described in [@zhao2012consistency], thus similar methods can be applied to show the consistency of community estimation. In addition, we need more regularity conditions for $F$, analogous to those in [@zhao2012consistency].
\[cond:lip\] Some regularity conditions hold for $F$.
1. $F$ is Lipschitz in its arguments and $F(cX_0,cY_0)=cF(X_0,Y_0)$ for constant $c\neq0$.
2. The directional derivatives $\frac{\partial^2F}{\partial\epsilon^2}(X_0+\epsilon(X_1-X_0),Y_0+\epsilon(Y_1-Y_0))|_{\epsilon=0+}$ are continuous in $(X_1,Y_1)$ for all $(X_0,Y_0)$ that is in a neighborhood of $(D(\boldsymbol{\pi})\bar{B}D(\boldsymbol{\pi})^T,\boldsymbol{\pi}\boldsymbol{\pi}^T)$.
3. Let $G(R,\bar{B})=F(R\bar{B}R^T,RJR^T)$. On $\mathcal{R}$, for all $\boldsymbol{\pi}$, $\bar{B}$ and some constant $C>0$, the gradient satisfies $\frac{\partial G((1-\epsilon)D(\boldsymbol{\pi})+\epsilon R,\bar{B})}{\partial \epsilon}|_{\epsilon=0+}<-C$.
Notice that the first condition in Condition \[cond:lip\] ensures that we could extract the common exponential factor. Thus we can ignore that term when we consider the population maximum in Condition \[cond:max\]. Naturally, the consistency of $\hat{\boldsymbol{\gamma}}$ is also required to ensure that the “sample version” is close to the “population version”. Now the main theorem is stated as follows.
\[thm:gene\] For all criteria $Q$ of the form (\[eq:cri\]), if $\boldsymbol{\pi}, \bar{B}, F$ satisfy Conditions \[cond:max\], \[cond:lip\] and $\hat{\boldsymbol{\gamma}}$ is consistent, then under PCABM, $Q$ is weakly consistent if $\varphi_n\to\infty$ and strongly consistent if $\varphi_n/\log n\to\infty$.
Consider a network generated by PCABM. If we ignore the covariates and use SBM instead, it is equivalent to fit the PCABM by setting $\hat{\boldsymbol{\gamma}}=0$. Suppose nodes $i$, $j$, and $k$ belong to the first community. We have $\lambda_{ik} = B_{11} e^{\mathbf{z}^T_{ik}\boldsymbol{\gamma}^0} \neq B_{11}e^{\mathbf{z}^T_{jk}\boldsymbol{\gamma}^0} = \lambda_{jk}$ unless $\mathbf{z}^T_{ik}\boldsymbol{\gamma}^0 = \mathbf{z}^T_{jk}\boldsymbol{\gamma}^0$. Therefore, the nodes in the same community could behave very differently depending on the value of the pairwise covariates. As a result, we would expect the regular SBM to have difficulty in detecting the communities when the magnitude of $\boldsymbol{\gamma}^0$ is large. We will demonstrate this point through a simulation example in Section \[sec:simu\] where we vary the magnitude of $\boldsymbol{\gamma}^0$.
We will show in the appendix that the log-likelihood satisfies the above conditions, and thus is consistent. The theorem is formally stated as follows.
\[thm:spec\] Under PCABM, the community label estimate $\hat{\boldsymbol{c}}$ is strongly consistent if $\varphi_n\to\infty$ and strongly consistent if $\varphi_n/\log n\to\infty$ under the MLE $\hat{\boldsymbol{\gamma}}$.
Since finding $\hat{\mathbf{c}}$ is a non-convex problem, we use tabu search [@beasley1998heuristic; @glover2013tabu] to find a solution. The detailed algorithm is described as follows.
Adjacency matrix $A$; pairwise covariates $Z$; initial assignment $\mathbf{e}$; number of communities $K$ Coefficient estimate $\hat{\boldsymbol{\gamma}}$ and community estimate $\hat{\mathbf{c}}$. Maximize $\ell_{\mathbf{e}}(\boldsymbol{\gamma})$ as in by some optimization algorithm (e.g., BFGS) to derive $\hat{\boldsymbol{\gamma}}$. Use tabu search to maximize $\ell_{\hat{\boldsymbol{\gamma}}}(\mathbf{e})$ as in to get $\hat{\mathbf{c}}$.
The idea of tabu search is to switch the class labels for a randomly-chosen pair of nodes. If the value of log-likelihood function increases after switching, we proceed with the switch. Otherwise, we ignore the switch by sticking with old labels. Because this algorithm is greedy, to avoid being trapped in local maximum, we “tabu" those nodes whose labels have been switched recently, i.e. we don’t consider switching the label of a node if it is in the tabu set. Though the global maximum is not guaranteed, tabu search usually gives satisfactory results from our limited numerical experience.
Spectral Clustering with Adjustment
===================================
\[sec:scwa\] Though the likelihood-based method has appealing theoretical properties, tabu search can sometimes be slow when the network size is large. In addition, the community detection results can be sensitive to the initial label assignments $\mathbf{e}$. As a result, we aim to propose a computationally efficient algorithm in the flavor of spectral clustering [@rohe2011spectral], which can also be used as the initial community label assignments for PCABM.
A Brief Review on Spectral Clustering
-------------------------------------
First, we introduce some notations and briefly review the classical spectral clustering with $K$-means for SBM. Let $\mathbb{M}_{n,K}$ be the space of all $n\times K$ matrices where each row has exactly one 1 and $(K-1)$ 0’s. We usually call $M\in\mathbb{M}_{n,K}$ a *membership matrix* with $M_{ic_i}=1$. Note that $M$ contains the same information as $\mathbf{c}$, and it’s introduced to facilitate the discussion of spectral methods.
From now on, we use PCABM$(M,B,Z,\boldsymbol{\gamma}^0)$ to represent PCABM generated with parameters in the parentheses. Let $G_k=G_k(M)=\{1\leq i\leq n:c_i=k\}$ and $n_k=|G_k|$ for $k=1,\cdots,K$. Let $n_{\min}=\min_{1\leq k\leq K} n_k$, $n_{\max}=\max_{1\leq k\leq K} n_k$ and $n'_{\max}$ is the second largest community size.
For convenience, we define matrix $P=[P_{ij}]\in [0,\infty)^{n\times n}$, where $P_{ij}=B_{c_ic_j}$, then it is easy to observe $P=MBM^T$. When $A$ is generated from a SBM with $(M,B)$, the $K$-dimensional eigen-decomposition of $P=UDU^T$ and $A=\hat{U}\hat{D}\hat{U}^T$ are expected to be close, where $\hat{U}^T\hat{U}=I_K$ and $D,\hat{D}\in\mathbb{R}^{K\times K}$. Since $U$ has only $K$ unique rows, which represent the community labels, the $K$-means clustering on the rows of $\hat{U}$ usually lead to a good estimate of $M$. We define $K$-means clustering as $$\label{eq:km}
(\hat{M},\hat{X})={\arg\min}_{M\in\mathbb{M}_{n,K},X\in\mathbb{R}^{K\times K}}\|MX-\hat{U}\|_F^2.$$ Though finding a global minimizer for the $K$-means problem (\[eq:km\]) is NP-hard [@aloise2009np], for any positive constant $\epsilon$, we have efficient algorithms to find an $(1+\epsilon)$-approximate solution [@kumar2004simple; @lu2016statistical]: $$\begin{aligned}
(\hat{M},\hat{X})&\in\mathbb{M}_{n,K}\times\mathbb{R}^{K\times K}\\
s.t. \quad\quad \|\hat{M}\hat{X}-\hat{U}\|_F^2&\leq(1+\epsilon)\min_{M\in\mathbb{M}_{n,K},X\in\mathbb{R}^{K\times K}}\|MX-\hat{U}\|_F^2.\end{aligned}$$
The goal of community detection is to find $\hat{M}$ that is close to $M$. To define a loss function, we need to take permutation into account. Let $\mathcal{S}_K$ be the space of all $K\times K$ permutation matrices. Following [@lei2015consistency], we define two measures of estimation error: the overall relative error and the worst case relative error: $$\begin{aligned}
L_1(\hat{M},M)&=n^{-1}\min_{S\in \mathcal{S}_K}\|\hat{M}S-M\|_0,\\
L_2(\hat{M},M)&=\min_{S\in \mathcal{S}_K}\max_{1\leq k\leq K}n_k^{-1}\|(\hat{M}S)_{G_k\cdot}-M_{G_k\cdot}\|_0.\end{aligned}$$ It can be seen that $0\leq L_1(\hat{M},M)\leq L_2(\hat{M},M)\leq2$. While $L_1$ measures the overall proportion of mis-clustered nodes, $L_2$ measures the worst performance across all communities.
Spectral Clustering with Adjustment
-----------------------------------
The existence of covariates in PCABM prevents us from applying spectral clustering directly on $A$. Unlike SBM where $A$ is generated from a low rank matrix $P$, $A$ consists of both community and covariate information in PCABM. Since $P_{ij}=\mathbb{E}[A_{ij}/e^{\mathbf{z}^T_{ij}\boldsymbol{\gamma}^0}]$, one intuitive idea to take advantage of the low rank structure is to remove the covariate effects, i.e. using the adjusted adjacency matrix $[A_{ij}/e^{\mathbf{z}^T_{ij}{\boldsymbol{\gamma}^0}}]$ for spectral clustering.
However, in practice, we don’t know what the the true parameter $\boldsymbol{\gamma}^0$ is, we naturally replace $\boldsymbol{\gamma}^0$ with empirical estimate $\hat{\boldsymbol{\gamma}}$. To this end, define the adjusted adjacency matrix as $A'=[A'_{ij}]$ where $A'_{ij}=A_{ij}\exp(-\mathbf{z}_{ij}^T\hat{\boldsymbol{\gamma}})$. By the consistency of $\hat{\boldsymbol{\gamma}}$ proved in Theorem \[CGM\], we can show $\|A'-P\|$ can be bounded in probability as in [@lei2015consistency; @yu2015useful], which is shown in Appendix \[appd:c\].
Based on this bound, we could then apply the standard spectral clustering algorithm on matrix $A'$ to detect the communities. We call this adjustment scheme the *Spectral Clustering with Adjustment* (SCWA) algorithm, which is elaborated in Algorithm \[alg:scwa\].
Adjacency matrix $A$; pairwise covariates $Z$; initial assignment $\mathbf{e}$; number of communities $K$; approximation parameter $\epsilon$ Coefficient estimate $\hat{\boldsymbol{\gamma}}$, community estimate $\hat{\mathbf{c}}$. Maximize $\ell_{\mathbf{e}}(\boldsymbol{\gamma})$ as in by some optimization algorithm (e.g., BFGS) to derive $\hat{\boldsymbol{\gamma}}$. Divide $A_{ij}$ by $\exp(\mathbf{z}_{ij}^T\hat{\boldsymbol{\gamma}})$ to get $A'_{ij}$. Calculate $\hat{U}\in\mathbb{R}^{n\times K}$ consisting of the leading $K$ eigenvectors (ordered in absolute eigenvalue) of $A'$. Calculate the $(1+\epsilon)$-approximate solution $(\hat{M},\hat{X})$ to the $K$-means problem (\[eq:km\]) with $K$ clusters and input matrix $\hat{U}$. Output $\hat{\mathbf{c}}$ according to $\hat{M}$.
To show the consistency of Algorithm \[alg:scwa\], one natural requirement is that $A'$ and $P$ should be close enough, which is stated rigorously in the following theorem.
\[THM:SC\] Let $A$ be the adjacency matrix generated by PCABM $(M,B,Z,\boldsymbol{\gamma}^0)$, and the adjusted adjacency matrix $A'$ is derived from the MLE $\hat{\boldsymbol{\gamma}}$. Assume Conditions \[cond:zbd\], \[cond:zpd\], and \[cond:e1\] hold and $\varphi_n\geq C\log n$. For any constant $r>0$, there exists a constant $C$ such that $\|A'-P\|\leq C\sqrt{\varphi_n}$ with probability at least $1-n^{-r}$.
With similar proof of Theorem 3.1 in [@lei2015consistency], we can prove the following Theorem \[main\] by combining Lemmas 5.1 and 5.3 in [@lei2015consistency], and Theorem \[THM:SC\]. Without loss of generality, we now assume $\|\bar{B}_{\max}\|\leq1$, which makes the statement of the theorem simpler.
\[main\] In addition to the conditions of Theorem \[THM:SC\], assume that $P=MBM^T$ is of rank K with the smallest absolute non-zero eigenvalue at least $\xi_n$. Let $\hat{M}$ be the output of spectral clustering using $(1+\epsilon)$ approximate $K$-means on $A'$. For any constant $r>0$, there exists an absolute constant $C>0$, such that, if $$\label{eq:cond}
(2+\epsilon)\frac{Kn\rho_n}{\xi^2_n}<C,$$ then, with probability at least $1-n^{-r}$, there exist subsets $H_k\subset G_k$ for $k=1,\cdots,K$, and a $K\times K$ permutation matrix $J$ such that $\hat{M}_{G\cdot}J=M_{G\cdot}$, where $G=\cup_{k=1}^K(G_k\setminus H_k)$, and $$\label{eq:error}
\sum_{k=1}^K\frac{|H_k|}{n_k}\leq C^{-1}(2+\epsilon)\frac{Kn\rho_n}{\xi^2_n}.$$
Inequality (\[eq:error\]) provides an error bound for $L_2(\hat{M},M)$. In sets $H_k$, the clustering accuracy of nodes can not be guaranteed.
Theorem \[main\] doesn’t provide us with an error bound in a straightforward form since $\xi_n$ contains $\rho_n$. The following corollary gives us a clearer view of the error bound in terms of model parameters.
\[cor:scwa\] In addition to the conditions of Theorem \[THM:SC\], assume $\bar{B}'s$ minimum absolute eigenvalue bounded below by $\tau>0$ and $\max_{kl}\bar{B}(k,l)=1$. Let $\hat{M}$ be the output of spectral clustering using $(1+\epsilon)$ approximate $K$-means. For any constant $r>0$, there exists an absolute constant $C$ such that if $$(2+\epsilon)\frac{Kn}{n^2_{\min}\tau^2\rho_n}<C,$$ then with probability at least $1-n^{-r}$, $$L_2(\hat{M},M)\leq C^{-1}(2+\epsilon)\frac{Kn}{n^2_{\min}\tau^2\rho_n},$$ and $$L_1(\hat{M},M)\leq C^{-1}(2+\epsilon)\frac{Kn'_{\max}}{n^2_{\min}\tau^2\rho_n}.$$
Compared with SCWA, the likelihood tabu search could lead to more precise results but takes a longer time. Also, the likelihood tabu search results are sensitive to the initial labels $\mathbf{e}$ in some settings. To combine the advantages of those two methods, we propose to use the results of SCWA as the initial solution for tabu search (PCABM.MLE as described in Algorithm \[alg:scwapcabm\]). We will conduct extensive simulation studies in Section \[sec:simu\] to study the performances of PCABM.SCWA and PCABM.MLE.
Adjacency matrix $A$; pairwise covariates $Z$; initial classes $\mathbf{e}$; number of communities $K$; approximation parameter $\epsilon$ Community estimate $\hat{\mathbf{c}}$ Use algorithm \[alg:scwa\] to get an initial community estimate $\tilde{M}$. Use $\tilde{M}$ as the initial communities for tabu search to find an optimal $\hat{\mathbf{c}}$. Output $\hat{\mathbf{c}}$ according to $\hat{M}$.
Simulations {#sec:simu}
===========
For all simulations, we consider $K=2$ communities with prior probabilities $\pi_1=\pi_2=0.5$. In addition, we fix $\bar{B} = \bigl( \begin{smallmatrix}2 & 1\\ 1 & 2\end{smallmatrix}\bigr)$. We generate data by applying the following procedure:
1. Determine parameters $\rho_n$ and $\mathbf{\gamma}^0$. Generate $\mathbf{z}_{ij}$ from certain distributions.
2. Generate adjacency matrix $A=[A_{ij}]$ from Poisson distribution with the parameters calculated using PCABM with parameters in step 1.
$\boldsymbol{\gamma}$ Estimation\[subsec::simu-gamma\]
------------------------------------------------------
For PCABM, estimating $\boldsymbol{\gamma}$ would be the first step, so we check the consistency of $\hat{\boldsymbol{\gamma}}$ claimed in our theory section.
The pairwise covariate vector $\mathbf{z}_{ij}$ has 5 variables, generated independently from $\text{Bernoulli}(0.1)$, $\text{Poisson}(0.1)$, $\text{Uniform}[0,1]$, $\text{Exponential}(0.3)$ and $N(0,0.3)$, respectively. The parameters for each distribution are chosen to make the variances of covariates similar.
We ran 100 simulations respectively for $n=100,200,300,400,500$. The parameters are set as $\rho_n=2(\log n)^{1.5}/n$, $\boldsymbol{\gamma}^0=(0.4,0.8,1.2,1.6,2)^T$. We obtained $\hat{\boldsymbol{\gamma}}$ by using BFGS to optimize likelihood function under random initial community assignments. We also repeated the experiment by ignoring the community structure, which leads to very similar results with the corresponding results in Table \[tab:gamma\_K=1\] of Supplementary Materials. This validates Condition \[cond:e1\], showing that estimating $\gamma$ and communities are decoupled. We list the mean and standard deviation of $\hat{\boldsymbol{\gamma}}$ in Table \[tab:gamma\]. It is clear that $\hat{\boldsymbol{\gamma}}$ is very close to $\boldsymbol{\gamma}^0$ even for a small network. The shrinkage of standard deviation implies consistency of $\hat{\boldsymbol{\gamma}}$.
$n$ $\boldsymbol{\gamma}^0_1=0.4$ $\boldsymbol{\gamma}^0_2=0.8$ $\boldsymbol{\gamma}^0_3=1.2$ $\boldsymbol{\gamma}^0_4=1.6$ $\boldsymbol{\gamma}^0_5=2$
----- ------------------------------- ------------------------------- ------------------------------- ------------------------------- -----------------------------
0.3986 0.797 1.1969 1.5964 1.9952
(0.0414) (0.0354) (0.0455) (0.0467) (0.0484)
0.4019 0.8049 1.2007 1.5991 2.0017
(0.0239) (0.0195) (0.0333) (0.0262) (0.033)
0.3986 0.801 1.1994 1.603 2.0003
(0.0205) (0.0151) (0.0227) (0.0217) (0.0234)
0.3976 0.7977 1.198 1.6 2.0015
(0.0157) (0.0129) (0.0202) (0.0162) (0.0197)
0.3952 0.799 1.1974 1.599 2.002
(0.0131) (0.0118) (0.0173) (0.014) (0.0147)
: Simulated results over 100 replicates of $\hat{\boldsymbol{\gamma}}$, displayed as mean (standard deviation).
\[tab:gamma\]
By taking a closer look at the network of size $n=500$, we compare the distribution of $\hat{\boldsymbol{\gamma}}$ with the theoretical asymptotic normal distribution derived in Theorem \[THM:ASY\]. We show the histogram for the first three coefficients in Figure \[fig:gamma\]. We can see that the empirical distribution matches well with the theoretical counterpart.
![Simulation results for $\hat{\boldsymbol{\gamma}}$ compared with theoretical values. []{data-label="fig:gamma"}](gamma1.png "fig:"){height="3cm"} ![Simulation results for $\hat{\boldsymbol{\gamma}}$ compared with theoretical values. []{data-label="fig:gamma"}](gamma2.png "fig:"){height="3cm"} ![Simulation results for $\hat{\boldsymbol{\gamma}}$ compared with theoretical values. []{data-label="fig:gamma"}](gamma3.png "fig:"){height="3cm"}
Community Detection\[subsec:simu-cd\]
-------------------------------------
After obtaining $\hat{\boldsymbol{\gamma}}$, we now move on to the estimation of community labels. Under PCABM, there are three parameters that we could tune to change the property of the network: $\boldsymbol{\gamma}^0$, $\rho_n$, and $n$. To illustrate the impact of these parameters on the performance of community detection, we vary one parameter while fixing the remaining two in each experiment. More specifically, we consider the form $\rho_n = c_{\rho}\times (\log n)^{1.5}/n$ and $\boldsymbol{\gamma}^0 = c_{\gamma}(0.4, 0.8, 1.2, 1.6, 2)$ in which we will vary the multipliers $c_{\rho}$ and $c_{\gamma}$. The detailed parameter settings for the three experiments are as follows.
1. $n\in\{100,200,300,400,500\}$, with $c_{\rho}=0.5$ and $c_{\gamma}=1.4$.
2. $c_{\rho}\in \{0.3,0.4,0.5,0.6,0.7\}$, with $n=200$ and $c_{\gamma}=1.4$.
3. $c_{\gamma} \in \{0.2,0.4,0.6,0.8,1.0,1.2,1.4,1.6\}$, with $n=200$ and $c_{\rho}=1$.
The results for the three experiments are presented in Figure \[fig:pcabm\]. Each experiment is carried out 100 times. The error rate is reported in terms of the average Adjusted Rand Index (ARI) [@hubert1985comparing], which is a measure of the similarity between two data clusterings. SBM.MLE and SBM.SC refer to the likelihood and spectral clustering methods under SBM, respectively. PCABM.MLE and PCABM.SCWA refer to Algorithms \[alg:scwapcabm\] and \[alg:scwa\] respectively.
[.45]{} ![Simulation results under PCABM for different parameter settings[]{data-label="fig:pcabm"}](sim1.png "fig:"){width=".9\linewidth"}
[.45]{} ![Simulation results under PCABM for different parameter settings[]{data-label="fig:pcabm"}](sim2.png "fig:"){width=".9\linewidth"}
[.45]{} ![Simulation results under PCABM for different parameter settings[]{data-label="fig:pcabm"}](sim3.png "fig:"){width=".9\linewidth"}
[.45]{} ![Simulation results under PCABM for different parameter settings[]{data-label="fig:pcabm"}](sim4.png "fig:"){width=".9\linewidth"}
When the number of nodes increases, it is clear from the first plot in Figure \[fig:pcabm\] that both algorithms under SBM provide us little more than a random guess. On the other hand, both PCABM-based algorithms perform quite well with PCABM.MLE having nearly perfect community detection performance throughout all $n$. Note that SCWA gradually catches up with MLE as we have more nodes. We observe a similar phenomenon when the sparsity level is changed. When the scale of $\boldsymbol{\gamma}^0$ is changed, both algorithms under PCABM still yield good results. As we know, when $\boldsymbol{\gamma}^0=\mathbf{0}$, our model reduces to SBM, so it is not surprising that SBM.MLE and SBM.SC both perform well when the magnitude of $\boldsymbol{\gamma}^0$ is relatively small and fail when the magnitude increases. Also, it appears that compared with the likelihood method, spectral clustering is more robust to model misspecification.
Impact of inaccurate $\boldsymbol{\gamma}$ estimate
---------------------------------------------------
In this section, we investigate how the accuracy of $\hat{\boldsymbol{\gamma}}$ affects the community label estimate. Here we use one dimensional covariate following $\text{Poisson}(0.1)$ and set the true coefficient $\gamma^0 = 2$, $n=500$ and $\rho_n = (\log n)^{1.5}/n$. We vary the coefficient estimates $\hat{\gamma}\in\{0,0.5,1,1.5,2,2.5,3,3.5\}$.
The community detection results are shown in the last plot of Figure \[fig:pcabm\] under different $\hat{\gamma}$ values. We can see that using a more accurate $\hat{\gamma}$ would yield more accurate community detection results. The maximum ARI is reached by PCABM.MLE when $\hat{\gamma}$ under the true $\gamma^0$ value. When we have an estimation bias of $\hat{\gamma}$, the community detection performance deteriorates. In particular, we see the choice of $\hat{\gamma}=0$ (i.e. SBM) leads to an average ARI around 0.4. This shows the importance of considering the pairwise covariates in the community detection procedure.
DCBM\[subsec::DCBM\]
--------------------
[.45]{} ![Simulation results under DCBM for different parameter settings[]{data-label="fig:deg"}](deg1.png "fig:"){width=".9\linewidth"}
[.45]{} ![Simulation results under DCBM for different parameter settings[]{data-label="fig:deg"}](deg2.png "fig:"){width=".9\linewidth"}
To show the robustness of our algorithms, we also apply Algorithms \[alg:scwa\] and \[alg:scwapcabm\] to networks generated by DCBM, which can be viewed as a misspecified case. The degree parameter for each node is chosen from $\{0.5,1.5\}$ with equal probability, $\bar{B} = \bigl( \begin{smallmatrix}2 & 1\\ 1 & 2\end{smallmatrix}\bigr)$, and $\rho_n = c_{\rho}\times (\log n)^{1.5}/n$. For covariates construction, we take $z_{ij}=\log d_i+\log d_j$, where $d_i$ is the degree of node $i$. As a comparison, we also implemented the likelihood method in [@zhao2012consistency] (DCBM.MLE) and the SCORE method in [@jin2015fast]. As in Section \[subsec:simu-cd\], we vary one parameter while fixing the remaining one in each experiment. More specifically, we consider the form $\rho_n = c_{\rho}\times (\log n)^{1.5}/n$ in which we will vary the multipliers $c_{\rho}$ and $n$. The detailed parameter settings for the two experiments are as follows with results presented in Figure \[fig:deg\].
1. $n\in\{100,200,300,400,500\}$, with $c_{\rho}=2$.
2. $c_{\rho}\in \{1.2, 1.6, 2.0, 2.4, 2.8, 3.2\}$, with $n=200$.
As expected, all algorithms perform better when the network is larger or denser. It is interesting to observe that SCWA is slightly better than SCORE in most cases. The two likelihood-based methods perform similarly and better than their spectral counterparts. The flexibility of PCABM allows us to model any factors that may contribute to the structure of network in addition to the underlying communities.
Correlated Covariates
---------------------
In the last simulation, we compare the two network models involving covariates introduced in Section \[sec:intro\]. In the covariates-adjusted model, we assume that the pairwise covariates are independent of the community labels. On the contrary, in the covariates-confounding model, the covariates’ distribution is governed by the community labels. An interesting question to ask is what will happen if the covariates used in the covariates-adjusted model are correlated with the community information.
![Simulation results for adding covariate of different correlation with community structure.[]{data-label="fig:cov"}](cov.png){width=".5\linewidth"}
To explore this, we first generate PCABM and then generate covariates correlated with the matrix $P_{n\times n}=[B_{c_ic_j}]$ which contains the community information. More specifically, we let $n=500$, $\bar{B} = \bigl( \begin{smallmatrix}2 & 1\\ 1 & 2\end{smallmatrix}\bigr)$, and $c_\rho=0.8$. We have one pairwise covariate $z_{ij}\sim N(0,0.3)$ with the corresponding $\gamma^0=2$. The “false" covariate $z_{ij}'$ is generated to be correlated with the $P$ but does not contribute to the generation of adjacency matrix $A$. For $i<j$, $z_{ij}'\sim N(0.6(\bar{B}_{c_ic_j}-1.5)r^{-1}(1-r^2)^{-1/2},0.09)$, which makes the Pearson correlation between $[z_{ij}', i<j]$ and $[P_{ij}, i<j]$ to be $r$. We evaluate the community detection performance of PCABM under three scenarios: (1) using only the true covariate $z_{ij}$, (2) using only the false covariate $z_{ij}'$, (3) using both covariates. We vary the correlation $r$ from 0 to 1 to see how it will change the community detection accuracy.
As shown in Figure \[fig:cov\], as the correlation increases, community detection performance becomes worse. The reason is that when the false covariate is correlated with the matrix $P$, it will contribute substantially to fitting the model. As a result, what we get after adjusting for such a covariate will contain less community information. This provides us some additional insights in that the community detection process in PCABM works in a “conditional" way, after taking away the contribution of the covariates.
Real Data Examples {#sec:realdata}
==================
Example 1: Political Blogs
--------------------------
The first real-world dataset we used is the network of political blogs created by [@adamic2005political]. The nodes are blogs about US politics, and edges represent hyperlinks between them. We treated the network as undirected and only focused on the largest connected component of the network, resulting in a subnetwork with 1,222 nodes and 16,714 edges.
ARI NMI Errors
-------------------------------------------------- ----------- ----------- --------
Karrer and Newman (2011) [@karrer2011stochastic] - 0.72 -
Zhao et al. (2012) [@zhao2012consistency] 0.819 - -
Jin (2015) [@jin2015fast] 0.819 0.725 58
PCABM.MLE **0.825** **0.737** **56**
: Performance comparison on political blogs data.
\[tab:politics\]
Because there are no other nodal covariates available in this dataset, we created one pairwise covariate by aggregating the degree information. We let $z_{ij}=\log(d_i\times d_j)$, where $d_i$ is the degree for the $i$-th node. The coefficient estimate for the covariate $\hat{\boldsymbol{\gamma}}$ is 1.0005 with the $95\%$ confidence interval being $(0.9898,1.0111)$. Table \[tab:politics\] summarizes the performance comparison of PCABM with some existing results on this dataset. Besides ARI, we also evaluated normalized mutual information (NMI) [@danon2005comparing], which is a measure of the mutual dependence between the two variables. It is observed that our model slightly outperforms all previous results, and the error rate is very close to the ideal results mentioned in [@jin2015fast], which is 55/1222. This shows that PCABM provides an alternative way to the DCBM by incorporating the degree information into a specific pairwise covariate. It also provides a significant improvement over the vanilla SBM, whose NMI is only 0.0001 reported in [@karrer2011stochastic].
Example 2: School Friendship
----------------------------
For real networks, people often use specific nodal covariates as the ground “truth" for community labels to evaluate the performance of various community detection methods. However, there could be different “true" community assignments based on different nodal covariates (e.g., gender, job, and age). [@peel2017ground] mentioned that communities and the covariates might capture various aspects of the network, which is in line with the idea presented in this paper. To examine whether PCABM can discover different community structures, in our second example, we treat one covariate as the indicator for the unknown “true" community assignments while using the remaining covariates as the known covariates in our PCABM model.
The dataset is a friendship network of school students from the National Longitudinal Study of Adolescent to Adult Health (Add Health). It contains 795 students from a high school (Grades 9-12) and its feeder middle school (Grade 7-8). The nodal covariates include grade, gender, ethnicity, and number of friends nominated (up to 10). We focused on the largest connected component with at least one covariate non-missing and treated the network as undirected, resulting in a network with 777 nodes and 4124 edges. For the nodes without gender, we let them be female, which is the smaller group. For those without grade, we generated a random grade according to their schools.
Unlike traditional community detection algorithms that can only detect one underlying community structure, PCABM provides us with more flexibility to uncover different community structures by controlling different covariates. Our intuition is that social network is usually determined by multiple underlying structures and cannot be simply explained by one covariate. Sometimes one community structure seems to dominate the network, but if we adjust the covariate associated with that structure, we may discover other interesting community structures.
In this example, we conducted three community detection experiments. In each experiment, out of the three nodal covariates (school, ethnicity, and gender), one was viewed as the proxy for the “true" underlying community, and community detection was carried out by using the pairwise covariates constructed using the other two covariates. For gender, school, and ethnicity, we created indicator variables to represent whether the corresponding covariate values were the same for the pair of nodes. For example, if two students come from the same school, the corresponding pairwise covariate equals 1; if they have different genders, the corresponding pairwise covariate equals 0. Also, we considered the number of nominated friends in all experiments and grade for predicting ethnicity and gender. For number of nominated friends, we used $\log(n_i+1)+\log(n_j+1)$ as one pairwise covariate, where $n_i$ is the number of nominated friends for the $i$-th student. “+1" was used here because some students didn’t nominate anyone. More specifically, consider the experiment where school is the targeted community to be learned by the model. For grades, we used the absolute difference to form a pairwise covariate. Using random initial community labels, we derived the estimates $\hat{\boldsymbol{\gamma}}$ in each experiment. In Tables \[tab:inf1\] and \[tab:inf2\], we show respectively the estimates when school and ethnicity are taken as the targeted community.
Covariate Estimate Std. Error $t$ value ($>|\text{t}|$)
------------ ---------- ------------ ----------- -----------------
White 1.251 0.043 29.002 $<0.001$\*\*\*
Black 1.999 0.051 38.886 $<0.001$\*\*\*
Hispanic 0.048 0.523 0.091 0.927
Others 0.019 0.543 0.035 0.972
Gender 0.192 0.034 5.620 $<0.001$\*\*\*
Nomination 0.438 0.024 18.584 $<0.001$\*\*\*
: Inference results when school is targeted community.
\[tab:inf1\]
Covariate Estimate Std. Error $t$ value ($>|\text{t}|$)
------------ ---------- ------------ ----------- -----------------
School 1.005 0.076 13.168 $<0.001$\*\*\*
Grade -1.100 0.028 -39.182 $<0.001$\*\*\*
Gender 0.198 0.034 5.813 $<0.001$\*\*\*
Nomination 0.498 0.023 21.679 $<0.001$\*\*\*
: Inference results when ethnicity is targeted community.
\[tab:inf2\]
School Race Gender
----------- ----------- ----------- -----------
PCABM.MLE **0.909** **0.914** [0.030]{}
SBM.MLE 0.048 0.138 -0.001
SBM.SC 0.043 -0.024 0.000
DCBM.MLE **0.909** 0.001 0.002
SCORE 0.799 0.012 0.011
: ARI comparison on school friendship data.
\[tab:friendship\]
[.9]{} ![Community detection with different pairwise covariates. From top to bottom, we present community prediction results for school, ethnicity and gender.[]{data-label="fig:friend"}](comm1.png "fig:"){width="\linewidth"} \[fig:sfig1\]
[.9]{} ![Community detection with different pairwise covariates. From top to bottom, we present community prediction results for school, ethnicity and gender.[]{data-label="fig:friend"}](comm2.png "fig:"){width="\linewidth"} \[fig:sfig2\]
[.9]{} ![Community detection with different pairwise covariates. From top to bottom, we present community prediction results for school, ethnicity and gender.[]{data-label="fig:friend"}](comm3.png "fig:"){width="\linewidth"} \[fig:sfig3\]
In both tables, the standard error is calculated by Theorem \[THM:ASY\], with the theoretical values replaced by the estimated counterparts. Thus, we can calculate the $t$ value for each coefficient and perform statistical tests. We can see that in both experiments, the coefficients for gender and the number of nominations are positive and significant in the creation of the friendship network. The significant positive coefficient of nominations shows that students with a large number of nominations tend to be friends with each other, which is intuitive. The positive coefficients of gender and school show students of the same gender and school are more likely to be friends with each other, which is in line with our expectations. The negative coefficient of grade means that students with closer grades are more likely to be friends. If we take a closer look at the coefficients of different ethnic groups in Table \[tab:inf1\], we find that only those corresponding to white and black are significant. This is understandable if we observe that among 777 students, 476 are white, and 221 are black. As for school and grade, students in the same school or grade tend to be friends with each other, as expected.
The network is divided into two communities each time (we only look at white and black students in the second experiment because the sizes for other ethnicities are very small). In Figure \[fig:friend\], school, ethnicity, and gender are targeted communities, respectively. We use different shades to distinguish true communities. Predicted communities are separated by the middle dash line so that the ideal split would be shades vs. tints on two sides. By these criteria, our model performs pretty well in the first two cases but fails to distinguish different genders. Indeed, many edges appear between the two detected communities in the third plot indicates bad performance. It can either be the existence of another unknown variable or gender’s contribution to the network structure is insignificant, given the covariates considered.
The results in terms of ARI are shown in Table \[tab:friendship\]. Note that, for all other methods, we would get only one community structure, whose performance is doomed to be bad for capturing different community structures. Also, to test the robustness of our method, in the experiment of detecting the ethnicity community, we tried to use the square of the grade difference, which led to almost the same ARI.
Discussion
==========
In this paper, we extended the classical stochastic block model to allow the connection rate between nodes not only depend on communities but also the pairwise covariates. We proved consistency in terms of both coefficient estimates and community label assignments for MLE under PCABM. Also, we introduced a fast spectral method SCWA with theoretical justification, which may serve as a good initial solution for the likelihood method.
There are many interesting future research directions on PCABM. Though in our paper we assumed the entries in the adjacency matrix are non-negative integers, it can be relaxed to be any non-negative numbers, and we expect similar theoretical results hold. It would also be interesting to consider highly imbalanced community sizes $n_{\min}/n_{\max}=o(1)$ and when the number of communities $K_n$ diverges. Also, for the bounded degree case where $\varphi_n=O(1)$, it is of great interest to study the properties of various estimators under PCABM. When we have high-dimensional pairwise covariates, adding a penalty term to conduct variable selection is also worth investigating.
Another interesting issue is the choice of the number of communities $K$ which is assumed to be known in this paper. However, in practice, it would be desirable to have an automatic procedure for choosing $K$. Some recent efforts toward this direction include [@saldana2017many; @le2015estimating; @wang2015likelihood; @lei2016goodness; @chen2016network; @li2016network; @yan2017exact]. It would be interesting to explore this direction under PCABM.
The python code for implementing the proposed algorithms is available on GitHub at <https://github.com/sihanhuang/PCABM>.
Proofs of theoretical results {#A}
=============================
The Appendix contains proofs of Theorems \[CGM\], \[THM:ASY\] and \[thm:gene\]. The proofs of Theorem \[THM:SC\] and some technical lemmas are collected in the supplementary materials.
Proof of Theorem \[CGM\] and Theorem \[THM:ASY\]
------------------------------------------------
In the following proof, we will use $\hat{\boldsymbol{\gamma}}$ instead of $\hat{\boldsymbol{\gamma}}(\mathbf{e})$ for simplicity. Define the empirical version of $\theta(\boldsymbol{\gamma})$, $\boldsymbol{\mu}(\boldsymbol{\gamma})$ and $\Sigma(\boldsymbol{\gamma})$ as $$\begin{aligned}
\hat{\theta}^{kl}(\boldsymbol{\gamma})=&\sum_{(u,v)\in s_{\mathbf{e}}(k,l)}e^{\mathbf{z}_{uv}^T\boldsymbol{\gamma}}/|s_{\mathbf{e}}(k,l)|,\\
\hat{\boldsymbol{\mu}}^{kl}(\boldsymbol{\gamma})=&\sum_{(u,v)\in s_{\mathbf{e}}(k,l)}\mathbf{z}_{uv}e^{\mathbf{z}_{uv}^T\boldsymbol{\gamma}}/|s_{\mathbf{e}}(k,l)|,\\
\hat{\Sigma}^{kl}(\boldsymbol{\gamma})=&\sum_{(u,v)\in s_{\mathbf{e}}(k,l)}\mathbf{z}_{uv}\mathbf{z}_{uv}^Te^{\mathbf{z}_{uv}^T\boldsymbol{\gamma}}/|s_{\mathbf{e}}(k,l)|.\end{aligned}$$ For fixed $\boldsymbol{\gamma}$, by Chebyshev’s inequality, under Condition \[cond:e1\], for $\forall k,l\in[K]$, we know the weak law of large numbers holds, i.e., $\hat{\theta}^{kl}(\boldsymbol{\gamma})\xrightarrow{p}\theta(\boldsymbol{\gamma})$, $\hat{\boldsymbol{\mu}}^{kl}(\boldsymbol{\gamma})\xrightarrow{p}\boldsymbol{\mu}(\boldsymbol{\gamma})$ and $\hat{\Sigma}^{kl}(\boldsymbol{\gamma})\xrightarrow{p}\Sigma(\boldsymbol{\gamma})$.
First, we calculate the first and second order derivative of $\ell_{\mathbf{e}}$ w.r.t. $\boldsymbol{\gamma}$ as $$\begin{aligned}
\ell'_\mathbf{e}(\boldsymbol{\gamma})&= \frac{\partial\ell_{\mathbf{e}}}{\partial\boldsymbol{\gamma}}=\sum_{i<j}A_{ij}\mathbf{z}_{ij}-\sum_{i<j}A_{ij}\hat{\boldsymbol{\mu}}^{e_ie_j}(\boldsymbol{\gamma})/\hat{\theta}^{e_ie_j}(\boldsymbol{\gamma}),\\
\ell''_\mathbf{e}(\boldsymbol{\gamma})&= \frac{\partial^2\ell_{\mathbf{e}}}{\partial\boldsymbol{\gamma}\partial\boldsymbol{\gamma}^T}=\sum_{kl}\frac{O_{kl}(\mathbf{e})}{2\hat{\theta}^{kl}(\boldsymbol{\gamma})^2}\left[\hat{\boldsymbol{\mu}}^{kl}(\boldsymbol{\gamma})^{\otimes2}-\hat{\theta}^{kl}(\boldsymbol{\gamma})\hat{\Sigma}^{kl}(\boldsymbol{\gamma})\right].
\end{aligned}$$ Then conditioning on $Z$, we define the expectation and variance of $\ell'_{\mathbf{e}}(\boldsymbol{\gamma})$ as $\boldsymbol{\mu}_\ell(\mathbf{e},\boldsymbol{\gamma})=\mathbb{E}[\ell'_{\mathbf{e}}(\boldsymbol{\gamma})|Z]$ and $\Sigma_\ell(\mathbf{e},\boldsymbol{\gamma})=\text{var}[\ell'_{\mathbf{e}}(\boldsymbol{\gamma})|Z]$. $$\begin{aligned}
\frac{\mu_\ell(\mathbf{e},\boldsymbol{\gamma})}{N_n\rho_n}=&(N_n\rho_n)^{-1}\sum_{i<j}B_{c_ic_j}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\left[\mathbf{z}_{ij}-\frac{\hat{\boldsymbol{\mu}}^{e_ie_j}(\boldsymbol{\gamma})}{\hat{\theta}^{e_ie_j}(\boldsymbol{\gamma})}\right]\\
=&N_n^{-1}\sum_{i<j}\bar{B}_{c_ic_j}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\left[\mathbf{z}_{ij}-\frac{\boldsymbol{\mu}(\boldsymbol{\gamma})}{\theta(\boldsymbol{\gamma})}\right]+o(1)\\
=&N_n^{-1}\sum_{i<j}\bar{B}_{c_ic_j}[(e^{\mathbf{z}_{ij}^T\gamma^0}\mathbf{z}_{ij}-\boldsymbol{\mu}(\boldsymbol{\gamma}))+\frac{\boldsymbol{\mu}(\boldsymbol{\gamma})}{\theta(\boldsymbol{\gamma})}(\theta(\boldsymbol{\gamma})-e^{\mathbf{z}_{ij}^T\gamma^0})]+o(1),\\
\frac{\Sigma_\ell(\mathbf{e},\boldsymbol{\gamma})}{N_n\rho_n}=&(N_n\rho_n)^{-1}\sum_{i<j}B_{c_ic_j}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\left[\mathbf{z}_{ij}-\frac{\hat{\boldsymbol{\mu}}^{e_ie_j}(\boldsymbol{\gamma})}{\hat{\theta}^{e_ie_j}(\boldsymbol{\gamma})}\right]^{\otimes2}\\
=&N_n^{-1}\sum_{i<j}\bar{B}_{c_ic_j}e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\left[\mathbf{z}_{ij}-\frac{\boldsymbol{\mu}(\boldsymbol{\gamma})}{\theta(\boldsymbol{\gamma})}\right]^{\otimes2}+o(1)\\
=&(2N_n)^{-1}\sum_{kl}\bar{B}_{kl}\sum_{(i,j)\in s_{\mathbf{c}}(k,l)}[e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\mathbf{z}_{ij}^{\otimes 2}-2e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\mathbf{z}_{ij}\boldsymbol{\mu}(\boldsymbol{\gamma})^T/\theta(\boldsymbol{\gamma})\\
&+e^{\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\boldsymbol{\mu}(\boldsymbol{\gamma})^{\otimes 2}/{\theta(\boldsymbol{\gamma})}^2]+o(1).
\end{aligned}$$
By L.L.N., $\lim_{n\to\infty}\frac{\mu_\ell(\mathbf{e},\boldsymbol{\gamma})}{N_n\rho_n}$ and $\lim_{n\to\infty}\frac{\Sigma_\ell(\mathbf{e},\boldsymbol{\gamma})}{N_n\rho_n}$ exist, and we use $\boldsymbol{\mu}_\infty(\boldsymbol{\gamma})$ and $\Sigma_\infty(\boldsymbol{\gamma})$ to denote them respectively, where $$\begin{aligned}
\boldsymbol{\mu}_\infty(\boldsymbol{\gamma})=&\sum_{ab}\bar{B}_{ab}\pi_a\pi_b\left[\boldsymbol{\mu}(\boldsymbol{\gamma}^0)-\frac{\boldsymbol{\mu}(\boldsymbol{\gamma})}{\theta(\boldsymbol{\gamma})}\theta(\boldsymbol{\gamma}^0)\right],\\
\Sigma_\infty(\boldsymbol{\gamma})=&\sum_{ab}\bar{B}_{ab}\pi_a\pi_b\left[\Sigma(\boldsymbol{\gamma}^0)-\frac{2\boldsymbol{\mu}(\boldsymbol{\gamma}^0)\boldsymbol{\mu}(\boldsymbol{\gamma})^T}{\theta(\boldsymbol{\gamma})}+\frac{\theta(\boldsymbol{\gamma}^0)\boldsymbol{\mu}(\boldsymbol{\gamma})^{\otimes 2}}{{\theta(\boldsymbol{\gamma})}^2}\right],\end{aligned}$$ Specifically, at $\boldsymbol{\gamma}^0$, we have $\boldsymbol{\mu}_\infty(\boldsymbol{\gamma}^0)=\mathbf{0}$ and $\Sigma_\infty(\boldsymbol{\gamma}^0)=\sum_{ab}\bar{B}_{ab}\pi_a\pi_b[\Sigma(\boldsymbol{\gamma}^0)-\theta(\boldsymbol{\gamma}^0)^{-1}\boldsymbol{\mu}(\boldsymbol{\gamma}^0)^{\otimes 2}]$. By Condition \[cond:zpd\], $\Sigma_\infty(\boldsymbol{\gamma})$ is positive definite in a neighborhood of $\boldsymbol{\gamma}^0$.
Using Taylor expansion, $$\ell'_{\mathbf{e}}(\hat{\boldsymbol{\gamma}})-\ell'_{\mathbf{e}}(\boldsymbol{\gamma}^0)=\ell''_{\mathbf{e}}(\bar{\boldsymbol{\gamma}})(\hat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}^0),$$ where $\bar{\boldsymbol{\gamma}}=q\boldsymbol{\gamma}^0+(1-q)\hat{\boldsymbol{\gamma}}$ for some $q\in[0,1]$. Noticing that $\ell'_{\mathbf{e}}(\hat{\boldsymbol{\gamma}})=0$, so $$\hat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}^0=\left(\frac{\ell''_{\mathbf{e}}(\bar{\boldsymbol{\gamma}})}{N_n\rho_n}\right)^{-1}\frac{\ell'_{\mathbf{e}}(\boldsymbol{\gamma}^0)}{N_n\rho_n}\xrightarrow{p}\mathbf{0}.$$
To prove asymptotic normality, we rewrite the above formula as $$\sqrt{N_n\rho_n}(\hat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}^0)=-\left[\frac{1}{N_n\rho_n}\ell''_{\mathbf{e}}(\bar{\boldsymbol{\gamma}})\right]^{-1}\times\left[\frac{1}{\sqrt{N_n\rho_n}}\ell'_{\mathbf{e}}(\boldsymbol{\gamma}^0)\right].$$ Since $\hat{\boldsymbol{\gamma}}\xrightarrow{p}\boldsymbol{\gamma}^0$, we have $N_n\rho_n\ell''^{-1}_\mathbf{e}(\bar{\boldsymbol{\gamma}})\xrightarrow{p}\Sigma_2(\boldsymbol{\gamma}^0)$. Also, conditioning on $Z$, by C.L.T., $$\frac{1}{\sqrt{N_n\rho_n}}\ell'_{\mathbf{e}}(\boldsymbol{\gamma}^0)=\sqrt{N_n\rho_n}\left[\frac{1}{N_n\rho_n}\ell'_{\mathbf{e}}(\boldsymbol{\gamma}^0)\right]\xrightarrow{d} N(0,\Sigma_2(\boldsymbol{\gamma}^0)).$$ Thus, $$\sqrt{N_n\rho_n}(\hat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}^0)\xrightarrow{d} N(0,\Sigma_\infty^{-1}(\boldsymbol{\gamma}^0)).$$
Some Concentration Inequalities and Notations
---------------------------------------------
To prepare later proofs, we introduce some concentration inequalities and additional notations in this part.
One inequality that we will apply repeatedly is an extended version of Bernstein inequality for unbounded random variables introduced in [@wellner2005empirical].
\[lem:bernin\] Suppose $X_1,\cdots,X_n$ are independent random variables with $\mathbb{E}X_i=0$ and $\mathbb{E}|X_i|^k\leq\frac{1}{2}\mathbb{E}X_i^2L^{k-2}k!$ for $k\geq2$. For $M\geq\sum_{i\leq n}\mathbb{E}X_i^2$ and $x\geq0$, $$\emph{Pr}(\sum_{i\leq n}X_i\geq x)\leq\exp\left(-\frac{x^2}{2(M+xL)}\right).$$
To show that all Poisson distributions satisfy the above Bernstein condition uniformly under some constant $\bar{L}$, we give the following lemma.
\[lem:bern\] Assume $A\sim Pois(\lambda)$, let $X=A-\lambda$, then for any $0<\lambda<1/2$, there exists a constant $\bar{L}>0$ s.t. for any integer $k>2$, $\mathbb{E}[|X^k|]\leq\mathbb{E}[X^2]\bar{L}^{k-2}k!/2$.
$$\begin{aligned}
&\frac{2\mathbb{E}[|A-\lambda|^k]}{\lambda k!}=\frac{2}{\lambda k!}\mathbb{E}[(A-\lambda)^k|A\geq1]\text{Pr}(A\geq1)+\frac{2\lambda^{k-1}e^{-\lambda}}{k!}\\
\leq&\frac{2}{\lambda k!}\mathbb{E}[A^k|A\geq1]\text{Pr}(A\geq1)+e^{-\lambda}=\frac{2}{\lambda k!}\mathbb{E}[A^k]+e^{-\lambda}\\
=&\frac{2}{k!}\sum_{i=1}^k\begin{Bmatrix}k\\i\end{Bmatrix}\lambda^{i-1}+e^{-\lambda}\leq\frac{1}{k!}\sum_{i=1}^k{k \choose i}i^{k-i}\lambda^{i-1}+e^{-\lambda}\\
\leq&\frac{e^{k-1}}{k^k}\sum_{i=1}^k\left(\frac{ek}{i}\right)^ii^{k-i}\lambda^{i-1}+e^{-\lambda}=\sum_{i=1}^ke^{i+k-1}i^{k-2i}k^{i-k}\lambda^{i-1}+e^{-\lambda}\\
<&\sum_{i=1}^ke^{i+k-1}e^{-i}\lambda^{i-1}+e^{-\lambda}=e^{k-1}\frac{1-\lambda^k}{1-\lambda}+e^{-\lambda}\leq\frac{e^{k-1}}{1-\lambda}+1\\
\leq&\left(\frac{e^2+1}{1-\lambda}\right)^{k-2}.\end{aligned}$$
Notice that when $\lambda$ is bounded away from $1$, say $\lambda<1/2$, we can simply set $\bar{L}=2(e^2+1)$, then Bernstein condition is satisfied uniformly for all $\lambda$.
We introduce some notations. Let $|\mathbf{e}-\mathbf{c}| = \sum_{i=1}^n\mathbbm{1}(e_i\neq c_i)$. Given a community assignment $\mathbf{e}\in[K]^n$, we define $ V(\mathbf{e})\in\mathbb{R}^{K\times K}$ with their elements being $$\begin{aligned}
V_{ka}(\mathbf{e})=\frac{\sum_{i=1}^n\mathbbm{1}(e_i=k,c_i=a)}{\sum_{i=1}^n\mathbbm{1}(c_i=a)}=\frac{R_{ka}(\mathbf{e})}{\pi_a(\mathbf{c})}.\end{aligned}$$
One can view $R$ as the empirical joint distribution of $\mathbf{e}$ and $\mathbf{c}$, and $V$ as the empirical conditional distribution of $\mathbf{e}$ given $\mathbf{c}$. We can see that $V(\mathbf{e})=R(\mathbf{e})(D(\mathbf{c}))^{-1}$, where $D(\mathbf{c})=\text{diag}(\boldsymbol{\pi}(\mathbf{c}))$. Also, note that $V(\mathbf{e})^T\mathbf{1}=\mathbf{1}$, $V(\mathbf{e})\boldsymbol{\pi}(\mathbf{c})=\boldsymbol{\pi}(\mathbf{e})$ and $V(\mathbf{c})=I_K$. For the convenience of later proof, we also define $W(\mathbf{c})=D(\mathbf{c})\bar{B}D(\mathbf{c})$ and $$\begin{aligned}
\hat{T}(\mathbf{e})\triangleq& R(\mathbf{e})\bar{B}R(\mathbf{e})^T=V(\mathbf{e})W(\mathbf{c})V(\mathbf{e})^T,\\
\hat{S}(\mathbf{e})\triangleq& V(\mathbf{e})\boldsymbol{\pi}(\mathbf{c})\boldsymbol{\pi}(\mathbf{c})^TV(\mathbf{e})^T\end{aligned}$$
Replacing the empirical distribution $\boldsymbol{\pi}(\mathbf{c})$ by the true distribution $\boldsymbol{\pi}_0$, we define $W_0=D(\boldsymbol{\pi}_0)\bar{B}D(\boldsymbol{\pi}_0)$, where $D(\boldsymbol{\pi}_0)=\text{diag}(\boldsymbol{\pi}_0)$, and $T(\mathbf{e}), S(\mathbf{e})\in\mathbb{R}^{K\times K}$ as $$\begin{aligned}
T(\mathbf{e})\triangleq& V(\mathbf{e})W_0V(\mathbf{e})^T,\\
S(\mathbf{e})\triangleq& V(\mathbf{e})\boldsymbol{\pi}_0\boldsymbol{\pi}_0^TV(\mathbf{e})^T.\end{aligned}$$
The population version of $F\left(\frac{O}{2N_n\rho_n},\frac{E}{2N_n}\right)$ is $$F(\theta(\boldsymbol{\gamma}^0)T(\mathbf{e}),\theta(\hat{\boldsymbol{\gamma}})S(\mathbf{e})).$$ To measure the discrepancy between empirical and population version of $F$, we define $X(\mathbf{e}), Y(\mathbf{e},\hat{\boldsymbol{\gamma}})\in\mathbb{R}^{K\times K}$ to be the rescaled difference between $O, E$ and their expectations $$\begin{aligned}
X(\mathbf{e})\triangleq&\frac{O(\mathbf{e})}{2N_n\rho_n}-\theta(\boldsymbol{\gamma}^0)\hat{T}(\mathbf{e}),\\
Y(\mathbf{e},\hat{\boldsymbol{\gamma}})\triangleq&\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}-\theta(\hat{\boldsymbol{\gamma}})\hat{S}(\mathbf{e}).\end{aligned}$$
Before we establish bound for $Y(\mathbf{e},\hat{\boldsymbol{\gamma}})$, we need to consider the bound for $\hat{\boldsymbol{\gamma}}$. The following is a direct corollary of Theorem \[THM:ASY\]. Conditioned on $\|\hat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}^0\|_\infty\leq\phi$, we have $|e^{\mathbf{z}_{ij}\hat{\boldsymbol{\gamma}}}-\mathbb{E}e^{\mathbf{z}_{ij}\hat{\boldsymbol{\gamma}}}|\leq\exp\{p\alpha(\phi+\|\boldsymbol{\gamma}^0\|_\infty)\}\equiv\chi$ uniformly for any $i,j\in[n]$ and $\hat{\boldsymbol{\gamma}}$. Under this condition, we establish Lemma \[lem:con\] using Bernstein inequality.
\[lem:phi\] For any constant $\phi>0$, $\exists$ positive constants $C_\phi$ and $v_\phi$ s.t., $\emph{Pr}(\|\hat{\boldsymbol{\gamma}}-\boldsymbol{\gamma}^0\|_\infty>\phi)<C_\phi\exp(-v_\phi N_n\rho_n)$.
\[lem:con\] $$\begin{aligned}
\emph{Pr}(\max_{\mathbf{e}}\|X(\mathbf{e})\|_\infty\geq\epsilon)&\leq2K^{n+2}\exp(-C_1\epsilon^2N_n\rho_n)\label{eq:lm1}\end{aligned}$$ for $\epsilon<\beta_u\|\bar{B}\|_{\max}/\bar{L}$. $$\begin{aligned}
\label{eq:lm31}
\begin{split}
&\emph{Pr}(\max_{|\mathbf{e}-\mathbf{c}|\leq m}\|X(\mathbf{e})-X(\mathbf{c})\|_\infty\geq\epsilon)\\
\leq&2{n \choose m}K^{m+2}\exp\left(-\frac{C_3n}{m}\epsilon^2N_n\rho_n\right)
\end{split}\end{aligned}$$ for $\epsilon<\eta m/n$, where $\eta=2\beta_u\|\bar{B}\|_{\max}/\bar{L}$. $$\begin{aligned}
\label{eq:lm32}
\begin{split}
&\emph{Pr}(\max_{|\mathbf{e}-\mathbf{c}|\leq m}\|X(\mathbf{e})-X(\mathbf{c})\|_\infty\geq\epsilon)\leq2{n \choose m}K^{m+2}\exp\left(-C_4\epsilon N_n\rho_n\right)
\end{split}\end{aligned}$$ for $\epsilon\geq\eta m/n$. $$\begin{aligned}
\emph{Pr}(\max_{\mathbf{e}}\|Y(\mathbf{e},\hat{\boldsymbol{\gamma}})\|_\infty\geq\epsilon)&\leq2K^{n+2}\exp(-C_2\epsilon^2N_n\rho_n)\label{eq:lm2}\end{aligned}$$ for $\epsilon<\chi \kappa_2^2$. $$\begin{aligned}
\label{eq:lm41}
\emph{Pr}(\max_{|\mathbf{e}-\mathbf{c}|\leq m}\|Y(\mathbf{e},\hat{\boldsymbol{\gamma}})-Y(\mathbf{c},\hat{\boldsymbol{\gamma}})\|_\infty\geq\epsilon)&\leq2{n \choose m}K^{m+2}\exp\left(-\frac{C_5n}{m}\epsilon^2N_n\rho_n\right)\end{aligned}$$ for $\epsilon<\frac{2\chi m}{n}$. $$\begin{aligned}
\label{eq:lm42}
\emph{Pr}(\max_{|\mathbf{e}-\mathbf{c}|\leq m}\|Y(\mathbf{e},\hat{\boldsymbol{\gamma}})-Y(\mathbf{c},\hat{\boldsymbol{\gamma}})\|_\infty\geq\epsilon)&\leq2{n \choose m}K^{m+2}\exp\left(-C_6\epsilon N_n\rho_n\right)\end{aligned}$$ for $\epsilon\geq\frac{2\chi m}{n}$.
The proofs are all given conditioned on $|e^{\mathbf{z}_{ij}\hat{\boldsymbol{\gamma}}}-\mathbb{E}e^{\mathbf{z}_{ij}\hat{\boldsymbol{\gamma}}}|\leq\chi$. By combining Lemma \[lem:phi\], we could have the conclusion directly. For any $\hat{\boldsymbol{\gamma}}$, by Bernstein inequality, when $\epsilon<\chi \kappa_2^2$, $$\begin{aligned}
&\text{Pr}(|Y_{kl}(\mathbf{e},\hat{\boldsymbol{\gamma}})|\geq\epsilon)\leq2\exp\left(-\frac{\frac{1}{2}(2N_n\epsilon)^2}{|s_{\mathbf{e}}(k,l)|\chi^2+\frac{2}{3}\chi N_n\epsilon}\right)\\
\leq&2\exp\left(-\frac{6N_n\epsilon^2}{3\kappa_2^2\chi^2+2\chi\epsilon}\right)\leq2\exp\left(-\frac{6}{5\kappa_2^2\chi^2}\epsilon^2N_n\right).\end{aligned}$$
Let $X^1(\mathbf{e})=\frac{O(\mathbf{e})-\mathbb{E}[O(\mathbf{e})|Z]}{2N_n\rho_n}$ and $X^2(\mathbf{e})=X(\mathbf{e})-X^1(\mathbf{e})$, and we establish bound for $X^1(\mathbf{e})$ and $X^2(\mathbf{e})$ respectively. By Lemma \[lem:bernin\], for any $k,l\in[K]$, let $M=\beta_u\|\bar{B}\|_{\max}|s_{\mathbf{e}}(k,l)|\rho_n$, $L=\bar{L}$, and $x=2N_n\rho_n\epsilon$, then for $\epsilon<\beta_u\|\bar{B}\|_{\max}/\bar{L}$, $$\begin{aligned}
&\text{Pr}(X^1_{kl}(\mathbf{e})\geq\epsilon)\leq\exp\left(-\frac{4N_n^2\rho_n^2\epsilon^2}{2(\beta_u\|\bar{B}\|_{\max}|s_{\mathbf{e}}(k,l)|\rho_n+2N_n\rho_n\epsilon\bar{L})}\right)\\
\leq&\exp\left(-\frac{N_n\rho_n\epsilon^2}{\beta_u\|\bar{B}\|_{\max}+\epsilon\bar{L}}\right)\leq\exp\left(-\frac{\epsilon^2N_n\rho_n}{2\beta_u\|\bar{B}\|_{\max}}\right).\end{aligned}$$
Notice that $|X_{kl}^2(\mathbf{e})|/\|\bar{B}\|_{\max}\leq|Y_{kl}(\mathbf{e},\boldsymbol{\gamma}^0)|$. Thus, for $\epsilon<\chi \kappa_2^2\|\bar{B}\|_{\max}$, $$\begin{aligned}
\text{Pr}(|X^2_{kl}(\mathbf{e})|\geq\epsilon)\leq&\text{Pr}\left(|Y_{kl}(\mathbf{e},\boldsymbol{\gamma}^0)|\geq\frac{\epsilon}{\|\bar{B}\|_{\max}}\right)\\
\leq&2\exp\left(-\frac{6}{5\kappa_2^2\chi^2\|\bar{B}\|_{\max}}\epsilon^2N_n\right).\end{aligned}$$ Thus, the bound of $X(\mathbf{e})$ will be dominated by $X^1(\mathbf{e})$, and we will ignore the second term in the bound because it is just a small order and can be absorbed into the first one.
Similar to the arguments in [@zhao2012consistency], for $|\mathbf{e}-\mathbf{c}|\leq m$, $\epsilon<\frac{2\chi m}{n}$, $$\begin{aligned}
&\text{Pr}(|Y_{kl}(\mathbf{e},\hat{\boldsymbol{\gamma}})-Y_{kl}(\mathbf{c},\hat{\boldsymbol{\gamma}})|\geq\epsilon)\leq2\exp\left(-\frac{6N_n\epsilon^2}{6\chi^2mn/N_n+2\chi\epsilon}\right)\\
\leq&2\exp\left(-\frac{3(n-1)}{8\chi^2m}\epsilon^2N_n\right)\leq2\exp\left(-\frac{n}{4\chi^2m}\epsilon^2N_n\right).\end{aligned}$$ For $\epsilon\geq\frac{2\chi m}{n}$, $$\begin{aligned}
\text{Pr}(|Y_{kl}(\mathbf{e},\hat{\boldsymbol{\gamma}})-Y_{kl}(\mathbf{c},\hat{\boldsymbol{\gamma}})|\geq\epsilon)\leq&2\exp\left(-\frac{6N_n\epsilon^2}{6\chi^2mn/N_n+2\chi\epsilon}\right)\\
\leq&2\exp\left(-\frac{3}{4\chi}\epsilon N_n\right).\end{aligned}$$
Also, for $\epsilon<\frac{2\beta_u\|\bar{B}\|_{\max}m}{n\bar{L}}$, $$\begin{aligned}
&\text{Pr}(|X^1_{kl}(\mathbf{e})-X^1_{kl}(\mathbf{c})|\geq\epsilon)\leq\exp\left(-\frac{N_n\rho_n\epsilon^2}{\beta_u\|\bar{B}\|_{\max}mn/N_n+\epsilon\bar{L}}\right)\\
\leq&\exp\left(-\frac{n-1}{2\beta_u\|\bar{B}\|_{\max}m}\epsilon^2N_n\rho_n\right)\leq\exp\left(-\frac{n}{4\beta_u\|\bar{B}\|_{\max}m}\epsilon^2N_n\rho_n\right).\end{aligned}$$ For $\epsilon\geq\frac{2\beta_u\|\bar{B}\|_{\max}m}{n\bar{L}}$, $$\begin{aligned}
\text{Pr}(|X^1_{kl}(\mathbf{e})-X^1_{kl}(\mathbf{c})|\geq\epsilon)\leq&\exp\left(-\frac{N_n\rho_n\epsilon^2}{\beta_u\|\bar{B}\|_{\max}mn/N_n+\epsilon\bar{L}}\right)\\
\leq&\exp\left(-\frac{1}{3\bar{L}}\epsilon N_n\rho_n\right).\end{aligned}$$ We will omit the bound for $|X^2_{kl}(\mathbf{e})-X^2_{kl}(\mathbf{c})|$ since it’s a smaller order.
Proof of Theorem \[thm:gene\]
-----------------------------
We divide the proof into three steps.
*Step 1* : sample and population version comparison. We prove $\exists\ \epsilon_n\to0$, such that $$\begin{aligned}
\label{eq:sample}
{\rm Pr}\left(\max_{\mathbf{e}}\left|F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)-F(\theta(\boldsymbol{\gamma}^0)T(\mathbf{e}),\theta(\boldsymbol{\gamma}^0)S(\mathbf{e}))\right|\leq\epsilon_n\right)\to1,
\end{aligned}$$ if $\varphi_n\to\infty$ and $\hat{\boldsymbol{\gamma}}\xrightarrow{p}\boldsymbol{\gamma}^0$.
Since $$\begin{aligned}
&\left|F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)-\theta(\boldsymbol{\gamma}^0)F(T(\mathbf{e}),S(\mathbf{e}))\right|\\
\leq&\left|F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)-F(\theta(\boldsymbol{\gamma}^0)\hat{T}(\mathbf{e}),\theta(\hat{\boldsymbol{\gamma}})\hat{S}(\mathbf{e}))\right|\\
&+\left|F(\theta(\boldsymbol{\gamma}^0)\hat{T}(\mathbf{e}),\theta(\hat{\boldsymbol{\gamma}})\hat{S}(\mathbf{e}))-\theta(\boldsymbol{\gamma}^0)F(\hat{T}(\mathbf{e}),\hat{S}(\mathbf{e}))\right|\\
&+\theta(\boldsymbol{\gamma}^0)\left|F(\hat{T}(\mathbf{e}),\hat{S}(\mathbf{e}))-F(T(\mathbf{e}),S(\mathbf{e}))\right|,
\end{aligned}$$ it is sufficient to bound these three terms uniformly. By Lipschitz continuity, $$\begin{aligned}
\begin{split}
&\left|F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)-\theta(\boldsymbol{\gamma}^0)F\left(\hat{T}(\mathbf{e}),\hat{S}(\mathbf{e})\right)\right|\\
\leq&M_1\|X(\mathbf{e})\|_\infty+M_2\|Y(\mathbf{e},\hat{\boldsymbol{\gamma}})\|_\infty,\label{eq:sm1}\\
\end{split}
\end{aligned}$$ $$\begin{aligned}
\begin{split}
&\left|F(\theta(\boldsymbol{\gamma}^0)\hat{T}(\mathbf{e}),\theta(\hat{\boldsymbol{\gamma}})\hat{S}(\mathbf{e}))-\theta(\boldsymbol{\gamma}^0)F(\hat{T}(\mathbf{e}),\hat{S}(\mathbf{e}))\right|\\
\leq&M_2|\theta(\hat{\boldsymbol{\gamma}})-\theta(\boldsymbol{\gamma}^0)\|\hat{S}(\mathbf{e})\|_\infty.\label{eq:sm2}
\end{split}
\end{aligned}$$ By (\[eq:lm1\]) and (\[eq:lm2\]), (\[eq:sm1\]) converges to $0$ uniformly if $\varphi_n\to\infty$. Since $\|\hat{S}(\mathbf{e})\|_\infty$ is uniformly bounded by $1$, (\[eq:sm2\]) also converges to $0$ uniformly. $$\begin{aligned}
\label{eq:sm3}
\begin{split}
&\left|F\left(\hat{T}(\mathbf{e}),\hat{S}(\mathbf{e})\right)-F\left(T(\mathbf{e}),S(\mathbf{e})\right)\right|\\
\leq& M_1\|\hat{T}(\mathbf{e})-T(\mathbf{e})\|_\infty+ M_2\|\hat{S}(\mathbf{e})-S(\mathbf{e})\|_\infty.
\end{split}
\end{aligned}$$ Since $\boldsymbol{\pi}(\mathbf{c})\xrightarrow{p}\boldsymbol{\pi}_0$, (\[eq:sm3\]) converges to $0$ uniformly. So we prove (\[eq:sample\]).
*Step 2* : proof of weak consistency. We prove that there exists $\delta_n\to0$, such that $$\begin{aligned}
\label{eq:weak}
{\rm Pr}\left(\max_{\{\mathbf{e}:\|V(\mathbf{e})-I_K\|_1\geq\delta_n\}}F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)<F\left(\frac{O(\mathbf{c})}{2N_n\rho_n},\frac{E(\mathbf{c},\hat{\boldsymbol{\gamma}})}{2N_n}\right)\right)\to1.
\end{aligned}$$
By continuity property of $F$ and Condition \[cond:max\], there exists $\delta_n\to0$, such that $$\begin{aligned}
\theta(\boldsymbol{\gamma}^0)F(T(\mathbf{c}),S(\mathbf{c}))-\theta(\boldsymbol{\gamma}^0)F(T(\mathbf{e}),S(\mathbf{e}))>2\epsilon_n
\end{aligned}$$ if $\|V(\mathbf{e})-I_K\|_1\geq\delta_n$, where $I_K=V(\mathbf{c})$. Thus, following (\[eq:sample\]), $$\begin{aligned}
{3}
&\text{Pr}\Bigg( &&\max_{\{\mathbf{e}:\|V(\mathbf{e})-I_K\|_1\geq\delta_n\}}F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)<F\left(\frac{O(\mathbf{c})}{2N_n\rho_n},\frac{E(\mathbf{c},\hat{\boldsymbol{\gamma}})}{2N_n}\right)\Bigg)\\
\geq &\text{Pr}\Bigg(&&\Bigg|\max_{\mathbf{e}:\|V(\mathbf{e})-I_K\|_1\geq\delta_n}\theta(\boldsymbol{\gamma}^0)F(T(\mathbf{e}),S(\mathbf{e}))\\
& &&-\max_{\mathbf{e}:\|V(\mathbf{e})-I_K\|_1\geq\delta_n}F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)\Bigg|\leq\epsilon_n,\\
& &&\Bigg|\theta(\boldsymbol{\gamma}^0)F(T(\mathbf{c}),S(\mathbf{c}))-F\left(\frac{O(\mathbf{c})}{2N_n\rho_n},\frac{E(\mathbf{c},\hat{\boldsymbol{\gamma}})}{2N_n}\right)\Bigg|\leq\epsilon_n\Bigg)\to1.
\end{aligned}$$
(\[eq:weak\]) implies $\text{Pr}(\|V(\mathbf{e})-I_K\|<\delta_n)\to1$. Since $$\begin{aligned}
&\frac{1}{n}|\mathbf{e}-\mathbf{c}|=\frac{1}{n}\sum_{i=1}^n\mathbbm{1}(c_i\neq e_i)=\sum_k\pi_k(1-V_{kk}(\mathbf{e}))\\
\leq&\sum_k(1-V_{kk}(\mathbf{e}))=\|V(\mathbf{e})-I_K\|_1/2,
\end{aligned}$$ weak consistency follows.
*Step 3* : proof of strong consistency.
To prove strong consistency, we need to show $$\begin{aligned}
\label{eq:strong}
{\rm Pr}\left(\max_{\{\mathbf{e}:0<\|V(\mathbf{e})-I_K\|_1<\delta_n\}}F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)<F\left(\frac{O(\mathbf{c})}{2N_n\rho_n},\frac{E(\mathbf{c},\hat{\boldsymbol{\gamma}})}{2N_n}\right)\right)\to1.
\end{aligned}$$
Combining (\[eq:weak\]) and (\[eq:strong\]), we have $${\rm Pr}\left(\max_{\{\mathbf{e}:\mathbf{e}\neq\mathbf{c}\}}F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)<F\left(\frac{O(\mathbf{c})}{2N_n\rho_n},\frac{E(\mathbf{c},\hat{\boldsymbol{\gamma}})}{2N_n}\right)\right)\to1,$$ which implies strong consistency.
By Lipschitz continuity and the continuity of derivative of $F$ w.r.t. $V(\mathbf{e})$ in the neighborhood of $I_K$, we have $$\begin{aligned}
\label{eq:st1}
\begin{split}
&F\left(\frac{O(\mathbf{e})}{2N_n\rho_n},\frac{E(\mathbf{e},\hat{\boldsymbol{\gamma}})}{2N_n}\right)-F\left(\frac{O(\mathbf{c})}{2N_n\rho_n},\frac{E(\mathbf{c},\hat{\boldsymbol{\gamma}})}{2N_n}\right)\\
=&\theta(\boldsymbol{\gamma}^0)F(\hat{T}(\mathbf{e}),\hat{S}(\mathbf{e}))-\theta(\boldsymbol{\gamma}^0)F(\hat{T}(\mathbf{c}),\hat{S}(\mathbf{c}))+\Delta(\mathbf{e},\mathbf{c}),
\end{split}
\end{aligned}$$ where $|\Delta(\mathbf{e},\mathbf{c})|\leq M_3(\|X(\mathbf{e})-X(\mathbf{c})\|_\infty)+M_4\|Y(\mathbf{e},\hat{\boldsymbol{\gamma}})-Y(\mathbf{c},\hat{\boldsymbol{\gamma}})\|_\infty)$, and $$F(T(\mathbf{e}),S(\mathbf{e}))-F(T(\mathbf{c}),S(\mathbf{c}))\leq-\bar{C}\|V(\mathbf{e})-I\|_1+o(\|V(\mathbf{e})-I_K\|_1).$$
Since the derivative of $F$ is continuous w.r.t. $V(\mathbf{e})$ in the neighborhood of $I_K$, there exists a $\delta'$ such that, $$\begin{aligned}
\label{eq:st2}
F(\hat{T}(\mathbf{e}),\hat{S}(\mathbf{e}))-F(\hat{T}(\mathbf{c}),\hat{S}(\mathbf{c}))\leq-(C'/2)\|V(\mathbf{e})-I\|_1+o(\|V(\mathbf{e})-I_K\|_1)
\end{aligned}$$ holds when $\|\boldsymbol{\pi}(\mathbf{c})-\boldsymbol{\pi}_0\|_\infty\leq\delta'$. Since $\boldsymbol{\pi}(\mathbf{c})\to\boldsymbol{\pi}_0$, (\[eq:st2\]) holds with probability approaching $1$. Combining (\[eq:st1\]) and (\[eq:st2\]), it is easy to see strong consistency follows if we can show $$\text{Pr}(\max_{\{\mathbf{e}\neq\mathbf{c}\}}|\Delta(\mathbf{e},\mathbf{c})|\leq C'\|V(\mathbf{e})-I_K\|_1/4)\to1.$$ Note $\frac{1}{n}|\mathbf{e}-\mathbf{c}|\leq\frac{1}{2}\|V(\mathbf{e})-I_K\|_1$. So for each $m\geq1$, $$\begin{aligned}
\label{eq:st3}
\begin{split}
&\text{Pr}\left(\max_{|\mathbf{e}-\mathbf{c}|=m}|\Delta(\mathbf{e},\mathbf{c})|>C'\|V(\mathbf{e}-I_K)\|_1/4\right)\\
\leq&\text{Pr}\left(\max_{|\mathbf{e}-\mathbf{c}|\leq m}\|X(\mathbf{e})-X(\mathbf{c})\|_\infty>\frac{C'm}{4M_3n}\right)(\equiv I_1)\\
&+\text{Pr}\left(\max_{|\mathbf{e}-\mathbf{c}|\leq m}\|Y(\mathbf{e},\hat{\boldsymbol{\gamma}})-Y(\mathbf{c},\hat{\boldsymbol{\gamma}})\|_\infty>\frac{C'm}{4M_4n}\right)(\equiv I_2).
\end{split}
\end{aligned}$$
Let $\eta_1=C'/4M_3$, if $\eta_1<\eta$, by (\[eq:lm31\]), $$\begin{aligned}
I_1\leq&2K^{m+2}n^m\exp(-\eta_1^2\frac{C_3m}{n}N_n\rho_n)=2K^2[K\exp(\log n-\eta_1^2C_3N_n\rho_n/n)]^m.
\end{aligned}$$ If $\eta_1>\eta$, by (\[eq:lm32\]), $$\begin{aligned}
I_1\leq&2K^{m+2}n^m\exp(-\eta_1\frac{C_4m}{n}N_n\rho_n)=2K^2[K\exp(\log n-\eta_1C_4N_n\rho_n/n)]^m.
\end{aligned}$$
Similar arguments hold for $I_2$ by using (\[eq:lm41\]) and (\[eq:lm42\]). In all cases, since $\varphi_n/\log n\to\infty$, $$\begin{aligned}
&\text{Pr}(\max_{\{\mathbf{e}\neq\mathbf{c}\}}|\Delta(\mathbf{e},\mathbf{c})|>C'\|V(\mathbf{e})-I_K\|_1/4)\\
=&\sum_{m=1}^{\infty}\text{Pr}(\max_{|\mathbf{e}-\mathbf{c}|=m}|\Delta(\mathbf{e},\mathbf{c})|>C'\|V(\mathbf{e})-I_K\|_1/4)\to0.
\end{aligned}$$ as $n\to\infty$. The proof is completed.
The Supplementary Material contains additional simulation results as well as the proof of Theorem \[THM:SC\].
Additional simulation results
=============================
Here, we present in Table \[tab:gamma\_K=1\] the simulation results on the estimation of $\boldsymbol{\gamma}$ for Section \[subsec::simu-gamma\] when we ignore the community structure. It is very similar to Table \[tab:gamma\] when the random initial community assignments are used.
$n$ $\boldsymbol{\gamma}^0_1=0.4$ $\boldsymbol{\gamma}^0_2=0.8$ $\boldsymbol{\gamma}^0_3=1.2$ $\boldsymbol{\gamma}^0_4=1.6$ $\boldsymbol{\gamma}^0_5=2$
----- ------------------------------- ------------------------------- ------------------------------- ------------------------------- -----------------------------
0.3933 0.796 1.2064 1.5963 2.0045
(0.0471) (0.0345) (0.056) (0.041) (0.0454)
0.3964 0.798 1.1977 1.6022 2.0027
(0.0251) (0.026) (0.0322) (0.0244) (0.0281)
0.3991 0.8011 1.1984 1.6026 2.0025
(0.0198) (0.016) (0.0256) (0.018) (0.0213)
0.3984 0.8008 1.1981 1.6018 2.0042
(0.0162) (0.0131) (0.018) (0.0149) (0.0188)
0.3985 0.7999 1.1971 1.5992 2.002
(0.0147) (0.0117) (0.0162) (0.0148) (0.0155)
: Simulated results over 100 replicates of $\hat{\boldsymbol{\gamma}}$, displayed as mean (standard deviation) for Section \[subsec::simu-gamma\] when we ignore the community structure.
\[tab:gamma\_K=1\]
Proof of Theorem \[THM:SC\] {#appd:c}
===========================
As a direct corollary of Theorem \[THM:ASY\], we have the following estimation error bound for $\hat{\boldsymbol{\gamma}}$.
\[lem:eta\] For any constant $\eta>0$, $\exists$ positive constants $C_\eta$ and $v_\eta$ s.t., $\emph{Pr}(\sqrt{n\rho_n}\|\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}}\|_\infty>\eta)<C_\eta\exp(-v_\eta n)$.
The following notations will be used in the proof.
- $W=A'-P$ and denote by $w_{ij}$ the $(i,j)$-th entry of $W$.
- For $t>0$, let $\mathcal{S}_t=\{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x}\|_2\leq t\}$ be the Euclidean ball of radius $t$ and set $\mathcal{S}=\mathcal{S}_1$.
The proof of Theorem \[THM:SC\] is adapted from [@lei2015consistency], so we will skip the common part and only clarify the modifications. The main idea is to bound $$\label{eq:bound}
\sup_{\mathbf{x}\in \mathcal{S}}|\mathbf{x}^T(A'-P)\mathbf{x}|.$$ The original proof consists of three steps: discretization, bounding the light pairs, and bounding the heavy pairs. Discretization is to reduce \[eq:bound\] to the problem of bounding the supremum of $\mathbf{x}^T(A'-P)\mathbf{y}$ for $\mathbf{x},\mathbf{y}$ in a finite set of grid points in $\mathcal{S}$. Then we divide $\mathbf{x},\mathbf{y}$ into light and heavy pairs, and bound them respectively. We will focus on the last two steps since the first step is the same as in [@lei2015consistency].
Discretization
--------------
For fixed $\delta_n\in(0,1)$, define $$\mathcal{T}=\{\mathbf{x}=(x_1,\cdots,x_n)\in \mathcal{S}:\sqrt{n}x_i/\delta_n\in\mathbb{Z},\forall i\},$$ where $\mathbb{Z}$ stands for the set of integers. The following lemma is the same as Lemma B.1 in [@lei2015consistency] and we will skip the proof.
$\mathcal{S}_{1-\delta_n}\subset convhull(\mathcal{T})$. As a consequence, for all $W\in\mathbb{R}^{n\times n}$, $$\|W\|\leq(1-\delta_n)^{-2}\sup_{\mathbf{x},\mathbf{y}\in \mathcal{T}}|\mathbf{x}^TW\mathbf{y}|.$$
For any $\mathbf{x},\mathbf{y}\in\mathcal{T}$, we have $$\mathbf{x}^T(A'-P)\mathbf{y}=\sum_{1\leq i,j\leq n}x_iy_j(A'_{ij}-P_{ij}).$$ We only need bound the above quantity now. We divide $(x_i,y_j)$ into *light pairs* $\mathcal{L}=\{(i,j):|x_iy_j|\leq\sqrt{\varphi_n}/n\}$ and *heavy pairs* $\mathcal{H}=\{(i,j):|x_iy_j|>\sqrt{\varphi_n}/n\}$. We will show that the tail for light pairs can be bounded exponentially while heavy pairs have a heavier tail. Thus, the rate of the latter one dominates.
Bounding the light pairs
------------------------
\[lem:light\] Under estimation error condition, for $c>0$, there exist constants $C_c, v_c>0$ s.t. $$P\left(\sup_{x,y\in \mathcal{T}}\left|\sum_{(i,j)\in\mathcal{L}(x,y)}x_iy_jw_{ij}\right|\geq c\sqrt{\varphi_n}\right)\leq C_c\exp\left[-\left(v_c-\log\left(\frac{7}{\delta}\right)\right)n\right].$$
Define $w'_{ij}=A_{ij}e^{-\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}-P_{ij}$ and $\delta_{ij}=A_{ij}e^{-\mathbf{z}_{ij}^T\hat{\boldsymbol{\gamma}}}-A_{ij}e^{-\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}$, then $w_{ij}=w'_{ij}+\delta_{ij}$, and notice that $$\text{Pr}(\sum w_{ij}>2t)\leq \text{Pr}(\sum w'_{ij}>t)+\text{Pr}(\sum \delta_{ij}>t),$$ so we could bound two parts respectively. Also, we need to keep in mind that $$\sum_{i<j}u_{ij}^2\leq\sum_{i<j}2(x_i^2y_j^2+x_j^2y_i^2)\leq2\sum_{1\leq i,j\leq n}x_i^2y_j^2=2\|x\|_2^2\|y\|_2^2\leq2.$$
*Step1 : Bound $w'_{ij}$.*
Since $$\begin{aligned}
\sum_{i<j}\mathbb{E}[(w'_{ij}u_{ij})^2|\mathbf{z}_{ij}]=&\sum_{i<j}u_{ij}^2\lambda_{ij}e^{-2\boldsymbol{z}_{ij}^T\boldsymbol{\gamma}^0}=\sum_{i<j}u_{ij}^2P_{ij}e^{-\boldsymbol{z}_{ij}^T\boldsymbol{\gamma}^0}\\
\leq&\rho_n\beta_l^{-1}\|\bar{B}\|_{\max}\sum_{i<j}u^2_{ij}\leq2\beta_l^{-1}\|\bar{B}\|_{\max}\rho_n,\end{aligned}$$ define $M=2\beta_l^{-1}\|\bar{B}\|_{\max}\rho_n$, $L=2\bar{L}\sqrt{\varphi_n}(n\beta_l)^{-1}$ and $x=c\sqrt{\varphi_n}$, we could applying Lemma \[lem:bernin\] to $u_{ij}w'_{ij}$ to get $$\begin{aligned}
\text{Pr}(\sum_{i<j}w'_{ij}u_{ij}\geq c\sqrt{\varphi_n})\leq&\exp\left(-\frac{c^2\varphi_n}{4c\bar{L}\varphi_n(n\beta_l)^{-1}+4\rho_n\beta_l^{-1}\|\bar{B}\|_{\max}}\right)\\
=&\exp\left(-\frac{c^2\beta_ln}{4c\bar{L}+4\|\bar{B}\|_{\max}}\right)\end{aligned}$$
*Step2 : Bound $\delta_{ij}$.*
We consider two cases $\|\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}}\|_{\infty}>\eta/\sqrt{n\rho_n}$ and $\|\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}}\|_{\infty}\leq\eta/\sqrt{n\rho_n}$ separately. Conditioning on the second case, by choosing $\eta<(p\zeta)^{-1}$, we have $$\begin{aligned}
u_{ij}\delta_{ij}=&u_{ij}[A_{ij}e^{-\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}(e^{\mathbf{z}_{ij}^T(\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}})}-1)]<2u_{ij}[A_{ij}e^{-\mathbf{z}_{ij}^T\boldsymbol{\gamma}^0}\mathbf{z}_{ij}^T(\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}})]\\
<&2\zeta\eta pu_{ij}A_{ij}/(\beta_l\sqrt{n\rho_n})<2u_{ij}A_{ij}/(\beta_l\sqrt{n\rho_n})\end{aligned}$$ The first inequality is due to $|e^t-1|<2|t|$ when $|t|<1$. Define $M=2\beta_u\|\bar{B}\|_{\max}\rho_n\geq\sum_{i<j}u_{ij}^2\lambda_{ij}=\text{var}(\sum_{i<j}u_{ij}A_{ij}|Z)$, $L=2\bar{L}\sqrt{\varphi_n}/n$ and $x=c\sqrt{\varphi_nn\rho_n}$, by Lemma \[lem:bernin\], $$\begin{aligned}
&\text{Pr}(\sum_{i<j}u_{ij}(A_{ij}-P_{ij})>c\sqrt{\varphi_nn\rho_n})\\
\leq&\exp\left(-\frac{c^2\varphi_nn\rho_n}{4\|\bar{B}\|_{\max}\beta_u\rho_n+4c\sqrt{n\rho_n}\varphi_n\bar{L}/n}\right)\\
=&\exp\left(-\frac{c^2n\sqrt{n\rho_n}}{4\|\bar{B}\|_{\max}\beta_u/\sqrt{n\rho_n}+4c\bar{L}}\right)\\
\leq&\exp\left(-\frac{c^2n\sqrt{n\rho_n}}{4\|\bar{B}\|_{\max}\beta_u+4c\bar{L}}\right)\end{aligned}$$
Because $$\sum_{i<j}u_{ij}P_{ij}\leq\rho_n\|\bar{B}\|_{\max}\beta_u\sum_{i<j}u_{ij}\leq\sqrt{2N_n}\rho_n\|\bar{B}\|_{\max}\beta_u\leq n\rho_n\|\bar{B}\|_{\max}\beta_u,$$ we have $$\begin{aligned}
\exp\left(-\frac{c^2n\sqrt{n\rho_n}}{4\|\bar{B}\|_{\max}\beta_u+4c\bar{L}}\right)\geq&\text{Pr}(\sum_{i<j}u_{ij}(A_{ij}-P_{ij})>c\sqrt{\varphi_nn\rho_n})\\
\geq&\text{Pr}(\sum_{i<j}u_{ij}A_{ij}>(c+\|\bar{B}\|_{\max}\beta_u)\varphi_n),\end{aligned}$$ which is equivalent to $\text{Pr}(\sum_{i<j}u_{ij}A_{ij}>c\varphi_n)\leq\exp(-C_cn\sqrt{\varphi_n})$, where $C_c$ is constant.
Thus, for $\eta<(p\zeta)^{-1}$, $$\begin{aligned}
&\text{Pr}\left(\sum_{i<j}\delta_{ij}u_{ij}>c\sqrt{\varphi_n}\right)\\
=&\text{Pr}\left(\sum_{i<j}\delta_{ij}u_{ij}>c\sqrt{\varphi_n}\middle\vert\|\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}}\|_{\infty}\leq\frac{\eta}{\sqrt{n\rho_n}}\right)\text{Pr}\left(\|\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}}\|_{\infty}\leq\frac{\eta}{\sqrt{n\rho_n}}\right)\\
&+\text{Pr}\left(\sum_{i<j}\delta_{ij}u_{ij}>c\sqrt{\varphi_n}\middle\vert\|\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}}\|_{\infty}>\frac{\eta}{\sqrt{n\rho_n}}\right)\text{Pr}\left(\|\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}}\|_{\infty}>\frac{\eta}{\sqrt{n\rho_n}}\right)\\
\leq&\text{Pr}\left(\sum_{i<j}\delta_{ij}u_{ij}>c\sqrt{\varphi_n}\middle\vert\|\boldsymbol{\gamma}^0-\hat{\boldsymbol{\gamma}}\|_{\infty}\leq\frac{\eta}{\sqrt{n\rho_n}}\right)+C_\eta\exp(-v_\eta n)\\
\leq&\text{Pr}\left(\sum_{ij}u_{ij}A_{ij}>c\beta_l\varphi_n/2\right)+C_\eta\exp(-v_\eta n)\\
\leq&\exp(-C_cn\sqrt{n\rho_n})+C_\eta\exp(-v_\eta n)\\
\leq& C_{c,\eta}\exp(-v_{c,\eta}n),\end{aligned}$$ where $C_{c,\eta}$ and $v_{c,\eta}$ are two constants determined by $c$ and $\eta$.
By a standard volume argument we have $|T|\leq e^{n\log(7/\delta)}$ (see Claim 2.9 of [@feige2005spectral]), so the desired result follows from the union bound.
Bounding the heavy pairs
------------------------
By the same argument as appendix B.3 of [@lei2015consistency], to bound $\sup_{\mathbf{x},\mathbf{y}\in \mathcal{T}}|\sum_{(i,j)\in\mathcal{H}(x,y)}x_iy_jw_{ij}|$, it suffices to show $$\sum\limits_{(i,j)\in\mathcal{H}}x_iy_jA'_{ij}=O(\sqrt{\varphi_n})$$ with high probability. Since $A'_{ij}=A_{ij}e^{-\mathbf{z}_ {ij}^T\boldsymbol{\gamma}^0}-(A_{ij}e^{-\mathbf{z}_ {ij}^T\boldsymbol{\gamma}^0}-A_{ij}e^{-\mathbf{z}_ {ij}^T\hat{\boldsymbol{\gamma}}})\leq A_{ij}\beta_l^{-1}(1+o(1))$, we only need to show $$\sum\limits_{(i,j)\in\mathcal{H}}x_iy_jA_{ij}=O(\sqrt{\varphi_n})$$ with high probability. Define $A''_{ij}=\max(A_{ij},1)$, then $A''_{ij}\sim B(1,1-e^{-\lambda_{ij}})$ and $A''_{ij}=A_{ij}(1-(1-A_{ij}^{-1})_+)=A_{ij}(1+o(1))$, so it’s sufficient to show $$\sum\limits_{(i,j)\in\mathcal{H}}x_iy_jA''_{ij}=O(\sqrt{\varphi_n}).$$ Because $\sum_{j}(1-e^{-\lambda_{ij}})=O(\sqrt{\varphi_n})$, so the problem is exactly equivalent to [@lei2015consistency]. We can directly get the following lemma.
(Heavy pair bound). For any given $r>0$, there exists a constant $C_r$ such that $$\sup_{x,y\in T}|\sum\limits_{(i,j)\in\mathcal{H}}x_iy_jw_{ij}|\leq C_r\sqrt{\varphi_n}$$ with probability at least $1-2n^{-r}$.
[^1]: The uniqueness is interpreted up to a permutation of the labels.
|
---
abstract: |
We prove an analogue of Sogge’s local $L^p$ estimates [@So15] for $L^p$ norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-Gérard-Tzvetkov [@BuGeTz], Hu [@Hu], and Chen-Sogge [@ChSo]. The improvements are logarithmic on negatively curved manifolds (without boundary) and by $o(1)$ for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get $o(1)$ improvements on $L^\infty$ estimates of Cauchy data away from a shrinking neighborhood of the corners, and as a result using the methods of [@GRS; @ZeJJ1; @ZeJJ], we get that the number of nodal domains of $2$-dimensional ergodic billiards tends to infinity as $\la \to \infty$. These results work only for a full density subsequence of any given orthonormal basis of eigenfunctions.\
We also present an extension of the $L^p$ estimates of [@BuGeTz; @Hu; @ChSo] for the restrictions of Dirichlet and Neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary. This part does not assume ergodicity on the manifolds.
address: 'Department of Mathematics, UC Irvine, Irvine, CA 92617, USA'
author:
- Hamid Hezari
title: Quantum ergodicity and $L^p$ norms of restrictions of eigenfunctions
---
Introduction
============
The quantum ergodicity results of [@Sh; @CdV; @Ze87], and the small scale quantum ergodicity results of [@Han; @HeRi] on negatively curved manifolds[^1] are recently shown to imply improvements for several measurements of eigenfunctions such as: $L^p$ norms [@HeRi; @So15], number of nodal domains [@ZeN], growth rates, the size of nodal sets, order of vanishing of eigenfunctions [@He16a], and also the inner radius of nodal domains [@He16b]. The purpose of this article is to prove another application of $L^2$ equidistribution of eigenfunctions. We will show that $L^p$ estimates of restrictions of eigenfunctions to submanifolds can be improved according to certain powers of $r(\la)$, where $r(\la)$ is the least radius of shrinking geodesic balls on which the eigenfunctions equidistribute uniformly.
Let $(X,g)$ be a compact connected smooth Riemannian manifold of dimension $n \geq 2$, with or without boundary. When $\partial X \neq \emptyset$, we assume that $\partial X$ is piecewise [^2] smooth. We denote the singular part of the boundary by $\mathcal S$. Let $\Sigma$ be a compact smooth submanifold [^3] of dimension $k$ of the interior $X \backslash \partial X$, or of the regular part of the boundary $\partial X \backslash \mathcal S$. The metric $g$ induces natural volume measures on $X$ and $\Sigma$, which we denote by $d_gv$ and $d_g \sigma$ respectively and denote $L^p(X)$ and $L^p(\Sigma)$ to be the corresponding $L^p$ spaces. Suppose $\Delta_g$ is the positive Laplace-Beltrami operator on $(X, g)$ (with Dirichlet or Neumann boundary conditions if $\partial X \neq \emptyset$). For $\lambda \geq 1$, we define the spectral cluster operators $\Pi_\lambda=\bigoplus_{ \sqrt{\lambda_j} \in\,[\sqrt{\lambda}, \sqrt{\lambda}+1]} \Pi_{E_{\lambda_j}}$, where $\Pi_{E_{\lambda_j}}$ is the orthogonal projection operator onto the eigenspace $E_{\lambda_j}=\, \text{ker}\, (\Delta_g - \lambda_j)$.
**Main tool: local $L^p$ restriction estimates**
-------------------------------------------------
The main lemma that enables us to prove our results is the following analogue of Sogge’s local $L^p$ estimates [@So15] for local $L^p$ norms of restrictions of eigenfunctions of $\Delta_g$ to $\Sigma$.
\[LocalLp\]Let $(X,g)$ and $\Sigma$ be as above and assume $\partial X$ is smooth if it is non-empty. Also let $2 \leq p \leq \infty$ and $r_0 = \text{inj}(X, g)$. Suppose there exists $\delta=\delta(n, k, p)$ and $C >0$ independent of $\lambda$, such that for all $\lambda \geq 1$ $$\label{NormOfSpectralCluster}
||\Pi_\lambda ||_{L^2(X) \to L^p(\Sigma)} \leq C \lambda^{\delta}.$$ Then for any eigenfunction $\psi_{\lambda_j} \in E_{\lambda_j}$ with $\lambda_j \geq 1$, any $x \in \Sigma$, and any $r \in [\lambda_j^{-1/2}, r_0/2]$, there exists a constant $C_1$, dependent only on $C$ and $\delta$, such that $$\label{LocalLpEstimates}
||\psi_{\lambda_j} ||_{L^p(B_{r/2}(x) \cap \Sigma)} \leq C_1 r^{-\frac{1}{2}} || \psi_{\lambda_j}||_{L^2(B_{r}(x))} \lambda_j^\delta .$$
Our first result is the following conditional theorem which will be obtained using the above lemma and a covering argument.
\[LpEstimatesSS\] Let $(X,g)$ and $\Sigma$ be as in Lemma \[LocalLp\] and $2 \leq p \leq \infty$. Suppose there exists $\delta=\delta(n, k, p)$ and $C$ independent of $\lambda$, such that for all $\lambda \geq 1$ $$\label{NormOfSpectralCluster1}
||\Pi_\lambda ||_{L^2(X) \to L^p(\Sigma)} \leq C \lambda^{\delta}.$$ Let $\psi_{\lambda_j} \in E_{\lambda_j}$ be an $L^2$(X)-normalized eigenfunction with $\lambda_j \geq 1$. Then there exists $r^*$ dependent only on $(X, g)$ and $\Sigma$, such that if for some $r \in [\lambda_j^{-1/2}, r^*]$ and for all geodesic balls $\{B_{r}(x)\}_{x \in \Sigma}$ we have $$\label{L2assumption2}
|| \psi_{\la_j}||^2_{L^2(B_{r}(x))} \leq K r^n,$$ for some constant $K$ independent of $x$, then $$|| \psi_{\la_j}||_{L^p(\Sigma)} \leq C_2 r^{\frac{n-1}{2}-\frac{k}{p}} \lambda_j^\delta.$$ Here $C_2$, depends only on $(X, g)$, $\Sigma$, p, $K$, and $C$.
As it is evident the above theorem is subject to two conditions. Under (\[NormOfSpectralCluster1\]) one gets the *trivial* $L^p(\Sigma)$ bounds $\la_j^ \delta$, but under (\[L2assumption2\]) one can improve the trivial bounds $\la_j^\delta$ by the small [^4] factor $r^{\frac{n-1}{2}-\frac{k}{p}}$. The condition (\[NormOfSpectralCluster1\]) on $\Pi_\la$ is proved in [@BuGeTz; @Hu; @ChSo] for smooth manifolds without boundary. The condition (\[L2assumption2\]) on $L^2$ norms on small balls is proved in [@Han; @HeRi] for negatively curved manifolds with $r= (\log \la_j)^{- \kappa}$, for any $\kappa \in (0, \frac{1}{2n})$. Putting these results and the above theorem together we obtain the following logarithmic improvements:
\[LpQENC\]
Let $(X, g)$ be a boundaryless compact connected smooth Riemannian manifold of dimension $n \geq 2$, with negative sectional curvature, and let $\Sigma$ be a compact submanifold of $X $ of dimension $k$. Let $\epsilon >0$ and let $\{ \psi_{\lambda_j}\}_{j \in \IN}$ be any ONB of $L^2(X)$ consisting of eigenfunctions of $\Delta_g$ with eigenvalues $\{\lambda_j\}_{j \in \IN}$. Then there exists $S \subset \IN$ of full density such that with the exception of $(k, p) =(n-1, 2)$ $$2 \leq p \leq \infty, j \in S: \quad || \psi_{\lambda_j}||_{L^p(\Sigma)} = O_{\epsilon}\left ( (\log \lambda_j)^{-(\frac{1}{2n})(\frac{n-1}{2}-\frac{k}{p})+\epsilon}\lambda_j^{\delta(n, k, p)} \right ),$$ where $$\label{delta}
\Small {\delta(n,k,p)= \begin{cases} \frac{n-1}{8}-\frac{n-2}{4p} & k=n-1, \; 2 \leq p < \frac{2n}{n-1} \; (\text{\cite{BuGeTz}}) \\
\frac{n-1}{4}-\frac{n-1}{2p} & k=n-1, \; \frac{2n}{n-1} \leq p \leq \infty \; (\text{\cite{BuGeTz}, and \cite{Hu} for} \; p=\frac{2n}{n-1} )\\
\frac{n-1}{4}-\frac{k}{2p} & k < n-2, \; 2 \leq p \leq \infty, \; \text{or} \; k=n-2, \;2<p \leq \infty \; (\text{\cite{BuGeTz}}) \\
\frac{1}{4} & k=n-2, n=3 , p=2 \; \; (\text{\cite{ChSo}}) \\
\text{see Remark \ref{log}} & k=n-2, n>3, p=2. \\
\end{cases} }$$ When $k=n-1$ and $\Sigma$ has a non-zero scalar second fundamental form, the exponent $\delta(n,n-1,p)$ can be improved to $$\label{deltatilde} \Small \tilde{\delta}(n,p)= \frac{n-1}{6}- \frac{2n-3}{6p}, \qquad 2 < p < \frac{2n}{n-1}. \qquad (\text{\cite{BuGeTz}})$$
In the the case of manifolds with ergodic geodesic flows (ergodic billiard flows when the manifold has a boundary), the improvements are given only by $o(1)$. We emphasize that the following theorem is only valid for interior submanifolds.
\[LpQE\] Let $(X, g)$ be a compact connected smooth Riemannian manifold of dimension $n \geq 2$, possibly with (piecewise smooth) boundary, and let $\Sigma$ be a compact submanifold of $X \backslash \partial X$ of dimension $k$. Suppose the geodesic flow on $(X, g)$ is ergodic when $\partial X = \emptyset $, or suppose the billiard flow on $(X,g)$ is ergodic when $\partial X \neq \emptyset$. Let $\{ \psi_{\lambda_j} \}_{j \in \IN}$ be any ONB of $L^2(X)$ consisting of eigenfunctions of $\Delta_g$ (with Dirichlet or Neumann boundary conditions when $\partial X \neq \emptyset$) with eigenvalues $\{\lambda_j\}_{j \in \IN}$. Then there exists $S \subset \IN$ of full density such that, with the exception of $(k, p) =( n-1, 2)$, we have $$2 \leq p \leq \infty, j \in S: \quad || \psi_{\lambda_j}||_{L^p(\Sigma)} = o\left (\lambda_j^{\delta(n, k, p)}\right ),$$ where $\delta$ is defined by (\[delta\]). When $k=n-1$ and $\Sigma$ has a non-zero scalar second fundamental form, the exponent $\delta(n,n-1,p)$ can be improved to $\tilde \delta$ denoted in (\[deltatilde\]).
Improved sup norms and the number of nodal domains
--------------------------------------------------
When $\partial X \neq \emptyset$, we get the following $o(1)$ improvements of the results of Grieser [@Gr02], Sogge [@So02], and [@Xu] for $L^\infty$ norms of QE eigenfunctions and their gradient, away from a shrinking neighborhood $\mathcal T_{{\varepsilon}_j} ( \mathcal S)= \{ x \in X; \; d(x, \mathcal S) \leq {\varepsilon}_j \}$ of the singular part of the boundary. In particular if $\Sigma \subset \partial X \backslash \mathcal S$ is a smooth compact submanifold of the regular part of the boundary, we get sup norms of the form $ o (\lambda_j^{\frac{n-1}{4}})$ for the boundary traces of QE eigenfunctions on $\Sigma$.
\[LinftyQE\] Let $(X, g)$ be a compact connected smooth Riemannian manifold of dimension $n \geq 2$ with piecewise smooth boundary and let $\mathcal S$ be the singular part of $\partial X$. Suppose the billiard flow on $(X,g)$ is ergodic. Let $\{ \psi_{\lambda_j} \}_{j \in \IN}$ be any ONB consisting of eigenfunctions of $\Delta_g$ (with Dirichlet or Neumann boundary conditions) with eigenvalues $\{\lambda_j\}_{j \in \IN}$. Then there exist $S \subset \IN$ of full density and $\{ {\varepsilon}_j \}_{j \in S}$ with ${\varepsilon}_ j \to 0^+$, such that for $j \in S$ $$\sup_{X \backslash \mathcal T_{{\varepsilon}_j}(\mathcal S)} | \psi_{\la_j} | = o\left (\lambda_j^{\frac{n-1}{4}}\right ) \quad \text{and} \quad \sup_{X \backslash \mathcal T_{{\varepsilon}_j}(\mathcal S)} \la_j^{- \frac12} |\nabla \psi_{\la_j} | = o\left (\lambda_j^{\frac{n-1}{4}}\right ).$$ Hence, in particular $$\text{in the Neumann case:} \quad \sup_{\partial X \backslash \mathcal T_{{\varepsilon}_j}(\mathcal S)} | \psi_{\la_j} | = o\left (\lambda_j^{\frac{n-1}{4}}\right ),$$ $$\qquad \text{in the Dirichlet case:} \quad \sup_{\partial X \backslash \mathcal T_{{\varepsilon}_j}(\mathcal S)} \la_j^{- \frac12}|\partial_n \psi_{\la_j} | = o\left (\lambda_j^{\frac{n-1}{4}}\right ).$$
As a corollary of the above sup norm estimates, and using the method [^5] of Jung-Zelditch [@ZeJJ], we get the following generalization of their results on the number of nodal domains for any ergodic billiard table with piecewise smooth boundary, including the Bunimowich stadium [@Bun], Sinai dispersive billiards [@Si], and their families.
\[NN\] Let $(X, g)$ be a compact connected smooth Riemannian manifold with piecewise smooth boundary of dimension $n=2$. Suppose the billiard flow on $(X,g)$ is ergodic. Let $\{ \psi_{\lambda_j} \}_{j \in \IN}$ be any ONB consisting of eigenfunctions of $\Delta_g$ (with Dirichlet or Neumann boundary conditions) with eigenvalues $\{\lambda_j\}_{j \in \IN}$. Then there exists $S \subset \IN$ of full density such that the number of nodal domains of $\psi_{\la_j}$ tends to infinity as $ \la_j \to \infty$ along $S$.
We recall that in [@ZeJJ], the above result was proved for non-positively curved manifolds with smooth concave boundary. It is known (see [@Si], [@ChSi], [@BuChSi]) that such billiard tables are ergodic. The main ingredients of the proof of [@ZeJJ] are quantum ergodicity theorems of [@GeLe; @Bu; @HaZe] for the boundary values of eigenfunctions, the so called “Kuznecov sum formula" for manifolds with smooth boundary [@HHHZ], and sup norm estimates of size $o(\la^{1/4})$ for the boundary values of eigenfunctions on positively curved manifolds with smooth concave boundary [@SoZe14]. For us to prove the above theorem, the first two ingredients are still available except that our boundary can have singular points. We will discuss in the proof that as long as we stay away from $\mathcal S$, the corners will not cause any problems. The last ingredient, which is the new ingredient, is the sup norms in Theorem \[LinftyQE\] that this paper provides.
$L^p$ restrictions for manifolds with boundary
-----------------------------------------------
In this section we show that as a corollary of the local $L^p$ restrictions estimates (Lemma \[LocalLp\]), one can extend the results of [@BuGeTz; @Hu; @ChSo] to manifolds with boundary.
\[BoundaryCor\] Let $(X, g)$ be a compact connected smooth Riemannian manifold of dimension $n$, with (piecewise smooth) boundary, and let $\Sigma$ be a compact smooth submanifold of $X \backslash \partial X$ of dimension $k$. Suppose $\psi_\la$ is an $L^2(X)$-normalized eigenfunction of $\Delta_g$ with Dirichlet or Neumann boundary conditions. Then for all $ 2 \leq p \leq \infty$ $$|| \psi_\la ||_{L^p( \Sigma)} \leq C_s \la^{\delta(n, k, p)},$$ where $C_s$ depends on $s= d( \Sigma, \partial X)$ and $\delta(n, k, p)$ is defined by (\[delta\]). When $k=n-1$ and $\Sigma$ has a non-zero scalar second fundamental form, the exponent $\delta(n,n-1,p)$ can be improved to $\tilde \delta(n, p)$ denoted in (\[deltatilde\]).
We emphasize that the above corollary holds for the eigenfunctions and not necessarily for the spectral cluster operators $\Pi_\la$. See the results of Blair[@Bla] where estimates on $||\Pi_\la ||_{L^2(X) \to L^p(\Sigma)}$ are given with a natural loss due to whispering gallery modes. We also underline that in the proof of this corollary, in addition to Lemma \[LocalLp\], we heavily use the results of [@BuGeTz; @Hu; @ChSo] on the spectral cluster operators $\Pi_\la$ on compact manifolds without boundary.
**Remarks**
-----------
\[Chen\] We point out that when $(X,g)$ is negatively curved, for $p= \infty$ the estimate in Theorem \[LpQENC\] is worse than Berard’s upper bound $ (\log \lambda_j)^{-\frac{1}{2}}{\lambda_j^{\frac{n-1}{4}}}$, which is valid on all non-positively curved compact manifolds [@Be]. In addition, for $k=n-1$ and the range $ p > \frac{2n}{n-1}$, and also $k \leq n-2$ and $ p > 2$, our estimates are weaker than the upper bounds $(\log \lambda_j)^{-\frac{1}{2}}{\lambda_j^{\delta(n, k, p)}}$ of [@Ch].
\[QER\] Our improved estimates do not include the case $(k, p)=(n-1, 2)$. However, by the quantum restriction theorems of [@TZ] and [@DZ], under a certain lack of microlocal symmetry assumption on $\Sigma$, the restrictions $\psi_\lambda |_\Sigma$ are QE on $\Sigma$. Hence in particular $ || \psi_\lambda ||^2_{L^2 (\Sigma)}$ are uniformly bounded. Of course one can interpolate these uniform upper bounds for $p=2$ with our estimates for $p=\frac{2n}{n-1}$ to get better estimates, however the full density subsequences that arise in the proofs of [@TZ] and [@DZ] depend on $\Sigma$, while the full desity subsequences that are chosen in our Theorems \[LpQE\] and \[LpQENC\] are independent of the choice of $\Sigma$.
\[log\] The exponents $\delta(k, n, p)$ in (\[delta\]), and $\tilde{\delta}(n,p)$ in (\[deltatilde\]) when $\Sigma$ is *curved*, hold more generally for the spectral cluster operators (\[NormOfSpectralCluster\]) and are sharp by [@BuGeTz] for the given range of parameters in (\[delta\]). The only case not covered in (\[delta\]) is the case $k=n-2$, $p=2$. It was proved in [@BuGeTz] that in this case for $\la \geq 2$ $$||\Pi_\lambda ||_{L^2(X) \to L^2(\Sigma)} \leq C \sqrt{ \log \la} \, \lambda^{\frac{1}{4}}.$$ It is believed that this estimate is not sharp and one should be able to remove the $\log$ term as was proved in [@ChSo] for $n=3$.
One might be able to generalize the $o(1)$ improvements in Theorem \[LpQE\] for semiclassical pseudodifferential operators whose principal symbols generate ergodic Hamiltonian systems. However, we point out that the local $L^p$ estimates we obtained used the finite speed propagation property of the even part of the wave group, something that we do not necessarily have in a more general framework. It is possible that a rescaling argument applied to the results of [@Ta12] and [@HaTa] would give local $L^p$ estimates for some range $r > h^\alpha$, which can still be useful and for example would give $o(1)$ improvements for QE eigenfunctions of semiclassical operators. In fact, such a rescaling argument with the help of semiclassical $L^p$ estimates of [@KTZ] was used in [@HeRi] to obtain local $L^p$ estimates, however the proof of [@HeRi] (although not explicitly stated) only worked for $r \geq \lambda^{-1/8}$. In [@So15], Sogge gave an elegant proof that works for $r \geq \lambda^{-1/2}$, which is the method we have adapted in the present paper.
Using the small-scale QE results of [@LeRu], and our Theorem \[LpEstimatesSS\], we can get polynomial improvements for toral eigenfunctions in dimensions $n \geq 3$.
One can also use the result of Blair [@Bla] to prove $o(1)$ improvements on his $L^p$ estimates for the boundary values of Neumann eigenfunctions, analogous to the sup norm estimates of Theorem \[LinftyQE\]. We underline that in this situation there will be a loss for $p \neq \infty$ due to whispering gallery modes near the boundary.
Using our method and also the spectral cluster estimates of Smith-Sogge [@SoSm] on manifolds with smooth boundary, one can give $o(1)$ improvements on $L^p$ norms of QE eigenfunctions on manifolds with piecewise smooth boundary, away from a shrinking neighborhood of the corners.
As a final remark, we must mention the recent works of Blair-Sogge [@BlSo], Xi-Zhang [@XZ], and Marshall [@Ma]. In [@BlSo], logarithmic improvements are given on the $L^2$ norms of restrictions to geodesics on non-positively curved surfaces. In [@XZ], inspired by the works of Blair-Sogge [@BlSo] and Sogge [@So16b] $\log \log$ improvements are obtained for $L^4$ geodesic restrictions on non-positively curved surfaces and $\log$ improvements in the case of compact hyperbolic surfaces. In [@Ma], $L^2$ geodesic restriction estimates are improved by a power of $\la$ for Hecke-Maass cusp forms.
\[SSQE\] **Background on small scale quantum ergodicity**
---------------------------------------------------------
First, we recall that the quantum ergodicity result of Shnirelman-Colin de Verdière-Zelditch [@Sh; @CdV; @Ze87] implies in particular that if the geodesic flow of a smooth compact Riemannian manifold without boundary is ergodic then for any ONB $\{ \psi_{\lambda_j} \}_{j=1}^\infty$ consisting of the eigenfunctions of $\Delta_g$, there exists a full density subset $S \subset \IN$ such that for any fixed $r < \text{inj}(X, g)$, independent of $\lambda_j$, one has $$\label{QE} || \psi_{\lambda_j}||^2_{L^2(B_{r}(x))} \sim \frac{\text{Vol}_g(B_r(x))}{\text{Vol}_g(X)}, \qquad \text{as}\quad \lambda_j \to \infty, \quad j \in S.$$ The analogous result on manifolds with piecewise smooth boundary and with ergodic billiard flows was proved by [@ZZ].
The small scale equidistribution problem asks whether (\[QE\]) holds for $r$ dependent on $\lambda_j$. A quantitative QE result of Luo-Sarnak [@LuSa] shows that the Hecke eigenfunctions on the modular surface satisfy this property along a density one subsequence for $r=\lambda^{- \kappa}$ for some small $\kappa>0$. Also, under the generalized Riemann hypothesis, Young [@Yo] has proved that small scale equidistribution holds for Hecke eigenfunctions for $r = \lambda^{-1/4 + \epsilon}$. In [@Han] and [@HeRi], this problem was studied for the eigenfunctions of compact negatively curved manifolds. To be precise, it was proved that on compact negatively curved manifolds without boundary, for any $\epsilon >0$ and any ONB $\{ \psi_{\lambda_j} \}_{j=1}^\infty$ consisting of the eigenfunctions of $\Delta_g$, there exists a subset $S_{\epsilon} \subset \IN$ of full density such that for all $x \in X$ and $j \in S_{\epsilon}$: $$\label{QENC} \quad K_1 r^n \leq || \psi_{\lambda_j}||^2_{L^2(B_{r}(x))} \leq K_2 r^n, \qquad \text{with} \;\; r=(\log \lambda_j)^{-\frac{1}{2n} +\epsilon},$$ for some positive constants $K_1, K_2$ which depend only on $(X, g)$ and $\epsilon$. [^6]
We also point out that although eigenfunctions on the flat torus ${{\mathbb R}}^n / {{\mathbb Z}}^n$ are not quantum ergodic, however they equidistribute on the configuration space ${{\mathbb R}}^n / {{\mathbb Z}}^n$ (see [@MaRu], and also [@Ri] and [@Taylor] for later proofs). So one can investigate the small scale equidistribution property for toral eigenfunctions. It was proved in [@HeRiTorus] that a commensurability of $L^2$ masses such as (\[QENC\]) is valid for a full density subsequence with $r = \lambda^{-1/(7n+4)}$. Lester-Rudnick [@LeRu] improved this rate of shrinking to $r= \lambda^{- \frac{1}{2n-2} +\epsilon}$, and in fact they proved that the stronger statement (\[QE\]) holds. They also showed that their results are almost sharp [^7]. The case of interest is $n=2$, which gives $r= \lambda^{-1/2 +\epsilon}$. A natural conjecture is that this should be the optimal rate [^8] of shrinking on negatively curved manifolds. A recent result of [@Han16] proves that random eigenbases on the torus enjoy small scale QE for $r= \lambda^{-\frac{n-2}{4n} +\epsilon}$, which is better than [@LeRu] for $n \geq 5$.
Definition of manifolds with piecewise smooth boundary {#PS2}
------------------------------------------------------
We follow the definition of [@HaZe], but we allow our Riemannian manifolds to be non-Euclidean.
\[PS\] Let $X$ be the closure of an open connected subset of a smooth compact connected boundaryless manifold $\tilde X$ of dimension $n \geq 2$. We say that $X \subset \tilde X$ is a piecewise smooth manifold if the boundary $\partial X$ is Lipschitz, and can be written as a finite disjoint union $$\partial X = H_1 \cup \dots \cup H_m \cup \mathcal S,$$ where each $H_i$ is an open subset of a smooth embedded hypersurface $S_i$, with $\text{int}(X)$ lying locally on one side of $H_i$, and where $\mathcal S$ is a closed subset that lies on a finite union of compact submanifolds of $\tilde X$ of dimensions $n-2$ or less. The sets $H_i$ are called boundary hypersurfaces of $X$. We call $\mathcal S$ the singular set (or sometimes corners), and write $\partial X \setminus \mathcal S$ for the regular part of the boundary.
Throughout with paper we assume that $g$ is a metric on $X$ that can be extended to a smooth metric $\tilde g$ on $ \tilde X$. A geodesic ball $B_r(x)$ in $X$ (centered at $x \in X$) is defined to be $\tilde B_r(x) \cap \tilde X$ where $\tilde B_r(x)$ is the geodesic ball in $(\tilde X, \tilde g)$ of radius $r$ centered at $x$. We also define $\text{inj}(X, g) = \inf_{x \in X} \text{inj}(x)$, where $\text{inj}(x)$ is the largest $R$ such that $\tilde B_R(x)$ is embedded in $\tilde X$. Note that this definition of injectivity radius is extrinsic and is smaller than or equal the intrinsic definition one can consider.
Proof of local $L^p$ restriction estimates
==========================================
As we discussed, the main ingredient is Lemma \[LocalLp\] whose proof is similar to Sogge’s local $L^p$ estimates [@So15] and follows by imitation. We give the proof since later in Section \[PSQE\] we need to make some modifications of this proof for manifolds with corners.
\[**Proof of Lemma \[LocalLp\]**\] First we choose a nonnegative function $\rho \in C^\infty ({{\mathbb R}})$ satisfying $$\rho(0)=1, \quad \text{and } \, \, \supp \hat \rho(t) \subset [-\frac12, \frac12].$$ Then we define the operator $$\begin{aligned}
A_{\la,r} & = \frac1\pi \int_{-\infty}^\infty r^{-1} \hat \rho(r^{-1}t) \, e^{it \sqrt{\la}}
\cos(t \sqrt{\Delta_g}) \, dt \label{A} \\ &=\rho(r(\sqrt{\la}-\sqrt{\Delta_g}))+\rho(r(\sqrt{\la}+\sqrt{\Delta_g})) \nonumber .\end{aligned}$$ By the properties of $\rho$, we have $$A_{\la,r} \psi_\la =\big (1+\rho(2r \sqrt{\la})\big ) \, \psi_\la.$$ Hence since $\rho$ is nonnegative we have $|\psi_\la| \leq |A_{\la,r} \psi_\la|$, which implies that $$\| \psi_\la \|_{L^{p}(B_{r/2}(x) \cap \Sigma)}\le \| A_{\la,r} \psi_\la \|_{L^{p}(B_{r/2}(x) \cap \Sigma)}.$$ To prove the lemma, it is enough to show that for all $f \in C^\infty (X)$ $$\|A_{\la,r}f\|_{L^p(B_{r/2}(x) \cap \Sigma)} \le C_1 r^{-\frac{1}{2}}\la^{\delta}\|f\|_{L^2(B_r(x))}.$$ To show this, we first observe that by the finite speed of propagation property of $\cos \big (t \sqrt{\Delta_g} \big )$, the integral kernel $\cos \big (t \sqrt{\Delta_g}\big )(x,y)$ vanishes if $d_g(x, y) > t$. Therefore, from the fact that $\supp \hat \rho (t) \subset [-\frac12, \frac12]$, the integral kernel $A_{\la,r}(x,y)$ of $A_{\la,r}$ satisfies $$A_{\la,r}(x,y)=0, \quad \text{if } \, \, d_g(x,y)> \frac{r}{2}.$$ This in particular shows that $$\|A_{\la,r}f\|_{L^p(B_{r/2}(x) \cap \Sigma)} = \|A_{\la,r}(f|_{B_r(x)})\|_{L^p(B_{r/2}(x) \cap \Sigma)} \leq \|A_{\la,r}(f|_{B_r(x)})\|_{L^p( \Sigma)}$$ As a result, our local $L^p$ restriction estimate is reduced to proving the global restriction estimate [^9] $$\|A_{\la,r}f\|_{L^{p}(\Sigma)}\le C_1r^{-\frac12}\la^{\delta}\|f\|_{L^2(X)},$$ for all $r \in [ \lambda^{-1/2}, \frac12 \,\text{inj}(X, g)]$ and $C_1$ that is uniform in $r$ and $\lambda$.
To prove this reduced estimate, we first recall that $\Pi_k=\bigoplus_{ \sqrt{\lambda_j} \in\,[\sqrt{k}, \sqrt{k}+1]} \Pi_{E_{\lambda_j}}$, where $\Pi_{E_{\lambda_j}}$ is the orthogonal projection operator onto the eigenspace $E_{\lambda_j}=\, \text{ker}\, (\Delta_g - \lambda_j)$, and that $$A_{\la,r}f=\sum_{j=0}^\infty \bigl[\rho(r( \sqrt{\la}-\sqrt{\la_j}))+\rho(r(\sqrt{\la}+\sqrt{\la_j}))\bigr] \, \Pi_{\lambda_j}f.$$ Because $\rho\in {\mathcal S}({{\mathbb R}})$, we have for every $N \in \IN$ and $\la\ge1$ $$|\rho(r(\sqrt{\la}-\sqrt{\la_j}))|+|\rho(r(\sqrt{\la}+\sqrt{\la_j}))|\le c_N(1+r|\sqrt{\la}-\sqrt{\la_j}|)^{-N}.$$ Therefore, $$\label{PiEstimate}
\|\Pi_k A_{\la,r}f\|_{L^2(X)}\le c_N(1+r|\sqrt{\la}-\sqrt{k}|)^{-N}\|\Pi_k f\|_{L^2(X)}, \quad N \in \IN.$$ Now, for each $m \in {{\mathbb Z}}$ let $I_m =\Bigl[\sqrt{\la} +\frac{2m-1}{r}, \sqrt{\la}+ \frac{2m + 1}{r}\Bigr).$ Since the number of intervals $[\sqrt{k}-1,\sqrt{k})$ that intersect $I_m$ as $\sqrt{k}$ varies in ${{\mathbb N}}$ is bounded by $\frac{2}{r}$, we can use the Cauchy-Schwarz inequality to obtain $$\Bigl\|\, \sum_{\{\sqrt{k} \in {{\mathbb N}}: \, [\sqrt{k}-1,\sqrt{k})\cap I_m \ne \emptyset\}}\Pi_k g
\Bigr\|_{L^{p}(\Sigma)}
\le 2 r^{-\frac12}\Bigl( \, \sum_{\{ \sqrt{k} \in \IN: \, [ \sqrt{k}-1, \sqrt{k})\cap I_m \ne \emptyset\}} \|\Pi_k g\|_{L^{p}(\Sigma)}^2
\, \Bigr)^{\frac {1}{2}}.$$ By using this inequality for $g= A_{\lambda, r}f$, using our assumption (\[NormOfSpectralCluster\]) that $$||\Pi_k ||_{L^2(X) \to L^p(\Sigma)} \leq C k^{\delta},$$ and using (\[PiEstimate\]), we see that $$\begin{aligned}
\Bigl\|\, \sum_{\{ \sqrt{k} \in \IN: \, [\sqrt{k}-1, \sqrt{k})\cap I_m \ne \emptyset\}} & \Pi_k A_{\la,r}f \Bigr\|_{L^{p}(\Sigma)}
\\
&\le 2C r^{-\frac12}\Bigl( \, \sum_{\{ \sqrt{k} \in \IN: \, [\sqrt{k}-1, \sqrt{k})\cap I_m \ne \emptyset\}} k^{2\delta} \|\Pi_k
A_{\la,r} f \|_{L^2(X)}^2\, \Bigr)^{\frac12} \notag
\\
&\le 2C c_N r^{-\frac12} \Big (1+|\sqrt{\la} +\frac{2m+1}{r}| \Big )^{2\delta}(1+|m|)^{-N}\| f \|_{L^2(X)}.\end{aligned}$$ Finally, by the above inequality $$\begin{aligned}
\|A_{\la,r}f\|_{L^{p}(\Sigma)} &\le \sum_{m \in {{\mathbb Z}}}
\bigl\|A_{\la,r}(\sum_{\la_j\in I_m}\Pi_{E_{\lambda_j}}f)\bigr\|_{L^{p}( \Sigma )}
\\
&=\sum_{m\in Z} \bigl\|\sum_{ \sqrt{k} \in \IN :[\sqrt{k}-1, \sqrt{k})\cap I_m \ne \emptyset} \Pi_k A_{\la,r}(\sum_{\la_j\in I_m} \Pi_{E_{\lambda_j}} f)\bigr\|_{L^{p}(\Sigma)}
\\
&\le 2C c_Nr^{-\frac12}\sum_{m \in {{\mathbb Z}}}(1+|m|)^{-N}\, \Big (1+| \sqrt{\la} + \frac{2m +1}{r}|\bigr)^{2\delta }\|f\|_{L^2(X)}
\\
&\le C_1 r^{-\frac12}\la^{\delta}\|f\|_{L^2(X)},\end{aligned}$$ where we have used $$| \sqrt{\la} +r^{-1}(2m+1)| \leq 2 \sqrt{\la} (1+|m|),$$ which is implied from the assumption $r \geq \la^{-1/2}$.
Proofs of global $L^p$ restrictions estimates
=============================================
In this section we prove Theorem \[LpEstimatesSS\] (and Theorems \[LpQENC\] and \[LpQE\] as its corollaries), using Lemma \[LocalLp\]. In the course of the proof we also show the following corollary which gives global $L^p$ estimates for restrictions of eigenfunctions to $\Sigma$ in terms of local $L^2$ norms on small balls in $(X,g)$ centered on $\Sigma$, and $L^2$ norm on a small tube around $\Sigma$.
\[GlobalLp\] Under the assumptions of Lemma \[LocalLp\], there exists $r^*$ dependent on $(X, g)$ and $\Sigma$, such that uniformly for all $r \in [\lambda_j^{-\frac12}, r^*]$: $$\label{GlobalLpEstimates}
||\psi_{\lambda_j} ||_{L^p(\Sigma)} \leq C_3 r^{-\frac{1}{2}} \left ( \sup_{x \in \Sigma} || \psi_{\lambda_j}||_{L^2(B_{r}(x))} \right)^{\frac{p-2}{p}} \left ( || \psi_{\lambda_j}||_{L^2(\mathcal T_{r}(\Sigma))} \right )^{\frac{2}{p}} \lambda_j^\delta,$$ where $\mathcal T_{r} (\Sigma)= \{ p \in X; d_g(p, \Sigma) < r \}$ is the tube of radius $r$ around $\Sigma$.
We approach as [@HeRi] and [@So15]. We fix $r < r_0/2=\text{inj}(X, g)/2$ and we choose a set of points $\mathcal J =\{ x_i \} \subset \Sigma$ such that $$\Sigma \subset \bigcup_{x_i \in \mathcal J} B_{r/2}(x_i),$$ in such a way that any point $p$ in $X$ belongs to at most $c$ many (or none) of the double balls $B_{r}(x_i)$, where $c$ only depends on $(X, g)$. We then select $r^* < \frac{r_0}{2}$ small enough so that for all $r \leq r^*$ $$\text{Vol}_g (\mathcal T_r (\Sigma)) \leq c_0 r^{n-k},$$ for some $c_0$ independent of $r$. This is possible because $\Sigma$ is a compact embedded smooth submanifold of $X \backslash \partial X$ or of $\partial X \backslash \mathcal S$ of dimension $k$. We also realize that by making $r^*$ even smaller we can make sure that for all $x \in \Sigma$ $$\text{Vol}_g (B_{r/2}(x)) \geq a_1 r^{n}$$ for some uniform $a_1$. This is obvious if $\Sigma$ is an interior submanifold. However, if $\Sigma \subset \partial X \backslash \mathcal S$, for a given $x \in \Sigma$ we first change our coordinates near $x$ to the upper half plane, then we realize that by choosing $r^*$ sufficiently small, for any $r < r^*$ we can fit a half Euclidean ball of radius $\frac{r}{3}$ inside the geodesic ball $B_{r/2}(x)$. We then have $$a_1 \, \text{card}( \mathcal J) r^n \leq \sum_{x_i \in \mathcal J} \text{Vol}_g(B_{r/2}(x_i)) \leq c \text{Vol}_g( \mathcal T_r(\Sigma)) \leq c c_0 r^{n-k} \, .$$ Therefore, we must have $ \text{card}( \mathcal J) \leq B r^{-k}$ for some $B$ that is independent of $r$.
Then using Lemma \[LocalLp\] we write $$\begin{aligned}
\| \psi _\la \|_{L^{p}(\Sigma)}^{p}
&\le \sum_{x_i \in \mathcal J} \| \psi_\la\|_{L^{p}(B_{r/2}(x_i) \cap \Sigma)}^{p}
\\
&\le C_1^p \la^{p \delta} \, r^{-\frac{p}{2}}\sum_{x_i \in \mathcal J }\| \psi_\la\|_{L^2(B_{r}(x_i))}^{p}
\\
&\le C_1^p \la^{\delta p} \, r^{-\frac{p}{2}}\, \Bigl(\sup_{x_i \in \mathcal J}
\| \psi _\la\|_{L^2(B_{r}(x_i))}^{p-2}\Bigr)\sum_{x_i \in \mathcal J}
\| \psi_\la\|_{L^2(B_{r}(x_i))}^2\end{aligned}$$ Corollary \[GlobalLp\] follows because $$\sum_{x_i \in \mathcal J}
\| \psi_\la\|_{L^2(B_{r}(x_i))}^2 \leq c \| \psi_ \la \| ^2 _{\mathcal T _r (\Sigma)}.$$ Theorem \[LpEstimatesSS\] follows by observing that under the $L^2$ assumption of this theorem and also our observation that $\text{card}(\mathcal J)< Br^{1-n}$, we have $$\begin{aligned}
\| \psi _\la \|_{L^{p}( \Sigma )}^{p}
&\le C_1^p \la^{p \delta} \, r^{-\frac{p}{2}} \sum_{x_i \in \mathcal J }\| \psi_\la\|_{L^2(B_{r}(x_i))}^{p} \\ &\leq C_2^p r^{-k} r^{\frac{p}{2}(n-1)} \lambda^{p \delta}. \end{aligned}$$
\[**Proof of Theorem \[LpQENC\]**\] As discussed in the introduction, this theorem is resulted immediately from Theorem \[LpEstimatesSS\] combined with the results of [@BuGeTz; @Hu; @ChSo] on the spectral cluster operators, and also the small scale QE results (\[QENC\]) of [@HeRi].
\[**Proof of Theorem \[LpQE\]**\] In the boundaryless case $\partial X =\emptyset$, this theorem follows easily from Theorem \[LpEstimatesSS\] and the below lemma combined with the quantum ergodicity results of [@Sh; @CdV; @Ze87]. We will prove this theorem for the case of $\partial X \neq \emptyset$ in Section \[Boundary\].
\[QElemma\] Let $(X,g)$ be a compact connected Riemannian manifold of dimension $n$, with or without boundary. When $\partial X \neq \emptyset$, we assume that $\partial X$ is piecewise smooth. Let $\{ \psi_{\la_j} \}_{j \in S}$ be a sequence of eigenfunctions of $\Delta_g$ with eigenvalues $\{\lambda_j\}_{j \in S}$ such that for some $r^*< \text{inj}(X,g)$ and for all $r \in (0, r^*)$ and all $x \in X$ $$\label{QEonX2}
\int_{B_r(x)} |\psi_{\la_j}|^2 \to \frac{\text{Vol}_g(B_r(x))}{\text{Vol}_g(X)}, \qquad \lambda_j \xrightarrow{j \in S} \infty.$$ Then there exists $r_0(g)$ such that for each $r \in (0, r_0(g))$ there exists $\Lambda_r$ such that for $ \lambda_j \geq \Lambda_r$ we have $$\int _{B_{r}(x)} | \psi_{\la_j}|^2 \leq K r^n,$$ uniformly for all $x \in X$. Here, $K$ is independent of $r$, $j$, and $x$.
We point out that this lemma is obvious when $x$ is fixed, however to have it work uniformly for all $x \in X$ we need to use a covering argument as follows.
First we choose $r_0(g) < \frac{r^*}{2}$ small enough so that for all $ r <r_0(g)$ $$\text{Vol}(B_{2r}(x)) \leq a r^n,$$ for some positive $a$ that is independent of $r$ and $x$. We cover $(X, g)$ using geodesic balls $\{B_{r/2}(x_i) \}_ {x_i \in \mathcal I}$ such that $\text{card}\, (\mathcal I)$ is at most $C_0r^{-n}$, where $C_0$ depends only on $(X, g)$. For the existence of such a covering see for instance Lemma 2 of [@CM]. Next for each $x_i \in \mathcal I$, by using (\[QEonX2\]) for balls $B_{2r}(x_i)$, we can find $\Lambda_{i, r}$ large enough so that for $\lambda_j \geq \Lambda_{i, r}$ $$\int _{B_{2r}(x_i)} | \psi_{\la_j}|^2 \leq K r^n,$$ with $K= \frac{2a}{\text{Vol}(X)}$. We claim that $ \Lambda_r =\max_{i \in \mathcal I} \{ \Lambda_{i, r} \}$ would do the job for all $x$ in $X$. So let $x$ be in $X$ and $r$ be as above. Then $x \in B_{r/2}(x_i) $ for some $i \in \mathcal I$ and clearly one has $B_r(x) \subset B_{2r}(x_i) $. This and the above inequality prove the lemma.
Improved supnorms and the number of nodal domains {#PSQE}
=================================================
In this section we prove Theorems \[LinftyQE\] and \[NN\]. The main technical obstacle is the presence of singular points on the boundary, which as we show can be overcome by the finite speed of propagation property of the solutions to the wave equation. Let us start with the proof of improved $L^\infty$ estimates.
**Proof of Theorem \[LinftyQE\]**
----------------------------------
First let us recall the following two results of [@So02] and [@Xu] on the supnorm of spectral clusters.
\[LinftyPi\] Let $(X,g)$ be a compact connected smooth Riemannian manifold of dimension $n$ with smooth boundary, and $\Pi_\la$ be the spectral cluster operator associated to $\Delta_g$ with Dirichlet or Neumann boundary conditions. Then $$\sup_X | \Pi_\la (f) | \leq C \la^{\frac{n-1}{4}} || f ||_{L^2(X)},$$ $$\sup_X | \la^{-\frac12} \nabla \, \Pi_\la (f) | \leq C \la^{\frac{n-1}{4}} || f ||_{L^2(X)},$$ where the constant $C$ depends only on $(X, g)$.
Now suppose $(X, g)$ is a manifold with piecewise smooth boundary, isomorphically embedded in a compact Riemannian manifold $(\tilde X, g)$ without boundary of the same dimension (see definition \[PS\]). For each $s \in [0, \text{inj}(X, g)/2]$ we choose a family $\{(X_s, g)\}$ of compact submanifolds of $( \tilde X, g)$, with smooth boundary for $s>0$, such that $$\label{Xs} \begin{cases} X_0=X, \quad X \subset X_s, \\
$$ X \backslash \mathcal T_s( \mathcal S) = X_s \backslash \mathcal T_s( \mathcal S), \end{cases}$$ where $\mathcal T_s( \mathcal S) =\{ x \in \tilde X; \, d(x, \mathcal S) < s \}$. This is possible by smoothing out $X$ in $\tilde X$ near its corners. We note that since the metrics on $X$ and $\{X_s\}$ are the restrictions of the metric $g$ on $\tilde X$, we have used the same notation for all of them. Let $\Delta_s$ be the Laplace-Beltrami operator on $(X_s, g)$ with Dirichlet or Neumann boundary conditions and consider the family of operators $A_{\la, r}(s)$ in (\[A\]) defined by $$A_{\la,r}(s) = \frac1\pi \int_{-\infty}^\infty r^{-1} \hat \rho(r^{-1}t) \, e^{it \sqrt{\la}}
\cos(t \sqrt{\Delta_s}) \, dt,$$ where $\rho \in C^ \infty({{\mathbb R}})$, $\rho(0)=1$, and $\supp \hat \rho(t) \subset [ -\frac12, \frac12]$. Now suppose $\psi_\la$ is an eigenfunction of $\Delta_0$. Then by repeating the argument of the proof of Lemma \[LocalLp\], we have $$A_{\la,r}(0) \psi_\la =\big (1+\rho(2r \sqrt{\la})\big ) \, \psi_\la,$$ and because $\rho \geq 0 $ we must have $$\label{above3} \| \psi_\la \|_{L^{\infty}(B_{r/2}(x))}\le \| A_{\la,r}(0) \psi_\la \|_{L^{\infty}(B_{r/2}(x))}.$$ Since the finite speed of propagation of $\cos \big (t \sqrt{\Delta_0} \big )$ also holds on manifolds with piecewise smooth boundary, $\cos \big (t \sqrt{\Delta_0}\big )(x,y)$ vanishes if $d_g(x, y) > |t|$. Hence because $\hat \rho (t) =0$ for $|t| \geq \frac{1}{2}$, we have $$A_{\la,r}(0)(x,y)=0, \quad \text{if } \, \, d_g(x,y)> \frac{r}{2}.$$ which implies that for all $ f \in C^\infty(X)$ $$\label{above4} \|A_{\la,r}(0)f\|_{L^\infty(B_{r/2}(x))} = \|A_{\la,r}(0)(f|_{B_r(x)})\|_{L^\infty(B_{r/2}(x))}.$$ Next we choose $s>0$ small enough so that $ X \backslash \mathcal T_{4s}(\mathcal S)$ has a non-empty interior. We claim that for $x \in X$ and $y \in X \backslash \mathcal T_{2s}(\mathcal S)$ as long as $d(x, y) \leq |t| < \frac{s}{2}$, we have $$\label{Claim} \cos \big (t \sqrt{\Delta_0} \big )(x, y) = \cos \big (t \sqrt{\Delta_s} \big )(x, y).$$ The proof of this, which we are about to present, is basically the same as the proof of finite speed of propagation. To do this, let $h$ be an arbitrary real-valued smooth function supported in $X \backslash \mathcal T_{2s}$ and define $$u(t, x) = \left ( \cos \big (t \sqrt{\Delta_0} \big ) - \cos \big (t \sqrt{\Delta_s} \big ) \right) (h).$$ By the properties of $X_s$, it is clear that $u$ satisfies $$\begin{cases} \partial_t^2 u(t, x) = -\Delta_0 u(t, x), \quad t \in \IR, \; x \in X, \\
u(0, x) =0, \quad \; x \in X,\\
\partial_t u(0, x) =0, \quad \; x \in X, \\
\forall t \in \IR, \; \forall x \in \partial X \backslash \mathcal T_{s} (\mathcal S): \; \begin{cases} u(t, x) =0 \; \text{(Dirichlet case)}, \\ \partial_n u(t, x) =0\; \text{(Neumman case)}. \end{cases} \end{cases}$$ Then for $t >0$ and $y$ in the support of $h$, consider the local energy $$E(t) = \int_{B_{s-t}(y)\cap X} |\partial_t u(t, x)|^2 + | \nabla_g u(t, x) |^2 \; d_gv,$$ where geodesic balls are defined in $\tilde X$. Differentiating $E(t)$, and using the coarea formula and the divergence theorem, we obtain $$\begin{aligned}
E'(t) & = - \int_{S_{s-t}(y)\cap X} |\partial_t u|^2 + | \nabla_g u|^2 \; d_g \sigma
\\ & \quad + 2 \int_{B_{s-t}(y)\cap X} (\partial_t^2 u)( \partial_t u) + < \nabla_g \partial_t u, \nabla_g u> \; d_gv\\ &= - \int_{S_{s-t}(y) \cap X} |\partial_t u|^2 + | \nabla_g u |^2 \; d_g \sigma + 2 \int_{\partial \big(B_{s-t}(y) \cap X \big )} (\partial_t u)(\partial_n u) \; d_g \sigma.\end{aligned}$$ We then note that $\partial \big(B_{s-t}(y) \cap X \big ) = (S_{s-t}(y) \cap X ) \cup (\partial X \cap B_{s-t}(y) )$. Since $y \in X \backslash \mathcal T_{2s}(\mathcal S)$, the set $\partial X \cap B_{s-t}(y)$ never intersects $ \mathcal T_s ( \mathcal S)$ and therefore we can use the boundary conditions to see that with either Dirichlet or Neumann boundary conditions, $ \partial_t u\, \partial_n u$ vanishes on $ \partial X \cap B_{s-t}(y)$. As a result, $$E'(t) = - \int_{S_{s-t}(y) \cap X} \left ( |\partial_t u|^2 + | \nabla_g u |^2 - 2(\partial_t u)(\partial_n u) \right ) \, d_g \sigma \leq 0 \; .$$ Since $E(0)=0$, we must have $E(t)=0$, which implies that $u(t, x)=0$ for all $t$, $x \in X $, and $y \in X \backslash \mathcal T_{2s}(\mathcal S)$ with $d(x, y) < s -|t|$. Since $h$ is arbitrary, if we choose $|t| < s/2$, the claim (\[Claim\]) follows. Consequently if $r <s/4$, we have $$A_{\la, r}(0)(x, y) = A_{\la, r}(s)(x, y), \quad x \in X, \, y \in X \backslash \mathcal T_{2s}(\mathcal S), \; d(x, y) < 2r,$$ $$\nabla_x A_{\la, r}(0)(x, y) = \nabla _x A_{\la, r}(s)(x, y), \quad x \in X, \, y \in X \backslash \mathcal T_{2s}(\mathcal S), \; d(x, y) < 2r.$$This identity tells us that if we choose any $x \in X \backslash \mathcal T_{3s}(\mathcal S)$, any $r < s/4$, and any $f \in C^\infty(\tilde X)$ we have $$\label{above1} \|A_{\la,r}(0)(f|_{B_r(x)})\|_{L^\infty(B_{r/2}(x))}=\|A_{\la,r}(s)(f|_{B_r(x)})\|_{L^\infty(B_{r/2}(x))},$$ $$\label{above2} \| \nabla \left ( A_{\la,r}(0)(f|_{B_r(x)}) \right )\|_{L^\infty(B_{r/2}(x))}=\| \nabla \left ( A_{\la,r}(s)(f|_{B_r(x)})\right )\|_{L^\infty(B_{r/2}(x))}.$$ However, since $X_s$ has a smooth boundary we can use Theorem \[LinftyPi\] and plug it in the argument of [@So15], or in the proof of our Lemma \[LocalLp\], to obtain $$\|A_{\la,r}(s)f\|_{L^ \infty (X_s)} \le C_s r^{-\frac{1}{2}}\la^{\frac{n-1}{4}}\|f\|_{L^2(X_s)},$$ $$\| \la^{-\frac12}\nabla A_{\la,r}(s)f\|_{L^ \infty (X_s)} \le C_s r^{-\frac{1}{2}}\la^{\frac{n-1}{4}}\|f\|_{L^2( X_s)},$$ for all $f \in L^2(X_s)$. If we use these inequalities with $f$ replaced by $f|_{B_{r/2}(x)}$ where $x \in X \backslash \mathcal T_{3s}(\mathcal S)$, then using (\[above3\]), (\[above4\]), (\[above1\]), and (\[above2\]) we arrive at $$||\psi_{\lambda} ||_{L^\infty(B_{r/2}(x))} \leq C_s r^{-\frac{1}{2}} || \psi_{\lambda}||_{L^2(B_{r}(x))} \lambda^\frac{n-1}{4},$$ $$|| \la^{-\frac12} \nabla \psi_{\lambda} ||_{L^\infty(B_{r/2}(x))} \leq C_s r^{-\frac{1}{2}} || \psi_{\lambda}||_{L^2(B_{r}(x))} \lambda^\frac{n-1}{4} .$$ We must emphasize that these estimates are for the eigenfunctions of $\Delta_0$ (with Dirichlet or Neumann boundary conditions) on $(X, g)$, which has a non-smooth boundary, and are only valid for $s<s_0$ (where $s_0$ depends only on $(X, g)$), $x \in X \backslash \mathcal T_{3s}(\mathcal S)$, and $\la^{-\frac12} \leq r <s/4$. The constant $C_s$ depends on $s$ and is independent of $r$, $\la$, and $x$.
Now let us assume the billiard flow on $(X, g)$ is ergodic. Then by the QE theorem of [@ZZ], for any ONB $\{ \psi_{\la_j} \}_{ j \in \mathbb N}$ of eigenfunctions of $\Delta_0$ with Dirichlet or Neumann boundary conditions, there exists $S \subset \IN$ of full density such that (\[QEonX2\]) holds. Hence by Lemma \[QElemma\], there exists $r_0$ dependent only on $(X, g)$ such that for each $r \in (0, r_0(g))$ there exists $\Lambda_r$ such that for $ \lambda_j \geq \Lambda_r$ ($j \in S$) $$\int _{B_{r}(x)} | \psi_{\la_j}|^2 \leq K r^n,$$ where $K$ is independent of $r$, $j$, and $x$. Applying these local $L^2$ estimates to the local $L^\infty$ estimates given above, we get for $j \in S$, $x \in X \backslash \mathcal T_{3s}(\mathcal S)$, and $\la_j^{-\frac12} \leq r <s/4$: $$\begin{aligned}
||\psi_{\lambda_j} ||_{L^\infty(B_{r/2}(x))} \leq K^{\frac12}C_s r^{\frac{n-1}{2}} \lambda_j^\frac{n-1}{4}, \label{local1} \\ \label {local2} || \la_j^{-\frac12} \nabla \psi_{\lambda_j} ||_{L^\infty(B_{r/2}(x))} \leq K^\frac12 C_s r^{\frac{n-1}{2}} \lambda_j^\frac{n-1}{4} . \end{aligned}$$ Without the loss of generality we assume that $C_s$ and $\Lambda_r$ are continuous monotonic functions with $C_s \to \infty$ as $s \to 0$ and $\Lambda_r \to \infty$ as $r \to 0$. We also assume that $\Lambda_r \geq r^{-2}$. Then we choose $r(s)$ a continuous monotonic function of $s$ on $(0, s_0)$ such that $r(s_0) < r_0$, $r(s) < s/4$ and $$r(s)^{\frac{n-1}{2}} C_s \to 0, \quad \text{as} \quad s \to 0.$$ We note that there exists $j_0 \in S$ such that for all $j > j_0$ we have $ \la_j \geq \Lambda_{r(s_0)}$. We discard all $ j \leq j_0$ from the set $S$, noting that the remaining set is still of full density (we call it $S$ again). Then for each $j \in S$ we define $s_j$ by $\Lambda_{r(s_j)}= \la_j$. By the choice of $s_j$ and $\Lambda_r$, the conditions $ \la_j \geq \Lambda_{r(s_j)}$ and $ \la_j^{-1/2} \leq r(s_j)$ are automatically satisfied, and Theorem \[LinftyQE\] follows by (\[local1\]) and (\[local2\]) if we choose ${\varepsilon}_j = 3 s_j$.
**Proof of Theorem \[NN\]; Number of nodal domains**
-----------------------------------------------------
We refer the reader to the paper [@ZeJJ] of Jung- Zelditch for the complete details of the facts we use in this section. Here we only present the necessary modifications that are needed to extend their results to all ergodic billiards with piecewise smooth boundary. Let us recall from [@ZeJJ] the following facts.
\[JJZ\] Let $(X, g)$ be a compact connected smooth Riemannian manifold of dimension $n=2$ with smooth boundary, parametrized with arc length. Let $\{ \psi_{\la_j} \}_{j \in S}$ be a sequence of eigenfunctions of $\Delta_g$ with Dirichlet or Neumann boundary conditions with eigenvalues $\{ \la_j \}_{j \in S}$, and $\{\psi^b_{\la_j}(s)\}_{j \in S}$ be its associated Cauchy data on $\partial X$ defined by $$\label{traces} \psi^b_{\la_j}(s)= \begin{cases} \psi_{\la_j}(s), \qquad \quad \quad \; \; \text{in the Neumann case},\\ \la_j^{-1/2} \partial_n \psi_{\la_j} (s), \quad \text{in the Dirichlet case} .\end{cases}$$ Let $ \Sigma \subset \partial \Omega$ be a closed connected arc. In addition, assume
- For all non-negative $f \in C^\infty_0( \partial X)$, with $\text{supp}(f) \subset \Sigma$, there exist $\Lambda_{f}$ and $c_1>0$ independent of $\Sigma$ and $\la_j$, such that $$\int_ \Sigma f(s) | \psi_{\la_j}(s) |^2 \, ds \geq c_1 \int_ \Sigma f(s) \, ds, \quad \la_j \geq \Lambda_{f}.$$
- For all $f \in C^\infty_0( \partial X)$, with $\text{supp}(f) \subset \Sigma$, there exist $\Lambda'_{f}$ and $c_2>0$ independent of $\Sigma$ and $\la_j$, such that $$\left |\int_ \Sigma f(s) \psi_{\la_j}(s) \, ds \right |^2 \leq c_2 \la_j^{-\frac12} \int_ \Sigma |f(s)|^2 \, ds, \quad \la_j \geq \Lambda'_{f}.$$
- For $j \in S$ [ $$\sup_\Sigma |\psi^b_{\la_j}| = o( \la_j^{\frac14}).$$]{}
Then for $j \in S$ $$\sharp \{s \in \Sigma; \; \psi^b_{\la_j}(s)=0 \} \to \infty \quad \text{as} \quad \la_j \to \infty.$$
See pages 818-819 of [@ZeJJ].
Also in [@ZeJJ], using a a topological argument based on [@GRS], the authors proved that:
\[Topological\] Under the initial assumptions of the above proposition (but not the additional assumptions (a)-(c)), one has $$\text{the number of nodal domains of} \; \psi_{\la_j} \; \geq \; \frac12 \sharp \{s \in \partial X; \; \psi^b_{\la_j}(s)=0 \} - c(X,g),$$ where $c(X,g)$ depends only on $(X,g)$.
See Theorem 6.3 of [@ZeJJ]. Although this result is only proved for surfaces with smooth boundary, it still follows even if $\partial X$ is piecewise smooth. The only difference is that one needs to assign a vertex to each singular point on the boundary, which can change the constant $c(X, g)$.
Let us now discuss the additional assumptions in Proposition \[JJZ\]. Assumption (a) holds in general for piecewise smooth ergodic manifolds by a result of Burq [@Bu] which shows a quantum ergodicity property for the boundary values $\psi^b_{\la_j}(s)$ (see also [@HaZe] where this is proved for Euclidean ergodic billiards). Hence given an ONB of eigenfunctions of $\Delta_g$ on such a manifold, we can find a full density subsequence for which (a) holds. Condition (c), with $\Sigma= \partial X$, was proved in [@SoZe14] for non-positively curved manifolds with smooth concave boundary, but in the more general case of ergodic manifolds with piecewise smooth boundary we can use our Theorem \[LinftyQE\]. In the case of manifolds with smooth boundary, condition (b) follows from:
\[[@HHHZ]\] \[HHHZ\] Let $(X, g)$ be a compact connected smooth Riemannian manifold of dimension $n \geq 2$ with smooth boundary. Let $\{ \psi_{\la_j} \}_{j \in \IN}$ be an ONB of eigenfunctions of $\Delta_g$ with Dirichlet or Neumann boundary conditions, and let $\{ \psi^b_{\la_j}(s) \}_{ j \in \IN}$ be its boundary traces defined in (\[traces\]). Suppose $\rho(t) \in C^\infty ({{\mathbb R}})$ is a function satisfying $$\rho(t) \geq 0, \quad \rho(0)=1, \quad \supp \hat \rho(t) \subset [-\frac12, \frac 12].$$ Then for $r_0$ sufficiently small and for all $ f \in C^\infty(\partial X)$ $$\label{completenessN} \frac{\pi}{2} \sum_{ j=1}^\infty \rho_0 \big (\sqrt{\la} - \sqrt{\la_j} \big ) \langle f, \psi^b_{\la_j} \rangle \psi^b_{\la_j}(x) = f(x) + o(1),$$ where $\rho_0$ is a function whose Fourier transform is $ \hat \rho (r_0^{-1} t)$. Also $\langle \, , \, \rangle$ is the natural inner product in $L^2(\partial X)$, and the above convergence is in $L^2(\partial X)$.
In fact in the Dirichlet case it is more convenient for us to use the following version $$\label{completenessD} \frac{\pi}{2} \sum_{ j=1}^\infty \rho_0 \big (\sqrt{\la} - \sqrt{\la_j} \big ) \langle f, \partial_n \psi_{\la_j} \rangle \partial_n \psi_{\la_j}(x) = f(x) \la + o(\la).$$ We prefer this version because its related to the operator $\cos (t \sqrt{ \Delta_g})$ which satisfies the finite speed of propagation, as opposed to the operator $\frac{\cos (t \sqrt{ \Delta_g})}{ \sqrt{ \Delta_g}}$ involved in (\[completenessN\]).
Before showing that Theorem \[HHHZ\] also works for manifolds with piecewise smooth boundary for $f$ with $\supp(f) \subset \Sigma \subset \partial X \backslash \mathcal S$, let us make the observation that since for all $N \in \IN$, $$\rho_0 \big (\sqrt{\la} + \sqrt{\la_j} \big )= O_N (\la^{-N} \la_j^{-N}),$$ and since $$\frac1\pi \int_{-\infty}^\infty \hat \rho(r_0^{-1}t) \, e^{it \sqrt{\la}}
\cos(t \sqrt{\Delta_g}) \, dt =\rho_0(\sqrt{\la}-\sqrt{\Delta_g})+\rho_0(\sqrt{\la}+\sqrt{\Delta_g}),$$ (\[completenessN\]) can be rewritten in the Neumann case as $$\label{N} \; \frac12 \int_{-\infty}^\infty \int_{\partial X} \hat \rho(r_0^{-1}t) \, e^{it \sqrt{\la}}
\cos(t \sqrt{\Delta_g})(x, y) f(y) \, d_g\sigma \, dt = f(x) + o(1),$$ and in the Dirichlet case (\[completenessD\]) can be written as $$\label{D} \frac12 \int_{-\infty}^\infty \int_{\partial X} \hat \rho(r_0^{-1}t) \, e^{it \sqrt{\la}} \left (\partial_{n_x} \partial_{n_y} \cos(t \sqrt{\Delta_g})(x, y) \right ) f(y) \, d_g\sigma \, dt = f(x) \la + o(\la).$$ Now suppose $(X, g)$ is a manifold with piecewise smooth boundary and $\Sigma \subset \partial X \backslash \mathcal S$ is a compact submanifold of the regular part of the boundary. Define $s = d_g(\Sigma, \mathcal S) / 4$ and let $(X_s, g)$ be the manifold with smooth boundary defined in (\[Xs\]). We saw in (\[Claim\]) that for $x \in X$ and $y \in X \backslash \mathcal T_{2s}(\mathcal S)$ as long as $d(x, y) \leq |t| < \frac{s}{2}$, we have $$\cos \big (t \sqrt{\Delta_g} \big )(x, y) = \cos \big (t \sqrt{\Delta_{g, s}} \big )(x, y).$$ Hence if we choose $r_0 <s$ we obtain (\[N\]) and (\[D\]), and therefore [^10] (\[completenessN\]) and (\[completenessD\]) for $(X, g)$ with non-smooth boundary as long as $\supp f \subset \Sigma$ and $x \in \Sigma$.
Finally, let us discuss how condition (b) can be obtained from these. Assume that $\rho$ satisfies the additional assumption $\rho(t) \geq \frac12$ on $[-\alpha ,\alpha]$ for some $\alpha >0$. Then using this property of $\rho$ and its non-negativity, after taking the inner product of both sides of (\[completenessN\]) and (\[completenessD\]) with $f \in C^\infty(\partial X)$ satisfying $\supp(f) \subset \Sigma$, we get $$\text{Neumann case:} \quad \sum_{|\sqrt{\la} - \sqrt{\la_j}| \leq \alpha/r_0 } | \langle f, \psi_{\la_j} \rangle |^2 \leq \frac{4}{\pi} \int_{\partial X} |f(x)|^2 + o_f(1)$$ $$\text{Dirichlet case:} \quad \sum_{|\sqrt{\la} - \sqrt{\la_j}| \leq \alpha/r_0 } | \langle f, \partial_n \psi_{\la_j} \rangle |^2 \leq \frac{4 \la}{\pi} \int_{\partial X} |f(x)|^2 + o_f(\la).$$ By applying an extraction procedure to the the above estimates, as performed in [@ZeJJ], one can see that for every $\tau \in (0, 1)$ there exists a subsequence of $\{ \psi_{\la_j} \}_{ j \in {{\mathbb N}}}$ of density $1 -\tau$ for which condition (b) of Proposition \[JJZ\] holds. Since, as we discussed, all other conditions are satisfied for a full density subsequence, Theorem \[NN\] follows by letting $\tau \to 0$.
\[finalremark\] In the above argument, when we were going back from (\[N\]) and (\[D\]) to (\[completenessN\]) and (\[completenessD\]) in the case of non-smooth boundary, we implicitly used the fact that if $\supp(f) \subset \Sigma$ then $$\| \sum_{ j=1}^\infty \rho_0 \big (\sqrt{\la} + \sqrt{\la_j} \big ) \langle f, \psi^b_{\la_j} \rangle \psi^b_{\la_j}(x) \|_{L^2(\Sigma)} = O (\la^{- \infty}).$$ In order to prove this we need to know that {$\psi^b_{\la_j}$ } is polynomially bounded on $\Sigma$. In fact if in the proof of Theorem \[LinftyQE\] we choose $r=\frac{s}{5}$, we get for $s>0$ $$\sup_{ X \backslash \mathcal T_s( \mathcal S)} | \psi_{\la_j}| \leq C'_s \la^{\frac{n-1}{4}} \quad \text{and} \quad \sup_{ X \backslash \mathcal T_s( \mathcal S)} | \la_j^{-\frac12} \nabla \psi_{\la_j} | \leq C'_s \la^{\frac{n-1}{4}}.$$
$L^p$ restrictions on manifolds with boundary {#Boundary}
=============================================
In this section we give a proof of Corollary \[BoundaryCor\] and also for the remaining part of Theorem \[LpQE\] when $\partial X \neq \emptyset$. Let $\Sigma \subset X \backslash \partial X$ be a compact submanifold and define $s = d ( \Sigma, \partial X) /4$. A proof similar to that of Theorem \[LinftyQE\] shows that for $x \in X$ and $y \in X \backslash \mathcal T_{2s}(\mathcal \partial X)$ as long as $d(x, y) \leq |t| < \frac{s}{2}$, we have $$\cos \big (t \sqrt{\Delta_g} \big )(x, y) = \cos \big (t \sqrt{\Delta_{\tilde g}} \big )(x, y),$$ where $(\tilde X, \tilde g)$ is a compact Riemannian manifold without boundary that contains $(X, g)$ as the closure of an open connected subset (see definition (\[PS2\])). By imitating the proof of Theorem \[LinftyQE\], we get that for $x \in X \backslash \mathcal T_{3s}(\partial X)$, and for $\psi_\la$ any eigenfunction of $\Delta_g$ on $(X, g)$ (with Dirichlet or Neumann boundary conditions) with eigenvalue $\la \geq 1$, and any $r$ satisfying $\la^{-\frac12} \leq r <s/4$, we have $$||\psi_{\lambda} ||_{L^p(\Sigma \cap B_{r/2}(x))} \leq C_s r^{-\frac{1}{2}} || \psi_{\lambda}||_{L^2(B_{r}(x))} \lambda^{\delta(n, k, p)}.$$ Corollary \[BoundaryCor\] follows by choosing $r =\frac{s}{5}$ and using a covering argument as in the proof of Theorem \[LpEstimatesSS\]. Theorem \[LpQE\] needs two additional ingredients: Lemma \[QElemma\] and the QE result of [@ZZ].
Acknowledgments {#acknowledgments .unnumbered}
===============
Th author would like to thank Gabriel Rivière for their comments on the first draft.
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[^1]: See [@LuSa; @Yo; @HeRiTorus; @LeRu] for parallel results in the arithmetic setting.
[^2]: See Section \[PS2\] for the precise definition.
[^3]: $\Sigma$ can have a boundary, in which case we assume its boundary is smooth.
[^4]: When $(k, p) =(n-1, 2)$ this small factor does not appear.
[^5]: See [@JaJu] for a different technique applied to even and odd QE eigenfunctions of surfaces with an isometric involution.
[^6]: The same result was proved in [@Han] for $r=(\log \lambda_j)^{-\frac{1}{3n} +\epsilon}$.
[^7]: That it fails for $r= \lambda^{- \frac{1}{2n-2} {\color{red}}{-} \epsilon}$ for a positive density subsequence of some ONB.
[^8]: We must mention a result of Ingremeau [@In], where small scale QE is proved for any $r \gg \la^{-1/2}$ for distorted plane waves on non-compact non-positively curved manifolds with Euclidean ends.
[^9]: One can probably use a quantitative version of [@Ta12] and [@HaTa] to give an alternate proof of this, since $A_{\lambda, r}$ is a quasimode.
[^10]: See Remark [^11].
|
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abstract: |
is traditionally evaluated by using short stimuli usually representing parts or *single* usage episodes. This opens the question on how the overall service perception involving *multiple* usage episodes can be evaluated—a question of high practical relevance to service operators. Despite initial research on this challenging aspect of multi-episodic perceived quality, the question of the underlying quality formation processes and its factors are still to be discovered.
We present a multi-episodic experiment of an service over a usage period of 6 days with 93 participants. Our work directly extends prior work investigating the impact of time between usage episodes. The results show similar effects — also the recency effect is not statistically significant. In addition, we extend prediction of multi-episodic judgments by accounting for the observed saturation.
author:
-
bibliography:
- 'Bibliography.bib'
title: |
Multi-episodic Perceived Quality\
of an Audio-on-Demand Service
---
Perceived quality, QoE, Audio streaming
(current page.north) node (anchor) ; ;
Introduction
============
Traditional research on perceptual investigate short time-scales spanning from several seconds up to several minutes and involving only judgments of single interaction. For these, it has been shown that later parts of the stimuli as well as the worst performance have a higher impact on post-experience ([i.e.]{}, retrospective) judgments [@weiss_modeling_2009]. These two effects are denoted as recency effect and peak effect, which are well-known from research on recall ([e.g.]{}, [@kahnemann_pain_1993]). For predicting retrospective judgments, it has been shown that a weighted average with a higher weighting on more recent momentary judgments or performance performs sufficiently. In this regard, *multi-episodic perceived quality* investigates the formation process of a subjective quality impression for a service or system that is used repeatedly.
However, the underlying formation process of perceived quality over several usage episodes ([i.e.]{}, multi-episodic perceived quality) is not well understood—especially considering usage periods of days, months, or even years. Following [@guse_multiepisodic_2017], a *usage episode* is defined as *a distinct and self-contained interaction by a user with a service or system to achieve his or her goal(s)*. Investigating and understanding the formation process of multi-episodic perceived quality is of high practical relevance to service operators as telecommunication services are prone to performance fluctuations ([e.g.]{}, varying network conditions). These fluctuations may be perceived by a user and therefore affect his/her instantaneous quality as well as episodic and multi-episodic quality.
This especially includes cloud-based multi-media services ([e.g.]{}, and streaming services that have become popular Internet applications). Research on multi-episodic perceived quality could show that a recency effect occurs ([e.g.]{}, [@guse_multiepisodic_2017; @guse_modelling_2014]) as well as a duration neglect [@guse_duration_2016]. Despite these first initial findings, the formation process remains far from being understood—rooted also in the high complexity to perform multi-episodic experiments given their duration and the between-subject design.
In this paper, we aim to better understand this formation process using an service. That is, if effects observed on multi-episodic perceived quality in one session can also be observed when the usage is extended to multiple usage episodes. Our experiment involves 93 participants using the service twice per-day over 6 days. We complement our study by also by applying the to investigate the impact on customer loyalty [@reichheld_one_2003]. While it is of questionable reliability [@npscritism] and not well-established the domain, it is popular in marketing and user retention analyses, e.g., as an service might apply in practice. [[**Structure.** ]{}]{} We first review related work in on which we base our study. We describe our hypotheses and research design in and then discuss the study results in . Finally, we conclude the paper and give an outlook on future work towards understanding multi-episodic QoE.
Related Work {#sec:rw}
============
[[**Multi-day experiments.** ]{}]{} Research on multi-episodic perceived quality emerged in 2011 with multi-day experiments. A first experiment evaluated Skype calls performed on a daily basis over 12 days [@moller_single-call_2011]. Each pair of subjects performed two video telephony calls per day while solving one task per call. This task-driven approach was selected to create a realistic usage situation as well as a comparable usage behavior ([i.e.]{}, one [@itu-t_recommendation_p.805_subjective_2007]). *Episodic judgments* ([i.e.]{}, perceived quality of one usage episode) were directly collected after finishing each call. *Multi-episodic judgments* ([i.e.]{}, all so far experienced usage episodes) were collected after the 2nd, 7th, and 12th day. Within each multi-episodic condition two performance levels limiting the overall transmission bandwidth were applied: and . Although the results were rather limited this experiment showed that multi-episodic perceived quality can be assessed successfully in a field experiment by applying a between-subject design—which we therefore adopt in this experiment. The results show that episodic judgments are reduced for usage episodes. Moreover, the results indicate that also subsequent episodic judgments are negatively affected even if these were presented in . Interestingly, a slight increase for episodic judgments was observed over the usage period. With regard to the formation process of multi-episodic judgments the results are rather limited. This is most likely due to the limited impact of usage episodes. Similar results were found for a service [@guse_macro-temporal_2013]. They further observed discrepancies between episodic judgments of usage episodes and the final multi-episodic judgment. Precisely, they observed that for a service providing mainly severely usage episodes, episodic judgments are more positive than multi-episodic judgments. However, the results are limited by the number of participants as well as that the defined performance levels could not be achieved.
Subsequently, an experiment with a service bundle consisting of an service and a service over a usage period of 15 days [@guse_macro-temporal_2013] was conducted. Combining their results with [@moller_single-call_2011] and [@guse_macro-temporal_2013], they presented initial models for predicting multi-episodic judgments based upon episodic judgments. They could show that a *linear moving average* outperforms a *windowed average*. Accounting for a peak effect resulted in decreased prediction performance. Overall, the results of multi-episodic perceived quality over usage periods spanning several days is rather limited. One reason might be that such experiments require to be conducted outside of the laboratory. Thus, the usage environment is uncontrolled and often an elaborate technical setup necessary. Also, the required between-subject design increases the effort. [[**Session quality.** ]{}]{} Multi-episodic perceived quality was further investigated in *individual sessions* ([i.e.]{}, continuous use of the same service with multiple usage episodes). This complements multi-day experiments as it is not yet known if and how the time between usage episodes affects multi-episodic judgments. In [@guse_duration_2016], an service was used to determine if the duration of a usage episode affects a subsequent multi-episodic judgments. It was observed that the duration of one usage episode does not affect the episodic and multi-episodic judgment. Subsequently, [@guse_multiepisodic_2017] conducted two experiments with overall 205 participants to investigate the impact of the usage situation in case of speech telephony. Both experiments consisted of six usage episodes that needed to be solved subsequently. In the first experiment, a pair of participants needed to solve one per usage episode together. The second experiment, was conducted by each participant alone simulating a 3rd-party listening situation. Here, recordings of the first experiment were used and participants needed to transcribe all information necessary to solve the . Here, the effects of presenting more usage episodes as well as presenting more usage episodes subsequently were investigated. Most notably, the results indicate that the usage situation has a very limited impact on episodic and multi-episodic judgments. Increasing the number of usage episodes resulted in a reduction of the subsequent multi-episodic judgment while remaining well above the episodic judgments of usage episodes. This indicates that previously experienced usage episodes still affect this judgment and that the formation process is not a pure average. Also a positional impact could be observed in both experiments: increasing the number of usage episodes following usage episode(s) before a multi-episodic judgment limits the observed reduction.
[[***Takeaway:***]{} *Despite first findings, the formation process of multi-episodic perceived quality remains far from being understood. While initial work in one session found some interesting insights, multi-day experiments so far remained mainly inconclusive. In this paper, we address this issue by extending [@guse_multiepisodic_2017] from one session to multiple days while using a similar experimental design.* ]{}
experiment {#sec:methodology}
==========
The goal of this experiment is to investigate if the effects observed on multi-episodic perceived quality in one session can also be observed if the usage period is extended to multiple days. This complements prior work and enables to improve prediction models for multi-episodic perceived quality.
Design
------
Our multi-episodic perceived quality experiment also follows a between-subject design ([i.e.]{}, only one multi-episodic condition was presented to each participant) in which participants use an service twice per day. We chose a usage period of 6 days to be able to investigate a higher number of multi-episodic conditions compared to prior work. To directly embed our experiment into related work, we follow the experimental design of [@guse_multiepisodic_2017].
We choose an service for two reasons. First, is a popular Internet service ([e.g.]{}, offered by popular apps such as Spotify or Audible). Second, an service enables a simple technical setup in which the experiment can be conducted by each participant alone. That is, no interaction (and thus no coupling) with other participants—as in typical interaction experiments—is needed. This reduces the experimental complexity, avoids social effects, and reduces the effort to conduct the experiment for the participants. We used an *audio book* as content to *i)* keep the experiment interesting for participants and *ii)* enable us to verify that the content was consumed. We chose the audio book *City of the Beasts* from Isabel Allende as it was used in [@guse_duration_2016]. While participants could not chose the content, it limits the effort to prepare the experiment and omits differences in content as source for noise. This audio book was cut into individual, self-contained parts of length. One part was presented in each usage episode. This should enable participants to focus on the content while limiting their effort. The content was presented in it’s chronological order.
In line with prior work, two performance levels and were applied. denotes the highest performance, yielding only very limited to no perceptible degradations. denotes the worst performance and is expected to provide a severely lower perceived quality. For , the source material (<span style="font-variant:small-caps;"></span>, , stereo) was encoded with MP3 (). The bitrate was selected to produce no audible impairments. For , the content was encoded with <span style="font-variant:small-caps;">LPC/10</span>[^1]. This codec was also used in prior work ([e.g.]{}, [@guse_multiepisodic_2017; @guse_duration_2016]) as it provides a severe degradations while providing speech intelligibility. We remark that the LP encoding is *unrealistic* for any multi-media streaming service. However, other quality degradation types beyond our scope (e.g., stalling) are expected in practice, yielding also quality fluctuations (LP/HP). We added the LP encoding as reference to prior work that used the same encoding to study the quality formation process of multi-episodic judgements. Participants used their own computer and pair of headphones to access the service via the Internet using a <span style="font-variant:small-caps;"></span>-capable web browser. The system was implemented using [@guse_thefragebogen_2019]. To exclude Internet-induced artifacts, the audio content was *preloaded* prior to starting each usage episode.
We apply the following hypotheses to evaluate multi-episodic judgments in one session [@guse_multiepisodic_2017].
- (*H1*) increasing the number of usage episodes leads to a higher reduction in multi-episodic judgments.\[hypo:number\]
- (*H2*) presenting usage episodes after usage episodes limits the reduction in multi-episodic judgments.\[hypo:position\]
- (*H3*) presenting usage episodes between usage episodes leads to a higher reduction than presenting the usage episodes consecutively.\[hypo:consecutive\]
Based upon the three hypotheses, seven multi/episodic conditions were created. Performance was only varied between days. Following [@guse_multiepisodic_2017], the first three days were presented in to provide a common baseline for the between-subject design. episodes were only presented from the 4...6th day.
\[tab:lab:hypothesesComparison\]
[0.6]{}[c|c|c|c|c|c]{} &\
& 1-3 & 4 & 5 & 6\
C0 & & & &\
C1 & & ****& &\
C3 & & & & ****\
C4 & & **** & **** &\
C5 & & & **** & ****\
C6 & & **** & **** & ****\
C8 & & **** & & ****\
The multi-episodic conditions are shown in . C1 and C3 present either the 4th or the 6th day in . C4, C5, and C8 present two days between the 4th and the 6th day in . C6 presents all usage episodes on these three days in . C0 ( only) was explicitly skipped in this experiment as [@guse_multiepisodic_2017] did not find evidence that the multi-episodic judgments would be affected. Also, the effect of a slight increase over the usage period reported by [@moller_single-call_2011] was rather small and was observed over a usage period of 14 days. Therefore, we use the multi-episodic judgment after the 3rd day as C0.
For the investigation of [**H1**]{}, the results of C0, C3, C5, and C6 can be compared. [**H2**]{}can be evaluated by comparing the results of C1 and C3 as well as C4 and C5. Finally, [**H3**]{}can be evaluated by comparing the results of C8 with C3 and C5.
Procedure
---------
[[**Introductory session.** ]{}]{} The experiment started with a introductory session. The goal was to explain the experimental procedure and to collect demographic data. Subsequently, a short training presenting short stimuli of typical audio degradations was conducted. Finally, two usage episodes with the service needed to be conducted to show participants how to use the service.
[[**Experiment.** ]{}]{} The multi-episodic part of this experiment began the day after the introductory session. Here, the usage episodes needed to be conducted daily between and as well as and , respectively. In line with prior work, judgments were taken on the 7/point continuous category rating scale (see ). Episodic judgments were taken after every usage episode and multi-episodic judgments after the second usage episode of the 3rd day and the 6th day.
![Continuous 7-point scale defined in [@itu-t_recommendation_p.851_subjective_2003].[]{data-label="fig:quality_scale"}](figure/quality_scale){width="0.99\columnwidth"}
[[**Control questions.** ]{}]{} As participants could not be supervised during the experiment, we presented two content-related questions after every usage episode. For every questions the correct answer out of three options needed to be selected. This allows to evaluate if a participant had experienced the content. This was inspired by cheating prevention approaches for crowdsourcing [@hossfeld2014qoe].
[[**Final assessment.** ]{}]{} On the day after the 6th day usage period, a debriefing was conducted with every participants. Here, also the was assessed asking how likely it would be that the provided service would be recommended to friends or colleagues (0 *not likely at all* until 10 *extremely likely*).
Participants
------------
The experiment was conducted in Berlin from September until November 2015 with 57 female and 38 male participants aging from of 18 to 33 years ($\mu=25.8$, $\sigma=4.0$). Participants were required to have normal hearing capabilities. Also they needed to comply with the defined schedule and complete all questionnaires. As a reminder, participants were informed by email, when a usage episode should be conducted. Successful participation was compensated with .
Results {#sec:results}
=======
We next present the results of our experiment. First, the participants are screened for inconsistent judgments and error rate of the content-related questions is evaluated. Second, the potential impact of the between-subject design is evaluated. Then, the multi/episodic judgments are evaluated with regard to the three investigated hypotheses.
Plausibility Checks
-------------------
For the evaluation of consistent judgments, we use the episodic judgments and evaluate these individually. We consider a participant to be inconsistent if more than two episodic judgments exceed the 1.5 $\times$ *interquartile range* of the performance levels. None of the participants fulfilled this criteria. With regard to the content-related questions, we assume that participants should at least answer 50% correctly. Otherwise, a participant did not seem to follow the experimental instructions and thus would be excluded from further data evaluation. Out of the 24 questions, participants answered on average 20.8 questions ($\sigma = 2.7$) correctly. One participant was excluded.
Between-subject Design
----------------------
Given that a between-subject design is applied, we next investigate if this affects the episodic judgments between multi-episodic conditions. We show the per multi-episodic condition in ). For episodic judgments of , a significant difference is found ($H(5)=33.4978$, $p<0.001$). The post-hoc test shows that C5 is different to all other conditions ($p<0.05$) and that C3 is different to C4 and C8 ($p<0.02$). For , differences between conditions are also found ($H(5)=18.2748$, $p=0.0026$). The post-hoc test shows that C5 is different than C1, C4, and C8 ($p<0.05$). It must be noted that episodic judgments of C5 resulted in the highest for and in the lowest for . This is unexpected but not unlikely due to the between-subject design considering the number of multi-episodic conditions. A detailed analysis did not reveal reason(s) for this and we therefore presume that this does not prevent comparision between multi-episodic conditions.
For the multi/episodic judgment after the 3rd day, no significant differences between conditions are observed ($H(5)=5.2111$, $p=0.3907$). This indicates that as long as only episodes were presented, the between-subject design did not affect multi/episodic judgments. [[***Takeaway:***]{} *The between-subject design did not appear to affect the multi-episodic evaluation.*]{}
[0.85]{}[c|c|c|c|c]{} Condition & days & Participants & &\
C0 & - & - & -\
C1 & 4 & 16 & 4.5 (0.9) & 1.3 (0.8)\
C3 & 6 & 14 & 4.7 (0.7) & 1.3 (0.8)\
C4 & 4..5 & 13 & 4.5 (0.7) & 1.3 (0.5)\
C5 & 5..6 & 18& 5.0 (0.8) & 0.9 (0.5)\
C6 & 4..6 & 14 & 4.6 (0.6) & 1.2 (0.7)\
C8 & 4 and 6 & 15 & 4.5 (0.6) & 1.2 (0.7)\
Multi-episodic Judgments
------------------------
We next evaluate the three hypotheses stated in Section \[hypo:number\] using the final multi-episodic judgments.
[[**[**H1**]{}(increasing number of usage episodes).** ]{}]{} We first investigate if increasing the number of episodes *before* a multi/episodic judgment results in a decrease of this judgment ([i.e.]{}, a reduction in perceived quality is reported). This hypothesis is evaluated by comparing C0, C3, C5, and C6 ([i.e.]{}, 0-3 usage episodes). shows both multi-episodic judgments. C0, C3, C5, and C6, are significantly different ($H(3)=68.3657$, $p<0.001$). A post-hoc test finds that C0 is significantly different to all other conditions ($p<0.001$). Also, C3 and C5 ($p=0.002$) as well as C3 and C6 are significantly different ($p<0.001$). For C5 and C6, no significant difference is found ($p=0.366$).
[0.75]{}[c|c|c]{} Condition & episode(s) & Multi-episodic judgment\
C0 & ( only) & 4.7 (0.6)\
C3 & 6 & 3.6 (0.6)\
C5 & 5..6 & 2.5 (1.0)\
C6 & 4..6 & 2.4 (0.7)\
As a result, [**H1**]{}can only be partly accepted as the multi-episodic judgment decreased, but only for up to two usage episodes. The underlying reason for the observed saturation could not be derived from this experiment. [[***Takeaway:***]{} *We find that increasing the number of days directly before the multi/episodic judgment negatively affects this judgment. Here, we observe a reduction of approximately per usage episode for up to two episodes. No further decrease can be observed in case of three days. It must be noted that the multi/episodic judgment remains higher than the episodic judgments of usage episodes.*]{}
[[**[**H2**]{}(increasing number of usage episodes after ).** ]{}]{} We next evaluate the presence of a recency effect ([i.e.]{}, if presenting usage episodes after usage episodes limits the reduction in multi-episodic judgments). This hypothesis is investigated by comparing C1 vs. C3 ([i.e.]{}, one day ) and C4 vs. C5 ([i.e.]{}, two days ). shows the final multi/episodic judgment. With regard to the final multi/episodic judgment neither C1 and C3 ($W=138.50$, $p=0.136$, one-sided) nor C4 and C5 ($W=151.00$, $p=0.087$, one-sided) are significantly different. [[***Takeaway:***]{} *Unlike prior work which observed a recency effect on shorter usage periods, we did not find clear indications. As a service provider, usage episodes should thus be avoided as the subsequent usage episodes do not make up for prior experiences.*]{}
[0.75]{}[c|c|c]{} Condition & episode(s) & Multi-episodic judgment\
C1 & 4 & 4.1 (0.7)\
C3 & 6 & 3.6 (0.6)\
C4 & 4..5 & 3.0 (1.1)\
C5 & 5..6 & 2.5 (1.0)\
[[**[**H3**]{}(consecutive vs. non-consecutive usage episodes).** ]{}]{} Finally, we evaluate if usage episodes between usage episodes lead to a higher reduction than presenting the usage episodes consecutively. That is, do users prefer performance switches between usage episodes or rather continuous presentation of similar performing usage episodes. This hypothesis can be evaluated by comparing C4 and C5 with C8. C4 and C5 present each two days consecutively, whereas C8 presents the 4th and the 6th day in . shows the final multi/episodic judgment for these conditions. These conditions are not significantly different ($H(2)=2.3809$, $p=0.3041$). As a result, [**H3**]{}must be rejected. In fact, the slight improvement in the multi/episodic judgment of C8 compared to C5 might be explained by a recency effect. [[***Takeaway:***]{} *Performance switches between usage episodes do not seem affect the formation process.*]{}
[0.75]{}[c|c|c]{} Condition & episode(s) & Multi-episodic judgment\
C4 & 4..5 & 3.0 (1.1)\
C8 & 4 and 6 & 2.7 (0.4)\
C5 & 5..6 & 2.5 (1.0)\
[[**State of the art.** ]{}]{} Our results are in line with single-session results in prior work that investigated multi-episodic use in one sessions of one hour [@guse_multiepisodic_2017]. This indicates that the time between usage episodes in the studied experiments might only have a limited impact on multi-episodic judgments. We infer that it is very beneficial to investigate first multi-episodic perceived quality in one or more sessions of multiple usage episodes each. Then these findings can be verified and extended in multi-day experiments. This will also enable to create prediction models for multi-episodic perceived quality.
Net Promoter Score
------------------
The assesses how likely it would be that the provided service would be recommended to friends or colleagues (0 not likely at all to 10 extremely likely). It thereby divides participants into promoters (), passives (), and detractors () to determine the growth and churn of users of (non-technical) services. The results are in shown in . C1 and C3 (mainly passives) as well as C5, C6, and C8 (mainly detractors) achieve each a seemingly similar distribution while C4 stays between both groups. This indicates that the is negatively affected if more usage episodes are present. Notably, also a saturation is indicated as C5 and C6 are seemingly similar. However, the results contain outliers and the overall correlation coefficient of the with the final multi-episodic judgment is only 0.5. [[***Takeaway:***]{} *Multi-episodic judgments alone do not suffice to predict service recommendations captured by the NPS. We therefore assume that the NPS is affected by additional factors.*]{}
This highlights the need for future work to create holistic models capturing the overall service experience.
![Boxplot of the per multi-episodic condition.[]{data-label="fig:pred:E6nps"}](figure/plotE6NPS-1){width="\maxwidth"}
Quality Prediction
==================
We now evaluate different approaches to predict the multi-episodic judgments based upon the episodic judgments. Guse et al. [@guse_modelling_2014; @guse_multiepisodic_2017] proposed to predict a multi-episodic judgment by computing the *weighted average* of prior episodic judgments. The influence of each usage episode $e_i$ on the overall multi-episodic quality $m_n$ is expressed by its weight $a_i$. This was found sufficient considering the amount of data and noise. Using a weighted average also allows to account for a recency effect. We evaluate the *a)* [^2] and *b)* . Both functions are parametrized by the window parameter $\mathit{w}$. For , it is limited to $\mathit{w}~\in~\mathbb{N}$ and $0~<~\mathit{w}~\leq~\mathit{n}$. $$\label{eq:weight:window}
WF: a_i= \left\{
\begin{array}{ll}
1,& \text{if } i - n + w > 0 \\
0,& \text{otherwise}
\end{array}
\right.$$ $$\label{eq:weight:linear} LW: a_i= \left\{
\begin{array}{ll}
i - n + w,& \text{if } i - n + 2*w > 0 \\
0,& \text{otherwise}
\end{array}
\right.$$
We evaluate the prediction accuracy by computing the of the episodic ([i.e.]{}, input) and multi-episodic ([i.e.]{}, output).
[[**Prediction C0 (HP only).** ]{}]{} The prediction accuracy improves for an increasing $\mathit{w}$ ([cf.]{}, ). This is more prevalent for than for . WF achieves its minimal with $\mathit{w}=6$ ([i.e.]{}, all prior episodes). provides only a marginal decrease for $\mathit{w}~\geq~3$. [[***Takeaway:***]{} *The weighted average achieves a reasonable prediction accuracy while provides a slightly better, robust performance.*]{} This is in line with [@guse_multiepisodic_2017].
![Multi-episodic prediction accuracy for C0.[]{data-label="fig:pred:E6pred6"}](figure/plotE6BASE-1){width="\maxwidth"}
[[**Prediction C1-C8.** ]{}]{} With regard to the prediction of the multi/episodic judgment of the 6th day, both weight functions perform differently ([cf.]{}, ). While LF reaches a minimal at $\mathit{w}=4$ ([i.e.]{}, 0.15), WF not until $\mathit{w}=8$ ([i.e.]{}, 0.26). [[***Takeaway:***]{} *LF is preferable to WF, as a higher prediction accuracy is achieved. Also, LF requires a smaller $\mathit{w}$ while providing a higher robustness for choosing $\mathit{w}$.*]{}
![Multi-episodic prediction accuracy for all conditions (except C0).[]{data-label="fig:pred:E6pred12"}](figure/plotE6PRED-1){width="\maxwidth"}
[[**Accounting for Saturation.** ]{}]{} In line with Guse et. al [@guse_multiepisodic_2017], a saturation could be observed although not taken into account for prediction. For C6, the multi-episodic judgment remained at the same level as C5 although an additional usage episode was presented in . Also it remained approx. above the episodic judgment of . In fact, C5 and C6 only differ in the performance level of the 4th usage episode. As both were not judged differently, this suggests that this difference did not affect the formation process of the multi/episodic judgment. In case of C6, we prepose to adjust the episodic judgments of the usage episodes of the 4th day by the average of the usage episodes. Then the window function can be applied without further modification. For C6, this modification shifts the minimal from $\mathit{w}=9$ to $\mathit{w}=6$ for WF and for LW from $\mathit{w}=7$ to $\mathit{w}=4$ (see ). Also, the prediction performance of C6 (adjusted) and C5 resemble each other closely. [[***Takeaway:***]{} *This prefiltering approach for three consecutive usage episodes allows to account for the saturation as it increases prediction performance.*]{}
![Multi-episodic prediction accuracy for saturation.[]{data-label="fig:pred:SAT:E6"}](figure/plotE6SAT-1){width="\maxwidth"}
Conclusion
==========
We presented the results of a multi-episodic experiment of an service. The primary goal was to extend the work on multi-episodic perceived quality in one session [@guse_multiepisodic_2017] to several days. For this reason, the same hypotheses and also a very similar experimental design was applied. We made three observations. First, increasing the number of usage episodes decreases the directly following multi-episodic judgment ([**H1**]{}). Here, the multi-episodic judgment reaches saturation showing that prior usage episodes are still accounted for. Second, we could not find a significant impact of a recency effect ([**H2**]{}) that was observed in prior work [@guse_multiepisodic_2017] on single sessions. Third, consecutive vs. non-consecutive presentation of usage episode did not seem to affect multi-episodic judgments ([**H3**]{}). This is, interestingly, in line with multi-episodic experiments in one session [@guse_multiepisodic_2017] indicating that the time between usage episodes has a limited impact on the multi-episodic formation process.
[[**Future Work.** ]{}]{} Although our results are very promising, the formation process of multi-episodic perceived quality is still far from well understood. So far experiments forced participants to use service(s) in a certain manner by defining when and how to interact with them. Thus, intentionally preventing variation in usage behavior and interaction. However, in a normal setting a user has a motivation to interact with a service, sometimes a task, as well as a desired outcome and probably it’s importance. Therefore, (temporary) failures of services might prevent a user from fulfilling his/her task and therefore the multi-episodic formation process. Also, the attribution of reduced service performance or failures might affect multi-episodic judgments ([e.g.]{}, being in a remote area with limited mobile coverage). Moreover, it is still open how different modalities ([e.g.]{}, audio, video), service types ([e.g.]{}, web browsing), and service bundles are actually judged and how the formation process is determined.
[^1]: The LPC/10 encoded content was re-encoded with MP3 () for the actual transmission and reproduction.)
[^2]: Selecting $\mathit{w := n}$, this model type becomes the average over all prior episodic judgments as proposed by [@moller_single-call_2011].
|
---
author:
- 'Pavel Rojtberg [^1]\'
- 'Arjan Kuijper [^2]\'
bibliography:
- 'bibliography.bib'
title: Efficient Pose Selection for Interactive Camera Calibration
---
Camera calibration in the context of 3D computer vision is the process of determining the internal camera geometric and optical characteristics (intrinsic parameters) and optionally the position and orientation of the camera frame in the world coordinate system (extrinsic parameters) [@tsai1987versatile]. The performance of many 3D computer vision algorithms directly depends on the quality of this calibration. Furthermore, calibration is a recurring task that has to be performed each time the setup is changed. Even if a camera is replaced by an equivalent from the same series, the intrinsic parameters may vary due to build inaccuracies. The prevalent approach to camera calibration [@zhang2000flexible] is based on acquiring multiple images of a planar pattern of known size. However, there are degenerate pose configurations [@sturm1999plane] that lead to unreliable solutions. Therefore, the task of calibration cannot be performed by inexperienced users — even researchers working in the field often struggle to quantify what constitutes good calibration images.
There has been research on the effect of the angle between image plane and pattern on the estimation error; @triggs1998autocalibration related the angular spread to the error in focal length. He found a spread of more than $5^\circ$ necessary. @sturm1999plane further differentiated between estimating principal point and focal length. More importantly, they discussed possible singularities when using one and two planes for calibration and related them to the individual pinhole parameters; e.g. if the pattern is parallel to the image plane in every frame, the focal length cannot be determined. These findings were replicated in [@zhang2000flexible]. However, the effect of poses on the estimation of the distortion parameters or general camera-to-board poses have not been considered so far.
Another aspect is the quality and quantity of calibration data. @sun2005requirements evaluated the sensitivity of camera models to noise, training data quantity and the calibration accuracy in respect to model complexity. However, they only measured the residual error on the respective training set, which is subject to over-fitting. To overcome this, @richardson2013iros introduce the Max Expected Reprojection Error (Max ERE) metric that instead correlates with the testing error and thus allows a meaningful test for convergence.
![Exemplary calibration using 9 selected poses and the user guidance overlay, projecting for the bottom right camera.[]{data-label="fig:init_overlay"}](figure/poses.png "fig:"){width="24.00000%"} ![Exemplary calibration using 9 selected poses and the user guidance overlay, projecting for the bottom right camera.[]{data-label="fig:init_overlay"}](figure/overlay2.jpg "fig:"){width="24.00000%"}
Furthermore, they automatically compute a “best next pose” and use it for user guidance as an overlaid projection of the pattern. The poses are selected by performing an exhaustive search in a fixed set of about 60 candidate poses. For each pose a hypothetical calibration including this pose is performed and the pose that minimizes the Max ERE is selected. However the candidate poses are uniformly distributed in the field of view and do not explicitly consider the angular spread and degenerate cases [@sturm1999plane].
In the general context of user assistance for calibration tasks [@pankratz2015poster], camera calibration was not yet specifically considered.
We propose to analytically generate optimal pattern poses while explicitly avoiding the degenerate pose configurations. For this we relate poses to constraints on individual parameters, such that the resulting pose sequence constrains all calibration parameters and ensures an accurate calibration. This reduces the computation time from seconds to milliseconds compared to the exhaustive search of [@richardson2013iros].
The uncertainty of the calibration parameters is assessed using the covariance of the estimated solution. The pose sequence is then adapted such that more constraints are captured for the most uncertain parameter. The parameters covariance correlates with the testing error and therefore also serves as a convergence criterion.
Based on the above, our key contributions are;
1. Empirical evidence for the need of two distinct pose selection strategies and
2. an efficient pose selection scheme for implementing both of them.
This paper is structured as follows: in Section \[sec:immodel\] the used camera model and the uncertainty estimation method are introduced and the choice of a suitable calibration pattern is discussed. Section \[sec:poses\] motivates and describes our novel pose selection method, while Section \[sec:process\] describes the full calibration pipeline. In Section \[sec:evaluation\] the method is evaluated on real and synthetic data and compared with OpenCV [@opencv_library] and AprilCal [@richardson2013iros] calibration methods. Furthermore, the compactness of the resulting calibration is analyzed and an informal user survey is performed to show the usability of the method.
We conclude with Section \[sec:conclusion\] giving a summary of our results and discussing the limitations and future work.
Preliminaries {#sec:immodel}
=============
We will use the pinhole camera model that, given the camera orientation $\mathbf{R}$, position $\mathbf{t}$ and the parameter vector $\mathbf{C}$, maps a 3D world point $ \mathbf{P} = [X, Y, Z] $ to a 2D image point $\mathbf{p} = [x, y]$: $$\pi \left( \mathbf{P}; \mathbf{R}, \mathbf{t}, \mathbf{C} \right) = \mathbf{K} \, \Delta(\frac{1}{Z_c} \left[\mathbf{R} \; \mathbf{t} \right] \mathbf{P}).
\label{eq:cvcam}$$ Here $\left[\mathbf{R} \; \mathbf{t} \right]$ is a 3x4 affine transformation, $Z_c$ denotes the depth of $\mathbf{P}$ after affine transformation, and $\textbf{K}$ is the camera calibration matrix containing the focal lengths (and aspect ratio) $\left[ f_x, f_y \right]$ and the principal point $\left[c_x, c_y \right]$. @zhang2000flexible also includes a skew parameter $\gamma$ — however, for CCD cameras it is safe to assume $\gamma$ to be zero [@sun2005requirements; @hartley2005multiple]. $\Delta(\cdot)$ models the commonly used [@sun2005requirements] radial and tangential lens distortions (following [@heikkila1997four]) as
$$\begin{aligned}
\Delta(\mathbf{p}) =&\;\mathbf{p} \left( 1 + k_1 r^2 + k_2 r^4 + k_3 r^6 \right) \label{eq:raddist} \\
& +
\begin{pmatrix}
2p_1xy+p_2 \left( r^2 + 2x^2 \right) \\
p_1 \left( r^2 + 2y^2 \right) + 2p_2xy
\end{pmatrix}, \label{eq:tangdist}\end{aligned}$$
where $ r = \sqrt{x^2 + y^2} $.
Therefore $\textbf{C} = \left[ f_x, f_y, c_x, c_y, k_1, k_2, k_3, p_1, p_2 \right]$.
Estimation and error analysis
-----------------------------
Given $M$ images each containing $N$ point correspondences, the underlying calibration method [@zhang2000flexible] minimizes the geometric error
$$\epsilon_{res} = \sum_{i}^N \sum_{j}^M \parallel \mathbf{p}_{ij} - \pi \left( \mathbf{P}_i; \mathbf{R}_j, \mathbf{t}_j, \mathbf{C} \right) \parallel^2,
\label{eq:reperr}$$
where $\mathbf{p}_{ij}$ is an observed (noisy) 2D point in image $j$ and $\mathbf{P}_i$ is the corresponding 3D object point.
Eq. is also referred to as the reprojection error and often used to assess the quality of a calibration. Yet, it only measures the residual error and is subject to over-fitting. Particularly $\epsilon_{res} = 0$ if exactly $N = 10.5$ point correspondences are used [@hartley2005multiple §7.1].
The actual objective for calibration however, is the estimation error $\epsilon_{est}$, i.e. the distance between the solution and the (unknown) ground truth. @richardson2013iros propose the Max ERE as an alternative metric that correlates with the estimation error and also has a similar value range (pixels). However, it requires sampling and re-projecting the current solution. Yet for user guidance and monitoring of convergence only the relative error of the parameters is needed. Therefore, we directly use the variance $\bm{\sigma}_{C}^2 $ of the estimated parameters. Particularly, we use the index of dispersion (IOD) $\sigma^2_i / C_i$ to ensure comparability of the parameters among each other.
Given the covariance of the image points $\mathbf{\Sigma}_{p}$ the backward transport of covariance [@hartley2005multiple §5.2.3] is used to obtain $$\begin{aligned}
\mathbf{\Sigma}_{v} &= \left( \mathbf{J}^T \mathbf{\Sigma}_{p}^{-1} \mathbf{J} \right)^{+} \label{eq:variance} \\
\mathbf{J} &= \delta \pi / \delta \mathbf{v} \nonumber\end{aligned}$$ where $\mathbf{J}$ is the Jacobian matrix, $\mathbf{v} = [\mathbf{C}, \mathbf{R}_1, \mathbf{t}_1, \ldots ,\mathbf{R}_M, \mathbf{t}_M]$ is the vector of unknowns and $(\cdot)^{+}$ denotes the pseudo inverse. For simplicity and because of the lack of prior knowledge we assume a standard deviation of 1px in each coordinate direction for the image points thus $ \mathbf{\Sigma}_{p} = \mathbf{I} $.
The diagonal entries of $\mathbf{\Sigma}_{v}$ contain the variance of the estimated $\mathbf{C}$. $\mathbf{J}$ is already computed in Levenberg-Marquardt step of [@zhang2000flexible].
Calibration pattern {#sec:pattern}
-------------------
Our approach works with any planar calibration target e.g. the common chessboard and circle grid patterns. However, for interactive user guidance a fast board detection is crucial. Therefore, we use the self-identifying ChArUco [@charuco] pattern as implemented in OpenCV. This saves the time consuming ordering of the detected rectangles to a canonical topology when compared to the classical chessboard. However, one can alternatively use any of the recently developed self-identifying targets [@atcheson2010caltag; @birdal2016x; @fiala2008self] here.
The pattern size is set to 9x6 squares resulting in up to 40 measurements at the chessboard joints per captured frame. This allows to successfully complete the initialization even if not all markers are detected as discussed in section \[sec:heuristics\].
Pose selection {#sec:poses}
==============
The core idea of our approach is to explicitly specify individual key-frames which are used for calibration using the method of @zhang2000flexible. In this section first the relation of intrinsic parameters and board poses is discussed to motivate our split of the parameter vector into pinhole and distortion parameters. For each parameter group we then present our set of rules to generate an optimal pose while explicitly avoiding degenerate configurations.
Splitting pinhole and distortion parameters {#sec:singsample}
-------------------------------------------
Looking at eq. we see that both $\mathbf{K}$ and $\Delta(\cdot)$ are applied at post-projection and thus describe 2D-to-2D mappings. Therefore, one might consider estimating $\mathbf{C}$ just from one board pose that uniformly samples the image. However, as both intrinsic and extrinsic parameters are estimated simultaneously by [@zhang2000flexible], ambiguities arise.
Assuming $ \mathbf{R} = \mathbf{I} $ and the distortion parameters to be zero, by multiplying out we get $$\mathbf{p} = \begin{bmatrix} \dfrac{f_x (X + t_x)}{Z + t_z} + c_x \\
\dfrac{f_y (Y + t_y)}{Z + t_z} + c_y \end{bmatrix}\label{eq:ambiguity}$$ for all pattern points $\mathbf{P}$. In this case there are two ambiguities between
1. the focal length $f$ and the distance to camera $t_z$ and
2. the in-plane translation $[t_x, t_y]$ and principal point $[c_x, c_y]$.
These ambiguities can be resolved by requiring the pattern to be tilted towards the image plane such that there is only one $\mathbf{t}$ that satisfies eq. for all pattern points.
Considering the distortion parameters of $\Delta(\cdot)$ on the other hand, there are no similar ambiguities due to the non-linearity of the mapping. The parameters are rather determined by the maximal distortion strength evident in the image. Here it is more important to accurately measure the distortion in the corresponding image regions (see Figure \[fig:dist\_effect\]).
Therefore, we split the parameter vector $\textbf{C}$ into $\textbf{C}_K = \left[ f_x, f_y, c_x, c_y \right]$ and $\textbf{C}_\Delta = \left[ k_1, k_2, k_3, p_1, p_2 \right]$ and consider each group separately.
Avoiding pinhole singularities
------------------------------
While optimizing parameters in $\textbf{C}_K$, singular poses must be avoided. In addition to the case discussed above, we incorporate the cases identified in [@sturm1999plane]. Particularly, we restrict the 3D configuration of the calibration pattern as follows:
- The pattern must not be parallel to the image plane.
- The pattern must not be parallel to one of the image axes.
- Given two patterns, the “reflection constraint” must be fulfilled. This means that the vanishing lines of the two planes are not reflections of each other along both a horizontal and a vertical line in the image.
These restrictions ensure that each pose adds information that further constrains the pinhole parameters.
Pose generation {#sec:posegen}
---------------
![Exemplary pose selection state. *Top:* Index of dispersion. *Left:* Intrinsic calibration position candidates after one () and two () subdivision steps . *Right:* Distortion map with already visited regions masked out.[]{data-label="fig:posegen"}](figure/posegen.pdf){width="49.00000%"}
As described in Section \[sec:singsample\], each parameter group requires a different strategy to generate an optimal calibration pose.
For the intrinsic parameters $\textbf{C}_K$ we follow [@triggs1998autocalibration; @zhang2000flexible] and aim at maximizing the angular spread between image plane and calibration pattern. Accordingly, poses are generated as follows:
1. We choose a distance such that the whole pattern is visible, maximising the amount of observed 2D points.
2. Depending on the principal axis (e.g. $x$ for $f_y$) the pattern is tilted in the range of $(-70^\circ; 70^\circ)$ around that axis. The actual angle is interpolated using the sequence $\left[ 0.25, 0.75, 0.125, 0.375, \ldots \right]$ which corresponds to the binary subdivision of the $(0; 1)$ range (see Figure \[fig:posegen\]). This strategy, as desired, maximizes the angular spread.
3. The resulting pose would still be parallel to one of the image axes which prevents the estimation of the principal point along that axis [@sturm1999plane]. Therefore, the resulting view is rotated by $22.5^{\circ}$ which implements this requirement while keeping the principal orientation.
4. When determining $\left[c_x, c_y \right]$ the view is further shifted along the respective image axis by 5% of the image size. This increases the spread along that axis and leads to faster convergence in our experiments.
For the distortion parameters $\textbf{C}_\Delta$ the goal is to increase sampling accuracy in image regions exhibiting strong distortions. For this we generate a distortion map based on the current calibration estimate that encodes the displacement for each pixel. Using this map we search for the distorted regions as follows:
1. Threshold the distortion map (Figure \[fig:dist\_effect\]) to find the region with the strongest distortion.
2. Given the threshold image, an axis aligned bounding box (AABB) is fitted to the region, corresponding to a parallel view on the pattern. Note that the constraints for $\textbf{C}_K$ do not apply here.
3. The area covered by the AABB is excluded from subsequent searches (see Figure \[fig:posegen\]). Effectively, the distorted regions are thereby visited in order of distortion strength.
4. The pattern is aligned with the top-left corner of the AABB and positioned at a depth s.t. its projection covers 33% of the image width.
The angular range and width limits mentioned above were set such that the calibration pattern could be reliably detected using the Logitech C525 camera.
Initialization {#sec:init}
--------------
The underlying calibration method [@zhang2000flexible] requires at least two views of the pattern for an initial solution which we select as follows:
- For the parameters $\textbf{C}_K$ a pose tilted by $45^{\circ}$ around $x$ is selected (see Section \[sec:posegen\]). This particular angle was suggested by [@zhang2000flexible] and lies in between the extrema of $0^{\circ}$ where the focal length cannot be determined and $90^{\circ}$ where the aspect ratio and principal point cannot be determined.
- Without any prior knowledge we aim at an uniform sampling for estimating $\textbf{C}_\Delta$. To this end we compute a pose such that the pattern is parallel to the image plane and covers the whole view. While this violates the axis alignment requirements for $\textbf{C}_K$ poses, it still provides extra information as it is not coplanar to the first pose [@zhang2000flexible]. Furthermore, the reflection constraint is fulfilled.
To render an accurate overlay for the first pose without prior knowledge of the used camera, we employ a bootstraping strategy similar to [@richardson2013iros]; if the pattern can be detected, we perform a single frame calibration estimating the focal length only — the principal point is fixed at the center and $\textbf{C}_\Delta$ is set to zero.
Calibration process {#sec:process}
===================
In the following we present the parameter refinement and user guidance parts as well as any employed heuristics. This completes the calibration pipeline as used for the real data experiments.
Parameter refinement {#sec:param-refine}
--------------------
After obtaining an initial solution using two key-frames, the goal is to minimize the cumulated variance $ \sum_i \sigma_i^2 \mid \sigma_i^2 \in \bm{\sigma}^2_C $ of the estimated parameters $\mathbf{C}$. We approach this problem by targeting the variance of a single parameter $C_i \in \mathbf{C}$ at a time. Here we pick the parameter with the highest index of dispersion (MaxIOD) $\sigma^2_i / C_i$ ($\sigma^2_i $ iff $ C_i = 0$). Depending on the parameter group, a pose is then generated as described in Section \[sec:poses\].
For determining convergence, we use a ratio test of the parameter variance $r = \sigma^2_{i,n+1} / \sigma^2_{i,n}$. If the reduction $1 - r$ is below a given threshold, we assume the parameter to be converged and exclude it from further refinement. Here, we only consider parameters from the same group as there is typically only little reduction in the complementary group. The calibration terminates once all parameters $\mathbf{C}$ have converged.
User Guidance
-------------
To guide the user, the targeted camera pose is projected using the current estimate of the intrinsic parameters. This projection is then displayed as an overlay on top of the live video stream (See Figure \[fig:init\_overlay\] and the video in the supplemental material).
To verify whether the user is sufficiently close to the target pose we use the Jaccard index $J(A, B)$ (intersection over union) computed from the area covered by the projection of pattern from the target pose $T$ and the area covered by the projection from the current pose estimate $E$. We assume that the user has reached the desired pose if $J(T, E) > 0.8$.
Comparing the projection overlap instead of using the estimated pose directly is more robust since the pose estimate is often unreliable — especially during initialization.
Heuristics {#sec:heuristics}
----------
Throughout the process we enforce the common heuristic [@hartley2005multiple §7.2] that the number of constraints should exceed the number of unknowns by a factor of five. The used calibration method [@zhang2000flexible] not only estimates the intrinsic parameters $\mathbf{C}$, but also the relative pose of model plane and image plane i.e. the parameters $\mathbf{R}$, a 3D rotation, and $\mathbf{t}$, a 3D translation. When using $M$ calibration images we thus have $d = 9 + 6M$ unknowns and each point correspondence provides two constraints. For initialization ($M = 2$) we thus have $21$ unknowns, meaning $52.5$ point correspondences are needed in total or 27 correspondences per frame. For any subsequent frame only 15 points are required.
To prevent inaccurate measurements due to motion blur and rolling shutter artifacts the pattern should be still. To ensure this we require all points to be re-detected in the consecutive frame and the mean motion of the points to be smaller then $1.5$px (determined empirically).
\
Evaluation {#sec:evaluation}
==========
The presented method was evaluated on both synthetic and real data. The synthetic experiments aimed at validating the parameter splitting and pose generation rules presented in Section \[sec:poses\], while the real data was used for comparison with other methods. Furthermore, the compactness of the results with real data was estimated by optimizing directly on the testing set.
Synthetic data {#sec:validation}
--------------
We performed multiple calibrations, each using 20 synthetic images. The first two camera poses were chosen as described in section \[sec:init\] to allow a rough initial solution. The next 8 poses were chosen to optimize $\textbf{C}_\Delta$ while the last 10 poses were optimizing $\textbf{C}_K$ (and vice versa).
The camera parameters were based on the calibration parameters of a Logitech C525 camera $\textbf{C}_{real}$. However, the actual parameters were sampled around $\textbf{C}_{real}$ using a covariance matrix that allowed 10% deviation for each of the parameters $\Sigma = diag(0.1 \cdot \textbf{C}_{real})$ as $$\textbf{C} \sim \mathcal N(\textbf{C}_{real}, \Sigma).\label{eq:sample}$$ Therefore, each synthetic calibration corresponds to using a different camera $\textbf{C}$ with known ground truth parameters. To allow generalization to different camera models, we kept the above pose generation sequence, but used 20 different cameras $\textbf{C}$.
Figure \[fig:correl\] shows the mean standard deviation $\bm{\sigma}_C$ of the parameters. Notably there is a significant drop in $\sigma$ iff a pose matching the parameter group is used.
We also evaluated the usage of MaxIOD as an error metric by comparing it to MaxERE [@richardson2013iros] and a known estimation error $\epsilon_{est}$. Just as the MaxERE, the MaxIOD correlates with $\epsilon_{est}$ (see Figure \[fig:error-metrics\]). Additionally, as Figure \[fig:quality-level\] indicates, the IOD reduction is suitable for balancing calibration quality and the number of required calibration frames.
Real data
---------
For evaluating our method with real images, we recorded a separate testing set consisting of 50 images at various distances and angles covering the whole field of view. All images were captured using a Logitech C525 webcam at a resolution of 1280x720px. The autofocus was fixed throughout the whole evaluation, while exposure was fixed per sequence. Our method was compared to AprilCal [@richardson2013iros] and calibrating without any pose restrictions using OpenCV.
We used the pattern described in section \[sec:pattern\] that provides 40 measurements per frame for OpenCV as well as for our method. With AprilCal, we used the 5x7 AprilTag target that generates approximately the same amount of measurements.
The convergence threshold was set to 10% for our method and the stopping accuracy parameter of AprilCal was set to 2.0. As the OpenCV method does not provide convergence monitoring, we stopped calibration after 10 frames here.
Method mean $\epsilon_{est}$ frames used mean $\epsilon_{res}$
-------------------------------- ----------------------- ------------- -----------------------
Pose selection 0.518 9.4 0.470
OpenCV [@opencv_library] 1.465 10 0.345
AprilCal [@richardson2013iros] 0.815 13.4 1.540
Compactness test 0.514 7 0.476
: Our method compared to AprilCal and OpenCV on real data. Showing the average over five runs. Training on the testing set results in $\epsilon_{est} = 0.479$.[]{data-label="tbl:evaluation"}
Table \[tbl:evaluation\] shows the mean results over 5 calibration runs for each method, measuring the required number of frames, $\epsilon_{est}$ and $\epsilon_{res}$. Here our method requires only 70% of the frames required by AprilCal while arriving at a 36% lower $\epsilon_{est}$ (64% compared to OpenCV).
Analyzing the calibration compactness
-------------------------------------
The results in the previous section show that our method is able to provide the lowest calibration error $\epsilon_{est}$ while using fewer calibration frames then comparable approaches. However, it is not clear whether the solution is using the minimal amount of frames or whether it is possible to use a subset of frames while arriving at the same calibration error.
Therefore, we further tested the compactness of our calibration result. We used a greedy algorithm that, given a set of frames captured by our method, tries to find a smaller subset. It optimizes for the testing set, directly minimizing the estimation error.
The algorithm is computed as follows; given a set of training images (the calibration sequence)
1. the initialization frames as described in Section \[sec:init\] are added unconditionally;
2. each of the remaining frames is now *individually* added to the key-frame set and a calibration is computed.
3. For each calibration the estimation error $\epsilon_{est}$ is computed using the testing frames.
4. The frame that minimizes $\epsilon_{est}$ is incorporated into the key-frame set. Continue at step 2.
5. Terminate if $\epsilon_{est}$ cannot be further reduced or all frames have been used.
The greedy optimal solution requires 75% of the frames compared to the proposed method while keeping the same estimation error (see Table \[tbl:evaluation\]). This indicates that, while a significant improvement over [@richardson2013iros], our method is not yet optimal in the compactness sense. The greedy algorithm requires an a-priori recorded testing set and only finds a minimal subset of an existing calibration sequence, but cannot generate any calibration poses.
User survey
-----------
We performed an informal survey among 5 co-workers to measure the required calibration time when using our method. The tool was used for the first time and the only given instruction was that the overlay should be matched with the calibration pattern. The camera was fixed and the pattern had to be moved. On average the users required 1:33 min for capturing 8.7 frames at $\epsilon_{est} = 0.533$.
Conclusion and future work {#sec:conclusion}
==========================
We have presented a calibration approach to generate a compact set of calibration frames that is suitable for interactive user guidance. Singular pose configurations are avoided such that capturing about 9 key-frames is sufficient for a precise calibration. This is 30% less than comparable solutions. The provided user guidance allows even inexperienced users to perform the calibration in less than 2 minutes. Calibration precision can be weighted against the required calibration time using the convergence threshold. The camera parameter uncertainty is monitored throughout the processes, ensuring that a given confidence level can be reached repeatedly.
Our evaluation shows that the amount of required frames can still be reduced to speed up the process even more. We only use a widespread and simple distortion model, additional distortion coefficients like thin prism [@weng1992camera], rational [@ma2004rational] and tilted sensor are to be considered in future work. Eventually one could incorporate a detection of unused parameters. This would allow to start with the most complex distortion model which could be gradually reduced during calibration.
Furthermore the method needs adaptation to special cases like microscopy where the depth of field limits the possible calibration angles or calibration at large distance where scaling the pattern accordingly is not desirable.
The OpenCV based implementation of the presented algorithm is available open-source at <https://github.com/paroj/pose_calib>.
[^1]: [email protected]
[^2]: [email protected]
|
---
abstract: 'We consider the one-dimensional diffusion of a particle on a semi-infinite line and in a piecewise linear random potential. We first present a new formalism which yields an analytical expression for the Green function of the Fokker-Planck equation, valid for any deterministic construction of the potential profile. The force is then taken to be an asymmetric dichotomic process. Solving the corresponding energy-dependent stochastic Riccati equation in the space-asymptotic regime, we give an exact probabilistic description of returns to the origin. This method allows for a time-asymptotic characterization of the underlying dynamical phases. When the two values taken by the dichotomic force are of different signs, there occur trapping potential wells with a broad distribution of trapping times and dynamical phases may appear, depending on the mean force. If both values are negative, the time-asymptotic mean value of the probability density at the origin is proportional to the absolute value of the mean force. If they are both positive, traps no more exist and the dynamics is always normal. Problems with a shot-noise force and with a Gaussian white-noise force are solved as appropriate limiting cases.'
author:
- |
Petr CHVOSTA$\,\,$ and Noëlle POTTIER\
Groupe de Physique des Solides,\
Tour 23, 2 place Jussieu, 75251 Paris Cedex 05, France
date: 'November 16, 1998'
title: '[**One-Dimensional Diffusion in a Semi-Infinite Poisson Random Force**]{}'
---
23.0cm 14.0cm 1.0cm 1.0cm 0.5cm 0.0cm 0.0cm
3.3ex 5ex 2.5em 5ex 5ex 3ex 5ex 7ex 7ex
[*Heading:*]{} Statistical physics\
[*Short title:*]{} Diffusion in Poisson Quenched Disorder\
[*PACS number:*]{} 05 40 Fluctuation phenomena, random processes and Brownian motion.
Introduction
============
The study of dynamical features of diffusion or conductivity in random environments has been initiated in early eighties [@alexander] and since then intensively pursued [@havlin]-[@bouchaud]. The problem is usually formulated either in a discrete form, [*i.e.*]{} by means of the Pauli Master Equation [@vankampen], assuming the transfer rates to be random variables, or in a continuous formulation, [*i.e.*]{} using the Fokker-Planck equation [@gardiner]-[@horsthemke] and assuming the so-called drift-function to be a stochastic function in space. The transport properties are obtained through an average over the random parameters in the equation of motion.
The resulting dynamics in the time-asymptotic region may exhibit non-standard features. For instance, with a Gaussian drift function of non-zero mean value, a succession of dynamical phases is observed when the bias is varied. These phases are characterized by different drift and diffusion behaviours, normal ([*i.e.*]{} with a finite mobility and diffusion coefficient), or not. The existence of anomalous dynamical phases can be traced back to the existence of traps with a broad distribution of trapping times [@bouchaud].
Most of the standard treatments assume independent random transfer rates in the master equation [@bernasconi1; @bernasconi2], or, correspondingly, a white-noise drift function in the Fokker-Planck equation [@vinokur]-[@aslangulg3]. Nevertheless, as pointed out [*e.g.*]{} in [@derrida], a more realistic description should take into account the possibility of spatial correlations in the local transport parameters [@aslangulcorr1]-[@comtet].
Let us for instance consider a continuous medium and imagine first that the particle diffuses through an array of randomly positioned minima with randomly distributed depths. The simplest choice for a space-correlated bias yielding this scenario is that of a dichotomic noise [@horsthemke], with realizations alternately assuming two possible values of different signs. The potential then displays a succession of linear segments of random lengths and of alternately positive and negative slopes. One can well figure out that, in such a potential, the minima represent traps, some of them being deep traps with large trapping times. This case shares some similarities with the Gaussian-white-noise one, and should lead to results of the same type for the particle drift and diffusion properties ([*i.e.*]{} the existence of anomalous dynamical phases for certain values of the parameters of the model). But, interestingly enough, the choice of a dichotomic noise also allows to treat the case of a quenched random force taking alternately two values of the same sign. Since clearly the notion of deep traps—and even simply of traps— makes no more sense in this case, one does not expect the existence of anomalous dynamical phases.
More specifically, in the present paper, we assume that the lengths of the above described constant-force segments are independent and exponentially distributed, in which case the quenched random force is a Markovian Poisson process in space. We consider a particle diffusing on a semi-infinite line, [*i.e.*]{} we impose a reflecting boundary condition at the origin. These two hypotheses allow for exact analytical calculation and for the discussion of a rich variety of physically different situations.
The paper is organized as follows. In Section 2, our analysis begins with the Laplace transformation of the Fokker-Planck equation. One thus gets a differential equation in space, depending parametrically on the energy, as pictured by the Laplace variable $z$. The corresponding Green function in the presence of an arbitrary deterministic piecewise constant bias is derived. In Section 3, we introduce quenched disorder with general piecewise constant realizations. The localization probability of the particle at its (sharp) initial position satisfies a Riccati stochastic differential equation with a multiplicative noise (in space). Specifying further the noise to be the Markovian Poisson one, we are then able to derive a one-formula based (Eq. (\[ran24\])) parallel analysis of the disorder average of the localization probability of the particle at the origin on the one hand and the trapping time or the time-asymptotic average velocity on the other hand. In Section 4, we analyze various physical situations, according to the sign of the mean bias and to the presence or to the absence of traps. Finally, Section 5 contains our conclusions.
Generally speaking, the new results of our paper are the following. First, our procedure is exact for any fixed value of the Laplace variable $z$. On the one hand, this enables, at least in principle, the analysis of the disorder-averaged probability density at the origin for [*any*]{} time. On the other hand, we can carry out the small-$z$ analysis and derive the exact time-asymptotic formulae for this quantity in various physical regimes. Another specific feature of our work is the application of reflecting boundary conditions at the origin. Thus, clearly, in contrast to the equivalent problem on an infinite line, the situation with a positive or a negative mean bias [*are not*]{} equivalent. Indeed, with a negative mean bias the particle is in some sense stuck to the boundary at the origin or pushed back towards it, while with a positive mean bias it escapes towards infinity, the [*modus*]{} of its time-asymptotic motion being controlled by a parameter describing the typical depth of the potential traps.
Diffusion in a deterministic force
==================================
Homogeneous force
-----------------
Let us consider an overdamped Brownian particle acted upon by a standard white-noise Langevin force $\Gamma({\mbox{$\widetilde{t}$}})$ and by a position-dependent potential force $F({\mbox{$\widetilde{x}$}})$. Its dynamics is described by the viscous Langevin equation $$\label{det1}
\eta\frac{d}{d{\mbox{$\widetilde{t}$}}}\,{\mbox{$\widetilde{x}$}}({\mbox{$\widetilde{t}$}})=
F[{\mbox{$\widetilde{x}$}}({\mbox{$\widetilde{t}$}})]+\Gamma({\mbox{$\widetilde{t}$}})\,\,,$$ with $\eta$ being the viscosity. The correlation function of the Langevin force is equal to $2D_{0}\eta^{2}\delta({\mbox{$\widetilde{t}$}}-{\mbox{$\widetilde{t}$}}')$, where $D_{0}=k_{B}T/\eta$ is the diffusion constant in the absence of the potential force. The corresponding Fokker-Planck equation for the Green function ${\mbox{$\widetilde{P}$}}({\mbox{$\widetilde{x}$}},{\mbox{$\widetilde{y}$}};{\mbox{$\widetilde{t}$}})$ reads $$\label{det2}
\frac{\partial}{\partial{\mbox{$\widetilde{t}$}}}\,{\mbox{$\widetilde{P}$}}({\mbox{$\widetilde{x}$}},{\mbox{$\widetilde{y}$}};{\mbox{$\widetilde{t}$}})=
-\frac{\partial}{\partial{\mbox{$\widetilde{x}$}}}
\left[\,
-D_{0}\frac{\partial}{\partial{\mbox{$\widetilde{x}$}}}\,{\mbox{$\widetilde{P}$}}({\mbox{$\widetilde{x}$}},{\mbox{$\widetilde{y}$}};{\mbox{$\widetilde{t}$}})+
\frac{F({\mbox{$\widetilde{x}$}})}{\eta}{\mbox{$\widetilde{P}$}}({\mbox{$\widetilde{x}$}},{\mbox{$\widetilde{y}$}};{\mbox{$\widetilde{t}$}})\,\right]\,\,.$$ The bracketed expression represents the probability current ${\mbox{$\widetilde{J}$}}({\mbox{$\widetilde{x}$}},{\mbox{$\widetilde{y}$}};{\mbox{$\widetilde{t}$}})$. We assume the initial condition ${\mbox{$\widetilde{P}$}}({\mbox{$\widetilde{x}$}},{\mbox{$\widetilde{y}$}};0)=\delta({\mbox{$\widetilde{x}$}}-{\mbox{$\widetilde{y}$}})$ and the boundary conditions ${\mbox{$\widetilde{J}$}}({\mbox{$\widetilde{x}$}}_{0},{\mbox{$\widetilde{y}$}};t)=0$, ${\mbox{$\widetilde{J}$}}({\mbox{$\widetilde{x}$}}_{1},{\mbox{$\widetilde{y}$}};{\mbox{$\widetilde{t}$}})=0$. Consequently, the boundaries at ${\mbox{$\widetilde{x}$}}_{0}$ and ${\mbox{$\widetilde{x}$}}_{1}$ are reflecting and the probability density is always normalized to unity.
In order to make the following calculation more transparent, we introduce dimensionless variables. The potential force will be written in the form $F({\mbox{$\widetilde{x}$}})=F_{0}f({\mbox{$\widetilde{x}$}})$. The dimensionless coordinate is $x={\mbox{$\widetilde{x}$}}F_{0}/\eta D_{0}$, and the dimensionless time $t={\mbox{$\widetilde{t}$}}F_{0}^{2}/\eta^{2}D_{0}$. We thus get from Eq. (\[det2\]): $$\begin{aligned}
\label{det3}
\rule[-1ex]{0em}{4ex}\frac{\partial}{\partial t}P(x,y;t)&=&
-\frac{\partial}{\partial x}J(x,y;t)\,\,,\\
\label{det4}
\rule[-1ex]{0em}{4ex}J(x,y;t)&=&
-\frac{\partial}{\partial x}P(x,y;t)+f(x)P(x,y;t)\,\,.\end{aligned}$$ The original density and current are connected with their dimensionless counterparts [*via*]{} ${\mbox{$\widetilde{P}$}}({\mbox{$\widetilde{x}$}},{\mbox{$\widetilde{y}$}};{\mbox{$\widetilde{t}$}})=F_{0}P(x,y;t)/\eta D$ and ${\mbox{$\widetilde{J}$}}({\mbox{$\widetilde{x}$}},{\mbox{$\widetilde{y}$}};{\mbox{$\widetilde{t}$}})=F_{0}^{2}J(x,y;t)/\eta^{2}D$.
Let us assume for the moment a position-independent force: $f(x)=f$ for $x\in[x_{0},x_{1}]$. Performing the Laplace transformation, the Fokker-Planck equation (\[det2\]) yields a nonhomogeneous differential equation with constant coefficients, $$\label{det5}
\left[\frac{d^{2}}{dx^{2}}-f\frac{d}{dx}+z\right]\,P(x,y;z)=-\delta(x-y)\,\,.$$ We are using the same symbol for a given function $a(t)$ and for its Laplace transform $a(z)=\int_{0}^{\infty}dt\,\exp(-zt) a(t)$, the Laplace original (transform) being always indicated by writing the variable $t$ ($z$). Combining any particular solution of the nonhomogeneous equation with the general solution of the homogeneous equation, we have $P(x,y;z)=P_{N}(x,y;z)+P_{H}(x;z)$, with $$\begin{aligned}
\label{det6}
\rule[-1ex]{0em}{4ex}P_{N}(x,y;z)&=&
\frac{1}{2\alpha(z)}\left\{\Theta(y-x){\rm e}^{(x-y)\alpha^{+}(z)}+
\Theta(x-y){\rm e}^{-(x-y)\alpha^{-}(z)}\right\}\,\,,\\
\label{det7}
\rule[-1ex]{0em}{4ex}P_{H}(x;z)&=&
c^{+}(z){\rm e}^{x\alpha^{+}(z)}+c^{-}(z){\rm e}^{-x\alpha^{-}(z)}\,\,,\end{aligned}$$ where $\alpha(z)=\sqrt{z+f^{2}/4}$, $\alpha^{\pm}(z)=\sqrt{z+f^{2}/4}\pm f/2$, and $\Theta(x)$ is the Heaviside function. Having acquired the general solution for the density, we calculate the general expression for the probability current. In the last step, the two functions $c^{\pm}(z)$ are fixed from the reflecting-boundary conditions $J(x_{0},y;z)=0$, $J(x_{1},y;z)=0$.
The whole procedure is well known. However, in view of the following calculation, it is convenient to present the final result in a matrix form. We introduce a two-dimensional space with the basis $\{|1,0\rangle\,,\,|0,1\rangle\}$ and we express the pair density–current as two coordinates of a single state ket: $P(x,y;z)=\langle1,0|G(x,y;z)\rangle$ and $J(x,y;z)=\langle0,1|G(x,y;z)\rangle$. Adopting this convention, the result of the present simple example reads $$\label{det8}
|G(x,y;z)\rangle=
{\mbox{${\sf W}$}}(x_{1}-x;z)|1,0\rangle\,\Gamma(y;z)-\Theta(y-x){\mbox{${\sf W}$}}(y-x;z)|0,1\rangle\,\,.$$ Here we have introduced the abbreviation $$\label{det9}
\Gamma(y;z)=\frac{\langle0,1|{\mbox{${\sf W}$}}(y-x_{0};z)|0,1\rangle}
{\langle0,1|{\mbox{${\sf W}$}}(x_{1}-x_{0};z)|1,0\rangle}\,\,,$$ and the matrix $$\label{det10}
{\mbox{${\sf W}$}}(x;z)=\left(\begin{array}{cc}
\rule[-2ex]{0em}{5ex}
\frac{\displaystyle
\alpha^{-}(z){\rm e}^{x\alpha^{-}(z)}+\alpha^{+}(z){\rm e}^{-x\alpha^{+}(z)}}
{\displaystyle2\alpha(z)}&
\frac{\displaystyle
{\rm e}^{x\alpha^{-}(z)}-{\rm e}^{-x\alpha^{+}(z)}}
{\displaystyle2\alpha(z)}\\
\rule[-2ex]{0em}{5ex}
z\,\frac{\displaystyle
{\rm e}^{x\alpha^{-}(z)}-{\rm e}^{-x\alpha^{+}(z)}}
{\displaystyle2\alpha(z)}&
\frac{\displaystyle
\alpha^{+}(z){\rm e}^{x\alpha^{-}(z)}+\alpha^{-}(z){\rm e}^{-x\alpha^{+}(z)}}
{\displaystyle2\alpha(z)}
\end{array}\right).$$ Notice that, at given $z$, ${\mbox{${\sf W}$}}(x;z)$ satisfies the “dynamical equation” $$\label{det11}
\frac{d}{dx}{\mbox{${\sf W}$}}(x;z)={\mbox{${\sf H}$}}(z){\mbox{${\sf W}$}}(x;z)\hspace{2em},\hspace{2em}
{\mbox{${\sf H}$}}(z)=\left(\begin{array}{cc}-f&1\\z&0\end{array}\right)\,\,,$$ with the position $x$ playing here the role of time. Eq. (\[det10\]) gives ${\mbox{${\sf W}$}}(0;z)={\mbox{${\sf I}$}}$ (unity matrix), [*i.e.*]{} we have formally ${\mbox{${\sf W}$}}(x;z)=\exp[x{\mbox{${\sf H}$}}(z)]$.
Eq. (\[det8\]) yields the complete picture of the resulting motion ([*i.e.*]{} we can compute $P(x,y;z)$, $J(x,y;z)$ and these functions can be inverted into the time domain [@abramowitz]). For instance, taking $x_{0}=0$, $x_{1}=l$, $y\rightarrow 0^{+}$, and $x=0$, the probability density at the origin emerges as the ratio of two matrix elements: $$\label{det12}
P(0,0;z)=
\frac{\langle1,0|{\mbox{${\sf W}$}}(l;z)|1,0\rangle}{\langle0,1|{\mbox{${\sf W}$}}(l;z)|1,0\rangle}=
\frac{1}{z}\,\frac{\displaystyle
\alpha^{-}(z){\rm e}^{l\alpha^{-}(z)}+\alpha^{+}(z){\rm e}^{-l\alpha^{+}(z)}}
{\displaystyle{\rm e}^{l\alpha^{-}(z)}-{\rm e}^{-l\alpha^{+}(z)}}\,\,.$$ Moreover, for the semi-infinite line, we have $\lim_{l\rightarrow\infty}P(0,0;z)=\alpha^{-}(z)/z$. In this case, we get following picture. Having $f<0$, the force pushes the diffusing particle against the boundary. In this case, the time-asymptotic value of the probability density at the origin is $|f|$, the asymptotic value of the mean particle position is $|f|^{-1}$, [*i.e.*]{} the time-asymptotic velocity is zero. On the other hand, when $f>0$, $\lim_{l\rightarrow\infty}P(0,0;t)$ decreases exponentially to zero and the mean position increases linearly with time, the velocity being just $f$. Finally, in the marginal case $f=0$, $P(0,0;t)$ behaves asymptotically as $1/\sqrt{\pi t}$, the mean position increases as $\sqrt{\pi t}$, and the asymptotic velocity is zero.
Piecewise constant force
------------------------
Let the original interval $[x_{0},x_{N}]$ be divided into $N$ segments $[x_{k-1},x_{k}]$, $k=1,\ldots,N$, with lengths $l_{k}=x_{k}-x_{k-1}$. Let $f_{k}$ be the constant force in the $k$-th subinterval. We assume that the particle has been originally placed in the $M$-th segment, [*i.e.*]{} $P(x,y;0)=\delta(x-y)$ with $y\in[x_{M-1},x_{M}]$. The boundary conditions at $x_{0}$ and $x_{N}$ are again reflecting.
The procedure for solving the Fokker-Planck equation will be parallel to that in the above simple example. The general solution in the $M$-th segment consists of two parts. First, the particular solution of the nonhomogeneous equation (\[det5\]) (with $f_{M}$ instead of $f$) will assume the form (\[det6\]) with $\alpha_{M}(z)=\sqrt{z+f_{M}^{2}/4}$ instead of $\alpha(z)$ and $\alpha_{M}^{\pm}(z)=\alpha_{M}(z)\pm f_{M}/2$ instead of $\alpha^{\pm}(z)$. Second, the general solution of the homogeneous equation in the $M$-th subinterval assumes the form (\[det7\]), again with $\alpha_{M}^{\pm}(z)$ instead of $\alpha^{\pm}(z)$, and with two arbitrary functions $c_{M}^{\pm}(z)$ instead of $c^{\pm}(z)$. The general solution in the $k$-th subinterval, $k\ne M$, is also of the form (\[det7\]) with the substitutions $\alpha^{\pm}(z)\rightarrow\alpha_{k}^{\pm}(z)$ and $c^{\pm}(z)\rightarrow c_{k}^{\pm}(z)$. Altogether, the whole general solution depends on the $2N$ functions $c_{k}^{\pm}(z)$, $k=1,\ldots,N$. These are fixed from the requirements $J(x_{0},y;z)=J(x_{N},y;z)=0$ at the reflecting boundaries and from the continuity conditions for the probability density and for the probability current at the intermediate points $x_{1},x_{2},\ldots,x_{N-1}$.
The final result can be again expressed in matrix form. We designate the “evolution operator” for the $k$-th segment as ${\mbox{${\sf W}$}}_{k}(x;z)$—it is defined by the expression (\[det10\]) with the substitutions $\alpha(z)\rightarrow\alpha_{k}(z)$ and $\alpha^{\pm}(z)\rightarrow\alpha_{k}^{\pm}(z)$. Further, we introduce the notations $$\begin{aligned}
\label{det13}
\rule[-1ex]{0em}{4ex}{\mbox{${\sf W}$}}_{m,n}&=&
{\mbox{${\sf W}$}}_{m}(l_{m};z){\mbox{${\sf W}$}}_{m+1}(l_{m+1};z)\ldots{\mbox{${\sf W}$}}_{n}(l_{n};z)\,\,,\\
\label{det14}
\rule[-1ex]{0em}{4ex}\Gamma_{M,N}(y;z)&=&
\frac{\langle0,1|{\mbox{${\sf W}$}}_{1,M-1}{\mbox{${\sf W}$}}_{M}(y-x_{M-1};z)|0,1\rangle}
{\langle0,1|{\mbox{${\sf W}$}}_{1,N}|1,0\rangle}\,\,.\end{aligned}$$ Finally, let $|G_{k}(x,y;z)\rangle$ be the value of the state ket in the $k$-th segment, that is $|G(x,y;z)\rangle=|G_{k}(x,y;z)\rangle$ for $x\in[x_{k-1},x_{k}]$. The final result of the present Section then reads $$\begin{aligned}
\rule[-1ex]{0em}{4ex}|G_{1}(x,y;z)\rangle&=&
{\mbox{${\sf W}$}}_{1}(x_{1}-x;z){\mbox{${\sf W}$}}_{2,N}|1,0\rangle\,\Gamma_{M,N}(y;z)-\nonumber\\
\label{det15}
\rule[-1ex]{0em}{4ex}&-&
{\mbox{${\sf W}$}}_{1}(x_{1}-x;z){\mbox{${\sf W}$}}_{2,M-1}{\mbox{${\sf W}$}}_{M}(y-x_{M-1};z)|0,1\rangle\,\,,\\
\rule[-1ex]{0em}{4ex}&\ldots&\nonumber\\
\rule[-1ex]{0em}{4ex}|G_{M-1}(x,y;z)\rangle&=&
{\mbox{${\sf W}$}}_{M-1}(x_{M-1}-x;z){\mbox{${\sf W}$}}_{M,N}|1,0\rangle\,\Gamma_{M,N}(y;z)-\nonumber\\
\label{det16}
\rule[-1ex]{0em}{4ex}&-&
{\mbox{${\sf W}$}}_{M-1}(x_{M-1}-x;z){\mbox{${\sf W}$}}_{M}(y-x_{M-1};z)|0,1\rangle\,\,,\\
\rule[-1ex]{0em}{4ex}|G_{M}(x,y;z)\rangle&=&
{\mbox{${\sf W}$}}_{M}(x_{M}-x;z){\mbox{${\sf W}$}}_{M+1,N}|1,0\rangle\,\Gamma_{M,N}(y;z)-\nonumber\\
\label{det17}
\rule[-1ex]{0em}{4ex}&-&
\Theta(y-x){\mbox{${\sf W}$}}_{M}(y-x;z)|0,1\rangle\,\,,\\
\label{det18}
\rule[-1ex]{0em}{4ex}|G_{M+1}(x,y;z)\rangle&=&
{\mbox{${\sf W}$}}_{M+1}(x_{M+1}-x;z){\mbox{${\sf W}$}}_{M+2,N}|1,0\rangle\,\Gamma_{M,N}(y;z)\,\,,\\
\rule[-1ex]{0em}{4ex}&\ldots&\nonumber\\
\label{det19}
\rule[-1ex]{0em}{4ex}|G_{N}(x,y;z)\rangle&=&
{\mbox{${\sf W}$}}_{N}(x_{N}-x;z)|1,0\rangle\,\Gamma_{M,N}(y;z)\,\,.\end{aligned}$$ It is easy to check that the boundary conditions and the continuity conditions are actually satisfied and that the resulting probability density is properly normalized.
Having at hand the complete Green function for the given composition of the segments, we now proceed to the analysis of some consequences. First, let $x_{0}=0$, $x_{N}=l$, $y\rightarrow 0^{+}$, and $x=0$. The probability density at the origin assumes a particularly simple form: $$\label{det20}
P(0,0;z)=
\frac{\langle1,0|{\mbox{${\sf W}$}}_{1}(l_{1};z){\mbox{${\sf W}$}}_{2}(l_{2};z)\ldots{\mbox{${\sf W}$}}_{N}(l_{N};z)
|1,0\rangle}
{\langle0,1|{\mbox{${\sf W}$}}_{1}(l_{1};z){\mbox{${\sf W}$}}_{2}(l_{2};z)\ldots{\mbox{${\sf W}$}}_{N}(l_{N};z)
|1,0\rangle}\,\,.$$ In Section 3, this expression will be taken as a starting point for the disordered-medium calculation.
Second, consider the value of the probability density at point $y$ in the $M$-th segment, [*i.e.*]{} at the initial position of the particle. Taking $x_{0}=0$ and $x\rightarrow y^{+}$, $P(x,x;z)$ equals the ratio $$\label{det21}
\frac{\langle0,1|{\mbox{${\sf W}$}}_{1,M-1}{\mbox{${\sf W}$}}_{M}(x-x_{M-1};z)|0,1\rangle\,\,
\langle1,0|{\mbox{${\sf W}$}}_{M}(x_{M}-x;z){\mbox{${\sf W}$}}_{M+1,N}|1,0\rangle}
{\langle0,1|{\mbox{${\sf W}$}}_{1}(l_{1};z)\ldots{\mbox{${\sf W}$}}_{M}(l_{M};z)\ldots{\mbox{${\sf W}$}}_{N}(l_{N};z)
|1,0\rangle}\,\,.$$ Due to the exponential nature of the operator ${\mbox{${\sf W}$}}_{M}(l_{M};z)$ in the denominator, we can split it as a product of two factors, ${\mbox{${\sf W}$}}_{M}(x-x_{M-1};z){\mbox{${\sf W}$}}_{M}(x_{M}-x;z)$. Further, we can insert in between the resolution of the unity operator, which yields $$\begin{aligned}
\rule[-1ex]{0em}{4ex}P(x,x;z)&=&\left[\frac{\langle0,1|
{\mbox{${\sf W}$}}_{k}(x_{k}-x;z){\mbox{${\sf W}$}}_{k+1}(l_{k+1};z)\ldots{\mbox{${\sf W}$}}_{N}(l_{N};z)|1,0\rangle}
{\langle1,0|
{\mbox{${\sf W}$}}_{k}(x_{k}-x;z){\mbox{${\sf W}$}}_{k+1}(l_{k+1};z)\ldots{\mbox{${\sf W}$}}_{N}(l_{N};z)|1,0\rangle}
+\right.\nonumber\\
\label{det22}
\rule[-1ex]{0em}{4ex}&+&\left.\frac{\langle0,1|
{\mbox{${\sf W}$}}_{1}(l_{1};z)\ldots{\mbox{${\sf W}$}}_{k-1}(l_{k-1};z){\mbox{${\sf W}$}}_{k}(x-x_{k-1};z)|1,0\rangle}
{\langle0,1|
{\mbox{${\sf W}$}}_{1}(l_{1};z)\ldots{\mbox{${\sf W}$}}_{k-1}(l_{k-1};z){\mbox{${\sf W}$}}_{k}(x-x_{k-1};z)|0,1\rangle}
\right]^{-1}.\end{aligned}$$ For any [*finite*]{} $x$, the small-$z$ limit of the second term turns out to be zero. Thus the quantity $\lim_{z\rightarrow0^{+}}P(x,x;z)$ is equal to the small-$z$ limit of the density at origin for a [*new*]{} medium of the total length $l-x$. The new interval consists of $N-k+1$ segments of the lengths $x_{k}-x,l_{k+1},\ldots,l_{N}$, the constant forces in these segments being $f_{k},\ldots,f_{N}$. This simple result will be of considerable value in the analysis of the disordered medium. Namely, imagine that the particle is launched from the point $x$. The total time $T(x)$ spent in the interval $(x,x+dx)$ is equal to $$\label{det23}
T(x)\,dx=\left[\int_{0}^{\infty}P(x,x;t)\,dt\right]\,dx=
\lim_{z\rightarrow 0^{+}}P(x,x;z)\,dx\,\,,$$ [*i.e.*]{} $T(x)$ can be related to the probability density at the origin for the above mentioned new interval. Of course, $T(x)$ can only be finite for an interval of infinite length. However, even for the semi-infinite interval, $T(x)$ is finite only if the particle can escape to infinity, that is, if the force in the last segment (which is necessarily of infinite length) is non-negative.
Third, let us consider the thermally-averaged position of the particle $M(l;t)$ (we take again $x_{0}=0$, $x_{N}=l$, and $y\rightarrow 0^{+}$). Introducing the Laplace transform $M(l;z)=\int_{0}^{l}dx\,xP(x,0;z)$ and integrating the Laplace transform of the Fokker-Planck equation, we have $$\begin{aligned}
\rule[-1ex]{0em}{4ex}M(l;z)&=&\left[x\int_{0}^{x}\,dx'P(x',0;z)\right]_{0}^{l}-
\int_{0}^{l}dx\,\int_{0}^{x}dx'\,P(x',0;z)=\nonumber\\
\label{det24}
\rule[-1ex]{0em}{4ex}&&=\frac{l}{z}-\int_{0}^{l}dx\,\frac{1}{z}[1-J(x,0;z)]=
\frac{1}{z}\int_{0}^{l}\,J(x,0;z)\,dx\,\,.\end{aligned}$$ Thereupon, taking the projection of the above Green function, we can write $$\label{det25}
M(l;z)=\frac{1}{z}\,
\frac{\displaystyle\sum_{k=1}^{N}\int_{x_{k-1}}^{x_{k}}\langle0,1|
{\mbox{${\sf W}$}}_{k}(x_{k}-x;z){\mbox{${\sf W}$}}_{k+1}(l_{k+1};z)\ldots{\mbox{${\sf W}$}}_{N}(l_{N};z)|0,1\rangle\,dx}
{\displaystyle\langle0,1|
{\mbox{${\sf W}$}}_{1}(l_{1};z){\mbox{${\sf W}$}}_{2}(l_{2};z)\ldots{\mbox{${\sf W}$}}_{N}(l_{N};z)|0,1\rangle}\,\,.$$ In the next Section, this result will allow to connect the thermally-averaged position $M(l;z)$ with the Laplace transform $P(0,0;z)$ of the probability density at the origin. We now turn to the detailed analysis of this latter quantity in the disordered medium.
Piecewise constant random force
===============================
The preceding calculation in valid for any [*deterministic*]{} piecewise constant force. Such a force can be conceived as a member of a randomly constructed family of functions, that is, as a realization of a stochastic process. An arbitrary fixed realization is associated with a given weight. Consequently, the same weight is attributed to the probability density at the origin $P(0,0;z)$ for this realization. We are thus guided to the question: what is the probability density of the random variable $P(0,0;z)$? This Section presents the exact answer for a semi-infinite medium and a special type of the stochastic process to be fully specified in Subsection 3.3. In the first and the second Subsection, our reasoning is valid for [*any*]{} piecewise constant random force.
Stochastic Riccati equation
---------------------------
Let us consider the following system of two stochastic differential equations, $$\label{ran1}
\frac{d}{d\lambda}|\psi(\lambda;z)\rangle=
{\mbox{${\sf H}$}}(\lambda;z)|\psi(\lambda;z)\rangle\hspace{2em},\hspace{2em}
{\mbox{${\sf H}$}}(\lambda;z)=
\left(\begin{array}{cc}-\phi(\lambda)&1\\z&0\end{array}\right)\,\,,$$ where $\phi(\lambda)$, $\lambda\ge0$, is a piecewise constant random process. Let us introduce the projections $R(\lambda;z)=\langle1,0|\psi(\lambda;z)\rangle$, $S(\lambda;z)=\langle0,1|\psi(\lambda;z)\rangle$, and take the initial conditions $R(0;z)=1$, $S(0;z)=0$. The system (\[ran1\]) can be solved for any specific realization of $\phi(\lambda)$. Actually, consider the composition described at the beginning of Subsection 2.2. ([*c.f.*]{} also Fig. 1), and let the “time” $\lambda$ equal $l-x$. We have $\phi(\lambda;z)=f_{N}$ for $\lambda\in[0,l_{N}]$. If $\lambda$ increases, the evolution is controlled by the operator ${\mbox{${\sf W}$}}_{N}(\lambda;z)$. At the end of this interval, the state ${\mbox{${\sf W}$}}_{N}(l_{N};z)|1,0\rangle$ represents the initial condition for the semigroup evolution in the succeeding interval $\lambda\in[l_{N},l_{N-1}+l_{N}]$. This evolution is governed by the operator ${\mbox{${\sf W}$}}_{N-1}(\lambda;z)$. Repeating this reasoning, the solution at the end of the $N$-th interval, [*i.e.*]{} at the point $\lambda=l$, reads $$\begin{aligned}
\label{ran2}
\rule[-1ex]{0em}{4ex}R(l;z)&=&\langle1,0|
{\mbox{${\sf W}$}}_{1}(l_{1};z){\mbox{${\sf W}$}}_{2}(l_{2};z)\ldots{\mbox{${\sf W}$}}_{N}(l_{N};z)|1,0\rangle\,\,,\\
\label{ran3}
\rule[-1ex]{0em}{4ex}S(l;z)&=&\langle0,1|
{\mbox{${\sf W}$}}_{1}(l_{1};z){\mbox{${\sf W}$}}_{2}(l_{2};z)\ldots{\mbox{${\sf W}$}}_{N}(l_{N};z)|1,0\rangle\,\,.\end{aligned}$$ By comparing with Eq. (\[det20\]), it can be seen that the random variable $P(0,0;z)$ is identical to the ratio $R(l;z)/S(l;z)$. From now on, the random variable $P(0,0;z)$ will be designated as $P(\lambda;z)$. We introduce its probability density $\pi(p,\lambda;z)=\left\langle\,\delta[p-P(\lambda;z)]\,\right\rangle$, where the symbol $\langle\ldots\rangle$ denotes the average over the quenched disorder. Taking the projections of Eq. (\[ran1\]), the random variables $R(\lambda;z)$ and $S(\lambda;z)$ obey the system of stochastic differential equations $$\label{ran4}
\frac{d}{d\lambda}R(\lambda;z)=-\phi(\lambda)R(\lambda;z)+S(\lambda;z)
\,\,\,\,,\,\,\,\,
\frac{d}{d\lambda}S(\lambda;z)=zR(\lambda;z)\,\,,$$ with $\lambda$ representing the total length of the interval accessible to the diffusing particle. Finally, on comparing the $\lambda$-derivative of the product $R(\lambda;z)=P(\lambda;z)\,S(\lambda;z)$ with the first equation (\[ran4\]) and on dividing by $S(\lambda;z)$, we obtain the stochastic Riccati differential equation obeyed by $P(\lambda;z)$: $$\label{ran5}
\frac{d}{d\lambda}P(\lambda;z)=-zP^{2}(\lambda;z)-\phi(\lambda)P(\lambda;z)+1
\hspace{1em},\hspace{1em}P(0;z)=+\infty\,\,.$$ For purely operational reasons, we introduce the random variable $Q(\lambda;z)=zP(\lambda;z)$. One gets from Eq. (\[ran5\]): $$\label{ran6}
\frac{d}{d\lambda}Q(\lambda;z)=-Q^{2}(\lambda;z)-\phi(\lambda)Q(\lambda;z)+z
\hspace{1em},\hspace{1em}Q(0;z)=+\infty\,\,.$$ The density $\kappa(q,\lambda;z)=\left\langle\,\delta[q-Q(\lambda;z)]\,\right\rangle$ is related with the density $\pi(p,\lambda;z)$ by $\pi(p,\lambda;z)=z\kappa(zp,\lambda;z)$. Regarding the relation $$\label{ran7}
\lim_{t\rightarrow\infty}
P(0,0;t)=\lim_{z\rightarrow0^{+}}zP(\lambda;z)=
\lim_{z\rightarrow0^{+}}Q(\lambda;z)\,\,,$$ the small-$z$ limit of the density $\kappa(q,\lambda;z)$ describes the time-asymptotic properties of the random variable $P(0,0;t)$.
Mean trapping time and mean velocity
------------------------------------
Let us now return to the reasoning associated with Eqs. (\[det21\])-(\[det23\]). We shall call the variable $T(x)=\lim_{z\rightarrow0^{+}}P(x,x;z)$ the [*trapping time*]{}. Presently, the density $P(x,x;z)$ and hence also the trapping time are random variables. Performing the small-$z$ limit in Eq. (\[det22\]), we have connected the trapping time with the particle-position probability density at the origin of an interval shorter than the original one. However, if the original interval is of [*infinite*]{} length, the same is true for the new one. Since the quenched random force is described by a [*stationary*]{} process, the probability density at the beginning of the new interval is stochastically equivalent with the density at the beginning of the original one. Summing up, one has: $$\label{ran8}
T(x)=\lim_{z\rightarrow0^{+}}\lim_{\lambda\rightarrow\infty}P(x,x;z)=
\lim_{z\rightarrow0^{+}}\lim_{\lambda\rightarrow\infty}P(\lambda;z)\,\,.$$ The trapping time is position-independent, [*i.e.*]{} $T(x)=T$, its mean value being $$\label{ran9}
\tau\stackrel{{\rm def}}{=}
\lim_{z\rightarrow0^{+}}\lim_{\lambda\rightarrow\infty}
\langle P(\lambda;z)\rangle=
\lim_{z\rightarrow0^{+}}\lim_{\lambda\rightarrow\infty}
\frac{1}{z}\langle Q(\lambda;z)\rangle\,\,.$$
In the preceding Section, we have introduced the Laplace transform of the thermally-averaged position, $M(\lambda;z)$. Presently, it is again a random variable. Using the definition of $S(\lambda;z)$, the probability current in Eq. (\[det25\]) can be rewritten as $J(x,0;z)=S(\lambda-x;z)/S(\lambda;z)$, which yields $$\label{ran10}
M(\lambda;z)=\frac{1}{z}\int_{0}^{\lambda}dx J(x,0;z)=
\frac{1}{z}
\frac{\displaystyle\int_{0}^{\lambda}dx\,S(x;z)}{S(\lambda;z)}\,\,,$$ where we have used the stationarity of the quenched random force. The derivative of this equation yields $$\label{ran11}
\frac{d}{d\lambda}M(\lambda;z)=\frac{1}{z}-zP(\lambda;z)M(\lambda;z)
\hspace{2em},\hspace{2em}M(0;z)=0\,\,.$$ Finally, introducing the Laplace transform of the thermally-averaged velocity, $V(\lambda;z)=zM(\lambda;z)$, the [*time*]{}-asymptotic velocity for the semi-infinite line is given by $$\label{ran12}
\lim_{t\rightarrow\infty}\lim_{\lambda\rightarrow\infty}V(\lambda;t)=
\lim_{z\rightarrow0^{+}}\lim_{\lambda\rightarrow\infty}
z\int_{0}^{\lambda}d\lambda'
\exp\left[-z\int_{\lambda'}^{\lambda}d\lambda''P(\lambda'';z)\right]\,\,,$$ where we have used $\lim_{t\rightarrow\infty}\lim_{\lambda\rightarrow\infty}V(\lambda;t)=
\lim_{z\rightarrow0^{+}}\lim_{\lambda\rightarrow\infty}zV(\lambda;z)$. The last formula can be rewritten in the form which reveals the well known [@bouchaud; @aslangulg1] [*self-averaging*]{} property of the time-asymptotic velocity, when it is nonzero. Indeed, [*assuming*]{} that the limit $\lim_{z\rightarrow0^{+}}\lim_{\lambda\rightarrow\infty}
\langle P(\lambda;z)\rangle$ is finite, we can write $$\begin{aligned}
\rule[-1ex]{0em}{6ex}
\lim_{t\rightarrow\infty}\lim_{\lambda\rightarrow\infty}V(\lambda;t)&=&
\lim_{z\rightarrow0^{+}}\lim_{\lambda\rightarrow\infty}
z\int_{0}^{\lambda}d\lambda'
\exp\left[-z(\lambda-\lambda')\langle P(\lambda;z)\rangle\right]
\times\nonumber\\
\label{ran13}
\rule[-1ex]{0em}{6ex}&\times&
\exp\left\{-z\int_{\lambda'}^{\lambda}d\lambda''
\left[P(\lambda'';z)-\langle P(\lambda;z)\rangle\right]\right\}\,\,.\end{aligned}$$ Due to the above assumption, if $\lambda\rightarrow\infty$, the integral in the second exponent represents a random variable which is finite (for high enough $\lambda''$, the typical trajectory of $P(\lambda'';z)$ swings around the mean value $\langle P(\lambda;z)\rangle$). Thereupon, for small $z$, the second exponential tends to unity and the remaining integration yields the reciprocal value of the non-random number (\[ran9\]). Thus, when the mean trapping time $\tau$ is finite, the asymptotic velocity is a self-averaging quantity equal to $\tau^{-1}$ . If the mean trapping time diverges, the asymptotic velocity vanishes. In this latter case, the disorder-averaged time-asymptotic mean position either tends to a constant (for a negative mean force), or increases slower than linearly.
For the sake of completeness, the [*second*]{} thermally-averaged moment $N(\lambda;z)=\int_{0}^{\lambda}dx\,x^{2}P(x,0;z)$ can be also connected to the probability density at the origin. Actually, first, the Fokker-Planck equation implies $N(\lambda;z)=(2/z)\int_{0}^{\lambda}dx\,xJ(x,0;z)$. Thereupon, on deriving this expression, we get $$\label{ran11a}
\frac{d}{d\lambda}N(\lambda;z)=2M(\lambda;z)-zP(\lambda;z)N(\lambda;z)
\hspace{2em},\hspace{2em}N(0;z)=0\,\,.$$ Summing up, the first and the second thermally-averaged moments obey a system of stochastic differential equations (\[ran11\]), (\[ran11a\]), with $P(\lambda;z)$ playing the role of the “input” noise.
Dichotomic random force
-----------------------
Let the forces in the individual segments assume alternately just two values, $F_{\pm}=F_{0}f_{\pm}$. We shall always take $f_{-}<f_{+}$; the equality would imply a position-independent constant force. Let the lengths of the constant-force segments be independent random variables. The generic probability density for the (dimensionless) lengths of the constant-force segments will be taken of the form $\rho_{\pm}(\lambda)=n_{\pm}\exp(-\lambda n_{\pm})$, where $1/n_{\pm}$ denotes the mean length of the segments with the force $f_{\pm}$. Due to this assumption, the resulting four-parameter stochastic process $\phi(\lambda)$ is Markovian and it usually referred to as the [*asymmetric dichotomic noise*]{} [@horsthemke]. We shall always work with a [*stationary*]{} dichotomic noise. We can set $$\label{ran14}
\mu\stackrel{{\rm def}}{=}\langle\phi(\lambda)\rangle=
\frac{f_{-}n_{+}+f_{+}n_{-}}{n_{-}+n_{+}}\,\,\,\,,\,\,\,\,
\langle\,\phi(\lambda)\phi(\lambda')\,\rangle=
\frac{\sigma}{\lambda_{c}}\exp\left(-\frac{|\lambda-\lambda'|}{\lambda_{c}}
\right)\,\,,$$ where we have introduced the intensity $\sigma\stackrel{{\rm def}}{=}n_{-}n_{+}(f_{+}-f_{-})^{2}/(n_{-}+n_{+})^{3}$, and the correlation length $\lambda_{c}\stackrel{{\rm def}}{=}(n_{-}+n_{+})^{-1}$. The statistical properties of the stationary noise are invariant with respect to the inversion $\lambda\rightarrow-\lambda$ and to the translation $\lambda\rightarrow\lambda-l$.
Let us now focus on Eq. (\[ran6\]). It is convenient to associate with the variable $Q(\lambda;z)$ an overdamped motion of a hypothetical particle. While the “time” $\lambda$ increases, this particle moves alternately under the influence of the “forces” $$\label{ran15}
K_{\pm}(q;z)=-q^{2}-f_{\pm}q+z=-[q-q_{\pm}(z)]\,[q-q_{\pm}'(z)]\,\,,\\$$ where we have introduced the four quantities $$\label{ran16}
q_{\pm}(z)=\sqrt{z+f_{\pm}^{2}/4}-f_{\pm}/2
\hspace{1em},\hspace{1em}
q_{\pm}'(z)=-\sqrt{z+f_{\pm}^{2}/4}-f_{\pm}/2\,\,.$$ Notice the ordering $q_{+}'(z)<q_{-}'(z)<0<q_{+}(z)<q_{-}(z)$, valid for any real positive $z$ and for any values of the parameters $f_{-}<f_{+}$. The corresponding “potentials” $U_{\pm}(q;z)=-\int\,K_{\pm}(q;z)\,dq$ display minima at $q_{\pm}(z)$ and maxima at $q_{\pm}'(z)$. Starting from its initial “position” at infinity, the particle always slides either towards $q_{-}(z)$ or towards $q_{+}(z)<q_{-}(z)$. For any fixed $\lambda$ the particle can only be found between the coordinate valid for the potential $U_{+}(q;z)$ and that valid for $U_{-}(q;z)$. Accordingly, for an arbitrary nonzero $z$, the probability density $\kappa(q,\lambda;z)$ vanishes outside the finite interval $$\label{ran17}
\left[
\frac{\displaystyle
q_{+}(z){\rm e}^{\lambda q_{+}(z)}-q_{+}'(z){\rm e}^{\lambda q_{+}'(z)}}
{\displaystyle{\rm e}^{\lambda q_{+}(z)}-{\rm e}^{\lambda q_{+}'(z)}}
\,\,,\,\,
\frac{\displaystyle
q_{-}(z){\rm e}^{\lambda q_{-}(z)}-q_{-}'(z){\rm e}^{\lambda q_{-}'(z)}}
{\displaystyle{\rm e}^{\lambda q_{-}(z)}-{\rm e}^{\lambda q_{-}'(z)}}
\right]\,\,,$$ and it displays two $\delta$-function contributions at the edges of this support, their weights being $n_{\mp}\exp(-\lambda n_{\pm})/(n_{-}+n_{+})$. This singular part describes an exponentially decreasing probability of having just one segment in the whole interval of length $\lambda$. Obviously, in the limit $\lambda\rightarrow\infty$, the support is simply $[q_{+}(z),q_{-}(z)]$ and the singular part is missing.
In order to solve Eq. (\[ran6\]) for the dichotomic noise in question we follow the standard steps as described in [@horsthemke]. First, we introduce the joint densities $$\label{ran18}
\kappa_{\pm}(q,\lambda;z)\,dq=
{\rm Prob}\left\{\,Q(\lambda;z)\in(q,q+dq)\,\,{\rm and}\,\, \phi(\lambda)=
f_{\pm}\right\}\,\,.$$ One has $\kappa(q,\lambda;z)=\kappa_{-}(q,\lambda;z)+\kappa_{+}(q,\lambda;z)$. These densities obey the coupled partial differential equations $$\begin{aligned}
\rule[-1ex]{0em}{6ex}\frac{\partial}{\partial\lambda}
\left[\begin{array}{c}
\kappa_{-}(q,\lambda;z)\\\kappa_{+}(q,\lambda;z)
\end{array}\right]&=&-
\frac{\partial}{\partial q}
\left[\begin{array}{c}
K_{-}(q;z)\kappa_{-}(q,\lambda;z)\\K_{+}(q;z)\kappa_{+}(q,\lambda;z)
\end{array}\right]-\nonumber\\
\label{ran19}
\rule[-1ex]{0em}{6ex}&-&
\left[\begin{array}{cc}n_{-}&-n_{+}\\-n_{-}&n_{+}\end{array}\right]\,
\left[\begin{array}{c}
\kappa_{-}(q,\lambda;z)\\\kappa_{+}(q,\lambda;z)
\end{array}\right]\,.\end{aligned}$$ We are looking for the stationary solution $\kappa_{\pm}(q;z)=\lim_{\lambda\rightarrow\infty}\kappa_{\pm}(q,\lambda;z)
$. Hence we remove the $\lambda$-derivative on the l.h.s. of Eq. (\[ran19\]). Introducing the two new functions $$\begin{aligned}
\label{ran20}
\rule[-1ex]{0em}{6ex}\xi(q;z)&=&
\frac{K_{-}(q;z)K_{+}(q;z)}{n_{-}K_{+}(q;z)+n_{+}K_{-}(q;z)}
\left[n_{-}\kappa_{-}(q;z)-n_{+}\kappa_{+}(q;z)\right]\,\,,\\
\label{ran21}
\rule[-1ex]{0em}{6ex}\eta(q;z)&=&
K_{-}(q;z)\kappa_{-}(q;z)+K_{+}(q;z)\kappa_{+}(q;z)\,\,,\end{aligned}$$ and carrying out the corresponding substitution in Eq. (\[ran19\]), we arrive at two independent equations: $$\label{ran22}
\frac{d}{dq}\eta(q;z)=0\hspace{1em},\hspace{1em}
\frac{1}{\xi(q;z)}\frac{d}{dq}\xi(q;z)=-
\left(\frac{n_{-}}{K_{-}(q;z)}+\frac{n_{+}}{K_{+}(q;z)}\right)\,\,.$$ Hence the function $\eta(q;z)$ is simply a constant, which in fact is equal to zero. Actually, Eq. (\[ran15\]) yields $K_{\pm}[q_{\pm}(z);z]=0$, and conservation of probability entails $\kappa_{\pm}[q_{\mp}(z);z]=0$. As for the differential equation for $\xi(q;z)$, its solution reads $$\label{ran23}
\xi(q;z)=\frac{1}{C(z)}\left[\frac{q_{-}(z)-q}{q-q_{-}'(z)}\right]^{\nu_{-}(z)}
\left[\frac{q-q_{+}(z)}{q-q_{+}'(z)}\right]^{\nu_{+}(z)}
\hspace{1em},\hspace{1em}
\nu_{\pm}(z)=\frac{n_{\pm}}{\sqrt{4z+f_{\pm}^{2}}}\,\,,$$ where $C(z)$ is a normalization constant. Finally, inverting the transformation $\{$(\[ran20\]), (\[ran21\])$\}$, we get $\kappa_{\pm}(q;z)=\mp\xi(q;z)/K_{\pm}(q;z)$ and the stationary density $\kappa(q;z)=\lim_{\lambda\rightarrow\infty}\kappa(q,\lambda;z)$ reads $$\begin{aligned}
\rule[-1ex]{0em}{6ex}\kappa(q;z)&=&\frac{1}{C(z)}
\left\{
\frac{1}{[q_{-}(z)-q][q-q_{-}'(z)]}+\frac{1}{[q-q_{+}(z)][q-q_{+}'(z)]}
\right\}\times\nonumber\\
\label{ran24}
\rule[-1ex]{0em}{6ex}&\times&
\left[\frac{q_{-}(z)-q}{q-q_{-}'(z)}\right]^{\nu_{-}(z)}
\left[\frac{q-q_{+}(z)}{q-q_{+}'(z)}\right]^{\nu_{+}(z)}
\Theta[q;q_{+}(z),q_{-}(z)]\,\,,\end{aligned}$$ where we have denoted $\Theta(q;x,y)=\Theta(q-x)\Theta(y-q)$. In the final step, the constant $C(z)$ is fixed from the normalization condition $\int_{q_{+}(z)}^{q_{-}(z)}\kappa(q;z)\,dq=1$, that is $$\label{ran25}
C(z)=(f_{+}-f_{-})\displaystyle\int_{q_{+}(z)}^{q_{-}(z)}\,q\,
\frac{[q_{-}(z)-q]^{\nu_{-}(z)-1}\,[q-q_{+}(z)]^{\nu_{+}(z)-1}}
{[q-q_{-}'(z)]^{\nu_{-}(z)+1}\,[q-q_{+}'(z)]^{\nu_{+}(z)+1}}\,dq\,\,.$$ Some steps of the underlying calculation are given in the Appendix—the resulting form of the above integral can be written as $$\begin{aligned}
\rule[-1ex]{0em}{6ex}C(z)&=&
\frac{1}{\left\langle\sqrt{4z+\phi^{2}(\lambda)}\right\rangle}\,\,
\frac{[q_{-}(z)-q_{+}(z)]^{\nu_{-}(z)+\nu_{+}(z)}}
{[q_{+}(z)-q_{-}'(z)]^{\nu_{-}(z)}[q_{-}(z)-q_{+}'(z)]^{\nu_{+}(z)}}
\times\nonumber\\
\label{ran26}
\rule[-1ex]{0em}{6ex}&\times&
{\mbox{${\cal B}$}}[\nu_{-}(z),\nu_{-}(z)]\,\,
{\mbox{${\cal F}$}}\left[\nu_{-}(z),\nu_{+}(z),\nu_{-}(z)+\nu_{+}(z)+1;-u(z)\right].\end{aligned}$$ Here ${\mbox{${\cal B}$}}(x,y)=\Gamma(x)\Gamma(y)/\Gamma(x+y)$ denotes the Euler beta function, ${\mbox{${\cal F}$}}(a,b,c;x)$ is the Gauss hypergeometric function [@abramowitz], and we have used the abbreviation $$\label{ran27}
u(z)=\frac{[q_{-}(z)-q_{+}(z)][q_{+}'(z)-q_{+}'(z)]}
{[q_{-}(z)-q_{+}'(z)][q_{+}'(z)-q_{-}'(z)]}\,\,.$$ Eqs. (\[ran24\]), (\[ran26\]) for the stationary density represent the main result of the present Section. The corresponding moments can be obtained by the usual integration. Again, the ensuing calculation is commented on in the Appendix.
Discussion
==========
Our general four-parameter description of the dichotomic force provides a rich spectrum of special regimes, which can be analyzed using Eq. (\[ran24\]), where, as indicated above, the probability density $\kappa(q;z)$ stands for $\lim_{\lambda\rightarrow\infty}\kappa(q,\lambda;z)$. In the same way, we shall use the simpler designations $Q(z)$, $\pi(p;z)$ and $P(z)$ for the stationary values $Q(\lambda;z)$, $\pi(p,\lambda;z)$ and $P(\lambda;z)$.
Both forces are negative ($f_{-}<f_{+}<0$)
------------------------------------------
The mean force $\mu$ in Eq. (\[ran14\]) is negative and the potential consists of segments with a positive slope ([*c.f.*]{} Fig. 2). For any arbitrary realization of the quenched noise, the particle cannot escape to infinity—it can be found with probability one in a [*finite*]{} region. Intuitively, one expects a nonzero time-asymptotic mean value of the probability density at the origin, and a finite time-asymptotic value of the thermally-averaged mean position.
Consider the small-$z$ limit of the probability density (\[ran24\]). First, one has $\lim_{z\rightarrow 0^{+}}\nu_{\pm}(z)=n_{\pm}/|f_{\pm}|$. Second, the small-$z$ limits of the expressions $q_{\pm}(z)$ are $|f_{\pm}|$, whereas $q_{\pm}'(z)$ behave as $-z/|f_{\pm}|$. Analyzing the expressions in (\[ran17\]), the support of the probability density $\lim_{z\rightarrow 0^{+}}\kappa(q;z)$ is the interval $[|f_{+}|,|f_{-}|]$. Finally, we have $u(z)\rightarrow 0$, [*i.e.*]{} the hypergeometric function in Eq. (\[ran26\]) tends to unity. On collecting these observations, one gets $$\begin{aligned}
\rule[-1ex]{0em}{6ex}\lim_{z\rightarrow0^{+}}\kappa(p;z)&=&|\mu|\,
\frac{|f_{-}|^{\frac{n_{+}}{|f_{+}|}}|f_{+}|^{\frac{n_{-}}{|f_{-}|}}}
{(|f_{-}|-|f_{+}|)^{\frac{n_{-}}{|f_{-}|}+\frac{n_{-}}{|f_{-}|}+1}}
{\mbox{${\cal B}$}}^{-1}\left(\frac{n_{-}}{|f_{-}|},\frac{n_{+}}{|f_{+}|}\right)
\times\nonumber\\
\label{dis1}
\rule[-1ex]{0em}{6ex}&\times&
\frac{(|f_{-}|-q)^{\frac{n_{-}}{|f_{-}|}-1}(q-|f_{+}|)^{\frac{n_{+}}{|f_{+}|}
-1}}{q^{\frac{n_{-}}{|f_{-}|}+\frac{n_{+}}{|f_{+}|}+1}}\,
\Theta(q;|f_{+}|,|f_{-}|)\,\,.\end{aligned}$$ The corresponding moments $\lim_{z\rightarrow0^{+}}\left\langle Q^{k}(z)\right\rangle$ are all finite and they can be computed analytically by direct integration. In particular for $k=1$ one gets $\lim_{z\rightarrow0^{+}}\left\langle Q(z)\right\rangle=|\mu|$. Thus the time-asymptotic mean value $\lim_{t\rightarrow\infty}\langle
P(0,0;t)\rangle$ in the semi-infinite line is seen to be equal to the absolute value of the mean force, $|\mu|$. Note that this result cannot be obtained from the solution of the corresponding free-diffusion model on an infinite line ([*i.e.*]{} without the reflecting boundary condition at the origin). The higher moments are not so simply related to the properties of the random force. Further, in the present case, the mean trapping time (\[ran9\]) is infinite and the time-asymptotic velocity vanishes. The particle is in some sense stuck to the origin. If $f_{-}<f_{+}=0$, a slightly more complicated calculation reveals the same general conclusions.
Forces are of different signs ($f_{-}<0<f_{+}$)
-----------------------------------------------
The potential forms a system of traps ([*c.f.*]{} Fig. 1). The traps can only be efficient if the ratio $n_{+}/f_{+}$ is comparable with the ratio $n_{-}/|f_{-}|$. Otherwise, they are typically “shallow” and they do not represent sufficiently effective obstacles for the particle motion. The “trap-permeability” parameter, as defined by $\theta\stackrel{{\rm def}}{=}n_{-}/|f_{-}|-n_{+}/f_{+}$, will play an important role in the following discussion [@monthus1; @monthus2]. In fact, it is proportional to the mean force: $\theta=\mu(n_{-}+n_{+})/|f_{-}|f_{+}$. Having fixed the forces $f_{\pm}$, the mean force $\mu$ can be either positive or negative, depending on the parameters $n_{\pm}$, the value $\mu=0$ separating two regions with essentially different time-asymptotic dynamics.
Let us consider again the small-$z$ limit of Eq. (\[ran24\]). The quantities $q_{-}(z)$ and $q_{-}'(z)$ behave as in the preceding Subsection. Presently, however, one has $q_{+}(z)\sim z/f_{+}$ and $q_{+}'(z)\rightarrow-f_{+}$. Thereupon, the small-$z$ limit of the support (\[ran17\]) is now the interval $[0,|f_{-}|]$. Further, the variable (\[ran27\]) diverges and one has to use the analytic continuation of the hypergeometric function in Eq. (\[ran26\]).
### Negative mean force ($\mu<0$)
In the small-$z$ limit the normalization constant $C(z)$ alone converges to a finite number and we can safely carry out this limit separately in $C(z)$ and in the rest of the expression (\[ran24\]). The result reads $$\begin{aligned}
\rule[-1ex]{0em}{6ex}\lim_{z\rightarrow0^{+}}\kappa(q;z)&=&
\frac{n_{+}}{n_{-}+n_{+}}
\frac{f_{+}^{\frac{n_{-}}{|f_{-}|}}
\left(|f_{-}|+f_{+}\right)^{\frac{n_{+}}{f_{+}}-\frac{n_{-}}{|f_{-}|}+1}}
{|f_{-}|^{\frac{n_{+}}{f_{+}}-1}}
\frac{1}{\displaystyle{\mbox{${\cal B}$}}\left(
\frac{n_{-}}{|f_{-}|},\frac{n_{+}}{f_{+}}-\frac{n_{-}}{|f_{-}|}\right)}
\times\nonumber\\
\label{dis2}
\rule[-1ex]{0em}{6ex}&\times&
q^{\frac{n_{+}}{f_{+}}-\frac{n_{-}}{|f_{-}|}-1}
\left(|f_{-}|-q\right)^{\frac{n_{-}}{|f_{-}|}-1}
\left(q+f_{+}\right)^{-\frac{n_{+}}{f_{+}}-1}
\Theta(q;0,|f_{-}|)\,.\end{aligned}$$ All the moments of this limiting density exist and can be computed analytically. Let us just quote here the result for $k=1$: $\lim_{z\rightarrow0^{+}}\left\langle Q(z)\right\rangle=|\mu|$, [*i.e.*]{} we have again $\lim_{t\rightarrow\infty}\langle P(0,0;t)\rangle=|\mu|$. The mean trapping time diverges and the time-asymptotic velocity vanishes. As compared to the previous Subsection, the presence of traps does not modify the [*modus*]{} of the asymptotic dynamics.
### Zero mean force ($\mu=0$)
In this Sinai-like case, the small-$z$ analysis of the general expression for the first moment $\langle P(z)\rangle$ together with the Tauber theorem for the inverse Laplace transformation [@feller] yield the logarithmic decay $$\label{dis6}
\left\langle P(0,0;t)\right\rangle\stackrel{t\rightarrow\infty}{\approx}
\frac{|f_{-}|f_{+}}{n_{-}+n_{+}}\,
\frac{1}{\log t}\,\,.$$
### Positive mean force ($\mu>0$)
In the small-$z$ limit, the normalization constant $C(z)$ diverges as $z^{-\theta}$, $\theta>0$. The limiting density (\[ran24\]) is concentrated at one point: $\lim_{z\rightarrow0^{+}}\kappa(q;z)=\delta(q)$. All the moments $Q^{k}(z)$ are self-averaging quantities, their (non-random) limiting value being zero. On the other hand, the density $\lim_{z\rightarrow0^{+}}\pi(p;z)$ is a well behaved function, concentrated on the interval $[f_{+}^{-1},+\infty[$. Actually, if we first handle the substitution $\pi(p;z)=z\kappa(zp;z)$ in Eq. (\[ran24\]) and [*afterwards*]{} exercise the small-$z$ limit, we get $$\begin{aligned}
\rule[-1ex]{0em}{6ex}\lim_{z\rightarrow0^{+}}\pi(p;z)&=&
\frac{n_{-}}{n_{-}+n_{+}}\,
\frac{(|f_{-}|+f_{+})^{\frac{n_{-}}{|f_{-}|}-\frac{n_{+}}{f_{+}}+1}}
{|f_{-}|f_{+}^{\frac{n_{-}}{|f_{-}|}-\frac{n_{+}}{f_{+}}}}\,
{\mbox{${\cal B}$}}^{-1}\left(\frac{n_{+}}{f_{+}},\frac{n_{-}}{|f_{-}|}-
\frac{n_{+}}{f_{+}}\right)\times\nonumber\\
\label{dis3}
\rule[-1ex]{0em}{6ex}&\times&p
\left(p+\frac{1}{|f_{-}|}\right)^{-\frac{n_{-}}{|f_{-}|}-1}\,
\left(p-\frac{1}{f_{+}}\right)^{\frac{n_{+}}{f_{+}}-1}\,
\Theta\left(p;\frac{1}{f_{+}},+\infty\right)\,\,.\end{aligned}$$ Here we come to an essential conclusion [@monthus2]: the moment $\lim_{z\rightarrow0^{+}}\langle P^{k}(z)\rangle$ is only finite if $k<\theta$. Specifically, for $\theta\in]0,1[$, the limit $\lim_{z\rightarrow0^{+}}\langle P(z)\rangle$ is infinite, [*i.e.*]{} the mean trapping time is also infinite and the time-asymptotic velocity is zero. More precisely, using again the Tauber theorem, we get $$\label{dis4}
\left\langle P(0,0;t)\right\rangle\stackrel{t\rightarrow\infty}{\approx}
\frac{n_{-}}{n_{+}(n_{-}+n_{+})}\,
\frac{\Gamma^{2}\left(\displaystyle\frac{n_{-}}{|f_{-}|}\right)}
{\Gamma^{2}\left(\displaystyle\frac{n_{+}}{f_{+}}\right)\,\Gamma(\theta)}\,
\frac{f_{+}^{2(1-\theta)}(|f_{-}|+f_{+})^{2\theta}}{|f_{-}|^{2\theta}}\,
\frac{1}{t^{\theta}}\,\,.$$ If $\theta\ge1$, the first moment $\lim_{z\rightarrow0^{+}}\left\langle P(z)\right\rangle$ is finite and its reciprocal gives the (self-averaging) time-asymptotic velocity: $$\label{dis5}
\lim_{t\rightarrow\infty}\lim_{\lambda\rightarrow\infty}V(\lambda;t)=
(\theta-1)\frac{(n_{-}+n_{+})|f_{-}|f_{+}}
{(n_{-}+n_{+})^{2}-n_{-}|f_{-}|+n_{+}f_{+}}\,\,.$$ However, if $\theta\in[1,2]$, the small-$z$ limit of the second moment $\lim_{z\rightarrow0^{+}}\left\langle P^{2}(z)\right\rangle$ is infinite. This can be shown to imply the vanishing of the (static) diffusion constant for the disorder averaged dynamics (this quantity is not discussed here).
Both forces are positive ($0<f_{-}<f_{+}$)
------------------------------------------
In this case, the slope of the potential is always negative: the particle just slides towards infinity. The small-$z$ limit of the stationary probability density for the random variable $P(z)$ again follows from Eqs. (\[ran24\]), (\[ran26\]). The normalization constant alone tends to zero and one must operate with the whole expression (\[ran24\]). The result reads $$\begin{aligned}
\rule[-1ex]{0em}{6ex}\lim_{z\rightarrow0^{+}}\pi(p;z)&=&
\mu\left(\frac{f_{-}f_{+}}{f_{+}-f_{+}}\right)^
{\frac{n_{-}}{f_{-}}+\frac{n_{+}}{f_{+}}-1}
{\mbox{${\cal B}$}}^{-1}\left(\frac{n_{-}}{f_{-}},\frac{n_{+}}{f_{+}}\right)
\times\nonumber\\
\label{dis7}
\rule[-1ex]{0em}{6ex}&\times&p\,
\left(\frac{1}{f_{-}}-p\right)^{\frac{n_{-}}{f_{-}}-1}
\left(p-\frac{1}{f_{+}}\right)^{\frac{n_{+}}{f_{+}}-1}\,
\Theta\left(p;\frac{1}{f_{+}},\frac{1}{f_{-}}\right)\,\,.\end{aligned}$$ Obviously enough, all the moments of this density exist. For $k=1$, we get $$\label{dis8}
\tau=\lim_{z\rightarrow0^{+}}\langle P(z)\rangle=
\frac{n_{-}f_{-}+n_{+}f_{+}+(n_{-}+n_{+})^{2}}
{(n_{-}+n_{+})(n_{-}f_{+}+n_{+}f_{-}+f_{-}f_{+})}\,\,.$$ As expected, the mean trapping time is finite. The time-asymptotic velocity is self-averaging and equals the reciprocal of the above expression [@monthus1; @monthus2]. There is no anomalous dynamical phase in this case.
White shot-noise limit
----------------------
Originally, the dichotomic quenched force has been described by four parameters, $n_{\pm}\ge 0$ and $f_{\pm}$. Another convenient equivalent four-parameter set is the mean force $\mu$, the intensity $\sigma$, the correlation length $\lambda_{c}=1/(n_{-}+n_{+})$, already introduced in Eq. (\[ran14\]), and the “non-Gaussianity” parameter [@broeck1] $\gamma=|f_{+}-f_{-}|/(n_{-}+n_{+})$, the meaning of which being explained below. It is well known [@horsthemke; @broeck1] that an appropriate limit of the dichotomic noise yields the Poisson white shot-noise. Actually, consider the parametrization $$\label{dis9}
f_{-}=-\frac{\sigma-\gamma\mu}{\gamma}\,\,\,,\,\,\,
f_{+}=\xi\,\,\,,\,\,\,
n_{-}=\frac{\sigma}{\gamma^{2}}\,\,\,,\,\,\,
n_{+}=\frac{\xi}{\gamma}\,\,.$$ If we increase the parameter $\xi$, the force $f_{+}$ increases and the mean length of the segments with the force $f_{+}$ tends to zero. In the limit $\xi\rightarrow\infty$, the parameters $\mu$, $\sigma$, and $\gamma$ keep their values, whereas the correlation length $\lambda_{c}$ tends to zero. The limiting form of the correlation function in Eq. (\[ran14\]) is $2\sigma\delta(\lambda-\lambda')$. After the indicated limit, the quenched force displays an array of randomly positioned $\delta$-impulses on the constant background $-(\sigma-\gamma\mu)/\gamma$ ([*c.f.*]{} Fig. 3). The mean (dimensionless) distance between the $\delta$-impulses is $\gamma^{2}/\sigma$, their weights being [*randomly*]{} distributed with the probability density $\gamma^{-1}\exp(-w/\gamma)\Theta(w)$. Thus the parameter $\gamma$ represents the mean weight of the impulses. On the whole, in the present Subsection, the random potential is described by the three parameters $\sigma\ge 0$, $\gamma\ge 0$, and $\mu$. The potential wells only exist if $f_{-}<0$, [*i.e.*]{} if $\gamma\mu<\sigma$; Fig. 3 illustrates the typical form of the potential in this case.
Let us now carry out this limiting process in Eqs. (\[ran24\]) and (\[ran26\]). We get $q_{+}(z)\rightarrow 0$ and $\nu_{+}(z)\rightarrow1/\gamma$, [*i.e.*]{} the support of the density $\kappa(q;z)$ comes to be the interval $[0,q_{-}(z)]$. The density itself reads $$\begin{aligned}
\rule[-1ex]{0em}{6ex}\kappa(q;z)&=&
\frac{q_{-}(z)^{\nu_{-}(z)+1/\gamma}}{[q_{-}(z)-q_{-}'(z)]^{\nu_{-}(z)+1}}
{\mbox{${\cal B}$}}^{-1}\left[\nu_{-}(z),1+1/\gamma\right]\times\nonumber\\
\rule[-1ex]{0em}{6ex}&\times&
{\mbox{${\cal F}$}}^{-1}\left[\nu_{-}(z),\nu_{-}(z)+1,\nu_{-}(z)+\frac{1}{\gamma};
\frac{q_{-}(z)}{q_{-}(z)-q_{-}'(z)}\right]
\times\nonumber\\
\label{dis10}
\rule[-1ex]{0em}{6ex}&\times&
q^{1/\gamma}[q_{-}(z)-q]^{\nu_{-}(z)-1}[q-q_{-}'(z)]^{-\nu_{-}(z)-1}
\Theta\left[q;0,q_{-}(z)\right]\,\,. \end{aligned}$$ We are again interested in the small-$z$ limit of this probability density and in its moments. We shall restrict the discussion to the physically interesting case with traps, [*i.e.*]{} $\gamma\mu<\sigma$. In this case, the variable of the Gauss hypergeometric function in Eq. (\[ran26\]) tends to $1^{-}$ and we use an appropriate analytic-continuation formula [@abramowitz].
If $\mu<0$, the time-asymptotic value of the averaged density at origin is again $\lim_{t\rightarrow\infty}\left\langle P(0,0;t)\right\rangle=|\mu|$. If $\mu=0$ is zero, we observe the logarithmic decay $$\label{dis12}
\left\langle P(0,0;t)\right\rangle\stackrel{t\rightarrow\infty}{\approx}
\sigma\,\frac{1}{\log t}\,\,.$$ If $\mu>0$, the time-asymptotics is controlled by the trap-permeability parameter $\theta$. Presently, it can be written as $\theta=\mu/(\sigma-\mu\gamma)$. For $\theta\in]0,1]$, the trapping time diverges and the asymptotic velocity vanishes. More precisely, we have $$\label{dis13}
\left\langle P(0,0;t)\right\rangle\stackrel{t\rightarrow\infty}{\approx}
\frac{\Gamma^{2}\left(\displaystyle\theta+\frac{1}{\gamma}\right)}
{\Gamma^{2}\left(\displaystyle\frac{1}{\gamma}\right)\Gamma(\theta)}\,
\frac{\sigma\gamma^{2\theta}}{(\sigma-\mu\gamma)^{2\theta}}\,
\frac{1}{t^{\theta}}\,\,.$$ Finally, if $\theta\ge1$, the first moment $\lim_{z\rightarrow0^{+}}\left\langle P(z)\right\rangle$ is finite and its reciprocal gives the (self-averaging) time-asymptotic velocity: $$\label{dis14}
\lim_{t\rightarrow\infty}\lim_{\lambda\rightarrow\infty}V(\lambda;t)=
(\theta-1)\frac{\sigma-\mu\gamma}{1+\gamma}\,\,.$$
Gaussian-white-noise limit
--------------------------
It is well known [@horsthemke; @broeck1] that an appropriate limit of the dichotomic noise yields the Gaussian white noise. However, the Gaussian white noise can be also obtained as a limit of the above-described white shot noise: one simply sets $\gamma\rightarrow0^{+}$. This means that the mean weight of the $\delta$-impulses of the force tends to zero, and simultaneously their density $n_{-}=\sigma/\gamma^{2}$ increases, such that the product (density)$\times$(mean weight)$^{2}$ remains constant. The bias $\mu$ and the intensity $\sigma$ keep their values and the correlation function in Eq. (\[ran14\]) is again $2\sigma\delta(\lambda-\lambda')$. The quenched force displays an infinitely dense array of $\delta$ peaks in both directions, their weights being infinitely small. Notice that this limit can only be achieved if we start with the dichotomic random force taking two values of different signs.
The Gaussian-white-noise results simply emerge after we carry out the small-$\gamma$ limit in Eqs. (\[dis10\]). Particularly, we have $q_{-}(z)\rightarrow\infty$, [*i.e.*]{} the support of the probability density $\kappa(q;z)$ becomes the infinite interval $[0,+\infty[$. The density itself reads $$\label{dis15}
\kappa(q;z)=\frac{1}{2}\frac{z^{\theta/2}}
{\displaystyle{\mbox{${\cal K}$}}_{\theta}\left(\frac{2}{\sigma}\sqrt{z}\right)}
\frac{1}{q^{\theta+1}}
\exp\left[-\frac{1}{\sigma}\left(q+\frac{z}{q}\right)\right]
\Theta(q;0,+\infty)\,\,,$$ in accordance with the result found in Refs. [@bouchaud; @aslangulg2]. Presently, the trap-permeability parameter simply measures the ratio between the mean force and its intensity: $\theta=\mu/\sigma$.
The formula (\[dis15\]) is valid for arbitrary values of the parameters $\sigma$ and $\mu$. If $\mu<0$, we have again $\lim_{t\rightarrow\infty}\left\langle P(0,0;t)\right\rangle=|\mu|$. In the Sinai case, [*i.e.*]{} for $\mu=0$, the asymptotic behaviour is again given by Eq. (\[dis12\]). The corresponding result for the infinite line without the reflecting boundary condition at the origin is [@aslangulg2] $\left\langle P(0,0;t)\right\rangle\stackrel{t\rightarrow\infty}{\approx}
\sigma/(\log t)^{2}$. Thus the presence of the boundary slows down the decay of the disorder-averaged probability density at the origin. Having $\mu>0$ and $\theta\in]0,1[$, the asymptotic velocity vanishes and the averaged probability density at the origin decreases algebraically as $$\label{dis17}
\left\langle P(0,0;t)\right\rangle\stackrel{t\rightarrow\infty}{\approx}
\frac{\sigma^{1-2\theta}}{\Gamma(\theta)}\,\frac{1}{t^{\theta}}\,\,.$$ For example, $\theta=1/2$ yields the exact solution $\left\langle P(0,0;t)\right\rangle=1/\sqrt{\pi t}$, valid for any time. Finally, when $\theta\ge1$, the time-asymptotic disorder-averaged mean position linearly increases, the self-averaging velocity being $(\theta-1)\sigma$. The damping of the disorder-averaged probability density at the origin can be exemplified by taking $\theta=3/2$: we then get $\langle P(z)\rangle=2/(\sigma+2\sqrt{z})$, that is $$\label{dis18}
\langle P(0,0;t)\rangle=\frac{1}{\sqrt{\pi t}}-
\frac{\sigma}{2}\exp\left(\frac{1}{4}\sigma^{2}t\right)\,
{\rm erfc}\left(\frac{1}{2}\sigma\sqrt{t}\right)
\stackrel{t\rightarrow\infty}{\approx}
\frac{2}{\sigma^{2}\sqrt{\pi}}\,\frac{1}{t^{3/2}}\,\,.$$ This asymptotic behaviour should be contrasted with the exponential damping which takes place in the presence of a positive homogeneous deterministic force, as found in Subsection 2.1.
Conclusion
==========
In the present paper, a transfer-matrix-like method for solving diffusion problems in a piecewise linear random potential has been introduced. The formulae for the Green function derived in the second Section can be easily adapted to numerical simulation of the diffusive motion in [*any*]{} potential of the mentioned type. For example, the force can be assumed to be a semi-Markovian or a non-Markovian variant of the dichotomic noise [@araujo]-[@west], it can exhibit jumps of random magnitudes (kangaroo process [@feller]), etc. For any such process, our analysis is valid up to Subsection 3.3. Our subsequent choice of a Markovian dichotomic process has been dictated by a relatively direct possibility to get the asymptotic solution of the stochastic equations (\[ran5\]), (\[ran6\]).
Let us summarize the preceding discussion. The dynamical effects of the quenched disorder have been evinced by examining the varying stochastic features of a single random variable, namely, the probability density of the particle’s occurence at the origin. Having a negative mean bias, the time-asymptotic and disorder-averaged value of this quantity is proportional to the absolute value of the mean force. In the Sinai-like case, [*i.e.*]{} for the vanishing mean bias, we have given the exact asymptotic formula describing the decay of this quantity. The decay is slower than in the corresponding model without the reflecting boundary at the origin. Finally, having a positive mean bias, the particle escapes towards infinity, with a finite velocity or not, depending on the value of the trap-permeability parameter. The existence of deep traps with a long trapping time is crucial for the existence of anomalous dynamical phases.
In the present work, we have chosen to formulate the diffusion problem in the presence of reflecting boundary conditions. After a slight modification, the method can be adapted to other types of boundary conditions. For instance, taking a fixed probability density at two boundaries, our method can be used to the analysis of the stationary-flux distribution in the one-dimensional random medium [@monthus1].
We have not aimed at the exhaustive description of the particle dynamics. Instead, we have concentrated on features which can be directly related to the probability density at the origin. The detailed description of the “noise” $P(\lambda;z)$ allows, at least in principle, for an investigation of other aspects, such as the time-asymptotic thermally-averaged first moment of the particle’s position. Another example would be the disorder-averaged diffusion coefficient. Its analysis requires the calculation of the second thermally-averaged moment of the particle’s position on the one hand, and the higher-order (generally non-self-averaging) terms in the small-$z$ expansion of the first moment on the other hand. We have shown that the thermally-averaged moments obey a system of stochastic differential equations with $P(\lambda;z)$ playing the role of the “input” noise. The probabilistic description of the moments will be reported elsewhere.
Summing up, the paper presents an approximation-free study of the diffusive dynamics in an one-dimensional Markovian Poisson random potential. It provides a firm basis for the intuitive understanding of diffusion in more involved circumstances.
Acknowledgments. {#acknowledgments. .unnumbered}
----------------
One of the authors (P. Ch.) would like to thank the University Paris VII for financial support and to express his gratitude for the hospitality extended to him at the “Groupe de Physique des Solides”.
Appendix {#appendix .unnumbered}
========
Let us consider the integrals $$\begin{aligned}
\rule[-1ex]{0em}{6ex}I_{k}(z)&\stackrel{{\rm def}}{=}&
\int_{q_{+}(z)}^{q_{-}(z)}dq\,q^{k}\,\left\{
\frac{1}{[q_{-}(z)-q][q-q_{-}'(z)]}+\frac{1}{[q-q_{+}(z)][q-q_{+}'(z)]}
\right\}\times\nonumber\\
\label{app1}
\rule[-1ex]{0em}{6ex}&\times&
\left[\frac{q_{-}(z)-q}{q-q_{-}'(z)}\right]^{\nu_{-}(z)}
\left[\frac{q-q_{+}(z)}{q-q_{+}'(z)}\right]^{\nu_{+}(z)}\,\,.\end{aligned}$$ Hence $I_{0}(z)$ is equal to the integration constant $C(z)$ in Eq. (\[ran17\]). Having known $I_{0}(z)$ and $I_{k}(z)$ for some $k\ge1$, the $k$-th stationary moment $\langle Q^{k}(z)\rangle$ is simply given by the ratio $I_{k}(z)/I_{0}(z)$. In this Appendix, we want to display some intermediate steps in the calculation of the above integrals. In order to keep the following formulae in a reasonable shape, we shall write $\nu_{\pm}$ instead of the more descriptive designation $\nu_{\pm}(z)$, used in the main text. Similar remarks hold for the four $z$-dependent quantities $q_{\pm}$ and $q_{\pm}'$.
First, we notice the equality $$\label{app2}
q=\frac{(q_{-}-q)(q-q_{-}')+(q-q_{+})(q-q_{+}')}{f_{+}-f_{-}}\,\,,$$ and we introduce the substitution $x=(q-q_{+})/(q_{-}-q_{+})$. Thereupon, the integrals $I_{k}(z)$ assume the form $$\begin{aligned}
\rule[-1ex]{0em}{6ex}I_{k}(z)&=&
\frac{(q_{-}-q_{+})^{2k-1}}{(f_{+}-f_{-})^{k}}
\int_{0}^{1}dx\,\left[(1-x)(x+a)+x(x+b)\right]^{k+1}\times\nonumber\\
\label{app3}
\rule[-1ex]{0em}{6ex}&\times&
\frac{(1-x)^{\nu_{-}-1}}{(x+a)^{\nu_{-}+1}}
\frac{x^{\nu_{+}-1}}{(x+b)^{\nu_{+}+1}}\,\,,\end{aligned}$$ where $a=(q_{+}-q_{-}')/(q_{-}-q_{+})$ and $b=(q_{+}-q_{+}')/(q_{-}-q_{+})$.
Then, using the binomial theorem, we expand the $(k+1)$-th power in (\[app3\]) and write $I_{k}(z)$ as a sum of $k+2$ integrals: $$\begin{aligned}
\label{app4}
\rule[-1ex]{0em}{6ex}I_{k}(z)&=&
\frac{(q_{-}-q_{+})^{2k-1}}{(f_{+}-f_{-})^{k}}
\sum_{j=0}^{k+1}\frac{(k+1)!}{j!(k+1-j)!}J_{k,j}(z)\,\,,\\
\label{app5}
\rule[-1ex]{0em}{6ex}J_{k,j}(z)&=&\int_{0}^{1}dx\,
(1-x)^{\nu_{-}+j-1}(x+a)^{-\nu_{-}+j-1}x^{\nu_{+}+k-j}(x+b)^{-\nu_{+}+k-j}.
\,\,\,\,\,\,\end{aligned}$$ Finally, the integrals $J_{k,j}(z)$ are expressed through the Appell function ${\mbox{${\cal F}$}}_{1}$ [@appell] (the hypergeometric function of two variables [@abramowitz; @gradshteyn]). We have $$\begin{aligned}
\rule[-1ex]{0em}{6ex}
J_{k,j}(z)&=&a^{-\nu_{-}+j-1}b^{-\nu_{+}+k-j}
\frac{\Gamma(\nu_{-}+j)\Gamma(\nu_{+}+k-j+1)}{\Gamma(\nu+k+1)}\times
\nonumber\\
\label{app6}
\rule[-1ex]{0em}{6ex}&\times&
{\mbox{${\cal F}$}}_{1}\left(\alpha,\beta,\beta',\gamma;-a^{-1},-b^{-1}\right)\,\,,\end{aligned}$$ with $\alpha=\nu_{+}+k-j+1$, $\beta=\nu_{-}-j+1$, $\beta'=\nu_{+}-k+j$, $\gamma=\nu+k+1$, and $\nu=\nu_{-}+\nu_{+}$. In the small-$z$ limit, the behaviour of the result depends on the two variables $a$ and $b$; one can employ the functional relations between the Appell functions. Moreover, having a special relationship between the four parameters, the Appell function can be expressed in terms of the ordinary hypergeometric function. In our context, the case $k=0$ is exceptional due to the possibility of such reduction. Alternatively, this point can be seen by introducing the second substitution, $y=[x(1+b)]/(x+b)$. One then obtains $$\begin{aligned}
\rule[-1ex]{0em}{6ex}J_{0,0}(z)&=&
\frac{b}{a^{\nu_{-}+1}(1+b)^{\nu_{+}+1}}
\int_{0}^{1}dy\,y^{\nu_{+}}(1-y)^{\nu_{-}-1}(1+uy)^{-\nu_{-}-1}=\nonumber\\
\label{app7}
\rule[-1ex]{0em}{6ex}&=&
\frac{(1+a)^{-1}}{a^{\nu_{-}}(1+b)^{\nu_{+}}}
\frac{\Gamma(\nu_{-})\Gamma(\nu_{+}+1)}{\Gamma(\nu+1)}\,
{\mbox{${\cal F}$}}(\nu_{-},\nu_{+},\nu+1;-u)\,\,,\\
\rule[-1ex]{0em}{6ex}J_{0,1}(z)&=&
\frac{1}{ba^{\nu_{-}}(1+b)^{\nu_{+}}}
\int_{0}^{1}dy\,y^{\nu_{+}-1}(1-y)^{\nu_{-}}(1+uy)^{-\nu_{-}}=\nonumber\\
\label{app8}
\rule[-1ex]{0em}{6ex}&=&
\frac{b^{-1}}{a^{\nu_{-}}(1+b)^{\nu_{+}}}
\frac{\Gamma(\nu_{-}+1)\Gamma(\nu_{+})}{\Gamma(\nu+1)}\,
{\mbox{${\cal F}$}}(\nu_{-},\nu_{+},\nu+1;-u)\,\,,\end{aligned}$$ with $\nu=\nu_{-}+\nu_{+}$ and $u=(b-a)/[a(1+b)]$. Summing these two expressions, we acquire $$\label{app9}
I_{0}(z)=\frac{J_{0,0}(z)+J_{0,1}(z)}{q_{-}-q_{+}}=\frac{1}{\Omega}
\,\frac{1}{a^{\nu_{-}}(1+b)^{\nu_{+}}}\,
{\mbox{${\cal B}$}}(\nu_{-},\nu_{+})\,{\mbox{${\cal F}$}}(\nu_{-},\nu_{+},\nu+1;-u)\,,$$ where ${\mbox{${\cal B}$}}(x,y)$ is the Euler beta function [@abramowitz] and the factor $$\label{app10}
\Omega=\frac{n_{+}(q_{-}-q_{-}')+n_{-}(q_{+}-q_{+}')}{n_{-}+n_{+}}=
\frac{n_{+}\sqrt{4z+f_{-}^{2}}+n_{-}\sqrt{4z+f_{+}^{2}}}{n_{-}+n_{+}}$$ can be regarded as the mean value $\left\langle\sqrt{4z+\phi^{2}(\lambda)}\right\rangle$—[*c.f.*]{} Eq. (\[ran1\]). Finally, upon inserting the definitions of the $z$-dependent quantities $a=a(z)$, $b=b(z)$ and $u=u(z)$, one recovers the normalization constant (\[ran19\]) from the main text.
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Figure 1: {#figure-1 .unnumbered}
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=1.00mm
(140.00,140.00) (0.0,0.0)[(1,0)[140.0]{}]{} (140.0,0.0)[(0,1)[140.0]{}]{} (140.0,140.0)[(-1,0)[140.0]{}]{} (0.0,140.0)[(0,-1)[140.0]{}]{} (35,35)[(1,0)[84]{}]{} (120,35)[(1,0)[0]{}]{} (96.5,30)[(27.5,0)\[bl\][Coordinate $x$]{}]{} (35,91)[(1,0)[84]{}]{} (120,91)[(1,0)[0]{}]{} (96.5,86)[(27.5,0)\[bl\][Coordinate $x$]{}]{} (35,14)[(0,1)[49]{}]{} (35,64)[(0,1)[0]{}]{} (35,65)[(0,0)[Potential $U(x)$]{}]{} (35,73.5)[(0,1)[49]{}]{} (35,123.5)[(0,1)[0]{}]{} (35,124.5)[(0,0)[Force $f(x)$]{}]{} (35,77)(2,0)[43]{}[(1,0)[1]{}]{} (33,77)[(0,0)[$f_{-}$]{}]{} (33,91)[(0,0)[$0$]{}]{} (35,119)(2,0)[43]{}[(1,0)[1]{}]{} (33,119)[(0,0)[$f_{+}$]{}]{} (35,15)(0,1)[48]{}[(-1,-2)[2]{}]{} (49.3,92)[(1.6,0)\[bl\][$x_{1}$]{}]{} (77.3,92)[(1.6,0)\[bl\][$x_{2}$]{}]{} (84.3,92)[(1.6,0)\[bl\][$x_{3}$]{}]{} (105.3,92)[(1.6,0)\[bl\][$x_{4}$]{}]{} (41,94)[(1.57,0)\[bl\][$l_{1}$]{}]{} (62,94)[(1.57,0)\[bl\][$l_{2}$]{}]{} (93.5,94)[(1.57,0)\[bl\][$l_{4}$]{}]{} (111,94)[(1.57,0)\[bl\][$l_{5}$]{}]{} (49,14)(0,2)[53]{}[(0,1)[1]{}]{} (77,14)(0,2)[53]{}[(0,1)[1]{}]{} (84,14)(0,2)[53]{}[(0,1)[1]{}]{} (105,14)(0,2)[53]{}[(0,1)[1]{}]{} (49,90.5)[(0,1)[1]{}]{} (77,90.5)[(0,1)[1]{}]{} (84,90.5)[(0,1)[1]{}]{} (105,90.5)[(0,1)[1]{}]{} (35,77)[(1,0)[14]{}]{} (77,77)[(1,0)[7]{}]{} (105,77)[(1,0)[15]{}]{} (49,119)[(1,0)[28]{}]{} (84,119)[(1,0)[21]{}]{} (35,55)[(2,1)[14]{}]{} (49,62)[(1,-1)[28]{}]{} (77,34)[(2,1)[7]{}]{} (84,37.5)[(1,-1)[21]{}]{} (105,16.5)[(2,1)[14]{}]{}
Figure 2: {#figure-2 .unnumbered}
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=1.00mm
(140.00,140.00) (0.0,0.0)[(1,0)[140.0]{}]{} (140.0,0.0)[(0,1)[140.0]{}]{} (140.0,140.0)[(-1,0)[140.0]{}]{} (0.0,140.0)[(0,-1)[140.0]{}]{} (35,35)[(1,0)[84]{}]{} (120,35)[(1,0)[0]{}]{} (96.5,30)[(27.5,0)\[bl\][Coordinate $x$]{}]{} (35,91)[(1,0)[84]{}]{} (120,91)[(1,0)[0]{}]{} (96.5,86)[(27.5,0)\[bl\][Coordinate $x$]{}]{} (35,14)[(0,1)[49]{}]{} (35,64)[(0,1)[0]{}]{} (35,65)[(0,0)[Potential $U(x)$]{}]{} (35,73.5)[(0,1)[49]{}]{} (35,123.5)[(0,1)[0]{}]{} (35,124.5)[(0,0)[Force $f(x)$]{}]{} (35,77)(2,0)[43]{}[(1,0)[1]{}]{} (33,77)[(0,0)[$f_{-}$]{}]{} (33,91)[(0,0)[$0$]{}]{} (35,84)(2,0)[43]{}[(1,0)[1]{}]{} (33,84)[(0,0)[$f_{+}$]{}]{} (35,15)(0,1)[48]{}[(-1,-2)[2]{}]{} (47,92)[(1.6,0)\[bl\][$x_{1}$]{}]{} (75,92)[(1.6,0)\[bl\][$x_{2}$]{}]{} (82,92)[(1.6,0)\[bl\][$x_{3}$]{}]{} (103,92)[(1.6,0)\[bl\][$x_{4}$]{}]{} (41,94)[(1.57,0)\[bl\][$l_{1}$]{}]{} (62,94)[(1.57,0)\[bl\][$l_{2}$]{}]{} (93.5,94)[(1.57,0)\[bl\][$l_{4}$]{}]{} (111,94)[(1.57,0)\[bl\][$l_{5}$]{}]{} (49,14)(0,2)[39]{}[(0,1)[1]{}]{} (77,14)(0,2)[39]{}[(0,1)[1]{}]{} (84,14)(0,2)[39]{}[(0,1)[1]{}]{} (105,14)(0,2)[39]{}[(0,1)[1]{}]{} (49,90.5)[(0,1)[1]{}]{} (77,90.5)[(0,1)[1]{}]{} (84,90.5)[(0,1)[1]{}]{} (105,90.5)[(0,1)[1]{}]{} (35,77)[(1,0)[14]{}]{} (77,77)[(1,0)[7]{}]{} (105,77)[(1,0)[15]{}]{} (49,84)[(1,0)[28]{}]{} (84,84)[(1,0)[21]{}]{} (35,22)[(2,1)[14]{}]{} (49,29)[(4,1)[28]{}]{} (77,36)[(2,1)[7]{}]{} (84,39.5)[(4,1)[21]{}]{} (105,44.75)[(2,1)[14]{}]{}
Figure 3: {#figure-3 .unnumbered}
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(140.00,140.00) (0.0,0.0)[(1,0)[140.0]{}]{} (140.0,0.0)[(0,1)[140.0]{}]{} (140.0,140.0)[(-1,0)[140.0]{}]{} (0.0,140.0)[(0,-1)[140.0]{}]{} (35,35)[(1,0)[84]{}]{} (120,35)[(1,0)[0]{}]{} (106.5,30)[(27.5,0)\[bl\][Coordinate $x$]{}]{} (35,91)[(1,0)[84]{}]{} (120,91)[(1,0)[0]{}]{} (96.5,86)[(27.5,0)\[bl\][Coordinate $x$]{}]{} (35,14)[(0,1)[49]{}]{} (35,64)[(0,1)[0]{}]{} (35,65)[(0,0)[Potential $U(x)$]{}]{} (35,73.5)[(0,1)[49]{}]{} (35,123.5)[(0,1)[0]{}]{} (35,124.5)[(0,0)[Force $f(x)$]{}]{} (33,77)[(0,0)[$f_{-}$]{}]{} (33,91)[(0,0)[$0$]{}]{} (35,15)(0,1)[48]{}[(-1,-2)[2]{}]{} (49.3,92)[(1.6,0)\[bl\][$x_{1,2}$]{}]{} (70,92)[(1.6,0)\[bl\][$x_{3,4}$]{}]{} (84.3,92)[(1.6,0)\[bl\][$x_{5,6}$]{}]{} (105.3,92)[(1.6,0)\[bl\][$x_{7,8}$]{}]{} (41,94)[(1.57,0)\[bl\][$l_{1}$]{}]{} (62,94)[(1.57,0)\[bl\][$l_{3}$]{}]{} (79.5,94)[(1.57,0)\[bl\][$l_{5}$]{}]{} (93.5,94)[(1.57,0)\[bl\][$l_{7}$]{}]{} (114,94)[(1.57,0)\[bl\][$l_{9}$]{}]{} (49,14)(0,2)[39]{}[(0,1)[1]{}]{} (77,14)(0,2)[39]{}[(0,1)[1]{}]{} (84,14)(0,2)[39]{}[(0,1)[1]{}]{} (105,14)(0,2)[39]{}[(0,1)[1]{}]{} (35,77)[(1,0)[85]{}]{} (49,91)[(0,1)[7]{}]{} (49,99)[(0,1)[0]{}]{} (77,91)[(0,1)[28]{}]{} (77,120)[(0,1)[0]{}]{} (84,91)[(0,1)[14]{}]{} (84,106)[(0,1)[0]{}]{} (105,91)[(0,1)[21]{}]{} (105,113)[(0,1)[0]{}]{} (35,52)[(2,1)[14]{}]{} (49,59)[(0,-1)[7]{}]{} (49,52)[(2,1)[28]{}]{} (77,66)[(0,-1)[28]{}]{} (77,38)[(2,1)[7]{}]{} (84,41.5)[(0,-1)[14]{}]{} (84,27.5)[(2,1)[21]{}]{} (105,38)[(0,-1)[21]{}]{} (105,17)[(2,1)[15]{}]{}
|
---
author:
- 'D. Chicherin$^a$, E. Sokatchev$^{b}$'
title: ' Conformal anomaly of pseudo-scalar operators '
---
Introduction {#s1}
============
Conformal symmetry is a powerful constraint on the dynamics of quantum field theories. The theories with exact conformal symmetry have a vanishing beta function, e.g., ${{\cal N}}=4$ supersymmetric Yang-Mills theory (SYM). Conformal symmetry also has numerous applications in QCD, at leading order where the symmetry is unbroken, and beyond (see [@Braun:2003rp] for a review).
A key concept in conformal symmetry is that of a conformal primary operator $O(x)$ satisfying the defining condition $\mathbb{K}_\mu O(0)=0$, where $\mathbb{K}_\mu$ is the generator of special conformal transformations. The correlation functions of such operators are severely constrained by conformal invariance and by the requirement of crossing symmetry (the so-called ‘bootstrap’, see [@Poland:2018epd] for a review). In a four-dimensional field theory the operators are classified according to their parity properties into proper (parity-even) and pseudo (parity-odd) scalars, vectors, tensors, etc. In this note we study the conformal properties of pseudo-scalar operators. We show that a single operator of this type cannot be a conformal primary on its own, because of a specific conformal anomaly. This is only possible for certain anomaly-free mixtures of such operators, such as the Lagrangian of the ${{\cal N}}=4$ SYM theory. We also make contact between this new conformal anomaly and the well-known axial anomaly in non-supersymmetric theories.
The best known example of a pseudo-scalar operator is the topological term ${\Theta}= \frac{1}{2}{\epsilon}_{\mu\nu{\lambda}\rho} {\mbox{tr}}(F^{\mu\nu} F^{{\lambda}\rho}) \equiv F\tilde F$ built from the curvature $F_{\mu\nu}$ of a gauge theory, such as massless QCD/QED. Even though such theories have a non-vanishing beta function and hence broken conformal invariance, this effect will only show up at two loops (order $\sim g^4$ in the coupling). At one loop (order $\sim g^2$) the operator ${\Theta}$ may need renormalization (in the non-Abelian case) and may acquire an anomalous dimension. Yet, this is not in conflict with the renormalized operator being a conformal primary. What really makes this impossible is the parity-odd nature of the operator. Here is a simple argument explaining the point.
If an operator is a conformal primary (in a theory with a vanishing beta function), its correlation functions with other conformal operators should be conformally covariant. Examples of well behaved conformal operators are the conserved currents, e.g., electromagnetic current, energy-momentum tensor, etc. They are protected form renormalization and keep their canonical dimension. So, let us consider the three-point correlator $$\begin{aligned}
\label{1.1}
{\langle{V_\mu(x_1) V_\nu(x_2) {\Theta}(x_3) }\rangle} = {\epsilon}_{\mu\nu{\lambda}\rho} x^{\lambda}_{12} x^\rho_{13} \, F(x^2_{ij}) \,,\end{aligned}$$ where $V_\mu(x)= {\mbox{tr}}(\bar\Psi{\gamma}_\mu \Psi)$ is an electromagnetic current and $x_{ij} = x_i - x_j$. The expression on the right-hand side reflects Poincaré invariance as well as the parity property of the correlator. Indeed, ${\Theta}$ being parity-odd and $V$ being a proper vector (parity-even), the whole objects must be a pseudo-tensor of rank two, hence the presence of the Levi-Civita tensor. The further properties of the correlator, such as conservation at points 1 and 2 and its scaling dimension, have not yet been implemented in [(\[1.1\])]{}. For the purpose of our argument, we are only interested in its conformal covariance. If this object were conformal, we could choose the conformal frame $x_2=0$ and $x_3=\infty$ (after compensating the conformal weight at point 3 by an appropriate factor). Then we would be left with the single vector $x^\mu_1$, so the expression [(\[1.1\])]{} would vanish. Furthermore, if a conformal correlation function vanishes in a particular conformal frame, it vanishes in any frame. We conclude that the only way for the correlator [(\[1.1\])]{} to be conformal is to vanish identically.
Now, it is easy to see that in a theory with a gauge field and fermion matter, such as massless QED or QCD, the one-loop (order $\sim g^2$) three-point function [(\[1.1\])]{} does not vanish (see Section \[secBL3pC\] for a detailed calculation). This implies that the operator ${\Theta}$ is not a conformal primary. We could try to repair this ‘defect’ by allowing ${\Theta}$ to mix with some other operators with the same properties (pseudo-scalar of dimension 4). In QED/QCD there is such an operator, the divergence of the axial current $A_\mu = {\mbox{tr}}(\bar\Psi{\gamma}_\mu {\gamma}_5 \Psi)$. The two operators do indeed mix but only starting at two-loop level [@Larin:1993tq]. The situation improves in theories with fermion and scalar matter, such as ${{\cal N}}=4$ SYM. The latter theory has a vanishing beta function, so it makes sense to consider conformal primary operators at any perturbative level. In this case there exists another operator of the same type, the pseudo-scalar Yukawa coupling ${{\cal Y}}={\mbox{tr}}(\phi\bar\Psi{\gamma}_5 \Psi)$ where $\phi$ is the scalar matter field. As we show in Section \[secSusy\], a particular combination ${\Theta}+g{{\cal Y}}$ is indeed a well-defined conformal primary.
We can ask the question: What is the reason why the operator ${\Theta}$ cannot be a conformal primary? As noted earlier, at the lowest perturbative level this cannot be the breakdown of conformal invariance due to the beta function, nor an ultraviolet renormalization artifact. It turns out that the operator has a hidden singularity when inserted into a fermion propagator. The singularity becomes visible only if we make a conformal transformation of the Feynman diagram, and it has the effect of producing an [*anomalous conformal Ward identity*]{}. This somewhat subtle mechanism is analyzed in detail in Section \[secAnom\]. Such an anomaly is not limited to the three-point function [(\[1.1\])]{}, it is present in the correlator of ${\Theta}$ with any number of vector currents. In contrast, the anomaly does not occur for the parity-even scalar operator ${{\cal L}}= -\frac{1}{2}{\mbox{tr}}(F^{\mu\nu} F_{\mu\nu})$, which is the gauge field Lagrangian. The distinction between the scalar and pseudo-scalar operators is best seen using two-component spinor (chiral) notation (see Appendix \[AppConv\]).[^1] In it the chiral Lagrangian $L=-\frac{1}{2}{\mbox{tr}}(F^{{\alpha}{\beta}} F_{{\alpha}{\beta}})$ is complex, and its complex conjugate $\bar L = -\frac{1}{2}{\mbox{tr}}(\tilde F^{{{\dot\alpha}}{{\dot\beta}}} \tilde F_{{{\dot\alpha}}{{\dot\beta}}})$ is the anti-chiral Lagrangian. The real and imaginary parts $$\begin{aligned}
\label{}
{{\cal L}}= \frac1{2}(L+\bar L) \,, \qquad {\Theta}= \frac1{i}(L-\bar L)\end{aligned}$$ are the standard Yang-Mills Lagrangian and the topological term, respectively. The conformal anomaly occurs when inserting the chiral Lagrangian (or its anti-chiral conjugate) in a fermion line but it cancels if the real part is inserted (see Section \[secAnom\]). In a theory with scalar matter, such as ${{\cal N}}=4$ SYM, the insertion of the chiral Yukawa coupling $Y={\mbox{tr}}(\phi \psi^{\alpha}\psi_{\alpha})$ generates a similar anomaly, so that the chiral combination $L+gY$ is anomaly free. This combination (completed with the real $\phi^4$ term) happens to be the chiral on-shell Lagrangian of the ${{\cal N}}=4$ SYM theory, a component of the energy-momentum tensor supermultiplet.
Finally, in Section \[secChirAnom\] we comment on a possible link between our conformal anomaly and the well-known axial anomaly. The axial current $A_\mu$ is not conserved at loop level, therefore it acquires an anomalous dimension. The divergence ${\partial}^\mu A_\mu$ of a vector of non-canonical dimension is a conformal descendant, not a conformal primary. Since this divergence is related to the topological term ${\Theta}$ by the Adler-Bardeen theorem, the latter cannot be a conformal primary either.
Born-level correlator of two vector currents and the topological term {#s2}
=====================================================================
In this Section we calculate a three-point correlation function of gauge-invariant composite operators in the Born approximation (the lowest perturbative order) and find that it is not conformal despite of the classical conformal invariance of the underlying theory and of the composite operators. Our argument is valid for any 4D massless gauge theory involving fermions (e.g. massless QED, QCD, or super-Yang-Mills theory); non-abelian effects do not appear at this perturbative level. Only the following part of the Lagrangian is relevant in our Feynman graph calculations, $$\begin{aligned}
L_{\rm QCD} = - \frac{1}{2} {\mbox{tr}}\, F_{\mu \nu} F^{\mu \nu} + \frac{i}{2}\, \bar\Psi \gamma^\mu \overset{\leftrightarrow}{\cal D}_{\mu} \Psi \,,
\label{Lagr}\end{aligned}$$ where ${\cal D}_\mu = {\partial}_\mu - i g {{\cal A}}_\mu$ is the covariant derivative with $SU(N_c)$ gauge connection ${{\cal A}}_{\mu} = {{\cal A}}^a_{\mu} T_a$ and $F_{\mu \nu} = \frac{i}{g}[{\cal D}_\mu , {\cal D}_\nu]$ is the field-strength tensor. The color generators are normalized as ${\mbox{tr}}(T_a T_b) = \frac{1}{2}\delta_{ab}$. The Dirac spinor $\Psi_{i}$ and its conjugate $\bar\Psi^i$ are in the (anti)-fundamental representation of $SU(N_c)$, i.e. $\bar\Psi^i(T_a)_{i}{}^{j} \Psi_j$. The Lagrangian is invariant under conformal transformations classically; the UV renormalization of the fields and of the coupling constant do not appear at this perturbative level.
In view of the global $U(1)\times U(1)$ invariance of the Lagrangian [(\[Lagr\])]{} the electromagnetic vector $V_{\mu} ={\mbox{tr}}\, \bar\Psi\gamma_\mu \Psi$ and axial vector $A_{\mu} ={\mbox{tr}}\, \bar\Psi\gamma_\mu\gamma_5 \Psi$ currents are classically conserved. The vector current $V_{\mu}$ does not require infinite UV renormalizations and it is conserved at the quantum level as well. The conservation of the axial current at the quantum level is spoiled by the Adler-Bardeen anomaly which is one-loop exact.
Let us also introduce the dual field strength tensor $\tilde F_{\mu\nu} = \frac{1}{2}\epsilon_{\mu\nu\rho{\lambda}} F^{\rho{\lambda}}$ and consider the following pseudo-scalar gauge-invariant operator $$\begin{aligned}
\Theta = {\mbox{tr}}\, F_{\mu\nu} \tilde F^{\mu\nu} \,, \label{Theta}\end{aligned}$$ which is the well-know topological term, the divergence of a the gauge non-invariant Chern-Simons term.
[ccc]{}
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![Feynman diagrams for the Born-level correlator ${\langle{J_\mu(x_1)J_\nu(x_2)\Theta(x_3)}\rangle}_{g^2}$.[]{data-label="3pt"}](JJThfig1.eps "fig:"){width="4.8cm"}
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![Feynman diagrams for the Born-level correlator ${\langle{J_\mu(x_1)J_\nu(x_2)\Theta(x_3)}\rangle}_{g^2}$.[]{data-label="3pt"}](JJThfig2.eps "fig:"){width="4.2cm"}
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[c]{} ![Feynman diagrams for the Born-level correlator ${\langle{J_\mu(x_1)J_\nu(x_2)\Theta(x_3)}\rangle}_{g^2}$.[]{data-label="3pt"}](JJThfig3.eps "fig:"){width="4.2cm"}\
The currents $V_\mu$ and the pseudo-scalar $\Theta$ are classically conformally covariant operators. Let us consider their three-point correlation function in the lowest perturbative approximation $$\begin{aligned}
G^{(\rm odd)}_{\mu\nu}(x_1,x_2,x_3) ={\langle{V_\mu(x_1)\, V_\nu(x_2) \, \Theta(x_3) }\rangle}_{\rm Born} \,. \label{corrJJTh}\end{aligned}$$ The relevant Feynman diagrams are shown in Fig. \[3pt\]. In this case the lowest perturbative order is ${\cal O}(g^2)$. Despite of the fact that the diagrams involve space-time integration vertices ($x_0$ and $x_{0'}$) their sum turns out to be a rational function.[^2]
Chiral and anti-chiral Lagrangian insertions
--------------------------------------------
In the following we prefer to use two-component Lorentz spinor index notations, see App. \[AppConv\]. We decompose the Dirac fermion in a pair of Weyl fermions $\chi$ and $\psi$, $$\begin{aligned}
\Psi = (\chi_{{\alpha}} \, ,\, \bar\psi^{{{\dot\alpha}}}) \;\;\;,\;\;\;\;\;\;
\bar\Psi = (\psi^{{\alpha}} \, , \, \bar\chi_{{{\dot\alpha}}}) \,.\end{aligned}$$ The vector and axial vector currents are independent linear combinations of two Majorana currents, $$\begin{aligned}
& V_{\mu} = \psi^{{\alpha}} \sigma^\mu_{{\alpha}{{\dot\alpha}}} \bar\psi^{{{\dot\alpha}}} - \chi^{{\alpha}} \sigma^\mu_{{\alpha}{{\dot\alpha}}} \bar\chi^{{{\dot\alpha}}}= \bar\Psi{\gamma}_\mu\Psi\,, \label{2.5}\\
&A_{\mu} = \psi^{{\alpha}} \sigma^\mu_{{\alpha}{{\dot\alpha}}} \bar\psi^{{{\dot\alpha}}} + \chi^{{\alpha}} \sigma^\mu_{{\alpha}{{\dot\alpha}}} \bar\chi^{{{\dot\alpha}}} = i\bar\Psi{\gamma}_\mu{\gamma}_5\Psi \,. \label{2.6}\\end{aligned}$$ With a single Majorana spinor $\Psi^*=\Psi$ or equivalently $\chi=\psi$, only the axial vector combination [(\[2.6\])]{} is possible. For the purpose of the Feynman graph calculations in this section we use just the real current $J_{{\alpha}{{\dot\alpha}}} \equiv \sigma^{\mu}_{{\alpha}{{\dot\alpha}}} J_\mu = \psi_{{\alpha}} \bar\psi_{{{\dot\alpha}}}$ made of a single Majorana spinor $(\psi,\bar\psi)$; adding or subtracting the contribution of the other Majorana spinor $(\chi,\bar\chi)$ is straightforward.
We also split the real field strength tensor $F_{\mu\nu}$ into its chiral (or self-dual) $F_{{\alpha}{\beta}} = F_{{\beta}{\alpha}}$ and anti-chiral (or anti-self-dual) $\tilde F_{{{\dot\alpha}}{{\dot\beta}}} = \tilde F_{{{\dot\beta}}{{\dot\alpha}}}$ components $$\begin{aligned}
F_{\mu\nu}\, \sigma^{\mu}_{{\alpha}{{\dot\alpha}}} \sigma^{\nu}_{{\beta}{{\dot\beta}}} & = \epsilon_{{\alpha}{\beta}} \tilde F_{{{\dot\alpha}}{{\dot\beta}}} + \epsilon_{{{\dot\alpha}}{{\dot\beta}}} F_{{\alpha}{\beta}} \,, \\
\tilde F_{\mu\nu}\, \sigma^{\mu}_{{\alpha}{{\dot\alpha}}} \sigma^{\nu}_{{\beta}{{\dot\beta}}} & =-i \epsilon_{{\alpha}{\beta}} \tilde F_{{{\dot\alpha}}{{\dot\beta}}} +i \epsilon_{{{\dot\alpha}}{{\dot\beta}}} F_{{\alpha}{\beta}} \,,\end{aligned}$$ which are related by complex conjugation $\left[ F_{{\alpha}{\beta}} \right]^* = \tilde F_{{{\dot\alpha}}{{\dot\beta}}}$. Then both the Yang-Mills part of the Lagrangian [(\[Lagr\])]{} and the pseudo-scalar $F^{\mu\nu} \tilde F_{\mu\nu}$ split into chiral and anti-chiral pieces, $$\begin{aligned}
F_{\mu \nu} F^{\mu \nu} = \frac{1}{2} \left( F_{{\alpha}{\beta}} F^{{\alpha}{\beta}} + \tilde F_{{{\dot\alpha}}{{\dot\beta}}} \tilde F^{{{\dot\alpha}}{{\dot\beta}}} \right)\,, \\
F_{\mu \nu} \tilde F^{\mu \nu} = \frac{i}{2} \left( F_{{\alpha}{\beta}} F^{{\alpha}{\beta}} - \tilde F_{{{\dot\alpha}}{{\dot\beta}}} \tilde F^{{{\dot\alpha}}{{\dot\beta}}} \right)\,,\end{aligned}$$ related by complex conjugation. Thus, it is natural to introduce the chiral and anti-chiral forms of the YM Lagrangian $$\begin{aligned}
L = -\frac{1}{2}{\mbox{tr}}(F_{{\alpha}{\beta}})^2 \;, \quad \bar L = -\frac{1}{2}{\mbox{tr}}(F_{{{\dot\alpha}}{{\dot\beta}}})^2\,,
\label{chirYM}\end{aligned}$$ along with the real YM Lagrangian $$\begin{aligned}
{\cal L} = -\frac{1}{2}{\mbox{tr}}\, F_{\mu \nu} F^{\mu \nu} \,. \label{realYM}\end{aligned}$$ The chiral, anti-chiral and real Lagrangians differ by total derivatives, so they produce the same action $S_{\rm YM}= \int d^4x {{\cal L}}= \int d^4x L = \int d^4x \bar L $. We see that the imaginary part of the correlator with the chiral Lagrangian insertion $$\begin{aligned}
G^{(\rm chir)}_{\mu\nu}(x_1,x_2,x_3) ={\langle{J_{\mu}(x_1)\, J_{\nu}(x_2) \, L(x_3) }\rangle}_{\rm Born} \, \label{corrJJLchir}\end{aligned}$$ yields the correlator [(\[corrJJTh\])]{} involving the pseudo-scalar (or parity odd) topological term $\Theta$, $$\begin{aligned}
G^{(\rm chir)}_{\mu\nu} - \left[ G^{(\rm chir)}_{\mu\nu} \right]^* = i G^{(\rm odd)}_{\mu\nu}\,. \label{odd}\end{aligned}$$ Thus, it will be enough to evaluate the complex correlator [(\[corrJJLchir\])]{}. The Lagrangian insertions method is a powerful tool for generating multi-loop integrands of correlation functions in supersymmetric theories [@Eden:2000mv; @Eden:2010zz; @Eden:2011ku; @Eden:2012tu] and in massless QCD [@Chicherin:2020azt]. Let us note that by taking the real part of [(\[corrJJLchir\])]{} we find the real YM Lagrangian insertion [(\[realYM\])]{} in the correlator of two currents, $$\begin{aligned}
\frac{1}{2} \left( G^{(\rm chir)}_{\mu\nu} + \left[ G^{(\rm chir)}_{\mu\nu} \right]^* \right) = {\langle{J_{\mu}(x_1)\, J_{\nu}(x_2) \, {\cal L}(x_3) }\rangle}_{\rm Born} \equiv G^{(\rm even)}_{\mu\nu}(x_1,x_2,x_3)\,. \label{even}\end{aligned}$$
Now we turn to the calculation of the Feynman diagrams in Fig. \[3pt\] but with the chiral operator $L$ at point $x_3$. Even at the lowest perturbative level we have to carry out nontrivial space-time integrations.
Factorizable diagram: T-block $\times$ T-block
----------------------------------------------
The leftmost diagram in Fig. \[3pt\] factorizes in a product of two T-blocks depicted on the lhs of Fig. \[FigPsipsiF2\]. The T-block depends on three external points and involves one space-time integration vertex (here $({\beta}{\gamma})$ denotes weighted symmetrization), $$\begin{aligned}
&{\langle{\psi_{\alpha}(x_1) \bar\psi_{{\dot\alpha}}(x_2) F^{a}_{{\beta}{\gamma}}(x_3)}\rangle}_{g} = \frac{2i g\, T^a}{(2\pi)^6}
\int d^4 x_0 \, {\partial}_{{\alpha}\dot\delta}\frac{1}{x_{10}^2} {\partial}_{(\beta{{\dot\alpha}}} \frac{1}{x_{20}^2}
{\partial}^{\dot\delta}_{\gamma)}\frac{1}{x_{30}^2} \notag\\
&= \frac{4 g\, T^a}{(2\pi)^4}\left\{ \frac{(x_{12})_{{\alpha}{{\dot\alpha}}}}{x^4_{12}} \frac{(x_{31} {\tilde{x}}_{32})_{({\beta}{\gamma})}}{x^2_{13} x^2_{23}} + \frac{(x_{12} {\tilde{x}}_{23})_{{\alpha}({\beta}} (x_{32})_{{\gamma}){{\dot\alpha}}}}{x^2_{12} x^2_{13} x^4_{23}} \right\}\,.
\label{T-block}\end{aligned}$$ The rationality of the T-block [(\[T-block\])]{} follows from the ‘star-triangle’ identity $$\begin{aligned}
({\partial}_1)_{{\alpha}{{\dot\alpha}}} ({\partial}_2)^{{{\dot\alpha}}{\beta}} \int \frac{d^4 x_0}{x^2_{10} x^2_{20} x^2_{30} } = 4\pi^2 i\frac{(x_{13} {\tilde{x}}_{32})_{{\alpha}}{}^{{\beta}}}{x^2_{12} x^2_{13} x^2_{23}} \,, \label{str-trng}\end{aligned}$$ which is a consequence of the conformal covariance of this three-point integral (a Yukawa vertex).
Chiral insertion into the fermion propagator
--------------------------------------------
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![Building blocks of the correlator Feynman diagrams.[]{data-label="FigPsipsiF2"}](psibarpsiF.eps "fig:"){width="5.5cm"} ![Building blocks of the correlator Feynman diagrams.[]{data-label="FigPsipsiF2"}](psibarpsiF2.eps "fig:"){width="7cm"}
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The remaining two Feynman diagrams in Fig. \[3pt\] involve a genuine two-loop Feynman integral depicted on the rhs of Fig. \[FigPsipsiF2\]. It can be interpreted as the insertion of the chiral YM Lagrangian $L(x_3)$ in the fermion propagator ${\langle{\psi(x_1) \bar\psi(x_2)}\rangle}$, $$\begin{aligned}
&\Pi_{{\alpha}{{\dot\alpha}}}(x_1,x_2,x_3) \equiv {\langle{\psi_{\alpha}(x_1) \bar\psi_{{\dot\alpha}}(x_2)\, {\mbox{tr}}\, (F_{{\beta}{\gamma}}^2)(x_3)}\rangle}_{g^2} \notag\\
& = -\frac{4i g^2 C_F}{(2\pi)^{10}} \int d^4 x_0 d^4 x_{0'}\, {\partial}_{{\alpha}{{\dot\beta}}} \frac{1}{x_{10}^2} {\partial}_{({\beta}{{\dot\alpha}}} \frac{1}{x_{20'}^2} {\partial}_{\gamma) \dot\gamma} \frac{1}{x_{30'}^2} {\partial}^{\dot\gamma(\gamma} \frac{1}{x_{00'}^2} {\partial}^{{{\dot\beta}}\beta)} \frac{1}{x_{30}^2}\,.\label{eq1.15}\end{aligned}$$ We carry out one of the integrations, e.g. at point $x_{0'}$, in four dimensions by means of the star-triangle identity [(\[str-trng\])]{}. Then after simplifying the resulting integrand we arrive at $$\begin{aligned}
\Pi_{{\alpha}{{\dot\alpha}}}(x_1,x_2,x_3) = \frac{g^2 C_F}{(2\pi)^8}({\partial}_{1})_{{\alpha}{{\dot\beta}}}({\partial}_2)_{{\beta}{{\dot\alpha}}} & \biggl[ -3 {\partial}_{3}^{{{\dot\beta}}{\beta}} \int \frac{d^D x_0}{x_{10}^2 x_{20}^2 x_{30}^4} \notag\\
&+ \frac{(x_{23})_{\gamma\dot\gamma}}{2 x_{23}^2} \left( {\partial}_3^{\dot\gamma\beta} {\partial}_3^{{{\dot\beta}}\gamma} + {\partial}_3^{\dot\gamma\gamma} {\partial}_3^{{{\dot\beta}}{\beta}} \right) \int \frac{d^D x_0}{x_{10}^2 x_{20}^2 x_{30}^2} \biggr] \label{propcorDimReg}\end{aligned}$$ where we temporarily introduced dimensional regularization with $D=4-2{\epsilon}$. We expect the regular part of the Feynman integral [(\[eq1.15\])]{} to be finite (see below). Keeping $D=4$, in the first term we can write ${\partial}_3=-({\partial}_1+{\partial}_2)$ and $({\partial}_{1})_{{\alpha}{{\dot\beta}}}({\partial}_2)_{{\beta}{{\dot\alpha}}} ({\partial}_{3})^{{{\dot\beta}}{\beta}} = -\Box_1 ({\partial}_2)_{{\alpha}{{\dot\alpha}}} -\Box_2 ({\partial}_1)_{{\alpha}{{\dot\alpha}}} $. Taking into account that $\Box_{1} \frac{1}{x_{10}^{2}} = 4i\pi^2 \delta^{(4)}(x_{10})$ we lift the integration by these delta functions. The second integral in [(\[propcorDimReg\])]{} is done again by the star-triangle identity [(\[str-trng\])]{} with respect to points 1 and 3. Thus, the two space-time integrations in [(\[propcorDimReg\])]{} result in rational functions and we obtain $$\begin{aligned}
\Pi_{{\alpha}{{\dot\alpha}}}(x_1,x_2,x_3) =- \frac{4 i g^2 C_F}{(2\pi)^{6}} \frac{1}{x_{13}^4 x_{23}^4}\left\{ (x_{23})_{{\alpha}{{\dot\alpha}}} + \frac{(x_{12})_{{\alpha}{{\dot\alpha}}}}{2 x_{12}^4} \left[ -x_{13}^4 - x_{23}^4 + 2 x_{13}^2 x_{23}^2 + 4 x_{12}^2 x_{13}^2\right]\right\}.
\label{propCorr}\end{aligned}$$ We note that by hitting the Feynman integral [(\[eq1.15\])]{} with the Dirac operator ${\partial}_1^{{{\dot\beta}}{\alpha}}$, we lift the space-time integration at one of the interaction vertices and the remaining space-time integration is again reduced to the star-triangle identity [(\[str-trng\])]{}. Thus we derive a differential equation of the form ${\partial}^{{{\dot\alpha}}{\alpha}}_1 \Pi_{{\alpha}{{\dot\alpha}}} =$ ‘known rational function’. One can easily check that [(\[propCorr\])]{} satisfies this DE. An analogous DE with respect to $x_2$ is satisfied as well.
The regular part of [(\[eq1.15\])]{} is free from UV divergences, but they appear in the form of contact terms omitted in [(\[propCorr\])]{}. Indeed, the only possible source of divergences in [(\[propcorDimReg\])]{} is the integration region $x_0 \sim x_3$ in the first term. In order to extract the ${\epsilon}$-pole of the diagram, we expand the singular $D-$dimensional distribution $$\begin{aligned}
\frac{1}{x^4} \to \frac{i\pi^2}{{\epsilon}} \delta^{(4)}(x) + {\cal O}({\epsilon}^0) \label{delta}\end{aligned}$$ with $x = x_{30}$ in eq. [(\[propcorDimReg\])]{}, and find $$\begin{aligned}
\Pi_{{\alpha}{{\dot\alpha}}} (x_1,x_2,x_3)
= \frac{3 g^2 C_F}{2(2\pi)^4}\frac{1}{\epsilon} \left( -\delta(x_{13}) + \delta(x_{23}) \right) \frac{(x_{12})_{{\alpha}{{\dot\alpha}}}}{x_{12}^4}+ {\cal O}({\epsilon}^0) \,. \label{contact}\end{aligned}$$
Concluding this subsection we mention that the integration over the insertion point $x_3$ gives the one-loop correction to the fermion propagator: $$\begin{aligned}
\label{}
\lim_{{\epsilon}\to0} \int d^{D}x_3\, \Pi_{{\alpha}{{\dot\alpha}}}(x_1,x_2,x_3) = -\frac{3 g^2 C_F}{(2\pi)^{4}} \frac{(x_{12})_{{\alpha}{{\dot\alpha}}}}{x^4_{12}}\,.\end{aligned}$$ Note that the poles in the contact terms [(\[contact\])]{} cancel out and the correction is finite. This result differs from the familiar infinite propagator correction in the Feynman gauge or vanishing correction in the Landau gauge (see, e.g., [@Grozin:2005yg]). The explanation is that this quantity is not only gauge but also scheme dependent; our Lagrangian insertion procedure constitutess a different scheme.
Born-level three-point correlator {#secBL3pC}
---------------------------------
Using the expressions for the building blocks [(\[T-block\])]{} and [(\[propCorr\])]{} of the Feynman diagrams in Fig. \[3pt\], we find the Born-level correlator [(\[corrJJLchir\])]{} of two currents and the chiral YM Lagrangian $L$ [(\[chirYM\])]{}, $$\begin{aligned}
\label{JJLdiagr}
G^{(\rm chir)}_{{\alpha}{{\dot\alpha}}{\beta}{{\dot\beta}}}(x_1,x_2,x_3) = -\frac{4g^2 C_F}{(2\pi)^8} & \biggl[ \frac{(x_{12})_{{\alpha}{{\dot\beta}}} (x_{21})_{{\beta}{{\dot\alpha}}}}{x_{12}^4 x_{13}^4 x_{23}^4} + \frac{2(x_{13} \tilde x_{32})_{{\alpha}{\beta}} (\tilde x_{13} x_{32})_{{{\dot\alpha}}{{\dot\beta}}}}{x_{12}^2 x_{13}^6 x_{23}^6} \notag\\
&+ \frac{3{\epsilon}_{\alpha\beta}(\tilde x_{13} x_{32})_{\dot\alpha\dot\beta}}{x_{12}^4 x_{13}^4 x_{23}^4} + \frac{3{\epsilon}_{\dot\alpha\dot\beta}(x_{13} \tilde x_{32})_{\alpha\beta}}{x_{12}^4 x_{13}^4 x_{23}^4} \biggr].\end{aligned}$$ [One can easily check that the correlator satisfies current conservation at points $x_1$ and $x_2$.]{}
The Lorentz spinor notation makes it obvious that the first line in [(\[JJLdiagr\])]{} is real and conformally covariant, whereas the second line is imaginary and it breaks the conformal symmetry, see [(\[CC\])]{}, [(\[CT\])]{}.[^3] According to eq. [(\[odd\])]{} the latter corresponds to the correlator with the pseudo-scalar insertion $\Theta$ [(\[corrJJTh\])]{},[^4] $$\begin{aligned}
G^{(\rm odd)}_{\mu\nu}(x_1,x_2,x_3) \equiv {\langle{J_\mu(x_1)\,J_\nu(x_2)\, \Theta(x_3)}\rangle}_{\rm Born} = -\frac{24 g^2 C_F}{(2\pi)^8} \frac{{\epsilon}_{\mu\nu{\lambda}\rho} x^{\lambda}_{13} x^\rho_{23}}{x_{12}^4 x_{13}^4 x_{23}^4} \,.\label{LeviChivita}\end{aligned}$$ One can easily see that [(\[LeviChivita\])]{} satisfies the current conservation conditions ${\partial}^{\mu}_1 G^{(\rm odd)}_{\mu\nu} = {\partial}^{\nu}_2 G^{(\rm odd)}_{\mu\nu} = 0$ (up to contact terms). It is a finite[^5] rational function, which is not conformal. Thus the conformal symmetry is broken already at the lowest perturbative level.
On the contrary, the real part of [(\[JJLdiagr\])]{} yields the real Lagrangian insertion [(\[even\])]{} which is explicitly conformal, $$\begin{aligned}
G^{(\rm even)}_{\mu\nu}(x_1,x_2,x_3) = \frac{2g^2 C_F}{(2\pi)^8}\frac{1}{x_{13}^4 x_{23}^4}\left[ \frac{1}{x_{12}^2} I_{\mu\nu}(x_{12}) - 4 Z_\mu(x_1|x_2,x_3) Z_\nu (x_2|x_1,x_3) \right]. \end{aligned}$$ Here we employ the familiar conformal tensors (see, e.g., [@Schreier:1971um; @Erdmenger:1996yc]) $$\begin{aligned}
\label{2.27}
I^{\mu\nu}(x_{12}) = \eta^{\mu\nu} - \frac{2 x_{12}^\mu x_{12}^\nu}{x_{12}^2} \,,\qquad
Z_\mu (x_1|x_2,x_3) = \frac{x_{12}^\mu}{x_{12}^2} - \frac{x_{13}^\mu}{x_{13}^2} \,.\end{aligned}$$
Anomalous conformal Ward identity {#secAnom}
=================================
In the previous Section we verified by an explicit Feynman diagram calculation that the three-point correlator [(\[corrJJTh\])]{} involving the pseudo-scalar topological term $\Theta$ is not conformal already at the Born level. The interaction vertices, operator vertices and propagators respect the conformal symmetry, since they originate from the conformal Lagrangian [(\[Lagr\])]{}. The space-time integrations do not introduce divergences. Thus, we would naively expect unbroken conformal symmetry at this perturbative level. Now we explore in more details the mechanism leading to the conformal symmetry breaking and calculate the corresponding anomaly.
The breakdown of conformal symmetry could only come from hidden singularities in the Feynman integrals. They are revealed when we perform a conformal variation under the sign of the integral in dimensional regularization with $D =4-2{\epsilon}$ [@Braun:2003rp; @Drummond:2007au]. More precisely, we modify only the dimension of the measure but not that of the fields and coupling constant. Due to the mismatch of the conformal weights of the $D$-dimensional measure and the four-dimensional Lagrangian, we find the conformal variation (see [(\[Kboost\])]{}) of the action $$\begin{aligned}
\mathbb{K}^{\lambda}\int d^D x\, L_{\rm QCD}(x) = 2i(D- \Delta_{L}) \int d^D x \, x^{\lambda}\, L_{\rm QCD}(x) \,,\end{aligned}$$ where $\Delta_L =4$ is the conformal weight of the Lagrangian [(\[Lagr\])]{}. The variation is of order ${\cal O}({\epsilon})$. Thus, the conformal variation of a Feynman diagram amounts to inserting ${\epsilon}x^{{\lambda}}$ in the interaction vertices and promoting the space-time integrations to $D$ dimensions. If this modification leads to a UV (i.e., short distance) divergence in the space-time integrals, i.e. an ${\epsilon}$-pole, then the variation of the Feynman diagram is of $O({\epsilon}^0)$ as ${\epsilon}\to 0$ and the conformal symmetry is anomalous. The corresponding anomalous conformal Ward identity takes the form $$\begin{aligned}
&\mathbb{K}^{{\lambda}} \,{\langle{J_{\mu}(x_1) \, J_\nu(x_2)\, L(x_3)}\rangle}_{\rm Born} \notag\\
&= 4\lim_{{\epsilon}\to 0} {\epsilon}\int d^D x_0\, x_0^{\lambda}{\langle{J_{\mu}(x_1) \, J_\nu(x_2)\, L(x_3)\, L_{\rm QCD}(x_0)}\rangle}_{\rm Born} \,. \label{CAWI}\end{aligned}$$
In order to evaluate the rhs of the Ward identity we inspect the Feynman diagrams in Fig. \[3pt\] and successively insert an extra factor $x_0$ in the interaction vertices. The insertion of $x_0$ in the T-block integrations [(\[T-block\])]{} does not create an ${\epsilon}$-pole, so we can ignore the first diagram. However, the insertion of $x_0$ or $x_0'$ into the propagator correction type diagrams in Fig. \[3pt\] does create a UV divergence. Thus, only the Feynman integral $\Pi_{{\alpha}{{\dot\alpha}}}$ [(\[eq1.15\])]{} is responsible for the conformal anomaly. There are two contributions to the conformal anomaly: (i) insertion of $x_0^{\lambda}(\sigma_{\lambda})^{\dot\gamma\gamma}=x_0^{\dot\gamma\gamma}$ in the left interaction vertex in Fig. \[FigPsipsiF2\]; (ii) insertion of $x_{0'}^{\dot\gamma\gamma}$ in the right interaction vertex in Fig. \[FigPsipsiF2\]. In the following we ignore the regular part of the conformal variation, which is guaranteed to cancel in the sum of all diagrams, and keep only the anomalous part, which contributes to the rhs of [(\[CAWI\])]{}. In order to evaluate the first anomalous contribution we simplify the fermion propagator correction $\Pi_{{\alpha}{{\dot\alpha}}}$ [(\[eq1.15\])]{} as in [(\[propcorDimReg\])]{} and do the insertion of $x_0$ only in the first term in [(\[propcorDimReg\])]{}. The source of the UV pole is the singular distribution $1/x^4_{30}$ (see [(\[delta\])]{}). We find $$\begin{aligned}
\label{104}
\left[ \mathbb{K}_{\gamma\dot\gamma} \Pi_{{\alpha}{{\dot\alpha}}} \right]_{\rm anom\,1} &=-\lim_{\epsilon \to 0} 4i{\epsilon}\,
({\partial}_{1})_{{\alpha}{{\dot\beta}}}({\partial}_2)_{\beta{{\dot\alpha}}} {\partial}_3^{{{\dot\beta}}{\beta}} \frac{g^2 C_F }{(2\pi)^8}\int d^D x_0 \frac{ -3(x_{0})_{\gamma\dot\gamma}}{x_{10}^2 x_{20}^2 }\left[ \frac{i\pi^2}{{\epsilon}} \delta^{(4)}(x_{30}) + {\cal O}({\epsilon}^0) \right] \notag\\
&= -\frac{24 g^2 C_F}{(2\pi)^6} \frac{(x_{13})_{{\alpha}\dot\gamma}(x_{23})_{\gamma{{\dot\alpha}}}}{x_{13}^4 x_{23}^4} \,.\end{aligned}$$ We recall that without the insertion of $(x_{0})_{\gamma\dot\gamma}$ the integral produces the contact term [(\[contact\])]{}, which indicates the hidden singularity.
The second anomaly contribution is obtained by first integrating out the $x_0$ vertex in [(\[eq1.15\])]{} by means of the star-triangle identity [(\[str-trng\])]{}, and then rewriting the resulting integral with dimensional regularization (we relabel $x_{0'}$ to $x_0$): $$\begin{aligned}
\Pi_{{\alpha}{{\dot\alpha}}} = -\frac{2g^2 C_F}{(2\pi)^8} \int d^D x_0 \biggl[
\frac{3 (x_{10})_{{\alpha}{{\dot\beta}}}}{x_{13}^2 x_{10}^2} \, {\partial}_{{\beta}{{\dot\alpha}}} x_{20}^{-2} \,
{\partial}^{{{\dot\beta}}{\beta}} x_{30}^{-4}
+ \frac{(x_{13})^{{{\dot\beta}}(\delta}}{x_{13}^2} \left( {\partial}^{{\beta})\dot\delta} \frac{(x_{10})_{{\alpha}{{\dot\beta}}}}{x_{10}^2}\right) {\partial}_{{\beta}{{\dot\alpha}}} x_{20}^{-2}{\partial}_{\delta\dot\delta} x_{30}^{-4}
\biggr].\end{aligned}$$ One can easily verify that the integral is finite (up to contact terms). Inserting $x_0^{\dot\gamma\gamma}$ into it and extracting the ${\epsilon}$-pole by means of [(\[delta\])]{}, we find the second anomaly contribution $$\begin{aligned}
\left[ \mathbb{K}_{\gamma\dot\gamma} \Pi_{{\alpha}{{\dot\alpha}}} \right]_{\rm anom\,2} = -\frac{24 g^2 C_F}{(2\pi)^6}\frac{(x_{13})_{{\alpha}\dot\gamma}(x_{23})_{\gamma{{\dot\alpha}}}}{x_{13}^4 x_{23}^4} \,. \label{anom2}\end{aligned}$$ Collecting the anomalous contributions [(\[104\])]{} and [(\[anom2\])]{}, we derive the conformal anomaly of $G^{(\rm chir)}$ written in spinor notation [(\[JJLdiagr\])]{}, $$\begin{aligned}
\label{3.6}
\mathbb{K}^{\dot\gamma\gamma}\, (G^{(\rm chir)})^{{\alpha}{{\dot\alpha}}{\beta}{{\dot\beta}}} = \frac{24 i g^2 C_F}{(2\pi)^8} \frac{1}{x_{12}^4 x_{13}^4 x_{23}^4} \left[ x_{32}^{\dot\gamma\beta} x_{13}^{{{\dot\alpha}}\gamma} x_{21}^{{{\dot\beta}}{\alpha}} + x_{31}^{\dot\gamma{\alpha}} x_{23}^{{{\dot\beta}}\gamma} x_{12}^{{{\dot\alpha}}{\beta}} \right]\,.\end{aligned}$$
We recall that this anomaly originates from the insertion of the chiral Lagrangian [(\[corrJJLchir\])]{} into the two-point function of two real vector currents. Repeating the whole procedure with the anti-chiral Lagrangian, we get the [ complex conjugat ]{}of [(\[3.6\])]{}. In this way we see that the anomalies cancel for the real (parity even) Lagrangian insertion [(\[even\])]{}, while they add up for the imaginary (parity odd) insertion [(\[LeviChivita\])]{}. Converting [(\[3.6\])]{} to the vector notation, we find the conformal anomaly of [(\[LeviChivita\])]{} $$\begin{aligned}
\label{1011}
&\mathbb{K}_{\lambda}\, {\langle{J_{\mu}(x_1)\, J_{\nu}(x_2)\, \Theta(x_3)}\rangle}_{\rm Born} = \frac{12 i g^2 C_F}{(2\pi)^8}\frac{1}{x_{12}^4 x_{13}^4 x_{23}^4}\, \biggl[(x^2_{13} +x^2_{23} - x_{12}^2) {\epsilon}_{\mu\nu{\lambda}\rho} x^\rho_{12} \notag\\
&+ 2 (x_{13}+x_{23})_{\lambda}{\epsilon}_{\mu\nu\kappa\tau} x^\kappa_{13} x^\tau_{23} - 2 (x_{12})_\mu \epsilon_{\nu{\lambda}\kappa\tau} x_{13}^\kappa x_{23}^\tau -2 (x_{12})_\nu \epsilon_{\mu{\lambda}\kappa\tau} x_{13}^\kappa x_{23}^\tau \biggr].\end{aligned}$$ We have checked by an explicit calculation that the rhs of [(\[LeviChivita\])]{} does indeed verify the anomalous conformal Ward identity [(\[1011\])]{}.
In summary, we have shown that the conformal anomaly is due to hidden singularities in the Feynman integrals. The integral itself is finite (up to contact terms) but a conformal transformation changes the balance of powers and causes an UV divergence. Its effect is a non-vanishing conformal variation.
Conformal anomaly of the Yukawa operators
=========================================
The subtle effect of conformal symmetry breaking by pseudo-scalars operators is not limited to gauge theories and to the topological term ${\Theta}={\mbox{tr}}\,F_{\mu\nu}\tilde F^{\mu\nu}$. In this Section we show another, even simpler example of conformal symmetry breaking by the chiral Yukawa vertex in four dimensions.
Let us consider a massless complex scalars $\varphi$ coupled to a massless (anti)chiral fermion $\psi (\bar\psi)$ described by the real Lagrangian $$\begin{aligned}
L_{\rm Yuk} = {\partial}^{\mu} \bar \varphi \,{\partial}_{\mu} \varphi + \frac{i}{2} \psi^{\alpha}\overset\leftrightarrow{{\partial}}_{{\alpha}{{\dot\alpha}}} \bar\psi^{{\dot\alpha}}+ g \,\varphi\, \psi^{\alpha}\psi_{\alpha}+ g\, \bar\varphi \, \bar\psi_{{\dot\alpha}}\bar\psi^{{\dot\alpha}}\,. \label{YukLagr}\end{aligned}$$ The Lagrangian is classically conformal. Global $U(1)$ transformations are generated by the classically conserved current $$\begin{aligned}
J_{{\alpha}{{\dot\alpha}}} = \psi_{{\alpha}} \bar\psi_{{{\dot\alpha}}} - \frac{i}{2} \varphi \,{\partial}_{{\alpha}{{\dot\alpha}}} \bar\varphi + \frac{i}{2} \bar\varphi\, {\partial}_{{\alpha}{{\dot\alpha}}} \varphi\,, \qquad {\partial}^{{{\dot\alpha}}{\alpha}} J_{{\alpha}{{\dot\alpha}}} = 0 \,. \label{curr}\end{aligned}$$ The Lagrangian comprises chiral and antichiral Yukawa vertices, $$\begin{aligned}
Y = g \,\varphi\, \psi^{\alpha}\psi_{\alpha}\;,\qquad
\overline{Y} = g\, \bar\varphi \, \bar\psi_{{\dot\alpha}}\bar\psi^{{\dot\alpha}}\,, \label{YYbar}\end{aligned}$$ which are related by complex conjugation. We can form a real (it appears in the Lagrangian [(\[YukLagr\])]{}) and an imaginary combinations out of them, $$\begin{aligned}
\widehat{Y} \equiv Y + \overline{Y} \;,\qquad
\widetilde Y \equiv Y - \overline{Y} = g \,\varphi\, \psi^{\alpha}\psi_{\alpha}- g\, \bar\varphi \, \bar\psi_{{\dot\alpha}}\bar\psi^{{\dot\alpha}}\,, \label{YY}\end{aligned}$$ which are parity even and odd, respectively.[^6]
-------------------------------------------------------------------------------------------------------------------------- --
![Insertion of the chiral $Y$ in the fermion propagator.[]{data-label="FigPsiPsiY"}](psibarpsiY.eps "fig:"){width="7cm"}
-------------------------------------------------------------------------------------------------------------------------- --
Now we consider the three-point correlator of two currents [(\[curr\])]{} and a chiral Yukawa operator, $$\begin{aligned}
{\langle{J_{\mu}(x_1)\, J_{\nu}(x_2)\, Y(x_3)}\rangle}_{\rm Born} \,. \label{JJYL}\end{aligned}$$ In the Born approximation it is a rational function of order ${\cal O}(g^2)$. There are several Feynman diagrams contributing to [(\[JJYL\])]{}. We restrict our attention to the diagrams with the insertion of $Y$ into the fermion propagator (see Fig. \[FigPsiPsiY\]). The remaining diagrams contributing to [(\[JJYL\])]{} are conformal[^7] and can be easily evaluated by means of the star-triangle identity [(\[str-trng\])]{}. We find $$\begin{aligned}
& \Pi_{{\alpha}{{\dot\alpha}}} \equiv {\langle{\psi_{{\alpha}}(x_1)\, \bar\psi_{{{\dot\alpha}}}(x_2)\,Y(x_3)}\rangle}_{g^2} \notag\\
& = -\frac{4g^2}{(2\pi)^{8}} {\partial}_{{\beta}{{\dot\alpha}}} x_{32}^{-2} \int d^4 x_0 \, x_{30}^{-2}\, {\partial}^{{{\dot\beta}}{\beta}} x_{30}^{-2} \, {\partial}_{{\alpha}{{\dot\beta}}} x_{10}^{-2} = \frac{4g^2}{(2\pi)^6} \frac{(x_{23})_{{\alpha}{{\dot\alpha}}}}{x_{13}^4 x_{23}^4}\,, \label{propY}\end{aligned}$$ where we used the identity $x_{30}^{-2}\, {\partial}^{{{\dot\beta}}{\beta}} x_{30}^{-2} = \frac{1}{2}{\partial}^{{{\dot\beta}}{\beta}} x_{30}^{-4}$ and integrated by parts, thus producing $\Box x_{10}^{-2} = 4 i \pi^2 \delta^{(4)}(x_{10})$. The complex conjugate gives the antichiral insertion $\overline{Y}$ into the fermion propagator, $$\begin{aligned}
\label{4.7}
{\langle{\psi_{{\alpha}}(x_1)\, \bar\psi_{{{\dot\alpha}}}(x_2)\,\overline{Y}(x_3)}\rangle}_{g^2} = -\frac{4i g^2}{(2\pi)^6} \frac{(x_{13})_{{\alpha}{{\dot\alpha}}}}{x_{13}^4 x_{23}^4}\,.\end{aligned}$$ Combining the two insertions we find that the real one $\widehat{Y}$ is conformal, $$\begin{aligned}
\label{4.8}
{\langle{\psi_{{\alpha}}(x_1)\, \bar\psi_{{{\dot\alpha}}}(x_2)\,\widehat{Y}(x_3)}\rangle}_{g^2}= -\frac{4ig^2}{(2\pi)^6} \frac{(x_{12})_{{\alpha}{{\dot\alpha}}}}{x_{13}^4 x_{23}^4}\,,\end{aligned}$$ while the imaginary one $\widetilde Y$ is not. Thus, the Born-level three-point correlator involving the parity-even operator $\widehat{Y}$ is conformal, $$\begin{aligned}
\mathbb{K}_{\lambda}\,{\langle{J_{\mu}(x_1)\, J_{\nu}(x_2)\, \widehat{Y}(x_3)}\rangle}_{\rm Born} = 0 \,, \label{confJJYhat}\end{aligned}$$ while the correlator ${\langle{J_\mu J_\nu \widetilde{Y}}\rangle}$ involving the pseudo-scalar $\widetilde Y$ is not conformal.
The mechanism of conformal symmetry breaking is the same as in the gauge sector in Section \[secAnom\]: A singular distribution $1/x^4$ in the integral at the interaction vertex in Fig. \[FigPsiPsiY\] causes a pole which results in an anomaly term in the conformal variation. For the chiral insertion $Y$, the analog of eq. [(\[CAWI\])]{} takes the following form $$\begin{aligned}
& \mathbb{K}^{\lambda}\,{\langle{J_{\mu}(x_1)\, J_{\nu}(x_2)\, Y (x_3)}\rangle}_{\rm Born} = \notag\\
&= 4\lim_{{\epsilon}\to 0} {\epsilon}\int d^D x_0\, x_0^{\lambda}{\langle{J_{\mu}(x_1) \, J_\nu(x_2)\, Y(x_3)\, L_{\rm Yuk}(x_0)}\rangle}_{\rm Born} \,. \label{CAWI2}\end{aligned}$$ Only the propagator correction type diagram contributes to the anomaly. The remaining diagrams are conformal and can be ignored. Then according to [(\[CAWI2\])]{} the anomalous part of the conformal variation of $\Pi_{{\alpha}{{\dot\alpha}}}$ [(\[propY\])]{} is given by $$\begin{aligned}
\left[ \mathbb{K}^{\dot\gamma\gamma} \, \Pi_{{\alpha}{{\dot\alpha}}} \right]_{\rm anom} = \frac{16i g^2}{(2\pi)^8}\lim_{{\epsilon}\to 0} {\epsilon}\int d^D x_0 \, x_{0}^{\dot\gamma\gamma} x_{30}^{-2} {\partial}^{{{\dot\beta}}{\beta}} x_{30}^{-2} {\partial}_{{\alpha}{{\dot\beta}}} x_{10}^{-2} {\partial}_{{\beta}{{\dot\alpha}}} x_{32}^{-2}\,.\end{aligned}$$ In order to extract the pole we proceed as in [(\[propY\])]{} and replace $x_{30}^{-4}$ by its pole part [(\[delta\])]{}, $$\begin{aligned}
\left[ \mathbb{K}^{\dot\gamma\gamma} \, \Pi_{{\alpha}{{\dot\alpha}}} \right]_{\rm anom} & = \frac{8ig^2}{(2\pi)^8}\lim_{{\epsilon}\to 0} {\epsilon}\int d^4 x_0\, {\partial}^{{{\dot\beta}}{\beta}} \left[ x_{0}^{\dot\gamma\gamma} {\partial}_{{\alpha}{{\dot\beta}}} x_{10}^{-2} \right] \frac{i\pi^2}{{\epsilon}} \delta^{(4)}(x_{30})\, {\partial}_{{\beta}{{\dot\alpha}}} x_{32}^{-2} \notag\\
& = -\frac{16g^2}{(2\pi)^6}\frac{(x_{13})_{{\alpha}}^{\dot\gamma} (x_{32})_{{{\dot\alpha}}}^{\gamma}}{x_{13}^4 x_{23}^4} \,,\end{aligned}$$ where we omitted the contact term. Then the anomalous conformal Ward identity for the three-point correlator with the chiral $Y$ takes the following form $$\begin{aligned}
&\mathbb{K}^{\dot\gamma\gamma} \, {\langle{J^{{\alpha}{{\dot\alpha}}}(x_1)\, J^{{\beta}{{\dot\beta}}}(x_2)\, Y (x_3)}\rangle}_{\rm Born} \notag\\
&= \frac{16 i g^2}{(2\pi)^8} \frac{1}{x_{12}^4 x_{13}^4 x_{23}^4} \left[ x_{21}^{{{\dot\beta}}{\alpha}} x_{13}^{{{\dot\alpha}}\gamma} x_{32}^{\dot\gamma{\beta}} + x_{12}^{{{\dot\alpha}}{\beta}} x_{31}^{\dot\gamma{\alpha}} x_{23}^{{{\dot\beta}}\gamma} \right]\,. \label{JJYanom}\end{aligned}$$ The anomaly of ${\langle{J_{\mu} J_{\nu} \overline{Y} }\rangle}$ has the opposite sign, so the two anomalies cancel in the real combination $\widehat{Y}$ and double in the pseudo-scalar combination $\widetilde Y$. Remarkably, the expression for the anomaly [(\[JJYanom\])]{} is identical (up to normalization) with the anomaly in the gauge sector [(\[3.6\])]{}.
Conformal anomaly cancellation in ${{\cal N}}=4$ SYM {#secSusy}
====================================================
So far we have observed that the pseudo-scalar operators $\Theta$ [(\[Theta\])]{} and $\widetilde Y$ [(\[YY\])]{} from the gauge and Yukawa sectors, respectively, break the conformal symmetry at the lowest perturbative level. Now we are going to show that supersymmetry helps to restore conformal symmetry.
We consider the maximally supersymmetric ${{\cal N}}=4$ Yang-Mills theory. It comprises: gauge bosons, pseudo-real scalars $\phi_{AB}=\frac1{2} {\epsilon}_{ABCD} \bar\phi^{CD}$ in the antisymmetric ${\bf 6}$ representation of the R-symmetry $SU(4)$, (anti)chiral fermions $\psi^A_{{\alpha}}$ and $\bar\psi_A^{{{\dot\alpha}}}$ in the (anti)fundamental representation of $SU(4)$. All the fields are massless and they transform in the adjoint representation of the color group $SU(N_c)$. The theory is conformal at the quantum level, i.e. $\beta(g) = 0$ to all orders in the coupling. The gauge-invariant composite operators form multiplets of supersymmetry. The stress-tensor multiplet is of particular interest, being a protected half-BPS multiplet. The operators in the multiplet are not renormalized and keep their canonical dimensions. The multiplet contains all the conserved currents of the theory, as well as the chiral on-shell Lagrangian $$\begin{aligned}
\label{A.11}
L_{{\cal N} = 4}
& = {\mbox{tr}}\left\{- \frac12 F_{\alpha\beta}F^{\alpha\beta} + {\sqrt{2}} g \psi^{\alpha A} [\phi_{AB},\psi_\alpha^B] - \frac18 g^2 [\phi^{AB},\phi^{CD}][\phi_{AB},\phi_{CD}] \right\} \end{aligned}$$ and its anti-chiral conjugate. The real part of $L_{{\cal N} = 4}$ is a conformal parity-even scalar. Its imaginary part is a parity-odd scalar including the gauge sector pseudo-scalar $\Theta$ [(\[Theta\])]{} and the pseudo-scalar Yukawa term $\widetilde{Y}$ [(\[YY\])]{}, $$\begin{aligned}
\label{5.2}
{\rm Im}\,L_{{\cal N} = 4} = {\mbox{tr}}\left\{ \frac{i}{4} \left( F_{\alpha\beta}F^{\alpha\beta}- \bar F_{\dot\alpha\dot\beta}\bar F^{\dot\alpha\dot\beta} \right) - \frac{ig}{\sqrt{2}} \psi^{\alpha A} [\phi_{AB},\psi_\alpha^B] - \frac{ig}{\sqrt{2}} \bar\psi_{{{\dot\alpha}}A} [\phi^{AB},\bar\psi^{{\dot\alpha}}_B] \right\}\,.\end{aligned}$$ Using the equations of motion we can rewrite the Yukawa terms as a total derivative, $$\begin{aligned}
\label{A.13}
{\rm Im}\,L_{{\cal N} = 4}\bigr|_{\rm EOM} = \frac{i}{4} {\mbox{tr}}\left( F_{\alpha\beta}F^{\alpha\beta}- \bar F_{\dot\alpha\dot\beta}\bar F^{\dot\alpha\dot\beta} \right) - \partial^{\dot\alpha \alpha} {\mbox{tr}}(\bar\psi_{\dot\alpha A} \psi_\alpha^A) \,,\end{aligned}$$ therefore the action $S_{{\cal N} = 4} = \int d^4x\, L_{{\cal N} = 4} = \int d^4x\, \bar L_{{\cal N} = 4}$ is real. The fermion term in [(\[A.13\])]{} is the divergence of the $U(1)$ axial current which completes the R-symmetry $SU(4)$ to $U(4)$ in the free theory. In the interacting theory this $U(1)$ symmetry is broken by the Yukawa term because the real scalars $\phi$ have no $U(1)$ charge.
The main point we would like to make in this section is the cancellation of the conformal anomalies of $\Theta$ and $\widetilde{Y}$ in the particular combination that appears in [(\[5.2\])]{}. The same applies to the complex chiral Lagrangian [(\[A.11\])]{}. We conclude that $L_{{\cal N} = 4}$ is a conformal primary operator, as expected from a member of the super-conformal multiplet of the energy-momentum tensor.
Axial anomaly and conformal anomaly {#secChirAnom}
===================================
In the previous sections we have shown that the topological term ${\Theta}$ is not a conformal primary operator. This manifests itself in the fact that its correlation functions with other, conformal operators are not conformal. Here we give an alternative interpretation of this phenomenon, based on the relationship between ${\Theta}$ and the divergence of an axial current, the so-called axial anomaly [@Adler:1969gk; @Bell:1969ts; @Adler:1969er].
In the two-component spinor notation the vector and axial currents in QED/QCD are defined in [(\[2.5\])]{} and [(\[2.6\])]{}, respectively. Using the Feynman rules of Section \[s2\], we evaluate the mixed correlator of two vector and one axial currents, at Born (i.e., free) level, to be (see also [@Schreier:1971um; @Erdmenger:1996yc]) $$\begin{aligned}
\label{6.1}
&{\langle{V_\mu(x_1) V_\nu(x_2) A_{\lambda}(x_3)}\rangle}_{\rm Born} = \frac1{(2\pi^2)^3} \frac{M_{\mu\nu{\lambda}}}{x^4_{12} x^2_{13} x^2_{23}}\,, {\notag\\}&M_{\mu\nu{\lambda}} = {\epsilon}_{\mu' \nu' \rho{\lambda}}\, I^{\mu'}_\mu(x_{13}) \, I^{\nu'}_\nu(x_{23}) \, Z^\rho(x_3|x_1,x_2) \,,\end{aligned}$$ where the conformal tensors $I$ and $Z$ have been defined in [(\[2.27\])]{}. In the interacting theory the vector current is conserved and hence protected, while the axial one seizes to be conserved due to the axial anomaly. The properly renormalized axial current acquires anomalous dimension starting at two loops [@Larin:1993tq]: $$\begin{aligned}
\label{6.2}
{\gamma}= - \frac{3 C_F g^4}{2^7 \pi^4} + O(g^6) \,.\end{aligned}$$ We can predict the following form of the correlator that accounts for the anomalous dimension ${\gamma}(g)$ of $A$ at the point $x_3$ and also for the beta function: $$\begin{aligned}
\label{6.3}
{\langle{V_\mu(x_1) V_\nu(x_2) A_{\lambda}(x_3)}\rangle}_{\rm loop} = \frac{C(g)}{(2\pi^2)^3} \frac{M_{\mu\nu{\lambda}}}{(x^2_{12})^{2-{\gamma}/2} (x^2_{13})^{1+{\gamma}/2} (x^2_{23})^{1+{\gamma}/2}} +\frac{ {\beta}(g)}{g} \, \Delta_{\mu\nu{\lambda}}\,, \end{aligned}$$ where $C(g)=1+ O(g^2)$.
The form [(\[6.3\])]{} seems to contradict the literature [@Schreier:1971um; @Crewther:1972kn; @Erdmenger:1996yc] where it is claimed that the only allowed conformal form is [(\[6.1\])]{}. This assumes that the axial current has canonical dimension. What we have shown in [(\[6.3\])]{} are two ways of deviating from the form [(\[6.1\])]{}. The first is possible in a conformal theory with ${\beta}(g)=0$ but with a non-conserved axial current due to the axial anomaly. The second term $\Delta$ is the non-conformal correction due to the non-vanishing beta function.
[This three-point function satisfies the vector current conservation,]{}$$\begin{aligned}
\label{64}
{\partial}^\mu_1 \, {\langle{V_\mu(x_1) V_\nu(x_2) A_{\lambda}(x_3)}\rangle}_{\rm loop} =
{\partial}^\nu_2 \, {\langle{V_\mu(x_1) V_\nu(x_2) A_{\lambda}(x_3)}\rangle}_{\rm loop} = 0\,.\end{aligned}$$ However, due to the anomalous dimension [(\[6.2\])]{}, the axial current at point 3 is not conserved anymore: $$\begin{aligned}
\label{6.4}
{\langle{V_\mu(x_1) V_\nu(x_2)\, {\partial}^{\lambda}_{x_3} A_{\lambda}(x_3)}\rangle}_{\rm loop} = \frac{3 C_F g^4}{2^9 \pi^{10}}\, \frac{{\epsilon}_{\mu\nu{\lambda}\rho} x^{\lambda}_{13} x^\rho_{23}}{x_{12}^4 x_{13}^4 x_{23}^4} + O(g^6)\,.\end{aligned}$$ What about the conformal symmetry breaking term $\Delta$ on the rhs of [(\[6.3\])]{}? Without knowing its explicit form, we can argue that it must be conserved. Indeed, the all-order mechanism of non-conservation of the axial current [@Adler:1969er] relies on the presence of Adler’s fermion triangle subgraph. This has already been accounted for by the first term on the rhs of [(\[6.3\])]{}, so the same subgraph cannot contribute to the term $\Delta$.
Further, the Adler-Bardeen theorem [@Adler:1969er] tells us that the axial anomaly takes the form of an operator relation between the (properly renormalized) divergence of the axial current and the topological term: $$\begin{aligned}
\label{6.5}
{\partial}^{\lambda}A_{\lambda}= \frac{g^2}{8\pi^2}\, {\Theta}\,,\end{aligned}$$ where the coefficient is one-loop exact. Substituting this relation in [(\[6.4\])]{}, we find exact agreement with our result for the correlator [(\[LeviChivita\])]{}.
This simple argument not only confirms the well-know Adler-Bardeen relation [(\[6.5\])]{} but also gives us an alternative explanation why the correlator [(\[LeviChivita\])]{} cannot be conformal. Indeed, taking the divergence of a vector of non-canonical dimension in [(\[6.4\])]{} is not a conformal operation, hence the operator ${\Theta}$ cannot be a conformal primary, as we have shown earlier.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are indebted to S. Ferrara, G. Korchemsky, M. Porrati, I. Todorov and A. Zhiboedov for numerous discussions. E.S. is grateful to the MPP-Munich for hospitality during the work on this project.
Conventions and conformal properties in position space {#AppConv}
======================================================
We use the two-component spinor conventions of [@Galperin:2001; @Galperin:2001uw]. The relations between Lorentz four-vectors and $2\times 2$ matrices are defined by $$\begin{aligned}
\label{}
x_{{\alpha}{{\dot\alpha}}}=x^\mu(\sigma_\mu)_{{\alpha}{{\dot\alpha}}}\,, \qquad \tilde x^{{{\dot\alpha}}{\alpha}} = x^\mu(\tilde\sigma_\mu)^{{{\dot\alpha}}{\alpha}} = {\epsilon}^{{\alpha}{\beta}} {\epsilon}^{{{\dot\alpha}}{{\dot\beta}}} x_{{\beta}{{\dot\beta}}}\,,\end{aligned}$$ with the sigma matrices $\sigma_\mu = (1,\vec\sigma)$ and $\tilde\sigma_\mu = (1,-\vec\sigma)$. To raise and lower two-component indices we use the Levi-Civita tensors $$\begin{aligned}
\label{}
{\epsilon}_{12}=-{\epsilon}^{12}=
{\epsilon}_{\dot{1}\dot{2}}=-{\epsilon}^{\dot{1}\dot{2}}=1\, ,\qquad {\epsilon}^{{\alpha}{\beta}} {\epsilon}_{{\beta}\gamma}=\delta^{\alpha}_\gamma \,,\end{aligned}$$ satisfying the identities $$\begin{aligned}
\label{}
x_{{\alpha}{{\dot\alpha}}} \tilde y^{{{\dot\alpha}}{\beta}} + y_{{\alpha}{{\dot\alpha}}} {\tilde{x}}^{{{\dot\alpha}}{\beta}} = 2( x \cdot y) \delta_{\alpha}^{\beta}\,, \qquad\;
x_{{\alpha}{{\dot\alpha}}} {\tilde{x}}^{{{\dot\alpha}}{\beta}} = x^2 \delta_{\alpha}^{\beta}\,, \qquad
x^2 = \frac1{2} x_{{\alpha}{{\dot\alpha}}} {\tilde{x}}^{{{\dot\alpha}}{\alpha}}\,.\end{aligned}$$ The space-time derivative is defined as ${\partial}_{{\alpha}{{\dot\alpha}}} = \sigma^\mu_{{\alpha}{{\dot\alpha}}} {\partial}_\mu$ and has the property $$\begin{aligned}
\label{A4}
{\partial}_{{\alpha}{{\dot\alpha}}} {\tilde{x}}^{{{\dot\beta}}{\beta}} = 2 \delta_{\alpha}^{\beta}\delta_{{\dot\alpha}}^{{\dot\beta}}\,. \end{aligned}$$ Under complex conjugation the Lorentz tensors transform as follows, $$\begin{aligned}
\left[ {\epsilon}_{{\alpha}{\beta}} \right]^* ={\epsilon}_{{{\dot\alpha}}{{\dot\beta}}} \,,\quad \left[ x_{{\alpha}{{\dot\alpha}}} \right]^* = x_{{\alpha}{{\dot\alpha}}} \,,\quad
\left[ (x \tilde y)_{{\alpha}}{}^{{{\dot\alpha}}} \right]^* = - (\tilde x y)^{{{\dot\alpha}}}{}_{{\alpha}} \,. \label{CC}\end{aligned}$$
The easiest way to check conformal invariance is to make the discrete operation of conformal inversion $$\begin{aligned}
\label{}
I[x^\mu] =\frac{x^\mu}{x^2} \,, \qquad I^2 = \mathbb{I}\,. \label{inv}\end{aligned}$$ In the spinor notation the inversion acts as follows $$\begin{aligned}
I[x^{{\alpha}{{\dot\beta}}}_i] = (x^{-1}_i)^{{{\dot\alpha}}{\beta}}\,,\; \,
I[x_{ij}^{{\alpha}{{\dot\beta}}}] = - (x_i^{-1} x_{ij} x_j^{-1})^{{{\dot\alpha}}{\beta}} \,,\; \,
I[\, (x_{ij} \tilde x_{jk})_{{\alpha}{\beta}}\,] = (x_i^{-1} x_{ij} \tilde x_{jk} \tilde x_k^{-1})_{{{\dot\alpha}}{{\dot\beta}}} \,, \label{CT}\end{aligned}$$ where $x_{ij} \equiv x_i - x_j$. The basic fields in a $D=4$ conformal theory transform with specific conformal weights: $$\begin{aligned}
\label{B2}
I[\phi] = x^2 \phi\,, \quad I[\psi_{\alpha}] = x^2 {\tilde{x}}^{{{\dot\alpha}}{\alpha}} \psi_{\alpha}\,, \quad I[\bar\psi^{{\dot\alpha}}] = -x^2 x_{{\alpha}{{\dot\alpha}}} \bar\psi^{{\dot\alpha}}\,, \quad I[F_{{\alpha}{\beta}}] =x^2 {\tilde{x}}^{{{\dot\alpha}}{\alpha}} {\tilde{x}}^{{{\dot\beta}}{\beta}} F_{{\alpha}{\beta}} \,,\end{aligned}$$ namely $(+1)$ for a scalar $\phi$, $(+3/2)$ for a spinor $\psi$ and $(+2)$ for a field strength. These weights are chosen so that the free field equations are covariant.
The conformal boost generator $\mathbb{K}_\mu$ extends the Poincaré group to the conformal group. It can be represented as a sequence of an inversion [(\[inv\])]{}, an infinitesimal space-time translation, and another inversion, i.e. $\mathbb{K}_\mu = I\, \mathbb{P}_\mu\, I$. It acts on an $n$-point correlation function as the following differential operator $$\begin{aligned}
\mathbb{K}_{{\alpha}{{\dot\alpha}}} = i\sum_{i=1}^n \left[ x_i^2 {\partial}_{i,{\alpha}{{\dot\alpha}}} - x_{i,{\alpha}{{\dot\alpha}}} x_{i,{\beta}{{\dot\beta}}} {\partial}_{i}^{{{\dot\beta}}{\beta}} -2 \Delta_i x_{i,{\alpha}{{\dot\alpha}}} + R_{i,{\alpha}{{\dot\alpha}}} \right] \,, \label{Kboost}\end{aligned}$$ where the Lorentz rotation part $R_{{\alpha}{{\dot\alpha}}}$ of the generator acts on the dotted and undotted spinor indices as follows $$\begin{aligned}
&\left[ R_{{\alpha}{{\dot\alpha}}}\, \psi \right]_{\beta} =
\left[ R_{{\alpha}{{\dot\alpha}}} \right]_{\beta}{}^{\gamma} \psi_{\gamma}= \left[ x_{\beta{{\dot\alpha}}} \delta^{\gamma}_{{\alpha}} + {\epsilon}_{\beta\alpha} x_{\dot\alpha}^{\gamma}{} \right]\psi_{\gamma} \,,\notag\\
&\left[ R_{{\alpha}{{\dot\alpha}}}\, \bar\psi \right]_{\dot\beta} =
\left[ R_{{\alpha}{{\dot\alpha}}} \right]_{\dot\beta}{}^{\dot\gamma} \bar\psi_{\dot\gamma}= \left[ x_{\alpha{{\dot\beta}}} \delta^{\dot\gamma}_{{{\dot\alpha}}} + {\epsilon}_{\dot\beta\dot\alpha} x_{\alpha}^{\dot\gamma}{} \right]\bar\psi_{\dot\gamma}\,.\end{aligned}$$
[99]{}
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A. Grozin, “Lectures on QED and QCD,” In \*Grozin, Andrey: Lectures on QED and QCD\* 1-156 \[hep-ph/0508242\]. E. J. Schreier, “Conformal symmetry and three-point functions,” Phys. Rev. D [**3**]{} (1971) 980.
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[^1]: Whenever a regularization is needed, we use a version of the dimensional reduction scheme [@Siegel:1979wq; @Bern:1995db] in which the chiral notation is justified.
[^2]: Using the amplitude terminology we could say that the correlator is zero at the tree-level ${\cal O}(g^0)$, and its perturbative expansion starts at one-loop ${\cal O}(g^2)$.
[^3]: The two-component Levi-Chivita tensors in the second line are not covariant under conformal inversion.
[^4]: Here we are still using the generic notation $J$ for a current made from a single spinor. If specified to vector or axial currents as in [(\[2.5\])]{}, [(\[2.6\])]{}, it is easy to see that the correlator vanishes for the parity-even combination ${\langle{V A \Theta}\rangle}$ and has the form [(\[LeviChivita\])]{} for the parity-odd combinations ${\langle{VV\Theta}\rangle}$ and ${\langle{AA\Theta}\rangle}$.
[^5]: Up to contact terms, see [(\[contact\])]{}.
[^6]: The parity properties become more transparent if we use a four-component Majorana spinor $\Psi=(\psi, \bar\psi)$. The complex scalar field $\varphi=S+iP$ comprises a scalar $S$ and a pseudo-scalar $P$. Then the two combinations in [(\[YY\])]{} become $\widehat Y= ig\bar\Psi(S-{\gamma}_5 P)\Psi$ and $\widetilde{Y}=g\bar\Psi(P+{\gamma}_5 S)\Psi$. Also, the current [(\[curr\])]{} is an axial vector.
[^7]: This is in contrast with a gauge theory where the individual Feynman diagrams are gauge dependent and hence not conformal.
|
---
abstract: |
In geometry processing, numerical optimization methods often involve solving sparse linear systems of equations. These linear systems have a structure that strongly resembles to adjacency graphs of the underlying mesh. We observe how classic linear solvers behave on this specific type of problems. For the sake of simplicity, we minimise either the squared gradient or the squared Laplacian, evaluated by finite differences on a regular $1D$ or $2D$ grid.
We observed the evolution of the solution for both energies, in $1D$ and $2D$, and with different solvers: Jacobi, Gauss-Seidel, SSOR (Symmetric successive over-relaxation) and CG (conjugate gradient [@Shewchuk]). Plotting results at different iterations allows to have an intuition of the behavior of these classic solvers.
author:
- |
Nicolas Ray\
INRIA\
- |
Dmitry Sokolov\
Université de Lorraine\
bibliography:
- 'CGbehavior.bib'
title: Illustration of iterative linear solver behavior on simple $1D$ and $2D$ problems
---
Introduction {#introduction .unnumbered}
============
When facing an optimization problem, the simplest approach is often to iterate, locally improving current solution. For example, heat diffusion can be done by iteratively replacing each value by the average value of its neighbors. With more background on numerical optimization, one would like to formulate this optimization as a linear problem. With such abstraction, it is possible to use solvers as black boxes: we do not have an interpretation of the evolution of the solution during the iterations.
We are interested in observing the behavior of basic linear solvers for very simple cases. It does not necessarily impact the way to use them, but provides some intuition about the magic of linear solvers in the context of minimizing energies on a mesh.
Our meshes are regular $1D$ and $2D$ grids of size $n$, and variables are attached to vertices. We address the coordinates of the unknown vector $\mathbf{x}$ as follows: in $1D$ $\mathbf{x}_i$ is the $i^{th}$ element of $\mathbf{x}$ and in $2D$, the value $\mathbf{x}_{i,j}$ of the vertex located at the $i^{th}$ line and $k^{th}$ column is the $(n*i+j)^{th}$ element of $\mathbf{x}$.
Our tests are performed on the minimization of two energies:
- [*Gradient energy:*]{} The sum of squared difference between adjacent samples: $\Sigma_i(\mathbf{x}_i-\mathbf{x}_{i+1})^2$ in $1D$ and $\Sigma_{i,j}(\mathbf{x}_{i,j}-\mathbf{x}_{i+1,j})^2 +\Sigma_{i,j} (\mathbf{x}_{i,j}-\mathbf{x}_{i,j+1})^2$ in $2D$.
- [*Laplacian energy:*]{} The sum of squared difference between a sample and the average of its neighbors: $\Sigma_i (\mathbf{x}_i-\mathbf{x}_{i-1}/2-\mathbf{x}_{i+1}/2)^2$ in $1D$. In $2D$, it is the same on energy for boundary vertices, and $\Sigma_{i,j} (\mathbf{x}_{i,j}-\mathbf{x}_{i-1,j}/2-\mathbf{x}_{i+1,j}/2)^2 + (\mathbf{x}_{i,j}-\mathbf{x}_{i,j-1}/2-\mathbf{x}_{i,j+1}/2)^2$ over other vertices.
Note that for the Laplacian energy with the right number of constraints ($2$ in $1D$, and $3$ in $2D$), the minimization of the first energy is equivalent to the minimization of the second energy.
$1D$ problems
=============
Observing $1D$ problems allows to visualize the convergence by a $3D$ surface. In all figures in this section, the first axis (horizontal) represents the domain — a $1D$ grid, the second axis (vertical) represents the value associated to each vertex, and the third axis (depth) is the time.
Gradient energy
---------------
The $1D$ grid has $n$ vertices, here we develop the matrices for $n=5$. The problem is formalized by this set of equations $Ax=b$ : $$\begin{pmatrix}
0 & -1 & 0 & 0 & 0 \\
0 & 1 & -1 & 0 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 0 & 0 & 1 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1
\end{pmatrix}
\mathbf{x}=
\begin{pmatrix}
-2\\
0 \\
0 \\
6\\
2 \\
6
\end{pmatrix}$$
The $4$ first lines represent the objective. To enforce boundary constraints $\mathbf{x}_1=2$ and $\mathbf{x}_5=6$, we replaced coefficients applied to $\mathbf{x}_1$ and $\mathbf{x}_5$ by a contribution to the right hand side. We also add the last two lines to explicitly set $\mathbf{x}_1=2$ and $\mathbf{x}_5=6$. In the least squares sense, it produces the new system to solve $A^\top A\mathbf{x}=A^\top b$ i.e.
$$\begin{pmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 2 & -1 & 0 & 0 \\
0 & -1 & 2 & -1 & 0 \\
0 & 0 & -1 & 2 & 0 \\
0 & 0 & 0 & 0 & 1
\end{pmatrix}
\mathbf{x}=
\begin{pmatrix}
2\\
2 \\
0 \\
6\\
6
\end{pmatrix}$$
The condition number of $A^\top A$ is $5.9$ for $n=5$, and $1000$ for $n=50$. These numbers are not high enough to produce numerical instabilities.
This first experience (Fig. \[fig:1Dconvergence\]) shows the behavior of each solver with the same matrix. The boundary conditions (locked coordinates of $\mathbf{x}$) are set in the way that the solution is the straight line defined by $\mathbf{x}_i = i(\mathbf{x}_n-\mathbf{x}_1)/n$. We observe the expected relative speed of convergence i.e. Jacobi $<$ Gauss-Seidel $<$ SSOR $<$ Conjugate Gradient. We also visualize the impact of the order of the coordinates in Gauss-Seidel and SSOR: when only the left side is constrained (locked $\mathbf{x}_1$), the first iteration propagates the constraints to all coordinates whereas constraining only $\mathbf{x}_n$ take $n$ iteration to affect all the variables.
Both extremities are locked to strictly positive values.
The left extremity is locked to a strictly positive value, right to zero.
The right extremity is locked to a strictly positive value, left to zero.
Laplacian energy
----------------
As for the gradient energy case, the $1D$ grid has $n$ vertices, and we develop the matrices for $n=5$. The problem is formalized by this set of equations $A\mathbf{x}=b$ : $$\begin{pmatrix}
0 & 2 & -1 & 0 & 0 \\
0 & -1 & 2 & -1 & 0 \\
0 & 0 & -1 & 2 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1
\end{pmatrix}
\mathbf{x}=
\begin{pmatrix}
-2\\
0 \\
6\\
2 \\
6
\end{pmatrix}$$
The $3$ first lines represent the objective. As for the “gradient energy”, we replaced coefficients applied to $\mathbf{x}_1$ and $\mathbf{x}_5$ by a contribution to the right hand side to enforce $\mathbf{x}_1=2$ and $\mathbf{x}_5=6$. We also add the last two lines to explicitly set $\mathbf{x}_1=2$ and $\mathbf{x}_5=6$. In least squares sense, it produces the new system to solve $A^\top A\mathbf{x}=A^\top b$ i.e.
$$\begin{pmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 5 & -4 & 1 & 0 \\
0 & -4 & 6 & -4 & 0 \\
0 & 1 & -4 & 5 & 0 \\
0 & 0 & 0 & 0 & 1
\end{pmatrix}
\mathbf{x}=
\begin{pmatrix}
2\\
4 \\
-8 \\
12\\
6
\end{pmatrix}$$
The condition number of $A^\top A$ is $34.4$ for $n=5$, and $920000$ for $n=50$ coordinates. Such numbers produce numerical instabilities slowing down the solvers.
In Fig. \[fig:laplaceconvergence\], we observe the same expected relative speed of convergence i.e. Jacobi $<$ Gauss-Seidel $<$ SSOR $<$ Conjugate Gradient. Oscillations of the solution with Gauss-Seidel and SSOR are interesting to observe: it converges faster with higher local oscillations (in the solution space, not in the time dimension).
Another interesting observation is that CG required more than $20$ iterations to converge. This was unexpected because, while CG is mostly used as an iterative solver, it is also a direct solver that is supposed to converge in a worse $n$ iterations. Our interpretation is that the poor conditionning of the matrix generates numerical instabilities that slow down the solver. This phenomena is even worst with higher dimension problem ($n=100$) as illustrated in Fig. \[fig:CG1D\]–right.
iteration $40$
iteration $4000$
iteration $40000$
——————— Jacobi ———————— Gauss-Seidel ————- SSOR ———- Conjugate Gradient –
$2D$ problems
=============
The $2D$ problems are very similar to the $1D$ problems: we set the linear system $A\mathbf{x}=b$ with the gradient (resp. Laplacian) equations and add $3$ constraints ($2$ at corners of the grid and in the center). To solve it in the least squares senses, we have to solve the linear system $A^\top A\mathbf{x}=A^\top b$. We obtained results presented in Fig. \[fig:2Dgradient\] and \[fig:2DLaplacien\].
The observations are also very similar to $1D$: the speed of convergence is as usual i.e. Jacobi $<$ Gauss-Seidel $<$ SSOR $<$ CG as expected, and the non zero coordinates of $\mathbf{x}$ are “discovered” at each iteration in a front propagation fashion. Each constraint has an impact on the value of all vertices after $100$ iterations ($L^1$ distance of the diagonal). For the gradient energy, the CG method already have a fair solution and could be stopped at step $100$. For the Laplacian energy, that has a bad conditioning, the result at step $100$ is still far from the solution. Even at step $2400$, it is not converged... but at step $2500$, it’s done ! (as expected).
We can also observe that the conditioning affects all methods (not only CG).
Iteration 1
Iteration 5
Iteration 10
Iteration 50
Iteration 100
Iteration 2400
Iteration 2500
Iteration 10000
—————————— Jacobi ——– Gauss-Seidel ——– SSOR —— Conjugate Gradient –
Iteration 1
Iteration 5
Iteration 10
Iteration 50
Iteration 100
Iteration 2400
Iteration 2500
Iteration 10000
—————————— Jacobi ——– Gauss-Seidel ——– SSOR —— Conjugate Gradient –
Conclusion: what did I learn? {#conclusion-what-did-i-learn .unnumbered}
=============================
We observed two non obvious behaviors of the considered iterative linear solver:
- When starting with a null solution vector $\mathbf{x}$, all tested iterative solvers (including CG) remove the zero coefficient by front propagation from the constraints. It can be explained by the position of non zero coefficients in the matrix $A$ that is similar to the one of the vertices adjacency matrix.
- Ill-conditioned systems may take more than $n$ iterations to converge with the conjugate gradient algorithm. Stopping them earlier does not always give a good approximation of the solution e.g. $1D$ Laplacian.
|
---
abstract: 'In a recent letter \[Phys. Rev. Lett. [**104**]{}, 167201 (2010)\] we proposed a new confining method for ultracold atoms on optical lattices, which is based on off-diagonal confinement (ODC). This method was shown to have distinct advantages over the conventional diagonal confinement (DC), that makes use of a trapping potential, such as the existence of pure Mott phases and highly populated condensates. In this manuscript we show that the ODC method can also lead to lower temperatures than the DC method for a wide range of control parameters. Using exact diagonalization we determine this range of parameters for the hard-core case; then we extend our results to the soft-core case by performing quantum Monte Carlo (QMC) simulations for both DC and ODC systems at fixed temperature, and analyzing the corresponding entropies. We also propose a method for measuring the entropy in QMC simulations.'
address: 'Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA'
author:
- 'V. G. Rousseau, K. Hettiarachchilage, M. Jarrell, J. Moreno, D. E. Sheehy'
title: 'Using Off-diagonal Confinement as a Cooling Method'
---
Introduction
============
With recent experimental developments on cold atoms in optical lattices, the interest in the bosonic Hubbard model [@Fisher; @Batrouni1990] has dramatically increased. This model is characterized by a superfluid-to-Mott quantum phase transition for large onsite repulsion and integer values of the density of particles. In actual experiments the atoms are confined to prevent them from leaking out of the lattice. This is currently achieved by applying a spatially dependent magnetic field [@Greiner]. A parabolic potential is added into the Hubbard model [@Batrouni2002] to mimic the effect of the magnetic field. Therefore, the resulting model does not exhibit a true superfluid-to-Mott transition, since Mott regions always coexist with superfluid regions. This was predicted theoretically [@Batrouni2002], and later confirmed experimentally [@Folling].
Recently, we have proposed a new confining technique [@RousseauODC] where the atoms are confined via a hopping integral that decreases as a function of the distance from the center of the lattice. Since the confinement of the particles is due to the hopping or off-diagonal operators, we called it *Off-Diagonal Confinement* (ODC), as opposed to the conventional diagonal confinement (DC) which makes use of a parabolic confinement potential that is reflected in the density profile [@Batrouni2002]. For large on-site repulsion the ODC model exhibits pure Mott phases at commensurate filling while at other fillings it exhibits more populated condensates than the DC model. Another advantage of ODC is that simple energy measurements can provide insights into the Mott gap, while the presence of the harmonic potential may renormalize the value of the gap with respect to the uniform case [@Carrasquilla].
In this paper, we show that the ODC method can also lead to lower temperatures than the DC method for a wide range of parameters. Producing low temperatures in experiments is challenging, especially with fermions for which laser cooling is not as efficient as for bosons. In current experiments, fermions are cooled down by convection in the presence of cold bosons, leading to Bose-Fermi mixtures [@Ott; @Cazalilla; @Hebert; @Zujev]. Achieving lower temperatures for bosonic condensates will therefore result in colder Bose-Fermi mixtures. This manuscript is organized as follows. In section II we define our model and describe our methods. The hard-core limit is studied in section III in order to illustrate analytically how ODC produces temperatures that are lower than those obtained with DC. This will also serve to benchmark the quantum Monte Carlo (QMC) simulations we use for analyzing the general soft-core case. In section IV we present the algorithm we use for QMC simulations, and we propose a method for measuring the entropy with this algorithm. Results for the soft-core case are presented in section V. Finally we conclude in section VI.
Model and method
================
We consider bosons confined to a one-dimensional optical lattice with $L$ sites and lattice constant $a=1$. The Hamiltonian takes the form: $$\begin{aligned}
\label{Hamiltonian} \nonumber \hat\mathcal H &=& -\sum_{\big\langle i,j\big\rangle}t_{ij}
\Big(a_i^\dagger a_j^{\phantom\dagger}+h.c.\Big)+\frac{U}{2}\sum_i \hat n_i(\hat n_i-1)\\
& & +W\sum_i(i-L/2)^2\hat n_i\end{aligned}$$ The creation and annihilation operators $a_i^\dagger$ and $a_i^{\phantom\dagger}$ satisfy bosonic commutation rules, $\big[a_i^{\phantom\dagger},a_j^{\phantom\dagger}\big]=
\big[a_i^\dagger,a_j^\dagger\big]=0$, $\big[a_i^{\phantom\dagger},a_j^\dagger\big]=\delta_{ij}$, and $\hat n_i=a_i^\dagger a_i^{\phantom\dagger}$ is the number of bosons on site $i$. The sum $\sum_{\langle i,j\rangle}$ runs over all distinct pairs of first neighboring sites $i,j$, and $t_{ij}$ is the hopping integral between $i$ and $j$. The parameter $U$ is the strength of the local on-site interaction, and $W$ describes the curvature of the external trapping potential.
In this work we consider the grand-canonical partition function, $$\label{PartitionFunction} \mathcal Z=\textrm{Tr }e^{-\beta\big(\hat\mathcal H-\mu\hat\mathcal N\big)},$$ where $\displaystyle \beta=\frac{1}{k_B T}$, $k_B$ is the Boltzmann constant and $T$ the temperature. The chemical potential $\mu$ controls the average number of particles, $N=\langle\hat\mathcal N\rangle$, with $\hat\mathcal N=\sum_i\hat n_i$. The conventional DC model is obtained by setting $t_{ij}=1$ for all pairs of first neighboring sites $i,j$, and using $W>0$. For this model the value of $L$ is irrelevant as long as it is sufficiently large to contain the whole gas. The ODC model is obtained by setting $W=0$ and using a hopping integral $t_{ij}$ that decreases as a function of the distance from the center of the lattice, and vanishes at the edges. For this model, $L$ fully determines $t_{ij}$ as described below.
Typically the temperature is not a control parameter in cold-atoms experiments, and once laser cooling has been performed, the system has a fixed entropy which can be considered the control parameter. Then the temperature can be estimated numerically knowing the isentropies of the system [@Mahmud]. Therefore, our strategy for determining which of the two confining methods can achieve the lowest temperature is based on switching adiabatically from DC to ODC, so the entropy is conserved. Then we determine the temperatures $T_{dc}$ and $T_{odc}$ of the DC and ODC systems by equating the entropies
We will consider an experiment in which a *fixed* number $N$ of atoms is loaded into an optical lattice with a DC trap, described by Eq. (\[Hamiltonian\]) with parameters $t_{ij}=1$, $W=0.008$ (as in Ref. [@Batrouni2002]). We use $L=400$ in order to ensure the confinement of the whole gas. Then, we adiabatically switch to the ODC trap by slowly varying $t_{ij}$ and $W$ to $t_{ij}=(i+j+1)(2L-i-j-1)/L^2$ with $L=70$, and $W=0$ (as in Ref. [@RousseauODC]), keeping $N$ and $U$ the same.
However, in our calculation, it is actually more convenient to control the temperature than the entropy. Thus we consider both DC and ODC systems for a set temperatures $T$, and measure the corresponding entropies, $S_{dc}(T)$ and $S_{odc}(T)$. Then, knowing the initial temperature $T_{dc}$, the final temperature $T_{odc}$ can be extracted graphically by imposing the equality, $S_{dc}(T_{dc})=S_{odc}(T_{odc})$, as described in the next section.
The hard-core case: Exact analytical results
============================================
The hard-core limit ($U=+\infty$) of the model can be solved analytically. These exact results provide a solid benchmark for our study of the general soft-core case in the next section. We follow here the method used by Rigol [@Rigol2005]. In the hard-core limit, the $U$ term in (\[Hamiltonian\]) can be dropped if the standard bosonic commutation rules are replaced by $\big[a_i^{\phantom\dagger},a_j^{\phantom\dagger}\big]=\big[a_i^\dagger,a_j^\dagger\big]=
\big[a_i^{\phantom\dagger},a_j^\dagger\big]=0$ for $i\neq j$, and $a_i^{\phantom\dagger}a_i^\dagger+a_i^\dagger a_i^{\phantom\dagger}=1$, and $a_i^{\phantom\dagger2}=a_i^{\dagger2}=0$. With this algebra, the model (\[Hamiltonian\]) reduces to $$\label{HardcoreHamiltonian} \hat\mathcal H=-\sum_{\big\langle i,j\big\rangle}t_{ij}\Big(a_i^\dagger
a_j^{\phantom\dagger}+h.c.\Big)+W\sum_i(i-L/2)^2\hat n_i,$$ which describes hard-core bosons. By performing a Jordan-Wigner transformation, the hard-core creation and annihilation operators can be mapped onto fermionic creation and annihilation operators, $f_i^\dagger$ and $f_i^{\phantom\dagger}$, $$a_j^\dagger=f_j^\dagger \prod_{q=1}^{j-1}e^{i\pi f_q^\dagger f_q^{\phantom\dagger}},\quad a_j^{\phantom\dagger}=
\prod_{q=1}^{j-1}e^{-i\pi f_q^\dagger f_q^{\phantom\dagger}}f_j^{\phantom\dagger},$$ which satisfy the usual fermionic anticommutation rules, $\big\lbrace f_i^{\phantom\dagger},f_j^{\phantom\dagger}
\big\rbrace=\big\lbrace f_i^\dagger,f_j^\dagger\big\rbrace=0$, $\big\lbrace f_i^{\phantom\dagger},f_j^\dagger\big\rbrace=\delta_{ij}$. This leads to a model that describes free spinless fermions, $$\label{FermionicHamiltonian} \hat\mathcal H=-\sum_{\big\langle i,j\big\rangle}t_{ij}
\Big(f_i^\dagger f_j^{\phantom\dagger}+h.c.\Big)+W\sum_i(i-L/2)^2\hat n_i,$$ where $\hat n_i=f_i^\dagger f_i^{\phantom\dagger}$ represents the number of fermions on site $i$. Because the model (\[FermionicHamiltonian\]) is a quadratic form of $f_i^\dagger$ and $f_i^{\phantom\dagger}$, it can be solved by a simple numerical diagonalization of the $L\times L$ matrix. Denoting by $\epsilon_k$ with $k\in[1,L]$ the eigenvalues of this matrix, the partition function (\[PartitionFunction\]) takes the form $$\label{PartitionFunction2} \mathcal Z=\prod_{k=1}^L\Big(1+e^{-\beta(\epsilon_k-\mu)}\Big).$$ The entropy is defined as $S=-k_B\textrm{Tr }\mathcal D\ln\mathcal D$ with the density matrix $\displaystyle\mathcal D=\frac{1}{\mathcal Z}e^{-\beta(\hat\mathcal H-\mu\hat\mathcal N)}$. Working in a system of units where the Boltzmann constant $k_B=1$ and using the properties of the density matrix, it follows that $S=\ln\mathcal
Z+\beta\big\langle\hat\mathcal H\big\rangle-\beta\mu\big\langle\hat\mathcal N\big\rangle$. Substituting $\displaystyle\big\langle\hat\mathcal H\big\rangle-\mu\big\langle\hat\mathcal N\big\rangle=
-\frac{\partial}{\partial\beta}\ln\mathcal Z$ and using expression (\[PartitionFunction2\]) for $\mathcal Z$, the entropy takes the form $$\label{Entropy} S(\beta,\mu)=\sum_{k=1}^L\Big[\ln\Big(1+e^{-\beta(\epsilon_k-\mu)}\Big)+
\frac{\beta(\epsilon_k-\mu)}{e^{\beta(\epsilon_k-\mu)}+1}\Big].$$ The average number of particles $N$ is obtained by summing the Fermi-Dirac distribution, $$\label{FermiDirac} N(\beta,\mu)=\sum_{k=1}^L\frac{1}{e^{\beta(\epsilon_k-\mu)}+1}.$$
Fig. \[EntropyHardcoreN50\] shows the entropy (\[Entropy\]) as a function of temperature for both DC and ODC cases. The chemical potential $\mu$ is adjusted such that the average number of particles (\[FermiDirac\]) remains constant ($N=50$). An interesting feature is that the two curves cross at a temperature $T_c$, and that below $T_c$ the entropy of the ODC system is greater than the entropy of the DC system. Thus, if the initial temperature $T_{dc}$ is below $T_c$, then the final temperature $T_{odc}$ is lower when switching adiabatically from DC to ODC.
![ (Color online) The entropy as a function of temperature for 50 hard-core bosons, in the DC case (circles) and in the ODC case (triangles). There exists a critical temperature $T_c$ where the two curves cross. If the initial temperature $T_{dc}$ is below $T_c$, then the conservation of the entropy when switching adiabatically from DC to ODC implies that the final temperature $T_{odc}$ is lower. []{data-label="EntropyHardcoreN50"}](EntropyHardcoreN50.eps){width="45.00000%"}
Next we generalize our discussion by calculating, for a fixed number of particles $N$, the critical temperature $T_c$ below which the ODC method produces a temperature lower than the DC method when the confinement is switch adiabatically. In order to determine $T_c$ for given parameters $t^{dc}_{ij}$ and $W$ for the conventional DC system, and $t^{odc}_{ij}$ for the ODC model, one needs to solve for each value of $N$ a system of three coupled non-linear equations, $$\label{System}
\left\lbrace\begin{array}{l}
S_{odc}(\beta_c,\mu_{odc})=S_{dc}(\beta_c,\mu_{dc}) \\
N_{odc}(\beta_c,\mu_{odc})=N \\
N_{dc}(\beta_c,\mu_{dc})=N
\end{array}\right.$$ where $S_{odc}$ ($S_{dc}$) is given by Eq. (\[Entropy\]) and $N_{odc}$ ($N_{dc}$) is given by Eq. (\[FermiDirac\]), with $\mu=\mu_{odc}$ ($\mu=\mu_{dc}$), and $\beta=\beta_c$. The first equation corresponds to the conservation of the entropy when switching from DC to ODC, and the two others correspond to the conservation of the number of particles. Solving this system of equations determines the critical inverse temperature $\beta_c=1/T_c$, and the chemical potentials $\mu_{odc}$ and $\mu_{dc}$ that give the desired number of particles $N$.
For this purpose, we define an *error* function: $$\begin{aligned}
\nonumber \mathcal E(\beta_c,\mu_{odc},\mu_{dc}) &=& \big(S_{odc}(\beta_c,\mu_{odc})-
S_{dc}(\beta_c,\mu_{dc})\big)^2\\
\nonumber & & +\big(N_{odc}(\beta_c,\mu_{odc})-N\big)^2\\
& & +\big(N_{dc}(\beta_c,\mu_{dc})-N\big)^2.\end{aligned}$$ By construction, the solution of Eq. (\[System\]) minimizes this error function. Starting with an initial guess for $\beta_c$, $\mu_{odc}$, and $\mu_{dc}$, we calculate the error $\mathcal E(\beta_c,\mu_{odc},\mu_{dc})$ and its gradient $\vec\nabla\mathcal E=(\partial\mathcal E/\partial\beta_c,\partial\mathcal
E/\partial\mu_{odc},\partial\mathcal E/\partial\mu_{dc})$. Writing the initial guess as a vector, $\vec r=(\beta_c,\mu_{odc},\mu_{dc})$, we perform a correction $\Delta\vec r$ by following the opposite direction of the gradient $\vec\nabla\mathcal E$. Then we iterate until convergence.
Fig. \[TcVsN\] shows the critical temperature $T_c$ and the DC isotherms as functions of $N$. For a given number of particles and an initial temperature $T=T_c(N)$, the ODC and DC systems have the same temperature $T_{odc}=T_{dc}$ when the confinement is switch adiabatically. Below (above) $T_c$, the ODC system has a temperature $T_{odc}$ that is lower (higher) than the temperature $T_{dc}$ of the DC system. The point $P$ illustrates how the figure should be read: For a system with 34 particles and an initial DC temperature $T_{dc}=3$, the final ODC temperature is $T_{odc}\approx 1.5$. Note that $T_c$ vanishes when N=L=70. The resulting Mott phase found in the ODC case always has lower entropy than the mixed phases found in the DC case. This will be discussed in greater detail in the next section.
![ (Color online) The critical temperature $T_c$ and the DC isotherms as functions of the number of particles $N$. At $T=T_c$, there is no change in temperature when switching adiabatically from DC to ODC. Below $T_c$ (blue region), the ODC method gives a temperature $T_{odc}$ that is lower than the temperature $T_{dc}$ obtained with the DC method. Above $T_c$ (yellow region), it is the DC method that gives the lowest temperature. For example, the point $P$ corresponds to a system with 34 particles, an initial temperature $T_{dc}=3$, and a final temperature $T_{odc}\approx 1.5$. []{data-label="TcVsN"}](TcVsN.eps){width="45.00000%"}
Quantum Monte Carlo algorithm and the entropy
=============================================
For the treatment of soft-core interactions, we perform QMC simulations using the Stochastic Green Function (SGF) algorithm [@SGF] with tunable directionality [@DirectedSGF]. Although this algorithm was developed for the canonical ensemble, a trivial extension [@Wolak2010] allows us to simulate the grand-canonical ensemble. We propose a new method to measure the entropy by taking advantage of the grand-canonical ensemble. Our thermodynamic control parameters are the temperature $T$, the volume $V$ (number of sites $L$), and the chemical potential $\mu$.
![ (Color online) The entropy $S$ and the thermal susceptibility $\chi$ in the hard-core case. Comparison between results obtained with exact diagonalization using Eq. (\[Entropy\]) and QMC results with the SGF algorithm and Eq. (\[EntropyQMC\]). Two different temperatures are considered, $T=3$ and $T=0.25$, for both DC and ODC systems. []{data-label="Comparison"}](Comparison.eps){width="45.00000%"}
Unlike the analytical hard-core case, a direct measurement of the entropy is not possible with a single QMC simulation because the value of $\mathcal Z$ is unknown. However it is still possible to evaluate the entropy with a set of QMC simulations. For this purpose, we define the *thermal susceptibility* $\chi_{th}$ by the response of the number of particles $N$ to an infinitesimal change of the temperature $T$: $$\label{ThermalSusceptibility} \chi_{th}(T,V,\mu)=\frac{\partial N}{\partial T}\Big|_{V,\mu}$$ By substituting $N=\frac{1}{\mathcal Z}\textrm{Tr }\hat\mathcal N e^{-\beta(\hat\mathcal H-\mu\hat\mathcal N)}$ in expression (\[ThermalSusceptibility\]), we get an expression for the thermal susceptibility that can be directly measured in our simulations: $$\label{ThermalSusceptibilityQMC} \chi_{th}=\beta^2\Big[\big\langle\hat\mathcal N
\big(\hat\mathcal H-\mu\hat\mathcal N\big)\big\rangle-\big\langle\hat\mathcal N\big\rangle\big\langle
\big(\hat\mathcal H-\mu\hat\mathcal N\big)\big\rangle\Big]$$ Considering the energy $E=\langle\hat\mathcal H\rangle$ and the associated differential $dE=TdS-PdV+\mu dN$, where the pressure $P$ is defined as $\displaystyle P=-\frac{\partial E}{\partial V}\Big|_{S,N}$, and performing a Legendre transformation over the variables $S$ and $N$, we can define the grand-canonical potential $\Omega$ that depends only on our natural variables, $\Omega(T,V,\mu)=E-TS-\mu N=-PV$. Its differential takes the form $$d\Omega=-SdT-PdV-Nd\mu.$$ We can then extract a useful Maxwell relation, $$\frac{\partial S}{\partial\mu}\Big|_{V,T}=\frac{\partial N}{\partial T}\Big|_{V,\mu},$$ so the entropy can be easily obtained by integrating the thermal susceptibility over the chemical potential and keeping the temperature and the volume constant, $$\label{EntropyQMC} S(T,V,\mu)=\int_{\mu_0}^{\mu}\chi_{th}(T,V,\mu')d\mu',$$ where $\mu_0$ is the critical value of the chemical potential below which the average number of particles $N$ and the thermal susceptibility $\chi_{th}$ are vanishing.
![ (Color online) The entropy as a function of temperature for 50 soft-core bosons with $U=8$, in the DC case (circles) and in the ODC case (triangles). As in the hard-core case (Fig.\[EntropyHardcoreN50\]), there exists a critical temperature $T_c$ below which the entropy of the ODC system is higher than the entropy of the DC system, thus making the ODC method more efficient than the DC method for producing low temperatures. []{data-label="EntropySoftcoreN50"}](EntropySoftcoreN50.eps){width="45.00000%"}
In order to check the reliability of Eq. (\[EntropyQMC\]), we show on Fig. \[Comparison\] a comparison of the entropy of the hard-core case obtained with the SGF algorithm by integrating the thermal susceptibility (\[ThermalSusceptibilityQMC\]), and the entropy computed with Eq. (\[Entropy\]). The agreement is good for both DC and ODC systems at high ($T=3$) and low temperatures ($T=0.25$).
![ (Color online) The entropy as a function of the inverse onsite interaction $1/U$ for 50 soft-core bosons at $T=1$, in the DC (circles) and ODC cases (triangles). The entropy of the ODC system remains greater than the entropy of the DC system for any value of the onsite repulsion $U$, showing that our suggested cooling method works deep into the soft-core case. []{data-label="EntropySoftcoreN50VsU"}](EntropySoftcoreN50VsU.eps){width="45.00000%"}
We now release the hard-core constraint and set the onsite repulsion $U=8$. Fig. \[EntropySoftcoreN50\] shows the entropies for the DC and the ODC models as functions of temperature for $N=50$. The curves differ from the hard-core case only quantitatively, not qualitatively, showing that the method of cooling by switching from DC to ODC still works. Moreover, one notices that the critical temperature $T_c\approx3.5$ is higher than in the hard-core case ($T_c\approx 2.65$) which makes easier to access the regime where ODC is more efficient than DC.
Further, we extend our soft-core results to different values of the onsite repulsion. Fig. \[EntropySoftcoreN50VsU\] shows the entropy for the DC and the ODC models as function of the inverse onsite repulsion $1/U$ for $N=50$ and $T=1.0$. The curves show that the entropy of the ODC model is above the one of the DC model for any value of $U$. Thus, for this filling, the ODC method produce temperatures lower than the DC method for any value of the onsite repulsion. When $U$ is large, the results match with those obtained in the preceding section for the hard-core case (Fig. \[EntropyHardcoreN50\]).
However, the situation is different for $N=70$ as Fig. \[EntropySoftcoreN70VsU\] illustrates. At this integer filling, the entropy of the ODC model, which vanishes in the large $U$ limit, intersects the curve for the DC model. In this regime, the ODC model exhibits a pure Mott phase, hence with zero entropy. However the phase of the DC model has Mott regions coexisting with superfluid regions, so the entropy remains finite. Thus, ODC cannot be used to cool the system in this region.
![ (Color online) The entropy as a function of the inverse onsite interaction $1/U$ for 70 soft-core bosons at $T=1$, in the DC case (circles) and in the ODC case (triangles). For the ODC case the filling is commensurate and the system forms a pure Mott phase that cannot carry any entropy. Thus ODC cannot be used as a cooling method for commensurate fillings. []{data-label="EntropySoftcoreN70VsU"}](EntropySoftcoreN70VsU.eps){width="45.00000%"}
Concerning the experimental realization of our model, a holographic technique recently developed [@Greiner2] can be used to build the optical lattice with off-diagonal confinement. Using this method, an off-diagonal trap can be superposed to an existing diagonal trap. Then the diagonal trap can be turned off. The switching between the two traps can be in principle very fast, however the technical details of how this will work go beyond the scope of the present manuscript and must be developed by experimentalists. Nevertheless, a qualitative analysis reveals that three time scales must be considered. The time scale $\tau_m$ of the model system or roughly $\tau_m=1/t$ (in units where $\hbar=1$), the time scale of the experiment $\tau_e$, and the time scale $\tau_c$ which describes the coupling of the model system to its environment which includes the effects of the laser heating, evaporation, etc. In our proposal, it is important that the trap is adiabatically switch on the experimental time scale, but not on the time scale which describes the coupling of the trap to its environment, so that $\tau_m << \tau_e << \tau_c$.
Conclusion
==========
In this manuscript we propose that the adiabatic switch from the DC to the ODC method can produce lower temperatures for a wide range of initial temperatures and system parameters. In the hard-core limit, we determine the critical temperature $T_c$ for which the two methods have the same entropy. Below (above) $T_c$ and at constant entropy, the ODC method leads to temperatures that are lower (higher) than with the DC method. In order to extend our results to the soft-core case, we propose a simple method for evaluating the entropy with QMC, by measuring the thermal susceptibility $\chi_{th}$ in the grand-canonical ensemble and integrating it over the chemical potential $\mu$. Then we make use of the SGF algorithm [@SGF] with tunable directionality [@DirectedSGF], and show that the soft-core results are qualitatively the same as in the hard-core case.
This work was supported by the National Science Foundation through OISE-0952300, the TeraGrid resources provided by NICS under grant number TG-DMR100007, the high performance computational resources provided by the Louisiana Optical Network Initiative (http://www.loni.org), and the Louisiana Board of Regents, under grant LEQSF (2008-11)-RD-A-10.
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